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A non-hydrostatic model for free surface flows Thesis · January 1999

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Referent: Korreferent:

Prof. Dr.-Ing. Werner Zielke Prof. Dr.-Ing. habil. Rainer Helmig

Tag der Promotion:

12. November 1998

Institut fu omungsmechanik ¨ r Str¨ und Elektron. Rechnen im Bauwesen der Universit¨ at Hannover

Bericht Nr. 56/1999

Jacek A. Jankowski

A non-hydrostatic model for free surface flows

ISSN 0177 — 9028

Institut f¨ ur Str¨omungsmechanik und Elektronisches Rechnen im Bauwesen Universit¨at Hannover Appelstraße 9A, D – 30167 Hannover Tel.: Fax.: URL:

+49 – 511 – 762-3568 +49 – 511 – 762-3777 http://www.hydromech.uni-hannover.de

Von dem Fachbereich f¨ ur Bauingenieur- und Vermessungswesen der Universit¨at Hannover zur Erlangung des Grades eines Doktor – Ingenieurs genehmigte Dissertation.

Referent: Korreferent:

Prof. Dr.-Ing. Werner Zielke Prof. Dr.-Ing. habil. Rainer Helmig

Tag der Promotion:

12. November 1998

Abstract

i

Abstract An algorithm for solution of three-dimensional Navier-Stokes equations for incompressible free surface flows is developed. A decoupled algorithm based on the fractional step (operator-splitting) technique is applied. The solution is obtained in subsequent stages treating equations split into parts having well-defined mathematical properties, so that the most adequate methods for a given differential operator type can be used. The decoupled algorithm structure, which does not use the continuity equation explicitly, allows application of equal-order linear interpolation functions for all variables. The applied reference element type, a prism with six nodes and linear interpolation functions, is a compromise between the exactness of the interpolation, model complexity and computational cost. The finite difference method is applied for the time discretisation and the computational domain variability is taken into account by a standard σ-mesh structure which is well suited to most geophysical applications. The model is implemented in the framework of the Telemac system, developed in Laboratoire National d’Hydraulique in Chatou, Electricit´e de France (EDF) for modelling of free surface flows including transport phenomena. It is a far-reaching further development of existing three-dimensional code Telemac3D whose application domain is limited due to the hydrostatic approximation and because the free surface position is computed from shallow water equations. Therefore, the vertical acceleration is neglected and the free surface and bottom gradients must be small. In the new model, the vertical acceleration is taken into account by decomposing the global pressure into its hydrostatic (i.e. barotropic as well as baroclinic) and hydrodynamic components. In consequence, the hydrodynamic pressure is found from a pressure Poisson equation from fractional formulation, whereby the hydrostatic part is computed explicitly from the free surface elevation and density field. The final velocity is obtained under assumption of its incompressibility from the projection of the intermediate result, which is found without hydrodynamic pressure gradients. The free surface position can be found using height function methods based on the kinematic boundary condition or the free surface conservative equation (vertically integrated continuity equation). In the numerical solution algorithm the method of characteristics, streamline upwind Petrov-Galerkin FEM, or explicit formulation in finite elements can be chosen. The developed new algorithm steps have been verified separately, using analytical solutions. The verification of the algorithm as a whole has been performed using a number of benchmark test cases covering the targeted model application domain. Some of these allowed a formal comparison with an analytical problem solution. They include free surface and internal waves, sub- and supercritical channel flow over a steep ramp, windand buoyancy-driven currents. The implementation of the model within the well-validated Telemac system allows immediate application of the newly developed algorithm for practical hydraulic engineering problems and geophysical flows. Keywords: hydrodynamic-numeric model, non-hydrostatic, free surface flow.

ii

Zusammenfassung

Zusammenfassung Im Rahmen der Arbeit wurde ein Algorithmus zur L¨osung der dreidimensionalen NavierStokes Gleichungen f¨ ur inkompressible Fluide mit freier Oberfl¨ache entwickelt. Der Algorithmus basiert auf einer fractional step Methode, wobei die Grundgleichungen in Anteile mit gleichen mathematischen Eigenschaften aufgespalten werden, die dann mit optimal angepaßten Verfahren schrittweise gel¨ost werden (operator splitting). Es werden f¨ ur alle Variablen die gleichen linearen Ansatzfunktionen verwendet. Das prismatische 6-Knoten Element stellt einen Kompromiß zwischen Genauigkeit der Interpolation, Komplexit¨at des Modells und ben¨otigter Rechenzeit dar. Die zeitliche Diskretisierung wird mit einem Finite-Differenzen Verfahren unter Zugrundelegung der σ-Transformation durchgef¨ uhrt, was sich f¨ ur die meisten geophysikalischen Problemstellungen als geeignete Vorgehensweise erwiesen hat. Der neue Algorithmus ist Bestandteil des Programmsystems Telemac, das von der Electrit´e de France (EDF) im Laboratoire National d’Hydraulique in Chatou f¨ ur die Simulation der Str¨omungen mit freier Oberfl¨ache und Stofftransport entwickelt wurde. Das neue Modell stellt eine Weiterentwicklung des bereits existierenden dreidimensionalen Codes Telemac3D dar, dessen Anwendungsbereich infolge einer hydrostatischen Druckapproximation beschr¨ankt ist. Bislang konnten keine vertikalen Beschleunigungen ber¨ ucksichtigt werden und nur kleine Gradienten der freien Oberfl¨ache und des Bodens behandelt werden. Im neuen Algorithmus wird die vertikale Beschleunigung durch eine Zerlegung des globalen Drucks in einen hydrostatischen (d.h. barotropen sowie baroklinen) und einen hydrodynamischen Anteil bestimmt. Im Rahmen der fractional step Methode wird die hydrodynamische Komponente aus der L¨osung einer Poisson-Gleichung ermittelt, und der hydrostatische Druck explizit aus der Position der freien Oberfl¨ache und dem Dichtefeld berechnet. Das Geschwindigkeitsfeld wird unter Annahme der Inkompressibilit¨atbedingung aus einer Projektion des Zwischenergebnisses bestimmt, welches ohne hydrodynamischen Anteil berechnet wurde. Die Lage der freien Oberfl¨ache kann aus der kinematischen Randbedingung oder aus einer konservativen Formulierung (vertikal gemittelte Kontinuit¨atsgleichung) ermittelt werden. Zur numerischen L¨osung stehen wahlweise ein Charakteristiken Verfahren, eine semi-implizite SUPG FEM oder eine explizite Formulierung zur Verf¨ ugung. Alle entwickelten Teile des neuen Algorithmus’ sind getrennt mittels analytischer L¨osungen verifiziert worden. Der Gesamtalgorithmus ist anhand zahlreicher Testbeispiele aus dem erweiterten Anwendungsbereich des Modells verifiziert worden, wozu in vielen F¨allen analytische L¨osungen herangezogen werden konnten. Es sind unter anderem interne und Oberfl¨achenwellen, Kanalstr¨omungen u ¨ber eine Rampe im str¨omenden und schießenden Zustand sowie wind- und dichtegetriebene Str¨omungen behandelt worden. Die Implementierung des neuen Algorithmus’ innerhalb des etablierten Programmsystems Telemac erm¨oglicht einem großem Anwenderkreis die sofortige Nutzung f¨ ur praktische Problemstellungen des Ingenieurwesens und der Geophysik. Schlagworte: hydrodynamisch-numerisches model, nicht-hydrostatisch, Freispiegelstr¨omung.

Vorwort

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Vorwort Die vorliegende Arbeit ist in den Jahren 1995–1998 w¨ahrend meiner T¨atigkeit im Institut f¨ ur Str¨omungsmechanik und Elektronisches Rechnen im Bauwesen entstanden. Auf die M¨oglichkeit, ein nicht-hydrostatisches Modell auf der Basis von Bibliotheken des Programmsystems Telemac zu entwickeln, wurde ich erst im Mai 1995 aufmerksam gemacht. Im Herbst folgten die ersten Fortran-Zeilen und im Januar 1996 gab es schon eine erste lauff¨ahige Programmversion. Was ich zuerst als eine kleine und sehr unsystematisch ausge¨ ubte Nebenbesch¨aftigung bei der Bearbeitung eines BMBF-Projektes Mesoskalige Stofftransporte im Pazifik als Folge der technischen Eingriffe in die Tiefsee (1992-98) betrachtet habe, hat sich in 1996-97 mit jedem neuen berechneten Verifikationsbeispiel zu einer ruhigen Passion entwickelt. Der Text dieser Arbeit ist im wesentlichen auch in diesen Jahren entstanden. Im Winter und Fr¨ uhjahr 1998 wurde das Modell neu programmiert und mit vielen neuen zus¨atzlichen Verbesserungen ausgestattet, um anspruchsvollere Verifikationen durchf¨ uhren zu k¨onnen. Die Dissertation habe ich am letzten Tag im Juni 1998 abgegeben. Im ersten ruhigeren und zwanglosen R¨ uckblick auf meine Arbeit sehe ich als ihren Schwerpunkt die Er¨offnung der M¨oglichkeit der physikalisch und numerisch korrekten FEM-Modellierung der dreidimensionalen Freispiegelstr¨omungen mit einem entkoppelten Algorithmus. ¨ Diese Arbeit ist allen denen gewidmet, die mittels einer inneren Uberzeugung ihre bescheidenen Kr¨afte sammeln, um trotz des sie umgebenden Skeptizismus’ und Besserwisserei etwas Neues und Gutes auf diese Welt zu bringen. In diesem Zusammenhang ist die Bedeutung der auf der n¨achsten Seite zitierten uralten ¨ostlichen Weisheit zu verstehen. Obwohl ich die meisten Zeit als selbst¨andiger Einzelk¨ampfer mit den Problemen, die mir diese Arbeit bereitete, besch¨aftigt war, gibt es eine Reihe von Personen, bei denen ich mich bedanken m¨ochte. Die Danksagungen sind auf den letzten Seiten dieser Arbeit zu finden.

Hannover, Dezember 1998 Jacek A. Jankowski

iv

Kalama Sutra

From Kalama Sutra

The Buddha’s Advice to the Kalamas of Kesaputta [17]

[...] 3. The Kalamas who were inhabitants of Kesaputta sitting on one side said to the Blessed One: There are some monks and Brahmins, Venerable Sir, who visit Kesaputta. They expound and explain only their own doctrines; the doctrines of others they despise, revile, and pull to pieces. Some other monks and Brahmins too, Venerable Sir, come to Kesaputta. They also expound and explain only their own doctrines; the doctrines of others they despise, revile, and pull to pieces. Venerable Sir, there is doubt, there is uncertainty in us concerning them. Which of these reverend monks and Brahmins spoke the truth and which falsehood? 4. It is proper for you, Kalamas, to doubt, to be uncertain; uncertainty has arisen in you about what is doubtful. Come, Kalamas. Do not go upon what has been acquired by repeated hearing; nor upon tradition; nor upon rumour; nor upon what is in a scripture; nor upon surmise; nor upon an axiom; nor upon specious reasoning; nor upon a bias towards a notion that has been pondered over; nor upon another’s seeming ability; nor upon the consideration, ’The monk is our teacher.’ Kalamas, when you yourselves know: ’These things are bad; these things are blamable; these things are censured by the wise; undertaken and observed, these things lead to harm and ill,’ abandon them. [...] 10. Come, Kalamas. Do not go upon what has been acquired by repeated hearing; nor upon tradition; nor upon rumour; nor upon what is in a scripture; nor upon surmise; nor upon an axiom; nor upon specious reasoning; nor upon a bias towards a notion that has been pondered over; nor upon another’s seeming ability; nor upon the consideration, ’The monk is our teacher.’ Kalamas, when you yourselves know: ’These things are good; these things are not blamable; these things are praised by the wise; undertaken and observed, these things lead to benefit and happiness,’ enter on and abide in them. [...]

Contents Abstract . . . . . . . Zusammenfassung . . Vorwort . . . . . . . From Kalama Sutra .

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1 Introduction 1.1 Motivation and aim of the work . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Introducing remarks . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Why a non-hydrostatic code? . . . . . . . . . . . . . . . . . . . 1.1.3 Difficulties to be overcome . . . . . . . . . . . . . . . . . . . . . 1.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Models for three-dimensional incompressible shallow water flows 1.2.2 Available methods for incompressible free surface flows . . . . . 1.2.3 Non-hydrostatic free surface flow models . . . . . . . . . . . . . 1.3 Structure of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Solution methods for three-dimensional free surface flow equations 2.1 Formulation of governing equations . . . . . . . . . . . . . . . . . . . . . 2.1.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Incompressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Turbulent stress tensor . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Boussinesq approximation . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Non-inertial co-ordinate system fixed to the earth . . . . . . . . . 2.1.6 Equation set for geophysical, incompressible flows with Boussinesq approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Hydrostatic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Scaling the vertical equation of motion . . . . . . . . . . . . . . . 2.2.2 Hydrostatic approximation and its consequences . . . . . . . . . . 2.2.3 Three-dimensional shallow water equations . . . . . . . . . . . . . 2.3 Algorithm with the hydrostatic approximation . . . . . . . . . . . . . . . 2.3.1 Operator splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Advection and diffusion steps . . . . . . . . . . . . . . . . . . . . 2.3.3 Pressure – free surface – continuity step . . . . . . . . . . . . . . 2.4 Numeric considerations concerning the solution method . . . . . . . . . . 2.4.1 General solution methods for 3D Navier-Stokes equations . . . . . 2.4.2 Overview of the solution methods with FEM . . . . . . . . . . . . v

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Contents

2.5

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2.4.3 Equal and mixed interpolation . . . . . . . . . . . . . . . . . . . . 2.4.4 Method applied in the developed model . . . . . . . . . . . . . . . Pressure treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Deriving of the conventional pressure Poisson equation . . . . . . 2.5.2 General idea of the projection method . . . . . . . . . . . . . . . 2.5.3 Pressure equation from fractional step formulation . . . . . . . . . 2.5.4 Poisson equation for the hydrodynamic pressure in free surface flows 2.5.5 Treatment of pressure and buoyancy terms . . . . . . . . . . . . . Free surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Marker-and-cell method . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Volume of fluid method . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Height function methods . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Line segment method . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.5 Combinations of different methods . . . . . . . . . . . . . . . . . 2.6.6 Treating variable domain extents . . . . . . . . . . . . . . . . . . 2.6.7 Implemented free surface algorithms . . . . . . . . . . . . . . . . Initial and boundary conditions for the flow . . . . . . . . . . . . . . . . 2.7.1 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Dynamic boundary conditions . . . . . . . . . . . . . . . . . . . . 2.7.3 Impermeability: normal velocity at solid boundaries . . . . . . . . 2.7.4 Tangential velocity at solid boundaries . . . . . . . . . . . . . . . 2.7.5 Rigid lid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.6 Bottom and free surface stresses . . . . . . . . . . . . . . . . . . . 2.7.7 Open boundary conditions . . . . . . . . . . . . . . . . . . . . . . 2.7.8 Non-reflecting boundary conditions . . . . . . . . . . . . . . . . . 2.7.9 Absorbing boundary conditions . . . . . . . . . . . . . . . . . . . 2.7.10 Boundary conditions for the tracer . . . . . . . . . . . . . . . . . 2.7.11 Boundary conditions for the hydrodynamic pressure equation . . . Algorithm for the non-hydrostatic equations . . . . . . . . . . . . . . . . 2.8.1 Equation set to be solved . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Introducing remarks . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.3 Operator splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.4 Advection step . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.5 Diffusion step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.6 Continuity step . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.7 Free surface step . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.8 Free surface from vertically integrated momentum equations . . . 2.8.9 Projection-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Numeric algorithm description 3.1 Computational domain . . . . . . . . . . . 3.1.1 σ-mesh and σ-transformation . . . 3.1.2 Comments on the σ-transformation 3.1.3 Reference element and interpolation 3.1.4 Two-dimensional elements . . . . .

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3.2 Operator splitting and time discretisation . . . . . . . . . . . . . . . . . . 3.3 Continuity step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Poisson equation solver . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Formulation of the pressure Poisson equation in the finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Resulting divergence-free velocity field . . . . . . . . . . . . . . . 3.3.4 Source term of the pressure Poisson equation . . . . . . . . . . . . 3.4 Diffusion step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Diffusion step: formulation in finite elements . . . . . . . . . . . . 3.4.2 Boundary terms in the diffusion step . . . . . . . . . . . . . . . . 3.4.3 Boundary conditions for the diffusion step . . . . . . . . . . . . . 3.4.4 Computation of the source terms . . . . . . . . . . . . . . . . . . 3.5 Advection step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Advection with the method of characteristics . . . . . . . . . . . . 3.5.3 Advection with SUPG method . . . . . . . . . . . . . . . . . . . . 3.5.4 Boundary conditions for advection step . . . . . . . . . . . . . . . 3.6 Free surface step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Free surface with the method of characteristics . . . . . . . . . . . 3.6.2 SUPG formulation for the kinematic boundary condition . . . . . 3.6.3 Explicit formulation for the conservative free surface equation . . 3.6.4 Semi-implicit formulation for the conservative free surface equation 3.6.5 Free surface stabilisation . . . . . . . . . . . . . . . . . . . . . . . 4 Verification of algorithm stages 4.1 Diffusion step test: Ekman profiles . . . . . . . . 4.1.1 Slope current . . . . . . . . . . . . . . . . 4.1.2 Drift current . . . . . . . . . . . . . . . . . 4.1.3 Numerical test: drift and slope current . . 4.2 Tests of the Poisson equation step . . . . . . . . . 4.2.1 Testing the boundary conditions influence 4.2.2 Testing the source term influence . . . . . 4.3 Tests of the free surface step . . . . . . . . . . . . 4.3.1 Kinematic boundary condition . . . . . . . 4.3.2 Conservative free surface equation . . . . . 5 Model verification 5.1 Waves reflection . . . . . . . . . . 5.2 Wind-driven circulation . . . . . 5.3 Interfacial internal waves . . . . . 5.3.1 Theory . . . . . . . . . . . 5.3.2 Test realisation . . . . . . 5.4 Lock exchange flow . . . . . . . . 5.4.1 Hydrostatic interpretation 5.4.2 Lock exchange flow speed

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Contents

5.5

5.6 5.7 5.8 5.9

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5.4.3 Lock exchange flow – further considerations . . . . 5.4.4 Test realisation . . . . . . . . . . . . . . . . . . . . Standing wave in a closed basin . . . . . . . . . . . . . . . 5.5.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Deep water – short waves . . . . . . . . . . . . . . 5.5.3 Shallow water – long waves . . . . . . . . . . . . . 5.5.4 Comparison of different free surface algorithms . . . Subcritical and supercritical flow over a ramp . . . . . . . Channel with a bump . . . . . . . . . . . . . . . . . . . . . Waves travelling over an underwater channel . . . . . . . . Solitary wave in a long channel . . . . . . . . . . . . . . . 5.9.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . 5.9.2 Solitary wave propagation over a flat bottom . . . . 5.9.3 Solitary wave in a long channel with varying depth 5.9.4 Collision between two solitons . . . . . . . . . . . . Limits of application . . . . . . . . . . . . . . . . . . . . . 5.10.1 Free surface breaking . . . . . . . . . . . . . . . . . 5.10.2 Bottom gradients for supercritical flow . . . . . . . 5.10.3 Breaking of a solitary wave . . . . . . . . . . . . . .

6 Summary and Conclusions 6.1 Realised new developments . . . . . . 6.2 Result discussion . . . . . . . . . . . 6.3 Model applications and limitations . 6.4 Recommended further developments .

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A Various derivations A.1 Incompressibility . . . . . . . . . . . . . . . . . . . . . . . A.2 Coriolis terms . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Small amplitude free surface waves in a homogeneous fluid A.4 Equations of state for seawater . . . . . . . . . . . . . . . A.4.1 EOS–80: UNESCO equation of state . . . . . . . . A.4.2 Simple forms of the equation of state . . . . . . . . A.4.3 Suspended sediment influence on the water density A.5 Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Systematic approach to model development . . . . . . . . .

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Nomenclature

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Danksagung

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Chapter 1 Introduction 1.1 1.1.1

Motivation and aim of the work Introducing remarks

The physical modelling of free surface flows in a hydraulic laboratory has a very long tradition, but in the recent 30 to 40 years the numerical simulation of free surface flows gained more and more in importance. As numerical methods of computational fluid dynamics (CFD) developed to a considerable level and the computational power of computers compared to their price grew as well, it seems that the physical models are getting too expensive and outdated. Indeed, their role transformed to the provision of verification, calibration and support data sets for numerical modelling, especially for specific flow configurations or aspects. For modelling flows on the geophysical scale, such as whole rivers or lakes, or for assessing the results of anthropogenic activities in nature, the hydrodynamic-numerical models are nowadays indispensable. Supported by numerous advanced pre- and post-processing programs, the commercially available computational fluid dynamics codes became standard engineering tools. There are also some negative aspects of this situation. The rapid development of userfriendly interfaces makes even the most complex codes available also to non-specialists. Colourful animations help to obtain a wider public approval of the results. In effect, the risk of misuse and error has grown considerably, since the limitations of the numerical algorithms and other uncertainties are not adequately taken into consideration and clearly presented. The currently available tools are still far from being entirely satisfactory. There are still numerous flow cases, especially three-dimensional free surface flows, which cannot be simulated without a non-standard approach. The trend to take advantage of the possibilities of new computers yielding higher and higher accuracy, larger and larger storage and ever-growing computation speed is very important, but one cannot rely on them as the only source of improvements. Progress will be achieved when a considerable effort is also invested in improving, extending or even replacing basic algorithms by newer, better ones. 1

2

1.1 Motivation and aim of the work

Another major issue in computational fluid dynamics is securing the quality of the codes. The modern programs consist of large procedure or module libraries concerning various disciplines (e.g. hydrodynamics, water quality, sedimentology) and controlling the overall quality may be very difficult. Guidelines for widely acknowledged methodology for validation processes, which should be absolved by the computer codes, are presently being prepared by national and international associations connected with hydraulic research.

1.1.2

Why a non-hydrostatic code?

At present, one- and two-dimensional numerical models are widely used in engineering practice. As computer capacities grew, the application of computationally more intensive three-dimensional models became feasible. Some of these models are thought of as an extension of two-dimensional (vertically integrated) models in order to describe selected, but not all, three-dimensional properties of the flow. In most cases, where the vertical acceleration component is small, sufficient accuracy for simulating most geophysical free surface flows is obtained by using numerical models, which assume the hydrostatic pressure approximation. It simplifies the coupled three-dimensional set of equations to be solved and therefore allows development of computationally efficient three-dimensional codes. The hydrostatic models are very successful, for example, in describing horizontal velocity profiles in vertical direction or stratified currents in mostly horizontal flows. Nevertheless, these models cannot be pushed beyond their limits of application. There are situations, where the vertical acceleration, and therefore the non-hydrostatic pressure component cannot be neglected. This is the case when, for example, flows over abruptly changing bottom topography, orbital movements in short wave motions, or intensive vertical circulations are to be considered. Generally, the application of hydrostatic approximation is disputable, when the ratio of vertical to horizontal motion scales (velocities and distances) is not small. This approximation is neither appropriate in flows around obstacles, when the pressure at the boundary is not hydrostatic, nor when the flow is determined by stronger density gradients enhancing vertical movements. The motivation of this work is to describe properly these free surface flows, where the hydrostatic approximation is no longer valid. In many cases assumptions due to the basic equations form or to the applied numerical techniques (especially the mesh structure) cause limitations on the domain boundary geometry. A classical example is the severe constraint on the bottom or free surface steepness in models using shallow water equations. Other geometrical assumptions influence strongly the treatment of the moving domain boundaries or surfaces separating fluids with different properties. An additional consequence of using the shallow water equations is that they provide correct solutions for long waves only. The aim of this work is to develop a three-dimensional numerical model for these incompressible geophysical free surface flows, where the solution of entirely three-dimensional Navier-Stokes equations is required. The limitations of hydrostatic codes are to be overcome by taking the unchanged momentum equation for the vertical velocity component into consideration. The free surface movement computation should not be constrained

1.1 Motivation and aim of the work

3

to the long waves only. Development should take place in the framework of an existing, advanced, and well-validated model system, allowing immediate applications in complex geometries not only for the hydrodynamics, but also for e.g. transport phenomena and taking turbulence models of various complexity into consideration. Therefore, an existing finite element three-dimensional hydrostatic code, Telemac-3D, was taken as the starting point [74]. Precisely, it solves the three-dimensional, shallowwater, Reynolds-averaged, Navier-Stokes equations for incompressible, turbulent, free surface flows with Boussinesq approximation. To the typical application domain of this code belong geophysical free surface flows with complex geometry, where the full advantage of the finite element approach can be taken. Telemac system was developed by Electricit´e de France (Laboratoire National d’Hydraulique) using strict quality precautions. It has been well validated in numerous practical, mostly fluvial and marine applications including some advanced ones, as for example, the sediment transport in tidal estuaries [103, 110, 111]. Historically, Telemac3D code was developed from a two-dimensional code Telemac2D [52]. For instance, its two-dimensional routines are applied in order to compute the free surface movements in the three-dimensional code. Telemac3D is characterised by the operator-splitting technique of solving the hydrodynamic equations and the modular structure of the algorithm based on libraries of numerical procedures of a broad application scope which include modern equation solvers. This open structure allows efficient development of a new algorithm by replacing or building in additional steps to the existing algorithm frame. The main limitations of the Telemac3D code, encountered by applications at Institut f¨ ur Str¨omungsmechanik und ERiB were connected with hydrodynamics of areas with steep bottom gradients and buoyancy-driven currents, for example, in estuaries [110]. Similar, though less serious problems, were found when forecasting sediment transport by bottom-near deep ocean currents over variable bathymetry, where neglecting the vertical acceleration is not appropriate [84, 85]. Particular situations, where the models using the hydrostatic approximation are inadequate, are numerous. Examples include: interaction of currents with structures (e.g. pillars, pipelines); flows over larger bottom gradients; various aspects of sediment transport (erosion and consequent deposition, e.g. travelling bottom dunes or erosion behind bridge pillars); complex three-dimensional currents (e.g. secondary currents in river meanders) and finally a wider spectrum of internal and surface wave problems (short, non-hydrostatic waves, e.g. in connection with deposition and erosion of sediments in harbours). Most of the problems mentioned above concern flows in complex geometries, where the finite element method is the most effective one. A need to open the possibility of modelling these flows in the framework of the existing finite element models is clearly seen. The real challenge, however, is to provide a hydrodynamic-numeric model which could simulate the phenomena mentioned above simultaneously and still remain computationally feasible for practical applications.

4

1.1.3

1.2 Literature review

Difficulties to be overcome

The difficulties encountered by the development of fully three-dimensional numerichydrodynamic codes with a variable free surface are manifold. But three of them are the most important. First of all, in the past the prohibitive computation times reduced the application of the three-dimensional models only to academic test cases, hampering broader interest in them. Nowadays, new efficient algorithms for solving large equation systems and, above all, the available computational power allow application of these very complex models for scientific purposes. However, engineering practice also requires for those special cases that cannot be dealt with without an entirely three-dimensional approach, that the codes remain computationally feasible. Secondly, the complexity of these models is greater than awaited by a simple extension to a third dimension or by taking another equation into consideration. The hydrodynamic equations describing flows in domains with variable geometries are coupled in a complex way. Especially the coupling of the movements of the free surface and the interior solution brings a lot of difficulties. The changeable free surface not only defines the new computational geometry, but also influences the solution through boundary conditions for all flow variables. Last, but not least, the barotropic pressure gradients caused by free surface slopes are the main driving force for the shallow water flows. Additionally, the large diversity of presently available numerical methods makes it difficult to choose the optimal ones. The third source of difficulty is the fact, that not only equations, but also boundary conditions for flow variables are complex and coupled as well, and must be treated with the greatest care in order to secure a stable and physically meaningful solution. The boundary conditions for the pressure belong to the most difficult to be properly posed. Additionally, the boundary conditions must be applied on the moving boundaries. Last, but not least, the initial conditions must fulfil important mathematical constraints.

1.2

Literature review

In contrast to the flows with fixed boundaries, a basic requirement for analysing free surface flows is the determination of both velocity and pressure fields as well as the variable position of the free surface simultaneously. The shape of the deformable domain is a part of the solution to be achieved. The literature on the subject is large and a selection of relevant domains of interest as well as concentration on ideas and good examples, not a complete and chronological development description is necessary. Leaving specific details for next chapters, the following literature review concentrates on the most important and relevant aspects of the model presented in this work. For clarity and briefness, the vast field of internal (incompressible or compressible) flows and turbulence modelling is left (almost) untouched.

1.2 Literature review

1.2.1

5

Models for three-dimensional incompressible shallow water flows

One-dimensional (1D) and two-dimensional (2D, e.g. depth-integrated) models used in most geophysical and engineering applications are very well advanced and achieve satisfactory results in their domain of application [1, 166]. Their advantage is and will be computational efficiency and easy implementation with satisfactory accuracy for most (for one dimension, usually fluvial) practical applications. However, an important aspect that must be taken into consideration while applying these models is how to parameterise three-dimensional effects appropriately. As illustrative examples of well validated modern models for two-dimensional shallow water equations, the finite difference Trim2D model developed by Casulli [19], and finite element model Telemac2D developed by Galland, Goutal and Hervouet [52] or Tisat developed by Lang [97] should be mentioned. The first applicable three-dimensional (3D) models were developed as early as 1970 [34]. From this time on, 3D models found their way into various geophysical and engineering disciplines, but their application remained limited due to their usually high computational needs. For free surface flow, a well established type of model for three-dimensional shallow water flow (i.e. with hydrostatic approximation) exists [68, 120]. The majority of these models are developed as an extension of two-dimensional models in order to describe some chosen three-dimensional properties of the flow. The attractivity and strength of these models lies in the possibility of taking various boundary conditions at the free surface (wind, thermal radiation) and at the bottom (shear) into consideration in a natural way, so that bottom and free surface layers are resolved. They also allow modelling of stratified flows (baroclinic pressure gradients). A good example of a three-dimensional, finite difference (FD) model with σ-co-ordinates, turbulence modelling, the hydrostatic approximation and applied to coastal ocean problems was presented by Blumberg and Mellor (Princeton Ocean Model, POM) [11]. A special strategy, allowing separation of baroclinic and barotropic motion modes (mode splitting) is available to reduce computational cost. Another example of such a model is the one used to compute tidal flows with wind influence in the Irish Sea by Davies and Lawrence [33]. An extension of the Trim2D model to three dimensions, Trim3D by Casulli at al. [23, 27, 22] is probably the best example of an optimally developed FD model for threedimensional shallow water equations. It uses a semi-implicit formulation – the gradients of the free surface and velocity divergence are finite-differenced implicitly and the remaining terms explicitly. The convective terms are treated by a Langrangian-Eulerian method. This combination of semi-implicit formulation for propagation of gravity waves and Lagrangian methods yields an unconditionally stable model, well-suited for engineering purposes, especially for coastal regions with tidal flats. However, when baroclinic forcing (stratification) must be taken into consideration, there are limits to this model’s stability. The assumption of extremely simple, structured and rectangular mesh of equally-spaced squares allows a formulation of a very efficient and fully vectorisable code, but requires care when modelling flows in complex geometries.

6

1.2 Literature review

Telemac3D, developed in Laboratoire National d’Hydraulique in Chatou, France, (EDF), represents a state-of-the-art finite element (FE) model for three-dimensional shallow water equations [83]. It is based on the operator-splitting scheme, where the governing equations are split into parts having clear mathematical properties and treated using the most appropriate methods: e.g. semi-implicit FEM for diffusive parts, or a Lagrangian approach (method of characteristics) for advective parts. For computational efficiency, the simplest prismatic 3D finite elements with linear functions and unstructured mesh based on the σ-co-ordinate system are applied. For treatment of the resulting matrices, a fully vectorisable element-by-element method is implemented. For additional treatment of non-linearities iterative procedures are provided. The overall stability of this method is guaranteed, when the consecutive algorithm steps are also stable. The computational effort compared to the FDM-based models is surely larger, but FEM uses all the advantages of an unstructured mesh for resolving complex flow geometries. The 3D shallow water models use various formulations for finding free surface position – mostly from the vertically integrated continuity equation (conservative free surface equation), as Trim3D or POM, or from two-dimensional shallow water equations, as Telemac3D.

1.2.2

Available methods for incompressible free surface flows

The application limits of the three-dimensional shallow water models mentioned above are twofold. Firstly, the vertical accelerations in flows are neglected. Secondly, especially for the free surface schemes, the bottom and wave slopes in the main direction of the flow must be small. In order to overcome these limitations, an approach free of hydrostatic approximation is needed for the internal flow. Techniques as independent as possible from the bottom and free surface gradients are required for tracking the free surface movements (which also determine the variability of the computational domain). For this purpose, on the one hand the techniques developed for arbitrary incompressible three-dimensional internal flows and, on the other hand, schemes developed for more advanced free surface wave problems become attractive. Reviews of the present development and state-of-the-art in modelling incompressible flows are provided for example by L¨ohner [106] and by Gresho [58]. As examples of recent outstanding computations, the simulations of flows around cylinders in a duct by Glowinski and Pironneau [55], simulations of free surface flow extruding from a duct by Wambersie and Crochet [165] and flows in complex duct geometries by Tu and Fuchs [158] and Kwak et al. [93] can be mentioned. A discussion of the intensive computational demands of 3D-codes for fluid dynamics is provided by Nieuwstadt [119]. Four general methods of solving three-dimensional Navier-Stokes equations exist. First of all, from a direct discretisation of the governing equations a large global equation system for the primitive variables can be obtained. The resulting equation system cannot be solved using standard methods, and non-linear, iterative (e.g. Newton–Raphson) solvers

1.2 Literature review

7

must be applied [49, 50, 80, 168]. This direct solution method can be computationally very intensive, especially for three-dimensional problems. In a second group of methods, a decoupled approach is preferred, where instead of one large global equation system a few smaller, linear equation systems are to be solved. This is achieved by solving the governing equations in consecutive steps. Various related methods of decoupling exist; operator-splitting [52], pressure correction methods (SIMPLE) [124, 123], pressure methods [18, 24], projection methods [29, 57, 143], fractional step method [129], and finally Taylor methods [37, 38, 86]. A third group of methods, mostly for stationary flows, is based partially on developments in the domain of compressible flows and introduces artificial coefficients. In the artificial compressibility method, the set of equations for compressible flows is applied; when a steady state is achieved, the partial derivative of pressure in the compressible form of the continuity equation disappears [78, 148]. In the penalty function methods it is assumed that the pressure changes proportionally to the velocity divergence. The artificial proportionality parameter, bulk viscosity, must assume very large values. In theory, the numerical solution converges to the steady state for an infinite bulk viscosity [28, 133]. The last group of methods uses non-primitive variables, as a stream function or vorticity. Introducing them into the governing equations one obtains a set of transport and Poisson equations [3, 47]. The number of equations to be solved is actually reduced, but the form of boundary conditions may be very complex. For dealing with flows in stratified systems consisting of layers of water of different density and physiochemical properties (oceans, lakes) a very promising approach based on isopycnic co-ordinates has been developed. A model of this type consists of a set of interior layers of constant density, giving it the character of a stacked shallow water model. Special techniques are provided for lenses and outcrops of layers and for taking diapycnal mixing between the layers or up-welling/down-welling into account. An example is the MICOM model developed by Bleck et al. [9, 10] or a model by Casulli [21]. For simulations in a time-variable domain, the Arbitrary Lagrangian-Eulerian (ALE) Method has found greater popularity, especially for FEM applications. The main idea of this method is to adapt the mesh to the movements of the domain boundaries using an arbitrarily chosen mesh velocity [36]. The σ-transformation is a simple form of the ALE scheme, where the mesh spacing varies only in one direction (vertically) and is dictated by the free surface movements. Another approach is to use a general, deforming spacetime (FE) formulation in which the mesh movement is taken into account automatically [149]. Tracking of the free surface movements (or movements of an interface between fluids) can be performed using two general approaches: by balancing or tracking the fluid volume, or by controlling the free surface movements itself. Mixed methods are also available. Two methods for volume tracking exist. In principle, both allow arbitrary movements of the free surface. Massless marker particles moving passively with the fluid are applied to trace the fluid volume movements in a fixed mesh in the marker-and-cell (MAC) method,

8

1.2 Literature review

developed by Harlow and Welch [65]. The volume-of-fluid (VOF) method, introduced by Hirt and Nichols [79], uses a function which describes the filling of a given computational cell with a fluid. This function is advected with the fluid velocity. Both methods require additional geometrical algorithms to find the actual approximate shape of the interface in the partially filled, i.e. boundary cells. Method-specific additional techniques must be noted: an algorithm to redistribute the markers when required (MAC) or special bookkeeping adjustment to avoid over-filled or over-emptied cells (VOF). The surface-tracking algorithms use the kinematic boundary condition or the conservative free surface equation (being a vertically integrated continuity equation, and taking therefore also the fluid volume into consideration), e.g. [18, 22]. The searched variable is a height function, i.e. the position of the fluid surface (or interface) relative to a defined level plane. The main limitations of these methods are due to the mathematical requirement for the height function – it must be a single-valued function. In consequence, these methods cannot simulate breaking waves or very steep wave slopes, but allow tracking of a much sharper interface than VOF or MAC.

1.2.3

Non-hydrostatic free surface flow models

Not surprisingly, the first efforts to obtain free surface flow models which were free of limitations due to hydrostatic and/or shallow-water approximation, concerned entirely three-dimensional circulating flows, wave problems and larger movements of the free surface. Leschziner and Rodi computed in 1979 the 3D helicoidal secondary currents in a channel bend using k − ε turbulence model [104]. They applied a horizontal fictitious plane representing a free surface and applied known pressure gradients there in order to simulate free surface slopes in an indirect way. Frederiksen and Watts [50] developed in 1981 a finite element model with elements having dimensions in time and space to deal with transient free surface incompressible flow with examples for entrainment and circulating flows. Kawahara and Miwa [89] presented in 1984 a two-dimensional FEM model able to simulate travel and run-up of waves on a breakwater. Bulgarelli, Casulli and Greenspan [18] published in 1984 the pressure finite difference method for incompressible and compressible free surface flows with examples of stratified circulating flows. A very typical domain, where the non-hydrostatic conditions combined with a Coriolis effect are very important, are convective flows in the 0geophysical scale, for example down-welling and up-welling in oceans and lakes. Local convection events in the benthic ocean caused by episodic hydrothermal discharges from an oceanic ridge were modelled by Lavelle and Baker [98]. They used a primitive-equation, non-hydrostatic, finite difference model taking advantage of the axial symmetry of these plumes (confirmed by measurements), allowing the use of cylindrical co-ordinates (r, z). Therefore, the equations were integrated in a rectangular (r, z)-grid, reducing the problem to two dimensions.

1.2 Literature review

9

Deep water renewal in temperate lakes triggered by a storm surge (when some of the relatively cold upper water column is forced downwards through its compensation depth so that it becomes unstable and sinks) was simulated by Walker and Watts [163] with a high resolution, three-dimensional non-hydrostatic FD Boussinesq model. In order to model the density driven spiralling (rotating) plumes of the sinking cold surface water, not only the non-hydrostatic conditions had to be taken into consideration, but also the fully three-dimensional Coriolis force, i.e. both vertical and horizontal components of the earth rotation vector. The code was based on a revised semi-implicit method for pressure linked equations, as in the SIMPLE algorithm family. A similar approach in FEM was applied by Forkel [48] for investigation of material transport in lakes with complex bathymetry. Open ocean deep convection in convective cells due to strong cooling of the ocean surface were modelled by Sander et al. [136]. They also used a non-hydrostatic, staggered-grid, finite difference model with both vertical and horizontal Coriolis coefficients taken into consideration. The algorithm was, as in the previous case, based on pressure-linked equations. In almost all these applications, however, the movements due to the gravitational waves, treated as unimportant for the problem, were eliminated by ‘freezing’ the free surface. The rigid lid approximation was used, so that only the internal movement was resolved. The rigid lid approximation restricts therefore the applicability of these models to the flows where the barotropic component of the hydrostatic pressure can be neglected. For example, this is not valid in tidal flows. An algorithm using the idea of pressure decomposition into hydrostatic (i.e. including barotropic and baroclinic) and hydrodynamic parts for the geophysical flows in a finite difference implementation was presented by Casulli and Stelling [24]. In their code the free surface is found from the conservative free surface equation using provisional velocity fields resulting from the hydrostatic algorithm step and from the previous time-step solution. The hydrodynamic pressure is found from a Poisson equation. Verification results of models based on pressure methods were presented recently by Casulli [20], Casulli and Stelling [25] and Gaarthuis [51]. Developments using FEM-based projection (i.e. decoupled) methods have been reported very recently by Schr¨oder [140] and Chen [26]. They apply mixed-interpolation finite elements and use rigid-lid approximation or obtain the free surface from kinematic boundary condition. The hydrodynamic equations are decoupled by application of an equation for the global pressure. A fully three-dimensional oceanographic development for modelling ocean basins with movements of the free surface was presented by Mahadevan et al. [108]. They introduced the movements of the free surface in order to capture properly up-welling and downwelling in the oceanic mesoscale flows (models with open boundaries). They did this in order to study the thermohaline structure of the ocean that is sensitive to vertical circulations and bathymetry variability. In conclusion, it was found that it is essential to retain the commonly neglected component of the Coriolis acceleration in the vertical momentum equation. The application case for the Gulf of Mexico basin has shown that

10

1.3 Structure of the work

a non-hydrostatic model without the rigid-lid approximation usual in the oceanography and allowing faster waves performs well in reproducing a flow strongly influenced by the bathymetry [109]. Additionally, it was demonstrated that a non-hydrostatic model is well-posed and does not require a special treatment of the open boundaries like a hydrostatic one (e.g. enhanced viscosities at the inflow and outflow). Another method of modelling free surface flows with non-hydrostatic effects is to use a fully coupled method, which solves Navier-Stokes, continuity and free surface (kinematic boundary condition) equations. This method, implemented by Yost [168] as a finite element model, uses higher order interpolation functions and is associated with solution of large equation systems, making it computationally intensive. An important branch of non-hydrostatic models are two-dimensional models (in the vertical plane) aimed at wave problems and large free surface movements. Huerta and Liu [80] apply the the ALE method and mixed SUPG FEM for computation of flows with large surface movements – tsunami and dam breaks. For solving the resulting coupled equation system, they apply a predictor-multi-corrector scheme. Ramaswamy and Kawahara [131, 130] and Ramaswamy [132] develop a model based on the ALE technique which is able to deal with various wave problems and density currents with a free surface as well. They use a velocity correction method based on the Poisson pressure equation and find the free surface from the kinematic boundary condition, so that their algorithm belongs to the category of decoupled methods. In FDM, one of the most interesting applications of a decoupled method for numerical studies of breaking waves and hydraulic jumps was presented by Lemos [102]. The continuity is secured by iteration between pressure and velocity performed locally in each computational cell after obtaining an intermediate solution from the momentum equations, until the residual divergence disappears. The free surface is updated and defined by the VOF method and the turbulent processes are described by k − ε–model.

Tezduyar, Behr, et al. apply the general, deforming space-time (FE) formulation method for free surface flows and flows with an interface between two fluids [149, 150, 151]. Behr and Tezduyar also present special FE strategies allowing a feasible application of this intensively coupled method for larger scale applications using massively parallel computers [6].

1.3

Structure of the work

In the particular case of the model described in this work, the physical systems to be simulated are geophysical free surface flows in complex geometries. The model variables of interest are three or two-dimensional fields: velocity, free surface elevation and concentrations of transported substances. Due to the model assumption that the water in most geophysical flows can be treated as an incompressible, Boussinesq fluid, the pressure can be treated as a subsidiary variable. The transported substances can be divided into passive tracers, which move passively with the fluid, active tracers, which influence the fluid density and in consequence the flow, and particles with special properties, as sediments having their own settling velocity.

1.3 Structure of the work

11

The applied general numerical method, the finite element method, is one of the most appropriate methods to treat complex flow geometries. The algorithm structure allows a proper treatment of various boundary conditions appearing in nature. Due to the assumed structure of the mesh (σ-mesh) another important model assumption is met corresponding to the free surface (and the bottom as well). It must be represented by a single value function, excluding, for example, breaking waves problems from the model application domain. Additionally, the lateral boundaries are represented by vertically oriented surfaces. The conceptual model is highly influenced by the numerical method applied to solve the governing hydrodynamic equation set describing the physical system in a complex, timevariable geometry and with various boundary conditions. The algorithm of Telemac3D is based on the operator-splitting technique, resulting in a modular and open structure of the solution procedure. Parts or stages of the algorithm can be switched on or off according to the simulation requirements. When new methods are available, these stages can be consequently replaced, modified or even totally abandoned. The only disadvantage of this approach is a large number of algorithmic or software parameters to steer the complex program and forcing the user to perform test runs in order to find an optimal set for best numerical properties for a particular case. The hydrostatic Telemac3D algorithm assumed additionally that the hydrostatic approximation is valid and that the free surface can be obtained from shallow water equations. It limited the model application domain to shallow water flows without larger free surface and bottom gradients. In consequence, the development of the non-hydrostatic algorithm can be understood as an extension of the model application domain to more complex physical systems without changing the basic conceptual model properties (e.g. concerning the geometry or fluid properties) or algorithm structure. The existing previously hydrostatic algorithm has been thoroughly verified using various verification test cases covering the whole targeted application domain for this model, e.g. an inviscid channel flow with a bump, the Ekman spiral [41] test, an interfacial standing internal wave, and wind-driven currents, etc. These tests are very formally documented in a normalised way so that they can be also be used for example as implementation tests by installing the software on a new computer. Similar tests must vindicate all new developments to assure model quality. Therefore, the new algorithm steps have been carefully verified using analytical solutions. The verification of the algorithm as a whole has been done using test cases, which cover the targeted model application domain. The benchmark tests allow in most cases a formal comparison with an analytical problem solution. The tests include free surface and internal waves, sub- and supercritical channel flow over a steep ramp, wind- and buoyancy-driven currents. Accordingly, this work follows the strict development rules of a hydrodynamic-numeric model: • Chapter 1 (i.e. this chapter) defines the aim of the work, provides a literature review of the relevant state-of-the-art methods of modelling free surface flows and

12

1.3 Structure of the work

presents the structure of this work in the light of the systematic approach to a quality-ensuring hydrodynamic-numeric model development. • Chapter 2 reviews the basic theory of the incompressible flows as well as discusses methods of solving the Navier-Stokes equations in more detail with an emphasis on the finite element method. The techniques for representing and tracking the free surface and implementing of various boundary and initial conditions are presented. It supplies a description of the concepts and the algorithms of the previously existing model and the currently developed one abstracting from their numerical implementation. This chapter can be treated as a formal definition of the physical system and description of the conceptual model with all model parameters and data required. • Chapter 3 provides more detailed documentation of the applied numerical methods and software developments. • Chapter 4 consists of tests of the developed algorithm stages treated as separate units, due to the modular algorithm structure. It provides a strict software and algorithm verification with an heuristic investigation of the stability constraints. • Chapter 5 performs the verification of the model using tests showing its possibilities in reproducing non-hydrostatic free surface and internal flows and investigates application limits. Additionally, the test cases provide assessments of model accuracy and reliability. • Chapter 6 provides a summary and conclusions, as well as proposals for applications which would allow formal model validation. Finally, future research possibilities are sketched. • Appendix consists of a few sections providing additional information.

Chapter 2 Solution methods for three-dimensional free surface flow equations The pressure is a somewhat mysterious quantity in incompressible flows. [...] It is in one sense a mathematical artefact – a Lagrange multiplier that constrains the velocity field to remain divergence-free; i.e. incompressible – yet its gradient is a relevant physical quantity: a force per unit volume. It propagates at infinite speed in order to keep the flow always and everywhere incompressible; i.e. it is always in equilibrium with a time-varying divergence-free velocity field. [64] This chapter provides the definitions of the physical system and the conceptual model, as well as an outline of numerical algorithms applied to solve the three-dimensional free surface flow equations. It describes: • the governing equations with a stress on the approximations leading to the particular form of the basic equations applied in the model (which also define model limitations); • algorithms for solving Navier-Stokes equations and considerations concerning the relevant aspects of the finite element method for decoupled schemes; • free surface tracking algorithms; • required specific initial and boundary conditions; • algorithms of the hydrostatic (previous) and non-hydrostatic (new) models presented abstracting from their numeric implementation (described in chapter 3). 13

14

2.1 2.1.1

2.1 Formulation of governing equations

Formulation of governing equations Basic equations

As the starting point, the conventional set of the governing hydrodynamic equations for turbulent geophysical flows is considered [5, 53, 92, 128]: Conservation of mass:

1 d̺ +∇·u= 0 ̺ dt

(2.1)

∂u 1 + u · ∇u = ∇ · σ + g − 2Ω × u + f ∂t ̺

(2.2)

Conservation of momentum:

Transport equation for a tracer: ∂T + u · ∇T = ∇ · (ν T ∇T ) + qT ∂t

(2.3)

Equation of state (usually an empirical one): ̺ = ̺(Ti , p)

(2.4)

The equation set above is formulated for geophysical free surface flows in the non-inertial, orthogonal Cartesian co-ordinate system connected with the surface of the earth (Ω is the earth’s angular speed). For typical applications, x is directed eastward, y northward, and z points vertically upward in the direction of −g, the acceleration of gravity, figure 2.1. The velocity vector u has components (u, v, w). T describes a tracer: temperature, salinity, or passive effluent concentration transported by the flow. Tracers may influence the water density, as the pressure p does (active tracers, noted Ti in eq. 2.4), or not (passive tracers). The term f represents forces applied on the fluid, other than pressure and weight. The momentum conservation equations and the continuity equation (2.1–2.2) are written in a well-known form of Reynolds-averaged Navier-Stokes (NS) equations for Newtonian fluids [118, 146, 134], where the Boussinesq eddy-viscosity concept [12] is introduced. Strictly speaking, the velocity and pressure are stochastic variables devoid of turbulent fluctuations. The transport equation for the tracer(s) (2.3) coupled through the equation of state (2.4) with Navier-Stokes equations is introduced to deal with flows with variable density. The transport equation for tracers has undergone similar treatment concerning the turbulence concepts. In this formulation, the turbulent stress tensor σ has a form similar to the one valid for laminar flows of Newtonian fluids: σ = −pI + 2̺νD

(2.5)

ν is the turbulent eddy-viscosity tensor and ν T the turbulent diffusion tensor for tracers. They describe the effective viscosity and diffusion coefficients taking into consideration

2.1 Formulation of governing equations

15

free surface S(x, y, t) z, upwards

   =

domain Ω

6

g ?

O

 y, >    

North

*   

bottom B(x, y) -x,

QQ k Q Q

lateral boundary (vertical)

East

Figure 2.1: The computational domain. z-axis directed upwards, parallel to –g. the molecular effects and the fluid turbulence as well. D is the Navier-Stokes (deformation rate) tensor; p describes the global pressure. νD is a second order tensor multiplication with a contraction. The mathematical theory of the Navier-Stokes equations (2.1–2.2) is not complete, especially concerning the solution uniqueness, regularity and precise definition of boundary conditions [30]. The non-linear character of the advection term u · ∇u produces and maintains instabilities, for large Reynolds numbers Re exciting a wide scale of motions. The equations (2.1–2.4) are formulated for a domain Ω limited by three general types of boundaries: the bottom, the free surface and the lateral boundary. In the actual model implementation, they must fulfil a few important constraints. Namely, B(x, y), describing the bottom and S(x, y, t) representing the fluid free surface, are single-valued functions with respect to the horizontal surface OXY (2.1). The lateral boundaries are assumed to be vertical, so that they can be defined by a closed polygon L(x, y). The vertical projection of the domain on the horizontal surface OXY Ω0 is clearly defined and: Ω = {(x, y, z) ∈ R3 , (x, y) ∈ Ω0 ∧ B(x, y) ≤ z ≤ S(x, y, t)} (2.6) The lateral boundary can be a solid wall or a liquid, artificial open boundary introduced in order to truncate the domain. These constraints allow a clear and simple definition of the computational domain and the application of boundary conditions.

2.1.2

Incompressibility

One of the most widespread approximations in the geophysical flows is to treat water as an incompressible fluid. This approximation is based on a physically well-supported assumption that the water density is independent of pressure. The fluid density variations due to temperature, salinity and/or suspended matter are taken into account solely through the equation of state (2.4). A more precise discussion of the non-trivial conditions under which such a simplification is possible is given in the appendix, section A.1. The mathematical consequences of this approximation are paramount. When the

16

2.1 Formulation of governing equations

fluid is incompressible, the velocity field describing its movement is non-divergent (or solenoidal): ∇·u=0 (2.7) Incompressibility implies that fluid volume rather than fluid mass is conserved. As a further consequence, the hydrodynamic models based on this approximation assume that the volume must be conserved while the density (and so the mass) changes freely as a function of active tracers (temperature and salinity), but not pressure.

2.1.3

Turbulent stress tensor

The turbulent stress tensor is assumed to have a form analogous to the one proposed by Boussinesq (eddy-viscosity principle) [12]. Physically speaking, −σij is the amount of the i-th-component of the momentum flowing per unit time through a unit area normal to the jth direction. The equation 2.5 is obtained as follows. For laminar flows: σ = −pI + 2̺ν f D

(2.8)

The Navier–Stokes tensor D is: D=

 1 1 ∇u + (∇u)T − (∇ · u)I 2 3

(2.9)

The viscosity tensor ν f is assumed to be diagonal. For incompressible flows (∇ · u = 0), the components of the stress tensor σ are given by: σij = −pδij + ̺νf j (∂ui /∂xj + ∂uj /∂xi )

(2.10)

When the Navier-Stokes equations are Reynolds-averaged, an additional term appears in (2.10) due to turbulent fluctuations – the Reynolds stress tensor Rij = −̺u′ i u′ j . In the Boussinesq eddy-viscosity concept the Reynolds tensor is approximated to: Rij = −̺u′ i u′ j = +̺νtj (∂ui /∂xj + ∂uj /∂xi )

(2.11)

where ν t is the turbulent viscosity coefficient which varies in time and space and is obtained from a turbulence model. Adding (2.11) to (2.10) yields: σij = −pδij + ̺νj (∂ui /∂xj + ∂uj /∂xi )

(2.12)

where ν = ν f + ν t is the effective eddy-viscosity coefficient. For simplification, the turbulent eddy viscosity tensor ν is assumed to be diagonal, with three components νx , νy , νz . (The turbulent diffusion tensor for tracers assumes the same form.) It can be shown by introducing the stress tensor in the form (2.12) to the Reynolds-averaged incompressible Navier-Stokes equations (2.2) that for constant density a simplification equivalent to setting ̺ν(∇u)T = 0 is possible. Eventually, the stress tensor assumes a simple form: σij = −pin δij + ̺νj (∂ui /∂xj ) (2.13)

2.1 Formulation of governing equations

17

The term describing stresses in the momentum conservation equation (2.2) can therefore be written as follows: 1 1 ∇ · σ = − ∇p + ∇ · (ν∇u) (2.14) ̺ ̺

2.1.4

Boussinesq approximation

This approximation is based on the fact that in natural water bodies such as rivers, lakes and seas the variations of the density are relatively small. The variations of the relative density in the oceans are less than ±3% of the average density ̺0 . The concept of this approximation, formulated by Boussinesq [13, 145] is to take into account the variations of density only in those terms of the momentum conservation equation, where they give rise to buoyancy forces and to ignore them in those terms, where density appears as a parameter describing fluid inertia. It means that the variations of fluid mass are neglected, but their effect on fluid weight is retained. In the momentum conservation equation: ∂u 1 + u · ∇u + 2Ω × u = − ∇p + g + ∇ · (ν∇u) ∂t ̺

(2.15)

the density is assumed to be a sum of a constant reference density (average density, ̺0 ) and a variable density variance ∆̺ in the form ̺ = ̺0 + ∆̺. The reciprocal of the density in the pressure term −̺−1 ∇p can be linearised taking ∆̺/̺0 << 1 to: 1 1 1 = = ̺ ̺0 + ∆̺ ̺0 (1 +

∆̺ 1 1− ∆̺ ≈ ̺0 ̺0 ) ̺0

!

(2.16)

The pressure can also be written as the sum of the hydrostatic pressure pH resulting from the average density ̺0 : pH = ̺0 g(S − z) (2.17) where S(x, y) is the position of the free surface and the pressure variation ∆p, p = pH + ∆p. Therefore the pressure term in (2.15) can be transformed as following: !

