Keynote Merge

Keynote Lectures - Abstracts

Large-eddy simulations based on transported subgrid-scale energy Jos´e Carlos Fernandes Pereira IDMEC/IST, Mecˆ anica I, 1o andar/LASEF, Av. Rovisco Pais, 1049-001 Lisbon, PORTUGAL. [email protected]

Abstract In large-eddy simulations (LES) the large or grid scales (GS) which are responsible for the most important transfers of mass, momentum and heat are explicitly calculated while the effect of the small scales is modelled by a subgrid-scale (SGS) model. In many flow simulations the small scales of motion are statistically close to isotropic, carry a relatively small amount of the total kinetic energy, and adjust almost immediately to the dynamics of the large scales. However, in many engineering and Natural flows the isotropic assumption of the small scale motions is not observed, even at very high Reynolds numbers, particularly for the passive scalar field [1]. Furthermore, in many LES the SGS motions do possess a significant part of the total kinetic energy [2]. Moreover, for high Reynolds numbers and/or coarse meshes the SGS motions need a non-negligible time to adjust to local unsteadiness from the large scales i.e. the local equilibrium assumption between the large and small scales of motion - which is used in the great majority of SGS models - is not observed [3, 4]. One way to overcome these limitations, by discarding the local equilibrium assumption, consists in developing SGS models based on the transport equation of the SGS kinetic energy [5, 6, 7, 8, 9, 10]. The use of a transport equation for the SGS kinetic energy is interesting also to many hybrid RANS/LES and URANS/LES modelling strategies [11, 12, 13]. Similarly, in LES involving a passive or active scalar field several new unclosed terms arise, and one way to deal with these unknown terms is to solve an additional SGS scalar variance transport equation. For example, in LES of reacting flows the variance of the mixture fraction is very important and therefore some combustion models use an additional transport equation for the variance of the SGS mixture fraction [14]. The study of transport equations for the SGS kinetic energy and SGS scalar variance is thus of great relevance since in LES most of the terms from these equations are unknown and have to be modelled. This presentation focusses in the development of SGS models

based on modeled transport equations for the SGS kinetic energy and SGS scalar variance, by addressing three (3) topics. We start by analyzing the physical mechanisms associated with the dynamics of the SGS kinetic energy. The most intense kinetic energy exchanges between GS and SGS occur near the large flow structures and not randomly in space (Fig. 1). The GS kinetic energy is dominated by GS advection and GS pressure/velocity interactions, while the GS/SGS diffusion plays an important role to the local dynamics of both GS and SGS kinetic energy. The so-called local equilibrium assumption holds globally but not locally as most viscous dissipation of SGS kinetic energy takes place within the vortex cores whereas forward and backward GS/SGS transfer occurs at quite different locations (Fig. 2). Finally, it is shown that SGS kinetic energy advection may be locally large as compared to the other terms of the SGS kinetic energy transport equation [3, 4]. Next we address the effect of the existing SGS models on the vortices obtained from classical SGS models. One way to assess this issue, consists in analyzing the transport equation for the resolved enstrophy. Special emphasis is placed on the enstrophy SGS dissipation term, which represents the effect of the SGS models on the vortices computed from LES. When the filter is placed in the inertial range region the evolution of the vorticity norm is governed by the enstrophy production and enstrophy SGS dissipation, which represents, in the mean, a sink of resolved enstrophy. Thus the coherent vortices obtained from LES are subjected to an additional (nonviscous) dissipation mechanism. Extensive tests are conducted using several SGS models in order to analyze their ability to represent the enstrophy SGS dissipation. The models analyzed are the Smagorinsky, structure function, filtered structure function, dynamic smagorinsky, gradient, scale similarity, and mixed. It is shown, using both DNS and LES that the Smagorinsky, structure function, and mixed models cause excessive vorticity dissipation compared to the other models. An estimation of the ”vorticity error” and its wave number dependence is given, for each SGS model. Both DNS and LES show that the dynamic Smagorinsky and filtered structure function models seem to be the best suited to a correct prediction of the resolved vorticity filed (Fig. 3) [15, 16]. Finally, we discuss SGS models based on transport equations for the SGS kinetic energy and SGS scalar variance. In particular we analyze the modeling of the diffusion and dissipation terms. In virtually all models using these equations, the diffusion terms are lumped together, and their joint effect is modeled using a ”gradient-diffusion” model. It is shown that provided the implicit grid filter from the LES is in the dissipative range the diffusion terms pertaining to the SGS kinetic energy and SGS scalar variance transport equations are well represented by a gradient-diffusion model. However, this situation changes dramatically for both equations when considering inertial range filter sizes and high Reynolds numbers. The reason for this lies in part in a loss of local balance between the SGS turbulent diffusion and diffusion caused by GS/SGS interactions, which arises at inertial range filter sizes. Moreover, due to the deficient modeling of the diffusion by SGS pressure-velocity interactions, the diffusion terms in the SGS kinetic energy equation

