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Assessing the Performance of a Three Dimensional Hybrid Central-WENO Finite Difference scheme with Computation of a Sonic Injector in Supersonic Cross Flow

Published online by Cambridge University Press:  03 June 2015

Wai-Sun Don ,
Antonio de Gregorio ,
Jean-Piero Suarez  and
Gustaaf B. Jacobs
Wai-Sun Don*
Affiliation:
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
Antonio de Gregorio
Affiliation:
Department of Aerospace Engineering & Engineering Mechanics, San Diego State University, 5500 Campanile Drive San Diego, CA 92182, USA
Jean-Piero Suarez
Affiliation:
Department of Aerospace Engineering & Engineering Mechanics, San Diego State University, 5500 Campanile Drive San Diego, CA 92182, USA
Gustaaf B. Jacobs*
Affiliation:
Department of Aerospace Engineering & Engineering Mechanics, San Diego State University, 5500 Campanile Drive San Diego, CA 92182, USA
*
Corresponding author. Email: [email protected]
Email: [email protected]
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Abstract

A hybridization of a high order WENO-Z finite difference scheme and a high order central finite difference method for computation of the two-dimensional Euler equations first presented in [B. Costa and W. S. Don, J. Comput. Appl. Math., 204(2) (2007)] is extended to three-dimensions and for parallel computation. The Hybrid scheme switches dynamically from a WENO-Z scheme to a central scheme at any grid location and time instance if the flow is sufficiently smooth and vice versa if the flow is exhibiting sharp shock-type phenomena. The smoothness of the flow is determined by a high order multi-resolution analysis. The method is tested on a benchmark sonic flow injection in supersonic cross flow. Increase of the order of the method reduces the numerical dissipation of the underlying schemes, which is shown to improve the resolution of small dynamic vortical scales. Shocks are captured sharply in an essentially non-oscillatory manner via the high order shock-capturing WENO-Z scheme. Computations of the injector flow with a WENO-Z scheme only and with the Hybrid scheme are in very close agreement. Thirty percent of grid points require a computationally expensive WENO-Z scheme for high-resolution capturing of shocks, whereas the remainder of grid points may be solved with the computationally more affordable central scheme. The computational cost of the Hybrid scheme can be up to a factor of one and a half lower as compared to computations with a WENO-Z scheme only for the sonic injector benchmark.

Keywords

65P3077AxxWeighted essentially non-oscillatory central differencemulti-resolutionhybridshockinjector
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 6 , December 2012 , pp. 719 - 736
Copyright
Copyright © Global-Science Press 2012

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References

[1]Shu, C. W., High order weighted essentially non-oscillatory schemes for convection dominated problems, SIAM Rev., 51 (2009), pp. 82–126.Google Scholar
[2]Harten, A., High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49 (1983), pp. 357–393.Google Scholar
[3]Henrick, A. K., Aslam, T. D. and Powers, J. M., Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points, J. Comput. Phys., 207 (2005), pp. 542–567.Google Scholar
[4]Costa, B. and Don, W. S., High order Hybrid Central-WENO finite difference scheme for conservation laws, J. Comput. Appl. Math., 204(2) (2007), pp. 209–218.Google Scholar
[5]Costa, B. and Don, W. S., Multi-domain hybrid spectral-WENO methods for hyperbolic conservation laws, J. Comput. Phys., 224(2) (2007), pp. 970–991.Google Scholar
[6]Costa, B., Don, W. S., Gottlieb, D. and Sendersky, R., Two-dimensional multi-domain hybrid spectral-WENO methods for conservation laws, Commun. Comput. Phys., 1(3) (2006), pp. 548–574.Google Scholar
[7]Borges, R., Carmona, M., Costa, B. and Don, W. S., An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys., 227 (2008), pp. 3101–3211.CrossRefGoogle Scholar
[8]Castro, M., Costa, B. and Don, W. S., High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws, J. Comput. Phys., 230 (2011), pp. 1766– 1792.Google Scholar
[9]Gao, Z., Don, W. S. and Li, Z., High order weighted essentially non-oscillation schemes for one-dimensional detonation wave simulations, J. Comput. Math., 29(6) (2011), pp. 623–638.Google Scholar
[10]Gao, Z., Don, W. S. and Li, Z., High order weighted essentially non-oscillation schemes for two-dimensional detonation wave simulations, J. Sci. Comput., DOI: 10.1007/s10915-011-9569-0, 2012.Google Scholar
[11]Latini, M., Schilling, O. and Don, W. S., Effects of order of WENO flux reconstruction and spatial resolution on reshocked two-dimensional Richtmyer-Meshkov instability, J. Comput. Phys., 221 (2007), pp. 805–836.Google Scholar
[12]Johnsen, E., Larsson, J., Bhagatwala, A. V., Cabot, W. H., Moin, P., Olson, B. J., Rawat, P. S., Shankar, S. K., Sjogreen, B., Yee, H. C., Zhong, X. and Lele, S. K., Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves, J. Comput. Phys., 229 (2010), pp. 1213–1237.Google Scholar
[13]Fornberg, B., Calculation of weights in finite difference formulas, SIAM Rev., 40(3) (1998), pp. 685–691.Google Scholar
[14]Vasilyev, O., Lund, T. and Moin, P., A general class of commutative filters for LES in complex geometries, J. Comput. Phys., 146(1) (1998), pp. 82–104.CrossRefGoogle Scholar
[15]Adams, N. and Shariff, K., High-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems, J. Comput. Phys., 127 (1996), pp. 27–51.CrossRefGoogle Scholar
[16]Hill, D. and Pullin, D., Hybrid tuned center-difference-WENO method for large eddy simulations in the presence of strong shocks, J. Comput. Phys., 194 (2004), pp. 435–450.CrossRefGoogle Scholar
[17]Pirozzoli, S., Conservative hybrid compact-WENO schemes for Shock-Turbulence interaction, J. Comput. Phys., 178(1) (2002), pp. 81–117.Google Scholar
[18]Zingg, D. W., Comparison of high-accuracy finite difference methods for linear wave propagation, SIAM J. Sci. Comput., 22(2) (2000), pp. 476–502.CrossRefGoogle Scholar
[19]Bogey, C. and Bailly, C., A family of low dispersive and low dissipative explicit schemes for flow and noise computations, J. Comput. Phys., 194 (2004), pp. 194–214.Google Scholar
[20]Choen, A., Kaber, S. M., Muller, S. and Postel, M., Fully adaptive multiresolution finite volume schemes for conservation laws, Math. Comput., 72(241) (2001), pp. 183–225.Google Scholar
[21]Bihari, B. and Harten, A., Multiresolution schemes for the numerical solution of 2-D Conservation Laws, SIAM J. Sci. Comput., 18(2) (1997), pp. 315–354.Google Scholar
[22]Schetz, J. A., Hawkins, P. F. and Lehman, H., Structure of highly under-expanded transverse jets in a supersonic stream, AIAA J., 35(5) (1976), pp. 882–884.Google Scholar
[23]Viti, V., Numerical Analysis of the Jet Interaction Flowfield with a Main Jet and an Array of Smaller Jets, Ph.D. dissertation, Virginia Tech, 2002.Google Scholar