1 ∆̺ 1 1 ∆̺ − ∇p = − 1− ∇p = − ∇p + 2 ∇p = ̺ ̺0 ̺0 ̺0 ̺0 ∆̺ ∆̺ 1 − ∇p + 2 ∇pH + 2 ∇(∆p) = ̺0 ̺0 ̺0 ∆̺ ∆̺ ∆̺ 1 g− g∇S + 2 ∇(∆p) − ∇p + ̺0 ̺0 ̺0 ̺0

(2.18)

The idea of the Boussinesq approximation is to neglect the terms of the second order, namely the last two terms in (2.18): 1 ∆̺ 1 g − ∇p = − ∇p + ̺ ̺0 ̺0

(2.19)

18

2.1 Formulation of governing equations

The momentum conservation equation (2.2) transforms after applying the Boussinesq approximation to the following form: 1 ̺ ∂u + u · ∇u + 2Ω × u = − ∇p + g + ∇ · (ν∇u) ∂t ̺0 ̺0

(2.20)

It should be strongly stressed that while ̺0 is a constant, average fluid density in a given flow, ̺ is the variable, local density.

2.1.5

Non-inertial co-ordinate system fixed to the earth

The equation set (2.1–2.4) is formulated for a rotating Cartesian co-ordinate system fixed locally to the earth’s surface, as described in section 2.1.1. For this system the earth’s angular rotation velocity vector Ω at the geographic latitude ϕ is inclined towards the y-direction (North). While the centripetal acceleration appearing in this system can be treated as a small correction of the gravitational acceleration g, the Coriolis terms cannot be neglected in general: 







fH w − fV v 2wΩ cos ϕ − 2vΩ sin ϕ     2Ω × u =  2uΩ sin ϕ   =  fV u fH u 2uΩ cos ϕ

(2.21)

fH = 2Ω cos ϕ and fV = 2Ω sin ϕ are horizontal and vertical Coriolis coefficients. The gravitational influence of other heavenly bodies (the moon, the sun) is neglected. A more detailed discussion of the aspects connected with the non-inertial co-ordinate system is provided in appendix section A.2. For simulation of larger water bodies, such as seas and ocean parts using the rectangular co-ordinate system, the earth’s curvature is neglected. For instance, the upper limit for the model dimension using a Cartesian system is set in oceanography to one tenth of the earth’s radius [92]. Additionally, simplifying assumptions concerning the latitudinal variability of the Coriolis parameters can be made when required (appendix section A.2).

2.1.6

Equation set for geophysical, incompressible flows with Boussinesq approximation

The hydrodynamic equation set, considered as the basis for all further non-hydrostatic model developments consists of: (1) the three-dimensional equations of motion, with all directions equally treated and Boussinesq approximation, (2) continuity equation for incompressible fluids, (3) the transport equation for a tracer (temperature, salinity, passive effluent concentration), (4) equation of state. It can be written as follows: ∂u 1 ∂p + u · ∇u = − − wfH + vfV + ∇ · (ν∇u) ∂t ̺0 ∂x

(2.22)

1 ∂p ∂v + u · ∇v = − − ufV + ∇ · (ν∇v) ∂t ̺0 ∂y

(2.23)

2.2 Hydrostatic approximation

19

∂w 1 ∂p ̺ + u · ∇w = − − g + ufH + ∇ · (ν∇w) ∂t ̺0 ∂z ̺0

(2.24)

∇·u=0

(2.25)

∂T + u · ∇T = ∇ · (ν T ∇T ) + qT ∂t

(2.26)

̺ = ̺(T, s, c)

(2.27)

In the equation of state (2.27) the density is a function of temperature T , salinity s or suspended matter concentration c. An internationally acknowledged standard equation of state is the UNESCO EOS 80 equation of state for seawater [160], appendix section A.4. For the free surface flows, the equations (2.22–2.27) are associated with an equation for the free surface. A turbulence model can be applied in order to obtain ν and ν T .

2.2

Hydrostatic approximation

The main aim of this work is to develop a model for free surface flows which avoids the hydrostatic approximation. Therefore, this approximation and its consequences are discussed in detail in the following section.

2.2.1

Scaling the vertical equation of motion

The various terms of the vertical equation of motion can be examined applying dimensional analysis in order to make rough estimates of their magnitude. The vertical equation of motion (2.24) for constant and equal horizontal (νH ) and constant vertical (νz ) turbulent viscosities can be written in full as: dw ∂w ∂w ∂w ∂w 1 ∂p ̺ = +u +v +w =− − g + fH u dt ∂t ∂x ∂y ∂z ̺0 ∂z ̺0 ! 2 ∂ w ∂2w ∂2w +νH + + ν z ∂x2 ∂y 2 ∂z 2

(2.28)

Let us introduce the typical time, horizontal distance (length), depth and velocity component scales T , L, H, U, W and define the following non-dimensional variables (with asterisk) as: x∗ =

x L

y∗ =

y L

z∗ =

z H

t∗ =

Ut L

u∗ =

u U

v∗ =

v U

w∗ =

w W

(2.29)

As the characteristic pressure ̺U 2 is taken: p∗ =

p ̺U 2

(2.30)

20

2.2 Hydrostatic approximation

Setting (2.29) and (2.30) into (2.28) and dividing the result by the factor appearing in the pressure gradient term (U 2 /H) yields: HW L U

∂w∗ ∂w∗ ∂w∗ + u∗ + v∗ ∂t∗ ∂x∗ ∂y∗

!

W 2 ∂w∗ ∂p∗ H ̺ H w∗ =− − 2 g + fH u∗ 2 U ∂z∗ ∂z∗ U ̺0 U ! 2 2 W ∂ 2 w∗ HW ∂ w∗ ∂ w∗ + + ν +νH 2 2 z LU ∂x2∗ ∂y∗2 HU 2 ∂z∗2 +

(2.31)

Due to the continuity ∇ · u = 0 the following equation is approximately valid: W U = H L

(2.32)

Introducing this dependence into the viscous and advective terms in (2.31) one obtains: ∂w∗ ∂w∗ W 2 ∂w∗ H 2 dw∗ W 2 dw∗ H W ∂w∗ + u∗ + v∗ + 2 w∗ = 2 = 2 L U ∂t∗ ∂x∗ ∂y∗ U ∂z∗ L dt∗ U dt∗ ! ∂p∗ gH ̺ νz ∂ 2 w∗ fH H HW ∂ 2 w∗ ∂ 2 w∗ =− + + − 2 + u∗ + νH 2 2 ∂z∗ U ̺0 U LU ∂x2∗ ∂y∗2 LU ∂z∗2 !

(2.33)

Equation (2.33) is in the form, which allows formulation of conditions under which various terms are negligible compared to the vertical gradient of global pressure p. 1. The vertical acceleration dw/dt can be neglected when the horizontal scale of motion dominates the vertical one, which can be interpreted as H << L or W << U. Fortunately, in most of these geophysical flows, where hydrostatic approximation can be applied (shallow water, long waves), both these conditions are usually fulfilled. However, a very often overlooked consequence of the condition W << U is that steep free surface and bottom slopes must be avoided, with a probably acceptable upper limit of 1:10. 2. In the gravitational and buoyancy force term for the geophysical flows ̺/̺0 ≈ 1. The value of the ratio (gH/U 2 ) varies greatly. If the motion to be described has a character of a long wave, gH can be interpreted as squared celerity of the long waves c2 . Setting for U 2 the amplitude of the velocity in the long wave from equation (A.44) yields gH/U 2 = H/η0 , where η0 is the amplitude of the surface wave. As a consequence this term cannot be neglected. 3. The Coriolis force can be neglected when the coefficient U/fH H, playing the role of a Rossby number, is large. The horizontal Coriolis coefficient varies with the latitude, fH = 2Ω cos φ, and has its greatest value on the equator, 1.45842×10−4 s−1 and is equal to zero at the poles. For larger horizontal velocity and shallow waters the Coriolis term can be neglected. 4. The viscous terms can be neglected, when the coefficients L2 U 2 /νH HW and LU/νz , playing the role of Reynolds numbers, are large, which is the case for turbulent flows with dominating horizontal motions.

2.2 Hydrostatic approximation

2.2.2

21

Hydrostatic approximation and its consequences

Scaling the vertical equation of motion (2.24) shows that when the conditions listed above are fulfilled, the vertical acceleration component dw/dt, as well as Coriolis and dissipative terms can be neglected. The equation (2.24) reduces then to the simple and well-known hydrostatic equation: ∂p = −̺g (2.34) ∂z As a consequence, when the hydrostatic approximation is valid, the pressure in a free surface flow can be obtained directly from (2.34). The set of hydrodynamic equations is much simplified, because the pressure can be eliminated. When horizontal velocity components are known from solving (2.22–2.23) the vertical velocity component can be found directly by integrating the continuity equation (2.25). The hydrostatic pressure is not appropriate in cases where the vertical accelerations are important or the pressure field cannot be approximated by hydrostatic pressure, e.g. when: 1. the vertical accelerations caused in deeper waters by influence of the steep bottom slopes on the velocity field result in changing the flow direction (these influences are damped in shallow water by gravitational influence of the free surface movements); 2. the description of vertical orbital movements of the fluid particles in the wave problems is important (short waves); 3. the flows around structures are considered, where, e.g., the pressure field before the obstacle in a flow cannot be assumed hydrostatic; 4. flows in fluids with variable density, as buoyancy-driven or stratified flows, thermal convection, internal or lee waves are to be modelled.

2.2.3

Three-dimensional shallow water equations

When the hydrostatic approximation is valid, the global pressure can be obtained from equation (2.34): p=

Z

z

S

̺gdz =

Z

z

S

(̺0 + ∆̺)gdz = ̺0 g(S − z) + ̺0 g

Z

z

S

∆̺ dz ̺0

(2.35)

where a variation of density from its average value ̺0 for a given flow ∆̺ = ̺ − ̺0 is introduced. The gradients of this hydrostatic pressure can be set into the horizontal momentum equations (2.22–2.23) yielding a sum of terms representing gradients of the free surface and fluid density. For instance, in the x-direction: ∂S ∂ 1 ∂p = −g −g − ̺0 ∂x ∂x ∂x

"Z

z

S

∆̺ dz ̺0

#

(2.36)

22

2.3 Algorithm with the hydrostatic approximation

The resulting terms are often called barotropic (as independent of density and depth, gradients of a 2D function S(x, y)) and baroclinic (as depth- and density-dependent, gradients of a 3D function ∆̺(x, y, z)) parts of the hydrostatic pressure gradients. As the consequence, the pressure can be found from the density field, influenced by the concentration of active traces through the equation of state (2.4) and from the position of the free surface. The fields of the active tracers are described by the transport equations (2.3); for the free surface a separate (or derived) equation is required. Additionally, for reasons discussed in section A.2, the terms containing horizontal Coriolis parameters fH can be neglected for flows with dominating horizontal scales. Coriolis terms with the vertical coefficient fV remain unchanged. Finally, under incompressibility, Boussinesq and hydrostatic approximations, the following set of three-dimensional shallow water equations can be formulated as follows: ∂u ∂S ∂ + u · ∇u = −g −g ∂t ∂x ∂x

"Z

S

∆̺ dz + vfV + ∇ · (ν∇u) ̺0

∂S ∂ ∂v + u · ∇v = −g −g ∂t ∂y ∂y

"Z

S

∆̺ dz − ufV + ∇ · (ν∇v) ̺0

z

z

#

(2.37)

#

(2.38)

∂u ∂v ∂w + + =0 ∂x ∂y ∂z ∂T + u · ∇T = ∇ · (ν T ∇T ) + qT ∂t ̺ = ̺(s, T, c)

(2.39) (2.40) (2.41)

They are applicable for those shallow water flows, where some aspects of their vertical structure, e.g. stratification or vertical profiles of the horizontal velocity have to be resolved. The equivalency of shallow water and long-wave approximations are discussed in appendix section A.3. For closure, they need an additional equation for the free surface and a turbulence model yielding ν and ν T . The 3D shallow water equations (2.37–2.41) are the governing equations of the hydrostatic code Telemac3D.

2.3

Algorithm with the hydrostatic approximation

This section describes briefly the algorithm of the code Telemac3D, developed in Laboratoire National d’Hydraulique in Chatou, Electricit´e de France (EDF). The Telemac modelling system was developed for the two-dimensional case by Galland, Goutal and Hervouet [52] on the base of previous considerations of Benqu´e et al. [8]. The description of the three-dimensional, hydrostatic algorithm is given by Janin, Lepeintre and Pechon [83]. As an example of the applications of this model LeNormant et al. [103] and Malcherek [110] are to be mentioned. Leaving the numerical details to the chapter 3, let us concentrate on the main ideas of this algorithm, outlined in figure 2.2. The treatment of the free surface and pressure gradients in this model is characteristic for the techniques based on hydrostatic approximation.

2.3 Algorithm with the hydrostatic approximation

2.3.1

23

Operator splitting

The algorithm is based on the operator-splitting method, described in more detail in section 3.2. The leading idea is to split the differential operators appearing in equations into parts with respect to their mathematical and/or physical properties and treat them separately in consecutive fractional steps by applying the most appropriate numerical methods. The global algorithm is explicit in time, and consists of the sum of partial (intermediate) solutions for a variable f after advection f a , diffusion f d , and finally after pressure – free surface – continuity f n+1 steps. The explicitly formulated time derivative of the variable f is: f n+1 − f d f d − f a f a − f n ∂f = + + ∂t ∆t ∆t ∆t

(2.42)

f represents the horizontal velocity components u, v, or tracer concentrations T .

2.3.2

Advection and diffusion steps

In order to deal with the variability of the free surface, the σ-mesh with tiers of prismatic elements is used, described in full detail in section 3.1.1. It allows tracking the timevariable position of the free surface while formulating equations in the time-independent σ-transformed mesh. In the first step of the algorithm, the advection terms in the horizontal momentum conservation and transport equations (2.22-2.26) are treated: fa − fn + u · ∇f = 0 ∆t

(2.43)

This hyperbolic equation is treated using the method of characteristics (i.e. a Lagrangian method) in a separate step, or together with the diffusion step, basing on the streamline upwind Petrov-Galerkin (SUPG) method (section 3.5). In the diffusion (also parabolic) step the diffusive terms are treated: fd − fa = ∇ · (ν f ∇f ) + qf ∆t

(2.44)

where qf represents the source terms. The diffusion step is solved in the σ-mesh by applying the 3D finite element method. In treatment of the time derivative the advantages of the Crank-Nicolson formulation (semi-implicit) are used. In this step of the algorithm the solution of the transport equations ends. In this version of the algorithm the source terms qf for the velocity components are equal 0 for the velocity components; for tracers they represented mass or heat sources. The next step (pressure – free surface – continuity) influences the solution of the transport equation only through the actualisation of the vertical node co-ordinates after finding the new free surface position.

24

2.3 Algorithm with the hydrostatic approximation

2.3.3

Pressure – free surface – continuity step

2.3.3.1

Free surface and forcing

After the advection and diffusion steps, the following equations must be solved: ∂S ∂ un+1 − ud = −g −g ∆t ∂x ∂x

"Z

S

z

#

∆̺ dz + fV v n ̺0

S ∆̺ v n+1 − v d ∂S ∂ = −g −g dz − fV un ∆t ∂y ∂y z ̺0 ∂u ∂v ∂w + + =0 ∂x ∂y ∂z

"Z

#

(2.45) (2.46) (2.47)

The treatment of this set of equation is based on the fact that the hydrostatic approximation is equivalent to the long (shallow water) waves approximation (section A.3). Long waves can be described by vertically integrated Navier-Stokes equations, i.e. St.Venant equations [92, 53]. This step of the algorithm takes full advantages of this fact. However, all constraints resulting from the shallow water (hydrostatic) approximation should be born in mind. In the pressure – free surface – continuity step both barotropic and baroclinic parts of the hydrostatic pressure gradient as well as the Coriolis terms must be treated. Finally, the new free surface position has to be found. The equations to be solved in this step are integrated vertically between the bottom B(x, y) and the free surface S(x, y, t) to obtain Saint-Venant equations without diffusion and advection terms. They consist of the conservative free surface equation which is the vertically integrated continuity equation (section 2.6.3.2) and of two momentum conservation equations for mean horizontal velocity components: un+1 − ud ∂h ∂B +g = −g + Fx + fV v n ∆t ∂x ∂x v n+1 − vd ∂h ∂B +g = −g + Fy − fV un ∆t ∂y ∂y ∂h ∂(uh) ∂(vh) + + = FH ∂t ∂x ∂y

(2.48)

where h(x, y, t) = S(x, y, t) − B(x, y) is the water depth. The mean velocity components are computed as follows (for the numerical integration the trapezium rule is applied): 1 ui = h

Z

S

B

ui dz

(2.49)

The terms representing the buoyancy forces Fx and Fy are computed in an analogical way: 1 S Fx dz ′ = Fx = h B Z 1 S Fy = Fy dz ′ = h B Z

g S h B Z g S h B Z

∂ S ∂x z Z ∂ S ∂y z Z

∆ρ dzdz ′ ρ0 ∆ρ dzdz ′ ρ0

(2.50) (2.51)

2.3 Algorithm with the hydrostatic approximation

The source term FH is: FH =

25

h [w∗ (z∗ ) − w∗ (0)] z∗

(2.52)

where w∗ is the σ-transformed vertical velocity component. It is equal to zero at the surface and at the bottom due to the impermeability or kinematic boundary condition. Non-zero values appear e.g. in the case of discharges at the bottom or surface. In order to solve the equation system (2.48) the continuity equation is split into hyperbolic and parabolic parts, analogically to the global 3D-algorithm. The hyperbolic part: ∂h ∂h ∂h +u +v =0 (2.53) ∂t ∂x ∂y can be solved separately using the 2D method of characteristics, or treated using streamline upwind Petrov-Galerkin formulation. The remaining equations: ∂u ∂v ∂h +h +h = FH ∂t ∂x ∂y un+1 − ud ∂h ∂B +g = −g + Fx + fV vn ∆t ∂x ∂x v n+1 − v d ∂h ∂B +g = −g + Fy − fV un ∆t ∂y ∂y

(2.54)

are solved simultaneously in a coupled way for u, v and h using the 2D finite element method (on the basic mesh) with Crank-Nicolson (semi-implicit) treatment of the time derivative. For this purpose parts of the two-dimensional algorithm of the Telemac2D [52, 77] are applied. The pressure – free surface – continuity step yields the new (i.e. for tn+1 ) position of the free surface and new horizontal velocity components. The 3D horizontal velocity is finally computed as follows: un+1 = ud + un+1 3D − ud 3D − ∆tF u3D + ∆tFu v n+1 = v d + v n+1 3D − v d 3D − ∆tF v3D + ∆tFv

(2.55)

where Fu = Fx + fV v and Fv = Fy − fV u, i.e. three-dimensional source terms which were vertically integrated in the Saint-Venant equations to obtain Fu and Fv (eq. 2.50). Subscript 3D for a two-dimensional field means, that this field is three-dimensionalised by taking a value for a given 2D-node (x, y) for each σ-mesh level. 2.3.3.2

Vertical velocity component

The vertical velocity is found through integrating the continuity equation taking the impermeability condition at the bottom (section 2.7.3): w n+1(z) = w n+1 (B) −

Z

z B

∂un+1 dz − ∂x

Z

z

B

∂v n+1 dz ∂y

(2.56)

26

2.3 Algorithm with the hydrostatic approximation Telemac3D algorithm outline (hydrostatic) Operator-splitting (f = u, v): ∂f f n+1 − f d f d − f a f a − f n = + + ∂t ∆t ∆t ∆t Advection of horizontal velocity (hiperbolic): fa − fn + u · ∇f = 0 ∆t Diffusion of horizontal velocity (parabolic): fd − fa = ∇ · (ν∇f ) + qf ∆t ui =

Pressure and free surface (h = S − B,

1 RS h B

ui dz):

un+1 − ud ∂h ∂B +g = −g + Fu ∆t ∂x ∂x ∂h ∂B v n+1 − v d +g = −g + Fv ∆t ∂y ∂y ∂h ∂(uh) ∂(vh) + + =0 ∂t ∂x ∂y where the vertically integrated source terms are: g Fu = h

Z

S

g h

Z

S

Fv =

B

B

∂ ∂x

Z

S

∆ρ dzdz ′ + fV v n ρ0

∂ ∂y

Z

S

∆ρ dzdz ′ − fV un ρ0

z

z

3D horizontal velocity field: un+1 = ud + un+1 3D − ud 3D − ∆tF u3D + ∆tFu v n+1 = v d + v n+1 3D − v d 3D − ∆tF v 3D + ∆tFv Continuity (vertical velocity component): wn+1 (z) = wn+1 (B) −

Z

z

B

∂un+1 dz − ∂x

Z

z

B

∂v n+1 dz ∂y

Two-step tracer advection-diffusion (hyperbolic-parabolic): Tn − Td Td − Ta + = ∇ · (ν T ∇T ) + qT − u · ∇T ∆t ∆t

Figure 2.2: Telemac3D algorithm outline (hydrostatic)

2.4 Numeric considerations concerning the solution method

w n+1(B) = un+1

∂B ∂B + v n+1 ∂x ∂y

27

(2.57)

Effectively, the vertical component w is adjusted to the horizontal velocity components u, v so that the continuity equation is fulfilled and the resulting velocity field is divergencefree. This subordinate treatment is again well-supported by the hydrostatic approximation, valid when the vertical velocity component is much smaller than the horizontal ones.

2.4

Numeric considerations concerning the solution method

2.4.1

General solution methods for 3D Navier-Stokes equations

From the mathematical point of view, the hydrodynamic and transport equations to be solved are a set of non-linear equations of mixed parabolic-hyperbolic type. Three methods of numerical treatment of these equations have found widespread acceptance. Finite difference method (FDM) in the domain of free surface flows was first applied in oceanography [126]; it was quickly followed by finite element method (FEM) [31] and finite volume method (FVM) [123, 161], but mostly in engineering applications. Additionally, spectral methods exist [47, 68]. All these methods discretise the differential equation formulated for a continuous problem (with infinite number of degrees of freedom) into a discrete problem (with finite number of degrees of freedom) which can be solved using a computer. In FDM, the derivatives in the governing partial differential equations are converted in the process of discretisation into finite differences. Various approaches of different exactness are available. Consequently, the differential equations are converted into a system of algebraic equations for nodal values of variables, which can be solved using a suitable (linear or non-linear) equation solver in order to obtain an approximate solution. FDM is usually based on structured meshes and is most efficient when the spacing is equidistant and with quadrangle computational cells, leading to simple forms of algebraic equations to be solved. Good results are obtained when staggered grids are applied, with different nodes for velocity components and pressure, making it more similar to the finite volume method. The disadvantage of this method is a complicated implementation of boundary conditions in general-purpose codes, and problematic mass conservation properties. In FVM the discretisation method is based on balancing the fluxes between arbitrarily chosen control volumes which are basic computational cells. Instead of formulating the finite differences for separate nodes, it formulates balances of variable fluxes through boundaries of finite volumes, operating on nodal values. FVM can be adapted to more complex geometries in a relatively straightforward way. The resulting formulae and sets of algebraic equations can be further treated as in FDM. Because this method is based

28

2.4 Numeric considerations concerning the solution method

directly on the flow properties, it is commonly treated as the most natural method for fluid mechanics and shows much better conservative properties than FDM. In FEM, the discretisation is achieved by assuming that the value of a variable can be interpolated locally. The local approximation is substituted into a appropriately weighted integral of the governing equation (weak formulation), and the integrals evaluated so that a residue is minimised. This weighted residue method is the most appropriate one for the hydrodynamic equations (of a mixed type) when solutions by finding a minimum of an integral in an element (variational method) cannot be formulated in general. Similarly to FDM, a set of algebraic equations is obtained and solved. While with FDM the differential operators are approximated, in FEM the resulting function is discretised, leaving the differential operators unchanged. FEM and FVM show their strength in cases, where unstructured meshes are needed, allowing flexible adaptation to complex domain geometries and variable resolutions. The implementation of boundary conditions is also much easier. However, the mass conservation in FEM can be secured in a global system only, while in FVM it is guaranteed at the local, control volume level. In the spectral methods, the global procedure is similar to FEM, except that the assumed solution for a variable is given in the form of a series of known functions of spatial co-ordinates with coefficients to be determined. The approximation functions are not defined locally, as in FEM, but for the whole domain, which puts severe constraints on the domain geometry. Finally, a set of algebraic equations for the function coefficients is obtained. Combinations of various discretisation methods are also available. A very common approach is to use FEM for spatial, and FDM for time discretisation. This leads to schemes of an enhanced stability, as e.g. Taylor-Galerkin-FEM [37, 38]. FEM and FVM can be also coupled in order to obtain schemes, which are mass conservative not only in a global system, as in FEM, but also locally, as in FVM, i.e. Control Volume FEM [70, 71]. Independently from the used general discretisation method, FDM, FVM or FEM, in order to solve the equation set (2.22–2.27) four main groups of algorithms exist. They are distinguished and classified by their overall approach to the numerical solution of the governing equations. First, a coupled equation system can be obtained by direct discretisation (using finite difference or element method) of the governing equations in their primitive variable form (i.e. the momentum conservation and continuity equation and an equation for the free surface). The resulting global equation system is usually not easily solved using standard methods. Non-linear, iterative equation solvers (e.g. Newton-Raphson) or predictorcorrector methods with convergence being the greatest concern are used [49, 50, 80, 168]. In practice, only implicit formulation is successfully applied. This direct solution method of Navier-Stokes equations is computationally very intensive, especially for the 3D case. Second, decoupled methods for primitive variables exist, where momentum conservation equations for three velocity components and a (modified) continuity and/or a derived equation for pressure are solved in consecutive stages. The main idea of these methods is to solve sequentially a number of smaller, linear equation systems instead of an iterative

2.4 Numeric considerations concerning the solution method

29

solution of a larger, usually non-linear and slowly converging one. The free surface equation is also usually solved in a separate step. The decoupled methods differ mostly in their algorithm structure, but also in the in the treatment of the time derivatives, with an overall aim to avoid the solution of the fully implicit global equation system. They appear in the literature under various names, as splitting schemes [52], fractional step schemes [129] or projection methods [29, 57, 143]. Taylor-Galerkin-methods, where a more precise approximation of time derivatives is used before applying the finite element method, can be also assigned here [37, 38, 86]. Various pressure methods belong also to the decoupled methods [18, 24]; they are described in detail in the following sections. A special distinction between the two overall approaches – the direct, coupled one and the fractional, decoupled one – lies in this particular case in the treatment of the incompressibility constraint. While in the direct method the continuity equation appears explicitly as a constraint on the velocity field, in the decoupled pressure methods it is replaced by a derived Poisson equation for pressure. Thus, the momentum and continuity equations are decoupled, at least in the interior of the computational domain. The coupling appears by boundary conditions for pressure [64, 137]. The third group of methods takes advantage of techniques for compressible flows: 1. The penalty function method, where the continuity equation is replaced by a form stating that the pressure changes proportionally to the velocity divergence with a proportionality parameter called bulk viscosity [28]. The solution of a thus modified set of equations converges to the solution of the Navier–Stokes equations when the bulk viscosity ν → ∞, so that the scheme is exact enough only for very large ν. 2. The artificial compressibility method [78, 148], where a limited compressibility of the fluid is assumed and the continuity equation for compressible flows is applied. The term representing the partial derivative of pressure in the continuity equation disappears when steady state is achieved. 3. A combination of both [133]. The main disadvantage of these three methods is introduction of artificial, empirical variables or constants and the fact that they are applicable mainly to stationary flows. In the fourth general group of methods, Ψ − ω–methods, the stream function Ψ and vorticity ω are introduced into basic equations, yielding a Poisson equation for Ψ and a transport equation for ω in the two-dimensional case [47]. Using such non-primitive variables brings two advantages: it reduces the number of equations to be solved and does not require any pressure computation. A disadvantage is a complex form of the boundary conditions. For the 3D case, vector potentials Ψ and ω are defined (u = ∇ × Ψ and ω = ∇ × u). As a result, three Poisson equations for Ψ and three transport equations for ω components are obtained [3]. The computational effort grows considerably in comparison to other methods.

30

2.4.2

2.4 Numeric considerations concerning the solution method

Overview of the solution methods with FEM

While the mathematical theory for the solution of the Navier-Stokes equations with the finite element method (FEM) leading to a non-linear global algebraic equation system is well established [31, 129, 54], the theory for the decoupled methods is still matter for investigation [59, 57, 62, 86]. The main reason for the introduction of the decoupled methods is the computational efficiency of these codes compared to the cost of implicit, iterative solution of very large non-linear equation systems for coupled velocity and pressure in the direct methods. To obtain an algorithm, where this intense coupling can be replaced by a sequential solution of smaller and linear equation systems, various methods are applied. While the decoupled methods were successfully applied in the finite difference method [18, 24], their formal application in FEM has been hampered in the past by ambiguities caused by boundary conditions for the pressure [64, 137]. In order to approximately solve Navier-Stokes equations using the finite element method, the following steps must be followed: (1) reformulate the given problem using the weighted residue method (weak formulation); (2) perform discretisation in finite elements using finite-dimensional space of approximation functions, (3) solve the discrete problem through solution of a set of algebraic equations for the nodal values and finally (4) implement the method as a computer code. In following subsections the relevant FEM theory is briefly sketched. 2.4.2.1

Weighted residue method

According to the weighted residue method, an approximation of an unknown function u being a solution of a partial differential equation in the domain Ω(Γ) (with boundary Γ): L(u) = 0 (2.58) is searched for. The function u is approximated in the form of a linear combination of a finite number of base approximation functions ϕi being the functions of spatial coordinates: uh =

N X

ϕ i ci

(2.59)

i=1

The base functions ϕi must be chosen in such a way that the boundary conditions at Γ are satisfied for arbitrary real coefficients ci . When (2.59) is introduced to (2.58), a residual value ε appears due to the fact that uh is only an approximation of u: L(uh ) = ε 6= 0

(2.60)

In order to find an approximate solution of (2.58), this residue should be minimised in Ω. This is obtained by setting the scalar product of the residue and chosen weighting (test, trial) functions vi to zero, (vi , ε) = 0 (i.e. to assume that they are orthogonal). This is equivalent to setting (see section 2.4.2.5): Z

Ω

vj ε dΩ = 0

j = 1, ..., N

(2.61)

2.4 Numeric considerations concerning the solution method

31

where N is equal to the number of coefficients ci . The functions vi allow different error weighting in various parts of the domain Ω, and (2.61) must be evaluated only there, where vj are not equal to zero. Consequently, the approximate solution uh of the following integral equation is searched for: Z

Ω

vj ε dΩ =

Z

Ω

vj L(uh ) dΩ =

N X i=1

ϕ i ci

Z

Ω

vj L(ϕi )dΩ = 0

(2.62)

In the general formulation of the weighted residue method nothing is said about the interpretation of the base ϕi and test functions vi and the constants ci . According to the way in which the weighting functions are chosen and the general method in which the domain Ω is subdivided into a finite number of sub-domains (elements) different finite element methods are achieved. In the standard methods, ci are equal to ui = uh (xi ), i.e. function values at the defined finite element nodes and ϕi the interpolation (or shape) functions corresponding to a node i. The Lagrangian interpolation functions ϕ, which are applied in this work, are equal to 1 at the node i and to 0 at all other nodes. The required order of the interpolation functions ϕ depends on the order m of the differential operator L. In general, it is required that the (m − 1) order derivatives of ϕ must be continuous. 2.4.2.2

Standard Galerkin finite element method

In the standard Galerkin finite element method (called also Bubnov-Galerkin or standard FEM), which is applied e.g. in the Poisson equation or diffusion step of the presented algorithm (elliptic, parabolic), the weighting functions are chosen to be equal to the approximation functions v = ϕ. However, in order to approximate the advection-dominated problems properly, this method can be successfully applied only when the time derivative is approximated with Taylor series of higher order. This method, called Taylor-Galerkin FEM, has been developed by Donea et al. [37, 38]. 2.4.2.3

Streamline upwind Petrov-Galerkin finite element method

In Petrov-Galerkin finite element method, the weighting and approximation functions are different. This approach has been applied by Brooks and Hughes [16] in order to obtain stable solutions with FEM for strongly advective problems (hiperbolic). In FDM, the upwind (or upstream) differences are applied for discretisation of the advection equation in order to obtain stable schemes – only nodal values from the incoming flow direction are taken (and varying accordingly to the velocity vector direction). In the FEM, the weighting functions can be modified in such a way that more information is obtained from the upstream direction. This effect can be reached by application of appropriate weighting functions of higher order, or, as in the streamline upwind Petrov-Galerkin method (SUPG), by addition of polynomials of a lower order to the weighting function. In the SUPG, the weighting function is taken as: u ∇ϕi (2.63) vi = ϕi + α ||u||

32

2.4 Numeric considerations concerning the solution method

where α is an upwind parameter which can be set to a constant, or a variable calibrated according to a given problem [2] (see also section 3.6.2), u is the flow velocity. If linear approximation functions are applied, ∇ϕi is constant, and vi are equal to the approximation functions modified by a constant. The velocity norm is usually computed as the mean velocity in a given element. The SUPG method can be chosen in the advection step of the described algorithm. 2.4.2.4

Finite elements

In order to find a discrete solution uh of (2.62) in the domain Ω it is necessary to subdivide it into smaller sub-domains. In the process of subdivision of the bounded solution domain Ω into non-overlapping finite elements (called triangulation or partitioning) it is advisable to choose as simple geometric forms as possible in order to reduce computational cost. In 2D, usually triangles or quadrilaterals, in 3D prisms, tetrahedrals or hexaedrals are selected. The number of nodes should allow application of the chosen approximation functions, e.g. for polynomials of higher order the necessary number of nodes (being support points for them) grows. As the nodal values the variables themselves (Lagrange nodes) or their derivatives (Hermite nodes) can be chosen. The integrals resulting from the weak formulation are evaluated in the simplest, reference form of a given element type. The real values from elements adjusted to the domain geometry are transformed into the reference one to perform (simplified) computations. If the approximation functions used for approximation of the spatial co-ordinates are identical with them for the unknowns, the elements are called iso-parametric. A number of finite elements (which could be defined as a triplet (1) a geometric object, (2) a finite-dimensional space of functions defined on it and (3) a set of nodes and values on them) are available with various shape functions and different properties [31, 173]. The finite elements used in this model are discussed in chapter 3. 2.4.2.5

Hilbert and Sobolev spaces

For the formal analysis of the finite element method a linear function space L2 (Ω) is R defined being a space of all functions f → R, for which Ω f 2 dΩ < 0. Together with a scalar product defined as: Z (f, g) =

Ω

f g dΩ

(2.64)

(compare with equation 2.62) this linear vector space builds a Hilbert space V , natural for the problems considered here. The functions f are called square integrable functions. The norm in the space L2 (Ω) is defined as: ||u|| =

q

(u, u)

(2.65)

For analysis of the approximate solutions to differential equations, Sobolev spaces H k (Ω) are introduced. These Hilbert spaces consist of functions u all of whose partial derivatives

2.4 Numeric considerations concerning the solution method

33

up to the order k are square integrable: H k (Ω) = {u ∈ L2 (Ω) : D α u ∈ L2 (Ω), |α| ≤ k}

(2.66)

where the partial differentiation up to an order α is denoted as follows: ∂ |α| u ∂xα1 1 ...xαk k

Dαu =

|α| =

k X

αk

(2.67)

(D α u, D α v)L2 (Ω)2

(2.68)

i=1

and α = (α1 , α2 , ..., αk ). The scalar product in a Sobolev space is defined as: (u, v)H k (Ω) =

X

|α|≤k

and the norm as: ||u||H k (Ω) =

q

(u, u)H k (Ω) =

sX

|α|≤k

||(D αu)||L2 (Ω)2

(2.69)

For example, the scalar product in H 1 (Ω) is: (u, v)

H 1 (Ω)

=

Z

Ω

(uv + ∇u · ∇v) dΩ

(2.70)

From the mathematical point of view, the appropriate choice of the function space is important, because it allows proving the existence of a solution, estimate errors, etc. One speaks about the conform finite elements, when the interpolation functions are included in the same Sobolev space in which the weak formulation of the problem is stated. Therefore, the usual mathematical methodology for these elements is to tailor an appropriate space in order to investigate the method properties [87]. In FEM, the infinite dimensional space of the solution function V is reduced to a finite dimensional subspace Vh where (2.59) is valid, due to the limited number of the approximation functions. The conform approximation methods are based on the assumptions, that Vh ⊂ V and that the properties of V are valid also for Vh . 2.4.2.6

Weak formulation

The classical methodology of the more abstractly formulated FEM can best be illustrated by a simple example. The following boundary value problem for the Poisson equation (e.g. for pressure) in a bounded domain Ω(Γ) is to be considered: −∇2 u = f u=0

in at

Ω Γ

(2.71) (2.72)

As a classical solution one defines the solution of a partial differential equation of the second order which belongs to C 2 (Ω)∩C 0 (Γ) by Dirichlet boundary condition (BC) or to

34

2.4 Numeric considerations concerning the solution method

C 2 (Ω) ∩ C 1 (Γ) by Neumann BC. (C k is the set of functions with continuous derivatives up to order k). Let us apply the weighted residue methodology introduced in section 2.4.2.1. By multiplying the Poisson equation by an arbitrary test function v ∈ V and integrating it in Ω yields: Z Z − v∇2 u dΩ = vf dΩ (2.73) Ω

Ω

Applying Green’s formula (n is the normal vector to the boundary): Z

Ω

2

v∇ u dΩ = −

Z

Ω

∇v · ∇u dΩ +

Z

Γ

v∇un dΓ

(2.74)

to (2.73) and taking the boundary condition into consideration (the boundary integral vanishes), the problem considered can be formulated as follows: find such a u ∈ V that: a(u, v) = (f, v)

∀v ∈ V

(2.75)

∇v · ∇u dΩ

(2.76)

f v dΩ

(2.77)

where a is a bilinear form and: a(u, v) = (f, v) =

Z

ZΩ Ω

V is a space of continuous functions in Ω with ∂v/∂xi piecewise continuous in Ω and equal to zero at the boundary Γ. Such a description of V shows that the appropriate Hilbert space for this problem is H01 (Ω), defined as H01 (Ω) = {v ∈ H 1 (Ω) : v=0 on Γ}, and the weak formulation (2.75) reads now – find u ∈ H01 (Ω) that: ∀v ∈ H01 (Ω)

a(u, v) = (f, v)

(2.78)

The formulation (2.78) is named the weak formulation of (2.71) and its solution is named weak solution of (2.71). If a weak solution u exists, it is not automatically the classical solution of (2.71), because it must be regular enough that ∇2 u can also be defined in a classical way. The apparatus of functional analysis brings methods to find the solution of the weak formulation, and then show that it is regular enough to be the classical solution of the given problem. The usual way is to investigate the properties of the bilinear forms a(u, v) and to use theorems stating the solution existence, continuity and stability. The most useful for FEM considerations is that part of the theory which assigns linear operators L : U → V ′ to bilinear forms on elements from Hilbert spaces U, V such that a : U × V → R through: (L(u), v) = a(u, v) v∈V (2.79) V ′ = L(V, R) is a dual space to V , i.e. a set of all linear and continuous functions on V . The norm of this space is constructed using an operator norm, for l ∈ V ′ : ||l||V ′

(

|l(u)| = sup ||u||V u6=0

)

(2.80)

2.4 Numeric considerations concerning the solution method

35

For example, a dual space H −m (Ω) for a Sobolev space H m (Ω) has a following norm: (

(u, v)L2(Ω) ||u||−m = sup ||u||m v6=0

)

(2.81)

The problems considered here can be constructed using scalar products and have the following form: for a given f ∈ V ′ , a function u ∈ U is searched for, so that: a(u, v) = (f, v)

v∈V

(2.82)

The weak formulation (2.82) is therefore equivalent to finding the inverse operator to L, such that u = L−1 f . One of the most important theorems in the FEM theory states the conditions under which L is isomorphic, i.e. both L and L−1 are continuous. L : U → V ′ is isomorphic when the assigned form a : U × V → R fulfils the following conditions: • Continuity and boundedness (stability): for C ≥ 0: |a(u, v)| ≥ C · ||u||U · ||v||V

(2.83)

• Inf-sup-condition: for α > 0 and u ∈ U: (

a(u, v) inf sup u∈U v∈V ||U||U ||v||V

)

≥α>0

(2.84)

According to (2.80), (2.84) is equivalent to the statement for u ∈ U: ||L(u)||V ′ ≥ α||u||U

(2.85)

The bilinear forms fulfilling (2.84) so that for v ∈ V |a(v, v)| ≥ α||v||2V are called V -elliptic. For the weak formulation (2.82) one obtains from the inf-sup-condition (2.84) or (2.85): ( ( ) ) a(u, v) (f, v) (2.86) α||u||U ≤ sup = sup = ||f ||V ′ ||v||V ||v||V v∈V v∈V Therefore, for a bilinear and V -elliptic form a(u, v) the weak formulation (2.82) has a solution u = L−1 f for which ||u||U ≤ α−1 ||f ||V ′ . For FEM, as U and V finite dimensional spaces are taken. 2.4.2.7

Global equation system

In the finite element method, the solution is searched for in a subspace Vh of V , and the aim is to find such uh ∈ Vh that: a(uh , vh ) = (f, vh )

∀vh ∈ Vh

(2.87)

36

2.4 Numeric considerations concerning the solution method

In standard Galerkin FEM, the test functions are equal to the interpolation functions and: a(uh , ϕj ) = (f, ϕj ) j = 1..N (2.88) where N is the number of nodes. Because uh is approximated using Lagrangian interpolation functions so that: uh =

N X

ϕi u i

ui = uh (xi )

(2.89)

i=1

one may write the weak formulation in the form N X

ui a(ϕi , ϕj ) = (f, ϕj )

j = 1..N

(2.90)

i=1

being a system of N equations with N unknowns being the nodal values ui : Au = b

(2.91)

where uj is the vector of the unknown nodal values and bj = (f, ϕj ) is the load vector. The elements of the stiffness matrix aij = a(ϕi , ϕj ) are usually computed by summing the contributions from M different elements: a(ϕi , ϕj ) =

M X

am (ϕi , ϕj )

(2.92)

m=1

This matrix assembly can therefore be performed by computing element stiffness matrices separately and then summing them up in a loop over all elements. It must be noted that for Lagrangian interpolation functions am (ϕi , ϕj ) = 0 unless both nodes i and j are nodes belonging to a given element m. The load vector b is computed analogically. The solution of the equation system (2.91) yields the discrete solution of the given problem. The solution of the linear equation system (2.91) can be performed using various methods, from (exact and direct) Gaussian elimination for smaller problems to (approximate and iterative) minimising methods for a very large number of nodes, as, for example, with the conjugate gradient method. The iterative methods usually require pre-conditioning of the equation system matrices in order to reduce the number of iterations. For computations using a vector computer, a special technique for treatment of the resulting system matrix is advantageous to profit from the computational possibilities of these machines. By using iterative equation solvers, the operation of multiplying a matrix by a vector is very common and the most time-consuming computation part compared to other operations. Therefore, a special storage system can be introduced in order to make the matrix-vector product as effective as possible. For performing these products, the assembly of the global matrix is not required. In the element-by-element (EBE) method [81, 56, 73], the diagonal and extra-diagonal elements of the separate element stiffness matrices are stored in memory and special procedures for performing basic algebraic operations are provided. The main advantage is that the matrix-vector

2.4 Numeric considerations concerning the solution method

37

product is vectorisable and that the computationally intensive matrix assembly process is avoided. It should be noted that this storage method does not yield any adventages when using a scalar computer. The vectorisation idea is to work simultaneously in real time on a cluster containing a number of elements in a loop (e.g. vectorisation length of 64 or 128 on Cray YMP or 1024 elements on Fujitsu-Siemens VPP). Therefore, a loop over all elements is performed in chunks containing 1024 elements (for Fujitsu) significantly speeding up the computation process. However, the method requires sorting of the element numbers in a computational mesh after its generation in order to avoid vector dependencies. These vector dependencies do not occur, when an element of a processed vector appears only once in a cluster of 1024 elements. Namely, in a loop over element numbers in a given vectorisation length the situation must be avoided that two or more elements share the same node and this node has the same local numbers in the elements considered. The conditions for the mesh sorting get more severe as the targeted vectorisation length increases. The numerical libraries of Telemac3D contain numerous relevant modern iterative solvers and pre-conditioning procedures for large equation systems [73].

2.4.3

Equal and mixed interpolation

2.4.3.1

Stokes problem

Let us consider the stationary Stokes equations for an incompressible fluid with a constant dynamic viscosity µ contained in a domain Ω(Γ) ⊂ R3 : −µ∇2 u = −∇p + f in Ω ∇ · u = 0 in Ω u = 0 on Γ

(2.93) (2.94) (2.95)

where f are volume forces. The pressure is determined only up to a real constant C; if a pair u, p is a solution of (2.93) then a pair u, p + C is a solution as well. In order to obtain a unique solution for pressure an additional constraint should be posed, e.g. R Ω p dΩ = 0.

In order to obtain the weak formulation, (2.93) is multiplied by a test function v ∈ V = [H01 (Ω)]3 which is divergence-free, ∇ · v = 0 in Ω, integrate it in Ω, and apply Green’s formula: Z

vf dΩ = −µ

ΩZ

−

Ω

Ω

2

v∇ u dΩ +

v∇un dΓ + µ

ΓZ

=µ

Z

∇v · ∇u dΩ

Z

Ω

Z

Ω

v∇p dΩ =

∇v · ∇u dΩ +

Z

Γ

vpn dΓ −

Z

Ω

(∇ · v)p dΩ =

(2.96)

38

2.4 Numeric considerations concerning the solution method

The pressure term disappears from the problem because v = 0 on Γ and the incompressibility condition ∇ · v = 0 in Ω. The weak formulation can be stated as: find u ∈ V that: µ(∇u, ∇v) = (f, v) ∀v ∈ V (2.97) µ(∇u, ∇v) = µ (f, v) = V

Z

Z

Ω

Ω

∇v · ∇u dΩ

(2.98)

vf dΩ

(2.99)

= {v ∈ [H01 (Ω)]3 : ∇ · v = 0inΩ}

(2.100)

It means that when a finite element method is to be formulated for (2.98), a finitedimensional space Vh of V , the incompressibility condition ∇ · v = 0 must be fulfilled exactly. (It can easily be proved that this is the case, when v = ∇ × ψ, where ψ is a stream function of the velocity field). It can be shown that under this condition the (Galerkin) finite element method can yield the classical solution of (2.93). However, constraints are present – the incompressibility condition must be fulfilled exactly [87]. 2.4.3.2

Mixed interpolation

Mixed interpolation methods are applied in order to avoid the restriction to exactly divergence-free velocity. The solutions for u and p are searched for in different spaces V and Q, where V is not limited to divergence-free velocity. Formally, taking (2.96) into consideration, the weak formulation of (2.93) is set now as follows: find (u, p) ∈ V × Q that: µ(∇u, ∇v) − (p, ∇ · v) = (f, v) (q, ∇ · u) = 0 µ(∇u, ∇v) = µ (f, v) = V

Z

Z

Ω

Ω

∀v ∈ V ∀q ∈ Q

∇v · ∇u dΩ

(2.101)

(2.102)

v · f dΩ

(2.103)

= {v ∈ [H01 (Ω)]3 }

Q = {q ∈ [L2 (Ω)]3 :

(2.104) Z

Ω

q dΩ = 0}

(2.105)

(Note that (.,.) denotes a scalar product in the L2 space.). The finite element method, as previously discussed, is obtained by replacing V and P by finite-dimensional subspaces Vh and Qh . The mixed FEM can be formulated as follows: find (uh , ph ) ∈ Vh × Qh that: µ(∇uh , ∇vh ) − (ph , ∇ · vh ) = (f, vh ) (qh , ∇ · uh ) = 0

∀qh ∈ Qh

∀vh ∈ Vh

(2.106) (2.107)

2.4 Numeric considerations concerning the solution method

39

Introducing approximation functions ϕi for Vh (velocity) and ψi for Qh (pressure), the problem (2.106–2.107) can be written in a compact matrix form: Du − Cp = f CT u = 0

(2.108)

where D = (dij ), C = (cij ), F = (fj ) and: dij = a(ϕi , ϕj ),

cij = (ψi , ∇ · φj ),

fj =

Z

Ω

f ϕj dΩ

(2.109)

In this mixed interpolation method the possibility is opened to work with velocities which fulfil the incompressibility constraint only approximately, namely through the discrete formulation (2.107) or the second equation of (2.108). The subspaces Vh and Qh should be chosen in such a way that a compromise is achieved – the resulting method is stable and accurate as well. For choosing these spaces, a condition analogous to inf-sup-condition (2.84) is of paramount importance. The solution stability can be formulated as follows: there exists such a constant C that when (uh , ph ) ∈ Vh ×Qh is a solution of (2.106–(2.107) then ||uh||1 + ||ph || ≤ C||f ||−1 (2.110) where the notation introduced by (2.81) is applied. It can be shown that in order to fulfil (2.110) the following Ladyzhenskaya-Babuˇska-Brezzi condition (LBB-condition) for (2.106–2.107) must be satisfied [15]: there exists such a constant c that: sup vh ∈Vh

(qh , ∇ · vh ) ≥ c||qh ||0 > 0 ||vh ||1

∀qh ∈ Qh

(2.111)

The LBB condition (2.111) provides a means to construct spaces for velocity Vh and pressure Qh , so that the solution is stable. The most simple method is to choose Vh to be large enough. Obviously, for computational reasons it is advisable to choose it as small as possible, but in agreement with Qh . The LBB condition supplied the appropriate mathematical tool for understanding and removing the instabilities observed by practitioners when applying finite elements which do not fulfil this condition [14]. Largest problems were caused by parasitic modes or oscillations of the pressure [138, 164]. Various other methods to remove these instabilities have been applied previously, as applying of numerical filters or averaging of resulting values in a node patch, mass-lumping or setting un-physically high viscosity or diffusion coefficients [149]. These procedures have a disadvantage in common that a part of the physical information is simply damped in order to improve numerical stability. As a rule of thumb, the interpolation order for the velocity should be higher than for the pressure, however it is not enough to guarantee stability. For example, a quadrilateral element with linear approximation for velocity and piecewise constant functions for pressure proves to be unstable. Proofs of the LBB-condition for various element types are not trivial, but already a number of element families has been theoretically checked

40

2.4 Numeric considerations concerning the solution method

[54, 15, 87, 14, 173]. Another method to avoid instabilities is to define the pressure on a mesh which is coarser as the mesh for velocity [55]. Although the LBB condition has been formulated for elliptic problems, there is strong evidence of its validity also for hyperbolic problems, Navier-Stokes equations and shallow water equations [15, 54]. However, a strict mathematical proof of these hypotheses does not exist at present [2]. Stabilised solutions for Stokes problem can be achieved using the Petrov-Galerkin formulation [82]. Various other methods allow circumventing the LBB condition (2.111) for more advanced problems, as for example using enhanced assumed strain elements [164]. As an example of a 3D free surface model using the coupled, direct approach and where an element strictly fulfilling the LBB condition is applied (a cube with quadratic approximation functions for velocity, 27 nodes, and linear functions for pressure, 8 nodes) is the work of Yost [168]. The resulting global equation system is solved using the NewtonRaphson method. However, even in this case instabilities of the free surface have been observed. Huerta and Liu [80] in their model for 2D viscous flows with large movements of free surface, use a predictor-corrector method for solving the resulting equation system. Another example is a model developed by Frederiksen and Watts [50]. 2.4.3.3

Equal-order interpolation

For the decoupled methods, where the solution of the Navier-Stokes equations is achieved in separately performed steps and where a set of smaller equation systems are solved instead of a global one (section 2.4.1) a consistent FEM theory still does not exist. In the past, most researchers used elements where the pressure is approximated by functions of lower order than the velocity ones, as Gresho et al. [63, 57, 60] or Donea et al. [39], and succeeded even with elements which do not fulfil LBB conditions strictly. Larger efforts have been invested in better understanding of imposing various boundary conditions (especially pressure) [64, 137] or in stabilisation of the numerical integration in time [149, 150, 86, 89]. In search for a better approximation of pressure gradients in distorted elements, and in order to simplify the algorithms and improve the solution efficiency, equal order approximation for both pressure and velocity (bilinear) has been successfully (re)introduced by Gresho et al. [61, 62], although the equal-order elements also fail the LBB-test. Further investigation shows that when a weak formulation of a derived pressure Poisson equation (with automatically built-in boundary conditions) is solved in a separate step (sections 2.5, 3.3.1), stable and accurate solutions for incompressible flows are obtained [89, 66]. As an explanation why an equal-order approximation is appropriate, it is presumed, that the LBB condition is circumvented by the fact that the decoupled methods do not require an incompressibility condition in the form of an explicit equation ∇ · u = 0 occurring in the global equation set. The incompressibility is asymptotically achieved by the convergence of a pressure equation solution. As mentioned before, stability can be additionally enhanced by applying proper boundary conditions, improving the methods of integration in time and appropriate treatment of advection terms (SUPG).

2.5 Pressure treatment

41

For free surface flows, efficient equal-order finite element 2D models (in vertical plane, aimed for wave applications) have been developed by Ramaswamy and Kawahara [130, 132] and Jiang and Kawahara [86], both containing a solution of a pressure equation in a separate step. A 3D model with non-equal approximation based on a projection method for the global pressure [57] has been developed by Chen [26].

2.4.4

Method applied in the developed model

In the non-hydrostatic model described in this work, a decoupled method based on the previously existing operator-splitting approach is implemented. Linear interpolation functions in space are applied for all flow variables: for the velocity and the pressure in a prismatic 3D-element and for the water level in a triangular 2D-element (section 3.1.3). Therefore, this algorithm uses the equal-order interpolation functions in finite elements. Concerning integration in time, the overall algorithm is explicit. For the advection-diffusion and free surface steps semi-implicit or fully implicit approaches in the framework of the overall operator-splitting technique are possible (section 3.2). In the following sections the details of the applied method are provided, starting with an overview of the pressure treatment and pressure equations.

2.5

Pressure treatment

The characteristic feature of the implemented solution algorithm of the three-dimensional set of hydrodynamic equations without the hydrostatic approximation (2.22–2.27) is treatment of the pressure. Application of the hydrostatic approximation yields an explicit equation for the pressure (section 2.2) from the third (vertical) momentum conservation equation. For an arbitrary, three dimensional case without the hydrostatic approximation it is not possible. For compressible flows, the pressure appears in the continuity equation (2.1) because the fluid density is pressure-dependent. For the incompressible flows, in the continuity equation (2.25) only velocity components appear, and there exists no explicit equation for pressure. An equation for the pressure (or a potential function related to pressure) can be derived from the momentum conservation equations; in this case one speaks about a derived pressure equation. In the following paragraphs, the decoupled pressure methods for primitive variables using a derived pressure equation are shortly reviewed.

2.5.1

Deriving of the conventional pressure Poisson equation

Applying the divergence operator on the Navier-Stokes equations in the following general form: ∂u + u · ∇u = −∇P + ν∇2 u + f (2.112) ∂t

42

2.5 Pressure treatment

(where P = p/̺ is the kinematic pressure and f represents other force terms) one obtains: !

∂u ∇· + u · ∇u = −∇2 P + ν∇ · (∇2 u + f) ∂t

(2.113)

Assuming that divergence and ∂/∂t can be commuted and taking ∇ · u = 0, one obtains the general pressure Poisson equation: ∇2 P = ∇ · (ν∇2 u − u · ∇u) + ∇ · f

(2.114)

As a consequence of the incompressibility approximation, the global pressure p can be found as a solution of an elliptic partial differential equation. Each change of the pressure propagates at infinite speed in order to keep the flow incompressible everywhere. The boundary conditions for this separate equation for global pressure should be obtained from the momentum and continuity equations. The Neumann boundary conditions for this equation derived from the momentum equations (2.112) are: !

∂u +f = n · ∇P = n · ν∇ u − u · ∇u − ∂t ∂un − u · ∇un + fn ν∇2 un − ∂t 2

(2.115)

The Dirichlet boundary condition can be obtained from the dynamic boundary condition (2.178), i.e.: ∂un P =ν − Fn (2.116) ∂n The numeric solution of the thus stated Poisson–Neumann (2.114–2.115) or Poisson– Dirichlet (2.114–2.116) problem is not trivial, because in general it contains unknown velocities on the right hand side of the equation. The boundary conditions cannot be implemented in a straightforward way and are a source of numerous numerically caused ambiguities [64, 58].

2.5.2

General idea of the projection method

The large family of projection methods is based on the fact that any vector field v in a domain Ω with a smooth boundary δΩ can be uniquely decomposed into the following form [30]: v = vd + ∇φ (2.117) where vd is solenoidal (divergence-free) and parallel to the boundary δΩ: ∇ · vd = 0 vd · n = 0

in Ω at δΩ

(2.118)

2.5 Pressure treatment

43

and φ is a scalar function. ∇φ is therefore rotation-free and (2.117) is equivalent to splitting the vector v into divergence-free and rotation-free components. The momentum conservation and continuity equations can be written in the following way: ∂u + ∇P = S(u) ∂t

in Ω

(2.119)

where on the left side a sum of the local acceleration (velocity time partial derivative) and of the kinematic pressure gradient is present. The function of u on the right side appears as: S(u) = g + ν∇2 u − u · ∇u (2.120) Generally, S(u) is neither divergence-free nor rotation-free. Due to the incompressibility of the flow and from the vector analysis it can be remarked that: ∂u ∇· ∂t

!