are particularly difficult to reconcile with the gradient-diffusion assumption. In order to improve this situation, a new model, inspired by Clark’s SGS model, is developed for this term. The new model shows very good agreement with the exact SGS pressure-velocity term in a priori tests and better results than the classical model in a posteriori LES tests [17]. By far the greatest challenge for modelling in the transport equations for the SGS kinetic energy and SGS scalar variance comes from the viscous and the molecular SGS dissipation terms that represent the final (dissipation) stages of the ”energy cascade mechanism” whereby the SGS kinetic energy and SGS scalar variance are dissipated through the action of the molecular viscosity and diffusivity, respectively. We analyze the topology and spatial localisation of the viscous and the molecular SGS dissipation terms, and assess three models currently used for these terms. The models analysed here are the classical model used by e.g. Schumann [5] and Yoshizawa [6], the model used in hybrid RANS/LES by Paterson and Peltier [18], and by Hanjalic [19], and the model for the molecular SGS dissipation of SGS scalar variance from Jim´enez et al. [14]. The classical models for the molecular SGS dissipation give very good results. Moreover, the model constants approach asymptotically the theoretical values as the Reynolds number and filter sizes increases, which supports the use of a constant value in engineering and geophysical applications, instead of using a dynamic procedure for their computation as in Ghosal et al. [8]. For the molecular SGS dissipation of SGS scalar variance the model from Jim´enez et al. [14] performs even better than the classical model and should be the preferred model for this term when the Schmidt number is close to 1.0. Finally, all the tests showed that the models used in hybrid RANS/LES tested here give very poor. The reason behind this is connected with the deficient spectral representation of the exact molecular SGS dissipation term (Fig. 4) [20].

Keywords: large eddy simulation (LES), grid/subgrid-scale interactions, LES based on transport equations for the SGS kinetic energy and SGS scalar variance, diffusion and dissipation terms.

References [1] S. Grag and Z. Warhaft. On the small scale structure of simple shear flow. Phys. Fluids, 10:662, 1998. [2] E. Garnier, O. M´etais, and M. Lesieur. Synoptic and Frontal-Cyclone Scale Instabilities in Baroclinic Jet Flows. J. Atmos. Sci., 55:1316–1335, 1998. [3] C. B. da Silva and O. M´etais. On the influence of coherent structures upon interscale interactions in turbulent plane jets. J. Fluid Mech., 473:103–145, 2002. [4] C. B. da Silva and J. C. F. Pereira. On the local equilibrium of the subgrid-scales: The velocity and scalar fields. Phys. Fluids, 17:108103, 2005.

[5] U. Schumann. Subgrid-scale model for finite difference simulations of turbulent flows on plane channels and annuli. J. Comp. Phys., 18:376–404, 1975. [6] A. Yoshizawa. A statistically-derived subgrid model for the large-eddy simulation of turbulence. Phys. Fluids, 25(9):1532–1538, 1982. [7] V. C. Wong. A proposed statistical-dynamic closure method for the linear or nonlinear subgridscale stresses. Phys. Fluids, 4(5):1080–1082, 1992. [8] S. Ghosal, T. Lund, P. Moin, and K. Akselvol. A dynamic localisation model for large-eddy simulation of turbulent flows. J. Fluid Mech., 286:229–255, 1995. [9] A. Dejoan and R. Schiestel. Les of unsteady turbulence via a one-equation subgrid-scale transport model. Int. J. Heat Fluid Flow, 23:398, 2002. [10] S. Krajnovi´c and L. Davidson. A mixed one equation subgrid model for large-eddy simulation. Int. J. Heat and Fluid Flow, 23:413–425, 2002. [11] R. Schiestel and A. Dejoan. Towards a new partially integrated transport model for coarse grid and unsteady turbulent flow simulations. Theor. Comput. Fluid. Dyn., 18:443, 2005. [12] L. Davidson and S. Peng. Hybrid les-rans modelling: a one-equation sgs model combined with a k − w model for predicting recirculating flows. Int. J. Numer. Methods Fluids, 43:1003–1018, 2003. [13] C. de Langhe, B. Merci, and E. Dick. Hybrid rans/les modelling with an approximate renormalization group. i: Model development. Journal of Turbulence, 6(13), 2005. [14] C. Jim´enez, F. Ducros, B. Cuenot, and B. B´edat. Subgrid scale variance and dissipation of a scalar field in large eddy simulations. Phys. Fluids, 13(6):1748–1754, 2001. [15] C. B. da Silva and J. C. F. Pereira. The effect of subgrid-scale models on the vortices obtained from large-eddy simulations. Phys. Fluids, 16(12):4506–4543, 2004. [16] G. Hauet, C. B. da Silva, and J. C. F. Pereira. The effect of subgrid-scale models in the near wall vortices: a-priori tests. Phys. Fluids, 19:051702, 2007. [17] C. B. da Silva and J. C. F. Pereira. Analysis of the gradient-diffusion hypothesis in large-eddy simulations based on transport equations. Phys. Fluids, 19:035106, 2007. [18] E. G. Paterson and L. J. Peltier. Detached-eddy simulation of high-reynolds-number beveledtrailling-edge boundary layers and wakes. J. Fluids Eng., 127:897–906, 2005. [19] K. Hanjalic. Will RANS survive LES? a view of perspectives. J. Fluids Eng., 127:831–839, 2005. [20] C. B. da Silva S. Rego and J. C. F. Pereira. Analysis of the viscous/molecular subgrid-scale dissipation terms in les based on transport equations: A priori tests. Journal of Turbulence, (in Press), 2008.

