Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-05-24T20:44:27.234Z Has data issue: false hasContentIssue false

Refraction of a triple-shock configuration at planar fast–slow gas interfaces

Published online by Cambridge University Press:  04 April 2024

Enlai Zhang
National Key Laboratory of Shock Wave and Detonation Physics, Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang 621900, PR China
Shenfei Liao
National Key Laboratory of Shock Wave and Detonation Physics, Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang 621900, PR China
Liyong Zou*
National Key Laboratory of Shock Wave and Detonation Physics, Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang 621900, PR China
Zhigang Zhai*
Advanced Propulsion Laboratory, Department of Modern Mechanics, University of Science and Technology of China, Hefei 230026, PR China
Jinhong Liu
National Key Laboratory of Shock Wave and Detonation Physics, Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang 621900, PR China
Xinzhu Li
National Key Laboratory of Shock Wave and Detonation Physics, Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang 621900, PR China
Email addresses for correspondence:,
Email addresses for correspondence:,


This paper characterizes the refraction of a triple-shock configuration at planar fast–slow gas interfaces. The primary objective is to reveal the wave configurations and elucidate the mechanisms governing circulation deposition and velocity perturbation on the interface caused by triple-shock refraction. The incident triple-shock configuration is generated by diffracting a planar shock around a rigid cylinder, and four interfaces with various $Z_{{r}}$ (i.e. acoustic impedance ratio across the interface) are considered. An analytical model describing the triple-shock refraction is developed, which accurately predicts both the wave configurations as well as circulation deposition and velocity perturbation. Depending on $Z_{{r}}$, three distinct patterns of transmitted waves can be anticipated: a triple-shock configuration; a four-shock configuration; a four-wave configuration. The underlying mechanism for the formation of these wave configurations is elucidated through shock polar analysis. A novel physical insight into the contribution of triple-shock refraction to the interface perturbation growth is provided. The results indicate that the reflected shock in an incident triple-shock configuration makes significant negative contribution to both circulation deposition and velocity perturbation. This investigation elucidates the underlying mechanism responsible for the relatively insignificant contribution of baroclinic circulation to the Richtmyer–Meshkov-like instability induced by a non-uniform shock, and provides an explanation for the decrease in growth rate of interface perturbation amplitude with increasing Atwood number.

JFM Papers
© The Author(s), 2024. Published by Cambridge University Press

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.)


Abd-El-Fattah, A.M. & Henderson, L.F. 1978 a Shock waves at a fast-slow gas interface. J. Fluid Mech. 86 (1), 1532.CrossRefGoogle Scholar
Abd-El-Fattah, A.M. & Henderson, L.F. 1978 b Shock waves at a slow-fast gas interface. J. Fluid Mech. 89 (1), 7995.CrossRefGoogle Scholar
Abd-El-Fattah, A.M., Henderson, L.F. & Lozzi, A. 1976 Precursor shock waves at a slow-fast gas interface. J. Fluid Mech. 76 (1), 157176.CrossRefGoogle Scholar
