Skip to main content Accessibility help
×
Home
Hostname: page-component-768ffcd9cc-mqrwx Total loading time: 0.218 Render date: 2022-12-01T19:57:48.739Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

Osher Flux with Entropy Fix for Two-Dimensional Euler Equations

Published online by Cambridge University Press:  27 May 2016

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:hjzhu@skla.cardc.cn (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.

Type
Research Article
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. 4575.CrossRefGoogle Scholar
[2]Osher, S. and Solomon, F., Upwind difference schemes for hyperbolic systems of conservation laws, Math. Comput., 38 (1982), pp. 339374.CrossRefGoogle Scholar
[3]Roe, P., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43 (1981), pp. 357372.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. 447481.CrossRefGoogle Scholar
[5]Chakravarthy, S. and Osher, S., Numerical experiments with the Osher upwind scheme for the Euler equations, AIAA J., 21 (1983), pp. 12411248.CrossRefGoogle Scholar
[6]Osher, S., Riemann solvers, the entropy condition, and difference approximations, SIAM J. Numer. Anal., 21 (1984), pp. 217235.CrossRefGoogle Scholar
[7]Osher, S. and Chakravarthy, S., High resolution schemes and the entropy condition, SIAM J. Numer. Anal., 21 (1984), pp. 955984.CrossRefGoogle 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. 542565.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. 309341.CrossRefGoogle 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. 9497.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. 7194.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. 540565.CrossRefGoogle Scholar
[13]Qiu, J., Development and comparison of numerical fluxes for LWDG methods, J. Comput. Phys., 1 (2008), pp. 132.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. 807825.Google Scholar
[15]Felcman, J. and Havle, O., On a numerical flux for the shallow water equations, Appl. Math. Comput., 217 (2011), pp. 51605170.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. 2339.CrossRefGoogle Scholar
[18]Lin, H., Dissipation additions to flux-difference splitting, J. Comput. Phys., 117 (1995), pp. 2027.CrossRefGoogle 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. 511537.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. 271301.CrossRefGoogle Scholar
[21]Ren, Y., A robust shock-capturing scheme based on rotated Riemann solvers, Comput. Fluids, 32 (2003), pp. 13791403.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. 475493.Google Scholar
[23]Jacobs, P., Approximation Riemann solver for hypervelocity flows, AIAA J., 30 (1992), pp. 25582561.CrossRefGoogle Scholar
[24]Amaladas, J. and Kamath, H., Accuracy assessment of upwind algorithms for steady-state computations, Comput. Fluids, 27 (1998), pp. 941962.CrossRefGoogle Scholar
[25]Frink, N., Upwind scheme for solving the Euler equations on unstructured tetrahedral meshes, AIAA J., 30 (1991), pp. 7077.CrossRefGoogle Scholar
[26]Van Leer, B., Towards the ultimate conservative difference scheme IV. a new approach to numerical convection, J. Comput. Phys., 23 (1977), pp. 276299.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. 101136.CrossRefGoogle Scholar
[28]Toro, E., Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, 1997.CrossRefGoogle Scholar
[29]Woodward, P. and Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54 (1984), pp. 115173.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. 503509.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. 763776.CrossRefGoogle Scholar
[32]Kitamura, K., A further survey of shock capturing methods on hypersonic heating issues, AIAA paper, 2013–2698.Google Scholar
3
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Osher Flux with Entropy Fix for Two-Dimensional Euler Equations
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Osher Flux with Entropy Fix for Two-Dimensional Euler Equations
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Osher Flux with Entropy Fix for Two-Dimensional Euler Equations
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *