The nonlinear Galerkin method: A multiscale method applied to the simulation of homogeneous turbulent flows. (English) Zbl 0838.76060

Using results of direct numerical simulations in the case of two-dimensional homogeneous isotropic flows, we first analyze in detail the behavior of the small and large scales of Kolmogorov-like flows at moderate Reynolds numbers. We propose a multilevel scheme which treats the small and the large eddies differently. We derive estimates of all the parameters involved in the algorithm, which then becomes a completely self-adaptative procedure. Finally, we perform realistic simulations of (Kolmogorov-like) flows over several eddy-turnover times. The results are analyzed in detail, and a parametric study of the nonlinear Galerkin method is performed.


76M25 Other numerical methods (fluid mechanics) (MSC2010)
76F05 Isotropic turbulence; homogeneous turbulence
76D05 Navier-Stokes equations for incompressible viscous fluids
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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[1] P. Moin and J. Kim, Numerical investigation of turbulent channel flow, J. Fluid Mech., 118 (1982), 341-377. · Zbl 0491.76058 · doi:10.1017/S0022112082001116
[2] J. Kim, P. Moin, and R. Moser, Turbulence statistics in full developed channel flow at low Reynolds number, J. Fluid Mech., 177, (1987), 133-166. · Zbl 0616.76071 · doi:10.1017/S0022112087000892
[3] G. Erlebacher, M.Y. Hussaini, C.G. Speziale, and T.A. Zang, Toward the large-eddy simulation of compressible turbulent flows, J. Fluid Mech., 238 (1992), 155-185. · Zbl 0775.76059 · doi:10.1017/S0022112092001678
[4] S. Sarkar, G. Erlebacher, and M.Y. Hussaini, Compressible homogeneous shear: simulation and modeling, Turbulent Shear Flows 8, Springer-Verlag, Berlin (1993). · Zbl 0874.76030
[5] J.W. Deardorff, A. numerical study of three-dimensional turbulent channel flow at large Reynolds number, J. Fluid Mech., 41, (1970), 453-480. · Zbl 0191.25503 · doi:10.1017/S0022112070000691
[6] J.L. Lumley, Computational modeling of turbulent flows, Adv. in Appl. Mech., 18 (1978), 123-175. · Zbl 0472.76052 · doi:10.1016/S0065-2156(08)70266-7
[7] M. Lesieur, Turbulence in Fluids. Stochastics and Numerical Modeling, 2nd rev. ed., Kluwer, Dordrecht.
[8] C.G. Speziale, Analytical methods for the development of Reynolds stress closures in turbulence, Annual Rev. Fluid Mech., 23 (1991), 107-157. · Zbl 0723.76005 · doi:10.1146/annurev.fl.23.010191.000543
[9] D.C. Leslie and G.L. Quarini, The application of turbulence theory for the formulation of subgrid modeling procedures, J. Fluid Mech., 91(1) (1979), 65-91. · Zbl 0411.76045 · doi:10.1017/S0022112079000045
[10] M. Germano, U. Piomelli, P. Moin, and W.H. Cabot, A dynamic subgrid-scale eddy viscosity model, Phys. Fluids A, 3(7) (1991), 1760-1765. · Zbl 0825.76334 · doi:10.1063/1.857955
[11] C. Basdevant and R. Sadourny, Modélisation des échelles virtuelles dans la simulation numérique des écoulements turbulents bidimensionnels, J. Méc. Théor. Appl., (1983), 243-269.
[12] V. Yakhot and S.A. Orszag, Renormalization Group Analysis of Turbulence I. Basic theory, J. Sci. Comp., 1(1) (1986), 3-55. · Zbl 0648.76040 · doi:10.1007/BF01061452
[13] C. Foias, O. Manley, and R. Temam, Modeling of the interaction of small and large eddies in two dimensional turbulent flows, Math. Model. Numer. Anal., 22 (1988), 93-114. · Zbl 0663.76054
[14] M. Marion and R. Temam, Nonlinear Galerkin methods, SIAM J. Numer. Anal., 26 (1989), 1139-1157. · Zbl 0683.65083 · doi:10.1137/0726063
[15] M. Marion and R. Temam, Nonlinear Galerkin methods: the finite elements case, Numer. Math., 57 (1990), 205-226. · Zbl 0702.65081 · doi:10.1007/BF01386407
[16] R. Temam, Inertial manifolds and multigrid methods, SIAM J. Math. Anal., 21 (1990), 154-178. · Zbl 0715.35039 · doi:10.1137/0521009
[17] Y. Zhou, G. Vahala, and M. Hossain, Renormalization-group theory for the eddy-viscosity in subgrid modeling, Phys. Rev. A, 37(7) (1988), 2590-2598. · doi:10.1103/PhysRevA.37.2590
[18] Y. Zhou and G. Vahala, Reformulation of recursive-renormalization-group-based subgrid modeling of turbulence, Phys. Rev. E, 47(4) (1993), 2503-2519. · doi:10.1103/PhysRevE.47.2503
