Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T07:40:19.998Z Has data issue: false hasContentIssue false
  • English
  • Français

Osher Flux with Entropy Fix for Two-Dimensional Euler Equations

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  27 May 2016

Huajun Zhu ,
Xiaogang Deng ,
Meiliang Mao ,
Huayong Liu  and
Guohua Tu
Huajun Zhu*
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang, Sichuan 621000, China
Xiaogang Deng
Affiliation:
National University of Defense Technology, Changsha, Hunan 410073, China
Meiliang Mao
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang, Sichuan 621000, China Computational Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang, Sichuan 621000, China
Huayong Liu
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang, Sichuan 621000, China
Guohua Tu
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang, Sichuan 621000, China
*
*Corresponding author. Email:[email protected] (H. J. Zhu)
Get access

Abstract

We compare in this paper the properties of Osher flux with O-variant and P-variant (Osher-O flux and Osher-P flux) in finite volume methods for the two-dimensional Euler equations and propose an entropy fix technique to improve their robustness. We consider both first-order and second-order reconstructions. For inviscid hypersonic flow past a circular cylinder, we observe different problems for different schemes: A first-order Osher-O scheme on quadrangular grids yields a carbuncle shock, while a first-order Osher-P scheme results in a dislocation shock for high Mach number cases. In addition, a second-order Osher scheme can also yield a carbuncle shock or be unstable. To improve the robustness of these schemes we propose an entropy fix technique, and then present numerical results to show the effectiveness of the proposed method. In addition, the influence of grid aspects ratio, relative shock position to the grid and Mach number on shock stability are tested. Viscous heating problem and double Mach reflection problem are simulated to test the influence of the entropy fix on contact resolution and boundary layer resolution.