EΣ(K) , Eb(K) , EΣθ(K) , Eθb(K)

Figure 1: Ilustration of the influence of the flow coherent vortices (grey) on the evolution of the Figure 2: The failure of the local equilibrium SGS kinetic energy (red) in the far field of a assumption between the kinetic energy transfer plane jet into the small scales of motion Π and the viscous dissipation ε).

Σ ε∆b Σθ ε∆θb

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Figure 4: Comparison of spectra of the exact Figure 3: Comparison between the real and ∆ Σ and Σθ - and modeled - ε∆ b and εθb - viscous modeled enstrophy SGS dissipation in classical and molecular SGS dissipation terms using the a-priori tests. hybrid RANS/LES model).

Aspects of the Calculation of High-Reynolds-Number Flows with RANS solvers E¸ca L. Mechanical Engineering Department, IST, TU Lisbon, Portugal. [email protected]

Abstract In this presentation, the focus is on the Reynolds-Averaged Navier-Stokes (RANS) equations, which are still the most common model for the solution of engineering flow problems at high Reynolds numbers (above 106 ). It is well known that the RANS equations do not form a closed system due to the presence of the Reynolds stresses (produced by the statistical handling of the Navier-Stokes equations). In all the examples included in this presentation, we will model these terms by one or two-equation eddy-viscosity models. PARNASSOS [1] is a RANS solver for incompressible flows that has been developed in ongoing cooperation between the Maritime Research Institute of the Netherlands (MARIN) and IST. It uses an unusual solution strategy: the continuity and momentum equations are solved in a fully-coupled way with the continuity equation written in its original form (divergence of velocity equals 0). Although it requires structured grids, it has a multi-block capability and can compute free-surface flows [2] by a surface fitting method. One of the outstanding qualities of PARNASSOS is that it allows to compute high-Reynolds number flows (up to 109 ) without the use of wall functions and without performance deterioration. As an example of the potential of such code, figure 1 presents the friction, CF , and pressure, CP , resistance coefficients of the KVLCC2 tanker at model (5 × 106 ) and full (2.03 × 109 ) scale Reynolds numbers. Results are presented for sets of six grids with five eddy-viscosity turbulence models: three versions of the k − ω two equation model [3, 4] (TNT,BSL,SST); the one-equation model of Menter [5] (Mνt ) and the recently proposed √ k − kL two-equation model [6]. Although such CFD application might be considered fairly standard for these days, there is a strong emphasis on the importance of Code Verification and Solution Verification [7]. Addressing the problem of numerical uncertainty is mandatory to avoid misleading conclusions. Predicting scale effects with the coarsest grids data could well lead to the

Model Scale, 5 × 106 1.7

SST p= 4.9 p*=2 BSL p= 2.6 p*=2 TNT p= 3.3 p*=2 Mνt p= 2.4 p*=2 KSKL p= 3.0 p*=2

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0.3 CP×103

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SST p= 1.3 BSL p= 1.5 TNT p= 1.8 Mνt p= 2.2 p*=2 KSKL p= 2.0

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Figure 1: Convergence of the friction, CF , and pressure, CP , resistance coefficients with the grid refinement ratio, hi /h1 . KVLCC2 tanker. opposite tendency as found by using the finest grids results, if there is no estimate of the numerical uncertainty. Code Verification requires error evaluation, i.e. the knowledge of the exact solution. In the RANS context, this leads to the Method of Manufactured Solutions [8]. Such Manufactured Solutions should include the turbulence quantities transport equations [9], because it has been found that the observed order of accuracy may be affected by the turbulence model [10]. Verification of Calculations requires error estimation for an unknown exact solution. Although the numerical error of a RANS solution includes contributions from the roundoff, iterative and discretization errors, the latter is usually dominant. Most of the techniques available for the estimation of the discretization error rely on the existence of an ‘asymptotic range’, which is basically the dominance of the lowest order term of a power series expansion of the error as a function of the typical cell size. Attaining the ‘asymptotic range’ in RANS solutions of complex turbulent flows requires much denser grids than what is common practice nowadays. Therefore, developing a reliable error estimator for RANS solutions of practical complex turbulent flows remains an open challenge.