=

∂ (∇ · u) = 0 ∂t ∇ × ∇P = 0

(2.121)

Following the method described by Chorin [30], when u is given, then S(u) is also known and therefore can be projected simultaneously into subspaces of the divergencefree vectors ∂u/∂t and rotation-free vectors ∇P . This can be formally stated as: ∂u = P[S(u)] ∂t

∇P = Q[S(u)]

(2.122)

Operators defined this way, P and Q = I − P are called projection operators. P projects any vector v onto the null space of the divergence operator – ∇ · P(v) = 0, and Q projects any vector onto the null space of the rotation operator ∇ × Q(u) = 0. They fulfil the following dependencies: P 2 = P, Q2 = Q, PQ = QP = 0. Comparison of (2.122) and (2.119) brings the following form of these operators [57]: P = I − ∇(∇2 )−1 ∇ ·

Q = I − P = ∇(∇2 )−1 ∇·

(2.123)

Using these operators, the pressure and local acceleration can be decoupled from the Navier-Stokes equations. As the starting point, the numerical solution procedure takes the following approximate formulation: ∂˜ u = S(u) − ∇P˜ (2.124) ∂t where P˜ is a guessed or approximate pressure (from initial conditions, previous step). ˜ is not divergence-free, because P˜ 6= P in general. The resulting intermediate velocity u ˜ starts from the same initial condition as the final, real velocity u and can be projected u into the divergence-free subspace using operator P. Using this intermediate velocity given by the first step, a divergence-free solution can be approximated in the next stage by the projection: ˜ ud = P u (2.125)

44

2.5 Pressure treatment

It is assumed that ud is the final solution. However, the operator P is difficult to apply directly, because the operator ∇2 can be inverted formally only by Green function. In order to perform this inversion one must have information about the particular boundary conditions and domain geometry. Therefore, another approximation is met – recalling the equation (2.117) the projection step can be realised by the following decomposition: ˜ = ud + ∇φ u

∇ · ud = 0

(2.126)

where φ plays the role of a Lagrangian multiplier associated with the projection of the ˜ into the space of divergence-free vectors. The main idea of the intermediate solution u projection algorithm can be therefore sketched as follows: 1. Intermediate velocity step. Taking the divergence-free velocity from the initial condition u0 as well as an approximate initial pressure P0 , solve (2.124): ˜ ∂u = −∇P n + g + ν∇2 un − un · ∇un ∂t

(2.127)

where un = u0 and P n = P0 at t = 0. 2. Projection step. Perform the projection as follows: solve for φ and ud from (2.126): ˜ = ud + ∇φ u

∇ · ud = 0

(2.128)

in a two-step procedure: • Solve the pressure Poisson equation (PPE) achieved by applying the divergence operator on the first equation of 2.126: ˜ ∇2 φ = ∇ · u

(2.129)

• Determine the projected velocity, i.e. the final one: ˜ − ∇φ ud = u

(2.130)

3. Updating. Accepting the projected velocity as the final result un+1 = ud find the physical pressure P n+1 (e.g. by solving (2.114) – but it can be avoided as discussed further) and repeat the cycle. A number of algorithms based on the projection method, especially for internal flows, have been formulated since the first papers of Chorin [29] and Patankar [124, 123] (SIMPLE algorithm). In the description above, the non-trivial problems connected with the boundary conditions to be applied at different stages of this general algorithm, especially for the PPE were not mentioned. An extensive reviews of different variations of this method, various forms of PPE’s and, above all, boundary conditions are given by Gresho and Sani [64, 57, 58].

2.5 Pressure treatment

2.5.2.1

45

Projection-1 method

˜ is computed In the original method developed by Chorin [29] the intermediate velocity u from momentum equations without taking the pressure gradients into consideration. From this intermediate result, the global pressure and the divergence-free final velocity are found in the projection step. The function φ is in this case the physical global pressure. The realisation of this idea in a numerical scheme is described in section 2.5.3. 2.5.2.2

Projection-2 method

In the second version of the projection algorithm, the influence of the pressure gradients is not neglected in the solution of the momentum equations. The difference between ˜ and the final, divergence-free one un+1 is therefore smaller. the intermediate velocity u In the intermediate velocity step the pressure gradient from the previous step ∇P n is applied. In the projection step, a pressure correction φ and the final velocity are found. φ can be treated as a more abstract potential function related to pressure. In the original SIMPLE/SIMPLER algorithms [124, 123], in order to deal with the problems associated with the non-linearity of the momentum equations, an iterative procedure is implemented. The whole algorithm is repeated until the divergence of the intermediate velocity is approximately equal to zero. The pressure can be updated by solving the conventional pressure Poisson equation (2.114) from the known velocity field. 2.5.2.3

Projection-3 method

Gresho [57] proposed a projection-3 method as a possible improvement of the projection1 and projection-2 methods. In this scheme the rate of convergence is accelerated by taking into consideration the rate of change of the pressure gradients ∂∇P/∂t. A strict mathematical consideration presented by Shen [143] shows that the projection-3 method is not a proper approximation of the unsteady Navier-Stokes equations.

2.5.3

Pressure equation from fractional step formulation

The pressure Poisson equation used in the numerical realisation of the projection-1 method can be obtained from the time-discretised form of the Navier-Stokes equations. In the solution algorithm the velocity time derivative is treated explicitly and can be split into ∂u un+1 − u˜ u˜ − un = + (2.131) ∂t ∆t ∆t where u˜ is an intermediate solution for the velocity field, which does not need to satisfy the incompressibility condition. In this way the equation set (2.22–2.24) can be transformed into a set of equations containing all terms but pressure gradients and a second set containing them exclusively: u˜ − un + u · ∇u = −wfH + vfV + ∇ · (ν∇u) ∆t

(2.132)

46

2.5 Pressure treatment v˜ − v n + u · ∇v = −ufV + ∇ · (ν∇v) ∆t

(2.133)

w˜ − w n ̺ + u · ∇w = − g + ufH + ∇ · (ν∇w) ∆t ̺0

(2.134)

1 ∂p un+1 − u˜ =− ∆t ̺0 ∂x

v n+1 − v˜ 1 ∂p =− ∆t ̺0 ∂y

w n+1 − w˜ 1 ∂p =− ∆t ̺0 ∂z

(2.135)

Combining equations (2.135) and taking into consideration that the resulting field un+1 must fulfil the incompressible continuity equation (2.25): ∇ · un+1 = 0

(2.136)

another form of the Poisson equation for the pressure can be obtained: ∇2 p =

̺0 ˜ ∇·u ∆t

(2.137)

which is known as pressure Poisson equation from fractional step formulation. Finding the pressure from (2.137) and substituting it into equations (2.135) one obtains a divergence-free velocity field, i.e. satisfying the continuity equation. The appropriate boundary conditions for the different steps of this fractional algorithm are theoretically discussed in [64, 57, 58, 129]. According to Quartapelle [129] in the first part of the algorithm the Dirichlet and Neumann boundary conditions for the physical velocity should be applied. In the second part, the impermeability conditions must be fulfilled by the resulting field. The Neumann boundary condition in the form n · ∇p = 0 can be derived for this equation. Gresho et al. [64, 57, 58] discuss in detail consequences of applying various boundary conditions (and their simplifications), obtaining numerous variations of the projection algorithm. The advantage of the algorithm using equation (2.137) in comparison to the pressure ˜ on the right hand of Poisson equation (2.114) is based on the fact that the velocity u the equation (2.137) is now known from the first step of the algorithm. In the projection-2 method, the known pressure gradients from the previous step are taken into consideration when solving equations (2.132–2.134) and the pressure Poisson equation is formulated for a pressure correction. This scheme can be sketched as follows: ˜ − un u 1 ̺ + u · ∇u = − ∇pn + g + ∇ · (ν∇u) − 2Ω × u ∆t ̺0 ̺0 n+1 ˜ 1 u −u = − ∇(pn+1 − pn ) ∆t ̺0 n+1 ∇·u =0

(2.138)

2.5 Pressure treatment

2.5.4

47

Poisson equation for the hydrodynamic pressure in free surface flows

In contrast to the internal flows, for a free surface flow the pressure term in the momentum equations −̺−1 0 ∇p can be separated into terms consisting: (1) that part of pressure, which can be explicitly computed, and (2) that part, which can be found by solving a Poisson pressure equation (2.137). In the free surface flows, the hydrostatic pressure can be easily found. Therefore, in following, the pressure is decomposed (split) into a sum of the pressures p = pH + π, where the sum components can be physically interpreted as the hydrostatic pressure pH and hydrodynamic (dynamic, motion) pressure π. This fact can be used to develop an algorithm, where the equations (2.22–2.25) will be split in such a way, that the hydrodynamic pressure is treated as a correction to the hydrostatic component. A velocity field obtained from the gradients of the pressure correction field (i.e., the hydrodynamic pressure field) overlapped on the velocity field obtained using hydrostatic pressure, makes the resulting velocity field solenoidal, i.e. incompressible. The derivation follows the pattern described in the previous section: 1 ∂pH u˜ − un + u · ∇u = − − wfH + vfV + ∇ · (ν∇u) ∆t ̺0 ∂x

(2.139)

1 ∂pH v˜ − v n + u · ∇v = − − ufV + ∇ · (ν∇v) ∆t ̺0 ∂y

(2.140)

w˜ − w n 1 ∂pH ̺ + u · ∇w = − − g + ufH + ∇ · (ν∇w) ∆t ̺0 ∂z ̺0

(2.141)

∇2 π = un+1 − u˜ 1 ∂π =− ∆t ̺0 ∂x

̺0 ˜ ∇·u ∆t

1 ∂π v n+1 − v˜ =− ∆t ̺0 ∂y

(2.142) 1 ∂π w n+1 − w˜ =− ∆t ̺0 ∂z

(2.143)

This algorithm is analogous to the large projection algorithm family with this significant difference, that the split pressure components can be physically interpreted. In order to obtain the intermediate velocity field, the explicitly given hydrostatic pressure gradients are applied. In the projection step, the gradients of the hydrodynamic pressure obtained from a Poisson equation are used. Due to the fact that the free surface, as well as the tracer concentration field determining the fluid density are computed in a decoupled way, the hydrostatic pressure gradients are given at a time level n. For those rare cases, when the hydrodynamic pressure in comparison to the hydrostatic one is large in the whole computational domain, a version analogous to the projection-2 algorithm can be considered. In this case, in the momentum equations (2.139–2.141) not only the hydrostatic pressure gradients are applied, but also the gradients of the hydrodynamic pressure from the previous step are taken into consideration. In the second

48

2.5 Pressure treatment

part (projection), a pressure correction φ (treated as a correction to the hydrodynamic part of the pressure, φ = π n+1 − π n ) and the final velocity field is computed: ˜ − un u 1 1 ̺ + u · ∇u = − ∇pH n − ∇π n + g + ∇ · (ν∇u) − 2Ω × u ∆t ̺0 ̺0 ̺0 ˜ 1 un+1 − u (2.144) = − ∇φ ∆t ̺0 ∇ · un+1 = 0

2.5.5

Treatment of pressure and buoyancy terms

The main idea of the algorithm is to separate the hydrostatic pressure pH from the hydrodynamic pressure π and to treat them in different parts of the algorithm, whereby the hydrodynamic pressure is treated as a pressure correction to the hydrostatic one. The terms to be discussed (from eq. 2.22–2.24) are: 1 ∂pH 1 ∂π 1 ∂p =− − ̺0 ∂x ̺0 ∂x ̺0 ∂x 1 ∂p 1 ∂pH 1 ∂π = − =− − ̺0 ∂y ̺0 ∂y ̺0 ∂y ̺ 1 ∂pH 1 ∂π ̺ 1 ∂p − g=− − − g = − ̺0 ∂z ̺0 ̺0 ∂z ̺0 ∂z ̺0

Πx = − Πy Πz

(2.145)

There are two ways of separating (splitting) the pressures, discussed in following two sections. For simplicity, atmospheric pressure pa is set to zero and the hydrostatic pressure is assumed to be equal to zero at the free surface. 2.5.5.1

Hydrostatic pressure resulting from a constant mean density

The hydrostatic pressure can be defined by the following hydrostatic equation: ∂pH = −̺0 g ∂z

(2.146)

i.e. the hydrostatic pressure resulting from the constant mean fluid density for a given flow ̺0 (a parameter resulting from the Boussinesq approximation). This formulation is similar to the small perturbation equations. Integration in the water column yields: pH = −̺0 g(S − z) (2.147) pH this is the pressure, which would result if in equations (2.22–2.24) the hydrostatic approximation has been performed for a fluid with a constant density ̺0 . This is not the physical hydrostatic pressure, but a part of it resulting from splitting the equations.

2.5 Pressure treatment

49

The pressure and buoyancy terms are (for Πy analogically): # " 1 ∂π 1 ∂ ZS ∂S 1 ∂π 1 ∂p ̺0 gdz − =− = −g − Πx = − ̺0 ∂x ̺0 ∂x z ̺0 ∂x ∂x ̺0 ∂x

(2.148)

The vertical term is (introducing ∆̺ = ̺ − ̺0 ): Πz = −

1 ∂p ̺ 1 1 ∂π ̺ ∆̺ 1 ∂π − g = − (−̺0 g) − − g=− g− ̺0 ∂z ̺0 ̺0 ̺0 ∂z ̺0 ̺0 ̺0 ∂z

(2.149)

As a result, in the horizontal terms free surface gradients appear independent of the density (barotropic part) and gradients of the hydrodynamic pressure. The terms describing horizontal gradients of the density disappear. In the vertical term the buoyancy term representing density effects remains with the vertical gradient of the hydrodynamic pressure. 2.5.5.2

Hydrostatic pressure resulting from the local density

In this case the hydrostatic pressure is resulting from the local fluid density ̺(x, y, z) i.e. in the usual physical way, and defined by equation: ∂pH = −̺g ∂z

(2.150)

Integrating in the water column yields: pH =

Z

z

S

̺gdz =

Z

z

S

(̺0 + ∆̺)gdz = ̺0 g(S − z) + ̺0 g

Z

S

z

∆̺ dz ̺0

(2.151)

pH is now the physical hydrostatic pressure with taking the variable fluid density ̺ into consideration. ∆̺ = ̺ − ̺0 is a density variation from an average value for a given flow. The pressure and buoyancy terms are (for Πy analogically): "

#

S ∆̺ 1 ∂p 1 ∂ 1 ∂ 1 ∂π Πx = − ̺0 g =− [̺0 g(S − z)] − dz − = ̺0 ∂x ̺0 ∂x ̺0 ∂x ̺0 ̺0 ∂x z " # ∂S ∂ Z S ∆̺ 1 ∂π −g −g dz − ∂x ∂x z ̺0 ̺0 ∂x

Z

(2.152)

The vertical term is: Πz = −

1 ∂p ̺ 1 ∂π 1 ∂π ̺ ̺ − g= g− − g=− ̺0 ∂z ̺0 ̺0 ̺0 ∂z ̺0 ̺0 ∂z

(2.153)

As a result, free surface gradients independent of the density (barotropic part), horizontal gradients of the pressure resulting from density differences (baroclinic part) and gradients of the hydrodynamic pressure appear in the horizontal terms. In the vertical term only the vertical gradient of the hydrodynamic pressure remains. This is an important simplification and this treatment of the pressure terms is eventually implemented in the non-hydrostatic algorithm.

50

2.6

2.6 Free surface

Free surface

A free surface is a material surface across which the density is discontinuous. Calculating the position of the free surface in a hydrodynamic-numeric model provides a unique challenge. First of all, free surface and interfacial flows can be difficult to model numerically because the computational domain changes in time. The free surface must be discretely described (free surface representation) and advanced in time (free surface tracking or advancing). The application of diverse important boundary conditions at the free surface (surface tension, external pressure fields, wind stress, thermal radiation and boundary conditions for variables describing turbulence) is troublesome, because they must be applied at the eventual location of the free surface. This is difficult, especially when fluid regions may be coalescing, detaching or breaking up. All these problems are closely related – the algorithms used to advance or track the free surface depend on its discrete description or representation, and the boundary conditions can be applied only when the free surface position is known. Finally, the barotropic pressure gradients caused by free surface slopes are a driving force of the flow, especially in shallow waters, making the exact reproduction of free surface movements a crucial point in the modelling. A few methods of finding free surface position found wider acceptance. Two of them, marker-in-cell (MAC) and volume-of-fluid (VOF), and various mixed methods belong to schemes where the fluid regions or volume is tracked (figure 2.3). In the height function and line segment methods, the free surface itself is modelled. Although only the height function method has actually been used here, all these methods are described here in order to deliver a critical overview of the presently available methods in light of the finite element method. The position of the free surface is determined by the following equation: dxs = us dt

(2.154)

where xs describes a free surface point and us velocity at the free surface. The same equation applies for the fluid interior, and when there are no specific boundary conditions to be considered at the surface, its position can be found directly from tracking the fluid volume.

2.6.1

Marker-and-cell method

The Marker-and-cell method (MAC), developed 1965 by Harlow and Welch [65], uses massless marker particles distributed in the whole fluid volume to trace the free surface movements. The free surface is therefore tracked using a Lagrangian approach. In the original technique the computational grid is fixed in time and space. The method was improved later in several ways, e.g. by Miyata [115], Tom´e and McKee [156, 157] in the framework of the finite difference method, and Nakayama and Mori in finite elements [117].

2.6 Free surface

51

In this method the free surface is tracked by positions of fictitious and massless Lagrangian marker particles. Initially, they are equally distributed in the whole fluid volume (in order to distinguish the computational cells taken by the fluid) and are advected passively with the local fluid velocity. The fixed computational mesh covers the whole area of possible fluid movement, so that some cells remain empty. The instantaneous fluid configuration can be determined by finding the marker positions in subsequent time steps. Cells which have at least one empty neighbour cell are marked as boundary cells. In order to find the free surface position properly, a possibly uniform marker distribution in each time step and in each fluid cell is necessary (for instance, at least four markers for square 2D cells [65]). The marker distribution varies due to velocity gradients, inflows and outflows, and it is necessary to redistribute the particles evenly in all fluid cells when the they tend to spread disproportionately. In an optional step the position of the free surface inside the boundary cells can be determined from the marker distribution in them [115]. A great advantage of this method is the ability to handle complicated, general and arbitrary free surface problems: breaking surfaces [115]; dam breaking or splash of a falling column of water [65]; fluid detachment or coalescence (droplets); or to simulate filling complex moulding shapes [157]. This method is, however, computationally prohibitive for large-scale, three-dimensional applications, because of the need to use a large number of continually redistributed particles in order to find the surface shape properly from their positions. Even with a large number of marker particles, it is difficult to obtain sufficient information for determining the interface orientation in a cell. Because the number of particles is finite, false regions of void cells can be generated in regions with greater velocity shear (gradients). It is problematic to impose boundary conditions governing at the free surface, especially for pressure. Additionally, there are also severe restrictions relating the time step to the free surface wave speed, so that the computational grid must have sufficiently fine resolution. Finally, no three-dimensional applications of this method are known in the literature of the subject.

2.6.2

Volume of fluid method

Volume of fluid method (VOF) was introduced by Hirt and Nichols in 1981 [79]. It can be applied to problems with a number of fluids with different densities and can also handle breaking surfaces. A VOF algorithm consists generally of three parts: (1) an algorithm to track the volume and locate the free surface, (2) a scheme to track the surface as a sharp interface between fluids (e.g. water/air) moving through a given computational grid, and (3) means of applying boundary conditions at the surface. The main idea is to introduce a fractional volume (colour, filling fraction, fill state) function F (x, t) ∈ [0, 1] indicating the fraction of a mesh cell that is filled with a fluid of a particular type, with value exactly 1 in cells occupied by fluid and exactly 0 in void cells elsewhere. A fractional value in a cell indicates that this cell contains the free surface. For incompressible flow F can be regarded as a normalised density. As such, it must satisfy the following conservation equation: dF ∂F = + ∇ · (uF ) = 0 (2.155) dt ∂t

52

2.6 Free surface

real configuration

r rr rr rr rr rr rr rr rr rr rr rr

r r r r r r r r r r r r

r r r r r r r r r r r r

r r r r r r r r r r r r

r r r r r r r r r r r r

r r r r r r r r r r r r

r r r r r r r r r r r

r r r r r r r r r r r

r r r r r r r r r r r

Marker-in-Cell (MAC) r r r r r r r r r r

r r r r r r r r r

r r r r rr rr r rr rr rr r

Volume of Fluid (VOF)

Figure 2.3: Volume-tracking methods: MAC and VOF

2.6 Free surface

53

i.e. F is being advected in the velocity field u but not diffused. The volume tracking algorithm solves the advection equation for the function F in such a way that the interface is kept sharp. The interface position is recovered from F after solution. Applying the explicit integration scheme to equation (2.155) the following is obtained: F n+1 = F n +

∆t X n Q Vi k ik

(2.156)

where the superscripts denote the time steps, and Qnik represents the flow rate into element/cell i of a volume Vi through face k at the time-step n. In the computation of Qnik geometric factors must be taken into consideration: n Qnik = −αk qik = −αk

Z

k

unj nj dΩ

(2.157)

where αk represents the wet fraction of the cell face k, i.e. through which the fluid n flows. The fractional volume flux qik through the face k depends on the velocity field and direction of the unit normal vector n. There exists a number of different techniques based on purely geometrical considerations n to determine αk or qik directly according to the fill state of a given cell and its neighbours. It is equivalent to reconstructing the interface position from the eventual fluid configuration after each time step. Rudman [135] reviews and discusses advantages and disadvantages of the existing algorithms (Hirt-Nichols’ donor-acceptor [79], SLIC [121], FCT-VOF [170, 135] and Youngs’ method [169, 135]) in numerous test cases. Because the equation (2.156) does not guarantee that F remains bounded, i.e. F ∈ [0, 1], an additional flux-limiting (bookkeeping adjustment) algorithm must be applied in order to avoid over-filled or over-emptied elements. Depending on the fill states of the neighbouring cells and the values of F , the geometric factors αk are adjusted in such a way that F has the value exactly 1 in entirely filled cells and exactly 0 in empty ones. Applying boundary conditions with in the VOF method can be associated with difficulties, especially when dealing with a number of fluids with very different densities and viscosities and taking the surface tension into account. Examples of interesting free surface applications of this method include dam-breaking, Rayleigh-Taylor instability [79], wave breaking [102], gravity currents, collisions of droplets, impacting and splashing of liquid drops at a free surface, bubbles rising and bursting at a free surface [135], merging and fragmentation in multiphase flows [94]. In contrary to the Lagrangian MAC method, the VOF method represents an Eulerian approach. The advantage of VOF over the MAC method is in the storage requirements: only one word is required for one cell, compared to positions of a few marker particles. However, this method does not conserve mass and momentum exactly, as stated in [152, 153]. The computational costs of obtaining an accurate interface shape from geometrical considerations can be high.

54

2.6.3

2.6 Free surface

Height function methods

Other three-dimensional modelling techniques are based on treating the free surface directly as a moving boundary. The height function method requires that the free surface can be described in this formulation by a single-value function (height function) S(x, t) with respect to one of the co-ordinate directions. In this method the distance between the interface and a given reference level is calculated from a separate equation. This approach offers a simple and robust method of simulating free surface flows. However, the restriction to single valued functions exclude a wide class of flows, e.g. breaking surfaces, bubbles, drops. In contrast to the MAC and VOF methods, which can be described as volume-tracking methods, the height function methods are surface-tracking. The position of the advancing surface can be tracked using an Eulerian approach (in a fixed grid) or a Lagrangian one, where the mesh can be adapted to the position of the free surface at each time step (e.g. σ-mesh). The most widely used equations for tracking the free surface are the kinematic boundary condition and conservative free surface equation. Compared to the MAC and VOF methods the numerical implementation of the height function method is relatively simple. Therefore, for these flows, where the free surface can be described by a single-valued function, this method is to be preferred. This is the case in most fluvial and marine applications. An additional and very important advantage is that various boundary conditions at the free surface can be implemented without greater problems.

2.6.3.1

Kinematic boundary condition at the free surface

When the free surface can be represented by the equation F (x, y, z, t) = S(x, y, t)−z = 0, F will always be 0 for a particle at this surface and the rate at which the function F varies is given by its material derivative: dF ∂S ∂S ∂S = + us + vs − ws = 0 dt ∂t ∂x ∂y

(2.158)

The above equation is the well-known kinematic boundary condition at the free surface. The suffix s indicates the velocity components at the free surface. Equation (2.158) is a partial differential equation of a hyperbolic type. By requiring the continuity of motion, it can be shown that the surface satisfying (2.158) consists always of the same particles [96]. S(x, y, t) must be a single-valued function, so very steep surface gradients or wave-breaking effects cannot be reproduced. Although the evolution equation (2.158) is traditionally called a boundary condition, it can be used to determine and advance the position of the free surface, when all three components of the velocity at the surface are known. When the surface position is given for some reasons, the kinematic boundary condition can be used as a usual boundary condition, e.g., for the vertical component of the velocity.

2.6 Free surface

55

As an example of successful finite difference implementation, the models presented by Bulgarelli, Casulli and Greenspan [18] should be mentioned. Finite element implementations using kinematic boundary condition, as [90, 165], show the main limitations of this method, in the fact that the surface cannot break and the limited range of Reynolds numbers, for which the numerical solution remains stable. Similar problems are encountered by Yost [168] when testing the fully implicit finite element model mentioned above. Farmer et al. [43] use the FV method for computing the flow and the FD method for the kinematic boundary condition. 2.6.3.2

A conservative form of the free surface equation

Integration of the continuity equation (2.7) over the depth, i.e. from the bottom z = −B(x, y) to the free surface z = S(x, y, t), with usage of the Leibniz theorem1 in order to exchange the differentiation and integration sequence, yields: Z S Z S ∂v ∂w ∂u dz + dz + dz = −B ∂y −B ∂z −B ∂x Z Z ∂ S ∂S ∂ S ∂S ∂ ∂ + + ws − wb = 0 udz + ub (−B) − us vdz + vb (−B) − vs ∂x −B ∂x ∂x ∂y −B ∂y ∂y Z

S

The suffixes b and s indicate values on the bottom and surface, respectively. Using the impermeability condition at the bottom (2.185) and kinematic boundary condition at the free surface (2.158) one obtains the following conservative form of the free surface equation: ∂S ∂ ZS ∂ ZS udz + vdz = 0 (2.159) + ∂t ∂x −B ∂y −B This equation can be used to determine the free surface position instead of the kinematic boundary condition (2.158). The main advantage of this equation is that it includes the kinematic boundary condition and the impermeability condition at the bottom as well. This approach brings a method of finding a free surface while automatically satisfying the mass conservation criterion. It is applied in the pressure – free surface – continuity step of the hydrostatic Telemac3D algorithm. Casulli and Stelling [24] apply it in their 3D-non-hydrostatic code in an implicit formulation.

2.6.4

Line segment method

The line segment method is a generalisation of the height-function method and found its applications in 2D-models. The free surface (or any other interface) is defined as a chain of short line segments (shorter than mesh spacing) defined by an ordered set of points 1

Leibniz formula Z b(x,y) Z b(x,y) ∂b(x, y) ∂a(x, y) ∂f (x, y, z) ∂ dz + f (x, y, b) − f (x, y, a) f (x, y, z)dz = ∂x a(x,y) ∂x ∂x ∂x a(x,y)

56

2.6 Free surface

whose co-ordinates are moved according to the fluid velocity. Therefore, this method is not limited to single value surfaces. The position of the advancing surface points chain can be tracked using Eulerian and/or Lagrangian methods [144]. The most important difficulty appears when the chains intersect, fold over themselves or break up (merging or break up of fluid regions) – a procedure detecting such a situation and reordering the chain(s) is needed. Most applications of this method are found in the physicochemical simulations of two phase flows, crystal growth, forming of ice, melting, etc.

2.6.5

Combinations of different methods

Thomas et al. in search of mass- and energy-conserving free-surface computation developed a method based on a combination of the volume of fluid method with the kinematic boundary condition (FD) [153]. The method ensures a good mass conservation during long simulation times and smooth surface movements (wave breaking excluded). Limitations of this model are due to restrictions on the free surface movements ensuring numerical stability. An interesting treatment of the free surface in a fully conservative 3D FV code was developed by Lilek [105]. After solving the momentum equations using the specified pressure at the current free surface (the dynamic boundary conditions, see section 2.7.2), a pressure-correction scheme with zero pressure correction at the free surface is applied in order to enforce mass conservation. The prescribed pressure at the surface produces a velocity correction there, resulting in a non-zero mass flux. The new position of the free surface is determined by moving the boundaries of surface cells in such a way, that their movement compensates the flux obtained in the previous stage. These two steps are repeated iteratively until no compensation through movement of the free surface is needed and the obtained free surface position fulfils the kinematic boundary condition.

2.6.6

Treating variable domain extents

The movements of the free surface create a need to update the computational domain extents and, when needed, redefine node positions, boundary conditions, etc. For dealing with the variable domain extents, Eulerian approaches exist, where the free surface movement is described in a fixed mesh (mostly applied in FDM, e.g. [18], in FEM e.g. together with the MAC method [117]). Alternatively, with the Lagrangian approach the mesh moves with the flow (e.g. [131]). Both methods lead to different formulations of Navier-Stokes equations, whereby with the Lagrangian method the terms describing advection disappear. Lagrangian methods may lead to very distorted elements negatively influencing the accuracy. On the other hand, with the Eulerian approach difficulties in tracking the free surface and applying boundary conditions at the free surface exist. Therefore, a combined approach, known under a global name of Arbitrary Lagrangian-Eulerian (ALE) methods, has found great popularity in FEM applications, e.g. [36, 130, 80]. Globally

2.7 Initial and boundary conditions for the flow

57

speaking, in order to avoid larger element distortions, the ALE-mesh moves with an arbitrary velocity which is different from the local fluid velocity. When the mesh velocity is equal to 0, an Eulerian approach is obtained, when it is equal to the fluid velocity, the Lagrangian viewpoint is assumed. The equations are formulated and solved in a transformed co-ordinate system, and at each time step (or when the mesh becomes distorted enough) mesh re-zoning is performed in order to adapt it to the new domain geometry. For free surface flows it is very convenient to assign the arbitrary mesh velocity to the movements of the free surface [80, 26]. Tezduyar et al. apply a deforming space-time finite element formulation. In this case, the finite element formulation is written in the associated time-space domain; the spacetime finite element mesh covers the space-time domain of the problem. In this way, the mesh movement is taken into account automatically [149]. In general, this approach also allows an arbitrary definition of mesh movement, like ALE. Tezduyar, Behr, et al. successfully apply this method to free surface and interfacial flows [150, 151]. In the model described in this work, a well-known approach for the mesh re-zoning and adaptation to the movement of the free surface known as σ-transformation is assumed (described in detail in section 3.1). In principle, this method can be treated as a simplified ALE method applied to free surface geophysical flows, where mesh nodes can move only vertically with distances between the nodes relative to the water depth remaining constant in time. Telemac3D also provides a method to deal with tidal flats or flooding areas, which are not treated in this work [74].

2.6.7

Implemented free surface algorithms

Due to the σ–mesh structure of Telemac3D, requiring the free surface to be a singlevalued function, the most natural methods of finding the free surface are those based on the height function. Eventually, two methods based on the kinematic boundary condition and one based on the conservative free surface equation are implemented. Implementation of MAC or VOF methods can be considered, but without changing the mesh structure no special benefits can be expected due to the fact, that their most important advantages concern the ability to describe arbitrary free surface configurations.

2.7

Initial and boundary conditions for the flow

In this section the initial and boundary conditions are described. In order to set a well posed problem, initial and boundary data must be provided so that a unique solution of the governing partial differential equations exists and depends continuously on the data. The computational domain Ω for the geophysical flows with the free surface is bounded vertically between the bottom and the changeable free surface, and laterally by impermeable (solid walls, bottom) as well as open (inflow, outflow) boundaries. They represent physical boundaries associated with real limits of the domain as well as artificial boundaries due to the fact that the computational area can be truncated for practical reasons.

58

2.7 Initial and boundary conditions for the flow

The extents of these boundaries may change due to the movements of the free surface. While at the physical boundaries the boundary conditions (BC) can be obtained from the flow physics, the artificial boundaries require care in providing information so that a unique solution can be obtained. In the theory of the incompressible flows, on the domain boundary Γ one or more velocity components w can be prescribed (imposed as a Dirichlet boundary condition), so that [64]: u(x, t) = w(x, t) x∈Γ t≥0 (2.160) Z

Ω

∇ · wdΩ =

Z

Γ

w · ndΓ = 0

t≥0

(2.161)

The equation (2.161) results from application of the divergence theorem to the continuity equation, which must be fulfilled by w. Neumann boundary conditions for velocity on Γ are obtained from the normal or tangential stress components: σn = (σ · n) · n σt = (σ · n) · t

(2.162) (2.163)

As it will be shown in the next sections, most of the Neumann BC at the boundaries for a flow variable f can be formulated in the following form: k

∂f = n · ∇f = af + b ∂n

(2.164)

where k, a, b are constants or functions independent of f , what simplifies the implementation of these BC in numerical schemes (chapter 3). Both types of boundary condition are described in detail in following sections, preceded by a discussion of properly set initial conditions.

2.7.1

Initial conditions

As an initial condition (IC) in the computational domain interior as well as on the boundaries all the independent variables such as density, velocity, free surface position, temperature and tracer concentrations must be given. Additionally, for incompressible flows, it is of the greatest importance that the initial velocity field u0 must be solenoidal and fulfils initial boundary conditions as well: u(x, t) = u0 (x, 0) ∇ · u0 = 0 u0 · n = w(x, 0) · n

x∈Ω∪Γ x∈Ω x∈Γ

t=0 t=0 t=0

(2.165) (2.166) (2.167)

In practice, very often a null flow field is convenient. However, it is also convenient to initialise the flow at the velocity values equal to or interpolated from the inflow/outflow boundary conditions. In some cases it is also important to define the pressure field at t = 0.

2.7 Initial and boundary conditions for the flow

59

However, there are cases in which an initialisation of the model starting from a given divergent initial velocity U0 may be dictated by practice. In this case, a projection onto the space of divergence-free velocities (solenoidal) can be performed, treating the field U0 given as the initial one as an intermediate velocity (see section 2.5.3). First, the following Poisson equation is solved: ∇2 φ = ∇ · U0

(2.168)

an then, the projection is performed: u0 = U0 − ∇φ

(2.169)

obtaining a divergence-free initial field u0 . This field must fulfil all the initial boundary conditions. As the initial pressure the hydrostatic pressure resulting from the initial free surface configuration and density field is quite appropriate. (If the projection-2 method is chosen, the initial hydrodynamic pressure can be set to zero in most cases.)

2.7.2

Dynamic boundary conditions

The dynamic boundary conditions are defined as the BC obtained from the stress continuity at the boundaries of the computational domain. The boundary is oriented by a normal vector. For a boundary described by an equation F (x, y, z) = 0: n=

∇F |∇F |

(2.170)

As a convention in fluid mechanics, the normal vector points out of the computational domain. As the kinematic boundary condition represents the continuity of the velocity at the free surface, the dynamic boundary conditions are the expression of the requirement of the stress vector σ · n continuity across the free surface. However, the dynamic boundary conditions can be applied at all kinds of boundaries. In the normal direction to the boundary: σ · n = (−pin + 2µD) · n = (−pout I + τ ) · n (2.171) pin and pout are the pressures inside and outside the domain boundary oriented by the normal vector n (and two tangential vectors t1 and t2 ). τ the outside boundary stress tensor (e.g. wind or bottom stresses). The dynamic viscosity µj = ̺νj , where νj are the components of the variable eddy viscosity. σ is the turbulent stress tensor and D is the Navier-Stokes tensor, given by (2.9). For Reynolds-averaged incompressible Navier-Stokes equations for turbulent flows (2.2), with introduction of the Boussinesq eddy-viscosity concept [12] (note section 2.1.3) the turbulent stress tensor reads: σij = −pin δij + µj (∂uj /∂xi )

(2.172)

60

2.7 Initial and boundary conditions for the flow

From conditions (2.171) one normal and two tangential conditions can be obtained by projection of these three conditions in the direction of the normal or a tangential vector. This is a convenient way of treating these conditions in domains with complicated boundary geometry. The projection is realised as follows: σn = (σ · n) · n σt = (σ · n) · t

(2.173) (2.174)

The projections in normal and tangential directions of (2.172) yield: σn = −pin + µx

∂unx ∂unz ∂unz + µy + µz = −pout + τnout ∂x ∂y ∂z

σt = µx

∂utz ∂utz ∂utx + µy + µz = τtout ∂x ∂y ∂z

(2.175)

(2.176)

where fnout and ftout are the normal and tangential outside boundary stress force (traction force) components. Using the sloppy, but short and convenient notation ∂f /∂n = n · ∇f the conditions (2.175 – 2.176) can be written in a compact way: −pin + µ

∂un = −pout + τnout ∂n

µ

∂ut = τtout ∂n

(2.177)

For simple geometries it may be convenient to let normal and tangential directions coincide with the co-ordinate system axes. For the free surface given by the equation z = S(x, y, t) from (2.171) one obtains: ∂u τxz ps ∂S ∂u ∂S ∂u ∂S = − + νx + νy ∂z ̺ ̺ ∂x ∂x ∂x ∂y ∂y τyz ps ∂S ∂v ∂S ∂v ∂S ∂v = − + νx + νy νz ∂z ̺ ̺ ∂y ∂x ∂x ∂y ∂y ∂w τzz ps pa ∂w ∂S ∂w ∂S νz = + − + νx + νy ∂z ̺ ̺ ̺ ∂x ∂x ∂y ∂y νz

(2.178)

where the outside stress components multiplied by surface gradients are already neglected, which is a good approximation in the case of wind stresses. ps and pa are the pressure at the surface and the atmospheric pressure, respectively. From equations (2.178) simplified conditions can be obtained for these applications, where the gradients of the free surface are small enough that they can be neglected and n ≈ (0, 0, 1): νz

τxz ∂u = ∂z ̺

νz

∂v τyz = ∂z ̺

ps = pa − τzz + ̺νz

∂w ∂z

(2.179)

For the bottom, similar conditions can be obtained. At the lateral boundaries n = (nx , ny , 0) and a two-dimensional form of (2.175) can be applied. For example, in a

2.7 Initial and boundary conditions for the flow

61

simple case when the lateral boundary is represented by n = (nx , 0, 0), one obtains the following conditions: pin = ̺νx

∂u + τxx ∂x

νx

∂v = τxy ∂x

νx

∂w = τxy ∂x

(2.180)

The dynamic boundary conditions are usually applied in order to take the wind stress on the surface and friction on the impermeable boundaries into consideration. They provide a Neumann boundary condition for the velocity parallel to the boundary. For inviscid flows, the dynamic boundary conditions reduce to the equilibrium of pressures at both sides of the boundary: pin = pout

(2.181)

In the case of the free surface, (2.181) is called the inviscid free surface normal stress boundary condition and is a very good approximation of the dynamic boundary conditions also for small movements of the free surface for flows with low vertical viscosity νz . In most of the simple application cases presented in this work it proves to be exact enough. The dynamic boundary conditions also yield conditions for a pressure equation derived from Navier-Stokes equations (section 2.7.11).

2.7.3

Impermeability: normal velocity at solid boundaries

Impermeability conditions at the bottom and (impermeable) lateral boundaries represented by a normal vector n = (nx , ny , nz ) for the velocity at the boundary ub read: ub · n = 0

(2.182)

This condition expresses the fact that no mass flux is allowed through the impenetrable boundary and therefore the normal velocity must vanish there. The impermeability condition can be implemented in different ways depending on the numerical method applied. The first method is to correct the resulting velocity field (which generally does not fulfil the impermeability condition exactly) at the boundary ub by setting the normal velocity at this boundary to null: un = (ub · n) · n = 0

(2.183)

Effectively, from each resulting velocity component u, v, w the appropriate x, y and z-components of the resulting normal velocity un are subtracted, so that the impermeability condition is eventually fulfilled: ub = u − un2x − vnx ny − wnx nz vb = v − unx ny − vn2y − wny nz

wb = w − unx nz − vny nz −

wn2z

(2.184)

62

2.7 Initial and boundary conditions for the flow

Another method is to obtain a condition in the form of an equation analogical to the kinematical boundary condition (2.158). E.g., for the bottom, described by the equation F (x, y, z) = B(x, y) + z the direction of the normal vector is given by (∂B/∂x, ∂B/∂y, 1) and from (2.182) one obtains: ub

∂B ∂B + vb + wb = 0 ∂x ∂y

(2.185)

This equation can be used e.g. to impose such a vertical velocity component which fulfils the impermeability condition by given horizontal velocity components and bottom gradients. In this way one component (w) plays a subordinate role to the two others (u, v).

2.7.4

Tangential velocity at solid boundaries

The dynamic boundary conditions (2.177) yield Neumann boundary conditions for the component of the velocity tangent to a solid boundary. If they cannot be applied for some reason, they must be replaced by a Dirichlet boundary condition for the tangential velocity (slip condition): ut = u0

(2.186)

This boundary condition is usually connected with zero Neumann boundary condition for the normal velocity component. u0 can be chosen to be equal 0 (no-slip condition) or given a non-zero value. By setting the tangential velocity to zero it should be remembered that the computational mesh must be fine enough to resolve the velocity profile between the boundary and free stream velocity.

2.7.5

Rigid lid

The rigid lid boundary condition at the free surface is an example of a very simplified treatment of the free surface. This boundary condition descends from the well-known rigid lid approximation in oceanography [53]. This approximation is widely used for investigation of flows in separated barotropic and baroclinic modes. It takes advantage of the fact that for the baroclinic mode the displacements of the free surface are small compared with the movements of internal interfaces. In consequence, the vertical component of the velocity at the surface is set to zero and for the horizontal components zero Neumann boundary conditions are applied. Obviously, at the free surface (usually assumed to be flat) pressure gradients can be given representing gradients or movements of the free surface (or atmospheric pressure gradients) for the depth-independent barotropic mode. In practice, for purely density-driven flows with Boussinesq approximation (small density variations), a horizontal free surface is appropriate.

2.7 Initial and boundary conditions for the flow

2.7.6

63

Bottom and free surface stresses

Bottom and free surface stress (shear) is taken into consideration by applying appropriate Neumann boundary conditions at the bottom and/or free surface (section 2.7.2). By application of the shear conditions defined by the outside stress tensor, in order to obtain proper velocity profiles in the turbulent bottom or surface boundary layer, the eddy-viscosity coefficients should be in general obtained from an appropriate turbulence model. 2.7.6.1

Bottom shear

The bottom boundary condition for a rough bottom is given by an empirical formulation in the form: √ ∂u u u2 + v 2 τb = νz = (2.187) ρ ∂z C2 where τb is the bed shear stress and νz the vertical turbulent viscosity coefficient. The roughness coefficient C is computed by assuming the existence of a logarithmic velocity profile up to at least one tenth of the distance between the first two planes of the computational mesh in the lowest part of the bottom boundary layer [103]. 2.7.6.2

Free surface (wind) shear

The surface wind stress is in most cases computed according to the empirical formula τw ̺water

= νz

∂uH ̺air cf uw |uw | = ∂n ̺water

(2.188)

where uH is the horizontal velocity at the surface, νz vertical turbulent viscosity coefficient and uw the wind velocity. Equation (2.188) states a Neumann boundary condition for the horizontal velocity components. The empirical coefficient cf is given e.g. by Flather [45]: 0.565 · 10−3 |uw | ≤ 5m/s −3 5m/s < |uw | < 19.22m/s cf =  (−0.12 + 0.137|uw | · 10  2.513 · 10−3 |uw | ≥ 19.22m/s   

2.7.7

(2.189)

Open boundary conditions

The open boundary conditions appear due to the fact that the flow domain must be truncated for practical reasons. The mathematics forces us to select boundary conditions there, which are physically artificial. Their form depends strongly on the type of the mathematic character (hyperbolic, elliptic, or parabolic) of the equations to be solved. At open boundaries the boundary conditions for the independent variables must be given, providing appropriate information to obtain a stable and unique solution in the

64

2.7 Initial and boundary conditions for the flow

domain. Actually, the solutions computational fluid dynamics are usually driven by the boundary conditions, and it is of extreme importance that well-posed and physically realistic boundary conditions are applied. The truncation of the computational domain results in two types of boundaries, namely, the inflow and outflow boundaries. At the inflow boundaries the conditions are usually well known and are easy to implement as imposed values. All components of the velocity, density, free surface, pressure, temperature and tracer concentrations have to be specified. However, for many applications the theory provides hints in order to reduce the number of needed boundary conditions, e.g. in the case of the channel flow it is enough to provide a given flux through the inflow boundary and require a zero normal gradient of the depth. The outflow boundary conditions require special care and are more difficult to apply because they cannot be predicted a priori for unsteady flows. Theoretically, the stress continuity, i.e., the dynamic boundary conditions (2.177) are to be provided. Commonly, the outflow sections are positioned where the character of the flow is well known, e.g., approximately unidirectional or where the stresses are known. For example, for a fully developed channel flow, or away from obstacles, there is no change of the velocity components across the boundary and fn = −p as well as ft = 0. This yields the set of the outflow boundary conditions [58, 161]: p = pout

or S = Sout

∂T =0 ∂n

(2.190)

These conditions specify the pressure (or the water depth) and let the normal gradients of other variables than the normal velocity be null. Such boundary conditions are implemented in a very natural way in the finite element (and volume) methods. For the normal velocity, there are theoretical concerns regarding outflow boundary conditions in the form: ∂un =0 (2.191) ∂n are set, as stated by [58, 137]. This boundary condition is in theory under-determined, resulting in general in an infinite number of solutions. For incompressible fluids, requirement ∇ · u = 0 by valid ∂un /∂n = 0 is over-restrictive for the tangential component, causing uτ = 0. For viscous flows, the most proper boundary conditions are given by dynamic boundary conditions, e.g. equation (2.180) set for all velocity components. In practice however, ∂un /∂n = 0 is usually used successfully. It should be noted, however, that the boundary conditions in the form (2.190) can be used only when all flows entering the domain are defined by imposed inflow boundary conditions. By applying outflow boundary conditions special care must be taken when positioning the outflow sections. Sensitivity studies should be made in order to demonstrate that the interior solution is not affected by the outflow boundary. When the simple forms of outflow boundary influence negatively the interior solution, especially for unsteady flows, some kind of more complex boundary conditions must be imposed, such as non-reflecting or absorbing boundary conditions (sections 2.7.8, 2.7.9).

2.7 Initial and boundary conditions for the flow

65

A usual, simple method to avoid applying these special and imperfect cases of boundary conditions is to make the computational domain so large that the area of interest can be treated as practically unbounded. Another method, e.g. for phenomena developing in the domain interior, is to stop the simulation when the boundaries start to influence the internal solution significantly. In some situations, e.g., for the channel flow, the inflow and outflow can be specified according to the type of the flow in order to provide a unique solution. E.g. one specifies the flux (and initial depth only) at the inflow boundary and water depth at the outflow one. It should be noted that in some applications the inflow boundary may become an outflow one during the simulation (e.g. tidal flows), and vice versa (reverse flows), or even a simultaneous inflow/outflow boundary (estuaries) – and the algorithm must react appropriately. Sometimes advantage can be taken from special geometrical aspects of the flow allowing cyclic or periodic boundary conditions where the outflow values are given as inflow values or symmetry boundary conditions where the domain can be truncated along lines of symmetry. A specific type of the boundary conditions are the conditions for the global pressure or the hydrodynamic (motion) pressure at these boundaries (section 2.7.11).

2.7.8

Non-reflecting boundary conditions

For flow problems dominated by advection or wave motion a special case of open boundary conditions is needed at the open/liquid boundaries which artificially truncate the computational domain. A boundary condition is needed which allows phenomena generated in the domain of interest to pass through the artificial boundary without undergoing any significant distortion or without influencing the interior solution. The intrinsic difficulty in imposing the boundary conditions of this kind arises due to the lack of knowledge of the environmental behaviour outside the domain. Therefore, some extrapolation taking into account the mathematical character of the equations to be solved must be used. Actually, most effective non-reflecting boundary conditions (NRBC) assume some specified behaviour or character of the flow (for free surface flows, mostly hyperbolic) just outside the given boundary. In cases where the simulation starts from an artificial or an awkwardly posed initial condition (e.g. flows over obstacles starting with a uniform velocity field) the non-reflecting boundary conditions are needed in order to allow the initially appearing wiggles or erroneous flow structures to exit the domain without causing any instabilities. For simulations aiming at the achievement of a steady-state solution, the boundary conditions appropriate for the final state are often applied. When the scheme is not dissipative (physically or numerically), such boundary conditions may cause serious difficulties concerning the solution convergence. A very important aspect of non-reflecting boundary conditions is that they must be also applied at the inflow boundaries with imposed values for flow variables.

66

2.7 Initial and boundary conditions for the flow

Most widespread non-reflecting boundary conditions for equations of hyperbolic nature are based on the Sommerfeld radiation condition. It is valid for the wave equation in the form: ∂2ϕ − c2 (ω)∇2ϕ = 0 (2.192) ∂t2 The Sommerfeld radiation condition brings a simple linear relationship between the partial time derivative of a variable in a wave motion and its spatial derivatives: ∂ϕ + c(ω) · ∇ϕ = 0 ∂t

(2.193)

where c is a frequency-dependent wave phase velocity. It essentially states that the (outgoing) wave propagates in the positive direction without changing its form. Orlanski [122] transposed the Sommerfeld radiation condition from the frequency to the time domain replacing the frequency-dependent wave phase velocity by a time-varying, propagation velocity, i.e. coefficient c(t): ∂ϕ + c(t) · ∇ϕ = 0 ∂t

(2.194)

As can be easily seen, a single wave component of the form ϕ(x, t) = ϕ0 exp(k · x − ωt) fulfils (2.194) when ci (t) = (ω/ki). In a finite difference stream function – vorticity model applied for oceanic mesoscale flows, Orlanski [122] computed numerically the coefficient c in the x-direction at an open boundary from the model spacing ∆x and ∆t and derivative values at previous time-step levels: cx = ∆x/∆t if cx = −(∂ϕ/∂t)/(∂ϕ/∂x) if cx = 0 if

−(∂ϕ/∂t)/(∂ϕ/∂x) > ∆x/∆t 0 < −(∂ϕ/∂t)/(∂ϕ/∂x) < ∆x/∆t −(∂ϕ/∂t)/(∂ϕ/∂x) < 0

(2.195)

assuring model stability and that no information (waves) from outside can enter the computational domain. From (2.194) a Dirichlet BC for the new time level is obtained [122]. The Orlanski condition is local both in time and space and works well when the incident waves are monochromatic (ω = const) or for non-dispersive long waves, where c is constant and independent of ω. For unsteady, dispersive waves and nonmonochromatic wave motions problems arise. However in many applications for e.g. internal turbulent flows, the boundary condition in form (2.194) where c can be so calibrated that it is independent on the outflow surface position and guarantees overall conservation (the outflow mass flux equal to the incoming one), proved to yield good results [44]. Enquist and Majda [42] developed a perfectly non-reflecting boundary condition for the wave equation (2.192) that is non-local in time and space (they called it absorbing BC). It required the complete time history of variables along the boundary in order to update them in the next time step. This boundary condition, very awkward for numerical computation, can be replaced by a derived local approximation to the general solution

2.7 Initial and boundary conditions for the flow

67

up to a required order of accuracy n. Higdon [75, 76] developed a more general form of these boundary condition for a 2D finite difference wave model (including the angle of incidence αj of the incoming waves):  

n Y

j=1

!

c ∂  ∂ + ϕ=0 ∂t cos αj ∂x

(2.196)

This boundary condition is perfectly non-reflecting for a plane wave hitting the boundary at one of the a priori given angles αj . Its first approximation n = 1 and for α1 = 0 is the well-known Sommerfeld radiation condition. Hedstrom [69] developed non-reflecting boundary conditions for non-linear hyperbolic systems derived from the characteristic form of the hydrodynamic equations with an application for gas dynamics (shock waves). Thompson [154, 155] developed this idea further to multidimensional boundary conditions describing incoming and outgoing waves as well. An application of so obtained boundary conditions in a wave model for shallow and deep water waves is provided by Schr¨oter [141, 142]. For this method, the set of n governing equations is written in the following form: ∂U ∂U +A +C=0 ∂t ∂x

(2.197)

where U is a vector of n primitive variables, A is a n × n matrix and C is a term which does not contain any derivatives of the variables appearing in U. When the system (2.197) is hyperbolic, the matrix A has n real eigenvalues λi . The diagonal matrix of eigenvalues Λ, Λii = λi is obtained from similarity transformation: SAS−1 = Λ

(2.198)

where S contains as rows the left (row) eigenvectors li of A defined by li A = λi li and the columns of S−1 are the right (column) ri eigenvectors of A defined by Ari = λi ri . These vectors are normalised, so that li rj = δij . Multiplying (2.197) by S yields the following characteristic form of the governing equations: ∂U ∂U S + ΛS + SC = 0 (2.199) ∂t ∂x or, using the eigenvectors, li

∂U ∂U + λi li + li C = 0 ∂t ∂x

(2.200)

If a function V can be defined, so that dVi = li dU + li Cdt

(2.201)

∂Vi ∂Vi + λi =0 ∂t ∂x

(2.202)

then (2.200) transforms to

68

2.7 Initial and boundary conditions for the flow ν(x)

6

ν(x)



-

no damping

-

-x

damping zone

Figure 2.4: Defining a damping zone for absorbing boundary conditions.

The equation above is a set of wave equations stating, that the functions Vi (Riemann invariants) do not change along a characteristic curve defined by dx/dt = λi in the plane xt. It plays an analogous role to the Sommerfeld radiation condition (2.193). According to the idea of Hedstrom [69], it is assumed, that the functions Vi for the incoming waves are independent of time at the boundaries. For the waves described by incoming characteristics it is assumed that ∂Vi /∂t = 0 (or by setting an imposed Dirichlet value there). For the outgoing waves of known λi , the values at the boundary nodes can be found by the method of characteristics from (2.202). The main difficulty in applying this method of determining non-reflecting boundary conditions at the boundaries is that Vi can be determined from (2.201) when A and C are constant in the whole domain and (2.201) is integrable. However, arguing that the form (2.200) can be obtained for most practical cases, Thompson [154, 155] develops a general boundary condition formalism for hyperbolic systems. The non-reflecting boundary conditions based on the Thompson method were also developed for the two-dimensional code Telemac2D (shallow water equations) for advectiondominated flows [73]. The various source terms represented by C in (2.197) are treated as known and computed explicitly in the framework of the operator splitting. For the outgoing waves, the method of characteristics for the boundary nodes is applied, exactly as for the internal nodes in the advection step (section 3.5.2). As the advection velocity the eigenvalues λi (i.e. for shallow water equations (u, u + c, u − c) for the direction normal to a given boundary node are assumed. The contributions from relevant Riemann invariants Vi are added to obtain boundary values for u, v, h and tracer concentrations. For Telemac3D the non-reflecting boundary conditions are currently being developed elsewhere (J.-M. Hervouet, LNH/EDF, 1998, personal communication). For entirely barotropic flows it is theoretically possible to use the routines developed for Telemac2D. Consequently, in this work, no specific developments in this direction are made and only the simplest form of non-reflecting boundary conditions is applied (section 5.7).

2.7 Initial and boundary conditions for the flow

2.7.9

69

Absorbing boundary conditions

The absorbing boundary conditions are an effective method of truncating the computational domain. The idea of this approach is to ensure that waves generated inside the domain are effectively absorbed at a given boundary or some buffer area before it. Typical examples of an application are numerical wave models for wave tanks or channels, where waves generated at a wave making surface run through an area of interest and then are absorbed at a numerical beach, sponge layer or a piston-like absorbing surface. In the numerical beach or sponge method, an additional dissipative term is added to the dynamic free surface condition or to the kinematic condition or to both [32]. A coefficient playing the role of a viscosity coefficient is set to zero all over the area, except in a zone before the artificial boundary where it gets non-zero values. According to the particular application, it damps the wave elevation, fluid velocity, wave potential, or a combination of them. Water waves, passing through the region, where the damping is applied, systematically lose their energy (figure 2.4). When the damping (sponge) zone is sufficiently long compared to the water wave length, the waves may be entirely absorbed. This approach works effectively as a low-pass filter for the waves and the absorbing coefficient can be tuned to the spectrum of the incoming waves. The piston-like absorbing boundary condition is a Neumann boundary condition for the wave potential motivated by properties of the incident waves to be absorbed and derives from a study of the control of the physical wave absorbing devices in wave tanks [32]. For a 3D hydrodynamic model, this method can be most effectively applied for the free surface determination with the kinematic boundary condition and computed with the SUPG method, as described in section 3.6.2.1. It may be used together with setting an analogical damping zone for the velocity components through viscosity coefficient values near the boundaries.