[19] F. Jauberteau, C. Rosier, and R. Temam, A nonlinear Galerkin method for the Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 80 (1990), 245-260. · Zbl 0722.76039 · doi:10.1016/0045-7825(90)90028-K
[20] F. Jauberteau, C. Rosier, and R. Temam, The nonlinear Galerkin method in computational fluid dynamics, Appl. Numer. Math., 6 (1990), 361-370. · Zbl 0702.76077 · doi:10.1016/0168-9274(90)90026-C
[21] T. Dubois, F. Jauberteau, and R. Temam, Solution of the incompressible Navier-Stokes equations by the nonlinear Galerkin method, J. Sci. Comput., 8(2) (1993), 167-194. · Zbl 0783.76068 · doi:10.1007/BF01060871
[22] G.K. Batchelor, The Theory of Homogeneous Tubulence, Cambridge University Press, Cambridge (1953). · Zbl 0053.14404
[23] C.G. Speziale and P.S. Bernard, The energy decay in self-preserving isotopic turbulence revisited, J. Fluid Mech., 241 (1992), 645-667. · Zbl 0756.76030 · doi:10.1017/S0022112092002180
[24] J.R. Herring, S.A. Orszag, R.H. Kraichnan, and D.G. Fox, Decay of the two-dimensional homogeneous turbulence, J. Fluid Mech., 66(3) (1974), 417-444. · Zbl 0296.76026 · doi:10.1017/S0022112074000280
[25] M.E. Brachet, M. Meneguzzy, H. Politano, and P.L. Sulem, The dynamics of freely decaying two-dimensional turbulence, J. Fluid Mech., 194 (1988), 333-349. · doi:10.1017/S0022112088003015
[26] T. Dubois and R. Temam, Separation of Scales in Turbulence Using the Nonlinear Galerkin Method, Advances in Computational Fluid Dynamics, W.G. Habashi and M. Hafez, eds., Gordon and Breach, New York (1995).
[27] C. Foias, O. Manley, and R. Temam, Approximate inertial manifolds and effective viscosity in turbulent flows, Phys. Fluids A, 3(5) (1991), 898-911. · Zbl 0732.76001 · doi:10.1063/1.858212
[28] F. Jauberteau, Résolution numérique des équations de Navier-Stokes instastionnaires par méthodes spectrales. Méthode de Galerkin nonlinéaire, Thèse, Université de Paris-Sud, 1990.
[29] C. Canuto, Y. Hussaini, A. Quarteroni, and T. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York (1988). · Zbl 0658.76001
[30] R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam (1977).
[31] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York (1988). · Zbl 0662.35001
[32] R.H. Kraichnan, Inertial ranges in two-dimensional turbulence, Phys. Fluids, 10, (1967), 1417-1423. · doi:10.1063/1.1762301
[33] R.H. Kraichnan and D. Montgommery, Two-dimensional turbulence, Rep. Progr. Phys., 43 (1980), 547-619. · doi:10.1088/0034-4885/43/5/001
[34] S. Chen, G.D. Doolen, J.R. Herring, R.H. Kraichnan, S.A. Orszag, and Z.S. She, Far dissipation range of turbulence, Phys. Rev. Lett., 70, (1993), 3051. · doi:10.1103/PhysRevLett.70.3051
[35] C. Foias, O. Manley, and L. Sirovich, Empirical and Stokes eigenfunctions and the far dissipative turbulent spectrum, Phys. Fluids A, 2(3) (1990), 464-467. · Zbl 0704.76025 · doi:10.1063/1.857744
[36] O.P. Manley, The dissipation range spectrum, Phys. Fluids A, 4(6) (1992), 1320-1321. · doi:10.1063/1.858408
[37] L.M. Smith and W.C. Reynolds, The dissipation range spectrum and the velocity-derivative skewness in turbulent flows, Phys. Fluids A, 3(5) (1991), 992. · doi:10.1063/1.857979
[38] P. Constantin, C. Foias, M. Manley, and R. Temam, Determining modes and fractal dimension of turbulent flows, J. Fluid Mech., 150 (1985), 427-440. · Zbl 0607.76054 · doi:10.1017/S0022112085000209
[39] R. Temam, Induced trajectories and approximate inertial manifolds, Math. Model. Numer. Anal., 32, (1988), 163. · Zbl 0688.58036
[40] R. Temam, Attractors for the Navier-Stokes equations, localization and approximation, J. Fac. Sci. Tokyo Sect. IA Math., 36 (1989), 629. · Zbl 0698.58040
[41] A. Debussche and R. Temam, Convergent families of approximate inertial manifolds, J. Math. Pure Appl., 73 (1994), 485-522. · Zbl 0836.35063
[42] A. Debussche and T. Dubois, Approximation of exponential order of the attractor of a turbulent flow, Phys. D, 72 (1994), 372-389. · Zbl 0814.76030 · doi:10.1016/0167-2789(94)90239-9
[43] C. Badevant and F. Pascal, Nonlinear Galerkin method and subgrid-scale model for two-dimensional turbulent flows, Theoret. Comput. Fluid Dynamics, 3(5) (1992), 267-284. · Zbl 0775.76084 · doi:10.1007/BF00717644
[44] S.A. Orszag, In Proc. 5th Internat. Conf. on Numerical Methods in Fluids Dynamics, Lecture Notes in Physics, Vol. 59, (1977), p. 32.
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