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Numerical simulation of water flow around a rigid fishing net. (English) Zbl 1173.76410
Summary: This paper is devoted to the simulation of the flow around and inside a rigid axisymmetric net. We describe first how experimental data have been obtained. We show in detail the modelization. The model is based on a Reynolds Averaged Navier-Stokes turbulence model penalized by a term based on the Brinkman law. At the out-boundary of the computational box, we have used a “ghost” boundary condition. We show that the corresponding variational problem has a solution. Then the numerical scheme is given and the paper finishes with numerical simulations compared with the experimental data.

76S05 Flows in porous media; filtration; seepage
76F99 Turbulence
76M30 Variational methods applied to problems in fluid mechanics
76M10 Finite element methods applied to problems in fluid mechanics
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI arXiv
[1] Allaire, G., Homogenization of the navier – stokes equations and derivation of brinkman’s law, (), 7-20 · Zbl 0759.76072
[2] Angot, P.; Bruneau, C.H.; Fabrie, P., A penalization method to take into account obstacles in viscous flows, Numer. math., 81, 497-520, (1999) · Zbl 0921.76168
[3] Botsford, L.; Castilla, J.; Peterson, C., The management of fisheries and marine ecosystems, Science, 277, 509-515, (1997)
[4] Boccardo, L.; Gallouët, T., Nonlinear elliptic and parabolic equations involving measure data, J. funct. anal., 87, 149-169, (1989) · Zbl 0707.35060
[5] Brézis, H., Analyse fonctionnelle, (1993), Masson
[6] Bruneau, C.-H.; Fabrie, P., New efficient boundary conditions for incompressible navier – stokes equations: a well-posedness result, RAIRO modél. math. anal. numér., 30, 815-840, (1996) · Zbl 0865.76016
[7] Dauge, M.; Bernardi, C.; Maday, Y., Spectral methods for axisymmetric domains, (1999), Gauthier Villars
[8] Dauge, M.; Bernardi, C.; Maday, Y., Polynomials in the Sobolev world, (2003), Publications du Laboratoire J.-L. Lions
[9] G. Germain, J.V. Facq, D. Priour, Flow characterization around a cod-end, IMAM congress, Portugal, 2005.
[10] Girault, V.; Raviart, P-A., Finite element methods for navier – stokes equations, (1986), Springer-Verlag · Zbl 0413.65081
[11] P. Grisvard, Singularités des solutions du problème de Stokes dans un polygone, Univ. de Nice, 1978.
[12] Hecht, F.; Pironneau, O.; Le Hyaric, A.; Ohtsua, K., Freefem++ manual, (2006), Laboratoire Jacques Louis Lions Paris
[13] Khadra, K.; Angot, P.; Parneix, S.; Caltagirone, J.P., Fictitious domain approach for numerical modelling of navier – stokes equations, Int. J. numer. methods fluids, 34, 651-684, (2000) · Zbl 1032.76041
[14] Lederer, J.; Lewandowski, R., On the RANS 3D model with unbounded eddy viscosities, Ann. IHP an. nonlin., 24, 413-441, (2007) · Zbl 1132.35069
[15] Lewandowski, R., The mathematical analysis of the coupling of a turbulent kinetic energy equation to the navier – stokes equation with an eddy viscosity, Nonlin. anal. TMA, 28, 2, 393-417, (1997) · Zbl 0863.35077
[16] Lewandowski, R., Analyse mathématique et océanographie, (1997), Masson · Zbl 1347.86001
[17] Lewandowski, R., Vorticities in a LES model for 3D periodic turbulent flows, J. math. fluid mech., 8, 398-422, (2006) · Zbl 1099.76027
[18] Le Dret, H.; Lewandowski, R.; Priour, D.; Chagneau, F., Numerical simulation of a cod end net. part 1: equilibrium in a uniform flow, J. elasticity, 76, 139-162, (2004) · Zbl 1060.74026
[19] Lions, J.-L.; Magenes, E., Problèmes aux limites non homogènes et application, vol. 1, (1968), Dunod
[20] Mittal, R.; Iaccarino, G., Immersed boundary methods, Annual rev. fluid mech., 37, 239-261, (2005) · Zbl 1117.76049
[21] Mohammadi, B.; Pironneau, O., Analysis of the k-epsilon turbulence model, (1994), Masson, Springer
[22] F. Murat, Solutiones renormalizadas de EDP elipticas no lineales, Lectures at Sevilla’s University, 1990.
[23] O’Neill, F.-G., Axisymmetric trawl cod-ends made from netting of a general mesh shape, IMA J. appl. math., 62, 245-262, (1999) · Zbl 1053.74567
[24] Peskin, C.-S., Flow patterns around heart valves: a numerical method, J. comput. phys., 10, (1972) · Zbl 0244.92002
[25] G. Pichot, Modélisation et analyse numérique du couplage filet-écoulement hydrodynamique en vue d’estimer la forme de la prise dans une poche chalut, Ph.D. Thesis of Rennes 1 University, to be defended, 2007.
[26] Priour, D., Calculation of net shapes by the finite element method with triangular elements, Comm. numer. meth., 15, 755-763, (1999) · Zbl 0945.74074
[27] B. Vincent, Etude numérique et expérimentale des écoulements guidés par une paroi perméable axisymétrique, Application à la modélisation des chaluts pour en améliorer la sélectivité, Thèse Ecole Centrale Nantes, 1996.
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