Keywords: RANS solvers, Verification, Numerical Errors.

References [1] van der Ploeg A., E¸ca L., Hoekstra M. - Combining Accuracy and Efficiency with Robustness in Ship Stern Flow Computations - 23rd Symposium on Naval Hydrodynamics, Val re Reuil, France, September 2000. [2] Raven H.C., van der Ploeg A., Starke B. - Computation of free-surface viscous flows at model and full scale by a steady iterative approach - 25th Symposium on Naval Hydrodynamics, St. John’s, Newfoundland, August 2004.

[3] Kok J.C. - Resolving the Dependence on Free-stream values for the k − ω Turbulence Model - NLR-TP-99295, 1999. [4] Menter F.R. - Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications - AIAA Journal, Vol.32, August 1994, pp. 1598-1605. [5] Menter F.R. - Eddy Viscosity Transport Equations and Their Relation to the k − ǫ Model - Journal of Fluids Engineering, Vol. 119, December 1997, pp. 876-884. √ [6] Menter F.R., Egorov Y., Rusch D. - Steady and Unsteady Flow Modelling Using the k− kL Model - Turbulence Heat and Mass Transfer 5, 2006. [7] Roache P.J. - Verification and Validation in Computational Science and Engineering Hermosa Publishers, 1998. [8] Roache P. - Code Verification by the Method of Manafuctored Solutions - ASME Journal of Fluids Engineering, Vol. 114, No 1, March 2002, pp. 4-10. [9] E¸ca L., Hoekstra M. - Verification of Turbulence Models with a Manufactured Solution European Conference on Computational Fluid Dynamics, ECCOMAS CFD 2006, September 2006, Netherlands. [10] E¸ca L., Hoekstra M., Hay A., Pelletier D. - Verification of RANS solvers with Manufactured Solutions - Engineering with Computers, Vol. 23, No 4, December 2007, pp. 253-270

CFD for Environment and Civil Engineering applications: Vofor2DV model Rita F. Carvalho IMAR, Department of Civil Enginering,University of Coimbra, Portugal [email protected]

Abstract The use of CFD and efficient results visualization is increasing in civil-hydraulics and environment engineering applications, even in areas traditionally based in physical experience like river hydraulics, dams and hydroelectric projects (see http://www.aldenlab.com/index.cfm/news?NID=107). This presentation aims to present an overview of the fluid equations of motion, simplified equations and auxiliary models most used in civil and environment engineering field and the associated numerical models. A numerical model for studying two-dimensional flows in a vertical plane based on a volume of fluid and a fractional volume and area obstacle representation techniques is also presented emphasizing some details of numerical approaches, auxiliary models, recent developments and potential future refinements. Starting by the fluid equations of motion, the well known Navier-Stokes equations, details, usual simplifications, as well as the characteristics of the flow behavior in specific cases in these fields are pointed. Therefore, several models like Saint-Venant, Boussinesq type and VOF type are briefly described, emphasizing the main limitations and developments. Some applications that reinforce the aim of each kind of model are presented. Special care is given to the results analysis and their comparisons between physical, model and numerical representations. The VOFOR2DV model is based on the Reynolds-averaged Navier-Stokes equations governing the motion of the mean 2D incompressible flows in the x  z plane, in which the free surface is described using a refined Volume-Of-Fluid (VOF) method [4]. The internal obstacles are described by means of the Fractional Area-Volume Obstacle Representation (FAVOR) algorithm [5]. The governing equations are discretized using a finite difference staggered grid system of rectangular cells (control volume) with variables width ' x i and height ' z j . The porosity function T is defined at the horizontal and vertical faces of each control volume using the mesh-wide arrays AT and AR and at the centre of the control volume using another array AC allowing the treatment of curved boundaries in cartesian meshes without stair-step instabilities. The details of the VOFOR2DV can be found in [1]. Figure 1 illustrates some hydraulic structures applications. Numerical simulations using different mathematical models are then compared with some experimental work: a partial submerse break water and landslides into the water. Finally, recent advances related with auxiliary model inclusion in Vofor2dv are presented. Keywords: CFD, free-surface, numerical model, VOF, FAVOR

a)

b)

c) Figure 1: a) Hydraulic jump stilling basin [1]; b) Hydraulic jump stilling basin with a baffle wall [1]; c) Stepped spillway with small slope: numerical results vs photo [3]; d) Stepped spillway with large slope numerical results vs PIV data [2]