2.7.10

Boundary conditions for the tracer

Fluxes or imposed values of the tracers must be given at the domain boundaries. For example, in case of the temperature T : λ

∂T = −qb ∂n

or T = Tb

(2.203)

where qb is the heat flux at the given boundary and λ the thermal conductivity. The values are usually obtained from various empirical formulations or for instance, in the case of pollutants, according to discharge characteristics. For many applications the no-flux condition is appropriate: ∂T =0 (2.204) ∂n

70

2.7.11

2.7 Initial and boundary conditions for the flow

Boundary conditions for the hydrodynamic pressure equation

Boundary conditions for hydrodynamic pressure can be obtained from purely physical deliberations or from more abstract mathematical reasoning based on the particular features of the numerical algorithm. In the case of a Poisson pressure equation from fractional formulation written for the hydrodynamic pressure (section 2.5.3), the particular type of boundary conditions depend strongly on the physical system to be modelled. As a starting point it should be realised that setting the imposed (Dirichlet) value of null for hydrodynamic pressure at a boundary has the physical meaning of applying purely hydrostatic pressure there. Hydrostatic pressure means no fluid motion, or that hydrostatic approximation is assumed to be valid. As a further consequence, on these boundaries, where an imposed value of zero dynamic pressure appears, it is automatically ˜ is needed in order to fulfil the incompressibility required that no velocity correction for u condition. Therefore, hydrostatic approximation π = 0 is an acceptable boundary condition for these open inflow boundary sections, where the velocity is thoroughly defined and its divergence is assured to be equal to zero. For the viscous flows, at the open boundaries and at the free surface the dynamic boundary conditions (2.177) can be implemented. At the free surface, the following Dirichlet BC is valid: ∂un πs = ̺ν (2.205) ∂n where for simplicity the atmospheric pressure is set to zero and no outside stress (e.g. wind) is present. When the gradients of the free surface are small enough, this boundary condition can be simplified to: ∂w (2.206) πs = ̺νz ∂z It can be seen that in the inviscid case ν = 0 an imposed value of π = 0 at the free surface is appropriate. However, when the rigid lid approximation for the free surface is applied, a Neumann BC should be provided (section 2.7.5). For all open vertical sections (2.177) yields for π: π = ̺νn · ∇un

(2.207)

From the equation above it can be immediately seen that for these outflow sections, where boundary conditions (2.190) apply (i.e., ∂un /∂n = 0): ∂π =0 ∂n

(2.208)

The zero Neumann boundary condition may also be appropriate for the inflow boundaries.

2.8 Algorithm for the non-hydrostatic equations

71

A Neumann boundary condition for the pressure Poisson equation (2.142) at the solid walls or at the bottom can be obtained from the equations (2.143) and the impermeability condition for the final velocity n · un+1 = 0: 1 1 ˜ ) = − n · ∇π (n · un+1 − n · u ∆t ̺0

(2.209)

∂π ̺0 ˜n = n · ∇π = u ∂n ∆t

(2.210)

obtaining:

Condition (2.210) yields another constraint on the hydrodynamic pressure field to be obtained from (2.142). Namely, it should provide not only the correction of the intermediate velocity field in the domain interior, but also guarantee the impermeability ˜ n have already been set to zero by condition fulfilment at the solid walls. If the values of u securing the impermeability boundary condition in the hydrostatic part of the algorithm (2.139-2.141), equation (2.210) yields ∂π/∂n = 0. Boundary condition (2.209) is valid ˜ differs for some reason from a also in all those cases, when the intermediate velocity u prescribed Dirichlet value at the boundary and must be corrected. The methods of assuring impermeability or fulfilment of a Dirichlet BC for the velocity by obtaining specific hydrodynamic pressure gradients require a very good approximation of the derivatives at a given boundary (e.g. regular elements and fine spacing). This can be avoided when the Dirichlet and impermeability conditions are already fulfilled by the intermediate velocity field. In this case, instead of (2.209) a simple zero Neumann boundary condition for the hydrodynamic pressure appears, which is naturally implemented in the method of finite elements. ∂π/∂n = 0 means only that no velocity correction is needed in the normal direction to the boundary. Nothing is given, for example, in the tangential direction and the zero derivative is only approximately fulfilled. In practice, the impermeability condition (2.184) can additionally be set. In the case of projection-2 method, instead of hydrodynamic pressure a correction to its value is found through the pressure Poisson equation. In this case in practice most simple boundary conditions are adequate [57]: a zero Neumann BC at the solid and open boundaries and zero Dirichlet BC at the free surface.

2.8 2.8.1

Algorithm for the non-hydrostatic equations Equation set to be solved

For the sake of clarity, a set of three-dimensional equations (2.22–2.27) is written here in the form applied in the non-hydrostatic algorithm, with pressure terms explicitly introduced into them: ∂u ∂S ∂ + u · ∇u = −g −g ∂t ∂x ∂x

"Z

S z

#

∆̺ 1 ∂π dz − − wfH + vfV + ∇ · (ν∇u) ̺0 ̺0 ∂x

72

2.8 Algorithm for the non-hydrostatic equations

"Z

#

S ∆̺ ∂v 1 ∂π ∂S ∂ dz − + u · ∇v = −g −g − ufV + ∇ · (ν∇v) ∂t ∂y ∂y z ̺0 ̺0 ∂y 1 ∂π ∂w + u · ∇w = − + ufH + ∇ · (ν∇w) ∂t ̺0 ∂z ∂u ∂v ∂w + + =0 ∂x ∂y ∂z ∂T + u · ∇T = ∇ · (ν T ∇T ) ∂t ̺ = ̺(s, T )

(2.211)

Additionally, an equation for the free surface position must be solved.

2.8.2

Introducing remarks

The non-hydrostatic version is a far-reaching further development of the existing code Telemac3D (see section 2.3). The new algorithm is coded in a parallel way to the previous one, so that direct comparison between the new non-hydrostatic and the old hydrostatic versions is possible by simply changing a steering variable. The new algorithm is outlined in figure 2.5. For the treatment of the governing equations (2.211) the decoupled method based on the operator splitting, and a projection method based on splitting the global pressure into hydrostatic and hydrodynamic part is chosen. The hydrodynamic pressure component is found by applying the pressure equation from the fractional step formulation. Height function methods for representing and tracking of the free surface are selected. Additionally, the appropriate boundary conditions are implemented. The developments are optimal for the existing solution algorithm structure of Telemac3D. Compared to the structure of the previously existing hydrostatic algorithm, briefly described in section 2.3, the two-dimensional pressure – free surface – continuity step was removed. In consequence, the terms representing the gradients of the free surface, the buoyancy and Coriolis forces are moved to the diffusion step where they appear as source terms of the diffusion equation. The modified diffusion and convection steps are performed not only for the horizontal components of the velocity, but obviously also for the vertical one. They yield the intermediate solution for the velocity field, which does not have to fulfil the incompressibility condition (fractional step intermediate solution). The removed pressure – free surface – continuity step is replaced by two new steps. First, the continuity step is implemented, where the pressure Poisson equation is solved and the projection of the velocity is performed. This step yields the solenoidal, three dimensional velocity field. Then, in the free surface step the position of the free surface is found. The most important difference is in the treatment of the vertical component of the velocity. In the previous algorithm the vertical component of the velocity is adjusted to the horizontal velocity ones by means of integrating the continuity equation (2.56) so that the incompressibility condition is satisfied. In this code, the vertical component

2.8 Algorithm for the non-hydrostatic equations

73

of the velocity is obtained from the vertical Navier-Stokes equation. Due to the hydrostatic approximation, the previous algorithm did not take into consideration vertical accelerations, whereas the new algorithm does. The characteristic feature of the non-hydrostatic algorithm is splitting the pressure into two components: (1) the hydrostatic pressure resulting from the density in situ and (2) the hydrodynamic pressure, as defined and described in section (2.5.5.2). For flows, where the dynamic pressure gradients are large, the projection-2 method can be considered, where in the continuity step a pressure correction to the global pressure is found and the hydrodynamic pressure gradients are already taken into consideration in diffusion step. Another typical feature is the free surface algorithm, where three numerical methods of finding the free surface are provided for the user to choose from. The structure of the algorithm and its software realisation allows its easy modifications in future. In this section the outline of the implemented algorithm is provided, whereby the numerical details are presented in chapter 3. For an overview, Table 2.1 is provided. The developments described in this work do not concern the transport and turbulence modelling as well as the equation solvers – they are listed in Table 2.1 for completeness.

2.8.3

Operator splitting

The time derivatives in the Navier-Stokes equations as well as transport equations for the tracers are treated explicitly in the framework of the operator splitting (2.3.1). Therefore, the fractional step method of solving three-dimensional hydrodynamic equations described in subsection 2.5.4 can be applied. The equations are split into fractional steps and treated separately by appropriate numerical methods. The explicitly formulated time derivative of the variable f is: f n+1 − f d f d − f a f a − f n ∂f = + + ∂t ∆t ∆t ∆t

(2.212)

In the equation above, f stands for the components of the velocity u, v, w and superscripts denote the intermediate levels of advection a and diffusion d steps, as well as the time step levels n and n + 1. The final value f n+1 is obtained in the continuity step. For the tracers (f represents then temperature, salinity, pollutants, etc.) the continuity step does not apply and: ∂f f n+1 − f a f a − f n = + (2.213) ∂t ∆t ∆t

2.8.4

Advection step

As previously, the advection (also hyperbolic) step is: fa − fn + u · ∇f = 0 ∆t

(2.214)

74

2.8 Algorithm for the non-hydrostatic equations

Table 2.1: Basic characteristics of the solution algorithm Computational aspect (algorithm step) Mesh Advection step Diffusion step

Pressure treatment

Continuity Velocity projection Free surface representation Free surface updating

Free surface BC

Bottom BC

Lateral wall BC Lateral open BC

Turbulence modelling

Transport

Equation solvers

Selected approach

Remarks

σ-mesh Characteristics or SUPG-FEM Standard Galerkin FEM

only vertically variable entirely 3D entirely 3D hydrostatic and Coriolis forcing as source terms

Split: – hydrostatic: – hydrodynamic: projection method explicitly in FEM height function (a) from kinematic BC – with characteristics – with SUPG (b) from conservative FS eq. (c) from St.-Venant eqs. normal: – viscous/inviscid tangential: – viscous tangential: – viscous – inviscid impermeability (a) free outflow (b) inflow (c) non-reflecting BC (d) absorbing BC – constant eddy-viscosity – mixing length – k−ε – active tracers – passive tracers – sediment – particle tracking iterative, preconditioned conjugate gradient methods

obtained explicitly obtained from PPE (projection-1) from hydrodynamic pressure only single-valued no breaking waves unlimited stability limited stability conservative shallow water only inviscid: p = pout wind stress (empiric) bottom stress (Chezy) impermeability, adherence Neumann imposed/Neumann for simple cases only wave problems damping functions by stratification influence density deposition/erosion Element-by-element technique

2.8 Algorithm for the non-hydrostatic equations

75

where f denotes velocity components as well as tracers. The non-linear terms of the equation set (2.211) are treated using the method of characteristics, or SUPG method. The only difference to the hydrostatic algorithm is that the vertical velocity component is treated as well. When SUPG is applied, the advection step is not treated separately, but together with the diffusion step.

2.8.5

Diffusion step

In the diffusion (also parabolic) step the diffusive terms (2.211) are treated using the finite element method: ud − ua = ∇ · (ν f ∇u) + (FCor + F∇S + F∆̺ ) + qu ∆t

(2.215)

Fi are different forcing terms in equations (2.211): Fcor Coriolis terms, F∇S free surface gradients, and F∆̺ density gradients. FCor

−fH w + fV v  −fV u  =  fH u

F∇S

F∆̺





−g ∂S ∂x   =  −g ∂S ∂y  0 



S ∂ −g ∂x z  ∂ RS =  −g ∂y z 0



R

∆̺ dz ̺0 ∆̺ dz ̺0

(2.216)

(2.217)    

(2.218)

qu represent other forcing accelerations, e.g. non-inertial ones. For tracers T , the diffusion step takes the following form: Td − Ta = ∇ · (ν T ∇T ) + qT ∆t

(2.219)

where qT describes the source terms needed to simulate discharge of effluents or heat sources, for example. ˜ as As the result of the diffusion and advection steps the intermediate velocity field u described in subsection 2.5.4 is found. In this stage, the found horizontal velocity components u˜ = ud and v˜ = v d are comparable with the results for un+1 and v n+1 which would result from solving three-dimensional shallow water equations (2.37–2.41) using the previous hydrostatic Telemac3D algorithm as described in section 2.3. The difference is that also the vertical component w˜ = w d is simultaneously computed from the third, vertical momentum conservation equation. The continuity constraint is not fulfilled by this field.

76

2.8.6

2.8 Algorithm for the non-hydrostatic equations

Continuity step

In the first part of the continuity step the 3D elliptic Poisson equation for the hydrodynamic pressure π (2.142) is solved using the finite element method (analogically to the diffusion step): ̺0 ∇2 π = ∇ · ud (2.220) ∆t and the resulting solenoidal velocity is found explicitly from the gradients of the hydrodynamic pressure, i.e. the projection of the intermediate velocity is performed (equations 2.143): ∆t un+1 = ud − ∇π (2.221) ̺0 After this stage the resulting three-dimensional velocity field (u, v, w) satisfies the incompressibility condition. When the projection-2 method is chosen, instead of hydrodynamic pressure π a more abstract variable, namely the pressure correction to the pressure applied in the advectiondiffusion step is found (section 2.8.9).

2.8.7

Free surface step

Using the resulting, final velocity, the position of the free surface in the next step can be found. There exist three versions of this stage. The first two versions use the kinematic boundary condition (2.158): ∂S ∂S S n+1 − S n = −us − vs + ws (2.222) ∆t ∂x ∂y and the third one the conservative formulation of the free surface equation (2.159): S n+1 − S n ∂ =− ∆t ∂x

Z

Sn

−B

un+1 dz −

∂ ∂y

Z

Sn −B

v n+1 dz

(2.223)

Both equations are partial differential equations of a hyperbolic type. Equation (2.222) solved by the semi-implicit Streamline Upwind Petrov-Galerkin method (SUPG), or using the method of characteristics. Equation (2.223) is solved explicitly in finite elements or semi-implicitly using SUPG, while the velocity is integrated using the free surface elevation at the time level n. For these flows, where only baroclinic motions are of importance, the position of the free surface can be frozen. The appropriate boundary conditions for velocity (rigid lid) have to be provided.

2.8 Algorithm for the non-hydrostatic equations

77

Non-hydrostatic algorithm outline Operator-splitting ∂u un+1 − ud ud − ua ua − un = + + ∂t ∆t ∆t ∆t Source terms (2D/3D, in FE): Fu = −g∇H S n − g∇H

  Sn Z n ∆̺  dz  − 2Ω × un + qu

̺0

z

Advection step (3D, hyperbolic): ua − un + u · ∇u = 0 ∆t Diffusion step (3D, parabolic): ud − ua = ∇ · (ν∇u) + Fu ∆t Poisson equation (3D, elliptic): ∇2 π =

̺0 ∇ · ud ∆t

Velocity projection (3D, in FE): un+1 = ud −

∆t ∇π ̺0

Free surface step (2D, hyperbolic) in two versions: ∂S ∂S S n+1 − S n = −us − vs + ws ∆t ∂x ∂y S n+1 − S n ∂ =− ∆t ∂x

ZS

−B

∂ udz − ∂y

ZS

vdz

−B

Two-step tracer advection-diffusion (hyperbolic-parabolic): Tn − Td Td − Ta + = ∇ · (ν T ∇T ) + qT − u · ∇T ∆t ∆t

Figure 2.5: Non-hydrostatic algorithm outline.

78

2.8.8

2.8 Algorithm for the non-hydrostatic equations

Free surface from vertically integrated momentum equations

In many geophysical flows the position of the free surface can be treated as a result of purely barotropic motions, e.g. can be found using the vertically integrated St.-Venant equations in the form analogical to (2.54). It means that the free surface can be found using the shallow water approximation, but the internal motion modes are to be resolved as well. It must be mentioned, that the restriction on the shortness of the free surface waves that can be modelled remains. In this case the advantages of the robust Telemac2D algorithm can be taken in finding the free surface. In this version the advection and diffusion stages are virtually the same as described in previous sections. In the diffusion step, however, the terms representing the free surface gradients are set to zero and treated in the free surface step. Consequently, the baroclinic (density gradients) part of the hydrostatic pressure is treated in the diffusion step and the barotropic (free surface gradients) in the vertically integrated equations. After the diffusion step the free surface step follows, in which, using the Telemac2D, ˜ and the following equations are solved in order to find the intermediate velocity field u the new free surface elevation: ∂h ∂hu ∂hv + + =0 ∂t ∂x ∂y usurf − ud ∂h ∂B +g = −g ∆t ∂x ∂x vsurf − v d ∂h ∂B +g = −g ∆t ∂y ∂y

(2.224) (2.225) (2.226)

The solution procedure is described in section 2.3.3. After solving this set of equations, the three dimensional velocity field is obtained by applying equations (2.55). After this step the continuity step follows, where the resulting, solenoidal velocity field is obtained.

2.8.9

Projection-2

For these flows, where the gradients of the hydrodynamic pressure are large, i.e. comparable or even greater than the gradients of the hydrostatic pressure, a version of the non-hydrostatic algorithm based on the projection-2 method can be thought of (section 2.5.4). In this method in the diffusion step not only gradients of hydrostatic pressure, but also the known hydrodynamic pressure gradient from the previous time step appears, eq. (2.215). In this way too large differences between the divergent intermediate and final divergence-free velocity are avoided. This method requires not only that the initial velocity field must be divergence-free, but also that the hydrodynamic pressure field must be properly initialised as well. Usually, a zero-field is appropriate, or a known analytical solution. However, when this is not the

2.8 Algorithm for the non-hydrostatic equations

79

case, an initialisation step for velocity and hydrodynamic pressure must be performed, as described in section 2.7.1 and gradients of the initial hydrodynamic pressure computed. After initialisation, the hydrodynamic pressure gradients are treated in the diffusion step. In the continuity step a pressure correction to the hydrodynamic pressure φ = π n+1 − π n is found and the projection step for the velocity is executed. The hydrodynamic pressure for the next step is simply π n+1 = π n + φ. However, π n+1 is not required in the next time step – only its already computed gradients are needed and consequently updated.

80

2.8 Algorithm for the non-hydrostatic equations

Chapter 3 Numeric algorithm description The difference between theory and practice is smaller in theory than in practice. Anonymous This chapter provides a description of the numeric realisation of the non-hydrostatic algorithm presented in chapter 2. Due to operator splitting, the algorithm has a modular structure with distinctive parts with clear mathematical properties: advection, diffusion, continuity, free surface. Verification and performance tests of these steps treated as separate units are given in chapter 4. Verification of the complete algorithm is presented in chapter 5.

3.1

Computational domain

In this section the assumptions taken in order to deal with the time-variable extents of the computational domain are discussed.

3.1.1

σ-mesh and σ-transformation

The three-dimensional mesh structure known as σ-mesh, was presented first by Phillips [125]. The computational domain Ω is bounded by the bottom z = B(x, y), time-variable free surface z = S(x, y, t) (both single-valued functions) and vertical lateral boundaries (figure 2.1). The bottom surface is discretised by a mesh of triangles, forming the twodimensional base mesh. The three-dimensional mesh is obtained by duplicating it in the vertical direction in such a way that the z co-ordinates of the nodes are defined by: z = B(x, y) + z∗ (S(x, y, t) − B(x, y)),

z∗ ∈ [0, 1]

(3.1)

At various mesh levels defined by a discrete set of z∗ values, the x, y node co-ordinates remain the same as at the bottom base mesh. In consequence, the domain is divided into 81

