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A simple and stable scale-similarity model for large eddy simulation: Energy balance and existence of weak solutions. (English) Zbl 1039.76027

Summary: In averaging the Navier-Stokes equations, the problem of closure arises. Scale-similarity models address closure by (roughly speaking) extrapolation from the (known) resolved scales to the (unknown) unresolved scales. In a posteriori tests, scale-similarity models are often the most accurate but can prove to be unstable when used in a numerical simulation. In this report, we consider the scale-similarity model given by \(\nabla \cdot w=0 \text{ and } w_t+\nabla \cdot (\overline{ww})-\nu\Delta w+\nabla p=f.\) We prove that it is stable (the solution satisfies an energy inequality), and deduce from that the existence of weak solutions of the model.

MSC:

76F65 Direct numerical and large eddy simulation of turbulence
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
35Q35 PDEs in connection with fluid mechanics
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[1] Mohammadi, B.; Pironneau, O., Analysis of the K-Epsilon Turbulence Model (1994), John Wiley and Sons
[2] Sagaut, P., Large Eddy Simulation for Incompressible Flows (1998), Springer: Springer New York · Zbl 1020.76001
[3] Germano, M., Differential filters of elliptic type, Phys. Fluids, 29, 1757-1758 (1986) · Zbl 0647.76042
[4] Mullen, J. S.; Fischer, P. F., Filtering techniques for complex geometry fluid flows, (Technical Report (2000), Argonne National Lab) · Zbl 0931.76067
[5] W.J. Layton and R. Lewandowski, Analysis of an eddy viscosity model for large eddy simulation of turbulent flows, J. Math. Fluid Mechanics; W.J. Layton and R. Lewandowski, Analysis of an eddy viscosity model for large eddy simulation of turbulent flows, J. Math. Fluid Mechanics · Zbl 1021.76020
[6] Bardina, J.; Ferziger, J. H.; Reynolds, W. C., Improved subgrid scale models for large eddy simulation, AIAA, 80, 1-10 (1980)
[7] Sarghini, F.; Piomelli, U.; Balaras, E., Scale-similar models for large eddy simulation, Phys. of Fluids, 11, 1596-1607 (1999) · Zbl 1147.76488
[8] Carati, D.; Winckelmans, G. S.; Jeanmart, H., On the modeling of the subgrid-scale and filtered-scale stress tensors in large-eddy simulation (2000), Univ. Libre de Bruxelles: Univ. Libre de Bruxelles Belgium, Technical report
[9] Winckelmans, G. S.; Lund, G. S.; Carati, D.; Wray, D., A priori testing of subgrid-scale models for the velocity-pressure and vorticity-velocity formulations, (Proc. Summer Res. Progn-Center for Turb. Research (1996)), 309-328, Stanford
[10] Layton, W., Approximating the larger eddies in fluid motion V: Kinetic energy balance of scale similarity models, Mathl. Comput. Modelling, 31, 8/9, 1-7 (2000) · Zbl 1042.76537
[11] Horiuti, K., The role of the Bardina model in large eddy simulation of turbulent channel flow, Phys. Fluids A1, 2, 426-428 (1989)
[12] Liu, S.; Menereau, C.; Katz, J., On the properties of similarity subgridscale models as deduced from measurements in a turbulent jet, J.F.M., 275, 83-119 (1994)
[13] Galdi, G. P., Lectures in Mathematical Fluid Dynamics (2000), Birkhäuser-Verlag
[14] Galdi, G. P., An introduction to the mathematical theory of the Navier-Stokes equations, (Linearized Steady Problems, Volume 38 (1994), Springer Tracts in Natural Philosophy) · Zbl 0828.76006
[15] Leray, J., Sur le mouvement d’un fluide visqueux emplissant l’espace, Acta Math., 63, 193-248 (1934)
[16] Lions, J. L., Quelques Méthodes des Résolutions de Problèmes aux Limites Non Linéaire (1969), Gauthier Villard · Zbl 0189.40603
[17] Constantine, P.; Foias, C., Navier Stokes Equations (1988), University Chicago Press · Zbl 0687.35071
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