References [1] Carvalho, R. F. 2002, “Acções hidrodinâmicas em estruturas hidráulicas: Modelação computacional do ressalto hidráulico”, Tese de Doutoramento, Faculdade de Ciências e Tecnologias da Universidade de Coimbra, Coimbra [2] Carvalho, R. F. & Amador, A.T., “Flow field over a stepped spillway: Physical and numerical approach”, IAHR – Symposium on Hydraulic Structures (IAHR), 12-13 October, 2008, China [3] Carvalho, R. F. & Martins, R. – “Stepped spillway with hydraulic jumps: scale and numerical models of a conceptual prototype”: Journal of Hydraulic Engineering da ASCE (submitted and partial accepted 2007); [4] Hirt & Nichols, 1981. ”Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries”. J. Comp. Phys. 39, 201–225. [5] Hirt, C. W. and Sicilian, J., “A porosity technique for the definition of obstacles in rectangular cell meshes”, Fourth International Conference on Ship Hydrodynamics, Washington, DC, September 1985.

An Overview of CFD Applications in Industrial Aerodynamics D. X. Viegas

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ADAI/LAETA, Departamento de Engenharia Mecˆ anica, Universidade de Coimbra. [email protected]

Abstract An overview of CFD applications to practical situations created by industrial aerodynamics and environmental problems that are dealt with by ADAI team is presented. In the general case CFD is used in parallel with experimental methods to study problems that are formulated by the industry; experimental results are used to validate the numerical model and then the model is used to analyze other situations that cannot be covered easily by the experimental program. Examples of case studies related to some applications that were developed recently or that are under development at ADAI are presented. Wind engineering: wind shelter of a rowing track; wind flow around buildings; wind modeling over complex topography. Industrial aerodynamics: transport of particles by a flow; flow in a discharge valve. Car aerodynamics: flow modeling around a car. Climate and comfort: efficiency of an air curtain to preserve isothermal conditions in a compartment; characterizing and optimizing the behavior of hygroscopic wheels. Forest fires: modeling of fire propagation to improve safety and to support decision making process. The methodology employed combining CFD and experimental methods has proven to be robust and reliable to provide solutions to several applications created by industry and environment protection in the deadlines requested.

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Given the nature of its contents this paper was prepared in collaboration with several colleagues from ADAI. Their names will be given in the full presentation

Large and small scale aspects of the turbulent entrainment in jets Carlos B. da Silva IDMEC/IST, Mecˆ anica I, 1o andar/LASEF, Av. Rovisco Pais, 1049-001 Lisbon, PORTUGAL. [email protected]

Abstract This work analysis several large and small scale aspects of the turbulent entrainment mechanism that exists in mixing layers, wakes, and jets. In these flows flow field can be divided into two regions. In one region the flow is turbulent (T) and its vorticity content is high, while in the other region the flow consists of largely irrotational (nonturbulent - NT) flow. The two flow regions are divided by the turbulent/nonturbulent (T/NT) interface where the turbulent entrainment mechanism takes place.The physical mechanisms taking place at the T/NT interface is important in many natural and engineering flows since important exchanges of mass, momentum and passive or active scalar quantities take place across the T/NT interface. It was assumed in the past that the turbulent entrainment mechanism is mainly driven by ”engulfing” motions caused by the large scale flow vortices, but recent experimental and numerical works give more support to the original model of Corrsin and Kistler [1] where the entrainment is primarily associated with small scale (”nibbling”) eddy motions (Mathew and Basu [2], Bisset et al.[3], Westerweel et al.[4], Holzner et al. [5]). However, it is assumed that the entrainment and mixing rates are largely determined by the large scales of motion. The present work uses a direct numerical simulation (DNS) of a turbulent plane jet at Reλ ≈ 120 in order to analyze several large and small scale aspects of the turbulent entrainment and particularly their interplay. Figure 1 shows contours of vorticity modulus for this simulation. The T/NT interface is detected using a similar procedure than the one described in previous works (figure 2). Conditional statistics of the vorticity norm and vorticity components in relation to the distance to the T/NT interface are given in figure 3. The vorticity is zero in the irrotational flow region, rises steeply across the T/NT interface, and is more or less constant inside the turbulent region. Recently da Silva and Pereira [6] analyzed the invariants of the velocity-gradient, rateof-strain, and rate-of-rotation tensors across the T/NT interface in order to characterize several aspects of the small scale dynamics near the T/NT interface. In the present work

we focus on the intense vorticity structures (IVS) from the flow, as defined by Jim´enez [7], in order to analyze the interplay between the large and small scales of the flow during the turbulent entrainment. Notice that right at the T/NT interface the flow lacks these large scale structures, in agreement with the analysis from da Silva and Pereira [6] (see figure 4). An interesting result is the existence of non negligible viscous dissipation rate outside the turbulent region. It turns out that this interesting phenomena is caused the the presence of IVS near the T/NT interface (Fig. 5). The presentation will focus on how the presence of these IVS commands the evolution of many small scale quantities and ultimately imposes the entrainment rate.

