Skip to main content Accessibility help

On the late-time growth of the two-dimensional Richtmyer–Meshkov instability in shock tube experiments

  • Robert V. Morgan (a1), R. Aure (a1), J. D. Stockero (a1), J. A. Greenough (a2), W. Cabot (a2), O. A. Likhachev (a1) and J. W. Jacobs (a1)...

In the present study, shock tube experiments are used to study the very late-time development of the Richtmyer–Meshkov instability from a diffuse, nearly sinusoidal, initial perturbation into a fully turbulent flow. The interface is generated by two opposing gas flows and a perturbation is formed on the interface by transversely oscillating the shock tube to create a standing wave. The puncturing of a diaphragm generates a Mach $1. 2$ shock wave that then impacts a density gradient composed of air and SF6, causing the Richtmyer–Meshkov instability to develop in the 2.0 m long test section. The instability is visualized with planar Mie scattering in which smoke particles in the air are illuminated by a Nd:YLF laser sheet, and images are recorded using four high-speed video cameras operating at 6 kHz that allow the recording of the time history of the instability. In addition, particle image velocimetry (PIV) is implemented using a double-pulsed Nd:YAG laser with images recorded using a single CCD camera. Initial modal content, amplitude, and growth rates are reported from the Mie scattering experiments while vorticity and circulation measurements are made using PIV. Amplitude measurements show good early-time agreement but relatively poor late-time agreement with existing nonlinear models. The model of Goncharov (Phys. Rev. Lett., vol. 88, 2002, 134502) agrees with growth rate measurements at intermediate times but fails at late experimental times. Measured background acceleration present in the experiment suggests that the late-time growth rate may be influenced by Rayleigh–Taylor instability induced by the interfacial acceleration. Numerical simulations conducted using the LLNL codes Ares and Miranda show that this acceleration may be caused by the growth of boundary layers, and must be accounted for to produce good agreement with models and simulations. Adding acceleration to the Richtmyer–Meshkov buoyancy–drag model produces improved agreement. It is found that the growth rate and amplitude trends are also modelled well by the Likhachev–Jacobs vortex model (Likhachev & Jacobs, Phys. Fluids, vol. 17, 2005, 031704). Circulation measurements also show good agreement with the circulation value extracted by fitting the vortex model to the experimental data.

Corresponding author
Email address for correspondence:
Hide All

Present address: Roxar Flow Measurement, Gamle Forusvei 17, 4065 Stavanger, Norway.


Present address: Lockheed Martin Aeronautics, 1011 Lockheed Way, Palmdale, CA 93599.

