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Entropy splitting for high-order numerical simulation of compressible turbulence. (English) Zbl 1139.76332
Summary: A stable high-order numerical scheme for direct numerical simulation (DNS) of shock-free compressible turbulence is presented. The method is applicable to general geometries. It contains no upwinding, artificial dissipation, or filtering. Instead the method relies on the stabilizing mechanisms of an appropriate conditioning of the governing equations and the use of compatible spatial difference operators for the interior points (interior scheme) as well as the boundary points (boundary scheme). An entropy-splitting approach splits the inviscid flux derivatives into conservative and nonconservative portions. The spatial difference operators satisfy a summation-by-parts condition, leading to a stable scheme (combined interior and boundary schemes) for the initial boundary value problem using a generalized energy estimate. A Laplacian formulation of the viscous and heat conduction terms on the right hand side of the Navier-Stokes equations is used to ensure that any tendency to odd-even decoupling associated with central schemes can be countered by the fluid viscosity. The resulting methods are able to minimize the spurious high-frequency oscillations associated with pure central schemes, especially for long time integration applications such as DNS. For validation purposes, the methods are tested in a DNS of compressible turbulent plane channel flow at low values of friction Mach number, where reference turbulence data bases exist. It is demonstrated that the methods are robust in terms of grid resolution, and in good agreement with published channel data. Accurate turbulence statistics can be obtained with moderate grid sizes. Stability limits on the range of the splitting parameter are determined from numerical tests.

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76M20 Finite difference methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
76F50 Compressibility effects in turbulence
Full Text: DOI
[1] Yee, H.C.; Sandham, N.D.; Djomehri, M.J., Low dissipative high order shock-capturing methods using characteristic-based filters, J. comput. phys., 150, 199, (1999) · Zbl 0936.76060
[2] Sjögreen, B.; Yee, H.C., multi-resolution wavelet based adaptive numerical dissipation control for shock-turbulence computations, (2000), NASA Ames Research Center
[3] H. C. Yee, and, B. Sjögreen, Designing adaptive low-dissipative high order schemes for long-time integrations, in, Turbulent Flow Computation, edited by, D. Drikakis and B. Geurts, Kluwer, 2002.
[4] Deleted in proof.
[5] Yee, H.C.; Vinokur, M.; Djomehri, M.J., Entropy splitting and numerical dissipation, J. comput. phys., 161, 1, (2000) · Zbl 0987.65086
[6] Olsson, P.; Oliger, J., energy and maximum norm estimates for nonlinear conservation laws, (1994), NASA Ames Research Center
[7] Gerritsen, M.; Olsson, P., Designing an efficient solution strategy for fluid flows. I. A stable high order finite difference scheme and sharp shock resolution for the Euler equations, J. comput. phys., 129, 245, (1996) · Zbl 0899.76281
[8] M. Vinokur, and, H. C. Yee, Extension of efficient low dissipation high order schemes for 3-D curvilinear moving grids, in, Proceedings of the Computing the Future. III: Frontiers of Computational Fluid Dynamics-2000, June 26-28, Half Moon Bay, CA, NASA/TM-2000-209598. · Zbl 1047.76559
[9] Harten, A., On the symmetric form of systems of conservation laws with entropy, J. comput. phys., 49, 151, (1983) · Zbl 0503.76088
[10] Strand, B., Summation by parts for finite difference approximations for d/dx, J. comput. phys., 110, 47, (1994) · Zbl 0792.65011
[11] P. Olsson, Summation by Parts, Projections and Stability, RIACS Technical Report NASA Ames Research Center, 1994-1995. · Zbl 0848.65064
[12] Carpenter, M.H.; Nordstrom, J.; Gottlieb, D., A stable and conservative interface treatment of arbitrary spatial accuracy, J. comput. phys., 148, 341, (1999) · Zbl 0921.65059
[13] N. D. Sandham, and, H. C. Yee, Entropy splitting for high-order numerical simulation of compressible turbulence, in, Proceedings of First International Conference on Computational Fluid Dynamics, Kyoto, Japan, July 10-14, 2000, edited by, N. Satofuka, Springer, 2001. · Zbl 1139.76332
[14] Coleman, G.N.; Kim, J.; Moser, R., A numerical study of turbulent supersonic isothermal-wall channel flow, J. fluid mech., 305, 159, (1995) · Zbl 0960.76517
[15] Huang, P.G.; Coleman, G.N.; Bradshaw, P., Compressible turbulent channel flows: DNS results and modeling, J. fluid mech., 305, 185, (1995) · Zbl 0857.76036
[16] N. D. Sandham, and, R. J. A. Howard, Direct simulation of turbulence using massively parallel computers. in, Parallel Computational Fluid Dynamics, edited by, D. R. Emerson, et al., Elsevier, Amsterdam/New York, 1998.
[17] Kim, J.; Moin, P.; Moser, R.D., Turbulence statistics in fully-developed channel flow at low Reynolds number, J. fluid mech., 177, 133, (1987) · Zbl 0616.76071
[18] Li, Q., direct numerical simulation of compressible turbulent channel flow, (2002), University of Southampton
[19] M. Ashworth, D. R. Emerson, N. D. Sandham, Y. F. Yao, and, Q. Li, Parallel DNS using a compressible turbulent channel flow benchmark, in, Proc. ECCOMAS CFD Conference, Swansea, Wales, 4-7 Sept. 2001.
[20] H. C. Yee, and, P. K. Sweby, Dynamics of Numerics & Spurious Behaviors in CFD Computations, RIACS Technical Report 97.06, NASA Ames Research Center, 1997.
[21] M. Alam, and, N. D. Sandham, DNS of transition near the leading edge of an aerofoil, in, Direct and Large-Eddy Simulation IV, edited by, B. J. Geurts, R. Friedrich, and O. Métais, Kluwer Academic, Dordrecht/New york, 2001.
[22] A. A. Lawal, and, N. D. Sandham, Direct simulation of transonic flow over a bump, in, Direct and Large-Eddy Simulation IV, edited by, B. J. Geurts, R. Friedrich, and O. Métais, Kluwer Academic, Dordrecht/New york, 2001.
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