82

3.1 Computational domain

z

6 ``` ``` ```

XX X S    S S XX X QQ   PP  

``` ``` ```  ```

  

PP PP PP PP

z∗ 6 1

   

σ-

  

Ω

0 ∗

Ω

Figure 3.1: σ-transformation. Side view of the real and transformed mesh. a number of tiers formed by prismatic elements with 6 nodes, whose lateral, quadrilateral faces are oriented vertically. Triangular bases of the prisms follow the mesh surfaces. The vertical projection of this mesh on a horizontal surface builds a two-dimensional mesh of triangles, called further the base mesh Ω0 . The vertical co-ordinates of the real mesh nodes change with the time-variable position of the free surface, but the horizontal coordinates remain constant. The z∗ values for each mesh level are also constant in time, which allows a transformation of the time-variable real mesh into a σ-mesh, fixed in time (figure 3.1):     x∗ = x y∗ = y (3.2)  z−B(x,y)   z∗ = S(x,y,t)−B(x,y) As it will be shown later, in computation of various matrices resulting from FEM formulation advantages are taken of the stationary σ-mesh geometry.

3.1.2

Comments on the σ-transformation

The σ-transformation defined by (x, y, z) → (x, y, z∗) transforms the time-dependent real mesh into a stationary σ-mesh, where all mesh levels are parallel and horizontal, what is a very important advantage. As it can be easily proved, the Jacobian of this transformation is equal to the water depth: |J| = det|∂xj /∂x∗i | = h(x, y, t) = S(x, y, t) − B(x, y). The surfaces defining the free surface and the bottom, where important boundary conditions are applied, are discretised by triangles of the base mesh, following the bottom topography or time-variable free surface configuration. In σ-transformation these boundaries are horizontal. The lateral boundaries consist of quadrilaterals oriented vertically. In σ-transformation all these elements are rectangular. The main disadvantage of such a mesh structure is the fact that with a constant number of horizontal mesh levels deeper parts of the domain can be under-discretised and shallower ones over-discretised. In most practical cases some compromise must be met. The lateral boundaries have to be represented by vertical walls. For modelling of flooding and drying areas as e.g. tidal flats special strategies are needed (masking of elements, [74]).

3.1 Computational domain

83

Another well-known disadvantage of the σ-transformation is the condition of hydrostatic consistency, especially important for the finite difference models [110]. For very steep bottom gradients (or free surface ones) a situation may occur where one or more of the nodes at the bottom of a real mesh element are situated above some nodes at the element top. For the reasons mentioned in section 3.4.4.2 this situation is not so dangerous as in FDM, but ambiguities by computation of horizontal gradients in elements may occur, leading to errors. This situation can be avoided already during mesh generation by finer horizontal discretisation of greater bottom slopes or by prescribing lower discretisation in the vertical. Steep free surface slopes cause similar problems, which cannot be avoided without mesh adaptivity. An advantage of the σ-transformation is the fact that the form of time derivative of a variable f (x, y, z, t) does not change during transformation to f (x, y, z∗, t), and for the advection equation: df (x, y, z, t) = dt

∂f ∂t

!

∂f +u ∂x x,y,z∗

!

∂f +v ∂y y,z∗ ,t

!

+ w∗

x,z∗ ,t

∂f ∂z∗

!

=0

(3.3)

x,y,t

where w∗ is the transformed vertical velocity component w in σ-co-ordinates (x, y, z∗): dz∗ w∗ = = dt

∂z∗ ∂t

!

∂z∗ + u ∂x x,y,z

!

∂z∗ + v ∂y y,z,t

!

∂z∗ + w ∂z x,z,t

!

(3.4) x,y,t

The horizontal components do not change. When a σ-transformed advection velocity u∗ = (u, v, w∗) is defined, the σ-transformation does not change the form of the advection equation. Applying σ-transformation to the diffusion equation leads to lengthy and complicated expressions, which are useless in practice [110]. While in the advection step realised with the method of characteristics in the σ-mesh full advantage can be taken of its simple geometry (section 3.5.2), in the diffusion step (and/or advection with the SUPGmethod) another approach is chosen (sections 3.4, 3.5.3). By computation of the matrices resulting from FEM-formulation, the appearing interpolation functions in the real mesh elements are transformed into σ-mesh in order to simplify computations of various matrix elements. A practical simplification is that in the σ-mesh both the impermeability condition at the bottom (2.185) and kinematic boundary condition at the surface (2.158) are equivalent to setting the transformed vertical velocity component w∗ to null at the bottom and surface.

3.1.3

Reference element and interpolation functions

The elements of the real mesh Ω are prisms with 3 quadrilateral faces oriented vertically and two triangular faces at the top and at the bottom, which are not horizontal in general

84

3.1 Computational domain

6 @ @ @ @

5

4

1

3 \   \  ``` ``` \ `` \ 2

Figure 3.2: The prismatic element of the real mesh. Local numbering of nodes is indicated. (figure 3.2). The model variables are computed at the nodes i = 1, ..., N of the mesh Ω being the vortices of the prisms. Each function f is approximated by: f=

N X

fi ϕi

(3.5)

i

where fi are function values at the nodes and ϕi the Lagrangian basis interpolation functions corresponding to a node i, which equal 1 at the node i and 0 at all other nodes. In order to simplify the computations of different matrices resulting from FEM formulations, they are performed in reference elements of a simple form and with reference base functions. Each real element is transformed into a reference one, and the elementary contributions are computed in each element in a decoupled way. In Telemac3D, the Element-by-Element technique is applied and a classical global matrix assembly is not performed. The diagonal and extra-diagonal elements of the element matrices are separately stored and vectorised procedures for performing algebraic operations are provided. The reference element in Ωref is shown in figure 3.3. The basis functions ψi corresponding to the 6 vortices (nodes) i of the reference elements are listed as follows: ψ1 = (1 − α − β)(1 − γ)/2 ψ4 = (1 − α − β)(1 + γ)/2 ψ2 = α(1 − γ)/2 ψ5 = α(1 + γ)/2 ψ3 = β(1 − γ)/2 ψ6 = β(1 + γ)/2

(3.6)

The basis functions ϕi in the real mesh Ω are obtained by an iso-parametric transformation of ψi . The vortices of the prismatic elements in the real mesh Ω have co-ordinates (xi , yi, zi ). The transformation F of the local co-ordinates of a given point P (α, β, γ) within a given element into its real co-ordinates P (x, y, z) is given by: x=

6 X i=1

xi ψi (α, β, γ)

y=

6 X i=1

yiψi (α, β, γ)

z=

6 X i=1

zi ψi (α, β, γ)

(3.7)

3.1 Computational domain

85 γ

6

4

! !! ! ! !! ! !! !!

1

5

0

6

1

-

β

1

α



!

1

! !! ! ! !! ! ! !!

–1

3

2 Figure 3.3: The prismatic reference element. Local numbering of nodes and the local axes are shown. The formula for (x, y, z) can be simplified because the lateral faces of prisms are vertical, and x1 = x4 , y1 = y4 ; x2 = x5 , y2 = y5 ; x3 = x6 , y3 = y6 . In consequence, one obtains: x = (1 − α − β)x1 + αx2 + βx3 (3.8) y = (1 − α − β)y1 + αy2 + βy3 1 1 [(1 − α − β)z1 + αz2 + βz3 ](1 − γ) + [(1 − α − β)z4 + αz5 + βz6 ](1 + γ) z = 2 2 The Jacobian |J| of this transformation is:

1 |J| = [(x2 − x1 )(y3 − y1 ) + (x1 − x3 )(y2 − y1 )] 2 [(1 − α − β)z4 + αz5 + βz6 − (1 − α − β)z1 − αz2 − βz3 ]

(3.9)

The functions ϕi are given by: ϕi (x, y, z) = ψi (F −1 (x, y, z))

(3.10)

For computation of the diffusion matrix (3.28) in section 3.3.1 using transformation from the real to the reference element a difficulty arises. The transformation of the expressions typical for the diffusion matrix into the reference element yields: Z

Ω

∂ϕi ∂ϕj dxdydz = (3.11) ∂x ∂x ! ! Z ref ∂ϕj ∂α ∂ϕj ∂β ∂ϕj ∂γ ∂ϕi ∂α ∂ϕi ∂β ∂ϕi ∂γ |J| dαdβdγ + + + + ∂α ∂x ∂β ∂x ∂γ ∂x ∂α ∂x ∂β ∂x ∂γ ∂x Ω

86

3.1 Computational domain

η

η 1

6

6

@ @ 3@

1 0

1

4 @ @ @ @ @ @ @ 2@ @

1

3 -

–1

-

ξ

1 1

ξ

2 –1

Figure 3.4: The two-dimensional reference elements. where each expression in form ∂α/∂x brings a factor 1/|J| (obtained from Jacobi matrix inversion). As a result a polynomial expression divided by |J| is obtained for integration. With the Jacobian given by (3.9) the denominator of the expression to be integrated in (3.11) is not constant. The analytical integration of such a rational fraction leads to very complex logarithmic expressions, which are useless for practical purposes. Therefore, advantage is taken of the simpler geometry of the σ-mesh Ω∗ (details in section 3.4). As the result of the σ-transformation, the triangular faces of the real elements at the bottom and top become horizontal, and z1∗ = z2∗ = z3∗ , z4∗ = z5∗ = z6∗ . The Jacobian of the transformation F ∗ of the co-ordinates of a point P (α, β, γ) to P (x, y, z∗) appears to be constant: 1 |J∗ | = [(x2 − x1 )(y3 − y1 ) + (x1 − x3 )(y2 − y1 )] (3.12) 2 and the expressions like (3.11) can be integrated analytically without difficulty by applying mathematical expert systems [72, 91]. The main advantages of this arduous procedure (the formulas obtained tend to be very lengthy) are: (1) the obtained formulas are exact compared to numeric integration, (2) they are vectorised in a straightforward way.

3.1.4

Two-dimensional elements

In some parts of the algorithm a two-dimensional approach is needed, as e.g. in the free surface step or by computation of the boundary terms in the diffusion step (section 3.4). In the case of the free surface and bottom, triangular elements with linear interpolation are applied; for lateral boundaries quadrilaterals. The reference elements are: a triangle with vortices in (0, 0), (0, 1), (1, 0) and a square comprised of co-ordinate points (−1, −1), (1, −1), (1, 1), (−1, 1), figure 3.4. The linear interpolation functions are listed as follows:

3.2 Operator splitting and time discretisation

triangle ψ1 = (1 − ξ − η) ψ2 = ξ ψ3 = η

87

square ψ1 = (1 − ξ − η + ξη)/4 ψ2 = (1 + ξ − η − ξη)/4 ψ3 = (1 + ξ + η + ξη)/4 ψ4 = (1 − ξ + η − ξη)/4

(3.13)

For further details the reference [73] is recommended.

3.2

Operator splitting and time discretisation

The numerical algorithm presented here is based on the operator splitting technique. The main idea is based on splitting the operators in the governing equations into parts of clearly distinguishable mathematical properties. The resulting equation parts are treated separately and subsequently in stages using the most appropriate and advantageous methods according to the types of the obtained split operators. Let us consider a partial differential equation for a variable f in the form analogical to (2.22-2.26): ∂f + Af = 0 (3.14) ∂t where A is a differential operator, which could be split linearly into a few parts, e.g. A = A1 +A2 . In the particular case of equations (2.22-2.26), A1 represents the advection and A2 the diffusion operator. In contrast to the space discretisation, the time discretisation in Telemac3D follows the finite difference method. The time derivatives of model variables f are approximated by one-step forward differences. These differences are split into a sum of contributions due to the subsequent application of the operators A1 and A2 : ∂f (x, y, z, t) f n+1 − f n f n+1 − f˜ f˜ − f n = = + ∂t ∆t ∆t ∆t

(3.15)

where f n+1 , f n represent model variable values at times tn+1 and tn , the time-step ∆t = tn+1 − tn and f˜ is an intermediate (fractional step) solution due to the operator splitting. In the following, the equation (3.14) is solved in two steps: f˜ − f n + A1 f = 0 ∆t f n+1 − f˜ + A2 f = 0 ∆t

(3.16) (3.17)

The time variability of the variable f is assumed to be linear and in the following the semi-implicit formulation (Crank-Nicolson, [46]) is applied: f (x, y, z, t) = θf n+1 + (1 − θ)f n

(3.18)

88

3.3 Continuity step

which allows weighing of the approximate value of f between the time levels tn+1 and tn with the Crank-Nicolson implicitness factor θ. By θ = 1 one obtains an implicit, by θ = 0 an explicit formulation of the time derivative (3.15). One may choose between the explicit and implicit formulations by changing the value of θ. The explicit formulation is usually computationally advantageous, but with a time-step value limited by stability criterion. The implicit formulations are computationally expensive and characterised by a relatively greater numerical diffusion, but stable where the explicit formulation does not work [8]. With the value of θ = 0.5 this formulation has an error O(∆t2 ), i.e. consistency order two, what is an important advantage. For practical reasons the value of θ is often taken slightly higher than the optimal value of 0.5, between 0.55 and 0.65, in order to take advantage of the stabilising influence of the implicitness. The stability of the operator splitting method is achieved, when all the fractional steps of the resulting algorithm are stable [110]. For entirely explicit formulation the consistency order is one, so that the accuracy depends linearly on the time step. The main advantage of the operator splitting technique is that various methods of integration can be applied in different algorithm parts according to the mathematical nature of the equations.

3.3 3.3.1

Continuity step Poisson equation solver

The kernel of the non-hydrostatic algorithm consist of the Poisson equation for the hydrodynamic pressure. The solution of this step of the algorithm gives the hydrodynamic pressure field, allowing an immediate computation of the final, solenoidal velocity field. The treatment of this elliptic equation follows clearly all the characteristic steps of the finite element method (section 2.4.2) and is the best introduction to the particular techniques implemented in Telemac-3D. Therefore, this step is to be described first, ignoring the sequence of the algorithm stages (advection, diffusion, continuity, free surface).

3.3.2

Formulation of the pressure Poisson equation in the finite element method

In order to simplify the notation, the Poisson equation (2.142) is written for a variable p(x, y, z) and the right hand side is taken to be a known scalar field R(x, y, z). In the implemented non-hydrostatic algorithm, p represents the hydrodynamic pressure π, and ˜ , section 2.5: R(x, y, z) = ̺0 ∆t−1 ∇ · u ∇2 p(x, y, z) = R(x, y, z)

(3.19)

The functions p and R are interpolated using Lagrangian interpolation functions ϕi for a node i: X pi ϕ i (3.20) p= i

3.3 Continuity step

89

The finite element procedure follows in this clearly elliptic equation case the (standard) Galerkin method: the equation (3.19) multiplied by the test functions (equal here to the interpolation functions mentioned above) and then integrated in the domain Ω with the boundary Γ(Ω): ∂2p ϕi 2 dΩ + ∂x Ω

Z

∂2p ϕi 2 dΩ + ∂y Ω

Z

Z

∂2p ϕi 2 dΩ = ∂z Ω

Z

Ω

ϕi R dΩ

(3.21)

Integrating the terms of the second order by parts, i.e. using Green’s formula1 one obtains for the first term on the left side of (3.21): Z

Ω

ϕi

∂2p dΩ = ∂x2

Z

Γ

ϕi

∂p nx dΓ − ∂x

Z

Ω

∂ϕi ∂p dΩ ∂x ∂x

(3.22)

and analogical formulations for derivatives according to y and z. (nx , ny , nz ) is the normal vector to the boundary. Introducing terms in the form (3.22) to (3.21) yields: !

∂ϕi ∂p ∂ϕi ∂p ∂ϕi ∂p − dΩ + + ∂x ∂x ∂y ∂y ∂z ∂z Ω ! Z Z ∂p ∂p ∂p + ϕi nx + ny + nz dΓ = ϕi R dΩ ∂x ∂y ∂z Γ Ω Z

(3.23)

As a consequence of the integration, the application of linear interpolation functions ϕi in a second-order problem (diffusion) is possible. The boundary integrals allow imposition of the boundary conditions (Dirichlet and Neumann). Setting the interpolation formula (3.20) to (3.23) yields: X j

pj

Z

−

Ω

X

∂ϕi ∂ϕj ∂ϕi ∂ϕj ∂ϕi ∂ϕj + + ∂x ∂x ∂y ∂y ∂z ∂z pk

k

Z

Γ

!

dΩ

(3.24)

∂ϕk ∂ϕk ∂ϕk nx + ϕi ny + ϕi nz ϕi ∂x ∂y ∂z

!

dΓ = −

X j

Rj

Z

Ω

ϕi ϕj dΩ

where j are internal and k boundary nodes of the computational domain. In Telemac3D the flux through the boundary represented by the boundary integral is approximated in a linear way. At all boundaries: ∂pk ∂pk ∂pk nx + ny + nz = ak pk + bk ∂x ∂y ∂z

(3.25)

whereby the coefficients a and b are defined per boundary node k. In the eventual code realisation they are separated into surface, bottom and lateral boundary nodes. It 1

Green’s formula

Z

Ω

αvi,i dΩ =

Z

Γ

αvi ni dΓ −

Z

α,i vi dΩ Ω

90

3.3 Continuity step

is a convenient way to impose Neumann boundary conditions, although the boundary geometry is treated in a simplified way. The normal vector (nx , ny , nz ) must also be defined per node, which may be a source of errors for complicated boundary shapes. Taking (3.25) into consideration yields: X j

pj

Z

Ω

∂ϕi ∂ϕj ∂ϕi ∂ϕj ∂ϕi ∂ϕj + + ∂x ∂x ∂y ∂y ∂z ∂z −

X j

!

Rj

dΩ −

Z

Ω

X

pk

k

ϕi ϕj dΩ +

Z

Γ

ak ϕi ϕk dΓ =

X

pk

k

Z

Γ

(3.26)

bk ϕk dΓ

For simplification of this equation system, the set of coefficients of the type gij =

Z

Ω

ϕi ϕj dΩ

(3.27)

is interpreted as a mass matrix G and the set of coefficients cij =

Z

Ω

∂ϕi ∂ϕj ∂ϕi ∂ϕj ∂ϕi ∂ϕj + + ∂x ∂x ∂y ∂y ∂z ∂z

!

dΩ

(3.28)

is interpreted as a diffusion matrix C. Following this notation, the linear equation system to be solved for the unknown vector p = [pj ] can be written as: Ap = B

(3.29)

where A is the C matrix corrected by a known boundary term due to coefficients ak , and B is equal to −GR, where R = [Rj ], corrected by a boundary term due to the coefficients bk in (3.26). The imposed (Dirichlet) boundary conditions are taken into consideration through modification of the global equation system matrix A and the right hand side vector B, excluding the nodes with imposed, known values from consideration. It must be mentioned that in the eventual implementation in the code the equation (3.29) is solved in the σ-mesh, which does not change in time, and taking all advantages from the geometrical simplifications taking place there (a constant transformation Jacobian, section 3.1.1). The transformation of the equations and resulting terms and matrices are thoroughly analogous to the descriptions depicting the diffusion step provided in detail in section (3.4) and omitted here for the sake of clarity. Integration of all matrices is carried out analytically, using mathematical expert software. The resulting linear equation system (3.29) is solved using an effective linear equation solver (a preconditioned conjugate gradient solver). The verification of the Poisson equation solver using analytical solutions is given in section 4.2.

3.3.3

Resulting divergence-free velocity field

Having computed the dynamic pressure, the final, divergence-free (solenoidal) velocity field un+1 is computed in a straightforward way, using equations (2.221). In this way the j

3.4 Diffusion step

91

projection step of the non-hydrostatic algorithm is realised: un+1 = udj − j

∆t ∂π ̺0 ∂xj

(3.30)

The 3D spatial derivatives of the dynamic pressure are computed for each node using the weak formulation in finite elements. The derivative of a given scalar function f (x, y, z) is computed locally in the direct surrounding Ω0 of the node i in (patch) the following way: ! R ∂ϕj R ∂f PN ∂f j fj Ω0 ∂x ϕi dΩ0 Ω0 ∂x ϕi dΩ0 = PN R ∂ϕj = R (3.31) ∂x i Ω0 ϕi dΩ0 j xj Ω0 ∂x ϕi dΩ0 In practice, the computation of (3.31) is performed in a loop over all elements by computing contributions originating from each element to a node i. Finally, these contributions are assembled by summing up contributions from all elements in a vectorised loop. This method of derivative computation has the only disadvantage that the exactness of the computation can be different at the boundaries than in the domain interior. Finally, the impermeability boundary conditions (section 2.7.3) are imposed on the resulting velocity field. In practice, it is necessary even when the dynamic pressure field has been obtained using boundary conditions ensuring impermeability.

3.3.4

Source term of the pressure Poisson equation

As the source term denoted as R(x, y, z) in section 3.3.1 the divergence of the interme˜ . The diate velocity must be taken, multiplied by a factor: R(x, y, z) = ̺0 ∆t−1 ∇ · u divergence, being a sum of partial derivatives of velocity components, is computed using the weak formulation in finite elements (3.31).

3.4 3.4.1

Diffusion step Diffusion step: formulation in finite elements

In a sharp contrast to the σ-transformation of the advection equation, the diffusion equation cannot be transformed into an equivalent or simple form in σ-co-ordinates due to the nonconformity of this transformation [110]. Because the Jacobian of the transformation from the real to the reference element is not constant, the analytical integration of the diffusion matrix formulated in the real mesh is not possible. If simplifying approximations are made, they lead to parasitic numerical diffusion [112]. In order to avoid both these ambiguities, the diffusion step is formulated directly in the σ-mesh. The treatment of the diffusion step follows patterns analogous to the Poisson equation treatment (section 3.3.1). In contrary to the description of the Poisson equation solver, the derivations in this section take the σ-transformation into consideration immediately.

92

3.4 Diffusion step

A verification of this stage of the algorithm using analytical solutions for Ekman profiles is given in section 4.1. The equations to be solved for a variable f (x, y, z, t) (one of the velocity components u, v, w or a tracer concentration c) in the diffusion (parabolic) step are: fd − fa = ∇ · (ν f ∇f ) + qf ∆t

(3.32)

where f a are results of the advection step, and f d the searched solution of the diffusion step. The approximation of f on the right hand side of this equation follows the semiimplicit (Crank-Nicolson) formulation: f = θf d + (1 − θ)f n

(3.33)

In order to avoid the difficulties by transforming the diffusion equation (3.32) directly, the σ-transformation is applied after formulating the variational form of this equation in the real mesh. The finite element procedure follows the standard Galerkin method, applicable for parabolic equations. The equation (3.32) is multiplied by the test functions ϕ equal to the interpolation ones and then integrated in the real domain Ω with the boundary Γ(Ω): 1 ∆t

Z

Ω

ϕi f d dΩ −

1 ∆t

Z

Ω

ϕi f a dΩ =

Z

Ω

ϕi ∇ · (ν f ∇f ) dΩ +

Z

Ω

qf dΩ

(3.34)

It is advantageous to treat the diffusion step in the transformed, σ-mesh Ω∗ (section 3.1.1). Equation (3.34) transforms to: Z Z 1 Z 1 Z ∗ d ∗ ∗ a ∗ ∗ ϕi f h dΩ − ϕi f h dΩ = ϕi ∇ · (ν f ∇f )h dΩ + qf h dΩ∗ (3.35) ∗ ∗ ∗ ∗ ∆t Ω ∆t Ω Ω Ω

where h(x, y, t) is the Jacobian of the σ-transformation (section 3.1.1), equal to the water depth. The variable f is interpolated using Lagrangian interpolation functions ϕ∗i for a node i, in the transformed mesh, i.e.: X fj ϕ∗j (3.36) f= j

Introducing (3.36) into (3.35) yields: 1 X d f ∆t j j

Z

X

ϕ∗i ∇ · (ν f ∇ϕ∗j )h dΩ∗ +

j

fj

Z

Ω∗

Ω∗

ϕ∗i ϕ∗j h dΩ∗ −

1 X a f ∆t j j X j

Z

ϕ∗i ϕ∗j h dΩ∗ =

qf j

Z

Ω∗

Ω∗

ϕ∗i ϕ∗j h dΩ∗

(3.37)

3.4 Diffusion step

93

After taking (3.33) into consideration and some algebra we obtain the following set of equations: X

fjd

j



1 ∆t

X

Z

Ω∗

ϕ∗i ϕ∗j h

∆tqf j + fja

j



∗

dΩ + θ 1 ∆t

Z

Ω∗

Z

Ω∗

ϕ∗i ∇

·

(ν f ∇ϕ∗j )h

ϕ∗i ϕ∗j h dΩ∗ + (θ − 1)

∗

dΩ

X



fjn

j

=

Z

Ω∗

(3.38) ϕ∗i ∇ · (ν f ∇ϕ∗j )h dΩ∗

In the equation (3.37) the following matrices can be easily distinguished: the mass matrix M (divided by ∆t): 1 Z mij = ϕ∗ ϕ∗ h dΩ∗ (3.39) ∆t Ω∗ i j and the generic diffusion matrix E: Z

eij =

Ω∗

Z

Ω∗

ϕ∗i ∇ · (ν f ∇ϕ∗j )h dΩ∗ =

ϕ∗i

"

! ∗

∂ϕj ∂ νx ∂x ∂x

+

(3.40) ! ∗

∂ϕj ∂ νy ∂y ∂y

+

!# ∗

∂ϕj ∂ νz ∂z ∂z

h dΩ∗

The computation of the mass matrix M is straightforward, but this is not the case for the diffusion matrix E. For the sake of clarity, only the diffusion along the x-co-ordinate will be considered (y analogically): exij

=

Z

Ω∗

ϕ∗i

∂ϕ∗ ∂ νx j h dΩ∗ ∂x ∂x !

(3.41)

Applying the σ-transformation, the derivative according to the x-co-ordinate transforms to: ∂f ∂f ∂z∗ ∂f = + (3.42) ∂x ∂x ∂x ∂z∗ The diffusion term along the x-axis transforms to the following expression: exij

∂ϕ∗j ∂z∗ ∂ϕ∗j ∂ νx + + = ∂x ∂x ∂x ∂z∗ Ω∗ " !#) ∗ ∗ ∂ϕ ∂ϕ ∂z ∂ ∂z ∗ ∗ j j hϕ∗i νx dΩ∗ + ∂x ∂z∗ ∂x ∂x ∂z∗ Z

(

hϕ∗i

"

!#

(3.43)

where all derivatives are expressed in terms of the transformed co-ordinates (x, y, z∗ ). The term exij is transformed using Green’s formula2 , i.e. integrated per partes. The first term in equation 3.43 is integrated according to x, the second according to z∗ . The result 2

Green’s formula

Z

Ω

αvi,i dΩ =

Z

Γ

αvi ni dΓ −

Z

α,i vi dΩ Ω

94

3.4 Diffusion step

is a sum of a volume and a boundary integrals: exij

=

dxij

+

sxij

∂ϕ∗j ∂ϕ∗i ∂z∗ ∂ϕ∗i ∂z∗ ∂ϕ∗j hνx − dΩ∗ + + ∂x ∂x ∂z∗ ∂x ∂x ∂z∗ Ω∗ ! ! Z ∂ϕ∗j ∂z∗ ∂ϕ∗j ∂z∗ ∗ ∗ ∗ dΓ∗ nz + + hϕi νx nx + ∗ ∂x ∂x ∂x ∂z∗ Γ !

Z

=

!

(3.44)

whereby the j index in the boundary integral describes only the boundary nodes at Γ∗ . The term h∂z∗ /∂x appearing in the equation above can be evaluated in the following way: # " ∂S ∂B ∂z∗ (3.45) = − z∗ + (1 − z∗ ) Kx = h ∂x ∂x ∂x and computed easily for each node. The volume integral from the equation (3.44), denoted Dx , takes the following form: dxij

=−

νx ∂ϕ∗ ∂ϕi h i + Kx h ∂x ∂z∗

Z

Ω∗

!

∂ϕ∗ ∂ϕj h j + Kx ∂x ∂z∗

!

dΩ∗

(3.46)

The volume integral D (the diffusion matrix) can be computed without greater problems in the reference element using mathematical software. The Jacobian of the transformation from the σ-mesh-element to the reference element is constant (see section 3.1.3), which is a great advantage when formulating the diffusion step in the σ-mesh. For the diffusion in the y-direction terms analogous to those equation 3.44 appear. In the z-direction, taking into consideration that ∂z∗ /∂z = 1/h: ezij = −

Z

Ω∗

Z νz ∂ϕ∗i ∂ϕ∗j νz ϕ∗j dΩ∗ + dΓ∗ ϕ∗i n∗z ∗ h ∂z∗ ∂z∗ h ∂z Γ ∗

(3.47)

which also can be integrated without problems. The global equation set to be solved (3.38) can be written shortly in the matrix notation: [M + θ(D + S)]f d = M(∆tqf + f a ) + (θ − 1)(D + S)f n

(3.48)

whereby D is the diffusion matrix, and S the boundary integral, as stated in equations (3.44) and (3.47). M is the mass matrix divided by the time step ∆t.

3.4.2

Boundary terms in the diffusion step

Denoting the boundary node numbers as k, the boundary integrals in all three directions can be written as follows: ∂ϕ∗k ∂ϕ∗k dΓ∗ + h + Kx sik = ∗ h ∂x ∂z∗ Γ ! Z Z ∗ ∂ϕk ∂ϕ∗k νz ϕ∗ ∗ ∗ ∗ νy dΓ∗ + + Ky ϕ∗i n∗z j dΓ∗ ϕi (hny + Ky nz ) h h ∂y ∂z∗ h ∂z∗ Γ∗ Γ∗ Z

νx ϕ∗i (hn∗x

Kx n∗z )

!

(3.49)

3.4 Diffusion step

95

In order to understand the physical meaning of (3.49) the flux of the variable f through a real surface Γ oriented by a normal vector n = (nx , ny , nz ) is considered: Φf = (ν∇f ) · n = νx

∂f ∂f ∂f nx + νy ny + νz nz ∂x ∂y ∂z

(3.50)

Expressing this flux in the transformed co-ordinates one obtains: !

!

∂f νy ∂f νz ∂f ∂f ∂f νx h nx + h ny + + Kx + Ky nz Φ = h ∂x ∂z∗ h ∂y ∂z∗ h ∂z∗ f

(3.51)

In this stage the contributions to the boundary integral (3.49) generated at the bottom, surface and lateral boundaries are discussed separately. E.g. at the bottom (z∗ = 0), where n∗ = (n∗x , n∗y , n∗z ) = (0, 0, −1) (in the σ-mesh the bottom is transformed into a horizontal surface) and Kx = −∂B/∂x = nx /nz : sB ik

νx nx ∂ϕ∗k ∂ϕ∗k − = h + K x h nz ∂x ∂z∗ Γ∗B Z 1 − ∗ ϕ∗i Φf dΓ∗B nz ΓB Z

ϕ∗i

"

!

νy ny ∂ϕ∗ ∂ϕ∗ − h k + Ky k h nz ∂y ∂z∗

!

νz ∂ϕ∗k − h ∂z∗

#

dΓ∗B = (3.52)

At the surface (z∗ = 1), n∗ = (0, 0, 1) a similar expression is obtained by setting Kx = ∂S/∂x = nx /nz : Z 1 S sik = + ∗ ϕ∗i Φfk dΓ∗S (3.53) nz ΓS At the lateral boundaries (z∗ ∈ [0, 1]) n∗ = (n∗x , n∗y , 0) sLik

=+

Z

Γ∗L

hϕ∗i Φfk dΓ∗L

(3.54)

For the terms at the bottom and the free surface the transformation to a 2D reference element is very simple, the Jacobian being the given triangle surface in the σ-transformation (or the projection of the real triangular surface s of the prismatic 3D-element at the horizontal surface, s∗ = s/|nz |): sSij

=

Z

Γ∗S

ϕ∗i

1 f Φ dΓ∗S = nz k

Z

Γref S

ϕref i

s∗ f Φk dΓref S = nz

Z

ΓS

ϕi Φfk dΓS

(3.55)

Actually, when computing the boundary terms there is no special advantage from doing it in the σ-mesh. The situation is the same for the lateral boundaries (quadrilaterals).

96

3.4.3

3.4 Diffusion step

Boundary conditions for the diffusion step

The imposed boundary conditions can be taken into consideration in a very similar way to that described in section 3.3.1. The boundary fluxes, i.e. the Neumann boundary conditions are defined in the following way: (ν∇f ) · n = νi

∂fk k ∂fk k ∂fk k ∂fk = νx nx + νy ny + νz n = ak fk + bk ∂nk ∂x ∂y ∂z z

(3.56)

whereby the coefficients a and b are defined per boundary node k. For the determination of the numeric values of ak and bk in the eventual application, the horizontal viscosity (diffusivity) at the vertical lateral boundaries and the vertical one at the surface or at the bottom are used for νi . The computation of the boundary integrals is performed separately for the bottom, surface and lateral boundaries. The Neumann boundary conditions are taken into consideration through the boundary integrals influencing the linear system matrix (coefficients ak ) or the right hand side vector (coefficients bk ), in an analogous way to section 3.3.2. If ∂fk /∂nk = 0, ak and bk are simply set to 0 and boundary terms disappear. The Dirichlet boundary conditions are realised in this step through modification of the global equation system matrix and the right hand side vector, eventually excluding the nodes with imposed, known values from the consideration.

3.4.4

Computation of the source terms

A characteristic feature of the non-hydrostatic model in comparison with its hydrostatic complement is not only its three-dimensionality, but also the treatment of the forcing terms (i.e. density differences, free surface gradients and Coriolis force) in the threedimensional diffusion step of the algorithm (see section 2.8.5). In the hydrostatic model they are taken into consideration in the pressure – continuity – free surface step (section 2.3.3). In the non-hydrostatic one, they appear as a sum of terms forming the source term qf in the equation (3.32). 3.4.4.1

Free surface gradient terms

The terms representing the barotropic pressure gradients given by (2.217) are twodimensional and independent of depth. The spatial derivatives of the free surface position are computed for each node using the weak formulation in finite elements (3.31). In this formulation, the spatial derivatives of the free surface position in the previous step S(x, y, t) are computed locally in the 2D surrounding Ω0 of the node i (node patch) in the following way: ∂S ∂x

!

= i

R

∂S 2D dΩ0 Ω0 ∂x ϕi R 2D dΩ0 Ω0 ϕ i

=

where ϕ2D are the 2D interpolation functions. i

∂ϕ2D j ϕ2D i ∂x R ∂ϕ2D PN j 2D j xj Ω0 ∂x ϕi

PN j

fj

R

Ω0

dΩ0 dΩ0

(3.57)

3.4 Diffusion step

3.4.4.2

97

Buoyancy terms

The buoyancy terms, i.e. the accelerations which arise due to density differences, defined by eq. (2.218) can be computed in two different ways. To obtain these terms, horizontal derivatives of a vertically integrated relative density difference field ∆̺/̺0 must be computed:   ∂ R S ∆̺n dz −g ∂x z ̺ 0   ∂ R S ∆̺n  F∆̺ =  (3.58)  −g ∂y z ̺0 dz  0 The first method, which seems to be natural, is to compute the relative density gradients using the weak formulation in finite elements. The relative density field is integrated first, and then the derivatives are computed according to the formula (3.31). However, an analysis of this method shows that it causes spurious numerical density currents over a non-horizontal bottom, even when the hydrostatic consistency condition is fulfilled [110, 112] (see section 3.1.2). The clue to this problem is the succession of the integration and differentiation of a variable interpolated by linear base functions. When the relative density is being integrated, terms of the second order appear. Therefore, the integrals appearing in the weak derivative formulation (3.31) must be also computed with an exactness of the second order in the whole domain. For the formula (3.31) with linear interpolation functions this is not the case at the boundaries. For a non-horizontal bottom, errors appear even for stable density fields with a linear vertical profile [110]. Another formulation of the source term resulting from the density gradients must be applied in this case. The idea is to transform the density term into an expression that can be differentiated without problems using a first-order procedure. Naturally, this is the case when the term is differentiated first and the result integrated next. The integration follows the trapezium rule (NL is the number of mesh levels/planes and j is the plane number corresponding to z value): Fx∆̺

∂ = −g ∂x

∂ −g ∂x

Z

z

S

L −1 ∆̺n ∂ NX dz = −g ̺0 ∂x i=j

Z

zi+1

zi

∆̺n dz = ̺0

" NX L −1 1

!#

"

!

i=j

∆̺ni+1 ∆̺ni (zi+1 − zi ) + 2 ̺0 ̺0

L −1 ∂ ∆̺ni+1 ∆̺ni 1 NX (zi+1 − zi ) + − g 2 i=j ∂x ̺0 ̺0

=

∂ + (zi+1 − zi ) ∂x

(3.59) ∆̺ni+1 ∆̺ni + ̺0 ̺0

!#

In this way only the terms of the first order are differentiated and the weak derivative is accurate enough in the whole domain. This formulation is exact for the linearly changing density fields. Due to the obvious advantages of the second formulation, it is implemented in the model. However, for non-linear density fields, errors can still appear. In order to deal with this problem, higher order interpolation functions would have to be introduced.

98

3.4.4.3

3.5 Advection step

Coriolis terms

The computation of the Coriolis terms defined by equation (2.216) is performed without any difficulties using the velocity component values from the previous step: FCor

3.4.4.4

−fH w n + fV v n  −fV un  =  n fH u 



(3.60)

Other forcing

The term qf in equation (2.215) can also include all other forcing accelerations, for example those appearing in non-inertial systems (e.g. sloshing of a fluid in tanks) or a prescribed pressure gradient, etc.

3.5 3.5.1

Advection step Introduction

The advection step of the new algorithm is identical with the previous version. It is presented here for completeness of the description. It should be noted that virtually the same methods (characteristics and SUPG), but in two dimensions, are applied to solve the equations for the free surface and are tested and discussed in section 3.6. The non-linear advection (or hyperbolic) terms in the equations (2.22-2.26): fa − fn + u · ∇f = 0 ∆t

(3.61)

can be treated using the method of characteristics or the streamline upwind PetrovGalerkin (SUPG) method. The first is realised in the σ-mesh, where a transformed velocity field is advected in the transformed velocity field, the latter in the real mesh, but taking advantages of the σ-transformation when computing various matrices.

3.5.2

Advection with the method of characteristics

The description of the treatment of the advection (hyperbolic) step with the method of characteristics is given in [110]. It is presented here shortly for completeness. It illustrates well the philosophy of the operator splitting technique – treating the operators appearing in the hydrodynamic equations separately with most appropriate methods. In this particular case it is the Lagrangian approach to the advection. The σ-transformation does not change the form of the advection equation (section 3.1.2). Therefore, in this step the transformed velocity and tracer concentration values are

3.5 Advection step

99

advected in the transformed velocity field u∗ = (u, v, w∗), according to equation (3.3) in the time-independent σ-mesh of a simple geometry. The non-linear advection terms in the equations (2.22-2.26): fa − fn ∂f ∂f ∂f fa − fn =0 + u∗ · ∇f = +u +v + w∗ ∆t ∆t ∂x ∂y ∂z∗

(3.62)

are treated using the Lagrange method. In this method the time variability of a variable f is observed along a streamline: df ∂f = + u∗ · ∇f dt ∂t

(3.63)

The equation (3.62) states also that the variable f does not change along the streamline (characteristic). In order to discretise the Lagrange time derivative (3.63), the beginning (the base) at time tn and the end of the streamline at time tn+1 must be known. In this method as the end points of the streamlines (characteristics) at time level tn+1 the mesh nodes are taken. The base points xbi at time level tn are found following the streamlines backward in time using the differential equations for the streamlines: dxi = ui dt

(3.64)

Integration of (3.64) between the time levels tn and tn+1 yields xbi = xn+1 − ui∆t i

(3.65)

which is equivalent to the Euler explicit method (or 1st order Runge-Kutta method). Having found the base points, it remains to interpolate the function values fk from the surrounding nodes of elements in which the base points have been found. This interpolated value is equal to the value of f at the node xi at time level tn+1 , because f is constant along the streamline. At boundaries, a few extraordinary situations are met. When a characteristic curve, followed backwards in time, leaves the computational domain at an open boundary, as the base point its intersection with the boundary is assumed. When an impermeable boundary is hit, the streamline is followed further along the boundary in the direction of the velocity component parallel to the given boundary section. In this step of the algorithm Dirichlet boundary conditions are also taken into consideration – the values at given nodes are set to the imposed values. The main advantage of this method is a natural treatment of the non-linear terms of the advection equation (Lagrange method) and theoretically no limits on the time-step according to the Courant number. The method of characteristics is namely the only explicit method that is unconditionally stable according to the chosen time-step. The main disadvantage is the fact that for the interpolation of the function values at the position of the found base points, the linear base functions in prismatic elements are used. Therefore serious damping (numerical diffusion) occurs due to interpolation. Using interpolation functions of higher order may improve the exactness of this step. Another disadvantage are problems with mass conservation using this scheme [110].

100

3.5 Advection step

3.5.3

Advection with SUPG method

The advection step in which the hyperbolic advection equation (2.214) is to be solved, can also be treated using the Streamline Upwind Petrov-Galerkin (SUPG) method in finite elements [16, 116]. The idea of this method is shown for the advection of the v velocity component and, for simplicity, in the real mesh. In contrast to the method of characteristics, the advection equation is formulated in the real co-ordinates. The σ-transformation appears in the stage, when the resulting matrices are treated. Using the explicit formulation for the time partial derivative in the real mesh as in (2.214), one obtains: va − vn ∂v ∂v ∂v +u +v +w =0 (3.66) ∆t ∂x ∂y ∂z v can be weighted between the levels n and a using the implicitness parameter θ: v = θv a + (1 − θ)v n

(3.67)

Therefore, for θ = 1 a fully implicit, and for θ = 0 a fully explicit advection schema is obtained. v(x, y, z, t) is interpolated using Lagrangian interpolation functions ϕj for a node j, i.e.: v=

X

vj ϕj

(3.68)

j

The finite element procedure follows the SUPG method. The equation (3.66) is multiplied by the test functions ψi and then integrated in the domain Ω: 1 ∆t

1 ψi v dΩ− ∆t Ω

Z

a

∂v dΩ+ ψi v dΩ+ ψi u ∂x Ω Ω

Z

Z

n

∂v dΩ+ ψi v ∂y Ω

Z

Z

ψi w

Ω

∂v dΩ = 0 (3.69) ∂z

Introducing (3.68) to 3.69 yields: X

vja

j

X Z X Z 1 Z ∂ϕj ∂ϕj vj ψi v vj ψi u dΩ + dΩ + ψi ϕj dΩ + ∆t Ω ∂x ∂y Ω Ω j j X

vj

j

Z

Ω

ψi w

X ∂ϕj 1 vjn dΩ = ∂z ∆t j

Z

Ω

(3.70)

ψi ϕj dΩ

or shortly: X j

vja



1 ∆t

Z

Ω



ψi ϕj dΩ +

X j

vj

Z

Ω



ψi u · ∇ϕj dΩ =

X j

vjn



1 ∆t

Z

Ω



ψi ϕj dΩ

(3.71)

where u = (u, v, w). In the Upwind Streamline Petrov-Galerkin Method the test functions ψi in the advection terms of (3.71) are replaced by: ψi = ϕi + α

u ∇ϕi |u|

(3.72)

3.5 Advection step

101

where α is a real constant (upwind parameter). In this way for the advective mass transport the upstream nodes are stronger weighted than the downstream ones. In all other terms ψi = ϕi is set, as in the standard Galerkin method. Realising this idea in (3.71) yields: X

vja

j



+α

X 1 Z vj ϕi ϕj dΩ + ∆t Ω j 

X j

vj

Z

Ω

Z

Ω



ϕi u · ∇ϕj dΩ !

X u 1 Z vjn ∇ϕi u · ∇ϕj dΩ = ϕi ϕj dΩ |u| ∆t Ω j 



(3.73)

Introduction of (3.67) into (3.73) yields: X

vja

j

=

"

#

Z Z 1 Z u ∇ϕi u · ∇ϕj dΩ ϕi ϕj dΩ + θ ϕi u · ∇ϕj dΩ + θα ∆t Ω Ω Ω |u|

X j

vjn

"

(3.74) #

Z Z 1 Z u ∇H ϕi u · ∇ϕj dΩ ϕi ϕj dΩ + (θ − 1) ϕi u · ∇ϕj dΩ + (θ − 1)α ∆t Ω Ω Ω |u|

In order to solve the linear equation system (3.74) the following matrices of coefficients must be computed: Mass matrix M (divided by ∆t): 1 ∆t

mij = Advection matrix K:

Z

kij = SUPG matrix U: uij = α

Ω

Z

Ω

Z

ϕi ϕj dΩ

(3.75)

ϕi u · ∇ϕj dΩ

(3.76)

Ω

u ∇ϕi u · ∇ϕj dΩ |u|

(3.77)

And, by implicit treatment of the advection terms the following set of linear equations must be solved in order to find the unknown vector va : [M + θ(K + U)]va = [M + (θ − 1)(K + U)]vn

(3.78)

The computation of the matrices can be performed in the σ-mesh exactly, therefore the integrals appearing in the mass, advection and SUPG matrix are transformed into σmesh (like in the diffusion step, section 3.4) and using the definition of w∗ (equation 3.4). Z 1 ϕ∗i ϕ∗j h dΩ∗ (3.79) mij = ∗ ∆t Ω kij = uij = α

Z

Z

Ω∗

Ω∗

ϕ∗i u∗ · ∇ϕ∗j h dΩ∗

u∗ ∇ϕ∗i u∗ · ∇ϕ∗j h dΩ∗ |u∗ |

(3.80) (3.81)

102

3.6 Free surface step

where u∗ is the transformed velocity (u, v, w∗). Eventually, in contrast to the implemented method of characteristics, the nodal values are the real velocity values, but the advection field is the transformed velocity u∗ . The integration of the matrices is carried out analytically. By the eventual implementation in the fully three-dimensional code, a constant value α = ∆t/2 is assumed for the upwind parameter (see also section 3.6.2). The Dirichlet boundary conditions are taken into consideration by modification of the global equation set (3.78). As it can be seen after the description of the diffusion step, which leads to solution of a linear set of equations like (3.78) as well, both steps can be treated simultaneously. A common set of linear equations for advection and diffusion, resulting from addition of appropriate matrices and vectors is solved for each velocity component and tracer. Comparing (3.78) and (3.48) the equation system in the coupled advection-diffusion step for a resulting variable vd reads: {M + θ[(D + S) + (K + U)]} vd = M(∆t qf ) + {[M + (θ − 1)[(D + S) + (K + U)]} vn

3.5.4

(3.82)

Boundary conditions for advection step

If the advection step is realised with the method of characteristics, only Dirichlet boundary conditions are imposed on the resulting field f a . If the SUPG method is applied, the boundary conditions are imposed as described in section 3.4.

3.6 3.6.1

Free surface step Free surface with the method of characteristics

The solution method for the kinematic boundary condition (2.158) follows exactly the description in section 3.5.2 for the advection equation, but in two dimensions. For interpolating the free surface position at the bases of the characteristics the two-dimensional linear interpolation in triangles are applied (section 3.1.4).

3.6.2

SUPG formulation for the kinematic boundary condition

The kinematic boundary condition (2.158) is a hyperbolic partial differential equation which can be solved in the finite element method using the Streamline Upwind PetrovGalerkin (SUPG) method [16, 116]. The kinematic boundary condition can be treated as a two-dimensional advection equation with a source term: ∂S ∂S ∂S +u +v =w ∂t ∂x ∂y

(3.83)

3.6 Free surface step

103

Using the explicit formulation for the time partial derivative in (3.83) one obtains: ∂S ∂S S n+1 − S n +u +v =w ∆t ∂x ∂y

(3.84)

S can be weighted between the time level n and n + 1 using the implicitness parameter θ: S = θS n+1 + (1 − θ)S n (3.85) For θ = 1 a fully implicit, and for θ = 0 a fully explicit schema is obtained. The function S(x, y, t) is interpolated using 2D Lagrangian interpolation functions ϕi for a node i, i.e.: X

S=

Sj ϕ j

(3.86)

j

As in section 3.5.3, the finite element formulation follows the SUPG method. The equation (3.84) is multiplied by the test functions ψ and then integrated in the domain Ω: 1 ∆t

Z

1 ∆t

ψi S n+1 dΩ −

Ω

Z

Ω

Z

ψi S n dΩ +

Ω

ψi u

∂S dΩ + ∂x

Z

Ω

∂S dΩ = ∂y

ψi v

Z

Ω

ψi w dΩ (3.87)

Introducing (3.86) to 3.87 one obtains: X

Sjn+1

j

1 ∆t

Z

ψi ϕj dΩ +

Ω

X j

X ∂ϕj Sj Sj ψi u dΩ + ∂x Ω j Z

X

wj

j

Z

ψi ϕj dΩ +

Ω

X

Z

ψi v

∂ϕj dΩ = ∂y

Sjn

1 ∆t

Z

Ω

j

Ω

(3.88)

ψi ϕj dΩ

or: X

Sjn+1

j



X

1 ∆t

Z

Ω

∆twj

j



ψi ϕj dΩ + 

1 ∆t

Z

Ω

X

Sj

j

Z

Ω



ψi ϕj dΩ +



ψi uH · ∇H ϕj dΩ = Sjn

X j



1 ∆t

Z

Ω

(3.89)



ψi ϕj dΩ

The SUPG method the test functions in the advective terms are to be replaced by: u (3.90) ψi = ϕi + α ∇ϕi |u| where α is a real constant and ϕi interpolation functions. In all other terms ψi = ϕi is set. Introducing so defined test functions to (3.89) yields: X

Sjn+1

j

X j

X j



Sj

1 Z ϕi ϕj dΩ + ∆t Ω 

Z

∆twj

Ω

(3.91) 

ϕi uH · ∇H ϕj dΩ + α



1 ∆t

Z

Ω



ϕi ϕj dΩ +

X j

X

Sj

j

Sjn



1 ∆t

!

Z

uH ∇H ϕi uH · ∇H ϕj dΩ = |uH |

Z

ϕi ϕj dΩ

Ω

Ω



104

3.6 Free surface step

Introduction of (3.85) into (3.91) yields: X

Sjn+1

j

"

= ∆t

1 ∆t

X

Z

ϕi ϕj dΩ + θ

wj



Ω

j

X j

Sjn

"

1 ∆t

1 ∆t

Z

Ω

Z

Ω

Z

ϕi uH · ∇H ϕj dΩ + θα

Ω

Z

Ω



#

uH ∇H ϕi uH · ∇H ϕj dΩ |uH |

ϕi ϕj dΩ +

ϕi ϕj dΩ + (θ − 1)

Z

ϕi uH · ∇H ϕj dΩ + (θ − 1)α

Ω

Z

Ω

#

uH ∇H ϕi uH · ∇H ϕj dΩ |uH |

In order to solve the linear equation system (3.92) the following matrices of coefficients must be computed: mass matrix M divided by ∆t: mij = advection matrix K: kij = SUPG matrix U: uij = α

Z

Ω

Z

Ω

1 ∆t

Z

Ω

ϕi ϕj dΩ

(3.92)

ϕi uH · ∇H ϕj dΩ

uH ∇H ϕi uH · ∇H ϕj dΩ |uH |

(3.93)

(3.94)

Finally, the following set of linear equations must be solved in order to find the unknown vector Sn+1 = [sn+1 ij ]: ASn+1 = B (3.95) where A = M + θ(K + U) and the vector B = M(Sn + ∆tW) + (θ − 1)(K + U)Sn , where W = [wi ]. The Dirichlet boundary conditions are taken into consideration by modification of the global equation set (3.95). The value of upwind parameter α influences strongly the properties of the SUPG algorithm. A constant value of α = ∆t/2 and the horizontal velocity vector itself uH is taken instead of the unit vector in the direction of the horizontal velocity uH /|uH |. This leads to the following form of the SUPG matrix (3.94): ∆t Z uij = uH ∇H ϕi uH · ∇H ϕj dΩ 2 Ω

(3.96)

A classical SUPG scheme is also available, where instead of (3.90) the following expression is assumed ψi = ϕi + K∇ϕi (3.97) The vector K has the following components: K=

∆x u , 2||u||

∆y v 2||u||

!

(3.98)

3.6 Free surface step

105

The velocity norm is computed as the mean velocity in an element, ||u|| = (u2 + v 2 )1/2 and the ∆x and ∆y are the maximum extents of an element along the x and y-axis, respectively. The tests of the 2D-advection equation solvers are given in section 4.3.1. 3.6.2.1

Implementation of the absorbing boundary conditions

In order to implement the absorbing boundary conditions for the free surface (section 2.7.9), a dissipative term is introduced to the kinematic boundary condition for the free surface. Equation (3.83) transforms to: ∂S ∂S ∂S ∂S ∂ νx + us + vs = ws + ∂t ∂x ∂y ∂x ∂x

!

∂ ∂S + νy ∂y ∂y

!

(3.99)

which is equivalent to a transport equation for S with a source term w. The distribution of the values of νx and νy define the properties of damping zones. This advection-diffusion equation is treated in 2D in the same way as in the 3D case described in section 3.5.3. The resulting equation system can be sketched as follows: h i S n+1 + θ u · ∇S n+1 − ∇ · (ν∇S n+1 ) = ∆t Sn + ws + (θ − 1) [u · ∇S n − ∇ · (ν∇S n )] ∆t

3.6.3

(3.100)

Explicit formulation for the conservative free surface equation

In order to solve the conservative free surface equation (2.159) explicitly for a 2D node i, one may write: ! ) ∂(hni vn+1 ) Sin+1 − Sin ∂ Z Sn ∂ Z Sn ∂(hni un+1 i i =− udz − vdz = − + ∆t ∂x −B ∂y −B ∂x ∂y

(3.101)

where (un+1 , vn+1 ) are vertically integrated horizontal final velocity components (using the free surface elevation from the previous step S n ). The new free surface S n+1 is computed explicitly for each node in finite elements, i.e. without solving any equation system. The derivatives of vertically integrated velocity components are computed for each node using the weak formulation, equation (3.31). Because only horizontal gradients of the integrated velocities are needed, for simplification and in order to spare computation time two-dimensional interpolation functions ϕ2D (for a 2D-mesh of triangles) are used, calling for both numerator and denominator i of (3.31) a subroutine computing the vector ai : ai =

N X j

fj

Z

Ω0

∂ϕ2D j ϕ2D dΩ0 ∂x i

(3.102)

106

3.6 Free surface step

A version of this free surface computing method, basing on computing the derivative of the velocity integral locally in each 3D-element can be thought of as in (3.59). The integration in depth follows the trapezium rule (NL is the number of mesh levels/planes), e.g. for the term containing u velocity component: L −1 Z zi+1 ∂ ZS ∂ NX udz = udz = ∂x −B ∂x i=1 zi

L −1 1 ∂ NX (zi+1 − zi )(uni+1 + uni ) = ∂x i=1 2





(3.103)

L −1 1 NX ∂ ∂ (zi+1 − zi )(uni+1 + uni ) + (zi+1 − zi ) (uni+1 + uni ) 2 i=1 ∂x ∂x

"

3.6.4

#

Semi-implicit formulation for the conservative free surface equation

In the semi-implicit formulation for the conservative free surface equation (2.159), the velocity components at the time level n + 1, as well as the position of the free surface at the time level n and n + 1 are involved. The horizontal velocity components at the time level n + 1 can be easily integrated between the bottom B and the free surface elevation S n yielding (un+1 , v n+1 ). The scheme can be formulated as follows (using water depth h = S − B as the searched variable): hn+1 − hn ∂(hun+1 ) ∂(hv n+1 ) =− + ∆t ∂x ∂y ! n+1 n+1 ∂(u ) ∂(v ) ∂h n+1 ∂h n+1 u − v = −h − + ∂x ∂y ∂x ∂y !

(3.104)

where the water depth in the terms on the right hand side of this equation can be weighted between the time levels n and n + 1 using an implicitness factor θ: h = θhn+1 + (1 − θ)hn . The equation (3.104) is treated using SUPG FEM in exactly the same way as presented in section 3.6.2 (with the difference that the advection velocity is the mean one in depth), leading to a global matrix equation in the form: Ahn+1 = B

(3.105)

where A = M + θ(K + U + Q) and the vector B = [M + (θ − 1)(K + U + Q)]hn . The matrix Q contains the divergence of the mean horizontal velocity: qij =

Z

Ω

∇H · uH ϕi ϕj dΩ =

Z

Ω

∂u ϕi ϕj dΩ + ∂x

Z

Ω

∂v ϕi ϕj dΩ ∂y

(3.106)

The equation (3.105) is treated in exactly the same way as described in section 3.6.2.

3.6 Free surface step

3.6.5

107

Free surface stabilisation

The stability of the free surface algorithm is crucial for the stability of the global solution, because errors in computing the free surface gradients in FEM (barotropic part of hydrostatic pressure gradients) influence negatively and immediately the whole domain. The computation of the free surface based on the kinematic boundary condition, a hyperbolic equation, is over-sensitive for small errors or imperfectly set boundary conditions. Great care must be exercised in harmonising the boundary conditions for velocity, pressure and free surface. Due to its hiperbolic nature, the kinematic boundary condition does not include any dissipative terms which may smoothen the solution. Situations where some instabilities of the free surface occur are common in practice. When boundary conditions for the free surface are only approximately known, or awkwardly set, or a wave frequency dispersion occurs for waves with a length comparable with the spatial resolution, some stabilising techniques for the free surface are needed. The disadvantage of stabilising techniques is an artificial, non-physical smoothing of the free surface. The advantage is that the model can be applied in more complicated situations regarding properly set boundary conditions. In this work, a well-known filtering method in FEM, using the mass matrix M (equation 3.27) is applied [73]. In this technique, a vector V is modified according to the formula: V′ =

MV ML

(3.107)

where ML is a diagonal matrix obtained by mass-lumping of M, i.e. by obtaining its diagonal elements by totalling the terms in each row of M. The integral of the vector V is preserved, which in the case of the free surface as the filtered variable guarantees conservation of mass. Alternatively, the advection velocity in the free surface algorithms can be also filtered.

108

3.6 Free surface step

Chapter 4 Verification of algorithm stages Only a fool has no doubts. ˇ Karel Capek Due to the modular structure of the developed algorithm, it is possible and appropriate to test and verify its separate parts/stages. In this chapter the verification and performance tests of the diffusion step, Poisson equation solver and free surface step (2D advection equation) are presented. The theory is provided in chapter 3.

4.1

Diffusion step test: Ekman profiles

A particularly good example for testing the diffusion step is the reproduction of Ekman spirals. According to the theory of V.W. Ekman, characteristic velocity profiles appear in the stationary oceanic bottom and surface layers of frictional influence. In the deep ocean, these layers are relatively thin compared to the much thicker intermediate layer in mid-depths, where the current is not influenced by the bottom or wind shear, respectively. In this ocean interior, according to Ekman’s theory, the current is stationary and in geostrophic equilibrium. It means that the pressure gradient and Coriolis force cancel each other out in this layer. The spiral profiles appear due to the disturbance of the geostrophic equilibrium just below the surface by the wind shear (drift current) and just above the ocean floor by the bottom shear (slope current). Expressing it mathematically, they result from specific boundary conditions imposed on the momentum equations in layers of a defined thickness. The analytical solutions for these profiles were obtained for the stationary case and constant eddy viscosities by V.W. Ekman in 1905 [41]. The solution for the drift current is determined through a Neumann wind shear boundary condition at the surface. The solution for the slope current is determined by the driving pressure gradient, i.e. a source term in the equation, and the Dirichlet no-slip boundary condition at the bottom. Therefore, these examples are very well suited for verification of the numeric diffusion step implementation, treated as a separate program unit. 109

110

4.1 Diffusion step test: Ekman profiles

Short mathematical derivations of the solutions for the drift and slope currents are given in the following sections 4.1.1 and 4.1.2.

4.1.1

Slope current

Let us assume that a constant horizontal pressure gradient exists in the ocean. It can result from an external atmospheric pressure field or from a constant free surface slope prevailing for a long enough period of time. A stationary current in the interior of the ocean appears, when the pressure gradient force is balanced by the Coriolis force (geostrophic equilibrium). For simplicity, the co-ordinate system is so oriented, that ∇H p = (0, ∂p/∂y). The zaxis is oriented upwards, with z = 0 at the surface and z = −H at the bottom, H being the water depth. Assuming viscosity νz and the vertical Coriolis coefficient f to be constant for a stationary current in hydrostatic approximation, the momentum conservation equations reduce to: f v + νz

∂2u = 0, ∂z 2

−f u + νz

∂2v 1 ∂p = , 2 ∂z ̺ ∂y

∂p = ̺g ∂z

(4.1)

with the following boundary conditions at the surface and the bottom: ∂u ∂z ∂v τy = −νz ∂z u=v=0

τx = −νz

z=0

(4.2)

z=0 z = −H

Defining (uE , vE ) as the velocity field describing the deviation from the geostrophic equilibrium current (ug ) in the bottom layer, one obtains u = ug + uE and v = vE . The velocity ug satisfies the geostrophic equation for the given constant pressure gradient: −f ug −

1 ∂p =0 ̺ ∂y

(4.3)

Introducing a complex velocity V = uE + ivE into (4.1) and (4.2) and adding equations (4.1) one obtains the following problem: d2 V if − V =0 2 dz νz dV =0 dz V = −ug

z=0

(4.4)

z = −H

The global solution of the problem (4.4) has a form V = Aeβz + Be−βz . Setting this form into the equation (4.4) in order to obtain β yields: β = (1 + i)(f /2νz )1/2 = (1 + i)γ

(4.5)

4.1 Diffusion step test: Ekman profiles

111

where γ ∈ R. Taking the boundary conditions in (4.4) into consideration in order to find A, B one obtains a complex solution: V = −ug

cosh[(1 + i)γz] cosh [(1 + i)γH]

(4.6)

which can be transformed into the solutions for real parts of the velocity components (u, v): !

cosh[γ(H + z)] cos[γ(H − z)] + cosh[γ(H − z)] cos[γ(H + z)] u = ug 1 − cosh(2γH) + cos(2γH) ! sinh[γ(H + z)] sin[γ(H − z)] + sinh[γ(H − z)] sin[γ(H + z)] v = ug cosh(2γH) + cos(2γH)

(4.7)

For the deep ocean, an assumption of infinite depth is often appropriate. For simplicity of the solution, the z-axis is also oriented upwards, but this time z = 0 is taken at the bottom, and z → ∞ towards the surface: d2 V if − V =0 2 dz νz dV =0 dz V = −ug

z→∞

(4.8)

z=0

In this case the solution has a much simpler form: V = −ug e−γz [cos(γz) − i sin(γz)] u = ug [1 − e−γz cos(γz)] v = ug e−γz sin(γz)

(4.9)

In order to estimate the thickness of the layer of the frictional influence Ekman introduced a parameter called depth of frictional influence D defined as the distance above the bottom, where the value of the velocity deviation |ve | = ug e−γz has decreased to e−π = 1/23 of the geostrophic current: π 2νz D= =π γ f

!1/2

(4.10)

The vertical structure of the velocity profile depends strongly on the ratio H/D (figure 4.1). For water depths H < D the geostrophic balance is never attained, and the velocity has a component in the direction of the pressure gradient at every depth.

112

4.1.2

4.1 Diffusion step test: Ekman profiles

Drift current

The Ekman profiles for the wind-driven current in the surface layer of the ocean are obtained in an analogical way to the profiles in the bottom layer. Introducing the complex wind shear variable T = τx + iτy , where τx = ̺νz (∂u/∂z) and τy = ̺νz (∂v/∂z) one obtains the following problem to be solved for finite or infinite depth: if d2 V − V =0 2 dz νz dV T = ̺νz dz V =0 V =0

z=0

(4.11)

z = −H z→∞

finite depth infinite depth

where it is assumed that the z-axis points upwards and z = 0 corresponds to the position of the surface and z = −H is the flat bottom level or an infinite depth is assumed.

The solution for the infinite depth is:

π

V = V0 eγz ei(γz− 4 ) where V0 = u0 + iv0 =

T τwx τwy = +i 1/2 1/2 ̺(νz |f |) ̺(νz |f |) ̺(νz |f |)1/2

(4.12)

(4.13)

is the complex velocity at the surface. In order to obtain a simplified form of the analytical solution, the wind is assumed to blow in the y-direction, i.e., τx = 0, τy 6= 0. The solution is: π

V = v0 eγz ei(γz+ 4 )

π ) 4 π v = v0 eγz sin(γz + ) 4 u = v0 eγz cos(γz +

(4.14)

From this equation it is clear, that the drift current is deflected 45o to the right of the wind direction. The complex solution for the finite depth case is: V = V0

π sinh[(1 + i)γ(z + H)] exp(− i) cosh[(1 + i)γH] 4

(4.15)

Lengthy analytical formulae for the real velocity components u and v can be obtained from (4.15) using mathematical software. Assuming again that the wind blows in the y-direction, i.e., τx = 0, τy 6= 0, the solution is (ζ = z + H): √ ! 2 [sinh(γζ) cos(γζ) − cosh(γζ) sin(γζ)] cosh(γH) cos(γH) u = v0 (4.16) 2 cosh2 (γH) cos2 (γH) + sinh2 (γH) sin2 (γH) √ ! 2 [sinh(γζ) cos(γζ) + cosh(γζ) sin(γζ)] sinh(γH) sin(γH) +v0 2 cosh2 (γH) cos2 (γH) + sinh2 (γH) sin2 (γH)

4.1 Diffusion step test: Ekman profiles √

113

!

2 [sinh(γζ) cos(γζ) + cosh(γζ) sin(γζ)] cosh(γH) cos(γH) v = v0 2 cosh2 (γH) cos2 (γH) + sinh2 (γH) sin2 (γH) √ ! 2 [sinh(γζ) cos(γζ) − cosh(γζ) sin(γζ)] sinh(γH) sin(γH) −v0 2 cosh2 (γH) cos2 (γH) + sinh2 (γH) sin2 (γH)

4.1.3

(4.17)

Numerical test: drift and slope current

The Ekman spiral test for the drift current belongs to the standard tests of the Telemac-3D software. Therefore, very similar test parameters are chosen as in the standard test [83]. The basic mesh is shown in appendix, figure A.1. It measures 2000 m × 2000 m, giving a horizontal resolution of approximately 100 m; in the vertical direction 21 mesh levels are applied. The bottom is flat, whereby the depth varies from case to case in order to observe different velocity profiles. By the greatest depth applied (160 m), the vertical resolution is 8 m. In order to test the three-dimensional diffusion step exclusively, all other steps of the algorithm were switched off. In the case of the drift current, the stationary solution of a diffusion equation with a source term (the Coriolis force) for a zero initial condition for the velocity in the domain and a given boundary condition is achieved. For the case of the slope current, a solution of a diffusion equation with a source term (a sum of the Coriolis force and a constant pressure gradient), with a corresponding initial condition for the velocity and a given boundary condition is obtained. The solutions for the Ekman spirals are two-dimensional, for (u, v), with w = 0 when the result converges. For an additional test, if the stage is fully three-dimensional, the velocity diffusion subroutine was called for (u, v, w), (u, w, v) and (w, v, u) in the program code. i.e. with changing order of the velocity components, Coriolis forcing terms and boundary conditions. The final results were compared if they are identic. 4.1.3.1

Drift current

The surface wind stress is computed according to the empirical formula given by equations (2.188) and (2.189). The wind velocity, directed North, i.e. coincidental with the positive y–direction, is set at 32 m/s. At the assumed latitude of 50o North (Coriolis coefficient f = 1.1 × 10−4 s−1 and with the vertical viscosity νz = 0.1 m2 /s the wind shear yields a deflection 45o to the right from the wind direction when the stationary state is achieved. According to (4.13), the drift current velocity is v0 ≈ 1 m/s at the free surface. The depth of frictional influence in this case is D = 133.68 m (4.10). The only parameter varied in the test cases is the water depth H, corresponding to 0.1D, 0.25D, 0.5D and 1.2D, (13.37 m, 33,42 m, 66.84 m and 160 m). Horizontal diffusion coefficients are taken to equal 0. A no-slip boundary condition is set at the bottom. The initial condition is null velocity in the entire domain and a constant water depth. The time step is 50 s and in the integration time is 56 h, i.e. 4000 time steps. The results are presented in the figure 4.1. For depths 0.1D–0.5D, the numerical solutions converge to

114

4.1 Diffusion step test: Ekman profiles

Ekman spiral test - slope current 1.0 H=0.25D 0.8

H=1.2D

v [m/s]

0.6

0.4

H=0.5D

H=0.1D

0.2

0.0

-0.2 -0.2

0.0

0.2

0.4

0.6

0.8

1.0

u [m/s]

Ekman spiral test - drift current 0.03 H=0.25D

H=0.5D

0.025

v [m/s]

0.02

0.015

0.01 H=1.2D 0.005

0.0

-0.005 0.0

0.01

0.02

0.03

0.04

0.05

0.06

u [m/s]

Figure 4.1: Ekman spirals in the surface layer for finite water depth and various values of H/D Above: drift current. Below: slope current. Circles correspond to the numerical result and solid lines are the Ekman analytical solutions.

4.2 Tests of the Poisson equation step

115

1.0

u [m/s]

0.8

0.6

0.4

0.2

0.0 0

20

40

60

80

100

computation time [h]

Figure 4.2: Ekman test case, drift current. Convergence of the u velocity component at the surface to the analytical solution for H = 1.2D the analytical ones within the prescribed 56 h. The convergence behaviour for the most slowly converging case H = 1.2D is presented in figure 4.2 for an extended integration time of 96 h. 4.1.3.2

Slope current

For the same set of parameters and mesh, the bottom Ekman profiles for the slope current are computed for three different depths, 0.25D, 0.5D, 1.2D. The constant pressure gradient in the y–direction (North) corresponds for the assumed latitude with the constant geostrophic velocity ug = 0.05 m/s in the x–direction (East), equation (4.3). The constant pressure gradient is given as a constant source term in the diffusion step of the algorithm. The initial velocity is equal ug and directed East. A no-slip boundary condition at the bottom is applied. The results are presented in figure 4.1. Again, for the same integration time of 56 h, the numerical solutions converge to the analytical ones.

4.2

Tests of the Poisson equation step

Two tests of the Poisson equation stage are presented. Both are comparisons of an analytical solution with a numerical one. First, a solution of a problem thoroughly

116

4.2 Tests of the Poisson equation step

determined by boundary conditions is given in order to test the boundary conditions implementation. The source term of the Poisson equation is set to null. In the second case, the boundary conditions are simple, but the source term is not equal to zero in order to observe the behaviour of the numerical scheme in the this situation. In both cases the numerical solution was obtained using the mesh shown in fig. A.1 with 21 vertical levels, setting the absolute exactness of the linear equation solver (conjugate gradient method with diagonal pre-conditioning) to 10−6 . Because the applied analytical solutions are two-dimensional, for each case two runs were made: one in a vertical and another in a horizontal plane, respectively. All other stages of the algorithm were switched off, eventually reducing the program to a solver for the Poisson equation.

4.2.1

Testing the boundary conditions influence

The Poisson equation solver for the pressure (or the hydrodynamic pressure, or a pressure correction) must work appropriately with complicated boundary conditions given on various boundaries of the domain. In this section the numerical solution of the Poisson equation is compared with an analytical one with complex boundary conditions. In order to concentrate on the influence of the boundary conditions separately, the source term of the right hand side of the Poisson equation is set to null (Laplace equation), so that the problem solution depends thoroughly on the boundary conditions. The solution of the Laplace equation ∇2 T (x, y) = 0 in a two-dimensional rectangular area x ∈ [0, a] and y ∈ [0, b] is considered. The following boundary conditions are applied: 1. for x = 0, y ∈ [−b, b] : T = T0 (Dirichlet); 2. for x = a, y ∈ [−b, b] : ∂T /∂x = 0 (Neumann, set to 0); 3. for y = −b and y = b, x ∈ [0, a] : ∂T /∂x = −hT (Neumann). The solution follows the method of variable separation (Fourier method): T (x, y) = u(x)v(y) ∇2 T (x, y) = 0 ⇒ u′′ (x)v(y) + u(x)v ′′ (y) = 0 ⇒

(4.18) u′′ v ′′ = − = p2 u v

(4.19)

Taking into consideration the boundary conditions we obtain the following equation sets for u and v:  ′′ 2   u −p u=0 ′ u (a) = 0 (4.20)   u(0)v(y) = T0 v ′′ + p2 u = 0 v ′ (−b) = −hv(−b)   ′ v (0) = −hv(b)   

(4.21)

4.2 Tests of the Poisson equation step

The solution is:

(

117

u=cosh[p(a − x)] v=cos(py)

(4.22)

where(p = pk , k = 1..∞) are the successive solutions (eigenvalues) of: 1 pk = h tan(pk b)

(4.23)

The searched solution has the form of a sum: T (x, y) =

∞ X

k=1

Ak cosh[pk (a − x)] cos(pk y)

(4.24)

From the Dirichlet boundary condition one finds: T0 =

∞ X

Ak cosh(pk a) cos(pk y)

(4.25)

k=1

