Contributed Talks Merge B5

Contributed Lectures - Abstracts

A posteriori error estimates for the Mimetic Finite Difference approximation of the diffusion problem L. Beir˜ao da Veiga Department of Mathematics “F.Enriques”, University of Milan, Italy. [email protected]

G. Manzini IMATI-CNR, Pavia. [email protected]

Abstract The main characteristic of the Mimetic Finite Difference (MFD) method, when compared to a more standard finite element approach, is that the basis functions related to the discrete degrees of freedom are not explicitly defined. As a consequence, the operators and other quantities appearing in the problem must be approximated by discrete counterparts that satisfy finite dimensional analogs of some fundamental property. This approach allows for a greater flexibility of the mesh and the possibility to mimic intrinsic properties of the differential problem under study. In particular, general polyhedral (or polygonal in 2 dimensions) meshes, even with non convex and non matching elements, can be adopted. This flexibility makes the MFD method a very appealing ground for the application of mesh adaptivity. In the present talk we focus on the MFD scheme for the diffusion problem and derive local a posteriori error estimates for the method. The error estimator is shown to be both reliable and efficient with respect to an energy type norm involving a post-processed pressure [1]. Finally, the error indicator is combined with a simple adaptive process and a set of numerical tests is presented [2].

Keywords: Mimetic Finite Differences, diffusion problem, a posteriori error estimation.

References [1] L.Beir˜ao da Veiga, A residual based error estimator for the Mimetic Finite Difference method, Numer. Math 108: 387-406 (2008). [2] L.Beir˜ao da Veiga e G.Manzini, An a-posteriori error estimator for the mimetic finite difference approximation of elliptic problems with general diffusion tensor, in press on Int. J. Numer. Meth. Engrg.

Application of CFD in the Study of Supercritical Fluid Extraction with Structured Packing: Pressure Drop Calculations Jo˜ao B. Fernandes,1 Pedro C. Sim˜oes,2 Jos´e P. B. Mota3 REQUIMTE, Departamento de Qumica, Faculdade de Ciˆencias e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal 1 [email protected], 2 [email protected], 3 [email protected]

Esteban Saatdjian LEMTA-CNRS, 2 ave. de la forˆet de Haye, 54500 Vandoeuvre, France [email protected]

Abstract Supercritical fluid extraction (SFE) from the liquid phase is usually carried out in packed columns with structured packings, particularly those of the gauze type. Structured packing performs very well for SFE, mainly because of their relatively large surface area and free volume. Nevertheless, it also has important disadvantages such as high cost, low capacities at high flow rates, premature flooding, and entrainment of the liquid phase due to low density differences. The assessment of the real efficiencies of structured packings poses extreme difficulties related to the moderately high pressures involved in SFE processes. Recently, Computational Fluid Dynamics (CFD) has been used to characterize the complex multi-phase flow inside packed columns and to evaluate the influence of packing shape and geometry on the hydrodynamics and mass and heat transfer rates. Here, we present and discuss two different geometrical models that represent the structured gauze packing (Sulzer EX) that fills our pilot-scale SFE column. The first model consists of two contacting corrugated sheets enclosed in a box; this model focuses on the space between two packing sheets. The presence of neighboring packing sheets is accounted for by applying periodic conditions on the boundaries perpendicular to the main flow direction. The second geometric model consists of thirteen contacting packing sheets enclosed in a cylinder. In this case, the model tries to mimic a full packing element, even though we only simulate one third of the height of the actual packing element due to computational constraints. Periodicity is imposed on the flow in order to determine the fully developed flow field and estimate pressure drops for the whole column.

For both models, the flow field is computed using a standard k −  turbulence model. Our CFD models are validated by comparing simulation results and the pressure-drop data sets of Stockfleth and Brunner, Meyer (carbon dioxide at temperatures ranging from 313 to 393K and pressures ranging from 10.1 to 30 MPa) and Ola˜ no et al (data obtained with air at ambient pressure). The final objective of our work is to model the complex multi-phase transport phenomena present in SFE columns with structured packing in order to predict exchange rates and to optimize the process. The first stage, presented here, consists in an accurate modeling of the hydrodynamics inside the relatively complex geometry of the structured packing.

