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Size and shape of shock waves and slipline for Mach reflection in steady flow

Published online by Cambridge University Press:  29 March 2017

Chen-Yuan Bai  and
Zi-Niu Wu Open the ORCID record for Zi-Niu Wu [Opens in a new window]
Chen-Yuan Bai
Affiliation:
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China
Zi-Niu Wu*
Affiliation:
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PR China
*
†Email address for correspondence: ziniuwu@tsinghua.edu.cn
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Abstract

For Mach reflection in steady supersonic flow, the slipline and reflected shock wave from the triple point are disturbed by secondary Mach waves generated over the slipline and by the expansion fan from the rear wedge corner. Analytical expressions for the shape of the curved slipline and reflected shock wave are derived in this paper. It is found that, due to transmitted expansion waves from the expansion fan, the slipline has a slope discontinuity at the turning point, i.e., the intersection point of the slipline and the leading characteristics of the transmitted expansion wave. The hypothetical shock wave calculated by considering this slope discontinuity as flow deflection angle matches a similar wave observed in numerical results by computational fluid dynamics, suggesting the existence of a weak shock wave from this turning point. The effects of the secondary Mach waves upstream of the turning point and of the turning point weak shock wave mutually cancel out approximately so that the transmitted Mach waves can be approximated as straight characteristic lines. This simplification leads to a fast analytical model which can predict the Mach stem height and shape of the slipline and reflected shock wave with increasing accuracy for the decreasing deflection angle of the slipline at the triple point. The slipline slope discontinuity at the turning point and the hypothetical turning point weak shock wave are new phenomena found in this work.

JFM classification

Compressible Flows: Gas dynamics Aerodynamics: High-speed flow Compressible Flows: Shock waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 818 , 10 May 2017 , pp. 116 - 140
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Copyright
© 2017 Cambridge University Press 

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References

Azevedo, D. J. & Liu, C. S. 1993 Engineering approach to the prediction of shock patterns in bounded high-speed flows. AIAA J. 31, 83–90.CrossRefGoogle Scholar
Ben-Dor, G. 2007 Shock Wave Reflection Phenomena. Springer.Google Scholar
Ben-Dor, G., Ivanov, M., Vasilev, E. I. & Elperin, T. 2002 Hysteresis processes in the regular reflection—Mach reflection transition in steady flows. Prog. Aerosp. Sci. 38, 347–387.Google Scholar
Ben-Dor, G. & Takayama, K. 1992 The phenomena of shock wave reflection – a review of unsolved problems and future research needs. Shock Waves 2, 211–223.Google Scholar
Chpoun, A., Passerel, D., Li, H. & Ben-Dor, G. 1995 Reconsideration of the oblique shock wave reflection in steady flows. Part 1. Experimental investigation. J. Fluid Mech. 301, 19–35.CrossRefGoogle Scholar
Courant, R. & Friedrichs, K. O. 1948 Supersonic Flow and Shock Waves. Wiley-Interscience.Google Scholar
Emanuel, G. 1986 Gasdynamics: Theory and Applications. AIAA.CrossRefGoogle Scholar
Gao, B. & Wu, Z. N. 2010 A study of the flow structure for Mach reflection in steady supersonic flow. J. Fluid Mech. 656, 29–50.Google Scholar
Hasimoto, Z. 1964 Interaction of a simple expansion wave with a shock wave in two-dimensional flows of a gas. J. Phys. Soc. Japan 19, 1074–1078.Google Scholar
Henderson, L. F. & Lozzi, A. 1975 Experiments on transition of Mach reflection. J. Fluid Mech. 68, 139–155.CrossRefGoogle Scholar
Hornung, H. G. 2014 Mach reflection in steady flow. I. Mikhail Ivanov’s contributions, II. Caltech stability experiments. In AIP Conference Proceedings, vol. 1628, pp. 1384–1393. AIP Publishing.Google Scholar
Hornung, H. G., Oertel, H. & Sandeman, R. J. 1979 Transition to Mach reflection of shock waves in steady and pseudo-steady flows with and without relaxation. J. Fluid Mech. 90, 541–560.Google Scholar
Hornung, H. G. & Robinson, M. L. 1982 Transition from regular to Mach reflection of shock waves. Part 2. The steady-flow criterion. J. Fluid Mech. 123, 155–164.CrossRefGoogle Scholar
Ivanov, M. S., Ben-Dor, G., Elperin, T., Kudryavtsev, A. & Khotyanovsky, D. 2001 Mach-number-variation-induced hysteresis in steady flowshock wave reflections. AIAA J. 39, 972–974.CrossRefGoogle Scholar
Li, H. & Ben-Dor, G. 1995 Oblique-shock/expansion-fan interaction – analytical solution. AIAA J. 34, 418–421.CrossRefGoogle Scholar
Li, H. & Ben-Dor, G. 1996 Application of the principle of minimum entropy production to shock wave reflections. I. Steady flows. J. Appl. Phys. 80, 2027–2037.Google Scholar
Li, H. & Ben-Dor, G. 1997 A parametric study of Mach reflection in steady flows. J. Fluid Mech. 341, 101–125.Google Scholar
Liepmann, H. W. & Roshko, A. 1957 Elements of Gasdynamics. Wiley.CrossRefGoogle Scholar
Mach, E. 1878 Uber den verlauf von Funkenwellen in der Ebene und im Raume. Sitz.ber. Akad. Wiss. Wien 78, 819–838.Google Scholar
Mouton, C. A. & Hornung, H. G. 2007 Mach stem height and growth rate predictions. AIAA J. 45, 1977–1987.CrossRefGoogle Scholar
Mouton, C. A. & Hornung, H. G. 2008 Experiments on the mechanism of inducing transition between regular and Mach reflection Mach stem height and growth rate predictions. Phys. Fluids 20, 126103.Google Scholar
Rosciszewski, J. 1960 Calculations of the motion of non-uniform shock waves. J. Fluid Mech. 8, 337–367.Google Scholar
Tan, L. H., Ren, Y. X. & Wu, Z. N. 2006 Analytical and numerical study of the near flow field and shape of the Mach stem in steady flows. J. Fluid Mech. 546, 341–362.CrossRefGoogle Scholar
Teshukov, V. M. 1989 On stability of RR of shock waves. Zh. Prikl. Mekh. Tekh. Fiz. 2, 26–33.Google Scholar
von Neumann, J.1943 Oblique reflection of shock. Explos. Res. Rep. 12. Navy Department, Bureau of Ordinance, Washington, DC.Google Scholar
von Neumann, J.1945 Refraction, intersection and reflection of shock waves. NAVORD Rep. 203–245. Navy Department, Bureau of Ordinance, Washington, DC.Google Scholar
Vuillon, J., Zeitoun, D. & Ben-Dor, G. 1995 Reconstruction of oblique shock wave reflection in steady flows. Part 2. Numerical investigation. J. Fluid Mech. 301, 37–50.Google Scholar