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Evaluation of Riemann flux solvers for WENO reconstruction schemes: Kelvin-Helmholtz instability. (English) Zbl 1390.76617

Summary: Accurate and computationally efficient simulations of Euler equations are of paramount importance in both fundamental research and engineering applications. In this study, our main objective is to investigate the efficacy and accuracy of several Riemann solvers for high-order accurate weighted essentially non-oscillatory (WENO) reconstruction scheme as a state-of-the-art tool to study shear driven turbulence flows. The Kelvin-Helmholtz instability occurs when a perturbation is introduced to a continuous fluid system with a velocity shear, or where there is a velocity difference across the interface between two fluids. Here, we solve a stratified Kelvin-Helmholtz instability problem to demonstrate the performance of six different Riemann solvers’ ability to evolve a linear perturbation into a transition to nonlinear hydrodynamic two-dimensional turbulence. A single mode perturbation is used for our evaluations. Time evolution process shows that the vortices formed from the turbulence slowly merge together since both energy and enstrophy are simultaneously conserved in two-dimensional turbulence. Third-, fifth- and seventh-order WENO reconstruction schemes are investigated along with the Roe, Rusanov, HLL, FORCE, AUSM, and Marquina Riemann flux solvers at the cell interfaces resulting in 18 joint flow solvers. Based on the numerical assessments of these solvers on various grid resolutions, it is found that the dissipative character of the Riemann solver has significant effect on eddy resolving properties and turbulence statistics. We further show that the order of the reconstruction scheme becomes increasingly important for coarsening the mesh. We illustrate that higher-order schemes become more effective in terms of the tradeoff between the accuracy and efficiency. We also demonstrate that AUSM solver provides the least amount of numerical dissipation, yet resulting in a pile-up phenomenon in energy spectra for underresolved simulations. However, results obtained by the Roe solver agree well with the theoretical energy spectrum scaling providing a marginal dissipation without showing any pile-up at a cost of around 30% increase in computational time.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76Nxx Compressible fluids and gas dynamics
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[1] Lax, P. D., Weak solutions of nonlinear hyperbolic equations and their numerical computation, Commun Pure Appl Math, 7, 1, 159-193, (1954) · Zbl 0055.19404
[2] Godunov, S. K., A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Matematicheskii Sbornik, 89, 3, 271-306, (1959) · Zbl 0171.46204
[3] Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J Comput Phys, 43, 2, 357-372, (1981) · Zbl 0474.65066
[4] Harten, A., High resolution schemes for hyperbolic conservation laws, J Comput Phys, 49, 3, 357-393, (1983) · Zbl 0565.65050
[5] Sweby, P. K., High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J Numer Anal, 21, 5, 995-1011, (1984) · Zbl 0565.65048
[6] Anderson, W. K.; Thomas, J. L.; Van Leer, B., Comparison of finite volume flux vector splittings for the Euler equations, AIAA J, 24, 9, 1453-1460, (1986)
[7] Harten, A.; Osher, S.; Engquist, B.; Chakravarthy, S. R., Some results on uniformly high-order accurate essentially nonoscillatory schemes, Appl Numer Math, 2, 3, 347-377, (1986) · Zbl 0627.65101
[8] Yee, H. C., Construction of explicit and implicit symmetric TVD schemes and their applications, J Comput Phys, 68, 1, 151-179, (1987) · Zbl 0621.76026
[9] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S. R., Uniformly high order accurate essentially non-oscillatory schemes, III, J Comput Phys, 71, 2, 231-303, (1987) · Zbl 0652.65067