Keywords: Pressure drop, CFD modeling, structured packing, supercritical fluids

References [1] G. Brunner, Industrial process development in Countercurrent multistage gas extraction (SFE) processes, J. Supercrit.Fluids , Vol. 12, 283, 1998. [2] F. Haghshenas, M. Zivdar, R. Rahimi, M. N. Esfahany, A. Afacan, K. Nandakumar, K. T. Chuang, CFD simulation of mass transfer efficiency and pressure drop in a structured packed distillation column, Chem. Eng. Technol., Vol. 30(7), 854-861, 2007. [3] Jens-Torge Meyer, Druckverlust um Flutpunkte im Hochdruckgegenstromkolonen mit berkritischen Kohlendioxid, PhD Dissertation, Technische Universitt Hamburg-Harburg, Germany, 1998. [4] S. Olao, S. Nagura, H. Kosuge, K. Asano, Mass transfer in binary and ternary distillation by a packed column with structured packing, J. Chem. Eng. Japan, Vol. 28(6), 1995. [5] M. McHugh, V. Krukonis, Supercritical Fluid Extraction: Principles and Practice, Butterworths, Stoneham. [6] R. Stockfleth, G. Brunner, Hydrodynamics of a packed countercurrent column for the gas extraction, Ind. Eng. Chem., Vol. 38, 1999. [7] R. Stockfleth, G. Brunner, Holdup, pressure drop and flooding in packed countercurrent columns for the gas extraction, Ind. Eng. Chem., Vol. 40, 2001.

Stabilization Methods for Shallow-Water Equations Juha H. Videman Departamento de Matem´ atica/CEMAT, IST, Lisbon, Portugal [email protected]

Abstract We consider streamline diffusion, also known as SUPG (Streamline Upwind Petrov–Galerkin), methods applied to the time-dependent shallow-water equations. Streamline diffusion (SD) methods are finite element methods that combine good stability properties with high accuracy and are particularly suitable for hyperbolic and advection-diffusion equations. The SUPG method, introduced by Thomas Hughes and Alexander Brooks in 1979 [5], was applied and analysed intensively throughout the 80’s by Thomas Hughes and, in parallel, by Claes Johnson, and their co-workers, see, e.g., [9, 1, 12, 10, 8, 11, 6, 4, 7, 2, 3]. Claes Johnson adopted the name streamline diffusion method [9], extended it to the time–dependent problems and related the method, regarding the time discretization, to the discontinuous Galerkin method [12, 10]. Written in conservation form (mass/momentum flux), the shallow-water equations constitute a non-linear hyperbolic system, similar to the compressible Navier-Stokes equations, and their numerical approximation, either in conservative or non-conservative form, has been obtained by various finite difference and finite element methods, most recently by local discontinuous Galerkin methods. Rigorous error analyses have, however, been scarce and even more so for the fully discretized problem written in terms of the non-conservative variables – the depth-integrated horizontal velocities and the height of the free surface. In this talk, I will present some of our recent results on the application of SD methods, with time–space elements, to two–dimensional shallow-water equations written in a non–conservative form. We will prove error estimates of order hk and hk+1/2 using a suitably stabilized variational formulation. Our finite element approximation is continuous in space but possibly discontinuous in time and we use k th –order polynomials for the surface height and polynomials of order k or k + 1 for the velocities. This is a joint work with Clint Dawson from the Center for Subsurface Modeling at the Institute for Computational Engineering and Sciences at the University of Texas at Austin (USA).