Anderson, J.D. Jr. 2001 Fundamentals of Aerodynamics, 3rd edn. McGraw-Hill.Google Scholar
Arnett, W.D., Bahcall, J.N. & Kirshner, R.P. 1989 Supernova 1987A. Annu. Rev. Astron. Astrophys. 27, 629700.CrossRefGoogle Scholar
Bai, C. & Wu, Z. 2022 Type IV shock interaction with a two-branch structured transonic jet. J. Fluid Mech. 941, A45.CrossRefGoogle Scholar
Ben-Dor, G. 2007 Shock Wave Reflection Phenomena, 2nd edn. Springer.Google Scholar
Betti, R. & Hurricane, O.A. 2016 Inertial-confinement fusion with lasers. Nat. Phys. 12, 435448.CrossRefGoogle Scholar
Brouillette, M. 2002 The Richtmyer–Meshkov instability. Annu. Rev. Fluid Mech. 34, 445468.CrossRefGoogle Scholar
Brown, J.L. & Ravichandran, G. 2014 Analysis of oblique shock waves in solids using shock polars. Shock Waves 24, 403413.CrossRefGoogle Scholar
Bryson, A.E. & Gross, R.W.F. 1961 Diffraction of strong shocks by cones, cylinders, and spheres. J. Fluid Mech. 10, 116.CrossRefGoogle Scholar
Chaudhuri, A., Hadjadj, A. & Chinnayya, A. 2011 On the use of immersed boundary methods for shock/obstacle interactions. J. Comput. Phys. 230, 17311748.CrossRefGoogle Scholar
Colella, P. & Henderson, L.F. 1990 The von Neumann paradox for the diffraction of weak shock waves. J. Fluid Mech. 213, 7194.CrossRefGoogle Scholar
Gardner, J.H., Book, D.L. & Bernstein, I.B. 1982 Stability of imploding shocks in the CCW approximation. J. Fluid Mech. 114, 4158.CrossRefGoogle Scholar
Gounko, Y.P. 2017 Patterns of steady axisymmetric supersonic compression flows with a Mach disk. Shock Waves 27, 495506.CrossRefGoogle Scholar
de Gouvello, Y., Dutreuilh, M., Gallier, S., Melguizo-Gavilanes, J. & Mevel, R. 2021 Shock wave refraction patterns at a slow-fast gas-gas interface at superknock relevant conditions. Phys. Fluids 33, 116101.CrossRefGoogle Scholar
Guderley, K.G. 1947 Considerations on the structure of mixed subsonic-supersonic flow patterns. Tech. Rep. Air Materiel Command Technical Report No. F-TR-2168-ND, ATI No. 22780. GS-AAF-Wright Field No. 39. U.S. Wright–Patterson Air Force Base.Google Scholar
Hawley, J.F. & Zabusky, N.J. 1989 Vortex paradigm for shock-accelerated density-stratified interfaces. Phys. Rev. Lett. 63 (12), 12411244.CrossRefGoogle ScholarPubMed
He, Y., Peng, N., Li, H., Tian, B. & Yang, Y. 2023 Formation of the cavity on a planar interface subjected to a perturbed shock wave. Phys. Rev. Fluids 8, 063402.CrossRefGoogle Scholar
Henderson, L.F. 1966 The refraction of a plane shock wave at a gas interface. J. Fluid Mech. 26, 607637.CrossRefGoogle Scholar
Henderson, L.F. 1989 On the refraction of shock waves. J. Fluid Mech. 198, 365386.CrossRefGoogle Scholar
Henderson, L.F. 2014 The refraction of shock pairs. Shock Waves 24, 553572.CrossRefGoogle Scholar
Henderson, L.F. & Siegenthaler, A. 1980 Experiments on the diffraction of weak blast waves: the von Neumann paradox. Proc. R. Soc. Lond. A 369, 537555.Google Scholar
Hong, L., Bin, Y., Bin, Z. & Yang, X. 2022 On mixing enhancement by secondary baroclinic vorticity in a shock-bubble interaction. J. Fluid Mech. 931, A17.Google Scholar
Ishizaki, R. & Nishihara, K. 1997 Propagation of a rippled shock wave driven by nonuniform laser ablation. Phys. Rev. Lett. 78 (10), 19201923.CrossRefGoogle Scholar
