Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-24T20:14:56.264Z Has data issue: false hasContentIssue false

Instability evolution of a shock-accelerated thin heavy fluid layer in cylindrical geometry

Published online by Cambridge University Press:  10 August 2023

Ming Yuan
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, PR China
Zhiye Zhao*
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, PR China
Luoqin Liu
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, PR China
Pei Wang
Institute of Applied Physics and Computational Mathematics, Beijing 100094, PR China
Nan-Sheng Liu
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, PR China
Xi-Yun Lu
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, PR China
Email address for correspondence:


Instability evolutions of shock-accelerated thin cylindrical SF$_6$ layers surrounded by air with initial perturbations imposed only at the outer interface (i.e. the ‘Outer’ case) or at the inner interface (i.e. the ‘Inner’ case) are numerically and theoretically investigated. It is found that the instability evolution of a thin cylindrical heavy fluid layer not only involves the effects of Richtmyer–Meshkov instability, Rayleigh–Taylor stability/instability and compressibility coupled with the Bell–Plesset effect, which determine the instability evolution of the single cylindrical interface, but also strongly depends on the waves reverberated inside the layer, thin-shell correction and interface coupling effect. Specifically, the rarefaction wave inside the thin fluid layer accelerates the outer interface inward and induces the decompression effect for both the Outer and Inner cases, and the compression wave inside the fluid layer accelerates the inner interface inward and causes the decompression effect for the Outer case and compression effect for the Inner case. It is noted that the compressible Bell model excluding the compression/decompression effect of waves, thin-shell correction and interface coupling effect deviates significantly from the perturbation growth. To this end, an improved compressible Bell model is proposed, including three new terms to quantify the compression/decompression effect of waves, thin-shell correction and interface coupling effect, respectively. This improved model is verified by numerical results and successfully characterizes various effects that contribute to the perturbation growth of a shock-accelerated thin heavy fluid layer.

JFM Papers
© The Author(s), 2023. 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.)


