×

zbMATH — the first resource for mathematics

Effective numerical viscosity in spectral multidomain penalty method-based simulations of localized turbulence. (English) Zbl 1256.76050
Summary: The numerical dissipation operating in a specific spectral multidomain method model developed for the simulation of incompressible high Reynolds number turbulence in doubly periodic domains is investigated. The method employs Fourier discretization in the horizontal directions and the discretization in the vertical direction is based on a Legendre collocation scheme local to each subdomain. Both spatial discretizations are characterized by either no or near-negligible artificial dissipation. In high Reynolds number simulations, which are inherently under-resolved, stability of the numerical scheme is ensured through spectral filtering in all three directions and the implementation of a penalty scheme in the vertical direction. The dissipative effects of these stabilizers are quantified in terms of the numerical viscosity, using a generalization of the method previously employed to analyze numerical codes for the simulation of homogeneous, isotropic turbulence in triply periodic domains. Data from simulations of the turbulent wake of a towed sphere are examined at two different Reynolds numbers varying by a factor of twenty. The effects of the stabilizers are found to be significant, i.e. comparable, and sometimes larger, than the effects of the physical (molecular) viscosity. Away from subdomain interfaces, the stabilizers have an expected dissipative character extending over a range of scales determined by timestep and the degree of under-resolution, i.e. Reynolds number. At the interfaces, the stabilizers tend to exhibit a strong anti-dissipative character. Such behavior is attributed to the inherently discontinuous formulation of the penalty scheme, which suppresses catastrophic Gibbs oscillations by enforcing \(C_{0}\) and \(C_{1}\) continuity only weakly at the interfaces.