Ishizaki, R., Nishihara, K., Sakagami, H. & Ueshima, Y. 1996 Instability of a contact surface driven by a nonuniform shock wave. Phys. Rev. E 53 (6), R5592R5595.CrossRefGoogle ScholarPubMed
Jahn, R.G. 1956 The refraction of shock waves at a gaseous interface. J. Fluid Mech. 1 (5), 457489.CrossRefGoogle Scholar
Ji, J., Li, Z., Zhang, E., Si, D. & Yang, J. 2022 Intensification of non-uniformity in convergent near-conical hypersonic flow. J. Fluid Mech. 931, A8.CrossRefGoogle Scholar
Jones, M.A. & Jacobs, J.W. 1997 A membraneless experiment for the study of Richtmyer–Meshkov instability of a shock-accelerated gas interface. Phys. Fluids 9, 30783085.CrossRefGoogle Scholar
Lau-Chapdelaine, S.S.M. & Radulescu, M.I. 2016 Viscous solution of the triple-shock reflection problem. Shock Waves 26, 551560.CrossRefGoogle Scholar
Li, D., Guan, B. & Wang, G. 2022 b Effects of interface diffusion and shock strength on shock-accelerated $\textrm {SF}_6$ cylinder. Phys. Fluids 34, 076109.CrossRefGoogle Scholar
Li, D., Wang, G. & Guan, B. 2019 On the circulation prediction of shock-accelerated elliptical heavy gas cylinders. Phys. Fluids 31, 056104.CrossRefGoogle Scholar
Li, J., Ding, J., Luo, X. & Zou, L. 2022 a Instability of a heavy gas layer induced by a cylindrical convergent shock. Phys. Fluids 34, 042123.CrossRefGoogle Scholar
Li, Y., Samtaney, R. & Wheatley, V. 2018 The Richtmyer–Meshkov instability of a double-layer interface in convergent geometry with magnetohydrodynamics. Matt. Radiat. Extremes 3, 207218.CrossRefGoogle Scholar
Liang, Y., Ding, J., Zhai, Z., Si, T. & Luo, X. 2017 Interaction of cylindrically converging diffracted shock with uniform interface. Phys. Fluids 29, 086101.CrossRefGoogle Scholar
Liao, S., Zhang, W., Chen, H., Zou, L., Liu, J. & Zheng, X. 2019 Atwood number effects on the instability of a uniform interface driven by a perturbed shock wave. Phys. Rev. E 99, 013103.CrossRefGoogle ScholarPubMed
Lindl, J.D., Amendt, P., Berger, R.L., Glendinning, S.G. & Glenzer, S.H. 2004 The physics basis for ignition using indirect-drive targets on the National Ignition Facility. Phys. Plasmas 11 (2), 339491.CrossRefGoogle Scholar
Liu, H., Yu, B., Chen, H., Zhang, B. & Liu, H. 2020 Contribution of viscosity to the circulation deposition in the Richtmyer–Meshkov instability. J. Fluid Mech. 895, A10.CrossRefGoogle Scholar
Lodato, G., Vervisch, L. & Clavin, P. 2016 Direct numerical simulation of shock wavy-wall interaction: analysis of cellular shock structures and flow patterns. J. Fluid Mech. 789, 221258.CrossRefGoogle Scholar
Meshkov, E.E. 1969 Instability of a shock wave accelerated interface between two gases. Fluid Dyn. 4, 101104.CrossRefGoogle Scholar
Mostert, W., Pullin, D.I., Samtaney, R. & Wheatley, V. 2018 a Singularity formation on perturbed planar shock waves. J. Fluid Mech. 846, 536562.CrossRefGoogle Scholar
Mostert, W., Pullin, D.I., Samtaney, R. & Wheatley, V. 2018 b Spontaneous singularity formation in converging cylindrical shock waves. Phys. Rev. Fluids 3, 071401.CrossRefGoogle Scholar
Nishihara, K., Wouchuk, J.G., Matsuoka, C., Ishizaki, R. & Zhakhovsky, V.V. 2010 Richtmyer–Meshkov instability: theory of linear and nonlinear evolution. Phil. Trans. R. Soc. A 368, 17691807.CrossRefGoogle ScholarPubMed
Nourgaliev, R.R., Sushchikh, S.Y., Dinh, T.N. & Theofanous, T.G. 2005 Shock wave refraction patterns at interfaces. Intl J. Multiphase Flow 31, 969995.CrossRefGoogle Scholar
Olejniczak, J., Wright, M.J. & Candler, G.V. 1997 Numerical study of inviscid shock interactions on double-wedge geometries. J. Fluid Mech. 352, 125.CrossRefGoogle Scholar
Pellone, S., Stefano, C.A.D., Rasmus, A.M., Kuranz, C.C. & Johnsen, E. 2021 Vortex-sheet modeling of hydrodynamic instabilities produced by an oblique shock interacting with a perturbed interface in the HED regime. Phys. Plasmas 28, 022303.CrossRefGoogle Scholar
Peng, G., Zabusky, N.J. & Zhang, S. 2003 Vortex-accelerated secondary baroclinic vorticity deposition and late-intermediate time dynamics of a two-dimensional Richtmyer–Meshkov interface. Phys. Fluids 15 (12), 37303744.CrossRefGoogle Scholar
Peng, N., Yang, Y., Wu, J. & Xiao, Z. 2021 Mechanism and modelling of the secondary baroclinic vorticity in the Richtmyer–Meshkov instability. J. Fluid Mech. 911, A56.CrossRefGoogle Scholar
Polachek, H. & Seeger, R.J. 1951 On shock-wave phenomena-refraction of shock waves at a gaseous interface. Phys. Rev. 84, 922929.CrossRefGoogle Scholar
Ren, Z., Wang, B., Xiang, G., Zhao, D. & Zheng, L. 2019 Supersonic spray combustion subject to scramjets: progress and challenges. Prog. Aeronaut. Sci. 105, 4059.CrossRefGoogle Scholar
Richtmyer, R.D. 1960 Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Maths 13, 297319.CrossRefGoogle Scholar
Samtaney, R. & Zabusky, N.J. 1994 Circulation deposition on shock-accelerated planar and curved density-stratified interfaces: models and scaling laws. J. Fluid Mech. 269, 4578.CrossRefGoogle Scholar
Skews, B.W. & Ashworth, J.T. 2005 The physical nature of weak shock wave reflection. J. Fluid Mech. 542, 105114.CrossRefGoogle Scholar
Smalyuk, V.A. et al. 2020 Review of hydrodynamic instability experiments in inertially confined fusion implosions on National Ignition Facility. Plasma Phys. Control Fusion 62, 014007.CrossRefGoogle Scholar
Sun, M. & Takayama, K. 1999 Conservative smoothing on an adaptive quadrilateral grid. J. Comput. Phys. 150, 143180.CrossRefGoogle Scholar
Sun, M. & Takayama, K. 2003 Vorticity production in shock diffraction. J. Fluid Mech. 478, 237256.CrossRefGoogle Scholar
Taub, A.H. 1947 Refraction of plane shock waves. Phys. Rev. 72, 5160.CrossRefGoogle Scholar
Thomas, V.A. & Kares, R.J. 2012 Drive asymmetry and the origin of turbulence in an ICF implosion. Phys. Rev. Lett. 109, 075004.CrossRefGoogle Scholar
Toro, E.F. 2009 The MUSCL-Hancock method. In Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edn, pp. 429–432. Springer.CrossRefGoogle Scholar
Vasilev, E.I. 1999 Four-wave scheme of weak Mach shock wave interaction under von Neumann paradox conditions. Fluid Dyn. 34, 421427.Google Scholar
Vasilev, E.I., Elperin, T. & Ben-dor, G. 2008 Analytical reconsideration of the von Neumann paradox in the reflection of a shock wave over a wedge. Phys. Fluids 20, 046101.CrossRefGoogle Scholar
Velikovich, A.L. 2000 Richtmyer–Meshkov-like instabilities and early-time perturbation growth in laser targets and Z-Pinch loads. Phys. Plasmas 7 (5), 16621671.CrossRefGoogle Scholar
Velikovich, A.L., Schmitt, A.J., Zulick, C., Aglitskiy, Y., Karasik, M., Obenschain, S.P., Wouchuk, J.G. & Cobos Campos, F. 2020 Multi-mode hydrodynamic evolution of perturbations seeded by isolated surface defects. Phys. Plasmas 27, 102706.CrossRefGoogle Scholar
Von Neumann, J. 1943 Oblique reflection of shock. Tech. Rep. Explos. Res. Rep. 12. Navy Department, Bureau of Ordinance, Washington, DC.Google Scholar