The coefficients Ak can be obtained from the Fourier theorem: Rb

T0 cos(pk y)dy Ak cosh(pk a) = R0 b 2 0 cos (pk y)dy

which yields:

Ak =

2T0 sin(bpk ) [bpk + sin(bpk ) cos(bpk )] cosh(apk )

(4.26)

(4.27)

The final solution is: T (x, y) =

∞ X

2T0 sin(bpk ) cosh[pk (a − x)] cos(pk y) k=1 [bpk + sin(bpk ) cos(bpk )] cosh(apk )

(4.28)

In order to find the values of the solution (4.28) the eigenvalues pk of (4.23) are found numerically using the regula falsi algorithm [127]. The assumed values for the boundary conditions are T0 = 1 and h = 1, for the geometry a = 2 and b = 1. The comparison between the numerical and analytical solution is given in fig. 4.3, showing an excellent agreement.

4.2.2

Testing the source term influence

In this section a solution of the Poisson equation with a non-zero source term for simple, symmetrical Neumann boundary conditions is investigated. The solution of the Poisson equation ∂2T ∂2T + =q ∂x2 ∂y 2

(4.29)

118

y

4.2 Tests of the Poisson equation step

numeric solution

analytic solution

y

1.0

1.0 0.30

0.30

0.3

0.3

5

5

-0.5

0.40

0.45

0.50

0.65

0.60

0.75

0.70

0.85

0.80

0.95

0.90

0.70

0.65

0.0

0.55

0.40

0.45

0.50

0.55

0.60

0.75

0.80

0.85

0.0

0.90

0.5

0.95

0.5

-0.5 5

5

0.3

0.3

0.30

0.30

-1.0

-1.0 0.0

0.5

1.0

1.5

2.0

0.0

0.5

x

1.0

1.5

2.0

x

Figure 4.3: Poisson equation verification. Solutions determined by boundary conditions. is considered in a two-dimensional rectangular area x ∈ [−a, a] and y ∈ [−b, b]. The following symmetrical Neumann boundary conditions are applied: x = −a, x = a, y = −b, y = b,

y ∈ [−b, b] y ∈ [−b, b] x ∈ [−a, a] x ∈ [−a, a]

(∂T /∂x)x=−a + hT (−a, y) = 0 (∂T /∂x)x=a − hT (a, y) = 0 (∂T /∂y)y=−b + hT (x, −b) = 0 (∂T /∂y)y=b − hT (x, b) = 0

(4.30) (4.31)

Due to the symmetry of the boundary conditions it can additionally be stated that: (∂T /∂x)x=0 = 0

(∂T /∂y)y=0 = 0

(4.32)

As in the previous section the analytic solution method follows the method of variable separation (Fourier method), whereby the solution is searched for in the form: T (x, y) =

∞ X

Xm (x)Yn (y)

(4.33)

m,n=1

where Xm and Yn are linear combinations of the sin and cos functions: Xm (x) = Am cos(Mm x) + Bm sin(Mm y) Yn (x) = Cm cos(Nm y) + Dm sin(Nm y)

(4.34) (4.35)

From the boundary conditions (4.30) and (4.32) one finds: ^ m

µm a ^ γ n Yn′ (0) = Yn′ (b) + hYn (b) = 0 ⇒ Dm = 0 ∧ Nn = b n

′ ′ Xm (0) = Xm (a) + hXm (a) = 0 ⇒ Bm = 0 ∧ Mm =

(4.36)

4.3 Tests of the free surface step

119

where µm and γn are successive solutions (eigenvalues) of the following equations: µ tan µ = ah

γ tan γ = bh

(4.37)

Therefore, the sought solution has the following form: ∞ X

T (x, y) =

Hmn cos(

m,n=1

γn µm x) cos( y) a b

(4.38)

where Hmn = Am Cn . Substituting (4.38) into (4.29) one obtains: ∞ X

µ2n γn2 µm γn + cos( x) cos( y) = −q a2 b2 a b !

Hmn

m,n=1

(4.39)

The coefficients Hmn are to be found from the Fourier formula: Hmn

µ2n γn2 + 2 a2 b

!

=

Ra Rb

µm γn −a −b cos( a x) cos( b y)dxdy Ra Rb 2 γn 2 µm −a −b cos ( a x) cos ( b y)dxdy

−q

After computing integrals in (4.40) one obtains: −4q sin µm sin γn  Hmn =  µ2 2 γ n + bn2 (sin µm cos µm + µm )(sin γn cos γn + γn ) a2

(4.40)

(4.41)

The final solution is: T (x, y) = −4q

∞ X

m,n=1

sin µm sin γn cos( µam x) cos( γbn y) 

µ2n a2

+

2 γn b2



(sin µm cos µm + µm )(sin γn cos γn + γn )

(4.42)

where µm and γn are obtained from the equations (4.37) numerically using the regula falsi algorithm [127]. The assumed value for the Neumann boundary conditions is h = 1 and the source term is set as q = 1, for the geometry a = b = 1. The comparison between the numerical and analytical solution is given in fig. 4.4, showing a thoroughly acceptable overall agreement. The greatest discrepancies between the solutions are in the regions where the linear interpolation functions are not able to reproduce varying slopes of the solution function, especially in the middle of the domain.

4.3 4.3.1

Tests of the free surface step Kinematic boundary condition

In this section tests of the free surface algorithms based on the kinematic boundary condition are conducted. The kinematic boundary condition is identical to the twodimensional advection equation and therefore standard tests for this equation type are taken into consideration. The tests are performed for a constant, divergence-free velocity field, and the free surface algorithm is treated entirely separately, i.e. there is no coupling between the velocity field and free surface movements. The tests are performed, however, using the full 3D-mesh, so that the σ-transformation takes place.

120

4.3 Tests of the free surface step

numeric solution

y 1.0 0.65

0. 45

0.60

0. 50 0. 55

0.60

55

0.

analytic solution

0.70

45 0. 50 0. 55 0.

45 0. 50 0. 55 0.

0. 45 0. 50

y 1.0

0.65

0.70

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0 0.8

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-1.0

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0.

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45

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0.

45

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50

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50

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55

55

0. 0.65

55

0.

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-1.0 -1.0

-0.5

0.0

0.5

1.0

-1.0

-0.5

x

0.0

0.5

1.0

x

Figure 4.4: Poisson equation verification. Solution determined by a source term. 4.3.1.1

One-dimensional Gaussian profile advection

In the first test the performance of the algorithm is tested using a well-known example of the advection of a Gaussian profile along the x-axis, i.e. a virtually 1D-example. The numerical solution should not yield any deformation of the profile as it is advected in a constant velocity field. In this test a square (10 × 10 m) basic mesh was used with the resolution of ∆x = 0.2 m (figure A.1). 21 levels are used, resulting in the vertical mesh dimension of 10 m. The mesh resolution along x-axis is 0.2 m, and a constant advection velocity is taken: (u, v, w) = (0.2, 0, 0) m/s (which is not influenced by the profile shape). The Courant number, characterising the problem, C = u∆t/∆x, is changed by adjusting the time step ∆t only. As the initial condition a Gaussian profile y = exp[(x − m)2 /2σ 2 ] centred in m = 2 m and with σ = 0.25 m is imposed. Therefore, the amplitude of the profile is 10% of the vertical mesh dimension in the surrounding area. The performance of the algorithm based on the method of characteristics is presented in figure 4.5. The well-known feature of this method, yielding the best results for whole Courant numbers C and stability independent of C, is confirmed. For C in the range between the whole numbers, considerable numerical diffusion appears due to the spatial interpolation in the finite elements, eventually damping the advected profile. The curves in figure 4.5 are shown for time intervals 8 s, with total computation time of 40 s. The first curve represents the initial condition. Because only the time-step was changed, the total computation time is reached in different numbers of time-steps, from 400 for C = 0.1 down to 20 for C = 2.0. In order to check if the number of time-steps (i.e. time resolution) is additionally responsible for the numerical diffusion for the lower Courant numbers, a test was conducted for constant time-step of ∆t = 0.4 s, and different advection velocities u, equal to 0.05 m/s, 0.1 m/s, and 0.2 m/s (C = 0.4, 0.2 and 0.1). The results are shown in figure 4.6. The up-

4.3 Tests of the free surface step

121

per row of plots show the results for each 20 time-steps (i.e. 8 s) for the global advection time of 40 s. In the lower row, the total advection times are changed accordingly to the value of the advection velocity: 160 s, 80 s, and 40 s, so that a constant advection path length of 10 m is obtained. The profiles are shown after travelling constant advection paths, each 1.6 m long. It may be concluded that the number of time-steps affects the numerical diffusion, but this effect is much less important than the errors connected with the spatial interpolation. For the given initial profile (10% of the water depth) and parameters the semi-implicit SUPG (modified) algorithm for Courant numbers C > 1.6 becomes unstable. For a Gaussian profile of 1% of the water depth, it remains stable for approximately C < 2.0. A comparison of results for different Courant numbers is given in figure 4.7. The SUPG method shows much lower numerical diffusion for lower Courant numbers than the characteristics method. 4.3.1.2

Rotating cone test

A well-known benchmark test for the performance of numerical solutions of the unsteady two-dimensional advection equation is the rotating cone test, also known as the Molenkamp test [162]. In this test a smooth, two-dimensional Gaussian shape is rotated in a divergence-free, rotating velocity field without change of form. The parameters of this test are standardised in the literature. The advection equation for S(x, y, t) is solved in a domain x ∈ [−1, 1] and y ∈ [−1, 1], where the velocity field describes a constant rigid-body rotation, (u, v) = (−ωy, ωx), where ω = 2π. The axis of the rotation is in the middle of the computational domain. The initial condition is given by a Gaussian distribution which extends slightly outside the computational domain: 2

S(x, y, 0) = 0.014r ,

r 2 = (x + 0.5)2 + y 2

(4.43)

This means that the Gaussian hill of the amplitude equal 1 is centred in the position between the rotation axis and the domain edge, (x0 , y0 ) = (−0.5, 0). The velocity magnitude at the top of the hill is π m/s and at the edge 2π m/s. The boundary conditions are given at the boundaries by imposing the exact analytical solution of the problem (Dirichlet BC): 2

S(x, y, 0) = 0.014r ,

r 2 = (x + 0.5 cos ωt)2 + (y + 0.5 sin ωt)2

(4.44)

The numerical solution is required after one revolution, i.e. for t = 1 s. In this particular test a √ square (2 × 2 m) basic mesh is taken with resolution of ∆x = 0.04 m (or ∆x = 0.04 · 2 m diagonally, see figure A.1). 21 levels are used, with the bottom position −1 m (cone amplitude is also equal 1 m). As the characteristic Courant number for this problem, the one at the top of the Gaussian hill is assumed C = u∆t/∆x. The maximal Courant number in the domain equals 2C at

1.0

0.8

0.8

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0.6

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y [m]

C=2.0 C=1.0

0.0

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0.6

0.4

0.4

0.4

0.2

0.2

0.2

0.0 0.0

x [m]

4 2 0 10 8 6 4 2 0

x [m]

4.3 Tests of the free surface step

1.0

10 1.0

8

C=1.6

6 y [m] y [m]

10 8 6 4 2 0

x [m]

x [m]

10 x [m]

8 4

6 x [m]

122

1.0

Figure 4.5: Free surface with characteristics. Solution behaviour depending on value of the Courant number C. Initial profile centred by x = 2 m. Profiles shown for each 8 s for the global advection time of 40 s.

1.0

2 0 10 8 6 4 2 0 10 8 6 4 2 0

0.0 0.0 0.0

C=0.8 C=0.4 C=0.1

y [m] y [m] y [m]

8 x [m]

6 0.0 x [m] 0.4

0.2

0.0

0.4

0.2

0.0

8 6 0

2

4

x [m]

0.05 m/s, 400 t y [m]

y [m]

0

2

4

x [m]

0.05 m/s, 100 t

6

8

10

0.6 0.6

0

2

4

0.1 m/s, 200 t

0.8

0.0 0.0

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y [m]

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10

1.0 1.0

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4

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x [m]

6

8

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0.0 6

8

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1.0

y [m]

0

2

4

0.2 m/s, 100 t

x [m]

6

8

10

123 10

4.3 Tests of the free surface step

Figure 4.6: Free surface with characteristics. Solution behaviour depending on the number of constant time-steps (∆t = 0.4 s) for various advection velocities u = 0.05 m/s, 0.1 m/s, and 0.2 m/s (C = 0.4, 0.2 and 0.1). Initial profile centred by x = 2 m. The profiles shown: above – for each 20 time-steps (8 s), below – for each 1.6 m distance travelled (each 80, 40 and 20 timesteps, respectively).

C=0.2, characteristics

C=0.4, characteristics

y [m]

y [m]

1.0

1.0

1.0

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2

4

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8

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2

4

x [m]

6

8

10

0

C=0.1, SUPG

C=0.2, SUPG

0.8

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8

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8

10

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2

4

6 x [m]

8

10

0

2

4 x [m]

4.3 Tests of the free surface step

1.0

4

8

C=0.4, SUPG

1.0

0.0

6

y [m]

1.0

2

4 x [m]

y [m]

0

2

x [m]

y [m]

124

Figure 4.7: Comparison between the method of characteristics and SUPG. Solution behaviour depending on value of the Courant number C. Initial profile centred by x = 2 m. Profiles shown for each 8 s for the global advection time of 40 s.

C=0.1, characteristics y [m]

4.3 Tests of the free surface step

Characteristics C = u∆t/∆x 0.1 0.2 0.5 1.0 2.0

125

Semi-implicit Peak value C = u∆t/∆x 0.3070 0.1 0.3180 0.2 0.3648 0.5 0.5695 0.8 0.7072 0.9 1.0 1.6

SUPG Peak value 0.9668 0.9463 0.8886 0.8365 0.8190 0.8061 0.7354

Table 4.1: Rotating profile test. Reduction of the initial peak value 0.9927 after one revolution for methods based on kinematic BC.

the boundary, minimum 0 at the axis. The performance of both free surface algorithms based on the kinematic boundary condition is tested for different Courant numbers, varied between 0.1 and 2 by adjusting the time step only, resulting in time-steps varying between 0.00125 s and 0.025 s and time-step numbers between 40 and 800, respectively. The difficulty of this test is due to the following factors: (1) The Courant number changes along the distance between the axis of the rotation and the domain boundary, i.e. it is different in various parts of the advected profile; (2) the direction of advection relative to the mesh elements changes along the way; (3) the slope of the profile is relatively large; (4) the boundary conditions and the specific condition (u, v) = (0, 0) in the middle of the domain interfere with the solution as the numeric diffusion takes place. The worst and the best results for both methods are shown in figures 4.8 and 4.9 for the method of characteristics, and in figures 4.11 and 4.10 for modified semi-implicit SUPG (θ = 0.55), respectively. Additionally, a table with results for various C is provided (Table 4.1). The initial peak value by the given discretisation is 0.9927 m. Both methods for the given range of Courant numbers are free of oscillations and the general shape of the advected profile remains. However, large numerical diffusion for lower Courant numbers for the method of characteristics disqualifies it in comparison with SUPG. The SUPG method brings better results for lower Courant numbers and comparable results for higher ones, but with C > 1.6 the shape of the profile is no longer maintained and oscillations occur disqualifying the solution for C ≈ 2. For the implicit SUPG method, the peak value is much more damped, e.g. for C = 1.6, the peak value is ca. 50% lower, 0.3542. The practical test conclusions are that for the lower Courant numbers (and larger number of time-steps) the SUPG method is preferred. However, for larger Courant numbers the method of characteristics remains the only one available. Not surprisingly, the performance of this method is best for whole or higher Courant numbers and lower number of time-steps.

126

4.3 Tests of the free surface step

initial condition

after rotation

Figure 4.8: Rotating profile test, kinematic BC, method of characteristics, C = 0.1. The maximum value after one rotation is 0.307.

initial condition

after rotation

Figure 4.9: Rotating profile test, kinematic BC, method of characteristics, C = 2.0. The maximum value after one rotation is 0.707.

4.3 Tests of the free surface step

initial condition

127

after rotation

Figure 4.10: Rotating profile test, kinematic BC, SUPG, C = 0.1. The maximum value after one rotation is 0.967.

initial condition

after rotation

Figure 4.11: Rotating profile test, kinematic BC, SUPG, C = 1.6. The maximum value after one rotation is 0.736.

128

4.3 Tests of the free surface step

initial condition

after rotation

Figure 4.12: Rotating profile test, entirely explicit conservative free surface equation, C = 0.5. The maximum value after one rotation is 0.369, the minimum value is -0.122.

4.3.2

Conservative free surface equation

The algorithm for finding the free surface elevation from the conservative free surface equation was tested using the rotating cone test for identical parameters as in the previous section. The test disqualified the application of this algorithm in its entirely explicit version (section 3.6.3) in comparison to the kinematic boundary condition with the method of characteristics and SUPG. The scheme proves to be numerically very diffusive for lower Courant numbers (for C = 0.1 the peak was damped to 0.044 from the initial value 1.0, and for C = 0.2 to 0.107) and unstable beginning from Courant number C = 0.5 (peak value of 0.369 by oscillations amplitude down to -0.122, figure 4.12). For the implicit and semi-implicit formulation for the conservative free surface equation (section 3.6.4), the results are slightly better than obtained with the methods based on SUPG for kinematic boundary condition, table 4.2. The computations with semiimplicit (θ = 0.55) and implicit formulations remain stable for large Courant numbers, with growing damping of the amplitude and numerical oscillations. By θ = 0 (explicit formulation) strong oscillations appear already by C = 0.2 leading to a program breakup. As a consequence, only the semi-implicit SUPG version of the method based on the conservative free surface equation is adequate for practical purposes. Additionally, this scheme has better mass-conserving properties than the SUPG method based on kinematic boundary condition, as investigated further in section 5.5.4.

4.3 Tests of the free surface step

C = u∆t/∆x 0.1 0.2 0.5 0.8 1.0 2.0 5.0

129

Semi-implicit Peak value 0.9602 0.9467 0.8880 0.8369 0.8060 0.6993 0.5101

Implicit Peak value 0.8098

0.4705 0.4308 0.3213 0.2140

Table 4.2: Rotating profile test. Reduction of the initial peak value 0.9927 after one revolution for the conservative free surface equation with SUPG.

initial condition

after rotation

Figure 4.13: Rotating profile test, semi-implicit conservative free surface equation with SUPG, C = 0.1. The maximum value after one rotation is 0.960.

130

4.3 Tests of the free surface step

initial condition

after rotation

Figure 4.14: Rotating profile test, semi-implicit conservative free surface equation with SUPG, C = 2.0. The maximum value after one rotation is 0.699.

Chapter 5 Model verification Nothing shocks me, I am a scientist. Indiana Jones After the verification operator-splitting algorithm steps as separate units (previous chapter), this chapter considers the verification of the entire algorithm. The broad range of event-oriented test cases presented in this chapter is intended to cover the intended extension of the initial hydrostatic model application domain. The verification cases document the functionality of the developed model as well as provide the evidence for the assessment of realised numerical developments. The features of the new algorithm are illustrated in the most convincing way performing simulations of those physical systems, where a solution based on the hydrostatic approximation is not adequate (physically infeasible), though mathematically possible. Two examples of this type are provided without aiming for formal verification, but immediately showing the new model’s capabilities: waves reflection and interaction with topography, and a wind driven vertical circulation in a closed basin. For internal flows, the lock exchange flow as well as the interfacial internal waves of nonhydrostatic nature are well suited to verification of model performance. The obtained results are compared partially with the existing theory. The best benchmark tests allowing the strict verification using analytical solutions are probably small-amplitude short surface waves, impossible to reproduce properly with a hydrostatic model. The analytical solutions make it possible not only to compare the wave periods, but velocity and pressure profiles as well. As a standard test case, a standing wave in a closed basin is considered. This test is also a good example for the assessment of the model’s conservative properties and its stability. For completeness, the performance of the non-hydrostatic model is tested also for the hydrostatic (shallowwater) case of long standing waves. For channel flow, the subcritical and especially the supercritical flow with free surface over a ramp, provide an excellent verification case for the free surface algorithm. For this test case no solutions with the hydrostatic model can be obtained: in the subcritical 131

132

5.1 Waves reflection

case due to the bottom slope larger than 1:10, and for the supercritical case additionally due to the necessity of taking hydrodynamic pressure into consideration. The complex boundary conditions needed for the free surface computations for wave problems or where disturbances generated inside the domain must leave it through the boundaries (absorbing or non-reflecting BC) are tested in a channel flow example with a bump at the bottom and in case of waves travelling over an underwater channel. The ability of the model to describe the propagation of the non-linear waves of a finite amplitude is tested in a few examples concerning solitary waves. In ideal conditions, the solitary waves propagate over a flat bottom without changing their properties. The accuracy of the model is evaluated by comparison of the wave amplitude, shape and celerity as it travels. Wave-wave interactions and the influence of a changeable bottom are additionally tested. Additionally, a few examples testing the limits of applications (free surface and bottom steepness) are presented.

5.1

Waves reflection

This section yields an immediate check whether the non-hydrostatic algorithm reproduces short (deep water) motion in a fully three-dimensional problem. A square basin with a side of 100 m and a depth of 10 m is filled with a non-viscous fluid. The initial shape of the free surface S(x, y, t = 0) is a two-dimensional cosine profile with a peak of height η0 = 0.2 m and width of D = 40 m located exactly in the middle of the basin (x0 , y0 ) = (50, 50) m: S(x, y, 0) = η0 [1 + cos(2πr/D)]

r = [(x − x0 )2 + (y − y0 )2 )]1/2

(5.1)

The aim of this test is to observe the wave patterns developing for t > 0 and their reflection from the boundaries. The square basic mesh has 5000 regular triangular elements and a resolution of approx. 2m, and 21 mesh levels with equal distribution (figure A.1). In the hydrostatic case, the convection and diffusion steps of the algorithm are switched off, so that only the pressure-continuity step is executed. In the non-hydrostatic case, the diffusion coefficients are set to 0 and convection step is excluded. In consequence, Euler equations are solved. The results are shown in figures 5.1 and 5.2 for hydrostatic and non-hydrostatic case, respectively. The resulting wave patterns for 5, 6 and 10 s differ much due to various wave celerity in both models. The hydrostatic model treats all occurring waves as long waves with a celerity of c = (gh)1/2 ≈ 10.0 m/s, (gravitational acceleration g is set to 10.0 m/s2 for interpretation simplicity) whereby the non-hydrostatic one resolves the short wave motion with lower celerity (see appendix A.3). This simple test case immediately shows new properties of the non-hydrostatic model – the ability to simulate short free surface waves. One of the most interesting practical applications of this kind is the wave propagation in harbour basins, especially in connection with sediment erosion, deposition and transport

5.0s 100

S [m]

S [m]

y [m]

60

40

20

0.050 0.045 0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.005 -0.000 -0.005 -0.010 -0.015 -0.020 -0.025 -0.030 -0.035 -0.040 -0.045 -0.050

80

60 y [m]

0.200 0.180 0.170 0.160 0.150 0.140 0.130 0.120 0.110 0.100 0.090 0.080 0.070 0.060 0.050 0.040 0.030 0.020 0.010 0.000

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80

60 y [m]

0.050 0.045 0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.005 -0.000 -0.005 -0.010 -0.015 -0.020 -0.025 -0.030 -0.035 -0.040 -0.045 -0.050

80

y [m]

Figure 5.1: Wave patterns computed with hydrostatic approximation.

100

5.1 Waves reflection

initial condition

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initial condition

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0.200 0.180 0.170 0.160 0.150 0.140 0.130 0.120 0.110 0.100 0.090 0.080 0.070 0.060 0.050 0.040 0.030 0.020 0.010 0.000

80

0.050 0.045 0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.005 -0.000 -0.005 -0.010 -0.015 -0.020 -0.025 -0.030 -0.035 -0.040 -0.045 -0.050

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80

60 y [m]

0.050 0.045 0.040 0.035 0.030 0.025 0.020 0.015 0.010 0.005 -0.000 -0.005 -0.010 -0.015 -0.020 -0.025 -0.030 -0.035 -0.040 -0.045 -0.050

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100

5.1 Waves reflection

Figure 5.2: Wave patterns computed without hydrostatic approximation.

100

5.1 Waves reflection

135

x [m] 0

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0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 -0.20 -0.25 -0.30 -0.35 -0.40 -0.45 -0.50

0

Figure 5.3: Wave pattern in a harbour basin for t = 50 s, ∆x = 1 m. Waves with T = 4 s, λ = 24.7 m, c = 9.2 m/s. x [m] 0

50

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[m]

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0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 -0.20 -0.25 -0.30 -0.35 -0.40 -0.45 -0.50

0

Figure 5.4: Wave pattern in a harbour basin for t = 50 s, ∆x = 1 m. Waves with T = 10 s, λ = 92.4 m, c = 6.2 m/s.

136

5.2 Wind-driven circulation

by currents. (In principle, the newly developed algorithm allows the simulation of all these phenomena in one model.) A simple example of such an application is shown in figures 5.3 and 5.4. The mesh for this test case is obtained from the square mesh shown in figure A.1 by cutting appropriate areas out. In the first case, figure 5.3, sinusoidal waves of a period of T = 4 s and amplitude of η0 = 0.5 m enter a small harbour basin of a typical geometry through the southern entrance. The mesh resolution is ∆x = 1 m. The water depth is everywhere equal to 10 m. Under this conditions, the waves have the length λ = 24.7 m and phase speed c = 6.2 m/s (values computed with shallow water approximation are 39.6 m and 9.9 m/s, respectively). Water viscosity is assumed to be isotropic and equal to ν = 10−4 m2 /s. Dynamic BC are applied at the free surface. Figure 5.3 shows the free surface configuration at t = 50 s from the begin of the wave action. In figure 5.4 waves of a period of T = 10 s enter a five times larger basin (mesh resolution of ∆x = 5 m). This time, the wavelength is λ = 92.4 m and phase speed c = 9.2 m/s (values computed with shallow water approximation are 99.0 m and 9.9 m/s, respectively). The free surface configuration is shown at t = 50 s. In both cases the model reproduces well the typical phenomena such as: interaction of incident and reflected waves at the diagonal north-eastern wall forming a characteristic chessboard pattern, or wave diffraction at the western end of the entrance channel interfering with the incoming waves.

5.2

Wind-driven circulation

Another example illustrating the differences between the hydrostatic and non-hydrostatic computation is the well known example of a wind-driven circulation in a closed basin. In this case the vertical circulation in a deep water body caused by wind shear at the free surface is treated (i.e. a Neumann BC for the velocity is given). A pool measuring 10 m × 10 m and with a depth of 5 m is considered, with a strong wind blowing over the surface in the x-direction. The square basic mesh is shown in figure A.1; 21 evenly distributed levels are used. The horizontal resolution is 0.2 m and vertical 0.5 m. The wind stress is computed according to the empirical formulation (2.189). The wind velocity is taken to equal 10 m/s. The constant horizontal and vertical eddy viscosities are identical in value and equal to 0.001 m2 /s. The Coriolis force is neglected. As the initial condition null velocity in the entire domain is taken. The free surface is computed using the Telemac2D algorithm (section 2.8.8). The integration time is 1000 s (steady state achieved) with the time step of 1 s. The results are shown in figure 5.5. The hydrostatic computation yields an intensive vertical circulation near the wall. The flow seems to be strongly accelerated along the wall without presence of any vertical pressure gradient. This is an effect of obtaining the vertical velocity component from the continuity equation only. The fluid is accelerated along the fetch path of the wind (a Neumann BC) in the horizontal direction, but on the wall a Dirichlet impermeability condition sets the horizontal velocity component to zero. Due to the continuity equation ∇ · u = 0, large ∂u/∂x causes large ∂w/∂z, which

5.2 Wind-driven circulation

137

hydrostatic 0

-1

z [m]

-2

-3

-4

v [m/s] 0.1

-5 0

1

2

3

4

5

6

7

8

9

10

x [m]

non-hydrostatic 0

-1

z [m]

-2

-3

-4

v [m/s] 0.1

-5 0

1

2

3

4

5

6

7

8

9

x [m]

Figure 5.5: Wind-driven circulation test case.

10

138

5.3 Interfacial internal waves

explains the circulation pattern near the wall. This strong effect can be weakened when larger fluid viscosities are taken, smoothing the velocity gradients. The non-hydrostatic model results in a much more widespread circulation over the entire area of the basin. The presence of hydrodynamic pressure gradients before the wall decelerates the fluid in a much smoother way than in the previous case, when the only mechanism to diminish the velocity gradients was provided only by fluid viscosity. The hydrodynamic pressure controls also the vertical circulation pattern, which takes a much more natural form. This test case shows in a very illustrative way the role played by the hydrodynamic pressure when an interaction of fluid with a wall or structure and flows in the form of a vertical circulation is considered.

5.3

Interfacial internal waves

5.3.1

Theory

In this section waves appearing at the interface between two fluids of different density are considered. The stratification is not continuous, so that a sharp step in fluid densities appears at the interface. It is assumed that the stratification is stable, so that the density of the lower layer ̺d is greater than the density of the upper layer ̺u with a free surface (̺d > ̺u ). In the undisturbed state the thickness of the upper layer is Hu , the lower Hd and the position of the undisturbed free surface is z = 0, the undisturbed interface z = −Hu and of the flat bottom z = −H = −(Hu +Hd ). The problem was treated already by Stokes in 1847 [147]. The discussion of linearised, small perturbation equations for both layers under assumption of pressure and velocity continuity at the interface [147, 96, 92], yields a fourth-order dispersion relationship (characteristic equation for the wave celerity c) in the form (k = 2π/λ is the wave number): g g2 c4 [̺u +̺d coth(kHu ) coth(kHd )]−c2 ̺d [coth(kHu )+coth(kHd )]+(̺d −̺u ) 2 = 0 (5.2) k k which has two solutions (eigenvalues of the problem) c1,2 [92]: c1,2 =

g̺d [coth(kHu ) + coth(kHd )] 2k[̺u + ̺d coth(kHu ) coth(kHd )]

(5.3)

#1/2

(5.4)

̺2d [coth(kHu ) + coth(kHd )]2 ̺d − ̺u ± − 2 4[̺u + ̺d coth(kHu ) coth(kHd )] [̺u + ̺d coth(kHu ) coth(kHd )] "

One of the roots (5.3) is connected with the free surface movements (c1 ), the other (c2 ) with the interface wave. Assuming that (̺d −̺u )/̺d << 1 (as usually found in nature), the dispersion relationship (5.2) can be transformed into the following approximate form (exact to O((̺d − ̺u )/̺d ): 

"

g c − tanh[k(Hu + Hd )] k 2

#

g ̺d − ̺u =0 c − k ̺u coth(kHu ) + ̺d coth(kHd ) 2

(5.5)

5.3 Interfacial internal waves

139

showing two roots clearly. The first one is connected clearly with the gravity waves in the water of total depth H = Hu + Hd , unaffected by stratification (because (̺d − ̺u )/̺d << 1). The second root is the dispersion relationship of the waves at the interface. ci are the celerities (phase speeds) of these wave movements. For a rigid lid boundary condition at the surface this root is exact, but when movement of the free surface is allowed, it is correct only to O((̺d − ̺u )/̺d ).

Discussion of (5.2) and (5.3) can be conducted in categories of thickness of the layers, in typical configurations to be found in the nature, compared to the wave length at the interface (figure 5.6). 1. If the thicknesses of both layers Hd and Hu are much greater than the interface wave length λ = 2π/k, then coth(kHd ) ≈ 1 and coth(kHu ) ≈ 1 and from (5.2): c21 =

g k

c22 =

g ̺d − ̺u k ̺u + ̺d

(5.6)

It should be noted that in practice coth(kHd ) → 1 and coth(kHu ) → 1 already for kHu > π/2 and kHd > π/2. As a rule of thumb, Hd,u > 2λ. This case, with the interface away from the free surface and the bottom, is clearly the short wave or deep water approximation for interfacial waves. The short waves at the interface are dispersive. By absence of the free surface mode, the velocity values vanish exponentially with the vertical distance from the interface with an asymmetry depending on the density difference [92, 40]. 2. If the wave length λ is much greater than both layer thicknesses Hu and Hd (shallow water), then coth(kHu ) → 1/kHu and coth(kHd ) → 1/kHd and from (5.3) the following non-dispersive case can be obtained: c21 = g(Hd + Hu ) = gH

c22 = gHd Hu

̺d − ̺u ̺d Hd + ̺u Hu

(5.7)

The celerity of the internal mode c2 can be further simplified if ̺d ≈ ̺u : c22 = g

̺d − ̺u Hd Hu ̺d Hd + Hu

(5.8)

When the deeper layer is also much thicker than the upper one, Hd >> Hu : c22 = g

̺d − ̺u Hu ̺d

(5.9)

The last case is clearly the non-dispersive long wave (or shallow water) approximation. For practical applications of this approximation, it should be noted that coth(kHu,d ) ≈ 1/kHu,d already for λ > 20Hu,d. When the densities in both layers are similar, (̺d − ̺u )/̺d << 1 and the deeper layer is much thicker than the upper one (Hd >> Hu ), then: ̺d − ̺u Hu (5.10) c22 = g ̺u

140

5.3 Interfacial internal waves z

6

(1)

6

̺u

Hu

λ

0 λ << Hu λ << Hd

? 6

Hd

̺d

?

−(Hu + Hd ) λ

(2a)

6

Hu

? 6

λ >> Hu λ >> Hd

Hd

?

(2b)

6

Hu ? 6

λ >> Hu λ >> Hd Hd >> Hu

Hd

?

(3/4)

6

λ

Hu

λ << Hd Hu arbitrary

? 6

Hd

?

Figure 5.6: Various configurations of interfacial internal waves. (Explanations in text.)

5.3 Interfacial internal waves

141

and the long wave movement is analogous to the non-dispersive long surface wave in homogeneous water, but with reduced gravity acceleration g ′ = g(̺d − ̺u )/̺u . Accordingly, by absence of the free surface mode, the profiles of the horizontal velocity do not change with the depth. 3. If the thickness of the lower layer Hd is much greater than the interface wave length λ (Hu arbitrary), coth(kHd ) ≈ 1 and from (5.2): c21 =

g k

c22 =

g ̺d − ̺u k ̺u + ̺d coth(kHu )

(5.11)

4. If a very thin upper layer Hu < λ/20 lies over a thick, dense layer Hd > λ/2, coth(kHu ) → 1/kHu and coth(kHu ) ≈ 1, (5.2) yields: c21 =

g k

c22 = gHu

̺d − ̺u ̺d + ̺u kHu

(5.12)

It can be shown that using as a starting point the small perturbation equation with the hydrostatic approximation, the non-dispersive case of interfacial internal waves can be obtained [53], which proves the equivalency of long wave and hydrostatic approximations in this case. The celerity of the interface waves are always smaller than the celerity of the surface ones, with the most striking example of (5.10).

5.3.2

Test realisation

For testing the performance of the non-hydrostatic algorithm the dispersive, short interfacial wave example is taken. This is clearly the non-hydrostatic case, when both layer thicknesses, below and above the interface, are larger than the wave length. A standing uni-nodal interfacial wave in a closed basin of length 1 m (i.e. wave length 2 m) and water depth 10 m with both layer depths of 5 m has a sinusoidal initial shape with an amplitude of 1 m. The initial velocity is assumed to be zero everywhere. The lower, slightly denser layer has a salinity of 1 PSU and the layer above 0 PSU (see section A.4). The densities of both fluids are computed with a linear formula (5.31) and equal 999.972 kg/m3 and 1000.722 kg/m3 , respectively. The Boussinesq reference density is 1000.347 kg/m3 . The mesh used for computations has 2114 base elements and 1147 base nodes, resulting in horizontal resolution of ≈ 0.01333 m. 51 mesh levels have been taken, distributed more densely in the area of the interface movement. The viscosity of the fluid and salinity diffusion coefficients are equal in all directions, ν = 10−2 m2 /s and time-step 0.1 s. For hydrodynamic pressure, a zero Neumann BC is applied at all boundaries. ∂w/∂z = 0 is given at the free surface (computed from kinematic BC with semi-implicit SUPG method). The wave celerity computed with (5.6) is 0.0342 m/s and the period of oscillation is 58.5 s. When the formula (5.7) is used, values of 0.136 m/s and 14.74 s are obtained, respectively.

142

5.3 Interfacial internal waves

-2

-2

-4

-4

z [m]

0

z [m]

0

-6

-6

-8

-8 u [m/s] 0.5

-10

-10 0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

x [m]

0.6

0.8

1.0

x [m]

Figure 5.7: A short interfacial internal wave by t = 3.7 s (1/4 of the theoretical long wave period) computed with hydrostatic approximation (physically false). Left: Velocity iso-lines, u: dashed line, step value 0.005 m/s, w: chain-dotted line, step value 0.02 m/s. The position of the interface shown with solid lines. Right: velocity vectors.

-2

-2

-4

-4

z [m]

0

z [m]

0

-6

-6

-8

-8 u [m/s] 0.01

-10

-10 0.0

0.2

0.4

0.6 x [m]

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

x [m]

Figure 5.8: A short interfacial internal wave by t = 15 s (1/4 of the theoretical short wave period) computed without hydrostatic approximation (physically correct). Left: Velocity iso-lines, u: dashed line, step value 0.0005 m/s, w: chain-dotted line, step value 0.002 m/s. The position of the interface shown with solid lines. Right: velocity vectors.

5.4 Lock exchange flow

143

The results obtained with and without hydrostatic approximation for about 1/4 of the wave oscillation period (3.7 s and 15 s, respectively), are strikingly different. While in the (physically correct) non-hydrostatic case the velocity diminishes exponentially with the distance from the interface, this is not the case in the hydrostatic (physically false) case, compare figures 5.7 and 5.8. Neither model reproduces the wave period properly (the periods are about 15% larger), due to softening of the interface profile during simulation and viscosity influence. As a conclusion, it can be stated that the model reproduces satisfactorily the interfacial internal waves in the non-hydrostatic mode. However, for fully successful applications, techniques influencing the diffusion and viscosity coefficients, which reduce strongly the interface sharpness, should be investigated. First, it could be an appropriate turbulence model (constant diffusion coefficients are applied here); second, a method to follow the internal interface (e.g. based on separate σ-transformations below and above the interface).

5.4

Lock exchange flow

The lock exchange flow is a well-known test case to verify the modelling of buoyancydriven (gravity, density) flows or currents. Two incompressible fluids of slightly different density confined in a rectangular basin, with impermeable walls and a free surface, are initially divided by a very thin wall, figure 5.9. The left half of the basin is filled by fluid of higher density, and the right half by a fluid of lower density. In the initial moment of time the impermeable thin wall in the middle of the basin is removed and a buoyancydriven flow develops. A mutual intrusion of both fluids takes place. The denser fluid enters the area occupied initially by the lighter one along the bottom, and the lighter one does the same along the surface in the opposite direction. Eventually, after the interfacial wave has dissipated its energy, an equilibrium state will be achieved with two layers of equal thickness, the lighter fluid over the denser one. The name of this test case originates from the practical engineering problem concerning intrusion of a salt water wedge under fresh water when a lock gate is opened at the mouth of a fresh water channel leading to the sea. There are numerous other examples of gravity currents of this type in nature e.g., on a larger scale, motion of a cold front in a warmer atmosphere, or when a muddy stream enters a lake. Salt water wedges in estuaries are another example. The shape of the interface between the fluids is very different, when the flow simulation is carried out with or without hydrostatic approximation. The hydrostatic solution yields a characteristic rectangular pattern which is not observed in nature or in the laboratory, where the noses of density currents advancing along a flat bottom have a slope of approximately π/3 to the horizontal [159].

144

6

5.4 Lock exchange flow

z

̺2

t=0

̺1

6

x

z 

̺2

t>0

̺1 -

-

x

Figure 5.9: Lock exchange problem illustration.

5.4.1

Hydrostatic interpretation

The behaviour of the hydrostatic solution of this test case problem concerning a twofluid system can be understood by analysing the solution of the shallow water wave equation (i.e. hydrostatic) for an initial free surface displacement in the form of a step. This section explains why the interface shape in the hydrostatic solution takes the characteristic rectangular shape. Integration of the continuity equation in the vertical direction, as performed in subsection 2.6.3.2, leads to the conservative form of the free surface equation (2.159): ∂S ∂ + ∂t ∂x

∂ udz + ∂y −B

Z

S

Z

S

−B

vdz = 0

(5.13)

For long waves, the horizontal velocity component (A.44) is independent of depth (section A.3) and (5.13) transforms to: ∂S ∂ ∂ + [u(H + S)] + [v(H + S)] = 0 ∂t ∂x ∂y

(5.14)

For small perturbation of the free surface compared with the water depth H + S → H and one obtains: ∂S ∂ ∂ + (uH) + (vH) = 0 (5.15) ∂t ∂x ∂y The perturbation pressure in the long wave approximation is hydrostatic: p′ = ̺gS

(5.16)

5.4 Lock exchange flow

145

Introducing (5.16) to equations (A.20): ∂u/∂t = −g∂S/∂x ∂v/∂t = −g∂S/∂y

(5.17) (5.18)

Combining these equations with (5.15) leads to a wave equation: ∂S ∂ ∂2S gH − 2 ∂t ∂x ∂x

!

∂ ∂S − gH ∂y ∂y

!

=0

(5.19)

which in the particular case of constant water depth H transforms to a Laplace equation: 2 ∂2S ∂2S 2 ∂ S = c − ∂t2 ∂x2 ∂y 2

!

(5.20)

where c = (gh)1/2 is the celerity of long waves. The equation (5.20) has simple solutions, if there is no dependence on y. In this case for the initial free surface displacement being a function of x only, S = F (x) and the fluid being initially at rest, the time-dependent solution of (5.20) is: 1 S = [F (x + ct) + F (x − ct)] 2

(5.21)

1 u = − c−1 g[F (x + ct) − F (x − ct)] 2

(5.22)

and the velocity from (5.17):

Two important examples can illustrate the behaviour of the hydrostatic solution of the lock exchange flow. Both describe propagation of an initial displacement of the free surface in a form of a step. In case (a) the initial free surface displacement is described as a step function: F (x) = η0 sgn(x)

(5.23)

where sgn is the sign function: sgn(x) =

(

1, x>0 −1, x < 0

(5.24)

In this case a pair of wavefronts of height η0 are produced. In case (b) the initial free surface perturbation is confined to a limited area x ∈ [−L, L]: S(x) =

(

η0 , |x| < L 0, |x| > L

(5.25)

146

5.4 Lock exchange flow z

(a)

x

2η0 c

@

-c @

x

η0

z

(b)

η0 @

c

-c @ @

c

x

@ @

-c @

η0 /2 x

Figure 5.10: Solutions of the shallow water wave equations for (a) initial disturbance in the form of a single step function, (b) in the form of a confined displacement. Descriptions in the text.

producing two pairs of wavefronts. Both cases are shown schematically in fig. 5.10. These examples are especially important for understanding qualitatively the solution of the lock exchange problem in the hydrostatic case. In case (a) two wavefronts of relative height η0 move out in different directions from the position of the initial displacement with a celerity c = (gη0 )1/2 , leaving behind a region confined between x = −ct and x = ct with zero displacement, but with a steady motion with velocity u = c−1 gη0 directed to the right (from equation 5.22). In consequence u = c. In case (b) two pairs of wavefronts are produced, moving in opposite directions away from the positions of the initial free surface displacement. The velocity is zero except in those regions, where the surface elevation is η0 /2, and movement is directed away from the axis of symmetry, where a wall can be placed without influencing the movement. The characteristic rectangular shape of the interface is therefore a consequence of the solutions shown in fig. 5.10. If the lock exchange problem is solved using hydrostatic approximation, this shape of the interface should be observed.

5.4 Lock exchange flow

5.4.2

147

Lock exchange flow speed

The velocity with which the buoyancy-driven current advances can be estimated from energy conservation considerations. When the wall between two equal volumes of fluids of different density in a basin is removed, the instability causes a movement of the volumes toward a position of equilibrium, where a layer of lighter fluid lies above the denser one. Assuming the flow to be conservative in terms of the mechanical energy, the velocity of the motion can be estimated from the available potential energy being the difference between the potential energy of the initial state and the equilibrium. Assuming the base of the container (basin) to be a square of a width L and the water depth H (figure 5.11), the potential energy of the initial state per unit area of the base is: Z L/2 Z H 1 −1 ̺gzdxdz = (̺1 + ̺2 )gH 2 Ei = L (5.26) 4 −L/2 0 where ̺1 and ̺2 are densities of both fluids and the reference level for the potential energy is the bottom of the basin with the origin of the co-ordinate system at the interface. Computing the energy of the equilibrium state in the same way: 1 1 3 Ee = ( ̺1 + ̺2 )gH 2 4 2 2

(5.27)

The available potential energy is: 1 ∆E = Ei − Ee = (̺2 − ̺1 )gH 2 8

(5.28)

If all this energy can be converted into kinetic energy, the mean velocity of the movement will be equal to: 1 ̺2 − ̺1 gH (5.29) u2 = 2 ̺1 + ̺2 The above equation can be used to estimate approximately the velocity of the flow. Analogical treatment can be given a fully developed and therefore stationary lock exchange flow, when a perfect symmetry between the two layers of lighter and denser fluids of equal thickness can be assumed (figure 5.9). In a time interval ∆t, the wedges advance a distance of u∆t. In this time the lighter fluid gains potential energy amounting to ̺1 g(H/4)u∆t(H/2) per width unit of the channel; simultaneously, the heavier fluid loses energy amounting to ̺2 g(H/4)u∆t(H/2). The kinetic energy gained by both fluids is (1/2)(̺1 + ̺2 )u2 (H/2)u∆t. Again, when all available potential energy can be changed to the kinetic form, formula (5.29) is obtained [167] again: √ s s s q 2 ̺2 − ̺1 ̺2 − ̺1 ̺2 − ̺1 u= gH = 0.71 gH ≈ 0.5 gH = 0.5 g ′H (5.30) 2 ̺1 + ̺2 ̺1 + ̺2 ̺2 where g ′ = g∆̺/̺2 is the reduced gravity acceleration. This result should be compared with the formula for long, hydrostatic interfacial wave celerity (5.7) – it is virtually the same. This confirms also the considerations from the section 5.4.1.

148

5.4 Lock exchange flow



L

-

6

H

̺2

̺1

̺1

̺2

̺2

(ii)

(iii)

aa aa aa aa aa

̺1

?

(i)

Figure 5.11: Initial and equilibrium state for the lock exchange problem.

5.4.3

Lock exchange flow – further considerations

Numerous measurements have been made for the flow generated by removal of a vertical barrier separating two layers of equal depth and slightly different densities in completely enclosed conduits and with channels with a free surface [4, 167]. The laboratory experiments of Yih made in 1947 yield the stationary flow velocity as given by the formula (5.30) where instead of 0.71 the √ factor 0.67 appears [167]. Barr observed in a closed ′ H. When the free surface is present, the flow is conduit the velocity of u = 0.44 g√ √ slightly asymmetrical with u = 0.47 g ′H for the underflow and u = 0.59 g ′H for the overflow. Von K´arm´an [88] and Benjamin [7] computed the interface profiles showing the characteristic nose angle of π/3. Benjamin discusses the case of lock exchange flow showing that only in the case of a flow where both wedges have identical geometries a steady nondissipative flow can occur; otherwise mixing, waves, and/or internal hydraulic jumps can occur. The flow speed can be estimated from considerations concerning the mechanical energy conservation in the case of equal depths only, as presented in section 5.11.

5.4.4

Test realisation

The comparisons are carried out in two computational settings which correspond with both cases shown in fig. 5.10, in a channel-like basin. The salinity is taken as the (transported) active tracer influencing fluid density. The dependence of the density ̺ [kg/m3 ] on the salinity s [psu] is assumed to be linear (appendix section A.4.2): ̺ = 999.972 · (1 + 0.75 × 10−3 s)

(5.31)

The salinity is initially so distributed, that the left half of the basin is occupied by saline water with s = 1 psu, and the right one by fresh water (s = 0 psu). The density of both fluids and the mean reference density are 999.972 kg/m3 , 1000.722 kg/m3 , and 1000.347 kg/m3 respectively. In the initial condition the velocity in the whole domain

salinity

z [m]

s [PSU] -2 0.90 0.70 0.50 0.30 0.10

-3

-4 0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x [m] 0

tangential velocity component

-1 z [m]

u [m/s] 0.100 0.060 0.020 -0.020 -0.060 -0.100

-2

-3

-4 0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x [m] 0

vertical velocity component

-1 z [m]

Figure 5.12: Lock exchange problem. The results of the hydrostatic model.

-1

5.4 Lock exchange flow

0

w [m/s] -2 0.200 0.100 0.000 -0.100 -0.200

-3

-4 2

4

6

8

10

12

14

16 x [m]

18

20

22

24

26

28

30

149

0

150

0

salinity

z [m]

s [PSU] -2 0.90 0.70 0.50 0.30 0.10

-3

-4 0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x [m] 0

tangential velocity component

-1 z [m]

u [m/s] 0.100 0.060 0.020 -0.020 -0.060 -0.100

-2

-3

-4 0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x [m] 0

vertical velocity component

z [m]

-1 w [m/s] -2 0.040 0.020 -0.000 -0.020 -0.040

-3

-4 0

2

4

6

8

10

12

14

16 x [m]

18

20

22

24

26

28

30

5.4 Lock exchange flow

Figure 5.13: Lock exchange problem. The results of the non-hydrostatic model.

-1

salinity

z [m]

-1 s [PSU] -2 0.90 0.70 0.50 0.30 0.10

-3

5.4 Lock exchange flow

-4 0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x [m] 0

tangential velocity component

-1 z [m]

u [m/s] 0.100 0.060 0.020 -0.020 -0.060 -0.100

-2

-3

-4 0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x [m] 0

vertical velocity component

z [m]

-1 w [m/s] -2 0.200 0.100 0.000 -0.100 -0.200

-3

-4 0

2

4

6

8

10

12

14

16 x [m]

18

20

22

24

26

28

30

151

Figure 5.14: Lock exchange problem with two interfaces. The results of the hydrostatic model.

0

152

salinity

z [m]

-1 s [PSU] -2 0.90 0.70 0.50 0.30 0.10

-3

-4 0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x [m] 0

tangential velocity component

-1 u [m/s] z [m]

Figure 5.15: Lock exchange problem with two interfaces. hydrostatic model.

0

0.100 0.060 0.020 -0.020 -0.060 -0.100

-2

-3

-4 0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

x [m] 0

vertical velocity component

z [m]

w [m/s] -2 0.040 0.020 -0.000 -0.020 -0.040

-3

-4 0

2

4

6

8

10

12

14

16 x [m]

18

20

22

24

26

28

30

5.4 Lock exchange flow

The results of the non-

-1

5.4 Lock exchange flow

153

is set to zero, u = 0. At all lateral boundaries and at the bottom impermeability and a free slip boundary condition are set and no flux of salinity through the boundary is imposed. The turbulent diffusion and viscosity coefficients are set to zero (inviscid flow). The free surface is computed from long-wave equations in the hydrostatical case. In the non-hydrostatic case, the stabilised conservative free surface equation is applied. The channel measures 30 m in length, 3 m in width and 4 m depth. The basic mesh has 2416 nodes and 4500 elements, horizontal and vertical resolution is 0.2 m, 21 mesh levels are used (figure A.2). The time-step is 1 s. The results are presented in figures 5.12 and 5.13 as vertical cuts through the domain in the x-direction for t = 100 s. The differences in the interface shapes and velocity distribution are evident. The characteristic rectangular shape is observed with the hydrostatic approximation, whereas the non-hydrostatic solution yields a much more realistic density flow head shape. The domain is long enough for a stationary flow to establish in the middle of the domain when the heads of the wedges move away in opposite directions. For this setting the theory presented in the sections above can be applied in order to check the velocity of the flow. For the geometry and the densities of the fluids as mentioned above, the formula (5.30) yields a velocity of ca. 0.099 m/s. The time series of the horizontal velocity in points at the free surface and at the bottom and situated exactly in the middle of the domain (where the initial interface is situated), are shown in figure 5.16, for the nonhydrostatic and the hydrostatic cases. In the hydrostatic case a symmetric stationary flow develops, but the velocity is about 20% lower than theoretically expected. No free surface wave develops. In the non-hydrostatic case, the flows is also symmetric but with flow speed 10% lower as expected. The buoyancy-driven flow is modulated by a free surface standing wave motion with a period of about 9.85 s with a very small amplitude of 3 mm. The differences between hydrostatic and non-hydrostatic solutions are also evident in the case where, in the initial condition, the denser fluid occupies the central part of the channel between x = 7.5 m and x = 12.5 m. In this case one has two interfaces moving in different directions, figures 5.14 and 5.15 (integration time 50 s, time-step 1 s). The current will not change, if a wall is placed exactly in the middle of the channel. A good illustration of the different flow patterns in both the hydrostatic and nonhydrostatic cases is shown in figure 5.17. The computational domain is a cube 10 m × 10 m × 10 m; the basis mesh is shown in figure A.1. The time step is ∆t = 0.05 s, and the presented figure shows the flow at t = 30 s. The extreme difference between the vertical velocity component value as well as in the interface position is evident. As a conclusion, for proper simulation of density flows, the non-hydrostatic model should be preferred, because it yields a realistic interface shape as well as a more acceptable wedge intrusion speed. Additionally, the velocity distribution nearby the nose of the intruding dense flow is described much more realistically. As in the case of interfacial internal waves, improvements can be thought of which reduce the mixing at the interface.

154

5.4 Lock exchange flow

0.12

0.1 free surface

| u | [m/s]

0.08

0.06

0.04 bottom 0.02

0.0 0

20

40

60

80

100

120

140

160

180

200

120

140

160

180

200

t [s]

0.12

0.1 free surface

| u | [m/s]

0.08

0.06 bottom 0.04

0.02

0.0 0

20

40

60

80

100 t [s]

Figure 5.16: The lock exchange flow wedge speed time series for points at the free surface and at the bottom and situated exactly in the middle of the domain. Non-hydrostatic (above) and hydrostatic (below) solutions with free surface. The theoretical value for steady flow is 0.099 m/s.

5.5 Standing wave in a closed basin

155

z [m]

0

z [m]

0

v [m/s] 0.2

-10

-10 0

10 x [m]

0

10 x [m]

Figure 5.17: Flow patterns in a cubic domain 30 s after the initialisation; left: the nonhydrostatic solution, right: the hydrostatic one. Solid line: the interface between the fluids.

5.5

Standing wave in a closed basin

A small amplitude uni-nodal standing wave in a closed basin is a particularly good example for testing mass and energy conservation when computing transient flows. The test realisation is very simple, but the model properties to be tested are not trivial. As the initial condition a sloped free surface and no motion is taken – a configuration with maximum potential energy. A wave motion begins – and in the moment, when the free surface is horizontal, the whole available potential energy has been exchanged into kinetic one. When the highest water level at the opposite wall is reached, the maximum potential and the minimum kinetic energy state is achieved again, just as at the very beginning of the simulation. When there are no energy and mass losses (a free slip BC for velocity at all boundaries, no shear, no viscosity), such an oscillation in a numerical experiment should continue forever. Additionally, when the simulation is performed in a square basin with free surface slope parallel to one of the sides (i.e. motion takes place in a 2D vertical plane, figure 5.18), there should not be any parasitic motion perpendicular to the wave slope. An analytical solution of this wave problem exists. The resulting velocity patterns depend on the water depth, and the role of the vertical accelerations can be immediately seen in a situation corresponding to short (deep water) waves. Moreover, the deviations of the numerical result from an ideal movement can be especially well observed. The behaviour of the hydrostatic and non-hydrostatic models for short and long waves cases and various methods of computing the free surface movements can be compared clearly.

156

5.5 Standing wave in a closed basin z

6

η(x, y, t) 0

η0

-

–10 0

10

x

Figure 5.18: A uni-nodal standing wave in a closed basin.

5.5.1

Theory

The theory of the small amplitude progressive waves is shortly sketched in the appendix section A.3. A standing wave is caused by the superposition of two waves of the same period and amplitude η0 /2, but travelling in different directions: 1 η(x) = η0 [cos(kx − ωt) + cos(−kx − ωt)] = −η0 cos(kx) cos(ωt) 2

(5.32)

The inviscid fluid of constant density is confined in a closed basin with a square base L × L and with equilibrium depth H. z = 0 at the equilibrium surface level. The initial free surface elevation is given by: η(x) = η0 cos(kx) 0 < x < L

(5.33)

where k = 2π/nL, n = 2 for a uni-nodal standing wave. The frequency ω of the resulting wave is given by positive root of ω 2 = gktanh(kH), and the celerity of the wave is c = ω/k. According to the small amplitude wave theory (η0 << H) describing the resulting wave motion in the plane (x, z), the solutions for velocity potential ϕ, free surface elevation η, velocity components (u, v, w) and perturbation pressure p′ are [99, 100]: ch[k(z + H)] ϕ = cη0 cos(kx) sin(ωt) (5.34) ch(kH) η = η0 cos(kx) cos(ωt) u = ωη0

ch[k(z + H)] sin(kx) sin(ωt) sh(kH)

w = −ωη0

sh[k(z + H)] cos(kx) sin(ωt) sh(kH)

(5.35) (5.36) (5.37)

5.5 Standing wave in a closed basin

p′ = ̺gη0

157

ch[k(z + H)] cos(kx) cos(ωt) ch(kH)

(5.38)

(compare with section A.3) The perturbation pressure is defined as the difference between the global pressure and the equilibrium pressure p′ = p − p0 = ̺gη + π. Therefore, the hydrodynamic pressure π is given by: π = −̺gη + ̺gη0

ch[k(z + H)] cos(kx) cos(ωt) ch(kH)

(5.39)

The proper initial conditions for t = 0 are: η(x) = η0 cos(kx) u=0 w=0 ch[k(z + H)] cos(kx) π = −̺gη + ̺gη0 ch(kH)

(5.40)

The solutions given above are valid for a wave in a basin of an arbitrary depth. For the long wave (shallow water, H << L) the celerity c = (gH)1/2, the frequency ω = k(gH)1/2 and the following approximations are valid: u=

ωη0 sin(kx) sin(ωt) kH

w = −ωη0

5.5.2

z+H cos(kx) sin(ωt) H

(5.41) (5.42)

p′ = −̺gη0 cos(kx) cos(ωt)

(5.43)

π=0

(5.44)

Deep water – short waves

In a closed square pool of 10 m width and 10 m length an initial slope of the free surface in the form of (5.33) and a zero initial velocity is assumed. The initial condition corresponds therefore to the position of maximum potential energy and minimum kinetic energy. The equilibrium water depth is 10 m and the amplitude is set to η0 = 0.1 m, 1% of the water depth, fulfilling the assumptions of the small amplitude waves theory. The analytical solution of this problem is given in the previous section (5.5.1). For simplicity, g = 10 m2 /s and ̺ = 1000 kg/m3 , so that for the equilibrium water depth 10 m, the long wave celerity (from equation A.42) is exactly c = (gh)1/2 = 10 m/s and the period of oscillations is T = 2 s. However, the long-wave theory is not appropriate in this case, because the water depth is comparable with the wave length. Applying the general small amplitude wave theory, c = [(g/k) tanh(kh)]1/2 = 5.64 m/s and the period of oscillations is 3.55 s (from equation A.34 or A.37).

158

5.5 Standing wave in a closed basin

numeric

analytic

v [m/s]

v [m/s]

0.05

0.05

z [m]

500 x [m]

6

-8

-8

-10

-10

750

250 5

-6

7

8

9

10

-250

4

-6

-2

-4

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-10 3

-4

0

25

50

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1

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25

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-2 -2

0

750

0

0

-8

-10 0

1

2

3

4

5

6

7

8

9

10

x [m]

Figure 5.19: Comparison between numeric and analytic solution for t = 0.1 s. Iso-lines of dynamic pressure shown each 50 Pa. The basic mesh is shown in figure A.1. The time step is varied between 0.01 s and 0.1 s. Inviscid flow is assumed. The free surface is computed from the kinematic BC and the stabilised SUPG method is applied, when not stated otherwise. For velocity, at the impermeable boundaries impermeability or free slip conditions are given. At the free surface, the zero Neumann boundary condition is applied for all three velocity components. Hydrodynamic pressure is set to zero at the surface and impermeability conditions are imposed at all impermeable walls. The assumption of zero dynamic pressure is the only difference between the boundary conditions applied in the numerical model and in the analytical solution (see section 5.5.1). A comparison between the analytical and numerical solutions is given in vertical sections, figures 5.19, 5.20 and 5.21, for t =0.1 s, 5 s and 10 s respectively. All show a very good agreement of the numerical results with the analytical theory of short waves. The greatest differences occur in the pressure field, due to differences in the BC at the free surface. In order to demonstrate the behaviour of the solution after several periods of oscillation, time series for the free surface elevation at a boundary edges x = 0 m and x = 10 m for 30 s simulation time (over 8 periods, ∆t = 0.05 s) are presented in figure 5.22. The amplitude remains for a considerably long time constant and the period of oscillation gets only slightly longer: the time obtained for 8 full periods is about two time steps (0.1 s) longer than the theoretical one. The mass loss is about 0.03%. A very illustrative example is to compare the results of the hydrostatic and nonhydrostatic model for the same phase of motion, e.g. when the state of maximum kinetic energy is achieved, for one-fourth of the oscillation period. The results are shown for

5.5 Standing wave in a closed basin

159

numeric

analytic v [m/s]

v [m/s]

0.2

0.2 0

0

25

50

0

-2

-2

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250

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500

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z [m]

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8

9

10

750

-750

-10

-10 0

1

2

3

4

x [m]

5

6

7

8

9

10

x [m]

Figure 5.20: Comparison between numeric and analytic solution for t = 5 s. Iso-lines of dynamic pressure shown each 50 Pa.

numeric

analytic

v [m/s]

v [m/s]

0.2

0.2

0

0

-2

0

0

0

-2

-2

-2

-4

-4

-4

0

0 -8

-8

-10

-10

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0

-6

250

-6

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250

-250

z [m]

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-8

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1

2

3

4

5 x [m]

6

7

8

9

10

-8

-10 0

1

2

3

4

5

6

7

8

9

10

x [m]

Figure 5.21: Comparison between numeric and analytic solution for t = 10 s. Iso-lines of dynamic pressure shown each 50 Pa.

160

5.5 Standing wave in a closed basin

z [m]

0.0

3.552

7.104

10.656

14.208

17.76

21.312

24.864

28.416

0.1

0.1

0.05

0.05

0.0

0.0

-0.05

-0.05

-0.1

-0.1

0

5

10

15

20

25

30

t [s]

Figure 5.22: Time series for the free surface elevation at a boundary edge x = 0 m and x = 10 m (chain-dotted line) for 30 s simulation time. The oscillation period is 3.552 s, shown along the upper x-axis. 0.9 s for the non-hydrostatic and 0.5 s for the hydrostatic one. The difference between both solutions are striking (figure 5.23) – the hydrostatic model yields (unphysical in this case) long-wave velocity profiles compared to the exponentially diminishing ones for the (physical) non-hydrostatic case. Additionally, in figure 5.24 time series for surface elevation at the basin boundary with and without hydrostatic approximation are shown. The hydrostatic model reproduces perfectly the (unphysical) long wave period, although it does not conserve energy – the amplitude is almost completely damped after 30 s. On the other hand, the mass (volume) conservation is excellent (negligible, 2.6×10−4 m3 in a tank containing 1000 m3 ). Similar remarks can be made for the strongly damped dashed line in figure 5.24, representing the non-hydrostatic solution with application of the vertically integrated momentum equations for obtaining the free surface elevation and taking barotropic pressure gradients into consideration, as described in section 2.8.8. Although the mass is conserved and the wave period is reproduced properly, the energy is not conserved again.

5.5.3

Shallow water – long waves

In this verification case the setting is virtually the same, but the computational mesh represents now a long channel. The mesh shown in fig. A.2 is used. The channel depth is the same as of the pool, 10 m, width 15 m and length 150 m. This is approximately

5.5 Standing wave in a closed basin

161

hydrostatic

non-hydrostatic

v [m/s]

v [m/s]

0.2

0.2 0

0

0

-2

-2

-2

-2

-4

-4

-4

-4

-6

-6

-6

-6

-8

-8

-8

-8

-10

-10

z [m]

0

-10 0

1

2

3

4

5 x [m]

6

7

8

9

10

-10 0

1

2

3

4

5

6

7

8

9

10

x [m]

