Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T13:13:54.231Z Has data issue: false hasContentIssue false
  • English
  • Français

Modified Upwinding Compact Scheme for Shock and Shock Boundary Layer Interaction

Published online by Cambridge University Press:  20 August 2015

Chaoqun Liu ,
Ping Lu ,
Maria Oliveira  and
Peng Xie
Chaoqun Liu*
Affiliation:
Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019, USA
Ping Lu*
Affiliation:
Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019, USA
Maria Oliveira*
Affiliation:
Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019, USA
Peng Xie*
Affiliation:
Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019, USA
*
Corresponding author.Email:cliu@uta.edu
Email:ping.lu@mavs.uta.edu
Email:malubamol@gmail.com
Email:xiepeng.peng@gmail.com
Get access

Abstract

Standard compact scheme and upwinding compact scheme have high order accuracy and high resolution, but cannot capture the shock which is a discontinuity. This work developed a modified upwinding compact scheme which uses an effective shock detector to block compact scheme to cross the shock and a control function to mix the flux with WENO scheme near the shock. The new scheme makes the original compact scheme able to capture the shock sharply and, more importantly, keep high order accuracy and high resolution in the smooth area which is particularly important for shock boundary layer and shock acoustic interactions. Numerical results show the scheme is successful for 2-D Euler and 2-D Navier-Stokes solvers. The examples include 2-D incident shock, 2-D incident shock and boundary layer interaction. The scheme is robust, which does not involve case related parameters.

Keywords

47.11.-j47.11.BcCompact schemeWENOshock-boundary layer interactionshock detector
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 3 , March 2012 , pp. 1022 - 1042
Copyright
Copyright © Global Science Press Limited 2012

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]Costaa, B. and Don, W. S., High order hybrid central-WENO finite difference scheme for conservation laws, J. Comput. Appl. Math., 204 (2007), 209–218.Google Scholar
[2]Gaitonde, D. and Visbal, M., Pade-type high-order boundary filters for the Navier-Stokes equations, AIAA J., 38(11) (2000), 2103–2112.Google Scholar
[3]Gaitonde, D., Canupp, P. W. and Holden, M., Heat transfer predictions in a laminar hypersonic viscous/inviscid interaction, J. Thermophys. Heat Trans., 16(4) (2002), 481–489.Google Scholar
[4]Godunov, S. K., A difference scheme for numerical computation of discontinuous solution of hydrodynamic equations, Math. Sbornik., 47 (1969), 271–306, (in Russian) translated US Joint Publ. Res. Service, JPRS 7226, 1969.Google Scholar
[5]Harten, A., High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49 (1983), 357–393.Google Scholar
[6]Harten, A., Engquist, B., Osher, B. and Charkravarthy, Sr., Uniformly high order accurate essentially non-oscillatory schemes III, J. Comput. Phys., 71 (1987), 231–303.Google Scholar
[7]Jiang, G. S. and Shu, C. W., Efficient implementation of weighted ENO scheme, J. Comput. Phys., 126 (1996), 202–228.Google Scholar
[8]Jiang, L., Shan, H. and Liu, C., Weight compact scheme for shock capturing, Int. J. Comput. Fluid Dyn., 15 (2001), 147–155.CrossRefGoogle Scholar
[9]Kim, D. and Kwon, J., A high-order accurate hybrid scheme using a central flux scheme and a WENO scheme for compressible flowfield analysis, J. Comput. Phys., 210 (2005), 554–583.Google Scholar
[10]Lele, S. K., Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103 (1992), 16–42.Google Scholar
[11]Liu, C., Xie, P. and Olveira, M., High order modified compact scheme for high speed flow, Technical Report to Gaitonde at Air Force Research Lab.Google Scholar
[12]Liu, X.-D., Osher, S. and Chan, T., Weighted essentially nonoscilltory schemes, J. Comput. Phys., 115 (1994), 200–212.Google Scholar
[13]Oliveira1, M., Lu, P., Liu, X. and Liu, C., A new shock/discontinuity detector, Int. J. Comput. Math., 87 (2010), 3063–3078.Google Scholar
[14]Ren, Y., Liu, M. and Zhang, H., A characteristic-wise hybrid compact-WENO scheme for solving hyperbolic conservation laws, J. Comput. Phys., 192 (2003), 365–386.Google Scholar
[15]Roe, P. L., Approximate Riemann solvers, parameter vectors and difference schemes, J. Com-put. Phys., 43 (1981), 357–372.Google Scholar
[16]Sanyasiraju, Y. V. S. S. and Mishra, N., Exponential compact higher order scheme for nonlinear steady convection-diffusion equations, Commun. Comput. Phys., 9 (2011), 897–916.Google Scholar
[17]Shu, C. W. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing scheme, J. Comput. Phys., 77 (1988), 439–471.Google Scholar
[18]Shu, C. W. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes II, J. Comput. Phys., 83 (1989), 32–78.Google Scholar
[19]Su, J., Oliveira, M., Xie, P. and Liu, C., Error analysis for weighted higher order compact finite difference scheme, J. Appl. Math. Sci., 1 (2007), 2863–2881.Google Scholar
[20]Tu, S., Skelton, G. W. and Pang, Q., A compact high order space-time method for conservation laws, Commun. Comput. Phys., 9 (2011), 441–480.Google Scholar
[21]Van Leer, B., Towards the ultimate conservative difference scheme V: a second order sequel to Godunov’s scheme, J. Comput. Phys., 32 (1979), 101–136.Google Scholar
[22]Visbal, M. and Gaitonde, D., On the use of higher-order finite-difference schemes on curvilinear and deforming mashes, J. Comput. Phys., 181 (2002), 155–158.CrossRefGoogle Scholar
[23]Wadhams, T. P. and Holden, M. S., Summary of experimental studies for code validation in the LENS facility with recent Navier-Stokes and DSMC solutions for two- and three- dimensional separated regions in hypervelocity flows, AIAA 2004-917 42nd AIAA Aerospace Sciences Meeting and Exhibit, 5-8 January 2004, Reno, Nevada.Google Scholar
[24]Xie, P., Oliveira, M., Su, J. and Liu, C., High order weighted compact schemes for shock/boundary layer interaction, AIAA-2008-0754, January 2008.Google Scholar