[10] Lax, P. D.; Liu, X. D., Solution of two-dimensional Riemann problems of gas dynamics by positive schemes, SIAM J Sci Comput, 19, 2, 319-340, (1998) · Zbl 0952.76060
[11] Kurganov, A.; Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J Comput Phys, 160, 1, 241-282, (2000) · Zbl 0987.65085
[12] Fjordholm, U. S.; Mishra, S.; Tadmor, E., Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws, SIAM J Numer Anal, 50, 2, 544-573, (2012) · Zbl 1252.65150
[13] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics: a practical introduction, (2009), Springer · Zbl 1227.76006
[14] Bertschinger, E., Simulations of structure formation in the universe, Ann Rev Astron Astrophys, 36, 1, 599-654, (1998)
[15] Lee, T. K.; Zhong, X., Spurious numerical oscillations in simulation of supersonic flows using shock-capturing schemes, AIAA J, 37, 3, 313-319, (1999)
[16] Vos, J. B.; Rizzi, A.; Darracq, D.; Hirschel, E. H., Navier-Stokes solvers in European aircraft design, Prog Aerosp Sci, 38, 8, 601-697, (2002)
[17] Ekaterinaris, J. A., High-order accurate, low numerical diffusion methods for aerodynamics, Prog Aerosp Sci, 41, 3, 192-300, (2005)
[18] Tam, C. K.; Webb, J. C., Dispersion-relation-preserving finite difference schemes for computational acoustics, J Comput Phys, 107, 2, 262-281, (1993) · Zbl 0790.76057
[19] Rajpoot, M. K.; Sengupta, T. K.; Dutt, P. K., Optimal time advancing dispersion relation preserving schemes, J Comput Phys, 229, 10, 3623-3651, (2010) · Zbl 1190.65139
[20] Sengupta, T. K.; Rajpoot, M. K.; Saurabh, S.; Vijay, V., Analysis of anisotropy of numerical wave solutions by high accuracy finite difference methods, J Comput Phys, 230, 1, 27-60, (2011) · Zbl 1205.65239
[21] Wang, Z. J.; Fidkowski, K.; Abgrall, R.; Bassi, F.; Caraeni, D.; Cary, A., High-order CFD methods: current status and perspective, Int J Numer Meth Fluids, 72, 8, 811-845, (2013)
[22] Wang, Z. J., High-order methods for the Euler and Navier-Stokes equations on unstructured grids, Prog Aerosp Sci, 43, 1, 1-41, (2007)
[23] Shu, C. W., High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Rev, 51, 1, 82-126, (2009) · Zbl 1160.65330
[24] Pirozzoli, S., Numerical methods for high-speed flows, Annu Rev Fluid Mech, 43, 163-194, (2011) · Zbl 1299.76103
[25] Sengupta, T. K., High accuracy computing methods: fluid flows and wave phenomena, (2013), Cambridge University Press · Zbl 1454.76002
[26] Garnier, E.; Mossi, M.; Sagaut, P.; Comte, P.; Deville, M., On the use of shock-capturing schemes for large-eddy simulation, J Comput Phys, 153, 2, 273-311, (1999) · Zbl 0949.76042
[27] Tenaud, C.; Garnier, E.; Sagaut, P., Evaluation of some high-order shock capturing schemes for direct numerical simulation of unsteady two-dimensional free flows, Int J Numer Meth Fluids, 33, 2, 249-278, (2000) · Zbl 0977.76065
[28] Liska, R.; Wendroff, B., Comparison of several difference schemes on 1D and 2D test problems for the Euler equations, SIAM J Sci Comput, 25, 3, 995-1017, (2003) · Zbl 1096.65089
[29] Greenough, J. A.; Rider, W., A quantitative comparison of numerical methods for the compressible Euler equations: fifth-order WENO and piecewise-linear Godunov, J Comput Phys, 196, 1, 259-281, (2004) · Zbl 1115.76370
[30] Kurganov, A.; Lin, C. T., On the reduction of numerical dissipation in central-upwind schemes, Commun Comput Phys, 2, 1, 141-163, (2007) · Zbl 1164.65455
[31] Thornber, B.; Mosedale, A.; Drikakis, D., On the implicit large eddy simulations of homogeneous decaying turbulence, J Comput Phys, 226, 2, 1902-1929, (2007) · Zbl 1219.76027
[32] Klingenberg, C.; Schmidt, W.; Waagan, K., Numerical comparison of Riemann solvers for astrophysical hydrodynamics, J Comput Phys, 227, 1, 12-35, (2007) · Zbl 1128.85006
[33] Johnsen, E.; Larsson, J.; Bhagatwala, A. V.; Cabot, W. H.; Moin, P.; Olson, B. J., Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves, J Comput Phys, 229, 4, 1213-1237, (2010) · Zbl 1329.76138
