Hostname: page-component-76d6cb85b7-rxvq6 Total loading time: 0 Render date: 2026-07-11T04:52:28.279Z Has data issue: false hasContentIssue false
  • English
  • Français

Perturbation theory and numerical modelling of weakly and moderately nonlinear dynamics of the incompressible Richtmyer–Meshkov instability

Published online by Cambridge University Press:  23 June 2014

A. L. Velikovich ,
M. Herrmann  and
S. I. Abarzhi
A. L. Velikovich*
Affiliation:
Plasma Physics Division, Naval Research Laboratory, Washington, DC 20375, USA
M. Herrmann
Affiliation:
Arizona State University, Tempe, AZ 85287, USA
S. I. Abarzhi
Affiliation:
Carnegie Mellon University, Pittsburgh, PA 15213, USA, and Carnegie Mellon University Qatar, Education City, Doha, Qatar
*
†Email address for correspondence: sasha.velikovich@nrl.navy.mil
Get access Rights & Permissions [Opens in a new window]

Abstract

A study of incompressible two-dimensional (2D) Richtmyer–Meshkov instability (RMI) by means of high-order perturbation theory and numerical simulations is reported. Nonlinear corrections to Richtmyer’s impulsive formula for the RMI bubble and spike growth rates have been calculated for arbitrary Atwood number and an explicit formula has been obtained for it in the Boussinesq limit. Conditions for early-time acceleration and deceleration of the bubble and the spike have been elucidated. Theoretical time histories of the interface curvature at the bubble and spike tip and the profiles of vertical and horizontal velocities have been calculated and favourably compared to simulation results. In our simulations we have solved 2D unsteady Navier–Stokes equations for immiscible incompressible fluids using the finite volume fractional step flow solver NGA developed by Desjardins et al. (J. Comput. Phys., vol. 227, 2008, pp. 7125–7159) coupled to the level set based interface solver LIT (Herrmann, J. Comput. Phys., vol. 227, 2008, pp. 2674–2706). We study the impact of small amounts of viscosity on the flow dynamics and compare simulation results to theory to discuss the influence of the theory’s ideal inviscid flow assumption.

JFM classification

Interfacial Flows (free surface): Fingering instability Instability: Nonlinear instability Compressible Flows: Shock waves

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 751 , 25 July 2014 , pp. 432 - 479
Copyright
© Cambridge University Press 2014. This is a work of the U.S. Government and is not subject to copyright protection in the United States. 

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.)

Article purchase

Temporarily unavailable

Footnotes

‡

The original version of this article was published with the incorrect affiliation for S. I. Abarzhi. A notice detailing this has been published online and in print, and the error rectified in the online PDF and HTML copies.

References

Abarzhi, S. I. 2002 A new type of the evolution of the bubble front in the Richtmyer–Meshkov instability. Phys. Lett. A 294 (2), 95–100.CrossRefGoogle Scholar
Abarzhi, S. I. 2008 Review of nonlinear dynamics of the unstable fluid interface: conservation laws and group theory. Phys. Scr. T132, 014012.Google Scholar
Abarzhi, S. I. 2010 Review of theoretical modelling approaches of Rayleigh–Taylor instabilities and turbulent mixing. Phil. Trans. R. Soc. Lond. A 368 (1916), 1809–1828.Google ScholarPubMed
Abarzhi, S. I., Gauthier, S. & Sreenivasan, K. R. 2013 Turbulent mixing and beyond: non-equilibrium processes from atomistic to astrophysical scales. Phil. Trans. R. Soc. Lond. A 371, 20130268.Google Scholar
Abarzhi, S. I., Nishihara, K. & Glimm, J. 2003 Rayleigh–Taylor and Richtmyer–Meshkov instabilities for fluids with a finite density ratio. Phys. Lett. A 317 (5–6), 470–476.Google Scholar
Aglitskiy, Y., Karasik, M., Velikovich, A. L., Serlin, V., Weaver, J. L., Kessler, T. J., Nikitin, S. P., Schmitt, A. J., Obenschain, S. P., Metzler, N. & Oh, J. 2012 Observed transition from Richtmyer–Meshkov jet formation through feedout oscillations to Rayleigh–Taylor instability in a laser target. Phys. Plasmas 10 (10), 102707.Google Scholar
Aglitskiy, Y., Velikovich, A. L., Karasik, M., Metzler, N., Zalesak, S. T., Schmitt, A. J., Phillips, L., Gardner, J. H., Serlin, V., Weaver, J. L. & Obenschain, S. P. 2010 Basic hydrodynamics of Richtmyer–Meshkov-type growth and oscillations in the inertial confinement fusion-relevant conditions. Phil. Trans. R. Soc. Lond. A 368 (1916), 1739–1768.Google ScholarPubMed
Alon, U., Hecht, J., Mukamel, D. & Shvarts, D. 1994 Scale invariant mixing rates of hydrodynamically unstable interfaces. Phys. Rev. Lett. 72 (18), 2867–2870.Google Scholar
Alon, U., Hecht, J., Ofer, D. & Shvarts, D. 1995 Power-laws and similarity of Rayleigh–Taylor and Richtmyer–Meshkov mixing fronts at all density ratios. Phys. Rev. Lett. 74 (4), 534–537.Google Scholar
Anisimov, S. I., Drake, R. P., Gauthier, S., Meshkov, E. E. & Abarzhi, S. I. 2013 What is certain and what is not so certain in our knowledge of Rayleigh–Taylor mixing? Phil. Trans. R. Soc. Lond. A 371, 20130266.Google Scholar
Aprelkov, O. N., Igonin, V. V., Lebedev, A. I., Myshkina, I. Yu. & Ol’khov, O. V. 2010 Numerical and experimental study of Richtmyer–Meshkov instability in condensed matter. Phys. Scr. T142, 014025.CrossRefGoogle Scholar
Baker, G. A. & Grave-Morris, P. 1981 Padé Approximants, Part I: Basic Theory. Addison-Wesley.Google Scholar
Bakhrakh, S. M., Bezrukova, I. Yu., Kovaleva, A. D., Kosarim, S. S. & Ol’khov, O. V. 2006 Cumulative instability of the surface of condensed substances. Tech. Phys. Lett. 32 (2), 103–105.Google Scholar
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw Hill.Google Scholar
Berning, M. & Rubenchik, A. M. 1998 A weakly nonlinear theory for the dynamical Rayleigh–Taylor instability. Phys. Fluids 10 (7), 1564–1587.Google Scholar
Bodner, S. E., Colombant, D. G., Gardner, J. H., Lehmberg, R. H., McCrory, R. L., Seka, W., Verdon, C. P., Knauer, J. P., Afeyan, B. B. & Powell, H. T. 1998 Direct-drive laser fusion: status and prospects. Phys. Plasmas 5 (5), 1901–1918.Google Scholar
Brackbill, J. U., Kothe, D. B. & Zemach, C. 1992 A continuum method for modeling surface tension. J. Comput. Phys. 100 (2), 335–354.Google Scholar
Brouillette, M. 2002 The Richtmyer–Meshkov instability. Annu. Rev. Fluid Mech. 34, 445–468.Google Scholar
Buttler, W. T., Oró, D. M., Preston, D. L., Mikaelian, K. O., Cherne, F. J., Hixson, R. S., Mariam, F. G., Morris, C., Stone, J. B., Terrones, G. & Tupa, D. 2012 Unstable Richtmyer–Meshkov growth of solid and liquid metals in vacuum. J. Fluid Mech. 703, 60–84.Google Scholar