Keywords

Osher fluxentropy fixEuler equationfinite volume method“carbuncle” shock

MSC classification

Secondary: 65M06: Finite difference methods 65M70: Spectral, collocation and related methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 8 , Issue 4 , August 2016 , pp. 670 - 692
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Engquist, B. and Osher, S., Stable and entropy satisfying approximations for transonic flow calculations, Math. Comput., 34 (1980), pp. 45–75.Google Scholar
[2]Osher, S. and Solomon, F., Upwind difference schemes for hyperbolic systems of conservation laws, Math. Comput., 38 (1982), pp. 339–374.Google Scholar
[3]Roe, P., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43 (1981), pp. 357–372.CrossRefGoogle Scholar
[4]Osher, S. and Chakravarthy, S., Upwind schemes and boundary conditions with applications to Euler equations in general geometries, J. Comput. Phys., 50 (1983), pp. 447–481.CrossRefGoogle Scholar
[5]Chakravarthy, S. and Osher, S., Numerical experiments with the Osher upwind scheme for the Euler equations, AIAA J., 21 (1983), pp. 1241–1248.Google Scholar
[6]Osher, S., Riemann solvers, the entropy condition, and difference approximations, SIAM J. Numer. Anal., 21 (1984), pp. 217–235.Google Scholar
[7]Osher, S. and Chakravarthy, S., High resolution schemes and the entropy condition, SIAM J. Numer. Anal., 21 (1984), pp. 955–984.Google Scholar
[8]Lanser, D., Blom, J. and Verwer, J., Spatial discretization of the shallow water equations in spherical geometry using Osher's scheme, J. Comput. Phys., 165 (2000), pp. 542–565.CrossRefGoogle Scholar
[9]Fedkiw, R., Sapiro, G. and Shu, C., Shock capturing, level sets, and PDE based methods in computer vision and image processing: a review of Osher's contributions, J. Comput. Phys., 185 (2003), pp. 309–341.Google Scholar
[10]Qiang, L. and Zhao, W., Computation of turbulence flow on hybrid grids using k-w turbulence model and Osher scheme, Trans. Nanjing University Aeronaut. Astronaut., 21 (2004), pp. 94–97.Google Scholar
[11]Coclite, G., Mishra, S. and Risebro, N., Convergence of an Engquist-Osher scheme for a multi-dimensional triangular system of conservation laws, Math. Comput., 79 (2010), pp. 71–94.CrossRefGoogle Scholar
[12]Qiu, J., Khoo, B. and Shu, C., A numerical study for the performance of the Runge Kutta discontinuous Galerkin method based on different numerical fluxes, J. Comput. Phys., 212 (2006), pp. 540–565.CrossRefGoogle Scholar
[13]Qiu, J., Development and comparison of numerical fluxes for LWDG methods, J. Comput. Phys., 1 (2008), pp. 1–32.Google Scholar
[14]Lu, C., Qiu, J. and Wang, R., A numerical study for the performance of the WENO schemes based on different numerical fluxes for the shallow water equations, J. Comput. Math., 28 (2010), pp. 807–825.Google Scholar
[15]Felcman, J. and Havle, O., On a numerical flux for the shallow water equations, Appl. Math. Comput., 217 (2011), pp. 5160–5170.Google Scholar
[16]Peery, K. and Imlay, S., Blunt-body flow simulations, AIAA, (1988), 2904.Google Scholar
[17]Liou, M. and Steffen, C., A new flux splitting scheme, J. Comput. Phys., 107 (1993), pp. 23–39.Google Scholar
[18]Lin, H., Dissipation additions to flux-difference splitting, J. Comput. Phys., 117 (1995), pp. 20–27.Google Scholar
[19]Sanders, R., Morano, E. and Druguetz, M., Multidimensional dissipation for upwind schemes: stability and applications to gas dynamics, J. Comput. Phys., 145 (1998), pp. 511–537.CrossRefGoogle Scholar
[20]Pandolfi, M. and D'Ambrosio, D., Numerical instabilities in upwind methods: analysis and cures for the “carbuncle” phenomenon, J. Comput. Phys., 166 (2001), pp. 271–301.Google Scholar
[21]Ren, Y., A robust shock-capturing scheme based on rotated Riemann solvers, Comput. Fluids, 32 (2003), pp. 1379–1403.CrossRefGoogle Scholar
[22]Spekreijse, S., Multiple grid and Osher's scheme for the efficient solution of the steady Euler equations, Appl. Numer. Math., 2 (1986), pp. 475–493.Google Scholar
[23]Jacobs, P., Approximation Riemann solver for hypervelocity flows, AIAA J., 30 (1992), pp. 2558–2561.Google Scholar
[24]Amaladas, J. and Kamath, H., Accuracy assessment of upwind algorithms for steady-state computations, Comput. Fluids, 27 (1998), pp. 941–962.CrossRefGoogle Scholar
[25]Frink, N., Upwind scheme for solving the Euler equations on unstructured tetrahedral meshes, AIAA J., 30 (1991), pp. 70–77.Google Scholar
[26]Van Leer, B., Towards the ultimate conservative difference scheme IV. a new approach to numerical convection, J. Comput. Phys., 23 (1977), pp. 276–299.CrossRefGoogle Scholar
[27]Van Leer, B., Towards the ultimate conservative difference scheme V. a second-order sequel to Godunov's method, J. Comput. Phys., 32 (1979), pp. 101–136.Google Scholar
[28]Toro, E., Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, 1997.Google Scholar
[29]Woodward, P. and Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54 (1984), pp. 115–173.CrossRefGoogle Scholar
[30]Nishino, A., Ishikawa, T., Kitamura, K. and Nakamura, Y., Effect of the clearance between two bodies on TSTO aerodynamic heating, J. Japan Soc. Aer. Space Sci., 53 (2005), pp. 503–509.Google Scholar
[31]Kitamura, K., Shima, E., Nakamura, Y. and Roe, P. L., Evaluation of Euler fluxes for hypersonic heating computations, AIAA J., 48 (2010), pp. 763–776.Google Scholar
[32]Kitamura, K., A further survey of shock capturing methods on hypersonic heating issues, AIAA paper, 2013–2698.Google Scholar