[34] Gao, J.; Li, X.; Wang, Q., Effect of Riemann flux solver on the accuracy of spectral difference method for CAA problems, J Acoust Soc Am, 131, 4, 3430, (2012)
[35] Montecinos, G.; Castro, C. E.; Dumbser, M.; Toro, E. F., Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms, J Comput Phys, 231, 19, 6472-6494, (2012) · Zbl 1284.35268
[36] San, O.; Kara, K., Numerical assessments of high-order accurate shock capturing schemes: Kelvin-Helmholtz type vortical structures in high-resolutions, Comput Fluids, 89, 254-276, (2014) · Zbl 1391.76287
[37] Garnier, E.; Sagaut, P.; Deville, M., Large eddy simulation of shock/homogeneous turbulence interaction, Comput Fluids, 31, 2, 245-268, (2002) · Zbl 1059.76032
[38] Garnier, E.; Sagaut, P.; Deville, M., Large eddy simulation of shock/boundary-layer interaction, AIAA J, 40, 10, 1935-1944, (2002)
[39] Garnier, E.; Sagaut, P.; Deville, M., A class of explicit ENO filters with application to unsteady flows, J Comput Phys, 170, 1, 184-204, (2001) · Zbl 1011.76056
[40] Liu, X. D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J Comput Phys, 115, 1, 200-212, (1994) · Zbl 0811.65076
[41] Jiang, G. S.; Shu, C. W., Efficient implementation of weighted ENO schemes, J Comput Phys, 126, 202-228, (1996) · Zbl 0877.65065
[42] Titarev, V. A.; Toro, E. F., Finite-volume WENO schemes for three-dimensional conservation laws, J Comput Phys, 201, 1, 238-260, (2004) · Zbl 1059.65078
[43] Titarev, V. A.; Toro, E. F., WENO schemes based on upwind and centred TVD fluxes, Comput Fluids, 34, 6, 705-720, (2005) · Zbl 1134.65361
[44] Ha, Y.; Ho Kim, C.; Ju Lee, Y.; Yoon, J., An improved weighted essentially non-oscillatory scheme with a new smoothness indicator, J Comput Phys, 232, 1, 68-86, (2013) · Zbl 1291.65264
[45] Feng, H.; Huang, C.; Wang, R., An improved mapped weighted essentially non-oscillatory scheme, Appl Math Comput, 232, 453-468, (2014) · Zbl 1410.65306
[46] Fan, P.; Shen, Y.; Tian, B.; Yang, C., A new smoothness indicator for improving the weighted essentially nonoscillatory scheme, J Comput Phys, 269, 329-354, (2014) · Zbl 1349.65290
[47] Shen, Y.; Zha, G., Improvement of weighted essentially non-oscillatory schemes near discontinuities, Comput Fluids, 96, 254-276, (2014)
[48] Kraichnan, R. H., Inertial ranges in two-dimensional turbulence, Phys Fluids, 10, 1417-1423, (1967)
[49] Batchelor, G. K., Computation of the energy spectrum in homogeneous two-dimensional turbulence, Phys Fluids, 12, 233-239, (1969) · Zbl 0217.25801
[50] Leith, C. E., Atmospheric predictability and two-dimensional turbulence, J Atmos Sci, 28, 2, 145-161, (1971) · Zbl 0233.76109
[51] Danilov, S. D.; Gurarie, D., Quasi-two-dimensional turbulence, Phys Usp, 43, 9, 863-900, (2000)
[52] Tabeling, P., Two-dimensional turbulence: a physicist approach, Phys Rep, 362, 1, 1-62, (2002) · Zbl 1001.76041
[53] Kellay, H.; Goldburg, W. I., Two-dimensional turbulence: a review of some recent experiments, Rep Prog Phys, 65, 5, 845-894, (2002)
[54] Clercx, H. J.H.; van Heijst, G. J.F., Two-dimensional Navier-Stokes turbulence in bounded domains, Appl Mech Rev, 62, 2, 020802, (2009)
[55] Boffetta, G.; Ecke, R. E., Two-dimensional turbulence, Annu Rev Fluid Mech, 44, 427-451, (2012) · Zbl 1350.76022
[56] Shu, C. W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J Comput Phys, 77, 2, 439-471, (1988) · Zbl 0653.65072
[57] Gottlieb, S.; Shu, C. W., Total variation diminishing Runge-Kutta schemes, Math Comput, 67, 221, 73-85, (1998) · Zbl 0897.65058
[58] Gottlieb, S.; Shu, C. W.; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Rev, 43, 1, 89-112, (2001) · Zbl 0967.65098
[59] Borges, R.; Carmona, M.; Costa, B.; Don, W. S., An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J Comput Phys, 227, 6, 3191-3211, (2008) · Zbl 1136.65076