Carlès, P. & Popinet, S. 2001 Viscous nonlinear theory of Richtmyer–Meshkov instability. Phys. Fluids 13 (7), 1833–1836.Google Scholar
Chapman, P. R. & Jacobs, J. W. 2006 Experiments on the three-dimensional incompressible Richtmyer–Meshkov instability. Phys. Fluids 18 (7), 074101.Google Scholar
Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. Lond. A 286 (1335), 183–230.Google Scholar
Cotrell, D. L. & Cook, A. W. 2007 Scaling the incompressible Richtmyer–Meshkov instability. Phys. Fluids 19 (7), 078105.Google Scholar
Desjardins, O., Blanquart, G., Balarac, G. & Pitsch, H. 2008 High order conservative finite difference scheme for variable density low Mach number turbulent flows. J. Comput. Phys. 227 (15), 7125–7159.Google Scholar
Dhotre, A., Ramaprabhu, P. & Dimonte, G. 2008 A detailed numerical investigation of the single-mode Richtmyer–Meshkov instability. Bull. Am. Phys. Soc. 53 (15), 157.Google Scholar
Dimonte, G. 1999 Nonlinear evolution of the Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Plasmas 6 (5), 2009–2015.CrossRefGoogle Scholar
Dimonte, G., Morrison, J., Hulsey, S., Nelson, D., Weaver, S., Susoeff, A., Ron Hawke, R., Schneider, M., Batteaux, J., Dean Lee, D. & Ticehurst, J. 1996 A linear electric motor to study turbulent hydrodynamics. Rev. Sci. Instrum. 67 (1), 302–306.Google Scholar
Dimonte, G. & Ramaprabhu, P. 2010 Simulations and model of the nonlinear Richtmyer–Meshkov instability. Phys. Fluids 22 (1), 014104.CrossRefGoogle Scholar
Dimonte, G., Terrones, G., Cherne, F. J., Germann, T. C., Dupont, V., Kadau, K., Buttler, W. T., Oro, D. M., Morris, C. & Preston, D. L. 2011 Use of the Richtmyer–Meshkov instability to infer yield stress at high-energy densities. Phys. Rev. Lett. 107 (25), 264502.Google Scholar
Drake, R. P. 2009 Perspectives of high energy density physics. Phys. Plasmas 16 (5), 055501.Google Scholar
Emmons, H. W., Chang, C. T. & Watson, B. C. 1960 Taylor instability of finite surface waves. J. Fluid Mech. 7 (2), 177–193.CrossRefGoogle Scholar
Fraley, G. 1986 Rayleigh–Taylor stability for a normal shock-wave density discontinuity interaction. Phys. Fluids 29 (2), 376–386.CrossRefGoogle Scholar
Glendinning, S. G., Bolstad, J., Braun, D. G., Edwards, M. J., Hsing, W. W., Lasinski, B. F., Louis, H., Miles, A., Moreno, J., Peyser, T. A., Remington, B. A., Robey, H. F., Turano, E. J., Verdon, C. P. & Zhou, Y. 2003 Effect of shock proximity on Richtmyer–Meshkov growth. Phys. Plasmas 10 (5), 1931–1936.Google Scholar
Goncharov, V. N. 2002 Analytical model of nonlinear, single-mode, classical Rayleigh–Taylor instability. Phys. Rev. Lett. 88 (13), 134502.Google Scholar
Griffond, J. 2006 Linear interaction analysis for Richtmyer–Meshkov instability at low Atwood numbers. Phys. Fluids 18 (5), 054106.Google Scholar
Grove, J. W., Holmes, R., Sharp, D. H., Yang, Y. & Zhang, Q. 1993 Quantitative theory of Richtmyer–Meshkov instability. Phys. Rev. Lett. 71 (21), 3473–3476.CrossRefGoogle ScholarPubMed
Haan, S. W. 1991 Weakly nonlinear hydrodynamic instabilities in inertial fusion. Phys. Fluids B 3 (8), 2349–2355.Google Scholar
Hazak, G. 1996 Lagrangian formalism for the Rayleigh–Taylor instability. Phys. Rev. Lett. 76 (22), 4167–4170.Google Scholar
Hecht, J., Alon, U. & Shvarts, D. 1994 Potential flow models of Rayleigh–Taylor and Richtmyer–Meshkov bubble fronts. Phys. Fluids 6 (12), 4019–4030.CrossRefGoogle Scholar
Herrmann, M. 2008 A balanced force refined level set grid method for two-phase flows on unstructured flow solver grids. J. Comput. Phys. 227 (4), 2674–2706.Google Scholar
Herrmann, M. 2010 A parallel Eulerian interface tracking/Lagrangian point particle multi-scale coupling procedure. J. Comput. Phys. 229 (3), 745–759.Google Scholar
Herrmann, M., Moin, P. & Abarzhi, S. I. 2008 Nonlinear evolution of the Richtmyer–Meshkov instability. J. Fluid Mech. 612, 311–338.Google Scholar