References [1] Brooks, A, Hughes, T, Streamline Upwind/Petrov–Galerkin Formulations for Convection Dominated Flows with Particular Emphasis on the Incompressible Navier–Stokes Equations, Comp. Meth. Appl. Mech. Engrg 32 (1982), 199–259. [2] Hansbo, P, Szepessy, A, A Velocity–Pressure Streamline Diffusion Finite Element Method for the Incompressible Navier–Stokes Equations, Comp. Meth. Appl. Mech. Engrg 84 (1990), 175–192. [3] Hansbo, P, Johnson, C, Adaptive Streamline Diffusion Methods for Compressible Flow Using Conservation Variables, Comp. Meth. Appl. Mech. Engrg 87 (1991), 267–280. [4] Hughes, T, Recent Progress in the Development and Understanding of SUPG Methods with Special Reference to the Compressible Euler and Navier–Stokes Equations, Int. J. Num. Meth. Fluids 7 (1987), 1261–1275. [5] Hughes, T , Brooks, A, A Multi–Dimensional Upwind Scheme with No Crosswind Diffusion, in: Finite Element Methods for Convection Dominated Flows, T Hughes (ed.), ASME Monograph 34, (1979), pp. 19–35. [6] Hughes, T, Franca, L, Mallet, M, A New Finite Element Formulation for Computational Fluid Dynamics: VI. Convergence Analysis of the Generalized SUPG Formulation for Linear Time–Dependent Multidimensional Advective–Diffusive System, Comp. Meth. Appl. Mech. Engrg 63 (1987), 97–112. [7] Hughes, T, Franca, L, Hulbert , G, A New Finite Element Formulation for Computational Fluid Dynamics: VIII. The Galerkin/Least–Squares Method for Advective–Diffusive Equations, Comp. Meth. Appl. Mech. Engrg 73 (1989), 173–189. [8] Hughes, T, Mallet, M, A New Finite Element Formulation for Computational Fluid Dynamics: III. The Generalized Streamline Operator for Multidimensional Advective–Diffusive Systems, Comp. Meth. Appl. Mech. Engrg 58 (1986), 305–328. [9] Johnson, C, N¨ avert, U, An Analysis of Some Finite Element Methods for Advection–Diffusion Problems, in: Analytical and Numerical Approaches to Asymptotic Problems in Analysis, Axelsson, S et al (eds.), North– Holland, (1981), pp. 99–116. [10] Johnson, C, N¨ avert, U, Pitk¨ aranta, J, Finite Element Methods for Linear Hyperbolic Problems Comp. Meth. Appl. Mech. Engrg 45 (1984), 285–312. [11] Johnson, C, Saranen, J, Streamline Diffusion Methods for the Incompressible Euler and Navier–Stokes Equations, Math. Comp. 47 (1986), 1–18. [12] N¨ avert, U, A Finite Element Method for Convection–Diffusion Problems, PhD Thesis, Chalmers TU, G¨oteborg, (1982).

Convective dynamo in a rotating plane layer R. Chertovskih, S. Gama Departement of Applied Mathematics, University of Porto, Portugal. [email protected], [email protected]

O. Podvigina, V. Zheligovsky International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Moscow, Russian Federation. [email protected], [email protected]

Abstract Magnetic fields of planets and other astrophysical objects are often sustained by conducting fluid motions, driven by convection in their interior. A common feature of such objects is rotation. This work is aimed at investigation of magnetic field generation in an idealized setup. We consider a rotating conducting fluid heated from below in a plane horizontal layer (often regarded as representing a segment of a spherical shell in the interior of a planet). Flows and magnetic fields in square periodicity cells are examined for the aspect ratio, for which the trivial hydrodynamic steady state becomes unstable to square patterns at the minimal Rayleigh number. Thermal convection of electrically conducting fluid in a plane horizontal layer is considered here in the Boussinesq approximation. The fluid is heated from below in a plane horizontal layer rotating about the vertical axis, the stress-free isothermal horizontal boundaries being perfect electrical conductors. In the dimensionless form the CHM system is characterised by the Rayleigh (Ra, measuring the amplitude of buoyancy forces), Prandtl (P , the ratio of kinematic viscosity to thermal diffusivity), magnetic Prandtl (Pm , the ratio of kinematic viscosity to magnetic diffusivity) and Taylor (T a, measuring the rate of rotation) numbers. We investigate numerically the influence of rotation on the dynamo properties of the convective fluid flows by examining the structure of the hydrodynamic and CHM attractors for different values of T a, and by considering the kinematic dynamo problem for the hydrodynamic attractors. The existence regions (in T a) of the attractors of different geometry have been explored applying the technique of continuation in the parameter. √ Computations have been performed in square periodicity boxes of size L = 2 2 in a layer of depth 1 for Ra = 2300, P = 1 and P√ m = 8. The assumed spatial period corresponds to the horizontal wavenumber k = π/ 2 of the first unstable Fourier mode (in the absence of magnetic field) for stress-free boundaries [1]. For this particular Rayleigh

