On the Splitting Methods of Inviscid Fluxes for Implementing High-Order Weighted Compact Nonlinear Schemes

TU Guo-hua; CHEN Jian-qiang; MAO Mei-liang; ZHAO Xiao-hui; LIU Hua-yong

doi:10.21656/1000-0887.370518

Volume 37 Issue 12

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Article Navigation > Applied Mathematics and Mechanics > 2016 > 37(12): 1324-1344

TU Guo-hua, CHEN Jian-qiang, MAO Mei-liang, ZHAO Xiao-hui, LIU Hua-yong. On the Splitting Methods of Inviscid Fluxes for Implementing High-Order Weighted Compact Nonlinear Schemes[J]. Applied Mathematics and Mechanics, 2016, 37(12): 1324-1344. doi: 10.21656/1000-0887.370518

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On the Splitting Methods of Inviscid Fluxes for Implementing High-Order Weighted Compact Nonlinear Schemes

doi: 10.21656/1000-0887.370518

¹State Key Laboratory of Aerodynamics(China Aerodynamics Research andDevelopment Center), Mianyang, Sichuan 621000, P.R.China;²Computational Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang, Sichuan 621000, P.R.China

Funds: National Key Research and Development Project of China (2016YFA0401200);National Natural Science Foundation of China (11301525)

Received Date: 2016-11-16
Rev Recd Date: 2016-12-01
Publish Date: 2016-12-15

Abstract

Abstract

There is increasing popularity in using high-order weighted compact nonlinear schemes(WCNS) for complex flow simulations. The WCNS can be used in combination with many inviscid flux splitting methods. However, it is still uncertain which flux splitting is most suitable for the WCNS because most of the methods are devised on the basis of low-order discretization methods. It is also not very clear what will happen when these splitting methods are mounted directly in high-order accurate schemes. In order to provide some guide for selecting inviscid fluxes in the computation of surface heat transfer, the dissipations of the fluxes are studied. Every inviscid flux can be expressed as a summation of a central part and a dissipation part. All the fluxes have an identical central part which is very simple. However, different fluxes have different dissipation parts which are more or less complicated. The analysis on the source of flux dissipation shows that the dissipation is nearly proportional to flux jumps on grid interfaces. Numerical experiments show that high-order schemes usually produce far less flux jumps than low-order schemes in smooth regions, and logically the flux dissipations are quite lower. 3 canonical flows including hypersonic shock wave/boundary layer interactions(SWBLI) are simulated to show the influence of inviscid fluxes on heat transfer computing. Finally, a suggestion is given for selecting inviscid fluxes based on the dissipations and shock instabilities of van Leer’s flux splitting, the Steger-Warming(SW) flux splitting, the kinetic flux vector splitting (KFVS), Roe’s flux splitting, the AUSM(advection upwind splitting method)-type flux splitting and the HLL-type flux splitting.
- WCNS,
- shock wave/boundary layer interaction,
- Euler flux,
- high-order accuracy,
- heat transfer rate