Holmes, R. L., Dimonte, G., Fryxell, B., Gittings, M. L., Grove, J. W., Schneider, M., Sharp, D. H., Velikovich, A. L., Weaver, R. P. & Zhang, Q. 1999 Richtmyer–Meshkov instability growth: experiment, simulation and theory. J. Fluid Mech. 389, 55–79.Google Scholar
Holyer, J. Y. 1979 Large amplitude progressive interfacial waves. J. Fluid Mech. 93, 433–448.Google Scholar
Ingraham, R. L. 1954 Taylor instability of the interface between superposed fluids – solution by successive approximations. Proc. Phys. Soc. B 67 (10), 748–752.Google Scholar
Jacobs, J. W. & Catton, I. 1988 Three-dimensional Rayleigh–Taylor instability. 1. Weakly nonlinear theory. J. Fluid Mech. 187, 329–352.CrossRefGoogle Scholar
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, 23–32.Google Scholar
Jacobs, J. W. & Sheeley, M. 1996 Experimental study of incompressible Richtmyer–Meshkov instability. Phys. Fluids 8 (2), 405–415.Google Scholar
Jiang, G.-S. & Peng, D. 2000 Weighted ENO schemes for Hamilton–Jacobi equations. SIAM J. Sci. Comput. 21 (6), 2126–2143.Google Scholar
Kadau, K., Barber, J. L., Germann, T. C., Holian, B. L. & Alder, B. J. 2010 Atomistic methods in fluid simulation. Phil. Trans. R. Soc. Lond. A 368, 1547–1560.Google Scholar
Kivity, Y. & Hanin, M. 1981 Stability of interface and shock-wave driven by initial pressure discontinuity. Phys. Fluids 24 (6), 1010–1016.Google Scholar
Kotelnikov, A. D., Ray, J. & Zabusky, N. J. 2000 Vortex morphologies on reaccelerated interfaces: visualization, quantification and modeling of one- and two-mode compressible and incompressible environments. Phys. Fluids 12 (12), 3245–3264.Google Scholar
Layzer, D. 1955 On the instability of superposed fluids in a gravitational field. Astrophys. J. 122 (1), 1–12.Google Scholar
Likhachev, O. A. & Jacobs, J. W. 2005 A vortex model for Richtmyer–Meshkov instability accounting for finite Atwood number. Phys. Fluids 17 (3), 031704.Google Scholar
Matsuoka, C. 2010 Renormalization group approach to interfacial motion in incompressible Richtmyer–Meshkov instability. Phys. Rev. E 82 (3), 036320.Google Scholar
Matsuoka, C. & Nishihara, K. 2006 Vortex core dynamics and singularity formations in incompressible Richtmyer–Meshkov instability. Phys. Rev. E 73 (2), 026304.Google Scholar
Matsuoka, C., Nishihara, K. & Fukuda, Y. 2003 Nonlinear evolution of an interface in the Richtmyer–Meshkov instability. Phys. Rev. E 67 (3), 036301; erratum, Phys. Rev. E, 68(2) 029902.Google Scholar
Menikoff, R. & Zemach, C. 1983 Rayleigh–Taylor instability and use of conformal maps for ideal fluid flow. J. Comput. Phys. 51 (1), 28–64.Google Scholar
Meshkov, E. E. 1969 Instability of the interface of two gases accelerated by a shock. Fluid Dyn. 4 (5), 101–104.Google Scholar
Meshkov, E. E.2006 Studies of Hydrodynamic Instabilities in Laboratory Experiments (in Russian), Sarov, FGYC-VNIIEF, ISBN 5-9515-0069-9.Google Scholar
Meshkov, E. 2013 Some peculiar features of hydrodynamic instability development. Phil. Trans. R. Soc. Lond. A 371, 20120288.Google Scholar
Meyer, K. A. & Blewett, P. J. 1972 Numerical investigation of the stability of a shock accelerated interface between two fluids. Phys. Fluids 15 (5), 753–759.Google Scholar
Mikaelian, K. O. 1994 Comment on quantitative theory of Richtmyer–Meshkov instability. Phys. Rev. Lett. 73 (23), 3177.Google Scholar
Mikaelian, K. O. 1998 Analytic approach to nonlinear Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Rev. Lett. 80 (3), 508–511.Google Scholar
Mikaelian, K. O. 2003 Explicit expressions for the evolution of single-mode Rayleigh–Taylor and Richtmyer–Meshkov instabilities at arbitrary Atwood numbers. Phys. Rev. E 67 (2), 026319.CrossRefGoogle ScholarPubMed
Mikaelian, K. O. 2008 Limitations and failures of the Layzer model for hydrodynamic instabilities. Phys. Rev. E 78 (1), 015303.Google Scholar