number Ra = 2300, in the absence of rotation the hydrodynamic attractor is a travelling wave. This flow is a kinematic dynamo for Pm > 5.48 [2]. In our simulations Taylor number varied from T a = 0 (no rotation) to T a = 2000 (with the Coriolis force suppressing the fluid motion). The symmetry group of the convective hydromagnetic system is Z4 × T 2 × Z2 × Z2 , where Z4 is generated by rotation by π/2 about the vertical axis, the subgroups T are translations in horizontal directions, one Z2 is reflection about the horizontal midplane and the other one stems from magnetic field reversal. Attractors have been classified in particular by their symmetries, and bifurcations occurring in the system have been identified in terms of symmetry breaking. For parameter values under consideration hydrodynamic attractors (in the absence of magnetic field) of the convective system are steady rolls or travelling waves. For such flows the kinematic dynamo problem can be reduced to an eigenvalue problem; it has been solved using the algorithm [3]. The problem was studied numerically applying standard pseudospectral methods [4]. Most calculations were performed with the spatial resolution of 64 × 64 × 33 Fourier harmonics. For certain T a the computations were redone with the resolution of 128 × 128 × 65 harmonics to check the accuracy; the results remain qualitatively unaffected. Integration in time is done by a variant of the Adams-Bashforth method which reduces the stiffness of the system. In all the saturated nonlinear regimes that we have obtained, the flow has the structure of deformed rolls, magnetic energy remains much smaller than the flow kinetic energy, and magnetic field (when generated) is concentrated near the horizontal boundaries in halfropes, each spread along the flow streamlines (due to the flow advection) and cut into halves by the horizontal boundary along the rope axis. The work of RC is supported by the Funda¸c˜ao para a Ciˆencia e a Tecnologia under the grant SFRH/BD/23161/2005.

Keywords: magnetohydrodynamics, convective dynamo, kinematic dynamo, nonlinear dynamo, convection, rotating plane layer.

References [1] S. Chandrasekhar, Hydrodynamic and hydromagnetic stability, Oxford, 1961. [2] O. M. Podvigina, Magnetic field generation by convective flows in a plane layer, The European Physical Journal B, 50, 639–652, 2006. [3] V. A. Zheligovsky, Numerical solution of the kinematic dynamo problem for Beltrami flows in a sphere, Journal of Scientific Computing, 8 (1), 41–68, 1993. [4] J. P. Boyd, Chebyshev and Fourier spectral methods, Dover, 2000.

Particle tracking in fluid flow: a numerical approach for complex geometries L. A. Oliveira Mech. Eng. Dep. (FCTUC - Polo II), University of Coimbra, Portugal [email protected]

Abstract This talk reports a joint effort that is presently undertaken by four research teams, working at: (i) the “Laboratório de Aerodinâmica Industrial, LAI” of the University of Coimbra, Portugal; (ii) the University of Aveiro, Portugal; (iii) the University of McGill, Canada; (iv) the “Laboratoire d’Informatique pour la Mécanique et les Sciences de l’Ingénieur, LIMSI” of the University Pierre et Marie Curie, Paris, France. The basic aim is to numerically model dilute, three-dimensional, turbulent, incompressible fluid-solid particle flows that are bounded by impermeable walls of complex shape. The common motivation of the four teams is based on the wide variety of engineering applications involving particle dispersion in turbulent flows, including interior dust and particle pollutant control systems, separation processes, pneumatic transport systems, particle combustion in industrial furnaces or energy converters, sediment transport, erosion, some surface treatment procedures and development of new materials, safety and fire suppression systems, and food production processes, among others. In such applications, CFD is being increasingly used as an efficient, accessible and affordable way of making numerical predictions in support of design and optimization. In numerical simulations of fluid-particle flows, the continuous (fluid) phase is typically modelled via an Eulerian approach, while the dispersed (solid particle) phase is predicted using either an Eulerian or a Lagrangian approach. The Lagrangian approach is well suited for the description of the dispersed phase in the so-called dilute fluid-particle flows, in which the particle dynamics is controlled primarily by surface and body forces acting on the particle, rather than by particle-particle collisions or interactions. For the simulation of the continuous phase, control-volume finite element methods (CVFEMs) combine the merits of wellestablished finite-volume methods for regular geometries (easy interpretation of the formulation in terms of fluxes, forces, sources; satisfaction of local and global conservation requirements; and efficient techniques for handling the pressure-velocity coupling) and Galerkin finite element methods (mathematical models formulated in the Cartesian coordinate system even for irregularly shaped calculation domains). A formulation based on a CVFEM for the simulation of the carrier phase in a model for particle dispersion in dilute, two-dimensional, turbulent flows was recently reported by the present team (Oliveira et al. [1]). In that work, the motion of the solid (particulate) phase is simulated using a Lagrangian approach. An efficient algorithm is used for locating the particles in the finite element mesh. In the demonstration problem, which involves a particle-