Keywords: Turbulent entrainment, intense vorticity structures, kinetic energy and enstrophy dynamics, invariants of the velocity-gradient, rate-of-strain, and rate-of-rotation tensors.

References [1] S. Corrsin and A. L. Kistler. Free-stream boundaries of turbulent flows. Technical Report TN-1244, NACA, 1955. [2] J. Mathew and A. Basu. Some characteristics of entrainment at a cylindrical turbulent boundary. Phys. Fluids, 14(7):2065–2072, 2002. [3] D. K. Bisset, J. C. R. Hunt, and M. M. Rogers. The turbulent/non-turbulent interface bounding a far wake. J. Fluid Mech., 451:383–410, 2002. [4] J. Westerweel, C. Fukushima, J. M. Pedersen, and J. C. R. Hunt. Mechanics of the turbulent-nonturbulent interface of a jet. Phys. Review Lett., 95:174501, 2005. [5] M. Holzner, A. Liberzon, N. Nikitin, W. Kinzelbach, and A. Tsinober. Small-scale aspects of flows in proximity of the turbulent/nonturbulent interface. Phys. Fluids, 19:071702, 2007. [6] C. B. da Silva and J. C. F. Pereira. Invariants of the velocity-gradient, rate-of-strain, and rate-of-rotation tensors across the turbulent/nonturbulent interface in jets. Phys. Fluids, 055101, 2008. [7] J. Jimenez, A. Wray, P. Saffman, and R. Rogallo. The structure of intense vorticity in isotropic turbulence. J. Fluid Mech., 255:65–90, 1993.

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-1 -2 Figure 2: Sketch of the T/NT interface for the plane jet indicating the vorticity surface (solid -3 -2 -1 0 1 2 line) and the interface envelope (grey dashed x/H lines). The sketch shows also the coordinate system used in the computation of the plane jet Figure 1: Contours of vorticity modulus corre- (x, y) and the one used to analyse the T/NT sponding to Ω = 0.7U1 /H in a (x, y) plane of interface (xI , yI ). the jet at T /Tref = 27 <Ω>I <Ωz>I <|Ωx|>I <|Ωy|>I <|Ωz|>I

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Figure 4: Conditional mean profiles of radius, Figure 3: Conditional mean profiles of the vorlenght, and vorticity of the intense vorticity ticity norm and its components in relation to structures (IVS) in relation to the distance from the distance from the T/NT interface. the T/NT interface.

Figure 5: The role of the intense vorticity structures (grey) on the generation of irrotational viscous dissipation (contours) outside the turbulent region, near the T/NT interface (white). The plot shows only a small subdomain of the computation.

Hemorheology and hemodynamics: modeling and simulations Ad´elia Sequeira Department of Mathematics and CEMAT/IST, Technical University of Lisbon, Portugal. [email protected]

Abstract The blood circulation in the cardiovascular system depends not only on the driving force of the heart and the architecture and mechanical properties of the vascular system, but also on the mechanical properties of blood itself. Whole blood is a concentrated suspension of formed cellular elements including red blood cells (erythrocytes) white blood cells (leukocytes) and platelets (thrombocytes). Blood cells are suspended in plasma, an aqueous ionic solution. Experimental investigations over many years show that blood flow exhibits non-Newtonian characteristics such as shear-thinning viscosity, thixotropy, viscoelasticity and yield stress, and its rheology is largely due to three aspects of erythrocytes behavior: their ability to form a three dimensional (3D) microstructure at low shear rates, their deformability, and their tendency to align with the flow field at high shear rates. An understanding of the coupling between the blood composition and its physical properties is essential for developing suitable constitutive models to describe blood behavior. Hemodynamic factors such as flow separation, flow recirculation, and low and oscillatory wall shear stress are recognized as playing important roles in the localization and development of important vascular diseases. Therefore, mathematical and numerical simulations of blood flow in the vascular system can ultimately contribute to improve clinical diagnosis and therapeutic planning. However, meaningful hemodynamic simulations require constitutive models that can accurately model the rheological response of blood over a range of physiological flow conditions. Experimental evidence on the stability of the 3D microstructure of erythrocytes suggests that it is probably reasonable to treat the blood viscosity as constant in most parts of the arterial system of healthy individuals, due to the high shear rates found in these vessels and the length of time necessary for the blood microstructure to form. However, in diseases states in which the stability of the aggregates is enhanced or for diseases in which the arterial geometry has been altered to include regions of recirculation (e.g. saccular aneurysms), this simplifying assumption may need to be relaxed and a more complex blood constitutive model should be used. In addition, even in healthy patients, the non-Newtonian characteristics of blood can play an important role in parts of the venous system.

