Convergence of approximate deconvolution models to the mean Navier–Stokes equations
Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 2, p. 171-198

We consider a 3D Approximate Deconvolution Model ADM which belongs to the class of Large Eddy Simulation (LES) models. We aim at proving that the solution of the ADM converges towards a dissipative solution of the mean Navier–Stokes equations. The study holds for periodic boundary conditions. The convolution filter we first consider is the Helmholtz filter. We next consider generalized convolution filters for which the convergence property still holds.

DOI : https://doi.org/10.1016/j.anihpc.2011.10.001
Classification:  76D05,  35Q30,  76F65,  76D03
Keywords: Navier–Stokes equations, Large eddy simulation, Deconvolution models
@article{AIHPC_2012__29_2_171_0,
author = {Berselli, Luigi C. and Lewandowski, Roger},
title = {Convergence of approximate deconvolution models to the mean Navier--Stokes equations},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {29},
number = {2},
year = {2012},
pages = {171-198},
doi = {10.1016/j.anihpc.2011.10.001},
zbl = {1302.76083},
mrnumber = {2901193},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2012__29_2_171_0}
}

Berselli, Luigi C.; Lewandowski, Roger. Convergence of approximate deconvolution models to the mean Navier–Stokes equations. Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 2, pp. 171-198. doi : 10.1016/j.anihpc.2011.10.001. http://www.numdam.org/item/AIHPC_2012__29_2_171_0/

[1] N.A. Adams, S. Stolz, Deconvolution methods for subgrid-scale approximation in large eddy simulation, Modern Simulation Strategies for Turbulent Flow, R.T. Edwards (2001)

[2] L.C. Berselli, T. Iliescu, W.J. Layton, Mathematics of Large Eddy Simulation of Turbulent Flows, Scientific Computation, Springer-Verlag, Berlin (2006) | MR 2185509 | Zbl 1089.76002

[3] M. Bertero, B. Boccacci, Introduction to Inverse Problems in Imaging, IOP Publishing Ltd (1998) | MR 1640759 | Zbl 0914.65060

[4] A. Bhattacharya, A. Das, D. Moser, A filtered-wall formulation for large-eddy simulation of wall-bounded turbulence, Phys. Fluids 20 (2008), 115104 | Zbl 1182.76061

[5] A. Cheskidov, D.D. Holm, E. Olson, E.S. Titi, On a Leray-α model of turbulence, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005), 629-649 | MR 2121928 | Zbl 1145.76386

[6] P. Constantin, C. Foias, Navier–Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL (1988) | MR 972259 | Zbl 0687.35071

[7] A. Das, D. Moser, Filtering boundary conditions for les and embedded boundary conditions, C. Liu, L. Sakell, T. Beutner (ed.), Proceedings of the Third International Conference on DNS-LES, Greyden Press, Columbus (2001)

[8] C.R. Doering, J.D. Gibbon, Applied Analysis of the Navier–Stokes Equations, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge (1995) | MR 1325465 | Zbl 0838.76016

[9] A. Dunca, Y. Epshteyn, On the Stolz–Adams deconvolution model for the large-eddy simulation of turbulent flows, SIAM J. Math. Anal. 37 no. 6 (2006), 1890-1902 | MR 2213398 | Zbl 1128.76029

[10] C. Foias, D.D. Holm, E.S. Titi, The Navier–Stokes-alpha model of fluid turbulence, Physica D 152 (2001), 505-519 | MR 1837927 | Zbl 1037.76022

[11] B.J. Geurts, Elements of Direct and Large Eddy Simulation, Edwards Publishing, Flourtown, PA (2003)

[12] O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Mathematics and Its Applications vol. 2, Gordon and Breach Science Publishers, New York (1969) | MR 254401 | Zbl 0184.52603

[13] W. Layton, R. Lewandowski, A simple and stable scale similarity model for large eddy simulation: energy balance and existence of weak solutions, Appl. Math. Lett. 16 (2003), 1205-1209 | MR 2015713 | Zbl 1039.76027

[14] W.J. Layton, R. Lewandowski, Analysis of an eddy viscosity model for large eddy simulation of turbulent flows, J. Math. Fluid Mech. 4 no. 4 (2002), 374-399 | MR 1953786 | Zbl 1021.76020

[15] W.J. Layton, R. Lewandowski, On a well-posed turbulence model, Discrete Contin. Dyn. Syst. Ser. B 6 no. 1 (2006), 111-128 | MR 2172198 | Zbl 1089.76028

[16] W.J. Layton, R. Lewandowski, Residual stress of approximate deconvolution models of turbulence, J. Turbul. 7 (2006), 1-21 | MR 2254595 | Zbl 1273.76206

[17] K.L. Lele, Compact finite different schemes with spectral-like resolution, J. Comput. Phys. 103 (1992), 16-42 | MR 1188088 | Zbl 0759.65006

[18] J. Leray, Sur les mouvements dʼune liquide visqueux emplissant lʼespace, Acta Math. 63 (1934), 193-248 | MR 1555394

[19] M. Lesieur, O. Métais, P. Comte, Large-Eddy Simulations of Turbulence, Cambridge University Press, New York (2005) | MR 2173350 | Zbl 0910.76062

[20] R. Lewandowski, On a continuous deconvolution equation for turbulence models, Lecture Notes of Neças Center for Mathematical Modeling 5 (2009), 62-102 | MR 2962830

[21] R. Lewandowski, Approximations to the Navier–Stokes Equations, in preparation, release scheduled end 2012.

[22] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod (1969) | MR 259693 | Zbl 0189.40603

[23] D. Moser, N. Malaya, H. Chang, P. Zandonade, P. Veluda, A. Bhattacharya, A. Haselbacher, Theoretically based optimal large-eddy simulation, Phys. Fluids 21 (2009), 105104 | Zbl 1183.76363

[24] F. Murat, Compacité par compensation, Ann. Sc. Norm. Super. Pisa Cl. Sci. 4 no. 5 (1978), 489-507 | Numdam | MR 506997 | Zbl 0399.46022

[25] F. Murat, A survey on compensated compactness, L. Cesari (ed.), Contributions to Modern Calculus of Variations, Pitman Research Notes in Math. vol. 148, Longman, Harlow (1987), 145-183 | MR 894077

[26] U. Piomelli, E. Balaras, Wall-layer models for large-eddy simulations, Annu. Rev. Fluid Mech. 34 (2002), 349-374 | MR 1893771 | Zbl 1006.76041

[27] P. Sagaut, Large Eddy Simulation for Incompressible Flows — An Introduction, Scientific Computation, Springer-Verlag, Berlin (2001) | MR 1815221 | Zbl 0964.76002

[28] S. Stolz, N.A. Adams, An approximate deconvolution procedure for large-eddy simulation, Phys. Fluids 11 no. 7 (1999), 1699-1701 | Zbl 1147.76506

[29] S. Stolz, N.A. Adams, L. Kleiser, An approximate deconvolution model for large-eddy simulation with application to incompressible wall-bounded flows, Phys. Fluids 13 no. 4 (2001), 997-1015 | Zbl 1184.76530

[30] L. Tartar, Nonlinear partial differential equations using compactness method, Report 1584, Mathematics Research Center, University of Wisconsin, Madison, 1975.

[31] L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 115 no. 3–4 (1990), 193-230 | MR 1069518 | Zbl 0774.35008

[32] L. Tartar, Lecture Notes of the Unione Matematica Italiana, 7, Springer-Verlag/UMI, Berlin/Bologna (2009)

[33] R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI (2001) | MR 1846644 | Zbl 0981.35001