[60] Hu, X. Y.; Wang, Q.; Adams, N. A., An adaptive central-upwind weighted essentially non-oscillatory scheme, J Comput Phys, 229, 23, 8952-8965, (2010) · Zbl 1204.65103
[61] Kara, K.; Balakumar, P.; Kandil, O. A., Effects of nose bluntness on hypersonic boundary-layer receptivity and stability over cones, AIAA J, 49, 12, 2593-2606, (2011)
[62] Feng, H.; Hu, F.; Wang, R., A new mapped weighted essentially non-oscillatory scheme, J Sci Comput, 51, 2, 449-473, (2012) · Zbl 1253.65124
[63] Aràndiga, F.; Baeza, A.; Belda, A.; Mulet, P., Analysis of WENO schemes for full and global accuracy, SIAM J Numer Anal, 49, 2, 893-915, (2011) · Zbl 1233.65051
[64] Balsara, D. S.; Shu, C. W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J Comput Phys, 160, 2, 405-452, (2000) · Zbl 0961.65078
[65] Ruasnov, V. V., Calculation of intersection of non-steady shock waves with obstacles, USSR Comput Math Math Phys, 1, 267-279, (1961)
[66] LeVeque, R. J., Finite volume methods for hyperbolic problems, (2002), Cambridge University Press · Zbl 1010.65040
[67] Harten, A.; Lax, P. D.; Van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev, 25, 1, 35-61, (1983) · Zbl 0565.65051
[68] Davis, S. F., Simplified second-order Godunov-type methods, SIAM J Sci Stat Comput, 9, 3, 445-473, (1988) · Zbl 0645.65050
[69] Trangenstein, J. A., Numerical solution of hyperbolic partial differential equations, (2009), Cambridge University Press · Zbl 1187.65088
[70] Toro, E. F.; Hidalgo, A.; Dumbser, M., FORCE schemes on unstructured meshes I: conservative hyperbolic systems, J Comput Phys, 228, 9, 3368-3389, (2009) · Zbl 1168.65377
[71] Stecca, G.; Siviglia, A.; Toro, E. F., Upwind-biased FORCE schemes with applications to free-surface shallow flows, J Comput Phys, 229, 18, 6362-6380, (2010) · Zbl 1426.35192
[72] Liou, M. S.; Steffen, C. J., A new flux splitting scheme, J Comput Phys, 107, 1, 23-39, (1993) · Zbl 0779.76056
[73] Liou, M. S., A sequel to AUSM: AUSM^{+}, J Comput Phys, 129, 2, 364-382, (1996) · Zbl 0870.76049
[74] Wada, Y.; Liou, M. S., An accurate and robust flux splitting scheme for shock and contact discontinuities, SIAM J Sci Comput, 18, 3, 633-657, (1997) · Zbl 0879.76064
[75] Edwards, J. R.; Liou, M. S., Low-diffusion flux-splitting methods for flows at all speeds, AIAA J, 36, 9, 1610-1617, (1998)
[76] Hong Kim, K.; Ho Lee, J.; Hyun Rho, O., An improvement of AUSM schemes by introducing the pressure-based weight functions, Comput Fluids, 27, 3, 311-346, (1998) · Zbl 0964.76064
[77] Kim, K. H.; Kim, C.; Rho, O. H., Methods for the accurate computations of hypersonic flows: I. AUSMPW+ scheme, J Comput Phys, 174, 1, 38-80, (2001) · Zbl 1106.76421
[78] Liou, M. S., A sequel to AUSM, part II: AUSM-up for all speeds, J Comput Phys, 214, 1, 137-170, (2006) · Zbl 1137.76344
[79] Donat, R.; Marquina, A., Capturing shock reflections: an improved flux formula, J Comput Phys, 125, 1, 42-58, (1996) · Zbl 0847.76049
[80] Donat, R.; Font, J. A.; Ibánez, J. M.; Marquina, A., A flux-split algorithm applied to relativistic flows, J Comput Phys, 146, 1, 58-81, (1998) · Zbl 0930.76054
[81] Chandrasekhar, S., Hydrodynamic and hydromagnetic stability, (1961), Courier Dover Publications · Zbl 0142.44103
[82] Frank, A.; Jones, T. W.; Ryu, D.; Gaalaas, J. B., The magnetohydrodynamic Kelvin-Helmholtz instability: a two-dimensional numerical study, Astrophys J, 460, 777-793, (1996)
[83] San, O.; Staples, A. E., High-order methods for decaying two-dimensional homogeneous isotropic turbulence, Comput Fluids, 63, 105-127, (2012) · Zbl 1365.76064
[84] San, O.; Staples, A. E., Stationary two-dimensional turbulence statistics using a Markovian forcing scheme, Comput Fluids, 71, 1-18, (2014) · Zbl 1365.76070
[85] Stone, J. M.; Gardiner, T. A.; Teuben, P.; Hawley, J. F.; Simon, J. B., Athena: a new code for astrophysical MHD, Astrophys J Suppl Ser, 178, 1, 137-177, (2008)
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