In this talk we present a short overview of some constitutive models that can mathematically characterize the rheology of blood and some numerical simulations to illustrate its phenomenological behaviour [1, 2]. Some numerical simulations obtained in geometrically reconstructed real vessels will be shown to illustrate the hemodynamic behavior using Newtonian and non-Newtonian inelastic models under a given set of physiological flow parameters. Moreover, using a mesoscopic lattice Boltzmann flow solver for non Newtonian shear thinning fluids, we present a three-dimensional numerical study of the dynamics of leukocytes rolling and recruitment by the endothelial wall, based on in vivo experimental measurements in Wistar rat venules [3]. Preliminary numerical results obtained for a comprehensive model of blood coagulation and clot formation, that integrates physiologic, rheologic and biochemical factors will also be presented [4].

Figure 1: Wall shear stress distribution for a non-Newtonian Carreau-Yasuda model (left) and for a Newtonian model at high shear rate viscosity (right). Keywords: blood rheology, blood cells, non-Newtonian models, numerical simulations.

References [1] A.M. Robertson, A. Sequeira and M. Kameneva, Hemorheology. In:Hemodynamical Flows: Modelling, Analysis and Simulation. Series: Oberwolfach Seminars, G. P. Galdi, R. Rannacher, A. Robertson and S. Turek (eds.), Vol. 37, pp. 63-120, Birkhauser, 2008. [2] A.M. Robertson, A. Sequeira and R.G. Owens, Rheological models for blood. In: Cardiovascular Mathematics, A. Quarteroni, L. Formaggia and A. Veneziani (eds.), Springer-Verlag (2008), to appear. [3] Artoli A.M., Sequeira A., Silva A.S., and Saldanha C., Leukocyte rolling and recruitment by endothelial cells: hemorheological experiments and numerical simulations, Journal of Biomechanics, Vol. 40, Issue 15, pp. 3493-3502, 2007. [4] T. Bodn´ar and A. Sequeira, Numerical simulation of the coagulation dynamics of blood, Computational and Mathematical Methods in Medicine, Vol. 9, Issue 2, pp. 83-104, 2008.

Geometrical Multiscale Modelling of the Integrated Cardiovascular System Alexandra B. Moura CEMAT, Center for Mathematics and its Applications, Department of Mathematics, IST, Lisbon, Portugal. [email protected]

Abstract Over the past years, mathematical modelling and numerical simulation play a very important role in the understanding of the cardiovascular system, helping in the prediction of the origin and the development of pathologies, as well as the result of surgical interventions. Nevertheless, the functional and geometrical complexity of the human circulatory system make it a very difficult and challenging task to model. For instance, three-dimensional (3D) simulations are restricted to truncated regions of interest, due to their computational costs and the impossibility of having a 3D representation of the whole cardiovascular system. However, local realistic simulations can not be carried out without taking into account the global circulation, which is commonly neglected. Also, blood flow is a non-Newtonian fluid, presenting a shear-thinning behaviour [1], which should be taken into account, specially in small vessels or certain pathological cases. The geometrical multiscale modelling consists in coupling together different models with different levels of complexity and detail. In this way, the most adequate model can be applied to each part of the cardiovascular system, or type of investigation at hand, in an integrated manner. In this context, the remaining parts of the cardiovascular system in a local 3D simulation can be taken into account resorting to reduced one-dimensional (1D) or lumped parameters (also called zero-dimensional, 0D) models, Figure 1. We present the geometrical multiscale approach. We describe the different models, from 3D fluid-structure interaction (FSI) and generalized Newtonian models, to the 1D hyperbolic model, applied to study the pressure wave propagation in large networks of arteries, and lumped parameters models representing wide compartments, such as the heart or the resistance due to the venous bed. The different mathematical nature of all these models makes their integration the main challenge of this approach [3, 4, 5]. We discuss the techniques and strategies to couple them, both at the analytical and numerical levels, evidencing the main difficulties. In particular, the stability of the coupling between the 1D and the 3D FSI models is guaranteed through a non standard curl formulation of the 3D fluid equations. The effectiveness of the couplings and the geometrical multiscale approach is illustrated through numerical results.