Mueschke, N., Kraft, W. N., Dibua, O., Andrews, M. J. & Jacobs, J. W.2005 Numerical investigation of single-mode Richtmyer–Meshkov instability. In Proceedings of the ASME Fluids Engineering Division Summer Conference—2005, vol. 1, Pts A and B, pp. 185–193.Google Scholar
Neuvazhaev, V. Ye. & Parshukov, I. E. 1992 Study of the Richtmyer–Meshkov instability by the vortex method. In Mathematical Modelling and Applied Mathematics, Proceedings of the IMACS International Conference on Mathematical Modelling and Applied Mathematics, Moscow, USSR, 18–23 June 1990 (ed. Samarsky, A. A. & Sapagovas, M. P.), pp. 323–335. North-Holland.Google Scholar
Niederhaus, C. E. & Jacobs, J. W. 2003 Experimental study of the Richtmyer–Meshkov instability of incompressible fluids. J. Fluid Mech. 485, 243–277.Google Scholar
Nishihara, K., Wouchuk, J. G., Matsuoka, C., Ishizaki, R. & Zhakhovsky, V. V. 2010 Richmyer–Meshkov instability: theory of linear and nonlinear evolution. Phil. Trans. R. Soc. Lond. A 368 (1916), 1769–1807.Google Scholar
Olson, D. H. & Jacobs, J. W. 2009 Experimental study of Rayleigh–Taylor instability with a complex initial perturbation. Phys. Fluids 21 (3), 034103.Google Scholar
Oron, D., Arazi, L., Kartoon, D., Rikanati, A., Alon, U. & Shvarts, D. 2001 Dimensionality dependence of the Rayleigh–Taylor and Richtmyer–Meshkov instability late-time scaling laws. Phys. Plasmas 8 (6), 2883–2889.Google Scholar
Pelz, R. B. & Gulak, Y. 1997 Evidence for a real-time singularity in hydrodynamics from time series analysis. Phys. Rev. Lett. 79 (24), 4998–5001.Google Scholar
Peng, D., Merriman, B., Osher, S., Zhao, H. & Kang, M. 1999 A PDE-based fast local level set method. J. Comput. Phys. 155 (2), 410–438.Google Scholar
Remington, B. A., Drake, R. P. & Ryutov, D. D. 2006 Experimental astrophysics with high power lasers and Z-pinches. Rev. Mod. Phys. 78 (3), 755–807.Google Scholar
Richtmyer, R. D. 1960 Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Maths 13 (2), 297–319.Google Scholar
Rikanati, A., Oron, D., Sadot, O. & Shvarts, D. 2003 High initial amplitude and high Mach number effects on the evolution of the single-mode Richtmyer–Meshkov instability. Phys. Rev. E 67 (2), 026307.Google Scholar
Rosenhead, L. 1931 The formation of vortices from a surface of discontinuity. Proc. R. Soc. Lond. A 134 (823), 170–192.Google Scholar
Sadot, O., Erez, L., Alon, U., Oron, D., Levin, L. A., Erez, G., Ben-Dor, G. & Shvarts, D. 1998 Study of nonlinear evolution of single-mode and two bubble interaction under Richtmyer–Meshkov instability. Phys. Rev. Lett. 80 (8), 1654–1657.Google Scholar
Schwartz, L. W. 1974 Computer extension and analytic continuation of Stokes’ expansion for gravity waves. J. Fluid Mech. 62, 553–578.Google Scholar
Shu, C. W. 1988 Total-variation-diminishing time discretization. SIAM J. Sci. Stat. Comput. 9 (6), 1073–1084.Google Scholar
Sohn, S.-I. 2003 Simple potential-flow model of Rayleigh–Taylor and Richtmyer–Meshkov instabilities for all density ratios. Phys. Rev. E 67 (2), 026301.Google Scholar
Sohn, S.-I. 2004 Density dependence of a Zufiria-type model for Rayleigh–Taylor bubble fronts. Phys. Rev. E 70 (4), 045301.Google Scholar
Sohn, S.-I. 2008 Density dependence of a Zufiria-type model for Rayleigh–Taylor bubble fronts. Phys. Rev. E 78 (1), 017302.Google Scholar
Sohn, S.-I. 2009 Effects of surface tension and viscosity on the growth rates of Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Rev. E 80 (5), 055302.Google Scholar
Sreenivasan, K. R. 1999 Fluid turbulence. Rev. Mod. Phys. 71 (2), S383–S395.Google Scholar
Stanic, M., McFarland, J., Stellingwerf, R. F., Cassibry, J. T., Ranjan, D., Bonazza, R., Greenough, J. A. & Abarzhi, S. I. 2013 Non-uniform volumetric structures in Richtmyer–Meshkov flows. Phys. Fluids 25, 106107.Google Scholar