laden, turbulent plane mixing layer, a modified k-H turbulence model is used to characterize the velocity and length scales of the turbulent flow of the fluid phase. The effect of turbulence on the particle trajectories is accounted for through a stochastic model. The effect of the particles on the fluid time-mean velocity and turbulence (two-way coupling) is also addressed. The three-dimensional extension of the work reported in reference [1] is now available (see Oliveira et al. [2], [3]). In the present talk, a description of the numerical global procedure that is used in this research will be briefly presented, together with a demonstration problem that was selected for validation purposes. Even though the results reported in references [2] and [3] are rather encouraging, there is still obvious room for further improvement and development. The talk’s conclusion includes a brief mention to the main topics that should be addressed in the joint team’s future work. Keywords: CFD, CVFEM, multiphase flow, particle dispersion, Lagrangian-Eulerian, stochastic approach, two-way coupling. References [1] L.A. Oliveira, V.A.F. Costa, and B.R. Baliga, A Lagrangian-Eulerian model of particle dispersion in a turbulent plane mixing layer, Int. J. Numer. Meth. Fluids, vol 40, pp 639653, 2002. [2] L.A. Oliveira, V.A.F. Costa, and B.R. Baliga, A Lagrangian model for the dispersion of solid particles, in three-dimensional flow, using a CVFEM for the prediction of the continuous phase, Proc. RoomVent 2004, 9.th International Conference on Air Distribution in Rooms, University of Coimbra, Portugal, September 5-8, 2004. [3] L.A. Oliveira, V.A.F. Costa, and B.R. Baliga, Numerical model for the prediction of dilute three-dimensional, turbulent fluid-particle flows, using a Lagrangian approach for particle tracking and a CVFEM for the carrier phase, accepted by Int. J. Numer. Meth. Fluids (already available in the Journal’s site), 2008.

Numerical simulation of the windflow over complex topography: computational implementation and applications António M. Gameiro Lopes Dept. of Mechanical Engineering, University of Coimbra, Portugal [email protected]

Abstract The prediction of windfield over complex topography is of great value for several areas, such as the assessment of the eolic potential in a certain region, the evaluation of pollutant dispersion or the prediction of forest fires behaviour. The diagnostic model WindStation is a computer implementation of the Navier-Stokes solver CANYON [1], based on a graphical user interface. The original solver, written for a fully generalized coordinate system, was simplified to take advantage of the partial coordinate transformation in the two vertical coordinate planes, thus benefiting from improved run speed and lower memory storage requirements. The Reynoldsaveraged Navier-Stokes equations are solved in their steady-state formulation, using a control volume approach. The SIMPLEC algorithm [2] is employed for the coupling of momentum and continuity equations, while turbulence effects upon the mean flow field are taken into account with the k-H turbulence model [3]. Terrain roughness is modeled through a proper formulation of fluxes, adopting a logarithmic profile. Input data for the code consists on terrain elevation and terrain roughness description, stored in conventional ArcInfo ASCII grid files, and on wind data from meteorological stations. Alternatively, a wind profile may be specified. The software solves for wind speed and wind direction at the grid locations, along with turbulence quantities. Postprocessing tools allow the visualization of the wind field at several elevations above ground level (cf. figure 1 and figure 2), statistical analysis and data export, among other features. Comparison of computed data with experimental measurements is also presented in this talk. The present code will be, in the near future, included in a larger package for the simulation of forest fires.

Fig. 1 - Vectorial representation of the wind field at a fixed distance above ground level.

Fig. 2 - Colour-contour representation of the wind field at a fixed distance above ground level.

Keywords: Wind field, Complex topography, Navier-Stokes solver, Graphical interface

References [1]

A.M.G. Lopes, A.C.M. Sousa and D.X. Viegas, Numerical Simulation of Turbulent Flow and Fire Propagation in Complex Terrain, Numerical Heat Transfer, Part A, N. 27, pp. 229-253, 1995.

