zbMATH — the first resource for mathematics

A spectral vanishing viscosity method for large eddy simulations. (English) Zbl 0984.76036
Summary: A simulation approach for high Reynolds number turbulent flows is developed, combining concepts of monotonicity in nonlinear conservation laws with concepts of large eddy simulation. The spectral vanishing viscosity (SVV) is incorporated into Navier-Stokes equations for controlling high-wavenumber oscillations. Unlike hyperviscosity kernels, the SVV approach involves a second-order operator which can be readily implemented in standard finite element codes. In the work presented here, discretization is performed using hierarchical spectral/hp method accommodating effectively an ab initio intrinsic scale separation. The key result is that the monotonicity is enforced via SVV, leading to stable discretizations without sacrificing the formal accuracy, i.e. exponential convergence. Several examples demonstrate the effectiveness of the approach, including a comparison with eddy-viscosity spectral LES of turbulent channel flow.

76F65 Direct numerical and large eddy simulation of turbulence
76M22 Spectral methods applied to problems in fluid mechanics
76M10 Finite element methods applied to problems in fluid mechanics
Full Text: DOI
[1] Deardorff, J.W., A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers, J. fluid mech., 41, 453, (1970) · Zbl 0191.25503
[2] Moser, R.D.; Kim, J.; Mansour, N.N., Direct numerical simulation of turbulent channel flows up to reτ=590, Phys. fluids, 11, (1999) · Zbl 1147.76463
[3] Meneveau, C., Statistics of turbulence subgrid-scale stresses: necessary conditions and experimental tests, Phys. fluids, 6, 815, (1994) · Zbl 0825.76279
[4] Borue, V.; Orszag, S.A., Local energy flux and subgrid-scale statistics in three-dimensional turbulence, J. fluid mech., 366, 1, (1998) · Zbl 0924.76035
[5] Karniadakis, G.E.; Brown, G.L., Vorticity transport in modeling three-dimensional unsteady shear flows, Phys. fluids, 7, 688, (1995) · Zbl 1039.76503
[6] Lesieur, M.; Metais, O., New trends in large-eddy simulation, Annu. rev. fluid mech., 28, 45, (1996)
[7] Bardina, J., Improved turbulence models based on large eddy simulation of homogeneous incompressible turbulent flows, (1983), Stanford University
[8] Salvetti, M.V.; Banerjee, S., A priori tests of a new dynamic subgrid-scale model for finite-difference large-eddy simulations, Phys. fluids, 7, 2831, (1995) · Zbl 1026.76541
[9] Horiuti, K., A new dynamic two-parameter mixed model for large-eddy simulation, Phys. fluids, 9, 3443, (1997) · Zbl 1185.76760
[10] Dubois, T.; Jauberteau, F.; Temam, R., Solution of the incompressible navier – stokes equations by the nonlinear Galerkin method, J. sci. comp., 8, 167, (1993) · Zbl 0783.76068
[11] T. J. R. Hughes, L. Mazzei, and, K. E. Jansen, Large-eddy simulation and the variational multiscale method. Submitted for publication. · Zbl 0998.76040
[12] Porter, D.H.; Pouquet, A.; Woodward, P.R., Kolmogorov-like spectra in decaying three-dimensional supersonic flows, Phys. fluids, 6, 2133, (1994) · Zbl 0828.76042
[13] Boris, J.P.; Grinstein, F.F.; Oran, E.S.; Kolbe, R.J., New insights into large eddy simulation, Fluid dyn. res., 19, 19, (1992)
[14] Fureby, C.; Grinstein, F.F., Monotonically integrated large eddy simulation of free shear flows, Aiaa j., 37, 544, (1999)
[15] J.-L. Guermond, Subgrid stabilisation of Galerkin approximations of monotone operators, in, Proceedings of the European Science Foundation Conference on Applied Mathematics for Industrial Flow Problems (AMIF) (San Feliu de Guixols, Costa Brava, 1998).