Stanic, M., Stellingverf, R. F., Cassibry, J. T. & Abarzhi, S. I. 2012 Scale coupling in Richtmyer–Meshkov flows induced by strong shocks. Phys. Plasmas 19 (8), 082706.Google Scholar
Stokes, G. G. 1849 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Sussman, M., Smereka, P. & Osher, S. 1994 A level set method for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114 (1), 146–159.Google Scholar
Vandenboomgaerde, M., Gauthier, S. & Mügler, C. 2002 Nonlinear regime of a multimode Richtmyer–Meshkov instability: a simplified perturbation theory. Phys. Fluids 14 (3), 1111–1122.Google Scholar
Vandenboomgaerde, M., Mügler, C. & Gauthier, S. 1998 Impulsive model for the Richtmyer–Meshkov instability. Phys. Rev. E 58 (2), 1874–1882.Google Scholar
Van der Pijl, S. P., Segal, A., Vuik, C. & Wesseling, P. 2005 A mass-conserving Level–Set method for modelling of multi-phase flows. Intl J. Numer. Meth. Fluids 47 (4), 339–361.Google Scholar
Velikovich, A. L. 1996 Analytic theory of Richtmyer–Meshkov instability for the case of reflected rarefaction wave. Phys. Fluids 8 (6), 1666–1679.Google Scholar
Velikovich, A. L. & Dimonte, G. 1996 Nonlinear perturbation theory of the incompressible Richtmyer–Meshkov instability. Phys. Rev. Lett. 76 (17), 3112–3115.Google Scholar
Velikovich, A. & Phillips, L. 1996 Instability of a plane centered rarefaction wave. Phys. Fluids 8 (4), 1107–1118.Google Scholar
Volkov, N. B., Maǐer, A. E. & Yalovets, A. P. 2001 The nonlinear dynamics of the interface between media possessing different densities and symmetries. Tech. Phys. Lett. 27 (1), 20–24.Google Scholar
Volkov, N. B., Maǐer, A. E. & Yalovets, A. P. 2003 Nonlinear dynamics of the interface between continuous media with different densities. Tech. Phys. Lett. 48 (3), 275–283.Google Scholar
White, J., Oakley, J., Anderson, M. & Bonazza, R. 2010 Experimental measurements of the nonlinear Rayleigh–Taylor instability using a magnetorheological fluid. Phys. Rev. E 81 (2), 026303.Google Scholar
Wouchuk, J. G. 2001a Growth rate of the linear Richtmyer–Meshkov instability when a shock is reflected. Phys. Rev. E 63 (5), 056303.Google Scholar
Wouchuk, J. G. 2001b Growth rate of the Richtmyer–Meshkov instability when a rarefaction is reflected. Phys. Plasmas 8 (6), 2890–2907.CrossRefGoogle Scholar
Wouchuk, J. G. & Carretero, R. 2004 Linear perturbation growth at the trailing edge of a rarefaction wave. Phys. Plasmas 10 (11), 4237–4252.Google Scholar
Wu, S. J. 1999 Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Am. Math. Soc. 12 (2), 445–495.Google Scholar
Yang, Y., Zhang, Q. & Sharp, D. 1994 Small amplitude theory of Richtmyer–Meshkov instability. Phys. Fluids 6 (5), 1856–1873.Google Scholar
Zabusky, N. J., Kotelnikov, A. D., Gulak, Y. & Peng, G. 2003 Amplitude growth rate of Richtmyer–Meshkov unstable two-dimensional interface to intermediate times. J. Fluid Mech. 475, 147–162.Google Scholar
Zhang, Q. 1998 Analytical solutions of Layzer-type approach to unstable interfacial fluid mixing. Phys. Rev. Lett. 81 (16), 3391–3394.Google Scholar
Zhang, Q. & Sohn, S.-I. 1996 An analytical nonlinear theory of Richtmyer–Meshkov instability. Phys. Lett. A 212 (3), 149–155.Google Scholar
Zhang, Q. & Sohn, S.-I. 1997a Nonlinear theory of unstable fluid mixing driven by shock wave. Phys. Fluids 9 (4), 1106–1124.CrossRefGoogle Scholar
Zhang, Q. & Sohn, S.-I. 1997b Padé approximation to an interfacial fluid mixing problem. Appl. Math. Lett. 10 (5), 121–127.Google Scholar
Zhang, Q. & Sohn, S.-I. 1999 Quantitative theory of Richtmyer–Meshkov instability in three dimensions. Z. Angew. Math. Phys. 50, 1–46.Google Scholar
Zufiria, J. A. 1988 Bubble competition in Rayleigh–Taylor instability. Phys. Fluids 31 (3), 440–446.Google Scholar