Figure 1: Scheme of a geometrical multiscale model: coupling between a 3D carotid bifurcation (left), a 1d network representing the arteries from the heart to the Circle of Willis (middle) and a lumped parameters model representing the heart and terminal resistances (right). Keywords: Blood flow simulations, geometrical multiscale modelling, coupling strategies, fluid-structure interaction (FSI), hemorheology, 3D FSI model, reduced 1D and 0D models.

References [1] A. M. Robertson, A. Sequeira and M. Kameneva. Hemorheology, in: Hemodynamical Flows: Modeling, Analysis and Simulation, Series: Oberwolfach Seminars, Vol. 37, Galdi, G.P., Rannacher, R., Robertson, A.M., Turek, S., Birkhauser, 2008 [2] L. Formaggia, A. Quarteroni and A. Veneziani. Cardiovascular Mathematics. Springer, Berlin and Heidelberg, 2008. [3] L. Formaggia, A. Moura and F. Nobile. On the stability of the coupling of 3D and 1D fluidstructure interaction models for blood flow simulations. ESAIM: Mathematical Modelling and Numerical Analysis (M2AN), 41(4):743-769, 2007. [4] A. Moura, The Geometrical Multiscale Modelling of the Cardiovascular System: Coupling 3D FSI and 1D Models, PhD Thesis. [5] L. Formaggia, A. Moura, F. Nobile, T. Passerini. The interplay of different models in the simulation of the cardiovascular system. Procs. APCOM’07 in conjunction with EPMESC XI, December 3-6, 2007, Kyoto, Japan, 2007.

Physiological morphology of the right nasal cavity for three patient cases Alberto M. Gambaruto Dept. Mathematics, Instituto Superior T´ecnico, Lisbon, Portugal. [email protected]

Denis J. Doorly Dept. Aeronautics, Imperial College London, London, UK. [email protected]

Abstract The nasal airways accomplish a diverse range of functions: warming, humidifying and cleansing inspired air, and sampling for olfaction [Wolf, 2004]. Unsurprisingly, this brings about a complex the anatomical form of the nasal cavity. The Lagrangian dynamics of marker particles illustrate both the dynamics of the flow field and can be used to quantify the degree of convective mixing (or stirring) resulting from the complex morphology. Figure 1 illustrates a right nasal cavity surface geometry and the nomenclature of the distinct regions. Results of 3 patient cases are given for constant inspiration at rest (100 ml/s), equivalent to quiet, restful breathing. The Newtonian, incompressible, Navier-Stokes equations were solved to third order accuracy using the Fluent CFD code. Geometry deconstruction via Fourier descriptors [Gambaruto, 2008] is used to divide the nasal passage into regions and describe the flow features locally with aimed interpatient comparison. Measures obtained from the pathlines include, mixing based on the Shannon entropy [Kang, 2004] and mean transit paths. Vortical structures and recirculation zones are identified. For the patient cases studied, results indicate: the mean washout time (time for a particle to transit from inflow to outflow) ranges from 0.13s to 0.20s; relatively little mixing occurs, ˜30%, with respect to the theoretical maximum by the time that the flow reaches the nasopharynx. The portion of the flow that passes through the olfactory region originates from directly in front of the subject; quiet steady olfaction thus samples principally a specific region of the inspired volume, with relatively little mixing by the time the flow reaches the olfactory region (∼10%).

Keywords: nasal cavity, morphology, geometry deconstruction, Lagrangian dynamics, entropic mixing.

olfactory cleft septum

nasopharynx slice AC

middle meatus

middle turbinate

vestibule

medial passage inferior meatus inferior turbinate

slice PC slice MC

nasal valve region

(a)

naris

(b)

Figure 1: Nomenclature of right nasal cavity airway (patient case 3). The surface shown in (a) is the boundary of airway to surrounding solid structure. The location of illustrative slices taken in the anterior (AC), middle (MC) and posterior (PC) cavity regions are also shown. Slice MC is shown in (b) in the coronal plane.

References [1] Wolf M., Naftali S., Schroter R.C., Elad, D., Air Conditioning Characteristics of the Human Nose, The Journal of Laryngology and Otology, Vol. 118, pp. 87–92, 2004. [2] Gambaruto A.M., Taylor D.J., Doorly D.J., Modelling nasal airflow using a Fourier descriptor representation of geometry, International Journal for Numerical Methods in Fluids, in press, 2008. [3] Kang TG., Kuon T.H., Colored particle tracking method for mixing analysis of chaotic micromixers, Journal of Micromechanics and Microengineering, Vol. 14, pp. 891–899, 2004.