Figure 5.23: Comparison of solutions with (left) and without (right) the hydrostatic approximation, for the same phase of motion, T /4, equal 0.5 s and 0.9 s, respectively. the limit case for the application of the hydrostatic long wave theory, i.e. the ratio wave length to the depth 20:1. For the given channel it is 30:1. The initial free surface is given by (5.33) and a zero initial velocity is assumed. The amplitude is again η0 = 0.1 m and all other parameters are the same as before. Applying the general small amplitude wave theory to this case, the wave phase speed is c = [(g/k) tanh(kh)]1/2 = 9.93 m/s and the period of oscillations of 30.22 s. Long wave theory yields c = (gh)1/2 = 10 m/s and the period of oscillations T = 30.0 s, so it can be assumed that the hydrostatic approximation is almost entirely appropriate in this case. Figure 5.25 compares the free surface profile development over time for about 10 periods. The non-hydrostatic model reproduces the period of 30.22 s perfectly, while the hydrostatic one (only the pressure-continuity step switched on) yields exactly 30 s from the long wave (hydrostatic) theory. While the hydrostatic model shows large numerical diffusion (by the 9th period the amplitude is only 18% of the initial position), the nonhydrostatic one for the chosen time-step of 0.1 s is slightly unstable (see next section), yielding an increase in amplitude to 111% by the 9th period.

5.5.4

Comparison of different free surface algorithms

In this section different methods of computing the surface for the case of standing waves are compared.

162

5.5 Standing wave in a closed basin

z [m]

-0.05

0.0

0.05

0.1

-0.1

-0.05

0.0

0.05

0.1

0

0.0

-0.1

-0.15

0.15

-0.15 2

3.552

4 6

7.104

8

10.656

10 12

14.208

14 t [s] 16

17.76

18 20

21.312

22 26

24.864

24

28.416

28 30 0.15

Figure 5.24: Time series for the free surface elevation at a boundary edge x = 0 m. The oscillation period for the non-hydrostatic solution (chain-dotted line) is 3.552 s, for the hydrostatic one (solid line) 2.0 s. The dashed line: the non-hydrostatic solution with vertically integrated momentum equations for obtaining the free surface elevation.

5.5 Standing wave in a closed basin

163

non-hydrostatic

hydrostatic

t [s]

t [s]

0.00

0.00

-0.05

300

300

300 0.00

0.05

0.05

0.10

272.0

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0.00

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270

270

240

240

210

210

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150

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90

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60

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60

90 x [m]

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60

-0.05 0.05

30.2

90

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60.4

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90.7

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150

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0.05

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151.1

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0.00

270

0.00

0.00

0.0

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0.00

240

0.05

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211.5

0.00

30

-0.05

0.05

30

0.00

0.00 30

-0.05 -0.05 120

0 150

0.05

0 0

30

0.00

0.00 60

0.05 0

90

120

150

x [m]

Figure 5.25: Free surface oscillations for a channel of 150 m length and 10 m depth. Comparison between the non-hydrostatic and hydrostatic solutions. The changes of the free surface profile in the middle of the pool are shown as a function of pool length and time. The oscillation period for this wave is 30.2179 s, in the long wave approximation exactly 30.0 s.

164

5.5 Standing wave in a closed basin

0.0

3.552

7.104

10.656

14.208

17.76 0.15

0.1

0.1

0.05

0.05

0.0

0.0

z [m]

0.15

-0.05

-0.05

-0.1

-0.1

-0.15

-0.15 0

2

4

6

8

10

12

14

16

18

20

t [s]

Figure 5.26: Time series for the free surface elevation at a boundary edge x = 0 m for 20 s simulation time and different time steps: 0.01 s (solid line), 0.05 s (chain-dotted line) and 0.1 s (dashed line). The oscillation period is 3.552 s. 5.5.4.1

Dependency on the time step

First of all, not only the accuracy in reproducing the free surface oscillation period, but also the conservation of momentum depends on time step, as shown in figure 5.26. The computation has been performed using kinematic BC for the free surface with implicit SUPG method. The numerical result converges to the analytical one with diminishing time step. As expected for an operator-splitting algorithm with an overall explicit time discretisation, the consistency order of the technique is one and the accuracy depends approximately linearly on the time step. 5.5.4.2

Free surface stabilisation technique

The stability of the free surface algorithm (based on the kinematic boundary condition) for larger time steps can be enhanced using the stabilisation method described in section 3.6.5. Figure 5.27 shows the effects of stabilising the resulting free surface by filtering once and twice the results obtained by implicit SUPG method for ∆t = 0.05 s. This smoothing technique also brings some advantages by reducing the needed overall CPU time by 10%. 5.5.4.3

Comparison of the implemented algorithms

The properties of various free surface computation techniques are compared briefly in figure 5.28 for ∆t = 0.05 s. By 1000.0 m3 initial volume, the stabilised implicit methods

5.6 Subcritical and supercritical flow over a ramp

z [m]

0.0

3.552

7.104

165

10.656

14.208

17.76

0.15

0.15

0.1

0.1

0.05

0.05

0.0

0.0

-0.05

-0.05

-0.1

-0.1

-0.15

-0.15 0

2

4

6

8

10

12

14

16

18

20

t [s]

Figure 5.27: Time series for the free surface elevation at a boundary edge x = 0 m for 20 s simulation time and different filtering: none (dashed line), once (chain-dotted line) and twice (solid line). ∆t = 0.05 s. based on kinematic BC yield comparable results by volume loss at t = 30 s of −.094 m3 for SUPG, and growth of 0.27 m3 for the method of characteristics. The conservative equation method yields excellent volume conservation with a negligible mass increase, 0.74 × 10−11 m3 , whereby the fully explicit formulation (section 3.6.3) the momentum is not conserved – the amplitude diminishes in 30 s to ca. 65% of its initial value. By the implicit formulation (section 3.6.4), the results obtained with conservative equation are entirely comparable with the ones obtained with kinematic BC, with mass losses of 0.04 m3 . The overall computation times are comparable when applying of the method of characteristics or the explicit conservative free surface equation. When applying the SUPG method it is 5–10% greater. The implicit conservative free surface equation, based also on SUPG, needs less equation solver iterations yielding about 10% lower computation times then the equivalent kinematic BC method. Surprisingly in the light of test results mentioned in section 4.3.1, for the range of applied time-steps, the results do not depend on the implicitness of the free surface algorithms based on kinematic BC.

5.6

Subcritical and supercritical flow over a ramp

The channel flow has two characteristic regimes, namely subcritical and supercritical regimes. The characteristic number controlling these regimes is the Froude number: |u| |u| = Fr = √ gH c0

(5.45)

166

5.6 Subcritical and supercritical flow over a ramp

z [m]

0.0

3.552

7.104

10.656

14.208

17.76

0.15

0.15

0.1

0.1

0.05

0.05

0.0

0.0

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-0.1

-0.1

-0.15

-0.15 0

2

4

6

8

10

12

14

16

18

20

t [s]

Figure 5.28: Time series for the free surface elevation at a boundary edge x = 0 m for 20 s simulation time and different implicit free surface computation methods. Kinematic BC: characteristics (dashed line), SUPG (solid line). Conservative free surface equation: fully explicit (chain-dotted line), fully implicit (dotted line). √ where H is the channel depth and c0 = gH the long free surface wave phase speed. The supercritical regime is obtained, when F r > 1 and subcritical one for F r < 1 [171]. One of the features of the supercritical flow is the fact, that the flow velocity u is greater than the greatest possible velocity of the free surface waves c0 and waves cannot travel upstream. In the subcritical case, the waves can move downstream and upstream as well. As a consequence, the inviscid supercritical channel flow requires very simple boundary conditions: an imposed fluid velocity and depth at the inflow boundary and free outflow boundary conditions at the outflow. As an example of the free surface flow a supercritical flow over a ramp is chosen, with the geometry shown in figure 5.29. The slope of the ramp is 1:5, i.e. outside the validity domain for shallow water equations. The applied mesh is shown in figure A.2, the spacing is ∆x = 0.1 m. The inviscid free surface flow can be described by the Bernoulli equation valid along the streamline following the free surface from inflow and outflow sections and continuity equations stating mass conservation between these sections (where the flow is homogeneous): u2 + h + z = const 2g uh = q = const

(5.46)

For the supercritical case, as the initial condition, a free surface level of 1 m and an initial velocity of 1 m/s is given everywhere. At the inflow boundary Dirichlet BC for

5.6 Subcritical and supercritical flow over a ramp

167

1.2

[Pa]

z [m]

1.0 0.8 0.6 0.4 0.2 0.0 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

150 125 100 75 50 25 0 -25 -50 -75 -100 -125 -150

channel length [m]

Figure 5.29: The steady state (t = 100 s) hydrodynamic pressure (π) distribution for supercritical channel flow over a ramp with 1:5 slope (a vertical section along the channel). all flow variables are imposed (free surface level of 1.0 m and a constant velocity of u = 6 m/s) and zero dynamic pressure. At the outflow boundary, a zero Neumann BC for all variables is applied. The fluid speed is almost two times greater than the wave celerity c = 3.16 m/s (g = 10.0 m/s2 for simplicity). Solution of (5.46) for a given constant flux q = 6 m2 /s (i.e divided by channel width) yields a value for the water depth downstream from the ramp 1.0928 m and (for the water level 1.2928 m). Free surface is computed with the semi-implicit SUPG algorithm with θ = 0.55. The stabilisation of the free surface section in the middle of the channel in time for 10 s and with time-step ∆t = 0.01 s is shown in figure 5.30. The initially occurring wave travels out of the domain without causing any problems. The computed free surface level at the outflow is 1.2957 m, relative error to the analytical solution ≈ 0.22%. A simulation with ∆t = 0.1 s for 100 s brings steady state water level of 1.2971 m, relative error ≈ 0.33%. In the case of the subcritical flow, the waves generated over the ramp due to the inappropriate initial condition can travel in both directions. When the boundary conditions for the stabilised channel flow are applied, e.g. Dirichlet boundary conditions for all variables, or ∂h/∂x = 0 and q = const at the inflow section and a constant pressure at the outflow, the generated waves are reflected at the inflow and outflow sections and disperse over the ramp. When the algorithm does not provide a method to damp these oscillations or provide non-reflecting or absorbing boundary conditions, these waves can finally lead to breakup of the program. The computation for the subcritical flow over a ramp is performed for the initial velocity of 1 m/s everywhere and a flat free surface level of 1.0 m again. As the boundary conditions at the outflow boundary a constant dynamic pressure of 0 and a free outflow BC for all other variables are applied. The inflow BC are ∂h/∂x = 0 with the velocity changing according to the water depth so that a constant flux q = 1.0 m2 /s is provided. As previously, the inviscid flow is considered, whereby the free surface is computed with stabilised implicit SUPG method in order to provide short wave damping. Without damping the solution breaks up when the initial waves are reflected from boundaries.

168

5.7 Channel with a bump

10

1.00 9 8

1.00 7

1.00

time [s]

6 5 4

00

1.

0

1.3

1.00

3

1.20

00

2

1.

5

1.2

1

5

1.1

1.10 1.05

0 0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

length [m]

Figure 5.30: Stabilisation of the free surface profile for the supercritical flow over a ramp. The stabilisation of the free surface profile in time is shown in figure 5.31 for simulation time 100 s and ∆t = 0.1 s. The initially occurring waves are partially reflected from the boundaries, partially leave the domain, and finally are dissipated. The comparison between steady state free surface profiles is given in figure 5.32. For the subcritical flow the inflow water level stabilises at 1.12134 m and inflow velocity 0.891828 m/s, yielding q = 1.00004241 m2 /s. The outflow water level stabilises at 1.09897 m and inflow velocity 1.11229 m/s, yielding q = 0.99991534 m2 /s. This is an excellent agreement with theory – equations (5.46) yield a free surface level of 1.09927 m. The relative error is ≈ 0.03%. This tests shows clearly the ability of the newly developed model to reproduce supercritical and subcritical channel flows over a bottom topography with slopes excluding application of codes based on shallow water equations. The limits of application are discussed in section 5.10.2. Additionally, the stabilised method of computing the free surface is tested.

5.7

Channel with a bump

A good test case for the properties of the free surface algorithm is a subcritical stationary flow in a long channel with a bump in the middle of its length. For an inviscid case, the free surface profile can be computed easily from (5.46). As an initial condition a constant velocity and a flat free surface is usually assumed and computation is continued until a stable state is achieved. This artificial initial condition,

5.7 Channel with a bump

169

100 90 80 70

time [s]

60 50 40

1.100

30 20

1.075

10

1.050

1.025

0 0

1

2

3

4

5

1.025

6

7

8

9

10 11 12 13 14 15

length [m]

Figure 5.31: Stabilisation of the free surface profile for the subcritical flow over a ramp steered with constant inflow flux.

1.4

water level [m]

1.3

1.2

1.1

1.0

0.9 0

1

2

3

4

5

6

7

8

9

10 11 12 13 14 15

length [m]

Figure 5.32: The steady state (t = 100 s) free surface profiles for supercritical (solid line) and subcritical (dashed line) flows over a ramp.

170

5.7 Channel with a bump

0

z [m]

-2 -4 -6

u [m/s]

-8

1 0.1

-10 0

50

100

150

200

250

300

350

400

450

500

550

600

length [m]

Figure 5.33: A channel with a bump. Vertical section, steady state velocity. for a subcritical flow, produces free surface waves travelling downstream and upstream as well. The boundary conditions set at both open boundaries at the extremities of the channel are crucial for the stabilisation of the solution. The usual method is to impose boundary conditions that are appropriate for the targeted steady state. In this case, for a model computing the free surface form shallow water equations, they are a constant flux at the inflow open boundary and a constant water level at the outflow boundary. As a result, the waves occurring initially directly over the bump travel in both directions, partially leaving the domain, and partially being reflected backwards. The numerical diffusion dampens their amplitude, as well, and after some time the final state is achieved. For a 3D non-hydrostatic model there are various methods of setting the boundary conditions. For example, by imposing at both open boundaries a zero Dirichlet BC for the pressure and free outflow boundary conditions for the free surface and velocity, reflecting boundary conditions are obtained. This causes the waves induced by the artificial initial condition to bounce around the domain until they are dissipated – or cause numerical instabilities. Another method is to set absorbing or non-reflecting boundary conditions at the open boundaries to let the incoming waves be absorbed or leave the domain. The test case is realised in a channel of depth H = 10 m, length l = 600 m and width of B = 28 m, where a bottom profile (a bump) given by the formula: 2

zb = A sin

x − 0.5(l − l0 ) l0

!

(5.47)

with a height A = 3 m and a length of l0 = 200 m in centred in the middle (i.e. between xa = 200 m and xb = 400 m) is localised (figure 5.33, the applied basic mesh shown in figure A.3). For this case, when a constant flux of 10 m2 /s is given at the inflow boundary, the Bernoulli equation (5.46) yields for a stationary flow a slight submersion of the free surface profile with max. 0.0536 m discrepance from the initial zero level just above the highest point of the bump. The computations with a non-hydrostatic model are performed with a stabilised SUPG algorithm based on the kinematic BC for the free surface. For the advection step the method of characteristics is chosen. The flow is assumed to be inviscid. At channel

5.7 Channel with a bump

171

lateral walls, zero Neumann BC is applied. For open boundaries, two types of boundary conditions are chosen: • Boundary conditions corresponding to the stationary state, a zero Neumann BC at the inflow boundary and a Dirichlet one (a constant imposed water level) at the outflow. • Simple non-reflecting BC at inflow and outflow. The non-reflecting boundary conditions (section 2.7.8) are based on the properties of long waves imposed on a channel flow with a constant flux. They are realised in the form of Dirichlet BC for water level at the open boundaries Sb , whereby free outflow BC for the velocity and the hydrostatic pressure (i.e. imposed hydrodynamic pressure equal to 0) are assumed. For a channel aligned along the x-axis: Sb = Seq + m

heq (un+1 − ueq ) √ n gh

(5.48)

where un+1 is the vertical mean velocity at the time level n + 1 and hn the local water depth at the previous time step. The variables denoted by subscript eq concern the expected values at the open boundaries when the stable state is achieved. m = +1 at the outflow and m = −1 at the inflow boundary [141].

Imposing Dirichlet BC for water level at open boundaries, especially those that are only approximately valid, is connected with numerical instabilities assigned to reflections of waves at these points and other incompatibilities. In practice some stabilisation or filtering of the free surface is required. In this case, stabilisation by the method described in section 3.6.5 is applied. Without it, oscillations occur at those points, where Dirichlet boundary conditions are applied. The computation was performed with the overall horizontal mesh resolution of 8 m, 21 mesh levels (10 m water depth before and behind the bump) and the time step equal 1 s. The stabilisation of the free surface profile over time for reflecting (stationary state) and simple non-reflecting boundary conditions during 1000 s and during first 100 s (detailed) is shown in figures 5.34 and 5.35. Upon application of the non-reflecting BC the stable state is achieved quickly, for the reflecting BC the free surface does not stabilise in a comparable time. The steady state free surface profile achieved by application of the non-reflecting BC is presented in figure 5.36, the minimum value of the free surface level is 0.05451 m, with an error of 1.6% compared to the analytical solution. As a conclusion, for computations starting from simple homogeneous initial conditions and aimed at obtaining a stationary free surface flow, boundary conditions are crucial for the convergence efficiency to the stable state. Non-reflecting boundary conditions are required. Without them, the initially occurring waves cannot leave the domain. With boundary conditions corresponding to the stationary state, the convergence can be enhanced only by artificially increasing the dissipative properties of the free surface computation. However, it also shows that for practical purposes filtering of the free surface is required in order to obtain stable and converging solutions.

172

5.7 Channel with a bump

1000

1000

0.00

900

0.00

0.00

900

800

800

-0.05

0.00 700

700

-0.05

0.00

0

0.0 600

time [s]

500

0.00

05

-0.

0.00

400

400

0.00

0.05

0.0

300

500

-0.05

time [s]

00

0.

600

0.00

5

300

0.00

-0.0

5

200

200

-0.05

-0.05

0.00 100

0.05 0.00

0 0

100

100

0.00 0.00

0.00

0.10 200

0.00

0.10 0.00

0 300 length [m]

400

500

600

0

100

0.10 200

-0.05 300

400

500

600

length [m]

Figure 5.34: A channel with a bump. Stabilisation of the free surface profile for reflecting (left) and simple non-reflecting boundary conditions (right) during 1000 s of computation time.

5.8 Waves travelling over an underwater channel

100

173

100

90

0.05

90

00 0.

0.0

80

0.

00

5

80

-0.05 -0.05

0.10

0.00

0.10

-0.05 -0.05

0.00

0.10

0

50

0.0

0.00

5

5

-0.05

0.00

40

0.0

50

time [s]

60

0.0

time [s]

60

0 0.0

70

10 0. 0.15

0.05

70

40

10

0.

0.0

0.

00

0

-0 -0.0 .1 5 0

0 .0 -0 5 .1

10

5

00

0 0.1.05 0

0.

0.1

10

-0

05 20

0.00

5 0 05 0.0 0.1 .15 5 0.10 0. 0 .1 0

-0.

0.00

0

-0.1

20

-0.05

30

5

30

0 0

100

200

300 length [m]

400

500

600

0

100

200

300

400

500

600

length [m]

Figure 5.35: A channel with a bump. Stabilisation of the free surface profile for reflecting (left) and simple non-reflecting boundary conditions (right) for 100 s computation time (details).

5.8

Waves travelling over an underwater channel

A simple, but demanding test for free surface models is to simulate the propagation of waves over varying bathymetry. Additionally, the travelling waves should enter and leave the computational domain without artificial effects. In this case a typical situation which can be found in coastal regions is studied, with similar parameters and geometry to those used by Schr¨oter [142]. A wave-train with a small wave amplitude of η0 = 0.125 m and a period of T = 4 s enters a domain represented by a long channel from the left side (x = 0) in shallow water (equilibrium depth of 3 m) and then approaches a deep underwater channel. The waves leave the domain after passing a final shallow water part 3 m deep. Two geometries are chosen, shown in figure 5.38. They differ in length (600 m and 800 m), as well as in distance between the inflow and outflow domains and the channel edges (50 m and 50 m or 100 m and 200 m, respectively), but not in the underwater

174

5.8 Waves travelling over an underwater channel

0.0

-0.01

water level [m]

-0.02

-0.03

-0.04

-0.05

-0.06 0

100

200

300

400

500

600

length [m]

Figure 5.36: A channel with a bump. Steady state free surface profile. channel shape. The greatest depth of the 500 m long cosinusoidal channel cross-section is 43 m. The computational mesh structure is very similar in structure to the one shown in figure A.2, but 6 elements wide and 600 elements long, with a resolution in the direction parallel to the channel axis of 1 m or 1.33 m. In both cases the three-dimensional mesh has 21 levels distributed more densely nearby the surface. The time-step is 0.1 s and the simulation time 300 s. Viscous flow is assumed, with viscosity of ν = 10−4 m2 /s, but with no shear at all impermeable channel boundaries. Due to the small wave height non-linear effects are not to be awaited. For a wave of this period, at the entrance (water depth of 3 m) wave length of 19 m is to be observed. In the deep part of the channel, the deep water wavelength approaches 25 m (from the dispersion relation A.34). In the domain, the ratio between the equilibrium water depth and the wave length D/λ varies between 0.16 and 1.7. At the open outflow boundary (x = 600 m or x = 800 m), a simple non-reflecting boundary condition for the free surface elevation Sout with the incoming wave celerity and the mean velocity in the form: hn un+1 Sout = Seq + √ n gh

(5.49)

is imposed, with free outflow conditions for the velocity and zero Neumann boundary condition for the hydrodynamic pressure. Seq is the equilibrium free surface elevation (in this case it equals zero). At the inflow boundary (x = 0 m), the same boundary conditions for the velocity and the pressure are given, but the non-reflecting boundary condition takes the more complicated form imposing an incoming wave profile while

z [m]

0.0 -0.1 0

100

200

300

500

600

700

x [m]

0.1 z [m]

400

800

t=100s

0.0 -0.1 0

100

200

300

500

600

700

x [m]

0.1 z [m]

400

800

t=150s

0.0 -0.1 0

100

200

300

0.1 z [m]

400

500

600

700

x [m]

5.8 Waves travelling over an underwater channel

800

t=200s

0.0 -0.1 0

100

200

300

500

600

700

x [m]

0.1 z [m]

400

800

t=300s

0.0 -0.1 0

100

200

300

400

500

600

700

800

175

Figure 5.37: Water surface time development for the test case of waves travelling over an underwater channel.

t=50s

0.1

176

5.8 Waves travelling over an underwater channel

x [m] 200

300

400

500

600 0.2

0.0

0.0

-0.2

=25m

=19m

-0.2

=19m

0

0 -3m

-3m

z [m]

[m]

100

-20

-20

z [m]

[m]

0 0.2

-43m -40

-40

0

100

200

300

400

500

600

x [m] x [m] 200

300

400

500

600

700

800

0.1

0.1

0.0

0.0 =19m

=25m

-0.1

-0.1

=19m

0

0 -3m

z [m]

[m]

100

-3m

-20

-20

z [m]

[m]

0

-43m -40

-40

0

100

200

300

400

500

600

700

800

x [m]

Figure 5.38: Bottom and free surface profiles (t = 300 s) for the test case of waves travelling over an underwater channel.

5.8 Waves travelling over an underwater channel

177

simultaneously allowing waves arriving from the domain interior to leave it [142]: Sin = Simp −

hn un+1 − uimp (H + Simp ) ce

(5.50)

The imposed wave profile is: π + kx − ωt) 2 π 1 η0 gk (exp[k(H + Simp )])2 − 1 cos( + kx − ωt) uimp (H + Simp ) = 2 ω exp[k(H + Simp )] cosh(kH) 2 2 ω = gk tanh(kH) (5.51) ce = 2π/kT Simp = η0 cos(

where H = 3 m is the equilibrium water depth and uimp is the vertically integrated imposed velocity. The development of the wave profile as it enters the domain is shown in figure 5.37 for the longer domain. The frequency dispersion of the wave train front is the characteristic feature of the free surface profiles after 50 s and 100 s, with longer waves travelling quicker than the shorter ones. A reflection from the shoaling underwater channel slope is to be observed, as well as viscous damping of the amplitude. The observed wavelengths are in excellent agreement with the theory (appendix section A.3), yielding approximately λ = 19 m in the shallow parts before and after passing the deeper part, and λ = 25 m over the underwater channel (figure 5.38). In the case of the shorter channel, the reproduction of the wave length is also correct, but as the main wave train arrives at the steep slope between x = 300 m and x = 500 m, a strong reflection occurs causing a considerable increase of the wave amplitude. The boundary conditions, as stated above, allow the waves to enter and to exit the domain smoothly. However, their imperfect form (valid entirely for long waves only) causes a reflection. When the sections truncating the domain are located too near to the area of the main interest, the influence of the imperfect boundary conditions may interfere strongly with the solution in the domain interior. In the case of the shorter channel the reflection from the boundary could be misinterpreted as an influence of the shoaling bottom. Additionally, the boundary conditions are not perfectly mass conserving between the inflow and outflow sections. In the case of the shorter channel, during 300 s about 0.25% of the water mass leaves the domain added to numerical mass losses of 0.5%. In the case of the longer channel, mass losses are about 0.1%, and about 0.05% of the initial mass leaves the domain. As a conclusion, the model is able to deal with various wave problems over a changeable bathymetry reproducing typical phenomena properly. However, the greatest care is recommended by locating inflow and outflow sections and setting boundary conditions for waves entering and leaving the domain so that they are mass-conserving, non-reflective, numerically stable and do not negatively interfere with the solution in the domain interior.

178

5.9 5.9.1

5.9 Solitary wave in a long channel

Solitary wave in a long channel Theory

A solitary wave is a single elevation of water surface above an undisturbed surrounding, which is neither preceded nor followed by any free surface disturbances. Neglecting dissipation, as well as bottom and lateral boundary shear, a solitary wave travels over a horizontal bottom without changing its shape and velocity. Therefore, it is a good test case for testing capability of free surface algorithms to describe propagation of non-linear waves of finite amplitude. The accuracy of the model can be evaluated by comparing the amplitude and celerity of the wave with its theoretical values, as well as the deformation of the wave as it travels. There are numerous analytical studies of this form of non-linear finite-amplitude wave. The first approximation provided by Laitone [95, 132] is the most frequently used for comparative studies. For a vertical section of an infinitely long channel of an undisturbed depth h (z = 0 at the undisturbed surface), the following approximate formulae for velocity components u, w, free surface elevation η, pressure p and wave celerity c of a solitary wave with a height of H are valid: u=

w=

H 3gh h

q



3/2  

s



H 3H gd sech2  (x − ct) h 4 h3

q

s



s

(5.52) 

3H 3H z sech2  (x − ct) tanh  (x − ct) 3 h 4h 4 h3 s



3H (x − ct) η = h + Hsech2  4 h3 p = ̺g(η − z)

c=

q

g(H + h)

(5.53)

(5.54) (5.55) (5.56)

It is interesting that in this analytical approximation the vertical velocity component is not treated as small, as commonly taken, but the pressure can be assumed hydrostatic (equation 5.55) at the same level of exactness as the horizontal velocity (with O((H/h)2 ) [95]. Therefore, this initial condition is suitable for fair comparisons between models with and without hydrostatic approximation and the initial value of zero hydrodynamic pressure is assumed. Although the initial velocity field (5.52–5.53) is perfectly divergencefree, larger values of the hydrodynamic pressure appear immediately after the first time step (60% of the value of the hydrostatic pressure at the bottom). Following the test cases provided by Ramaswamy [132], a solitary wave described by the formulae (5.52–5.56) is applied in a long channel as an initial condition, and the behaviour of the solution is observed thereafter. Due to the fact that the simulation is performed in a finite domain, and Laitone’s formulae are valid for an indefinitely long channel, care must be taken choosing the initial position of the wave crest. In order to

5.9 Solitary wave in a long channel

179

deal with it, the use of the effective wave length λ concept is made. λ is equal to the doubled length between the wave crest and a point, where the free surface elevation is η(x) = 0.01H. According to Laitone: λ = 6.9

s

h3 H

(5.57)

For example, when H/h = 0.2, λ/2 ≈ 8D, and for a channel of 10 m depth, the initial distance between the solitary wave crest and a boundary should be at least 80 m. The steepness of the solitary wave grows with H/h. As the solitary wave travels along a shoaling bottom, its height H increases, until wave breaking appears. As the criterion for wave breaking the critical condition is assumed when the particle velocity at the crest equals the wave celerity. Various approximations yield values for maximum solitary wave height between H = 0.727h and H = 0.827h [113, 95].

5.9.2

Solitary wave propagation over a flat bottom

In order to test the free surface algorithm, the solitary wave propagation in a long channel is studied. In an ideal case, the wave should travel without changing its form and amplitude. A long channel 600 m long and 6 m wide, with a constant depth of 10 m is taken. The mesh is similar to the one shown in figure A.2, but with 6 elements wide and 600 long, with a resolution in the direction parallel to the channel axis of 1 m. The three-dimensional mesh has 11 equidistantly distributed levels. Inviscid flow without shear on walls and the bottom is assumed. All boundaries are impermeable. As the initial condition the hydrostatic approximation given by formulae (5.52–5.56) is applied, with a varying wave height of H = 2 m, 5 m, 7 m and 10 m (H = 0.2h, H = 0.5h, H = 0.7h, H = h), and the initial crest position at x = 80 m. The time step is taken as constant, ∆t = 0.1 s, and the simulation time 40 s (Courant number in the direction of wave propagation varies from 0.2 to about 1.0 at the wave crest). For computation of the free surface elevation in the non-hydrostatic case the SUPG method based on the kinematic boundary condition or the conservative free surface equation is applied. For hydrodynamic pressure, the zero value at the free surface is imposed. The results for comparison of free surface computation methods are summarised in figure 5.39, where semi-implicit (implicitness factor θ = 0.55) and implicit (θ = 1.0) formulations for both methods are shown for wave height H = 2.0 m. The development of free surface profiles in time is presented. For all methods the results are very good with amplitude error less than 5%. The influence of the implicitness factor is much larger in case of the kinematic boundary condition, which can be better seen with a larger wave height H = 5 m, figure 5.40. While the results with the conservative free surface equation remains relatively independent of the implicitness, the results with kinematic BC show large (up to 26%) dependency on it. With larger amplitude, the shape of the wave broadens while travelling. The mass losses also grow with the initial wave height. For kinematic BC, the mass loss for H = 2 m is

600 300 250 200 150 100 50 0

1

2

0

0

Ci

H = 2.0 m

50

Csi 1

x [m]

350

400

450

500

550

H = 1.95 m

600 100

150

200

250

300

350

400

450

500

550

H = 1.99 m 2

0

1

2

0

H = 2.0 m

50

Ki

H = 2.0 m

100

150

200

250

300

350

400

450

500

550

H = 1.89 m

550 500 450 400 350 300 250 200 150 100 50

Ksi

H = 2.0 m

0

z [m]

1

0

z [m]

2

0

z [m]

H = 2.05 m

600

5.9 Solitary wave in a long channel

600

180

z [m]

Figure 5.39: Solitary wave propagation in a long channel with a flat bottom for free surface computation methods based on SUPG: Ksi, Ki: kinematic boundary condition, semi-implicit and implicit; Csi, Ci: conservative free surface equation, semi-implicit and implicit. The free surface profiles shown for t = 0 s, 10 s, 20 s, 30 s, and 40 s.

600 300 250 200 150 100 0

2

4

6

0

0

Ci

50

H = 5.0 m

50

Csi 2

x [m]

350

400

450

500

550

H = 4.69 m

600 100

150

200

250

300

350

400

450

500

550

H = 5.05 m 4

6

0

2

4

6

0

2

H = 5.0 m

Ki

50

H = 5.0 m

100

150

200

250

300

350

400

450

500

550

H = 3.73 m

550 500 450 400 350 300 250 200 150 100

Ksi

50

H = 5.0 m

0

z [m]

4

0

z [m]

6

0

z [m]

H = 5.91 m

600

181

600

5.9 Solitary wave in a long channel

z [m]

Figure 5.40: Solitary wave propagation in a long channel with a flat bottom for free surface computation methods based on SUPG: Ksi, Ki: kinematic boundary condition, semi-implicit and implicit; Csi, Ci: conservative free surface equation, semi-implicit and implicit. The free surface profiles shown for t = 0 s, 10 s, 20 s, 30 s, and 40 s.

600 300 250 200 150 100 0

1

2

0

50

H = 2.0 m

50 0 0

2

x [m]

350

400

450

500

550

H = 1.89 m

600 100

150

200

250

300

350

400

450

500

550

H = 3.7 m 4

6

0

5

0

H = 5.0 m

50

H = 7.0 m

100

150

200

250

300

350

400

450

500

550

H = 4.2 m

550 500 450 400 350 300 250 200 150 100 50

H = 10.0 m

0

z [m]

5

0

z [m]

10

z [m]

H = 4.3 m

600

5.9 Solitary wave in a long channel

600

182

z [m]

Figure 5.41: Solitary wave propagation in a long channel with a flat bottom. Computed from implicit kinematic boundary condition with SUPG. Initial wave heights of H = 2 m, 5 m, 7 m and 10 m (H = 0.2h, H = 0.5h, H = 0.7h, H = h, h=10 m), note the changing vertical axis range. The free surface profiles shown for t = 0 s, 10 s, 20 s, 30 s, and 40 s.

5.9 Solitary wave in a long channel

183

2.0

z [m]

1.5 1.0 0.5 0.0 0

50

100

150

200

250

300

350

400

450

500

550

600

350

400

450

500

550

600

x [m]

2.0

z [m]

1.5 1.0 0.5 0.0 0

50

100

150

200

250

300 x [m]

Figure 5.42: Solitary wave propagation in a long channel with a flat bottom. Initial relative wave heights H = 0.2h. Above, the solution obtained with hydrostatic approximation, below, without it. The free surface profiles shown for t = 0 s, 10 s, 20 s, 30 s, and 40 s.

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Figure 5.43: Solitary wave profile for t = 40 s for various methods of free surface computation. less than 0.1%, and for H = 5 m increases to 0.2%. For conservative formulation, it is very small for H = 2 m (about 2×10−3 %) and increases to about 0.7% for H = 5 m. The dispersive effects behind the wave also diminish with the initial wave height. It is necessary for the two-dimensional velocity field at the surface to be filtered once as described in section 3.6.5 for the kinematic BC. Without filtering, the wave frequency dispersion appearing far behind the solitary wave crest leads to very short waves with wavelengths comparable with the mesh resolution (at t = 20 s), which causes growing numerical instabilities. (When instead of the velocity at the free surface, the free surface itself is filtered, then, for example, the wave height H diminishes from 2 m to 1.72 m for the implicit case.) For larger amplitudes, in the range of wave-breaking, the numerical oscillations grow quickly, leading to program break-up in the semi-implicit case. For the implicit algorithm, the results are not acceptable for the conservative equation, while the results with kinematic BC for all investigated wave heights are shown in figure 5.41. For the extreme initial wave height equal to the water depth (H = 10 m), i.e. already above the highest possible wave according to Laitone’s formula (and unphysical), the worst results are obtained. At t = 40 s, the amplitude diminishes to 4.3 m, 57% loss, with volume loss during computation of 0.5%. A very strong generation of secondary waves behind the travelling soliton and dispersive losses are observed. The results improve considerably with diminishing initial amplitude of the wave. For H = 7 m, H = 5 m, H = 2 m, the

5.9 Solitary wave in a long channel

185

amplitude decreases to 4.2 m, 3.7 m and 1.89 m, respectively, which are equivalent to 40%, 26%, and 5.5% amplitude loss, with volume (mass) error 0.2%, 0.1%, and 0.01%. The method of characteristics, applied for kinematic BC, yield the worst results for the same time-step (Courant numbers 0.2-1.0). While for initial wave height H = 2 m the mass loss is about 0.6% and the final height is H = 2.06 m, for initial wave height of 5 m a considerable growth to 6.99 m of final amplitude with a mass loss of 3% is obtained. The algorithm becomes quickly unstable for larger wave heights. For the case with H = 2 m, a comparative study with a solution obtained under hydrostatic approximation is performed, figure 5.42. In both approximations the semi-implicit formulation is applied. The hydrostatic (i.e. shallow water) solution shows a strong amplitude dispersion, resulting in a characteristic distorted wave form, where the wave crest moves with larger celerity than the wave slopes. The amplitude diminishes to H = 1.417 m (29% loss), compared to the non-hydrostatic final amplitude H = 1.89 m (5.5% amplitude loss) when kinematic BC is applied and H = 1.989 m (0.5% loss) when use is made of the conservative free surface equation (shown in figure 5.42). The non-hydrostatic wave crest travels with a mean celerity of 10.67 m/s or 10.77 m/s, respectively (figure 5.43), which is in good agreement with a theoretical speed of 10.85 m/s for the initial condition values and (5.56). The crest of the hydrostatic wave travels with a mean speed of 13.6 m/s. As a conclusion, in the newly developed algorithms the general configuration of the solitary wave remains unchanged, no amplitude dispersion can be observed. Much more care must be taken while calibrating models using the kinematic boundary condition than by the ones applying the conservative free surface equation. The worst results are obtained with the method of characteristics for kinematic BC. The frequency dispersion behind the wave appears due to the fact that the initial condition is not the exact solution of the solved equations (it is obtained with the hydrostatic approximation) and because of the interaction with the wall behind the wave. This dispersion effect increases considerably with initial height of the solitary wave. Not surprisingly, when this height is comparable with the water depth (unphysical case), the results are not acceptable anymore. The accuracy of the model grows for waves with smoother slopes and lower amplitudes. The results are thoroughly comparable, when instead of impermeable walls open boundaries with imposed hydrostatic pressure and simple non-reflecting boundary conditions for the free surface elevation at the boundary Sb are given at the channel ends: q

Sb = Seq ± (un+1 (h + η))/ g(h + η)

5.9.3

(5.58)

Solitary wave in a long channel with varying depth

In order to study the ability of the model to simulate finite amplitude wave interaction with varying bathymetry, a numerical experiment is performed in a very similar setting to before. In the long channel from the previous case the bottom profile is changed. The

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Figure 5.44: Profiles of solitary wave propagating in a long channel with variable depth. Initial relative wave height H = 0.2h. The non-hydrostatic solution (above) and the hydrostatic solution (below).

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Figure 5.45: Solitary wave propagation in a long channel with a bottom of variable depth. The free surface profiles shown for t = 0 s, 30 s, and 50 s. Iso-lines: vertical velocity component. Solution obtained with hydrostatic approximation.

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5.9 Solitary wave in a long channel

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Figure 5.47: Solitary waves profiles for t = 50 s for various methods of free surface computation. equilibrium water depth decreases now at the length between x = 160 m and x = 260 m from 10 m to 5 m. The initial solitary wave height is H = 2 m. The obtained free surface profiles are shown in figure 5.44. For the non-hydrostatic computation (using the semi-implicit conservative free surface equation), when the solitary wave passes through the area, where the water depth diminishes, the front of the wave steepens and then the wave divides into two waves. Further on, by t = 50 s, the wave separation becomes more evident, due to the fact that both waves move with different celerities, depending on their height. When the hydrostatic approximation is applied, no such effects are observed. The solutions for t = 0 s, t = 30 s and t = 50 s are additionally shown in figures 5.46 and 5.45. Comparison of the results when applying various free surface algorithms is shown in figure 5.47. The laboratory works of Madsen and Mei [107] yield for similar (but scaled) geometry the relative height of 160% for the larger wave and 71% for the smaller one, compared to the initial wave height; Hauguel [67] provides values 163% and 56%. Similar results (153% and 55%) were achieved in a numerical model by Schaper [139]. The results of the model developed here are best to be seen in figure 5.47. Various methods of computation yielded values in the range of 40-45% for the smaller wave and 110-210% for the larger one. In consequence, for modelling the influence of bottom topography on waves, the non-hydrostatic model yields very convincing results. However, when applying the kinematic BC for obtaining the free surface, greater care must be taken when calibrating the algorithm (figure 5.47).

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Figure 5.48: Collision between two solitons of equal amplitude; time series of free surface profiles obtained without (above) and without (below) hydrostatic approximation.

5.9 Solitary wave in a long channel

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Figure 5.49: Collision between two solitons of different amplitude; time series of free surface profiles. Initial wave heights H = 2 m, H = 4 m (H = 0.2h, H = 0.4h, h = 10 m).

5.9.4

Collision between two solitons

This test case is designed to test the capability of the model to describe wave-wave interactions. A head-on collision of two solitary waves in a channel with a flat bottom and water depth of h = 10 m is studied. During the collision, the vertical accelerations of the water particles are large and approximate analytical theories are no longer valid [102]; therefore this is a good example to test the properties of the non-hydrostatic model. For incident waves of equal amplitude, the highly reflective properties are observed, and after collision the wave characteristics should not change. The results are shown in form of a time series of water surface profiles. Only the results for the semi-implicit conservative free surface equation are shown; for other methods no new conclusions compared to these from previous sections can be formulated. The results are very good for the height of the waves H = 2 m (figure 5.48). At t ≈ 19 s, the maximum amplitude is 4.26 m, at t = 40 s, the amplitude of both waves equals 1.96 m, almost the same as in the case when the wave travels without any interaction, as in section 5.9.2. The wave profiles remain conserved. The results obtained with hydrostatic approximation are also shown in figure 5.48. In the case of solitons of different initial heights, the original waves after colliding are deformed due to dispersive effects and generation of secondary waves, especially behind the larger wave. The amplitude and celerity of the initial waves change after collision

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(in the real world cases, additional dissipation occurs). A collision between two solitons with initial wave heights H = 2 m and H = 4 m is shown in figure 5.49. As a conclusion, the newly developed model reproduces properly the wave-wave interaction of non-linear waves of a finite amplitude. The accuracy of the model diminishes with the incident wave steepness.

5.10

Limits of application

5.10.1

Free surface breaking

The algorithm developed in this work requires that the free surface and bottom must be represented by single-valued functions. A numerical experiment is performed in order to test the reaction of the developed algorithm situations when the physical conditions impose infinite free surface gradients, causing the free surface breaking. For example, such conditions are met in these flow regions, where transition between supercritical and subcritical free surface flow occurs. The experiment is performed in a similar setting to the supercritical inviscid channel flow, section 5.6. A ramp is built into a 15 m long channel. The bottom level is z = −2 m from the beginning of the channel to x = 7 m, then at a length of 1 m (between x = 7 m and x = 8 m) the bottom ascends up to a level z = 0 m, forming a very steep slope of 2:1 (200%), resolved by 10 nodes with a spacing of ∆x = 0.1 m. The initial free surface is flat and its level is at z = 1 m. The initial velocity is u = 6 m/s everywhere. 11 mesh levels with equal spacing is applied. The initial flow is supercritical, but an increase of depth over the value of 3.66 m immediately yields subcritical conditions (F r > 1). The change of the vertical profile of the initially occurring wave over time is shown in figure 5.50. When the subcritical conditions are achieved locally just before the ramp (x < 7 m), the occurring free surface distortion can travel upstream and conditions for wave breaking occur while the wave amplitude grows. (In the real world, a hydraulic jump eventually forms.) The algorithms based on the height function free surface tracking method (in this case SUPG, implicit) implemented in the model cannot describe such steep free surface slopes because they require the free surface to be a single-valued function. The computation breaks up a few time steps after the time the last profile is plotted, i.e. after t = 2.5 s. The break-up condition is achieved, when the free surface oscillations reach the bottom. A similar situation is observed, when the initial velocity is increased to u = 10 m/s everywhere. Then supercritical conditions are ensured in the whole area, even when the flow depth grows. The surface breaks by t = 2.0 s due to infinite free surface gradients, figure 5.51. Similar break-up of the free surface is obtained with 1:1 step (ascend from z = −1.0 m to 0 m by initial free surface at 1.0 m). Stable results are obtained (at this resolution) for a slope reduced to 1:2 (ascend from z = −0.5 m to 0 m), with a free surface slope of ca. 60%. Changing the number of the mesh levels does not improve the results.

5.10 Limits of application

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7

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Figure 5.50: Free surface profiles for a flow over a steep bottom step by conditions allowing appearance of a hydraulic jump. Profiles shown each 0.5 s. The free surface breaks by ca. 2.2 s. As a conclusion, model flow settings causing extremely steep free surface gradients must be avoided. As the heuristic tests show, the problems with obtaining stable solutions begin when the slopes reach and exceed π/4 (i.e. 1:1).

5.10.2

Bottom gradients for supercritical flow

In this section the stability of the numerical solution depending on the larger bottom (and free surface) gradients is investigated. When the triangles of the basis mesh are not far distorted from the ideal isosceles triangle for which the interpolation is most exact, the only source of the prismatic elements deformation are the free surface and the bottom. The computation of free surface gradients, which drive the flow, is erroneous in very distorted elements. In an analogous setting to the previous section, but with much larger initial water depth of 10 m and initial velocity of 20 m/s (supercritical conditions ensured everywhere), it is investigated at what maximum slope a steady flow can be achieved (step resolution as in previous chapter). Supercritical conditions are chosen, because in this case the free surface is much steeper than in the subcritical flow, which makes the test much more severe. It was found that a stable flow after an initial oscillation over the step can still be obtained for bottom gradients of 1:1 (it is, an ascent of the bottom from −1 m to 0 m level at a length of 1 m and resolved by 10 nodes in the flow direction). The free surface profile is shown in figure 5.52, for t = 60 s. By bottom gradient of 2:1, free surface gradients obtained during oscillations in the transient computation are too large, causing the program to break up. The hydrodynamic pressure distribution is similar

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8 7 6

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Figure 5.51: Free surface profiles for a flow over a steep bottom step by supercritical conditions. Profiles shown each 0.5 s. The free surface breaks by ca. 2.0 s.

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5.10 Limits of application

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Figure 5.53: Development of the free surface profile (shown each 1 s) for a solitary wave travelling from the left to the right with an initial relative height of H/h = 0.7. to the one shown in figure 5.29, but with much larger maximum values in the range of 10 kPa, i.e. comparable with the hydrostatic pressure values. As the heuristic tests show, the problems with obtaining stable solutions begin when the slopes exceed π/4. Additionally, the computation of the free surface gradients using the weak formulation in finite elements (3.31) can be erroneous in very distorted elements.

5.10.3

Breaking of a solitary wave

The solitary wave breaks, when its height exceeds 70-80% of the water depth (5.9.1). In a numerical model based on a height function free surface representation and which does not provide any artificial damping of the free surface, this condition will cause numerical oscillations, model break-up, mass losses, etc. For example, when as an initial condition in the test case for solitary wave propagating over a flat bottom (5.9.2) a wave height of H = 7 m by water depth of h = 10 m is given, wave-breaking occurs. The effects for the semi-implicit conservative free surface equation are shown in figure 5.53, large oscillations at the wave front cause the program to break up by t = 5.1 s. The model reacts properly for a physically infeasible initial condition. In the next test the limits of simulation of free surface flows are tested in a more natural setting, independent of initial conditions. A solitary wave whose initial profile is given by equations (5.52–5.56) propagates 200 m along a flat bottom in a depth of 10 m, and then approaches a shore along a shoaling bottom represented by a slightly sloped plane, with depth decreasing along a path of 400 m between 10 m to 0.1 m only. All other parameters are similar to those in section 5.9.2. The initial wave height is 5 m, also below

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the breaking condition in the water depth of 10 m. As the wave approaches the shore, at some point the conditions of wave breaking, when the wave height reaches about 80% of the equilibrium depth, are fulfilled (section 5.9.1). The model is unable to simulate the wave breaking due to the fact that the free surface is represented by a single-valued height function, but the model behaviour in this extreme case is to be tested. The results of this test are shown in the form of free surface profile time series in figure 5.54, for implicit kinematic BC and the implicit conservative free surface equation. As anticipated, when the wave height exceeds about 80% of the equilibrium water depth (wave breaking condition), the wave steepness is very large and the velocity at the wave crest is larger than the wave celerity. However, in the case of the kinematic BC, the model continues to compute further, but the wave peak of a height exceeding the breaking condition is simply cut out. By then, when large differences between the slopes of neighbour elements occur, considerable mass loss in the computational domain is observed. For the simulation time up to 30 s the mass loss is small, below 0.1%, further on (after 60 s) it reaches a large value of 10%. In the case of the conservative formulation, the program breaks up – the oscillations of the free surface reach the bottom. The model reacts as expected in a situation, when its limits of application are reached during the computation.

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Chapter 6 Summary and Conclusions Jede L¨osung eines Problems ist ein neues Problem. Johann Wolfgang von Goethe

6.1

Realised new developments

The newly developed algorithm, implemented in the Telemac system structure, addresses two important aspects. First of all, the vertical acceleration in free surface incompressible flows is taken into consideration by solving the vertical momentum conservation equation. Second, the free surface computation method allows simulation of its movements without limitation typical for shallow water equations – restriction to long waves as well as gentle free surface and bottom slopes. A few limitations remain due to the mesh structure (σ–mesh) and the linear interpolation in the finite elements, which e.g. do not allow breaking waves and introduce spurious errors by approximating stronger non-linearly varying functions. The new algorithm, whose theory is presented in chapter 2 and numeric implementation in chapter 3, is based consistently on the operator splitting scheme. This means that the operators in the hydrodynamic equations are split into parts and the resulting equations are treated in separate steps of the algorithm using the most appropriate methods for their mathematical character. Hydrodynamic variables are computed in subsequent stages (fractional step method). The overall algorithm stability and exactness is provided when all stages fulfil these requirements separately. The decoupled structure of the algorithm, which does not use the continuity equation explicitly, allows application of the equal-order interpolation functions in finite elements for both pressure and velocity components. Applying the equal order interpolation functions is not possible with coupled methods for incompressible flows due to the LBB condition. Additionally, a much simpler structure of the algorithm than for finite element methods with mixed interpolation functions results. The assumed reference element type, a prism with six nodes and linear approximation functions, is a good compromise between the exactness 199

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6.1 Realised new developments

of the solution, model complexity and computational cost. Compared to the widely used mixed interpolation, with linear interpolation functions for velocity and constant ones for pressure, much better resolution of the (hydrodynamic) pressure is obtained, which is very important in a domain of variable geometry. The main idea of the new non-hydrostatic model is to decompose the pressure into two physically interpretable parts, the hydrostatic and hydrodynamic pressures, the latter treated as a form of correction to the former. In consequence, the hydrostatic part is computed explicitly from the free surface elevation and density field, whereas the hydrodynamic pressure is found from a derived pressure Poisson equation (from fractional formulation). This elegant method is associated with well-known fractional step or projection methods of solving three-dimensional Navier-Stokes equations and has been applied before only in finite difference methods. The main advantage of this approach is that the final solution is much less sensitive to the well-posedness of the boundary conditions for the hydrodynamic pressure than in the case of the global one. In practice, very fine mesh spacing at boundaries is not required. In comparison to other projection methods the physical interpretation of the hydrodynamic pressure yields clearly defined and simple boundary conditions for the pressure Poisson equation. This circumvents obstacles which other researchers confronted. Because in most free surface flows the hydrostatic pressure gradient component usually dominates the hydrodynamic one, the requirement that the projection procedure should only be a correction to the intermediate solution for the velocity is fulfilled. This is a very important advantage, e.g. compared to the application of a Poisson equation for the global pressure. In the first step of the non-hydrostatic algorithm the hydrodynamic pressure is excluded from the consideration and the solution method follows the existing algorithm of Telemac-3D with the important exception that the vertical Navier-Stokes equation is treated as well. The advection and diffusion stages can be treated simultaneously (using SUPG formulation for advective terms) or decoupled (using the method of characteristics). In contrast to the previous hydrostatic algorithm, the driving forces such as free surface gradients (barotropic pressure gradients), the density gradients (baroclinic pressure gradients) and Coriolis force are treated in a fully three-dimensional manner in the diffusion step. The next two steps in the previous hydrostatic algorithm, the pressure-continuity steps, based on shallow water equations and yielding barotropic velocity component, free surface elevations and the vertical velocity component from the continuity equation, are abandoned. They are replaced by the continuity and free surface steps of the new algorithm. The new continuity step is based on the pressure Poisson equation from the fractional formulation. The provisional, intermediate solution, provided by the convection-diffusion steps is corrected by the non-hydrostatic component. This component is computed from the hydrodynamic pressure gradients under the assumption that the resulting final velocity must be divergence-free in the entire domain (incompressible flow) and satisfies appropriate boundary conditions. In this step the formal velocity projection is performed. The continuity step could be repeated iteratively, aiming for more exact fulfilment of the continuity equation and boundary conditions for flows with hydrodynamic pressure gra-

6.2 Result discussion

201

dients comparable to the hydrostatic gradients, but one iteration proved to be sufficiently exact. Several techniques for finding the free surface position are implemented. Two of them make use of the kinematic boundary condition, which is a hyperbolic equation for the free surface elevation using the velocity at the surface as the advecting field: (1) a scheme based on the method of characteristics and (2) on the semi-implicit SUPG method. Another method is based on the vertically integrated continuity equation (the free surface conservative equation) and solved using semi-implicit SUPG method or entirely explicit formulation. Finally, for applications where the free surface elevation should be obtained from the shallow water equations for some reason, but a non-hydrostatic correction for the flow is needed, a modification of the previous pressure-free-surface step can be applied.

6.2

Result discussion

The main attraction of the presented developments lie in the fact that they were made in the framework of an existing, well-verified modelling system, which makes immediate application possible in various related fields of the geophysical flows, such as water quality or sediment transport. The verification of the realised program developments which is documented in chapter 4, ensure the quality of the software. Furthermore, a broad range of event-oriented verification cases, presented in chapter 5, cover most of the intended extension of the application domain of the initial hydrostatic model. The test cases allow critical assessment of the properties of the new algorithm. The model is now capable of reproducing various physical phenomena, which were out of reach before. The main obstacles for flow simulations, which must be treated without limiting approximations, are removed. The provided evidence substantiate the claim that the developed model is an advanced, entirely threedimensional free surface model for incompressible flows with a broad application domain. The main theoretical difficulties in the development of this code were encountered when choosing the most appropriate approach for the pressure treatment and the free surface computation, discussed in chapters 1 and 2. The ambiguities due to providing well-posed boundary conditions excluded the application of an equation for the global pressure. Similar difficulties were encountered when trying a global iterative solution using a pressure correction term, as in the SIMPLE algorithm family. The finally successfully implemented method of splitting the pressure into physically interpretable hydrostatic and hydrodynamic parts (as e.g. applied by Casulli and Spalding [20]), which is a modification of the projection method idea, suits the operator-splitting procedure of Telemac3D well. The boundary conditions are clearly defined in this case. A projection-2 method was implemented, but it did not show any advantages and no test cases are presented. The method based on the pressure Poisson equation for the hydrodynamic pressure is robust enough to yield good results even when the hydrodynamic pressure influence is as large as the hydrostatic one.

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For the free surface, the methods based on the semi-implicit SUPG for the kinematic boundary condition and the conservative free surface equation proved to be fully satisfactory. In contrary, the computations based on explicit formulation for the conservative free surface show too large numerical damping or instabilities for larger Courant numbers. The method of characteristics applied for kinematic BC, however unconditionally stable, is the least mass conservative from all methods mentioned above, excluding it from applications with larger free surface variability. Methods based on marker-in-cell (MAC) or volume-of-fluid (VOF) approaches were not considered. Their greatest advantage is their ability to describe the most complex free surface geometries (e.g. breaking waves, droplets). Their disadvantage is larger demand for machine memory and processor time, as well as in difficulties in implementing specific boundary conditions such as surface stresses. For the σ-mesh structure of Telemac3D, which excludes free surface shapes which cannot be described by a single-valued function, the most attractive advantages of these methods cannot be exploited and only the disadvantages remain. As the verification cases show, the requirements for the boundary conditions for the free surface at an open out- or inflow boundary are severe. Special boundary condition types such as non-reflecting or absorbing boundary conditions are strongly recommended for practical applications. Other problems to be solved were connected with harmonisation of boundary conditions applied for velocity in different parts of the operator-splitting algorithm and those for free surface and the hydrodynamic pressure. For successful model applications the following requirements are important: 1. universality of application for various purposes; 2. user-friendly peripheries; 3. sufficient computation precision; 4. possibly low computational effort. From the point of view of model universality, the application domain of the developed model is much wider than previously. The non-hydrostatic algorithm presented in this work opens new possibilities for dealing with many free surface flow classes, which could not be modelled properly with a hydrostatic model. As it has been shown, a larger domain of free surface and internal waves processes is now included (waves of various wavelengths, dispersion, reflexion, diffraction, interference). Compared to the wave models (e.g. Boussinesq wave model [142]), the model yields not only the free surface configuration (and fluid flux) but also the three-dimensional current structures underneath. The density flows and various flows concerning vertical circulation, where vertical accelerations play an important role, can be better simulated as well. Additionally, the model delivers the global pressure field and therefore the interactions between free surface flows and structures can be modelled. Due to the fact that Telemac3D belongs to a larger, coupled family of models, using common subroutine libraries (Telemac–System, [73]), which is being continuously

6.3 Model applications and limitations

203

tested and developed in a number of institutes, the advantage of the non-hydrostatic algorithm in many application domains can be taken immediately. Telemac system includes libraries allowing modelling of transport (e.g. salinity, passive pollutants, etc.) and applying various turbulence models, therefore the way is open for more advanced applications. These include simulation of sediment transport, pollution, water quality, thermal impacts, more complex turbulence effects, particle-tracking and waves. Being embedded in the overall Telemac system means that user-friendly post- and pre-processing is available. However, through the new developments, the complexity of the models and therefore the number of possible user errors have grown. The code is written in such a way, that by switching one parameter in the input list, it produces hydrostatic or non-hydrostatic results, so that a direct comparison of results with identical input parameters (but, according to case, different boundary conditions) is possible. Several test applications are presented for illustration of the effect of taking into consideration or neglecting the hydrodynamic pressure and for model verification. The computational effort compared to the previous, hydrostatic version is greater. Although on one hand a larger algorithm step is abandoned (pressure – free surface – continuity), on the other hand the diffusion-convection stage for the vertical velocity component, the solution of the Poisson equation for the dynamic pressure and finding the new free surface position are introduced. The required computational time can be compared to the time consumed by the previous version with simultaneous transport of two tracers (e.g. temperature, salinity).