[16] Urbin, G.; Knight, D., Large eddy simulation of the interaction of a turbulent boundary layer with a shock wave using unstructured grids, Second AFSOR international conference on DNS and LES, (1999)
[17] LeVeque, R.J., Numerical methods for conservation laws, (1992) · Zbl 0847.65053
[18] Giannakouros, I.G.; Karniadakis, G.E., A spectral element-FCT method for the compressible Euler equations, J. comput. phys., 115, 65, (1994) · Zbl 0811.76058
[19] Oran, E.S.; Boris, J.P., Numerical simulation of reactive flow, (1987) · Zbl 0762.76098
[20] Lax, P.D., Hyperbolic systems of conservation laws and the mathematical theory of shock waves, ABMS-NSF regional conference series in applied mathematics 11, society for industrial and applied mathematics, (1972)
[21] Godunov, S.K., A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Mat. sb, 47, 271, (1959) · Zbl 0171.46204
[22] Tadmor, E., Convergence of spectral methods for nonlinear conservation laws, SIAM J. numer. anal., 26, 30, (1989) · Zbl 0667.65079
[23] Maday, Y.; Tadmor, E., Analysis of the spectral viscosity method for periodic conservation laws, Sinum, 26, 854, (1989) · Zbl 0678.65066
[24] E. Tadmor, Super viscosity and spectral approximations of nonlinear conservation laws, in, Numerical Methods for Fluid Dynamics, IV, edited by, M. J. Baines and K. W. Morton, Clarendon Press, Oxford, 1993, p, 69. · Zbl 0805.76057
[25] Chen, G.-Q.; Dy, Q.; Tadmor, E., Spectral viscosity approximations to multidimensional scalar conservation laws, Math. comp., 61, 629, (1993) · Zbl 0799.35148
[26] Ma, H.-P., Chebyschev – legendre spectral viscosity method for nonlinear conservation laws, SIAM J. numer. anal., 35, 901, (1998)
[27] Ma, H.-P., Chebyschev – legendre super spectral viscosity method for nonlinear conservation laws, SIAM J. numer. anal., 35, 903, (1998)
[28] Andreassen, φ.; Lie, I.; Wasberg, C.E., The spectral viscosity method applied to simulation of waves in a stratified atmosphere, J. comput. phys., 110, 257, (1994) · Zbl 0795.76056
[29] Kaber, S.M.O., A Legendre pseudospectral viscosity method, J. comput. phys., 128, 165, (1996) · Zbl 0863.65065
[30] Karniadakis, G.E.; Sherwin, S.J., Spectral/hp element methods for CFD, (1999) · Zbl 0954.76001
[31] Don, W.S., Numerical study of pseudospectral methods in shock wave applications, J. comput. phys., 110, 103, (1994) · Zbl 0797.76068
[32] Crandall, M.G.; Lions, P.L., Viscosity solutions of hamilton – jacobi equations, Trans. am. math. soc., 61, 629, (1983)
[33] Maday, Y.; Ould Kaber, S.M.; Tadmor, E., Legendre pseudospectral viscosity method for nonlinear conservation laws, SIAM J. numer. anal., 30, 321, (1993) · Zbl 0774.65072
[34] Tadmor, E., Total variation and error estimates for spectral viscosity approximations, Math. comp., 60, 245, (1993) · Zbl 0795.65064
[35] Lesieur, M.; Metais, O., New trends in large-eddy simulation, Ann. rev. fluid mech., 28, 45, (1996)
[36] Kraichnan, R.H., Eddy viscosity in two and three dimensions, J. atmos. sci., 33, 1521, (1976)
[37] J. P. Chollet, Two-point closures as a subgrid scale modelling for large eddy simulations, in, Turbulent Shear Flows IV, editd by, F. Durst and B. Launder, Springer-Verlag, Berlin/New York, 1984.
[38] Emmel, L., Methode spectrale multidomaine de viscosite evanescente pour des problems hyperboliques non lineaires, (1998), University of Paris 6 France
[39] Gottlieb, D.; Shu, C-W., On the Gibbs phenomenon and its resolution, SIAM review, 39, 644, (1998) · Zbl 0885.42003
[40] Gelb, A.; Tadmor, E., Detection of edges in spectral data, Appl. comput. harmonic anal., 7, 101, (1999) · Zbl 0952.42001
[41] Karamanos, G.-S., Large-eddy simulation using unstructured spectral/hp elements, (1999), Imperial CollegeDepartment of Aeronautics · Zbl 0948.76562
[42] Kim, J.; Moin, P.; Moser, R., Turbulence statistics in fully developed channel flow at low Reynolds number, J. fluid mech., 117, 133, (1987) · Zbl 0616.76071
[43] Piomelli, U., Models for large eddy simulations of turbulent flow including transpiration, (1987), Stanford University
[44] Wille, M., Large eddy simulation of jets in cross flows, (1997), Imperial CollegeDepartment of Chemical Engineering
[45] Panton, R.L., A Reynolds stress function for wall layers, J. fluids eng., 119, 325, (1997)
[46] Kreplin, H.P.; Eckelmann, H., Behaviour of the three fluctuating velocity components in the wall region of a turbulent channel flow, Phy. fluids, 22, 1233, (1979)
[47] Temam, R., Approximation of attractors, large eddy simulations and multiscale methods, Proc. R. soc. lond., 434, 23, (1991) · Zbl 0731.76018
[48] Tam, C.K.W.; Webb, J.C.; Dong, Z., A study of the short wave components in computational accoustics, J. comput. acoustics, 1, 1, (1993)
[49] Karniadakis, G.E.; Israeli, M.; Orszag, S.A., High-order splitting methods for incompressible navier – stokes equations, J. comput. phys., 97, 414, (1991) · Zbl 0738.76050
[50] Gottlieb, D.; Orszag, S.A., Numerical analysis of spectral methods: theory and applications, SIAM-cmbs, (1977) · Zbl 0412.65058
[51] G.-S. Karamanos, S. J. Sherwin, and, J. F. Morrison, Large-eddy simulation using unstructured spectral/hp elements, in, Recent Advances in DNS and LES, edited by, D. Knight & L. Sakell, Kluwer Academic, Dordrecht/Norwell, MA, 1999. · Zbl 0948.76562
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.