[2]

J.P., Van Doormaal and G.D. Raithby, Enhancements of the Simple Method for Predicting Incompressible Fluid Flows, Numerical Heat Transfer, Vol. 7, pp. 147-163, 1984.

[3]

B.E. Launder and D.B. Spalding, The Numerical Computation of Turbulent Flows, Computer Methods in Applied Mechanics and Engineering, Vol. 3, pp. 269-289, 1974.

Island wake dynamics: Madeira Archipelago case study Luis, E. A. CEMAT-IST, Technical University of Lisbon, Portugal. [email protected]

Caldeira, R. M. A. CIMAR-Centre of Marine and Environmental Research, Oporto University, Portugal. [email protected]

Santos, A. J. P. Department of Mechanical Engineering-IST, Technical University of Lisbon, Portugal. [email protected]

Videman, J. H. Department of Mathematics and CEMAT-IST, Technical University of Lisbon, Portugal. [email protected]

Abstract Madeira is a deep-sea island located in NE Atlantic (33◦ N ; 17◦ W ), its obstruction to the incoming oceanic and atmospheric flows induce leeward wake instabilities. The phenomena is frequently observed using remote sensing and field data [1]; [2]. Numerical models are often used to study the evolution of the leeward, mesoscale and sub-mesoscale, flows around the archipelagos [3]; [4]; [5]. The Regional Ocean Modeling (ROMS) is a free-surface, terrain-following, primitive equations ocean model. The hydrostatic primitive equations for momentum are solved using a split-explicit time-stepping scheme. A cosine shape time filter, centered at the new time level is used for the averaging of the barotropic fields. Time-discretized uses a third-order accurate predictor (Leap-Frog) and corrector (Adams-Molton) time-stepping algorithm. A third-order upstream biased was used for advection in order to allow for the generation of steep gradients in the solution. A methodology similar to [4] was followed to study the deep-sea island wake problem, in a three-dimensional mode. Nevertheless, unlike [4], a realistic representation of Madeira Archipelago bathymetry replaced the idealized cylinder. The depth was assumed uniform around the islands, in order to be able to isolate the effect of the islands per se, from the effect of the surrounding seamounts. The island was centered in a geostrophic channel like

configuration with a prescribed inflow at the upstream boundary such that the zonal current depended only on the vertical shear. East(E) and West(W) channel boundaries were set to slippery-tangential and zero normal conditions, whereas boundaries around the islands were set to zero-normal and no-slip flow. Results showed that oceanic wakes regimes were sensitive to three dimensionless parameters [6]: Reynolds number (Re), Rossby number (Ro), and Burger number (Bu). Von K´arm´ an vortex street generation was showed in regimes of Re≥100 .Wake asymmetries induce different behaviour for cyclonic and anticyclonic eddies than that showed by [4] . Multiple islands wake interferences affect eddy shedding behavior.

Keywords: island wakes, ROMS- Regional Ocean Modeling System, wake instability, eddy shedding, mesoscale flows, Madeira Archipelago.

References [1] Caldeira, R.M.A., S. Groom, P. Miller, D. Pilgrim and N. Nezlin, Sea-surface signatures of the island mass effect phenomena around Madeira Island, Northeast Atlantic, Remote Sensing of the Environment, 80, 336–360, (2002). [2] Caldeira, R. M. A., P. Marchesiello, N. P. Nezlin, P. M. DiGiacomo, and J. C. McWilliams, Island wakes in the Southern California Bight, J. Geophys. Res., 110, C11012, doi:10.1029/2004JC002675, (2005). [3] Dietrich, D. E., M. J. Bowman, C. A. Lin and A. Mestas-Nunez, Numerical studies of small island wakes, Geophysics Astrophysics and Fluid Dynamics, 83, 195–231, (1996). [4] Dong, C., J. C. McWilliams and A. F. Shchepetkin, Island Wakes in Deep Water, Journal of Physical Oceanography 37(4), 962–981, (2007). [5] Heywood K.J., D.P. Stevens, G.R. Bigg, Eddy formation behind the tropical island of Aldabra, Deep-Sea Res. I, 43(4), 555–578, (1996). [6] Tomczak, M., Island wakes in deep and shallow water, Journal of Geophysical Research, 93, 5153–5154, (1988).