MSC:
76M22 Spectral methods applied to problems in fluid mechanics
76F99 Turbulence
Software:
GASpAR; MPDATA
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bevilaqua, P.M.; Lykoudis, P.S., Turbulence memory in self-preserving wakes, J. fluid mech., 89, 589-606, (1978)
[2] Blackburn, H.M.; Schmidt, S., Spectral element filtering techniques for large eddy simulation with dynamic estimation, J. comp. phys., 186, 610-629, (2003) · Zbl 1047.76520
[3] Boris, J.P.; Grinstein, F.F.; Oran, E.S.; Kolbe, R.L., New insights into large eddy simulation, Fluid dyn. res., 10, 199-228, (1992)
[4] Boyd, J.P., Two comments on filtering (artificial viscosity) for Chebyshev and Legendre spectral and spectral element methods: preserving boundary conditions and interpretation of the filter as a diffusion, J. comp. phys., 143, 283-288, (1998) · Zbl 0920.65046
[5] Boyd, J.P., Chebyshev and Fourier spectral methods, (2001), Dover Mineola, New York · Zbl 0987.65122
[6] Cook, A.W.; Cabot, W.H., Hyperviscosity for shock-turbulence interactions, J. comp. phys., 203, 379-385, (2005) · Zbl 1143.76477
[7] Deville, M.O.; Fischer, P.F.; Mund, E.H., High order methods for incompressible fluid flow, (2002), Cambridge University Press · Zbl 1007.76001
[8] Diamessis, P.J.; Domaradzki, J.A.; Hesthaven, J.S., A spectral multidomain penalty method model for the simulation of high Reynolds number localized stratified turbulence, J. comp. phys., 202, 298-322, (2005) · Zbl 1061.76054
[9] Diamessis, P.J.; Redekopp, L.G., Numerical investigation of solitary internal wave-induced global instability in shallow water benthic boundary layers, J. phys. oceanogr., 36, 5, 784-812, (2006)
[10] P.J. Diamessis, G.R. Spedding, Scaling and structure of stratified turbulent wakes at high Reynolds numbers, in: Stratified Flows - Sixth International Symposium, Perth, Australia, 2006, pp. 183-188.
[11] P.J. Diamessis, G.R. Spedding, J.A. Domaradzki, Similarity scaling and vorticity structure in high Reynolds number stably stratified turbulent wakes. J. Fluid Mech. (in preparation). · Zbl 1225.76167
[12] Domaradzki, J.A.; Adams, N.A., Direct modelling of subgrid scales of turbulence in large eddy simulations, J. turbulence, 3, 1-19, (2002)
[13] Domaradzki, J.A.; Xiao, Z.; Smolarkiewicz, P.K., Effective eddy viscosities in implicit large eddy simulations of turbulent flows, Phys. fluids, 15, 3890-3893, (2003) · Zbl 1186.76146
[14] Domaradzki, J.A.; Radhakrishnan, S., Effective eddy viscosities in implicit modeling of decaying high Reynolds number turbulence with and without rotation, Fluid dyn. res., 36, 385-406, (2005) · Zbl 1153.76374
[15] Dommermuth, D.G.; Rottman, J.W.; Innis, G.E.; Novikov, E.A., Numerical simulation of the wake of a towed sphere in a weakly stratified fluid, J. fluid mech., 473, 83-101, (2002) · Zbl 1026.76026
[16] Don, W.S.; Gottlieb, D.; Jung, J.H., A multidomain spectral method for supersonic reactive flows, J. comp. phys., 192, 325-354, (2003) · Zbl 1047.76088
[17] Giraldo, F.X.; Hesthaven, J.S.; Warburton, T., Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations, J. comp. phys., 181, 499-525, (2002) · Zbl 1178.76268
[18] Giraldo, F.X.; Restelli, M., A study of spectral element and discontinuous Galerkin methods for mesoscale atmospheric modeling: equation sets and test cases, J. comp. phys., 227, 8, 3849-3877, (2008) · Zbl 1194.76189
[19] Gottlieb, D.; Hesthaven, J.S., Spectral methods for hyperbolic problems, J. comput. appl. math., 128, 83-131, (2001) · Zbl 0974.65093
[20] Grinstein, F.F.; Margolin, L.G.; Rider, W.J., Implicit large eddy simulation: computing turbulent fluid dynamics, (2007), Cambridge University Press New York · Zbl 1273.76213
[21] Grinstein, F.F.; Fureby, C., Recent progress on MILES for high Reynolds number flows, J. fluids eng., 124, 848-861, (2002)
[22] Guermond, J.L.; Shen, J., Velocity-correction projection methods for incompressible flows, SIAM J. numer. anal., 41, 1, 112-134, (2003) · Zbl 1130.76395
[23] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S.R., Uniformly high-order accurate essentially non-oscillatory schemes III, J. comp. phys., 71, 231, (1987) · Zbl 0652.65067
[24] Hesthaven, J.S., A stable penalty method for the compressible navier – stokes equations: II. one-dimensional domain decomposition schemes, SIAM J. sci. comput., 18, 3, 658-685, (1997) · Zbl 0882.76061
[25] Hesthaven, J.S.; Gottlieb, D., A stable penalty method for the compressible navier – stokes equations: I. open boundary conditions, SIAM J. sci. comput., 17, 3, 579-612, (1996) · Zbl 0853.76061
[26] Hesthaven, J.S.; Kirby, R.M., Filtering in Legendre computation, Math. comput., 77, 263, 1425-1452, (2008) · Zbl 1195.65138
[27] Hickel, S.; Adams, N.A.; Domaradzki, J.A., An adaptive local deconvolution method for implicit LES, J. comp. phys., 413, 436, (2006) · Zbl 1146.76607
[28] Hinze, J.O., Turbulence, (1975), McGraw-Hill Book Company · Zbl 0117.42904
[29] Karamanos, G.-S.; Karniadakis, G.E., A spectral vanishing viscosity method for large-eddy simulations, J. comput. phys., 163, 22-50, (2000) · Zbl 0984.76036
[30] Karniadakis, G.E.; Israeli, M.; Orszag, S.A., High-order splitting methods for the incompressible navier – stokes equations, J. comp. phys., 97, 414-443, (1991) · Zbl 0738.76050
[31] Kraichnan, R.H., Eddy viscosity in two and three dimensions, J. atmos. sci., 33, 1521, (1976)
[32] Lesieur, M., New trends in large-eddy simulations of turbulence, Annu. rev. fluid mech., 28, 45-82, (1996)
[33] Levin, J.G.; Iskandarani, M.; Haidvogel, D.B., A spectral filtering procedure for eddy-resolving simulations with a spectral element Ocean model, J. comp. phys., 137, 130-154, (1997) · Zbl 0898.76082
[34] Meneveau, C.; Katz, J., Scale-invariance and turbulence models for large eddy simulations, Annu. rev. fluid mech., 32, 1-32, (2000) · Zbl 0988.76044
[35] Meunier, P.; Diamessis, P.J.; Spedding, G.R., Self-preservation of stratified momentum wakes, Phys. fluids, 18, (2006), Article No. 106601 · Zbl 1138.76326
[36] A.S. Monin, A.M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence, vol. II. The MIT Press, Cambridge, Massachusetts, 1981.
[37] Piomelli, U., Large-eddy simulations: achievements and challenges, Prog. aero. sci., 35, 335, (1999)
[38] Pope, S.B., Turbulent flows, (2000), Cambridge University Press Cambridge · Zbl 0802.76033
[39] Rosenberg, P.; Fournier, A.; Fischer, P.; Pouquet, A., Geophysical-astrophysical spectral-element adaptive refinement (gaspar): object-oriented h-adaptive fluid dynamics simulation, J. comp. phys., 215, 16, 59-80, (2006) · Zbl 1140.86300
[40] Shahbazi, K.; Fischer, P.F.; Ethier, C.R., A high-order discontinuous Galerkin method for the unsteady incompressible navier – stokes equations, J. comp. phys., 222, 391-407, (2007) · Zbl 1216.76034
[41] Smolarkiewicz, P.K.; Margolin, L.G., MPDATA: a finite-difference solver for geophysical flows, J. comp. phys., 140, 459-480, (1998) · Zbl 0935.76064
[42] Spedding, G.R.; Browand, F.K.; Fincham, A.M., Turbulence, similarity scaling and vortex geometry in the wake of a towed sphere in a stably stratified fluid, J. fluid mech., 314, 53-103, (1996)
[43] Stastna, M.; Lamb, K.G., Vortex shedding and sediment resuspension associated with the interaction of an internal solitary wave and the bottom boundary layer, Geophys. rev. let., 14, 9, 2987-2999, (2002)
[44] Sweby, P.K., High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. numer. anal., 21, 995, (1984) · Zbl 0565.65048
[45] Tennekes, H.; Lumley, J.L., A first course in turbulence, (1972), The MIT Press · Zbl 0285.76018
[46] Townsend, A.A., The structure of turbulent shear flow, (1976), Cambridge University Press · Zbl 0325.76063
[47] Winters, K.B.; D’Asaro, E.A., Two-dimensional instability of finite amplitude internal gravity wave packets near a critical level, J. geophys. res., 94, C9, 12709-12719, (1989)
[48] Yang, X.; Domaradzki, J.A., Large eddy simulations of decaying rotating turbulence, Phys. fluids, 16, 4088-4104, (2004) · Zbl 1187.76576
[49] Zalesak, S.T., Fully multidimensional flux-corrected transport algorithms for fluids, J. comp. phys., 31, 335, (1979) · Zbl 0416.76002
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.