Amendt, P., Colvin, J.D., Tipton, R.E., Hinkel, D.E., Edwards, M.J., Landen, O.L., Ramshaw, J.D., Suter, L.J., Varnum, W.S. & Watt, R.G. 2002 Indirect-drive noncryogenic double-shell ignition targets for the National Ignition Facility: design and analysis. Phys. Plasmas 9 (5), 22212233.CrossRefGoogle Scholar
Arnett, W.D., Bahcall, J.N., Kirshner, R.P. & Woosley, S.E. 1989 Supernova 1987A. Annu. Rev. Astron. Astrophys. 27 (1), 629700.CrossRefGoogle Scholar
Balakrishnan, K. & Menon, S. 2010 On turbulent chemical explosions into dilute aluminum particle clouds. Combust. Theor. Model. 14 (4), 583617.CrossRefGoogle Scholar
Bell, G.I. 1951 Taylor instability on cylinders and spheres in the small amplitude approximation. Report No. LA-1321, LANL.Google Scholar
Bell, J.B., Day, M.S., Rendleman, C.A., Woosley, S.E. & Zingale, M. 2004 Direct numerical simulations of type Ia supernovae flames. II. The Rayleigh–Taylor instability. Astrophys. J. 608 (2), 883.CrossRefGoogle Scholar
Betti, R. & Hurricane, O.A. 2016 Inertial-confinement fusion with lasers. Nat. Phys. 12 (5), 435448.CrossRefGoogle Scholar
Buttler, W.T., et al. 2012 Unstable Richtmyer–Meshkov growth of solid and liquid metals in vacuum. J. Fluid Mech. 703, 6084.CrossRefGoogle Scholar
Chertkov, M., Lebedev, V. & Vladimirova, N. 2009 Reactive Rayleigh–Taylor turbulence. J. Fluid Mech. 633, 116.CrossRefGoogle Scholar
Chisnell, R.F. 1998 An analytic description of converging shock waves. J. Fluid Mech. 354, 357375.CrossRefGoogle Scholar
Ding, J., Li, J., Sun, R., Zhai, Z. & Luo, X. 2019 Convergent Richtmyer–Meshkov instability of a heavy gas layer with perturbed outer interface. J. Fluid Mech. 878, 277291.CrossRefGoogle Scholar
Ding, J., Si, T., Yang, J., Lu, X.-Y., Zhai, Z. & Luo, X. 2017 Measurement of a Richtmyer–Meshkov instability at an air-SF$_{6}$ interface in a semiannular shock tube. Phys. Rev. Lett. 119 (1), 014501.CrossRefGoogle Scholar
Epstein, R. 2004 On the Bell–Plesset effects: the effects of uniform compression and geometrical convergence on the classical Rayleigh–Taylor instability. Phys. Plasmas 11 (11), 51145124.CrossRefGoogle Scholar
Fu, C.-Q., Zhao, Z., Xu, X., Wang, P., Liu, N.-S., Wan, Z.-H. & Lu, X.-Y. 2022 Nonlinear saturation of bubble evolution in a two-dimensional single-mode stratified compressible Rayleigh–Taylor instability. Phys. Rev. Fluids 7 (2), 023902.CrossRefGoogle Scholar
Ge, J., Zhang, X.-T., Li, H.-F. & Tian, B.-L. 2020 Late-time turbulent mixing induced by multimode Richtmyer–Meshkov instability in cylindrical geometry. Phys. Fluids 32 (12), 124116.CrossRefGoogle Scholar
Guo, H.-Y., Wang, L.-F., Ye, W.-H., Wu, J.-F. & Zhang, W.-Y. 2017 Rayleigh–Taylor instability of multi-fluid layers in cylindrical geometry. Chin. Phys. B 26 (12), 125202.CrossRefGoogle Scholar
Hester, J.J. 2008 The Crab Nebula: an astrophysical chimera. Annu. Rev. Astron. Astrophys. 46 (1), 127155.CrossRefGoogle Scholar
Houseman, G.A. & Molnar, P. 1997 Gravitational (Rayleigh–Taylor) instability of a layer with non-linear viscosity and convective thinning of continental lithosphere. Geophys. J. Intl 128 (1), 125150.CrossRefGoogle Scholar
Hsing, W.W. & Hoffman, N.M. 1997 Measurement of feedthrough and instability growth in radiation-driven cylindrical implosions. Phys. Rev. Lett. 78 (20), 38763879.CrossRefGoogle Scholar
Isobe, H., Miyagoshi, T., Shibata, K. & Yokoyama, T. 2005 Filamentary structure on the Sun from the magnetic Rayleigh–Taylor instability. Nature 434 (7032), 478481.CrossRefGoogle ScholarPubMed
Jacobs, J.W., Jenkins, D.G., Klein, D.L. & Benjamin, R.F. 1995 Nonlinear growth of the shock-accelerated instability of a thin fluid layer. J. Fluid Mech. 295, 2342.CrossRefGoogle Scholar
Jacobs, J.W., Klein, D.L., Jenkins, D.G. & Benjamin, R.F. 1993 Instability growth patterns of a shock-accelerated thin fluid layer. Phys. Rev. Lett. 70 (5), 583586.CrossRefGoogle ScholarPubMed
Kane, J., Drake, R.P. & Remington, B.A. 1999 An evaluation of the Richtmyer–Meshkov instability in supernova remnant formation. Astrophys. J. 511 (1), 335340.CrossRefGoogle Scholar
Kishony, R. & Shvarts, D. 2001 Ignition condition and gain prediction for perturbed inertial confinement fusion targets. Phys. Plasmas 8 (11), 49254936.CrossRefGoogle Scholar
Lei, F., Ding, J., Si, T., Zhai, Z. & Luo, X. 2017 Experimental study on a sinusoidal air/SF$_{6}$ interface accelerated by a cylindrically converging shock. J. Fluid Mech. 826, 819829.CrossRefGoogle Scholar
Li, J., Ding, J., Luo, X. & Zou, L. 2022 Instability of a heavy gas layer induced by a cylindrical convergent shock. Phys. Fluids 34 (4), 042123.CrossRefGoogle Scholar