Von Neumann, J. 1945 Refraction, intersection and reflection of shock waves. Tech. Rep. NAVORD Rep. 203–245. Navy Department, Bureau of Ordinance, Washington, DC.Google Scholar
Wan, Q., Jeon, H., Deiterding, R. & Eliasson, V. 2017 Numerical and experimental investigation of oblique shock wave reflection off a water wedge. J. Fluid Mech. 826, 732758.CrossRefGoogle Scholar
Wang, H. & Zhai, Z. 2020 On regular reflection to Mach reflection transition in inviscid flow for shock reflection ona convex or straight wedge. J. Fluid Mech. 884, A27.CrossRefGoogle Scholar
Wang, H., Zhai, Z. & Luo, X. 2022 Prediction of triple point trajectory on two-dimensional unsteady shock reflection over single surfaces. J. Fluid Mech. 947, A42.CrossRefGoogle Scholar
White, D.R. 1952 An experimental survey of the Mach reflection of shock waves. PhD thesis, Princeton University.Google Scholar
Xiang, G. & Wang, B. 2019 Theoretical and numerical studies on shock reflection at water/air two-phase interface: fast-slow case. Intl J. Multiphase Flow 114, 219228.CrossRefGoogle Scholar
Yang, J., Kubota, T. & Zukoski, E.E. 1994 A model for characterization of a vortex pair formed by shock passage over a light-gas inhomogeneity. J. Fluid Mech. 258, 217244.CrossRefGoogle Scholar
Yang, X., Chern, I., Zabusky, N.J., Samtaney, R. & Hawley, J.F. 1992 Vorticity generation and evolution in shock-accelerated density-stratified interfaces. Phys. Fluids A 4, 15311540.CrossRefGoogle Scholar
Yang, Y., Li, S. & Wu, Z. 2013 Shock reflection in the presence of an upstream expansion wave and a downstream shock wave. J. Fluid Mech. 735, 6190.Google Scholar
Zabusky, N.J. 1999 Vortex paradigm for accelerated inhomogeneous flows: visiometrics for the Rayleigh–Taylor and Richtmyer–Meshkov environments. Annu. Rev. Fluid Mech. 31, 495536.CrossRefGoogle Scholar
Zaslavsky, B.I. & Safarov, P.A. 1975 Mach reflection of weak shock waves from a rigid wall. J. Appl. Mech. Tech. Phys. 14 (5), 624629.CrossRefGoogle Scholar
Zhai, Z., Li, W., Si, T., Luo, X., Yang, J. & Lu, X. 2017 Refraction of cylindrical converging shock wave at an air/helium gaseous interface. Phys. Fluids 29, 016102.CrossRefGoogle Scholar
Zhai, Z., Liang, Y., Liu, L., Ding, J., Luo, X. & Zou, L. 2018 Interaction of rippled shock wave with fast-slow interface. Phys. Fluids 30, 046104.CrossRefGoogle Scholar
Zhai, Z., Si, T., Luo, X. & Yang, J. 2011 On the evolution of spherical gas interfaces accelerated by a planar shock wave. Phys. Fluids 23, 084104.CrossRefGoogle Scholar
Zhang, E., Li, Z., Ji, J., Si, D. & Yang, J. 2021 Converging near-elliptic shock waves. J. Fluid Mech. 909, A2.CrossRefGoogle Scholar
Zhou, Y. 2017 a Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. I. Phys. Rep. 720–722, 1136.Google Scholar
Zhou, Y. 2017 b Rayleigh–Taylor and Richtmyer–Meshkov instability induced flow, turbulence, and mixing. II. Phys. Rep. 723–725, 1160.Google Scholar
Zou, L., Al-Marouf, M., Cheng, W., Samtaney, R. & Luo, X. 2019 Richtmyer–Meshkov instability of an unperturbed interface subjected to a diffracted convergent shock. J. Fluid Mech. 879, 448467.CrossRefGoogle Scholar
Zou, L., Liu, J., Liao, S., Zheng, X., Zhai, Z. & Luo, X. 2017 Richtmyer–Meshkov instability of a flat interface subjected to a rippled shock wave. Phys. Rev. E 95, 013107.CrossRefGoogle ScholarPubMed