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References

[1]	Ekaterinaris J A. High-order accurate, low numerical diffusion methods for aerodynamics[J].Progress in Aerospace Sciences,2005,41(3/4): 192-300.
[2]	DENG Xiao-gang, MAO Mei-liang, TU Guo-hua, ZHANG Han-xin, ZHANG Yi-feng. High-order and high accurate CFD methods and their applications for complex grid problems[J].Communications in Computational Physics,2012,11(4): 1081-1102.
[3]	DENG Xiao-gang, MAO Mei-liang, TU Guo-hua, LIU Hua-yong, ZHANG Han-xin. Geometric conservation law and applications to high-order finite difference schemes with stationary grids[J].Journal of Computational Physics,2011,230(4): 1100-1115.
[4]	DENG Xiao-gang, MAO Mei-liang, TU Guo-hua, ZHANG Yi-feng, ZHANG Han-xin. Extending weighted compact nonlinear schemes to complex grids with characteristic-based interface conditions[J].AIAA Journal,2010,48(12): 2840-2851.
[5]	TU Guo-hua, DENG Xiao-gang, MAO Mei-liang. Implementing high-order weighted compact nonlinear scheme on patched grids with a nonlinear interpolation[J].Computers & Fluids,2013,77: 181-193.
[6]	TU G, Deng X, Liu H, Zhao X. Validation of high-order weighted compact nonlinear scheme for heat transfer of complex hypersonic laminar flows[C]//The 4th Asian Symposium on Computational Heat Transfer and Fluid Flow.Hong Kong, 2013: ASCHT0095-T01-2-A.
[7]	Rizzetta D P, Viabal M R, Morgan P E. A high-order compact finite-difference scheme for large-eddy simulation of active flow control[J]. Progress in Aerospace Sciences,2008,44(6): 397-426.
[8]	Pirozzoli S. Numerical methods for high-speed flows[J].Annual Review of Fluid Mechanics, 2011,43: 163-194.
[9]	DENG Xiao-gang, ZHANG Han-xin. Developing high-order weighted compact nonlinear schemes[J].Journal of Computational Physics,2000,165(1): 22-44.
[10]	Cockburn B, Shu C. The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems[J].Journal of Computational Physics,1998,141(2): 199-224.
[11]	TU Guo-hua, YUAN Xiang-jiang, XIA Zhi-qiang, HU Zhen. A class of compact upwind TVD difference schemes[J].Applied Mathematics and Mechanics,2006,27(6): 675-682.
[12]	TU Guo-hua, YUAN Xiang-jiang, LU Li-peng. Developing shock-capturing difference methods[J].Applied Mathematics and Mechanics,2007,28(4): 433-440.
[13]	TU Guo-hua, YUAN Xiang-jiang. A characteristic-based shock-capturing scheme for hyperbolic problems[J].Journal of Computational Physics,2007,225(2): 2083-2097.
[14]	Shu C. High order weighted essentially nonoscillatory schemes for convection dominated problems[J].SIAM Review,2009,51(1): 82-126.
[15]	Wang Z J. High-order methods for the Euler and Navier-Stokes equations on unstructured grids[J].Progress in Aerospace Sciences,2007,43(1/3): 1-41.
[16]	Suresh A, Huynh H T. Accurate monotonicity-preserving schemes with Runge-Kutta time stepping[J].Journal of Computational Physics,1997,136(1): 83-99.
[17]	TU Guo-hua, DENG Xiao-gang, MAO Mei-liang. Assessment of two turbulence models and some compressibility corrections for hypersonic compression corners by high-order difference schemes[J].Chinese Journal of Aeronautics,2012,25(1): 25-32.
[18]	XU Chuan-fu, DENG Xiao-gang, ZHANG Li-lun, FANG Jian-bin, WANG Guang-xue, JIANG Yi, GAO Wei, CHE Yong-gang, WANG Zheng-hua, LIU Wei, CHENG Xing-hua. Collaborating CPU and GPU for large-scale high-order CFD simulations with complex grids on the TianHe-1A supercomputer[J].Journal of Computational Physics,2014,278(1): 275-297.
[19]	Kitamura K. Afurther survey of shock capturing methods on hypersonic heating issues[C]//21st AIAA Computational Fluid Dynamics Conference . San Diego, CA, 2013: 2013-2698.