Li, X., Fu, Y., Yu, C. & Li, L. 2021 Statistical characteristics of turbulent mixing in spherical and cylindrical converging Richtmyer–Meshkov instabilities. J. Fluid Mech. 928, A10.CrossRefGoogle Scholar
Liang, Y., Liu, L., Zhai, Z., Si, T. & Wen, C.-Y. 2020 Evolution of shock-accelerated heavy gas layer. J. Fluid Mech. 886, A7.CrossRefGoogle Scholar
Liang, Y. & Luo, X. 2021 On shock-induced heavy-fluid-layer evolution. J. Fluid Mech. 920, A13.CrossRefGoogle Scholar
Lindl, J., Landen, O., Edwards, J., Moses, E. & N.I.C. team 2014 Review of the national ignition campaign 2009–2012. Phys. Plasmas 21 (2), 020501.CrossRefGoogle Scholar
Lombardini, M., Pullin, D.I. & Meiron, D.I. 2014 Turbulent mixing driven by spherical implosions. Part I. Flow description and mixing-layer growth. J. Fluid Mech. 748, 85112.CrossRefGoogle Scholar
Luo, X., Li, M., Ding, J., Zhai, Z. & Si, T. 2019 Nonlinear behaviour of convergent Richtmyer–Meshkov instability. J. Fluid Mech. 877, 130141.CrossRefGoogle Scholar
Meshkov, E.E. 1969 Instability of the interface of two gases accelerated by a shock wave. Fluid Dyn. 4 (5), 101104.CrossRefGoogle Scholar
Mikaelian, K.O. 1982 Normal modes and symmetries of the Rayleigh–Taylor instability in stratified fluids. Phys. Rev. Lett. 48 (19), 13651368.CrossRefGoogle Scholar
Mikaelian, K.O. 1985 Richtmyer–Meshkov instabilities in stratified fluids. Phys. Rev. A 31 (1), 410419.CrossRefGoogle ScholarPubMed
Mikaelian, K.O. 1990 Rayleigh–Taylor and Richtmyer–Meshkov instabilities and mixing in stratified spherical shells. Phys. Rev. A 42 (6), 34003420.CrossRefGoogle ScholarPubMed
Mikaelian, K.O. 2005 Rayleigh–Taylor and Richtmyer–Meshkov instabilities and mixing in stratified cylindrical shells. Phys. Fluids 17 (9), 094105.CrossRefGoogle Scholar
Ott, E. 1972 Nonlinear evolution of the Rayleigh–Taylor instability of a thin layer. Phys. Rev. Lett. 29 (21), 14291432.CrossRefGoogle Scholar
Plesset, M.S. 1954 On the stability of fluid flows with spherical symmetry. J. Appl. Phys. 25 (1), 9698.CrossRefGoogle Scholar
Rayleigh, Lord 1883 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. s1–14 (1), 170177.Google Scholar
Richtmyer, R.D. 1960 Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Maths 13 (2), 297319.CrossRefGoogle Scholar
Schwendeman, D.W. & Whitham, G.B. 1987 On converging shock waves. Proc. R. Soc. Lond. A 413 (1845), 297311.Google Scholar
Sun, R., Ding, J., Zhai, Z., Si, T. & Luo, X. 2020 Convergent Richtmyer–Meshkov instability of heavy gas layer with perturbed inner surface. J. Fluid Mech. 902, A3.CrossRefGoogle Scholar
Taylor, G.I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A 201 (1065), 192196.Google Scholar
Walchli, B. & Thornber, B. 2017 Reynolds number effects on the single-mode Richtmyer–Meshkov instability. Phys. Rev. E 95 (1), 013104.CrossRefGoogle ScholarPubMed
Weir, S.T., Chandler, E.A. & Goodwin, B.T. 1998 Rayleigh–Taylor instability experiments examining feedthrough growth in an incompressible, convergent geometry. Phys. Rev. Lett. 80 (17), 37633766.CrossRefGoogle Scholar
Wu, J., Liu, H. & Xiao, Z. 2021 Refined modelling of the single-mode cylindrical Richtmyer–Meshkov instability. J. Fluid Mech. 908, A9.CrossRefGoogle Scholar
Yan, Z., Fu, Y., Wang, L., Yu, C. & Li, X. 2022 Effect of chemical reaction on mixing transition and turbulent statistics of cylindrical Richtmyer–Meshkov instability. J. Fluid Mech. 941, A55.CrossRefGoogle Scholar
Zhang, S., Liu, H., Kang, W., Xiao, Z., Tao, J., Zhang, P., Zhang, W. & He, X.-T. 2020 Coupling effects and thin-shell corrections for surface instabilities of cylindrical fluid shells. Phys. Rev. E 101 (2), 023108.CrossRefGoogle ScholarPubMed
Zhao, Z., Wang, P., Liu, N.-S. & Lu, X.-Y. 2020 Analytical model of nonlinear evolution of single-mode Rayleigh–Taylor instability in cylindrical geometry. J. Fluid Mech. 900, A24.CrossRefGoogle Scholar
Zhao, Z., Wang, P., Liu, N.-S. & Lu, X.-Y. 2021 Scaling law of mixing layer in cylindrical Rayleigh–Taylor turbulence. Phys. Rev. E 104 (5), 055104.CrossRefGoogle ScholarPubMed
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
Zhou, Y., et al. 2021 Rayleigh–Taylor and Richtmyer–Meshkov instabilities: a journey through scales. Physica D 423, 132838.CrossRefGoogle Scholar
Zhou, Y., Clark, T.T., Clark, D.S., Gail Glendinning, S., Aaron Skinner, M., Huntington, C.M., Hurricane, O.A., Dimits, A.M. & Remington, B.A. 2019 Turbulent mixing and transition criteria of flows induced by hydrodynamic instabilities. Phys. Plasmas 26 (8), 080901.CrossRefGoogle Scholar
Zou, L., Al-Marouf, M., Cheng, W., Samtaney, R., Ding, J. & Luo, X. 2019 Richtmyer–Meshkov instability of an unperturbed interface subjected to a diffracted convergent shock. J. Fluid Mech. 879, 448467.CrossRefGoogle Scholar