6.3

Model applications and limitations

In practice, the developed model can be applied to those geophysical or engineering problems, where a more proper reproduction of the physical flow properties, a high accuracy level and an entirely three-dimensional approach is required. The price for this precision is the higher computational cost. The applications are very important not only for practical reasons, but also for gaining experience and defining further development needs. The application of this model for the following fields should be straightforward: • Wind and density driven flows in larger water bodies such as lakes with taking complex bottom geometries (and steeper slopes) into consideration. • Current (or waves) interaction with structures, especially in harbours or coastal areas. • Wave-induced currents over complex bottom topography. • Density currents, for example the salt intrusions in estuaries, or convective circulations in lakes or oceans. • Internal waves of various kinds (e.g. for thermocline variability).

204

6.4 Recommended further developments

In this stage another, very important aspect of developing fully three-dimensional codes must be mentioned. Many three-dimensional features of the flow can be parameterised and introduced into two-dimensional models or even one-dimensional ones. Studies with three-dimensional models allow a very effective parameterisation of these features in form of coefficients or corrections introduced into models of lower complexity. In this way three-dimensional phenomena can be taken into consideration in computationally much more effective codes, which is of paramount importance for practical engineering applications. In the particular case of Telemac system, the interface between three-dimensional and two-dimensional models is straightforward, because both use the same basic mesh. The most important advantages of this approach can be found in the domain of sediment transport, pollutant spreading, the interaction structures-currents and wave problems.

6.4

Recommended further developments

Further developments of the model presented here are listed in the sequence of diminishing importance. They could include: • Mesh adaptivity. The developed algorithm is not exclusively applicable to the σ-mesh consisting of prismatic element tiers. The algorithm should work properly for elements allowing meshing of arbitrary three-dimensional domains, as e.g. tetrahedrons, which would open the possibility of mesh adaptation taking place in all three directions. In the framework of the operator-splitting algorithm consisting of steps which would require different mesh adaptivity from step to step, the adaptivity criteria should probably be dictated by gradients of the hydrodynamic variables. Development of efficient mesh generators or mesh refining/coarsening routines would be required in order to allow continuous mesh adaptation to the variable flow conditions and/or geometry. • Non-reflecting boundary conditions for the open (i.e. artificial) boundaries of the computational domain should be intensively investigated for various flow types. In the model described in this work, the simplest forms of NRBC are applied and absorbing boundary conditions were developed. The role of non-reflecting boundary conditions is important in securing stable solutions for transient flows, and for effectively converging computations of stationary flows as well. In more advanced applications in the field of free surface and internal waves some kind of these boundary conditions must always be imposed. • Variable domain boundaries. The σ-transformation is a traditional solution for the mesh adaptation to the movements of the free surface, which works well for most typical geophysical applications. Improvements allowing application of various vertical discretisation in different parts of a computational domain, nonvertical lateral boundaries or introducing mesh levels defined by surfaces of equal

6.4 Recommended further developments

205

density (isopycnal co-ordinates) would allow more accurate and efficient computations for complex bottom topography and coastal structures and for the presence of stratification. Moreover, solutions based on the Arbitrary Lagrangian-Eulerian (ALE) method would probably allow mesh adaptation to boundary movements in all three directions, opening the possibility of modelling interactions between fluid and moving, deformable structures and in an arbitrary geometry. This approach connected with specific free surface or volume tracking methods can also be applied for modelling of waves of larger amplitude, dam breaks, tidal flats, etc. The required computational effort may, however, grow considerably. • Stability estimates should be further investigated and built-in stability control (e.g. adapting the time step) considered for various algorithm steps. • Free surface algorithms. The free surface tracking algorithms are accurate, but very sensitive to the imposed boundary conditions and less mass conservative than volume-tracking methods. Alternative free surface algorithms, such as volumeof-fluids or marker-in-cell, or especially mixed methods, should be investigated in order to find equally accurate and stable solution methods comparable to or better than the height function methods. These special algorithms can also be applied to multiphase flows and for tracking internal interfaces between fluid/waters of different properties. • Overall algorithm structure. In this work, the decoupled method of solving the hydrodynamic equations for free surface flows is applied. However, for turbulence or transport modelling, the coupled approach may prove to be equally or more effective, maybe not for the global algorithm, but for its parts. This can be very attractive, especially if quickly converging methods for solving arbitrary and large (non-linear) equation systems become available. On the other hand, the projection-2 method should be further investigated. It is promising for flows where hydrodynamic pressure gradients are comparable or greater than the hydrostatic pressure ones. • Turbulence modelling. The developments presented in this work have purposefully been made independently of any actual turbulence modelling approach. However, they provide a good basis for further development in this field. Although FEM is still thought to be computationally too expensive for turbulence modelling methods requiring very fine meshes, the advantage of the decoupled method applied in this model is the possibility to use the simplest linear finite elements available. For modelling turbulent flows in complex geometries the FEM (or FVM) approach is thought to be the most appropriate one for unstructured meshes. The flexible and open Telemac algorithm structure provides a means to investigate various turbulence modelling methods. In Telemac3D constant eddy viscosity, mixing length and k-ε turbulence models are already available and there are no obstacles to developing further, more modern or application-specific ones. A decoupled finite element method has already been applied successfully with Large Eddy Simulation (LES), e.g. with an application for lakes [48].

206

6.4 Recommended further developments

Appendix A Various derivations A.1

Incompressibility

One of the most widely used approximations is to treat the water in the geophysical flows as an incompressible fluid, i.e. a fluid whose density depends on temperature and dissolved substances concentration, but not on the pressure. The most important consequence of this approximation is that the velocity field in the incompressible flows is non-divergent, i.e. solenoidal. The mass conservation equation in its general form is: 1 d̺ +∇·u=0 ̺ dt

(A.1)

Taking an equation of state ̺ = ̺(p, s, T ) into consideration, the density derivative in (A.1) can be expanded as a function of state variables: 1 ∂̺ dp 1 ∂̺ ds 1 ∂̺ dT 1 d̺ = + + ̺ dt ̺ ∂p dt ̺ ∂s dt ̺ ∂T dt which can be transformed to: 1 d̺ 1 dp α ds β dT = 2 + + ̺ dt ̺cs dt ̺ dt ̺ dt

(A.2)

(A.3)

where α and β are expansion coefficients with respect to the salinity and temperature and cs is the speed of sound. In the case of isentropic motion in the absence of viscous or diffusive effects, ds/dt = 0 and dT /dt = 0, and the mass conservation equation can be written as: (̺c2s )−1

dp +∇·u=0 dt

(A.4)

The simplest form of the isentropic motion is when the fluid element does not exchange heat with the surroundings and retains fixed chemical composition. After Batchelor [5] the motion is not isentropic when [5]: 207

208

A.2 Coriolis terms

1. Radiative energy exchange with the surroundings takes place. 2. Heat exchange through molecular conduction occurs: the flux of heat is given by the temperature gradient −k∇T (k is the thermal conductivity). 3. Heating or cooling due to change of phase, chemical reaction or to viscous dissipation takes place. Batchelor [5] gives the following three requirements to be fulfilled in order to neglect the pressure derivative in (A.4), i.e. to treat the fluid as incompressible (in which changes of the density with pressure are negligible): 1. The fluid particle velocity must be small compared with the speed of sound cs . 2. The phase speed (wavelength divided by period) of disturbances must be small compared to the speed of sound cs . 3. The vertical scale of motion must be small compared with the scale height defined as a mean value of ̺/|d̺/dz|. The scale height defined above is much greater than the depth for most geophysical flows. E.g. in the ocean this scale height is about 40 times greater than the mean ocean depth. Speed of sound in seawater is cs ≈ 1500 m/s. The greatest speed of disturbances, the phase speed of long gravitational waves, c = (gH)1/2 , is equal is about 200 m/s in the ocean at 4 km depth. The assumption of the incompressibility of the water is therefore appropriate for most hydrodynamic processes. The usual incompressibility approximation is therefore to assume that the density is independent of pressure and take account of the density changes due to temperature and salinity changes through the equation of state only, without taking these changes into consideration in the mass conservation equation. The velocity field in this approximation is non-divergent (or solenoidal): ∇·u=0 (A.5) So formulated, the approximation implies that volume rather than mass is conserved. Indeed, the volume is conserved while the density changes as a function of temperature and salinity (not pressure). Therefore this approximation should be removed from the basic equations when the effect of the thermal expansion is to be reproduced exactly, e.g. to cope with the water body surface position changes induced by heating or cooling [114].

A.2

Coriolis terms

The Coriolis term arises in the momentum equation, because the mathematical description of the flows is formulated in a non-inertial co-ordinate system fixed to the earth,

A.2 Coriolis terms

209

which rotates around its axis. The equation of motion F = ma is valid, when a is measured in an inertial system, i.e. in a co-ordinate system, whose origin is not accelerating. For practical reasons, as a reference inertial system, the system fixed relative to distant stars is applicable. The transformation from the fixed inertial system to the rotating system of the earth can be found in most authoritative books on mechanics. The derivation is shortly repeated here for completeness. If i, j, k is a triad of orthogonal unit vectors in the non-inertial system fixed in the rotating frame, then any vector s can be written in this system as s = s1 i + s2 j + s3 k. The rate of change of the vector s with respect to the time t appears as a result of change of its components in the moving system and due to the change of the unit vectors as the coordinate frame rotates with the angular velocity Ω. It is assumed that no translational acceleration of the non-inertial system appears. Ω is measured in the fixed, inertial system. For the observer in the inertial frame: ds dt

!

=

X i

dsi di i + si dt dt

!

=

X i

!

dsi i + si Ω × i = dt

ds dt

!e

+Ω×s

(A.6)

where the superscript e means relative to the rotating system. The same formula can be applied for the velocity u of a fluid element on the surface of the earth (measured in the inertial system): !e dR + Ω × R = ue + Ω × R (A.7) u= dt where, R the distance of the moving fluid element from the earth’s centre. Analogically, for the acceleration a, measured in the absolute frame: a=

du dt

!e

+ Ω × u = ae + Ω × u

(A.8)

It should be mentioned that Ω is the angular velocity of the earth’s rotation relative to the inertial system fixed to the distant stars (Ω = 2π/86164 rad s−1 = 7.292×10−5 rads−1 , whereby 86164 s = 23 h 56 min 4 s is one sidereal or star day). (Ω measured in the non-inertial system of the earth is obviously equal to 2π/86400 rad s−1 , i.e. 2π radians per solar day, 24 h). Introduction of (A.7) to (A.8): d a= dt

dR dt

!e

+Ω×R

!e

+Ω×

dR dt

!e

+Ω×R

!

(A.9)

yields after some algebra: a=

dua dt

!a

dΩ = ae + dt

!e

× R + 2Ω × ue + Ω × (Ω × R)

(A.10)

The terms on the right hand side of (A.10) represent: (1) acceleration in the system fixed to the earth, (2) apparent acceleration due to changes in the rotation velocity,

210

A.2 Coriolis terms

(3) apparent Coriolis acceleration, (4) apparent centripetal acceleration. It is usually assumed that the earth rotates steadily and the second term is therefore ignored. The equation of motion in an inertial system: du dt

!

i

1 = − ∇p + gi + f ̺

(A.11)

transforms into: due 1 = − ∇p − 2Ω × ue + gi − Ω × (Ω × R) + f dt ̺

(A.12)

The transformation yields two new apparent forces per unit mass, appearing in the noninertial system of the earth, relating to the Coriolis and centripetal accelerations. The true forces remain unchanged. The centripetal acceleration Ω × (Ω × R) = −Ω2 R, required to make a body rotate with the earth with the angular velocity Ω, and the gravitational acceleration gf have the same direction approximately on the equator only. However, the maximum value of the magnitude of the centripetal acceleration is only about 0.3% of the gravitational acceleration. Therefore, the centripetal acceleration is treated in practice, in the coordinate system fixed to the earth, as a small correction (in magnitude and direction) to the gravitational acceleration. The difference gi − Ω × (Ω × R) is introduced in the equations and is called the acceleration due to gravity g. The local orthogonal Cartesian co-ordinate system is assumed on the rotating earth, where x is directed eastward, y is northward, and z points vertically upward. The earth’s angular rotation velocity vector Ω is at the geographical latitude ϕ inclined to the y-direction (assumed here to be North). In such a system, the Coriolis acceleration has the following components: 







2wΩ cos ϕ − 2vΩ sin ϕ i j k     2Ω × u =  0 2Ω cos ϕ 2Ω sin ϕ  =  2uΩ sin ϕ  2uΩ cos ϕ u v w

(A.13)

Introducing horizontal and vertical Coriolis coefficients fH = 2Ω cos ϕ and fV = 2Ω sin ϕ: 



fH w − fV v   fV u 2Ω × u =   fH u

(A.14)

Usually, when the horizontal velocity scale is significantly larger than the vertical one and the hydrostatic approximation applies, just those terms, in which the horizontal Coriolis coefficient fH = 2Ω cos ϕ appears, are neglected. Only terms with the vertical one, fV = 2Ω sin ϕ remain. In this simplified case the Coriolis force has the following components (f = fV ): 







−f v 2wΩ cos ϕ − 2vΩ sin ϕ     2Ω × u =  2uΩ sin ϕ  ≈  fu  0 2uΩ cos ϕ

(A.15)

A.3 Small amplitude free surface waves in a homogeneous fluid

211

For simulation of fully three-dimensional flows, when all motion directions and scales are equally treated, the form of the Coriolis term (−2Ω × u) in the Navier Stokes equations in the rotating frame (2.2) must also be fully three-dimensional. It should be noted, that very close to the equator (ϕ = 0) the vertical Coriolis coefficient vanishes and the horizontal prevails and care is needed in treatment of the rotational effects. Because of this latitudinal dependence of the Coriolis parameter, the neglecting of terms with fH are also called upper ocean approximation. For simulation of larger areas (seas, ocean parts) using the rectangular co-ordinate system, appropriate assumptions depicting the latitudinal variability of the Coriolis parameters and the curvature of the earth are made. The usual approximations are the β-plane and the f -plane approximation. In both cases the earth’s curvature is neglected. In the β-plane approximation the origin of the Cartesian, rectangular co-ordinate system is centred at the latitude ϕ0 and a linear dependence of the vertical Coriolis parameter on the y-co-ordinate is assumed in the form fV = f0 + βy, where f0 = 2Ω sin ϕ0 , β = 2Ω cos ϕ0 /R and fH = 2Ω cos ϕ0 . In mid-latitudes (|ϕ| ≈ 45o ) |f0 | ≈ 10−4 rad/s and β ≈ 1.6 · 10−11 rad/s. In the f -plane approximation the Coriolis parameters are assumed to be constant in the entire computational domain. This approximation is valid as long as L cos ϕ0 /R << 1, L being the horizontal motion scale, the condition needed in order to neglect the parameter β. The application limits of these approximations are as follows. When the horizontal motion scale L is comparable with the radius R of the earth, the equations of motion in spherical co-ordinates and Coriolis terms without any approximations must be used. For intermediate scales up to L/R = O(10−1), the β-plane approximation is suitable, whereas for smaller scales f -plane approximation is appropriate.

A.3

Small amplitude free surface waves in a homogeneous fluid

In this section the linear theory of free surface waves is reviewed. For simplification, the non-rotating case (Ω = 0) is taken. A formal discussion of small amplitude wave theory can be found in many books treating water waves, e.g. [99, 100]. Here only a sketch of this theory is presented for completeness. Let us consider a incompressible and inviscid fluid of constant density ̺ in an equilibrium state and of uniform depth H (for instance, in a pool with a flat bottom). The position of the free surface is z = 0 and bottom co-ordinate is z = −H, i.e., z axis points upward and (x, y) axes form a horizontal plane. The atmospheric pressure is neglected, patm = 0. The equilibrium pressure p0 (z) is given by: p0 (z) = −̺gz

(A.16)

212

A.3 Small amplitude free surface waves in a homogeneous fluid

Let us assume a small perturbation of the equilibrium state of the free surface, described by a single-valued function: z = η(x, y, t) (A.17) The pressure is given by the sum of the hydrostatic and hydrodynamic parts π: p = ̺(η − z) + π

(A.18)

The perturbation pressure is given by: p′ = p − p0 = ̺gη + π

(A.19)

The perturbation causes a movement to appear. However, the perturbation quantities, e.g. velocities, are enough small that their products can be neglected. The linearised momentum conservation equations for small perturbation and continuity equation (2.7) reduce to the following set of equations: ̺∂u/∂t = −∂p′ /∂x ̺∂v/∂t = −∂p′ /∂y ̺∂w/∂t = −∂p′ /∂z

(A.20)

∂u/∂x + ∂v/∂y + ∂u/∂x = 0

(A.21)

Adding derivatives of the equations (A.20) and using (A.21) results in obtaining the Laplace equation for p′ : ∇2 p ′ = 0 (A.22) The boundary condition at the flat bottom is: w = 0 for z = −H

(A.23)

The kinematic boundary condition (2.6.3.1) at the surface applies (condition that a particle at the free surface z = η remains there): d(η − z) ∂η ∂η ∂η = +u +v −w =0 dt ∂t ∂x ∂y

(A.24)

(A.24) reduces for small perturbations to: w=

∂η ∂t

for z = η

(A.25)

When the surface tension is neglected (i.e. capillarity effects excluded), and the atmospheric pressure set to zero, the pressure must vanish at the surface. The boundary condition for the perturbation pressure at the free surface is: p = p0 + p′ = 0

⇒

p′ = ̺gη

for z = η

(A.26)

A.3 Small amplitude free surface waves in a homogeneous fluid

213

i.e. perturbation pressure near the surface is hydrostatic. Combining the third equation of (A.20) with (A.23) one obtains the boundary condition for perturbation pressure at the bottom: ∂p′ /∂z = 0 for z = −H (A.27) Because the difference between z = η and z = 0 is small, the boundary conditions at the free surface (A.25–A.26) can also be applied for z = 0. The solution of this problem depends on the initial perturbation of the free surface. In following it is assumed, that the basic disturbance has a sinusoidal form with an amplitude η0 , frequency ω and wavenumber vector k = (k1 , k2 ) (κ = |k| = 2π/λ, λ being the wavelength): η = η0 cos(k1 x + k2 y − ωt) = η0 cos φ (A.28) and arguing that an arbitrary disturbance can be described as a superposition of such sinusoidal waves. Assuming that the perturbation pressure π is proportional to η in the form of (A.28), the Laplace equation (A.22) yields ∂ 2 p′ − κ2 p′ = 0 ∂z 2

(A.29)

The solution of (A.29) is to be searched in the form of a combination of exponential functions in the imaginary domain. Taking the boundary conditions (A.26) and (A.27) one obtains: ch[κ(z + H)] p′ = ̺gη0 cos(k1 x + k2 y − ωt) (A.30) ch(κH) and from (A.20): u = gη0

k1 ch[κ(z + H)] cos(k1 x + k2 y − ωt) ω ch(κH)

(A.31)

v = gη0

k2 ch[κ(z + H)] cos(k1 x + k2 y − ωt) ω ch(κH)

(A.32)

κ sh[κ(z + H)] sin(k1 x + k2 y − ωt) ω ch(κH)

(A.33)

w = gη0

In order to satisfy the kinematic boundary condition at the free surface (A.25) for z = η ≈ 0, the following condition must be satisfied: ω 2 = gκ tanh(κH)

(A.34)

which is the well-known dispersion relation determining the frequency and phase speed (celerity) c = ω/κ of gravitational waves. As can be seen from (A.30), perturbation pressure decreases with depth, and: 1 π(x, −H, t) = π(x, 0, t) cosh(κH)

(A.35)

214

A.3 Small amplitude free surface waves in a homogeneous fluid

This ratio decreases rapidly with increasing κH. When κH << 1 perturbation pressure does not change with the depth. For k2 = 0, the movement takes place in the (x, z)-plane. The tip of the velocity vector describes an ellipse; the ratio between the vertical and horizontal axes of this ellipse is: |w| = tanh[κ(z + H)] |u|

(A.36)

The ellipse flattens with the depth and the ratio (A.36) vanishes at the bottom for z = −H as w = 0. For κH >> 1, i.e. in deep water |w|/|u| ≈ 1 and the ellipse reduces to a circle. Two very important approximations for gravitational waves apply. They are formulated with the length scale being the fluid equilibrium depth H. 1. Short waves for κ−1 << H, called also deep-water waves. The dispersion relation (A.34) can be approximated to: ω 2 = gκ (A.37) and the perturbation pressure is from (A.30): p′ = ̺gη0 exp(κz) cos(kx + ly − ωt)

(A.38)

The pressure perturbation, having a maximal value at the surface, vanishes exponentially with the depth. The phase speed c is a function of κ, i.e., short waves are dispersive. The velocity components are: u = gη0

k1 exp(κz) cos(k1 x + k2 y − ωt) ω

k2 exp(κz) cos(k1 x + k2 y − ωt) ω κ w = gη0 exp(κz) sin(k1 x + k2 y − ωt) ω

v = gη0

(A.39) (A.40) (A.41)

2. Long waves for κ−1 >> H, also called shallow-water waves. The dispersion relation (A.34) is again approximated to: ω 2 = κ2 gH

(A.42)

p′ = ̺gη0 cos(kx + ly − ωt)

(A.43)

The perturbation pressure is:

The pressure perturbation is independent of depth. The long waves are nondispersive, because their phase speed c = (gH)1/2 is independent of the wavenumber. The velocity components are: u = gη0

k1 cos(k1 x + k2 y − ωt) ω

(A.44)

A.4 Equations of state for seawater

215 k2 cos(k1 x + k2 y − ωt) ω

(A.45)

κ2 (H + z) sin(k1 x + k2 y − ωt) ω

(A.46)

v = gη0 w = gη0

The horizontal velocity component is independent of the water depth and the vertical component changes linearly with the depth. Because the considered fluid has a constant density ̺ (perturbation of density is zero), the perturbation pressure for the long wave case (A.43) has the same value that would be obtained if the pressure were calculated from the hydrostatic equation. The horizontal velocity component does not change with the water depth and the influence of the free surface movement reaches down to the bottom. Therefore, for the fluid of constant density, the hydrostatic approximation and the long-wave (or shallow-water) approximations are equivalent.

A.4

Equations of state for seawater

A.4.1

EOS–80: UNESCO equation of state

The UNESCO EOS–80 (International Equation of State for Seawater, 1980) equation of state for seawater is an internationally accepted standard for geophysical flows, defined in 1980 by the Joint Panel on Oceanographic Tables and Standards (UNESCO, 1981) [160]. The density of seawater ̺ (kg/m3 ) is treated as a function of the practical salinity s [PSU], temperature T (o C) and pressure p [bar] (e.g. for deep-sea applications). The equation is formulated in an algorithmic form; the relevant parts of the computations can be omitted when e.g. pressure can be assumed constant. The salinity is given as a dimensionless variable in practical salinity units, PSU 1 . This unit is for practical purposes almost identical with ppt (parts per thousand). This equation is valid for the following ranges of parameter variability: T s p 1

= −2 = 0 = 0

to to to

40o C, 42, 1000 bar,

’Practical salinity, 1978, definition: The practical salinity, symbol S, of a sample of seawater, is defined in terms of the ratio K15 of the electrical conductivity of the seawater sample at the temperature of 15o C and the pressure of one standard atmosphere, to that of a potassium chloride (KCl) solution, in which the mass fraction of KCl is 32.4356 × 10−3 , at the same temperature and pressure. The K15 value exactly equal 1 corresponds, by definition, to a practical salinity exactly equal to 35. The practical salinity is defined in terms of the ratio K15 by the following equation: 1/2 3/2 5/2 2 S = 0.0080 − 0.1692K15 + 25.3851K15 + 14.0941K15 − 7.0261K15 + 2.7081K15 , formulated and adopted by the Unesco/ICES/SCOR/IAPSO Joint Panel on Oceanographic Tables and Standards...etc. ...This equation is valid for a practical salinity S from 2 to 42.’ [160]

216

A.4 Equations of state for seawater

with an exactness of 3.5 · 10−6 (standard error). In the first step the reference density of pure water is computed as a function of temperature T : ̺w = a0 + a1 T + a2 T 2 + a3 T 3 + a4 T 4 + a5 T 5 . (A.47) a0 a1 a2 a3 a4 a5

= = = = = =

+ 999.842594, + 6.793952e-2, – 9.095290e-3, + 1.001685e-4, – 1.120083e-6, + 6.536332e-9.

Then the density by one standard atmosphere (i.e. by normal atmospheric conditions at the free surface, p = 0) is obtained as a function of salinity s: ̺(T, s, 0) = ̺w + (b0 + b1 T + b2 T 2 + b3 T 3 + b4 T 4 )s + (c0 + c1 T + c2 T 2 )s3/2 + d0 s2 . (A.48) b0 b1 b2 b3 b4

= = = = =

+ 8.24493e-1, c0 – 4.0899e-3, c1 + 7.6438e-5, c2 – 8.2467e-7, + 5.3875e-9, d0

= – 5.72466e-3, = + 1.0227e-4, = – 1.6546e-6, =

+ 4.8314e-4.

Finally, the influence of pressure p can be taken into consideration by: ̺(T, s, p) = ̺(s, T, 0)/(1 − p/K(s, T, p))

(A.49)

where K is the compressibility of seawater: (K = −1/V dV /dp = 1/̺d̺/dp ) which is obtained from a procedure containing polynomials with empirical constants. It should be noted that the reference densities used in the equation of state should not be identified with the reference average density ̺0 appearing due to the Boussinesq approximation in the momentum conservation equation (section 2.1.4).

A.4.2

Simple forms of the equation of state

The density variations can be treated as a function of active tracer concentrations: ∆̺ = f (T1 , ..., Tn ) ̺0

(A.50)

The tracer concentrations are transported according to the transport equation (2.3). For small variations of density and the tracer concentrations, a linearised form of the equation of state can be applied: X ∆̺ βi (T − T0 )i =− ̺0 i

(A.51)

A.4 Equations of state for seawater

217

Figure A.1: Mesh used in various verification cases for a square domain (Ekman profiles, free surface tests, waves reflection, wind-driven circulation, standing wave in a closed basin, etc.) with 2601 nodes and 5000 elements. The mesh size is scaled from case to case. where βi are dilatation coefficients (in the simplest case constant, for example positive for temperature or negative for salinity) and (T0 )i are reference values for tracers i. For particular applications, some optimised equations of state can be obtained. For example, for thermal transport in North-European stratified estuaries of lower salinity, the following equation of state in form ̺ = ̺(T, s) proved to be accurate enough [101]: ̺ = ̺0 [1 − αT (T − T0 )2 + αs s]

(A.52)

where ̺0 = 999.972 kgm−3 , T0 = 4o C, αT = 7.0 · 10−6 K−2 , αs = 0.75 · 10−3 PSU−1 .

218

A.4 Equations of state for seawater

Figure A.2: The channel basic mesh with 1060 nodes and 1806 elements applied in the lock exchange flow test, as well as standing wave in a channel and the flow over a ramp tests. Dimensions scaled from case to case.

Figure A.3: The channel mesh with 1147 nodes and 2114 elements used in the channel with a bump test.

A.5 Meshes

A.4.3

219

Suspended sediment influence on the water density

The suspended sediment concentration can also be taken as a parameter affecting water density and stratification. The dependence of the fluid density on the dry mass sediment concentration c [kg/m3 ] is taken as: ̺(p, S, T, c) = ̺(p, S, T ) +

̺sed − ̺(p, S, T ) c ̺sed

(A.53)

where ̺(p, S, T ) is the water density obtained from the EOS 80 equation of state and ̺sed is the sediment constituent density [kg/m3 ].

A.5

Meshes

The basic (2D) meshes applied in various test and verification cases presented in chapters 4 and 5 are shown in figures A.1, A.2 and A.3. For all test cases, the horizontal mesh spacing is uniform all over the computational domain. Only in three verification cases (Ekman profiles, section 4.1, interfacial internal waves, section 5.3 and waves over an underwater channel, section 5.8) the variable spacing in the vertical direction is used. In all other cases, the per cent distribution of the mesh levels is uniform, without larger discrepancies in the vertical and horizontal resolution.

A.6

Systematic approach to model development

A model, understood as an abstraction of reality, a mental, physical or purely mathematical representation or description of a system, can be implemented on a computer only if the model can be converted into an algorithm. The process of developing and implementation of a computer model consists of physical system definition, definition of the mental or conceptual model, its algorithmic implementation and finally, expressing it in terms of usable computer program, i.e. software. The definition of the physical system to be described is the fundamental step in the development process determining further capabilities of the model. It consists of collection and description of physical processes or phenomena relevant to the purpose of the model. The modelling process requires that these physical mechanisms and relevant effects of the model environment are formulated in terms of model variables. Usually in this stage, the formulation of the conceptual model being a mathematical (or verbal, logical) representation of the system, usually several approximations and model assumptions are introduced. The computer model requires that the conceptual model must be expressed in terms of a set of prescriptions or rules suitable for computation: an algorithm. The last development stage is the implementation of the algorithm in the form of a computer program, involving

220

A.6 Systematic approach to model development

not only coding of the algorithm using a programming language, but also designing data structures, and pre- and post processing of input and output data. The results of the development process determine the functionality of a computer model. The functionality is described usually in terms of how realistic the results in various applications are, and in terms of the response produced by the physical phenomena or events included in the model. Additionally, it concerns its ability to be adjusted for various applications. A computer or numeric model can be understood as a function transforming input data from a certain domain into output data. Input data can be categorised as: system parameters, describing the modelled physical system, algorithmic parameters which configure the algorithmic implementation (e.g. time-step) and software parameters (e.g. machine constants). Special attention must be paid to the system parameters. These include, for example, data which describe the configuration or properties of the modelled system. In a numeric hydrodynamic model these are: the geometry of the domain, fluid properties assumed to be fixed (e.g. fluid incompressibility), and parameters describing the way the boundary conditions are imposed. Data is incorporated at different development stages of the model, influencing the assumptions and approximations introduced in the conceptual model. These assumptions result in various constraints set upon the model parameters. For example, certain assumptions depicting the system geometry may lead, when the water depth is small compared to the horizontal scale of the domain, to the formulation of the hydrodynamic model in terms of vertically integrated shallow water equations. These require, in turn, that the data describing the boundary conditions must also be vertically integrated or that the bottom gradients must be small. A very important set of parameters includes those describing the initial condition or state of the system, as: initial velocities, free surface levels or concentrations of transported substances. Last, but not least is the data describing the variable boundary conditions, or environment of the model, such as wind influence or turbulence model parameters, applying on boundaries. Due to the high development costs most advanced hydrodynamic models are developed with the aim to cover an application domain as broadly as possible within the limits of their assumptions. They are usually designed to operate flexibly on a wide variety of system parameters. It is usually the case to obtain models for specific applications by narrowing the scope of a broader, complex, generic model with situation-specific datasets. However, for simple physical systems, models dedicated only to a specific situation are also developed and applied. The model validation process can be defined as a process of testing and documenting the quality and functionality of a developed computer model in relation to its intended applications and the physical system it represents [35]. The overall aim of the model validation process is therefore to determine the model domain of application, its reliability in describing the physical system it represents, and eventually to improve the model. The validation process begins during model development and continues until the model is operational. Broadly speaking, every activity involving a given model, every appli-

A.6 Systematic approach to model development

221

cation, yields information about the model’s quality. Usually, a documented validation process also describes the model’s functionality. So, broadly defined, the validation process begins with a series of verifications of model properties, assumptions, criteria, and results, together covering as much as possible of the intended application domain. Verification is a test of the model’s performance and accuracy in a well-defined situation or case, when the true, real results are known [172]. It shows whether the mathematically formulated problem is properly posed and solved. As mentioned above, the numeric model consists of three basic elements, a conceptual model and its algorithmic and software implementations. Leaving aside the software verification, concerned with testing if the software actually computes exactly what is prescribed and algorithm verification, being a series of tests whether the algorithm represents the conceptual model or not, let us concentrate on the third component: the conceptual model itself. Formally, the verification of the conceptual model can be divided into the event-oriented verification, where the physical phenomena included in the model are tested, and the application-oriented verification, where the applicability of the model in different operational situations or even real conditions is tested. The event-oriented verification tests are concerned with systematic modelling of separate physical processes, often in idealised conditions and with complicating factors eliminated, and with confirmation of whether or not the model behaves according to expectations. For example, the model is adapted to reproduce an analytical solution, or an experimental measurement series. A strict model verification must provide a comparison of the result of the numerical solution of the equations describing the physical system with an analytical solution of the given test problem or with a dataset obtained by an appropriate laboratory experiment or measurements in nature. The application-oriented verification is based on the testing of accuracy and quality of different model applications, where e.g. the entire scope of the model’s capabilities is used, and the results can be compared with measurements. It is also possible to verify a model comparing its results with the output of another, more general, complex and well validated model, representing the same system. One of the most typical parts of verification is sensitivity analysis, helping to assess the response of the model to the parameter variations. It helps to determine which parameters are crucial to the accuracy and functionality of the model, as well as to formulate practical guidelines for the applicability of the model. (This is also an important step of model calibration, understood as adjustment of model parameters to a given application in order to obtain improved results.) A true validation in the case of a hydrodynamic numeric model for geophysical flows, i.e. for natural flows, is based on tests which are designed to find out how closely the output of a verified and possibly calibrated model represents the real environment. It requires, that the model output is compared with field measurements in situ which have not been used in the calibration process. Therefore, validation is possible only if there is knowledge about the natural geophysical system, which has not already been built into the model. The validation activities are concerned in this case with the question

222

A.6 Systematic approach to model development

whether or not, for a given application, the conceptual model meets the requirements. The results of the overall validation process must provide the user with the ability to assess the model’s reliability and accuracy with respect to given applications as well as the model ability to be adjusted to specific conditions.

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Nomenclature

235

Nomenclature Abbreviations Symbol Meaning 1D, 2D, 3D one-, two-, three-dimensional BC boundary condition EBE element-by-element (method) EDF Electricit´e de France eq. equation FE, FD, FV finite elements, finite differences, finite volumes FEM, FDM, FVM finite element (difference, volume) method IC initial condition LBB Ladyzhenskaya-Babuˇska-Brezzi (condition) LNH Laboratoire National d’Hydraulique (EDF) NRBC non-reflecting boundary condition PSU practical salinity unit SUPG streamline upwind Petrov-Galerkin (scheme)

Geometrical coordinates Symbol i x, y, z x = (x1 , x2 , x3 ) z∗ B(x, y) S(x, y, t), s(x, y, t)

Meaning imaginary unit Cartesian coordinates, x eastwards, y northwards, z vertical upwards Cartesian coordinate vector Vertical coordinate in σ transformation Bottom coordinate (time-independent) Position of the free surface (time-dependent)

Mathematical and numerical symbols, variables and constants Symbol a(u, v) ch(x) f f˜ f a, f d fi fn ∂f /∂n n sech(x) sh(x)

Meaning bilinear form hyperbolic cosine, ch(x) = (ex + e−x )/2 a variable (in algorithms) intermediate value of variable f f variable value after addvection (a) or diffusion (d)stage a value of the f variable with the index i f variable value at the n-th time-step gradient of a variable normal to the boundary unit vector normal to a surface hyperbolic secant, 1/ch(x) hyperbolic sine, sh(x) = (ex − e−x )/2 Continuation on the next page

236

Nomenclature

Continuation from the previous page

C H k (Ω) H −m (Ω) I |J| L2 (Ω) R3 V′ Vh α, β, γ θ δΩ, Γ ξ, η λi ϕi ψi ∆t ∆x Ω Ω0 || · || (·, ·) A, L, P, Q A, u aij , ui

Courant number Sobolev space dual space for a Sobolev space H m (Ω) unit tensor Jacobian (of a transformation) space of all square-integrable functions f → R real, three-dimensional space dual space to V finite-dimensional subspace of V local coordinates in prismatic elements implicitness (Crank-Nicholson) coefficient domain boundary local coordinates in 2D elements eigenvalues interpolation functions (finite elements) test or weighting functions (finite elements) time-step spacing in the x-direction domain reference domain, reference element volume, basic mesh norm scalar product operators matrices (tensors), vectors elements of matrices (tensors), vectors

Physical variables and constants Symbol c c c0 f fH , fV g, g h k p pH p0 p′

Unit kg/m3 m/s m/s N kg−1 rad/s m/s2 m m−1 N/m2 N/m2 N/m2 N/m2

Meaning tracer concentration (e.g. pollutant) wave celerity, or concentration long (shallow water) wave celerity force per mass horizontal, vertical Coriolis coefficients gravitational acceleration, its value water depth, h = S − B wave vector global pressure hydrostatic pressure equilibrium pressure (wave theories) disturbance pressure (wave theories)

Continuation on the next page

Nomenclature

237

Continuation from the previous page

q s t u = (u, v, w) un ut D Fr H H, h N R R Re T Ti T η η0 λ µi ν νi π ̺ ̺0 ̺∗ ∆̺ σ τ τxy φ ω ω Ψ Ω

3

m /s PSU s m/s m/s m/s m/s2 1 m m rad/s m N/m2 1 o C kg/m3 s m m m kg/ms m2 /s m2 /s N/m2 kg/m3 kg/m3 kg/m3 kg/m3 N/m2 N/m2 N/m2 rad rad/s s−1 m2 /s rad/s

fluid flux salinity time velocity and its components normal velocity component tangential velocity component Navier-Stokes (deformation rate) tensor Froude number equilibrium water depth solitary wave: wave height and undisturbed water depth buoyancy (Brunt-V¨ais¨al¨a-) frequency Earth’s radius Reynolds stress tensor Reynolds number temperature tracer concentration wave period free surface variation from the equilibrium position wave amplitude wave length dynamic viscosity (component), i = x, y, z kinematic viscosity (or diffusivity) tensor kinematic viscosity (or diffusivity) (component), i = x, y, z hydrodynamic pressure fluid density reference fluid density mean fluid density (Boussinesq) density variation (̺ − ̺0 ) stress tensor shear stress stress force per surface xy phase angle velocity or radial frequency vorticity stream function rotation velocity of the Earth, 2π/86164 rad/s

Index acceleration centripetal, 208 Coriolis, 208 due to gravity, 208 gravity, 14, 208 reduced gravity, 139, 145 vertical, 2, 19–21, 72, 197 algorithm hydrostatic, 22, 26 advection step, 23 diffusion step, 23 pressure – free surface – continuity step, 24, 55, 198 non-hydrostatic, 41, 71, 77, 79 advection step, 74, 96, 98 continuity step, 75, 113 diffusion step, 74, 89, 107 free surface stabilisation, 105, 162, 165 free surface step, 76, 100, 117, 199 Poisson equation for hydrodynamic pressure, 75, 86, 198, 199 source terms, 74, 94, 107, 198 velocity projection, 75, 88, 199 regula falsi, 115 SIMPLE, 44 approximation β-plane, 209 f -plane, 209 Boussinesq, 17, 62 hydrostatic, 2, 5, 19, 21, 70, 139, 142, 208, 209 incompressibility, 15, 42, 205, 206 validity conditions, 206 rigid lid, 8, 9, 62

free surface, 15, 24, 55, 58 gradient, 20 lateral, 15, 58 normal vector, 59, 88 open, 70 stress, 62 bottom shear, 61, 63, 107 wind shear, 61, 63, 107, 111, 134 stress tensor, 59 boundary condition, 4, 57 absorbing, 68, 200 numeric realisation, 103 advection step, 97, 100 cyclic, 65 dynamic, 59, 70 hydrodynamic pressure, 69, 70, 114, 198 impermeability, 61, 70, 89 inflow, 63, 164, 169 inviscid free surface normal stress, 61 kinematic, 8, 54, 57, 76, 117, 162, 199, 210 numeric realisation, 100 numeric realization, 100 no-flux, 69 no-slip condition, 62 non-reflecting, 65, 169, 175, 200 Higdon, 66 Orlanski, 66 Sommerfeld radiation condition, 65 Thompson, 67 numeric implementation advection with characteristics, 97 advection with SUPG, 100 diffusion step, 94 Dirichlet, 88, 100, 102 Neumann, 87 open, 63 outflow, 64, 164, 169 periodic, 65

boundary, 15, 58 bottom, 15, 24, 55, 58 gradient, 3, 20, 21 bottom gradient, 164

238

Index

rigid lid, 8, 9, 62, 76, 137 slip condition, 62 tracer, 69 co-ordinate inertial system, 207 isopycnic, 7 non-inertial system, 14, 18, 206 rectangular system, 209 system transformed, 57 coefficient Coriolis, 18, 108, 208 condition for boundedness and continuity, 35 incompressibility, 38, 40 inf-sup, 35 LBB, 39, 40, 197 convection deep ocean, 9 current buoyancy-driven, 3, 21, 95, 141 drift, 107, 110, 111 gravity, 3, 21, 95, 141 in geostrophic equilibrium, 107 mesoscale ocean, 9, 209 secondary, 3, 8 slope, 107, 108, 113 wind-driven, 11, 110, 111, 134 deep water renewal, 8 density average, 17 equation of state, 14, 18, 139, 146 EOS–80, 213 sediment influence, 217 simple forms, 214 reference, 17 variance, 17 domain computational, 15, 79 variable extents, 7, 56 triangulation, 79 down-welling, 7, 8 eigenvalue, 67, 115 eigenvectors, 67 Ekman spiral, 11 energy

239

conservation, 153 kinematic, 145 kinetic, 155 potential, 145, 155 equation advection σ-transformed, 97 Bernoulli, 164 conservative free surface, 24, 54, 55, 57, 76, 126, 142, 163, 199 explicit numeric realisation, 103 semi-implicit numeric realisation, 104 continuity, 18 incompressible, 16, 46 integrated, 25 conventional pressure eq., 41 derived pressure eq., 41 diffusion σ-transformed, 89 geostrophic, 108 global system, 35, 36 element-by-element, 37 matrix assembly, 36 solution, 36 hydrodynamic set, 14 for incompressible flows, 18, 71 in characteristic form, 67 Laplace, 114, 143, 210 mass conservation, 14, 18 momentum conservation, 14, 18 with Boussinesq approximation, 18 Navier-Stokes, 2, 14 Reynolds-averaged, 16 Navier-Stokes eqs., 15 of state, 14, 15, 18, 139, 146 EOS–80, 213 Poisson, 33, 113 Poisson equation numeric formulation, 86 Poisson for hydrodynamic pressure, 47, 198 Poisson for pressure, 41, 44 Poisson for pressure from fractional formulation, 45, 198 shallow water eqs., 2, 21, 22 shallow water eqs., 2D, 24, 76, 199 St.-Venant, 24, 76 tracer transport, 14, 18

240

Index

transport, 14 wave eq., 65, 143 finite element, 32 2D reference element, 84 conform, 33 iso-parametric, 32 reference element, 81 flow around structures, 21 channel, 11 inviscid, 156, 164, 168 stationary, 165, 166 subcritical, 164, 166 supercritical, 164, 190, 191 fluid incompressible, 15, 205 Newtonian, 14 force buoyancy, 2, 17, 20, 24, 49, 95, 198 Coriolis, 9, 18, 20, 24, 96, 107, 198, 206 other forcing, 96 form bilinear, 34 V -elliptic, 35 formula Green’s, 34, 87, 91 formulation weak, 30, 32–35 derivative, 89, 94, 103 free surface, 2, 4, 50 conservative equation, 8, 55, 76, 126, 142, 163, 199 method height function, 8, 54, 190 line segment, 55 marker-in-cell (MAC), 8, 50, 200 volume of fluid (VOF), 8, 51, 200 representation, 50 tracking, 50 function approximation f., 30 base f., 30 Lagrangian interpolation f., 31, 36, 86, 98, 101 sign, 143 square integrable f., 32 step f., 143

test f., 30 weighting f., 30 hydrothermal discharges, 8 initial condition, 4, 57, 58 matrix advection, 99, 102 diffusion, 88, 91, 92 mass, 88, 91, 99, 102 SUPG, 99, 102 mesh σ-mesh, 7, 11, 57, 79, 197, 200 σ-transformation, 7, 23, 57, 79 diffusion step, 90 Jacobian, 80, 84, 92 method advection characteristics, 6, 23, 25, 67, 74, 76, 96, 100, 119, 123, 162 SUPG, 23, 25, 74, 76, 98, 119, 123, 162 Arbitrary Lagrangian-Eulerian (ALE), 7, 10, 56 deforming space-time FEM, 57 element-by-element, 6, 37 for Navier-Stokes eqs., 27 artificial compressibility, 7, 29 coupled, 6, 10, 28 decoupled, 7, 9, 29, 43 direct, 6, 28 fractional step, 7, 29 non-primitive variables, 7, 29 operator splitting, 7, 23, 29, 41, 74, 85, 197 penalty function, 7, 29 pressure correction, 7, 29 projection, 7, 29, 42 projection-1, 45 projection-2, 45, 46, 48, 78 projection-3, 45 Taylor, 7, 29 general finite difference, 27, 28 finite element, 3, 28, 28, 30 finite element, equal interpolation, 40 finite element, Galerkin, 31, 87, 90

Index

finite element, mixed interpolation, 38 finite element, Petrov-Galerkin, 31, 98, 101 finite element, standard, 31 finite element, SUPG, 31, 98, 101 finite volume, 27 space-time FEM, 10 spectral, 28 weighted residue, 30 Navier-Stokes eqs., 8 model, 217 application domain, 11, 199, 218 calibration, 1 conceptual, 11, 217 for 3D shallow water eqs., 5 functionality, 218 hydrostatic, 2 non-hydrostatic, 8 parameters algorithmic, 218 software, 218 system, 218 physical, 1 physical system, 217 three-dimensional, 2 two-dimensional, 2 validation, 219 process, 219 verification, 1, 129, 219 application-oriented, 219 event-oriented, 219 momentum conservation, 153 norm dual space, 34 Hilbert space, 32 Sobolev space, 33 number Courant, 118, 119, 119, 126 Froude, 163 Reynolds, 15, 20 Rossby, 20 operator inverse, 35 isomorphic, 35

241

projection, 43 splitting, 23, 41, 74, 85, 197 stability, 86 parameter Coriolis, 108, 208 depth of frictional influence, 109 upwind, 32, 102 pressure atmospheric, 48, 62, 70, 108 baroclinic, 5, 21, 22, 24, 49, 62, 76, 95, 198 barotropic, 4, 5, 21, 22, 24, 49, 50, 62, 76, 94, 198 decomposition, 9, 47, 198 equation boundary conditions, 42, 46, 69, 114, 198 conventional, 41 derived, 41 for hydrodynamic pressure, 47, 198 from fractional formulation, 45, 198 equilibrium, 209 gradient, 22, 48 hydrodynamic, 2, 47, 48, 155, 198, 210 hydrostatic, 21, 47, 210 from local density, 49 from mean density, 48 kinematic, 42 perturbation, 142, 154, 155, 210 Poisson equation, 9, 44 river meanders, 3 scalar product Hilbert space, 32 Sobolev space, 33 solution classical, 34 weak, 34 space dual, 34 for pressure, 38 for velocity, 38 Hilbert, 32 Sobolev, 33 Stokes problem, 37 subspace finite dimensional, 33, 38

242

Telemac, 201, 202 Telemac2D, 3, 5, 25, 76 Telemac3D, 3, 6, 11, 22, 26, 197, 201 tensor Navier-Stokes, 59 turbulent eddy-viscosity, 14 turbulent stress, 14, 16, 59 incompressible, 16 test bottom steepness, 130, 191 channel flow over a bump, 130, 166 sub- and supercritical over a ramp, 130 subcritical and supercritical over a ramp, 163 drift current, 110 free surface breaking, 130, 190 Gaussian profile advection, 118 interfacial internal waves, 129, 136 Laplace equation, 114 lock exchange flow, 129, 141 Molenkamp, 119 Poisson equation, 115 rotating cone, 119 slope current, 108 solitary wave, 130, 176 standing surface wave, 129, 153 waves over underwater channel, 130, 171 waves reflection, 129, 130 wind-driven circulation, 129, 134 theorem Fourier, 115 Leibniz, 55 time discretisation Crank-Nicolson, 25 explicit formulation, 86 implicit, 163 implicit formulation, 86 implicitness factor, 86, 98, 101 semi-implicit, 23, 25, 85, 90, 163 Tisat, 5 tracer active, 14, 214 passive, 14 salinity, 139, 146, 213

Index

sediment, 217 temperature, 214 transport sediment, 3, 201 triangulation, 32 Trim2D, 5 Trim3D, 5 turbulence Boussinesq eddy-viscosity, 14, 16, 20, 59 up-welling, 7, 8 vectorisation, 37, 89 velocity divergence-free, 44, 46, 78 Ekman profile, 107 Ekman spiral, 107 intermediate, 43–45, 75, 88, 89 potential, 154 solenoidal, 43, 44, 46, 47, 75, 78, 88 vertical component, 25, 72 wave breaking, 10, 177 dispersion relation, 211, 212 dispersion relationship, 136 internal celerity, 137 dispersive, 137 interfacial, 136 long, 137 non-dispersive, 137 short, 137 standing, 11 lee wave, 21 orbital movement, 21, 212 reflection, 130, 175 sinusoidal, 211 small amplitude, 209 solitary, 176 soliton, 176 surface celerity, 143, 164, 211 deep water, 3, 212 dispersive, 209, 212 long, 130, 142, 172, 209, 212 non-dispersive, 209, 212 shallow water, 212

Index

short, 3, 21, 130, 154, 172, 201, 209, 212 standing, 154 standing, long, 155 standing, short, 154 velocity, 154, 212 tsunami, 10 viscous damping, 175

243

244

Danksagung

Danksagung Obwohl ich die meisten Zeit als selbst¨andiger Einzelk¨ampfer mit den Problemen, die mir diese Arbeit bereitete, besch¨aftigt war, gibt es eine Reihe von Personen, bei denen ich mich bedanken m¨ochte. Ich hoffe ich vergesse niemandem. Zuerst Herr Prof. Werner Zielke: Im Oktober 1991 h¨atte er mich, zu dieser Zeit einen perfekten Outsider, nach einem Vorstellungsgespr¨ach wieder vor der T¨ ur setzen k¨onnen, was niemand bemerkt h¨atte. Er hat das aber nicht gemacht, und ist sp¨ater, sehr konsequent Chef und Lehrer geblieben, mit Gew¨ahrleistung der M¨oglichkeit zur Promotion an ¨ seinem Institut inbegriffen und schließlich der Ubernahme des Hauptreferats. Prof. Zielke hat eine Gabe, den Mitarbeitern eine gewisse Prise Freiheit zu geben, auch wenn die Resultate, wie immer in der Forschung, ohne Erfolgsgarantie sind. Ohne dieser Toleranz und Freiheit ist keine kreative T¨atigkeit m¨oglich, und diese Arbeit w¨are wahrscheinlich nicht entstanden. Danke! ¨ Unbedingt m¨ochte ich Prof. Rainer Helmig (den ich duzen darf!), f¨ ur die Ubernahme des Korreferates besonders herzlich danken. Er hat es auf besonders nat¨ urlich selbstverst¨andliche Weise gemacht und mich dabei ermuntert, ein bißchen selbstsicherer zu werden. Er hat zu meiner Dissertation so tiefgreifende, lehrreiche und allzu treffliche Anmerkungen formuliert, daß ich meine Arbeit in diesem str¨omungsmechanisch–numerischen Dschungel richtig zuordnen und die einfachsten Fragen (Was gibt es bereits? Was mache ich neu? Wof¨ ur ist das?) wirklich ganz klar beantworten konnte. Den ehrenw¨ urdigen Rest der zahlreicher Mitglieder meiner Pr¨ ufungskomission, n¨amlich (in alphabetischer Reihenfolge) den Herren Vorsitzenden Prof. Rudolf Damrath, Dr.-Ing. Olaf Kolditz, Dr.-Ing. Ralf Mahnken, Prof. Mark Markofsky, und Prof. Gustav Rosemeier danke ich f¨ ur ihr Engagement und eine ehrliche, gr¨ undliche und f¨ ur mich u berraschend sehr angenehme Pr¨ u fung. ¨ Andreas Malcherek hat im April oder Mai 1995 mich darauf aufmerksam gemacht, daß auf der Basis der Telemac-Bibliotheken ein Druck(korrektur)verfahren (in so unpr¨aziser Weise haben wir das zu dieser Zeit genannt) zu basteln sei, und dabei auf seine sehr direkte Weise gedroht, daß wenn ich das nicht mache, er das selber tun wird, obwohl er das Programmieren f¨ ur eine Sklavenarbeit h¨alt. F¨ ur diesen kr¨aftigen Ansporn muß ich dankbar sein. Andreas hat auch die urspr¨ unglichste Version dieses Textes (von 1996!) gelesen und seine Anmerkungen habe ich in vielen F¨allen dankbar ber¨ ucksichtigt. Reinhard ‘Phillip’ Hinkelmann danke ich f¨ ur seinen unglaublichen Einsatz beim Pr¨ uflesen und Formulieren der Vorschl¨age f¨ ur Verbesserungen in meiner schriftlichen Arbeit. Phillip geh¨ort zu den wenigen Menschen, die meine Dissertation wirklich sehr genau und sorgf¨altig gelesen haben. Er hat den gesamten Text auf solche konstruktiv kritische Art gelesen, die sich jeder Doktorand in den heimlichsten Tr¨aumen w¨ unscht. Phillip – Danke! Meine Zimmergenossin (und fr¨ uher auch HiWi!) Rebekka Kopmann hat mich immer auf ihre fr¨ohlich ungezwungene Weise in meiner t¨aglicher Arbeit bedinungslos und sofort unterst¨ utzt, insbesondere wenn eine gute Portion weiblicher Intuition notwendig war, um die f¨ ur m¨annlichen Wesen schwierigen Probleme zu l¨osen. Danke!

Danksagung

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Meinen ex-HiWis, Steffi R¨ uppel, und den beiden gerade Dipl.-Ing. gewordenen Hella Schwartkop und Martin Kohlmeier danke ich f¨ ur ihre Unterst¨ utzung. Ich habe von Euch auch viel gelernt! Ich bedanke mich auch bei meinen Kolleginnen und Kollegen am Institut, die direkt oder indirekt am Erfolg dieser Arbeit beteiligt waren, und eine ausgesprochen exzellente Arbeitsatmosph¨are. I would like to thank very much Michael Willey, whom I met by sheer chance in the internet and never in the so-called real world. Mike appreciated my humble advice concerning his planned bike trip to the Carpathians so much, that he felt obliged to proofread my thesis from the purely lingual point of view as a native speaker. Everything was arranged via internet... Finally, I received my LATEX files back from Helsinki transformed to HTML with all my mistakes corrected in red... Just in one week... What a quality of proofreader’s workmanship! – Mike, thanks a lot! Moim rodzicom i siostrze chcialbym podziekowa´ c za niezawodne podtrzymywanie mnie ֒ na duchu, oraz za ciag olnie w trakcie rozm´ow telefonicznych objawiana֒ troske, ֒ la, ֒ szczeg´ ֒ kiedy w ko´ ncu wreszcie zabiore֒ sie֒ powa˙znie za doktorat... Dzialalo bardzo motywujaco! ֒ Dziekuj e! ֒ ֒ Die beiden Frauen, die im Institut als ‘meine S¨ ußen’ oder einzeln als ‘meine Liebste’ und ‘mein Lieblingst¨ochterchen’ bekannt sind, sollen nicht vergessen werden! Es wird immer wieder bei den Promotionen den am n¨ahsten stehenden Familienmitglieder f¨ ur Geduld, das Vergeben der s¨ undhaften Vernachl¨assigung des famili¨aren Lebens, und so weiter, gedankt. Ich bedanke mich auch daf¨ ur, aber im besonderen Maße danke ich daf¨ ur, daß meine beiden S¨ ußen in dieser manchmal sehr nerv¨osen Zeit auch so viel gest¨ort haben, so daß ich ein Gleichgewicht zwischen der Arbeit und dem normalen Leben behalten konnte. Man sollte die materiellen Seiten nie vergessen. Ich bedanke mich bei allen Steuerzahlern in der Budesrepublik Deutschland, die das Budget vom Bundesministerium f¨ ur Bildung und Forschung realisiert haben, und den Menschen, die u ugen. Fast die ¨ber dies verf¨ ganze Zeit konnte ich aus diesem Budget meinen (und meiner Familie) Lebensunterhalt sichern, mal als Adressat einer Zuwendung, und mal als Bearbeiter eines Tiefseeumweltschutz–Projektes. Danke! Bei der bedrohten Spezies Yeti (und anderen eindeutig mehr real existierenden und mit wirklichen Problemen heimgesuchten Himalaja-V¨olker) m¨ochte ich mich f¨ ur die heimliche Unterst¨ utzung bedanken. Am Ende m¨ochte ich allen danken, bei denen ich vergessen habe, mich zu bedanken.

Verpflichtet, Jacek A. Jankowski Hannover, Dezember 1998

Liste der bisher erschienenen Institutsberichte (* = bereits vergriffen)

01/1970 * Holz, K.-P. Erg¨anzung des Verfahrens finiter Elemente durch Ecksingularit¨aten zur verbesserten Berechnung schiefwinkliger Platten. Dissertation, Techn. Univ. Hannover, 1970 02/1971 Ehlers, K.-D. Berechnung instation¨arer Grund- und Sickerwasserstr¨omungen mit freier Oberfl¨ache nach der Methode finiter Elemente. Dissertation, Techn. Univ. Hannover, 1971 03/1971 * Meissner, U. Berechnung von Schalen unter großen Verschiebungen und Verdrehungen bei kleinen Verzerrungen mit Hilfe finiter Dreieckselemente. Dissertation, Techn. Univ. Hannover, 1971 04/1972 Grotkop, G. Die Berechnung von Flachwasserwellen nach der Methode der finiten Elemente. Dissertation, Techn. Univ. Hannover, Sonderdruck aus dem Jahresbericht 1971 d. SFB 79, H. 2, 1972 05/1973 Schulze, K.-W. Eine problemorientierte Sprache f¨ur die Dynamik offener Gerinne. Dissertation, Techn. Univ. Hannover, Mitteil. d. SFB 79, Heft 1, 1973 06/1977 Beyer, A. Die Berechnung großr¨aumiger Grundwasserstr¨omungen mit Vertikalstruktur mit Hilfe der Finite–Element–Methode. Dissertation, Fortschrittberichte der VDI–Zeitschriften, Reihe 4, Nr. 34, 1977 07/1977 * Ebeling, H. Berechnung der Vertikalstruktur wind- und gezeitenerzeugter Str¨omungen nach der Methode der finiten Elemente. Dissertation, Fortschrittberichte der VDI–Zeitschriften, Reihe 4, Nr. 32, 1977 08/1977 * G¨ artner, S. Zur Berechnung von Flachwasserwellen und instationren Transportprozessen mit der Methode der finiten Elemente. Dissertation, Fortschrittberichte der VDI–Zeitschriften, Reihe 4, Nr. 30, 1977 09/1977 * Herrling, B. Eine hybride Formulierung in Wasserst¨anden zur Berechnung von Flachwasserwellen mit der Methode finiter Elemente. Dissertation, Fortschrittberichte der VDI–Zeitschriften, Reihe 4, Nr. 37, 1977 10/1979 Hennlich, H.-H. Aeroelastische Stabilit¨atsuntersuchung von Linientragwerken. Dissertation, Fortschrittberichte der VDI–Zeitschriften, Reihe 4, Nr. 49, 1979 11/1979 Kaloˇ cay, E. Zur numerischen Behandlung der Konvektions– Diffusions–Gleichung im Hinblick auf das inverse Problem. Dissertation, Univ. Hannover, 1979

12/1980

Januszewski, U. Automatische Eichung f¨ur ein- und zweidimensionale, hydrodynamisch-numerische Flachwassermodelle. Dissertation, Univ. Hannover, Fortschrittberichte der VDI–Zeitschriften, Reihe 4, Nr. 58, 1980 13/1982 * Carbonel Huam´ an, C.A.A. Numerisches Modell der Zirkulation in Auftriebsgebieten mit Anwendung auf die nordperuanische K¨uste. Dissertation, Univ. Hannover, 1982 14/1985 Tuchs, M. Messungen und Modellierung am Deep Shaft. Dissertation, Univ. Hannover, 1984 ¨ 15/1985 Theunert, F. Zum lokalen Windstau in Astuarien bei Sturmfluten – Numerische Untersuchungen am Beispiel der Unterelbe. Dissertation, Univ. Hannover, 1984 16/1985 Perko, H.-D. Gasausscheidung in instation¨arer Rohrstr¨omung. Dissertation, Univ. Hannover, 1984 17/1985 Crotogino, A. Ein Beitrag zur numerischen Modellierung des Sedimenttransports in Verbindung mit vertikal integrierten Str¨omungsmodellen. Dissertation, Univ. Hannover, 1984 18/1985 Rottmann–S¨ ode, W. Ein halbanalytisches FE–Modell f¨ur harmonische Wellen zur Berechnung von Wellenunruhen in H¨afen und im K¨ustenvorfeld. Dissertation, Univ. Hannover, 1985 19/1985 Nitsche, G. Explizite Finite–Element–Modelle und ihre Naturanwendungen auf Str¨omungsprobleme in Tidegebieten. Dissertation, Univ. Hannover, 1985 ¨ 20/1985 Vera Muthre, C. Untersuchungen zur Salzausbreitung in Astuarien mit Taylor’schen Dispersionsmodellen. Dissertation, Univ. Hannover, 1985 21/1985 Schaper, H. Ein Beitrag zur numerischen Berechnung von nichtlinearen kurzen Flachwasserwellen mit verbesserten Differenzenverfahren. Dissertation, Univ. Hannover, 1985 22/1986 Urban, C. Ein Finite–Element–Verfahren mit linearen Ans¨atzen f¨ur station¨are zweidimensionale Str¨omungen. Dissertation, Univ. Hannover, 1986 ¨ 23/1987 Heyer, H. Die Beeinflussung der Tidedynamik in Astuarien durch Steuerung – Ein Beitrag zur Anwendung von Optimierungsverfahren in der Wasserwirtschaft. Dissertation, Univ. Hannover, 1987 24/1987 G¨ artner, S. Zur diskreten Approximation kontinuumsmechanischer Bilanzgleichungen. Institutsbericht, davon 4 Abschnitte als Habilitationsschrift angenommen, Univ. Hannover, 1987 25/1988 Rogalla, B.U. Zur statischen und dynamischen Berechnung geometrisch nichtlinearer Linientragwerke unter Str¨omungs– und Wellenlasten. Dissertation, Univ. Hannover, 1988 ¨ 26/1990 * Lang, G. Zur Schwebstoffdynamik von Tr¨ubungszonen in Astuarien. Dissertation, Univ. Hannover, 1990

27/1990 28/1990

29/1991

30/1991 31/1991

dito 32/1992 33/1993

34/1993 35/1994

36/1994

37/1994

38/1994

39/1994

40/1994 41/1994

42/1995

Stittgen, M. Zur Fluid–Struktur–Wechselwirkung in flexiblen Offshore–Schlauchleitungen. Dissertation, Univ. Hannover, 1990 Wollrath, J. Ein Str¨omungs- und Transportmodell f¨ur kl¨uftiges Gestein und Untersuchungen zu homogenen Ersatzsystemen. Dissertation, Univ. Hannover, 1990 Kr¨ ohn, K.-P. Simulation von Transportvorg¨angen im kl¨uftigen Gestein mit der Methode der Finiten Elemente. Dissertation, Univ. Hannover, 1991 ¨ Lehfeldt, R. Ein algebraisches Turbulenzmodell f¨ur Astuare. Dissertation, Univ. Hannover, 1991 Pru ¨ser, H.-H. Zur mathematischen Modellierung der Interaktion von Seegang und Str¨omung im flachen Wasser. Dissertation, Univ. Hannover, 1991 Schr¨ oter, A. Das numerische Seegangsmodell BOWAM2 1990 – Grundlagen und Verifikationen – . Univ. Hannover, 1991 Leister, K. Anwendung numerischer Flachwassermodelle zur Bestimmung von Wasserlinien. Dissertation, Univ. Hannover, 1992 Ramthun, B. Zur Druckstoßsicherung von Fernw¨armenetzen und zur Dynamik von Abnehmeranlagen. Dissertation, Univ. Hannover, 1993 Helmig, R. Theorie und Numerik der Mehrphasenstr¨omungen in gekl¨uftet–por¨osen Medien. Dissertation, Univ. Hannover, 1993 Plu ¨ß, A. Netzbearbeitung und Verfahrensverbesserungen f¨ur Tidemodelle nach der Finiten Element Methode. Dissertation, Univ. Hannover, 1994 No Statistisch–numerische Beschreibung des Wellen¨thel, H. und Str¨omungsgeschehens in einem Buhnenfeld. Dissertation, Univ. Hannover, 1994 Shao, H. Simulation von Str¨omungs- und Transportvorg¨angen in gekl¨ufteten por¨osen Medien mit gekoppelten Finite–Element– und Rand–Element–Methoden. Dissertation, Univ. Hannover, 1994 ¨ Stengel, T. Anderungen der Tidedynamik in der Deutschen Bucht und Auswirkungen eines Meeresspiegelanstiegs. Dissertation, Univ. Hannover, 1994 Schubert, R. Ein Softwaresystem zur parallelen interaktiven Str¨omungssimulation und -visualisierung. Dissertation, Univ. Hannover, 1994 Alm, W. Zur Gestaltung eines Informationssystems im K¨usteningenieurwesen. Dissertation, Univ. Hannover, 1994 Benali, H. Zur Kopplung von FEM– und CAD–Programmen im Bauwesen ¨uber neutrale Datenschnittstellen. Dissertation, Univ. Hannover, 1994 Schr¨ oter, A. Nichtlineare zeitdiskrete Seegangssimulation im flachen und tieferen Wasser. Dissertation, Univ. Hannover, 1995

43/1995

44/1995

45/1995 46/1996 47/1996 48/1996

49/1996

50/1997 51/1997

52/1997 53/1997

54/1998 55/1998

55/1998

56/1999

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Blase, Th. Ein systemtechnischer Ansatz zur Modellierung von Hydraulik, Stofftransport und reaktionskinetischen Prozessen in Kl¨aranlagen. Dissertation, Univ. Hannover, 1995 Malcherek, A. Mathematische Modellierung von Str¨omungen ¨ und Stofftransportprozessen in Astuaren. Dissertation, Univ. Hannover, 1995 Lege, T. Modellierung des Kluftgesteins als geologische Barriere f¨ur Deponien. Dissertation, Univ. Hannover, 1995 Arnold, H. Simulation dammbruchinduzierter Flutwellen. Dissertation, Univ. Hannover, 1996 Kolditz, O. Stoff- und W¨armetransport im Kluftgestein. Habilitation, Univ. Hannover, 1996 Hunze, M. Numerische Modellierung reaktiver Str¨omungen in oberfl¨achenbel¨ufteten Belebungsbecken. Dissertation, Univ. Hannover, 1996 Wollschl¨ ager, A. Ein Random-Walk-Modell f¨ur Schwermetallpartikel in nat¨urlichen Gew¨assern. Dissertation, Univ. Hannover, 1996 Feist, M. Entwurf eines Modellierungssystems zur Simulation von Oberfl¨achengew¨assern. Dissertation, Univ. Hannover, 1997 Hinkelmann, R. Parallelisierung eines Lagrange–Euler–Verfahrens f¨ur Str¨omungs- und Stofftransportprozesse in Oberfl¨achengew¨assern. Dissertation, Univ. Hannover, 1997 Barlag, C. Adaptive Methoden zur Modellierung von Strofftransport im Kluftgestein. Dissertation, Univ. Hannover, 1997 Saberi-Haghighi, K. Zur Ermittlung der verformungsabh¨angigen Windbelastung bei H¨anged¨achern. Dissertation, Univ. Hannover, 1997 Kru ¨ger, A. Physikalische Prozesse im Nachkl¨arbecken – Modellbildung und Simulation. Dissertation, Univ. Hannover, 1998 Wolters, A. H. Zur Modellierung des instation¨aren thermohydraulischen Betriebsverhaltens von Fernw¨armeanlagen. Dissertation, Univ. Hannover, 1998 Wolters, A. H. Zur Modellierung des instation¨aren thermohydraulischen Betriebsverhaltens von Fernw¨armeanlagen. Dissertation, Univ. Hannover, 1998 Jankowski, J. A. A non-hydrostatic model for free surface flows. Dissertation, Univ. Hannover, 1999