[20]	Kitamura K, Shima E, Roe P L. Carbuncle phenomena and other shock anomalies in three dimensions[J].AIAA Journal,2012,50(12): 2655-2669.
[21]	Kitamura K, Roe P, Ismail F. Evaluation of Euler fluxes for hypersonic flow computations[J].AIAA Journal,2009,47(1): 44-53.
[22]	Kitamura K, Shima E, Nakamura Y, Roe P. Evaluation of Euler fluxes for hypersonic heating computations[J].AIAA Journal,2010,48(4): 763-776.
[23]	TU Guo-hua, ZHAO Xiao-hui, MAO Mei-liang, CHEN Jian-qiang, DENG Xiao-gang, LIU Hua-yong. Evaluation of Euler fluxes by a high-order CFD scheme: shock instability[J].International Journal of Computational Fluid Dynamics,2014,28(5): 171-186.
[24]	Pandolfi M, D’Ambrosio D. Numerical instabilities in upwind methods: analysis and cures for the “carbuncle” phenomenon[J].Journal of Computational Physics,2001,166(2): 271-301.
[25]	van Leer B. Flux-vector splitting for the Euler equation[M]// Upwind and High-Resolution Schemes . Berlin: Springer Berlin Heidelberg, 1997: 80-89.
[26]	Steger J L, Warming R F. Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods[J].Journal of Computational Physics,1981,40(2): 263-293.
[27]	Mandal J C, Deshpande S M. Kinetic flux vector splitting for Euler equations[J].Computers & Fluids,1994,23(2): 447-478.
[28]	Roe P L. Approximate Riemann solvers, parameter vectors, and difference schemes[J].Journal of Computational Physics,1981,43(2): 357-372.
[29]	Liou M S. Mass flux schemes and connection to shock instability[J].Journal of Computational Physics,2000,160(2): 623-648.
[30]	Kim K H, Lee J H, Rho O H. An improvement of AUSM schemes by introducing the pressure-based weight functions[J].Computers & Fluids,1998,27(3): 311-346.
[31]	Kim K H, Kim C, Rho O H. Methods for the accurate computations of hypersonic flows—I: AUSMPW+ scheme[J].Journal of Computational Physics,2001,174(1): 38-80.
[32]	Einfeldt B, Munz C D, Roe P L, Sjgreen B. On Godunov-type methods near low densities[J].Journal of Computational Physics,1991,92(2): 273-295.
[33]	Harten A, Lax P D, van Leer B. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws[M]// Upwind and High-Resolution Schemes.Berlin: Springer Berlin Heidelberg, 1997: 53-79.
[34]	Batten P, Clarke N, Lambert C, Causon D M. On the choice of wavespeeds for the HLLC Riemann solver[J].SIAM Journal on Scientific Computing,1997,18(6): 1553-1570.
[35]	Toro E F.Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction[M]. 3rd ed. Springer, 2009.
[36]	ZHANG Han-xin, ZHUANG Feng-gan. NND schemes and their applications to numerical simulation of two- and three-dimensional flows[J].Advances in Applied Mechanics,1991,29: 193-256.
[37]	TU Guo-hua, DENG Xiao-gang, MAO Mei-liang. A staggered non-oscillatory finite difference method for high-order discretization of viscous terms[J].Acta Aerodynamica Sinica,2011,29(1): 10-15.
[38]	DENG Xiao-gang, MAO Mei-liang, TU Guo-hua, LIU Hua-yong, ZHANG Han-xin. Geometric conservation law and applications to high-order finite difference schemes with stationary grids[J].Journal of Computational Physics,2011,230(4): 1100-1115.
[39]	Kopriva D A. Spectral solution of the viscous blunt-body problem[J].AIAA Journal,1993,31(7): 1235-1242.
[40]	Longo J M A, Hannemann K, Hannemann V. The challenge of modeling high speed flows[C]// The EUROSIM.2007.
[41]	Holden M S, Wadhams T P. A database of aerothermal measurements in hypersonic flow “building block” experiment for CFD validation[C]//41st AIAA Aerospace Sciences Meeting and Exhibit.Reno, Nevada, 2003: AIAA 2003-1137.
[42]	Kirk B S, Carey G F. Validation of fully implicit, parallel finite element simulations of laminar hypersonic flows[J].AIAA Journal,2010,48(6): 1025-1036.