Hide All
Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series.
Aure, R. & Jacobs, J. W 2008 Particle image velocimetry study of the shock-induced single mode Richtmyer–Meshkov instability. Shock Waves 18, 161167.
Badcock, K. J. 1992 A numerical simulation of boundary layer effects in a shock tube. Intl J. Numer. Meth. Fluids 14, 11511171.
Balakumar, B. J., Orlicz, G. C., Tomkins, C. D. & Prestridge, K. P. 2008 Dependence of growth patterns and mixing width on initial conditions in Richtmyer–Meshkov unstable fluid layers. Phys. Scr. T123, 014013.
Benjamin, R. F., Trease, H. E. & Shaner, J. W. 1984 Coherent density gradients in water compressed by a modulated shock wave. Phys. Fluids 27, 23902393.
Bonazza, R. & Sturtevant, B. 1996 X-ray measurements of growth rates at a gas interface accelerated by shock waves. Phys. Fluids 8, 24962512.
Brocher, E. F. 1964 Hot flow length and testing time in real shock tube flow. Phys. Fluids 7, 347351.
Collins, B. D. & Jacobs, J. W. 2002 PLIF flow visualization and measurements of the Richtmyer–Meshkov instability of an air/ ${\mathrm{SF} }_{6} $ interface. J. Fluid Mech. 464, 113136.
Cook, A. W. 2007 Artificial fluid properties for large-eddy simulation of compressible turbulent mixing. Phys. Fluids 19, 055103.
Cook, A. W. & Cabot, W. H. 2004 A high-wavenumber viscosity for high-resolution numerical methods. J. Comput. Phys. 195, 594601.
Cook, A. W. & Cabot, W. H. 2005 Hyperviscosity for shock–turbulence interactions. J. Comput. Phys. 203, 379385.
Dimonte, G. & Ramaprabhu, P. 2010 Simulations and model of the nonlinear Richtmyer–Meshkov instability. Phys. Fluids 22, 4014104.
Edwards, M. J., Lindl, J. D., Spears, B. K., Weber, S. V., Atherton, L. J., Bleuel, D. L., Bradley, D. K., Callahan, D. A., Cerjan, C. J., Clark, D., Collins, G. W., Fair, J. E., Fortner, R. J., Glenzer, S. H., Haan, S. W., Hammel, B. A., Hamza, A. V., Hatchett, S. P., Izumi, N., Jacoby, B., Jones, O. S., Koch, J. A., Kozioziemski, B. J., Landen, O. L., Lerche, R., MacGowan, B. J., MacKinnon, A. J., Mapoles, E. R., Marinak, M. M., Moran, M., Moses, E. I., Munro, D. H., Schneider, D. H., Sepke, S. M., Shaughnessy, D. A., Springer, P. T., Tommasini, R., Bernstein, L., Stoeffl, W., Betti, R., Boehly, T. R., Sangster, T. C., Glebov, V. Y., McKenty, P. W., Regan, S. P., Edgell, D. H., Knauer, J. P., Stoeckl, C., Harding, D. R., Batha, S., Grim, G., Herrmann, H. W., Kyrala, G., Wilke, M., Wilson, D. C., Frenje, J., Petrasso, R., Moreno, K., Huang, H., Chen, K. C., Giraldez, E., Kilkenny, J. D., Mauldin, M., Hein, N., Hoppe, M., Nikroo, A. & Leeper, R. J. 2011 The experimental plan for cryogenic layered target implosions on the national ignition facility: the inertial confinement approach to fusion. Phys. Plasmas 18 (5), 051003.
Glass, I. I. & Patterson, G. N. 1955 A theoretical and experimental study of shock-tube flows. J. Aero. Sci. 22, 73100.
Goncharov, V. N. 2002 Analytical model of nonlinear, single-mode, classical Rayleigh–Taylor instability at arbitrary Atwood numbers. Phys. Rev. Lett. 88, 134502.
Grinstein, F. F., Gowhardhan, A. A. & Wachtor, A. J. 2011 Simulations of Richtmyer–Meshkov instabilities in planar shock-tube experiments. Phys. Fluids 23, 034106.
Hahn, M., Drikakis, D., Youngs, D. L. & Williams, R. J. R. 2011 Richtmyer–Meshkov turbulent mixing arising from an inclined material interface with realistic surface perturbations and reshocked flow. Phys. Fluids 23, 046101.
Herrmann, M., Moin, P. & Abarzhi, S. I. 2008 Nonlinear evolution of the Richtmyer–Meshkov instability. J. Fluid Mech. 612, 311338.
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, 5579.
Huerte Ruiz de Lira, C., Velikovich, A. L. & Wouchuk, J. G. 2011 Analytical linear theory for the interaction of a planar shock wave with a two- or three-dimensional random isotropic density field. Phys. Rev. E 83, 056320.
Jacobs, J. W. 1993 The dynamics of shock accelerated light and heavy gas cylinders. Phys. Fluids A 5, 22392247.
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.
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, 583586.
Jacobs, J. W. & Krivets, V. V. 2005 Experiments on the late-time development of single-mode Richtmyer–Meshkov instability. Phys. Fluids 17, 034105.
Jacobs, J. W. & Sheeley, J. M. 1996 Experimental study of the Richtmyer–Meshkov instability of incompressible fluids. Phys. Fluids 8, 405415.
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.
Keane, R. D. & Adrian, R. J. 1990 Optimization of particle image velocimeters. Part 1. Double pulsed systems. Meas. Sci. Technol. 1, 12021215.
Keane, R. D. & Adrian, R. J. 1991 Optimization of particle image velocimeters. Part 2. Multiple pulsed systems. Meas. Sci. Technol. 2, 963974.
Kifonidis, K., Plewa, T., Sheck, L., Janka, H.-T. & Müller, E. 2006 Non-spherical core collapse supernovae. Astron. Astrophys. 453, 661678.
Kolev, T. V. & Rieben, R. N. 2009 A tensor artificial viscosity using finite element approach. J. Comput. Phys. 228, 83368366.
Krechetnikov, R. 2009 Rayleigh–Taylor and Richtmyer–Meshkov instabilities of flat and curved interfaces. J. Fluid. Mech. 625, 387410.
Layzer, D. 1955 On the instability of superposed fluids in a gravitational field. Astrophys. Rev. J. 122, 112.
Likhachev, O. A. & Jacobs, J. W. 2005 A vortex model for Richtmyer–Meshkov instability accounting for finite Atwood number. Phys. Fluids 17, 031704.
Liu, B. Y. H. & Lee, K. W. 1975 An aerosol generator of high stability. Am. Ind. Hyg. Assoc. 36, 861865.
Lombardini, M., Hill, D. J., Pullin, D. I. & Meiron, D. I. 2011 Atwood ratio dependence of Richtmyer–Meshkov flows under reshock conditions using large-eddy simulations. J. Fluid Mech. 670, 439480.
Mariani, C., Vandenboomgaerde, M., Jourdan, G., Souffland, D. & Houas, L. 2008 Investigation of the Richtmyer–Meshkov instability with stereolithographed interfaces. Phys. Rev. Lett. 100, 254503.
Matsuoka, C., Nishihara, K. & Fukuda, Y. 2003 Nonlinear evolution of an interface in the Richtmyer–Meshkov instability. Phys. Rev. E 67, 036301.
McFarland, J. A., Greenough, J. A. & Ranjan, D. 2011 Computational parametric study of a Richtmyer–Meshkov instability for an inclined interface. Phys. Rev. E 84, 026303.
Meshkov, E. E. 1969 Instability of the interface of two gases accelerated by a shock wave. Izv. Akad. Nauk. SSSR Maekh. Zhidk. Gaza. 4, 151157.
Meyer, K. A. & Blewett, P. J. 1972 Numerical investigation of the stability of a shock-accelerated interface between two fluids. Phys. Fluids 15, 753759.
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, 026319.
Mirels, H. 1956 Attenuation in a shock-tube due to unsteady-boundary-layer action. NACA TN3278.
Mirels, H. & Braun, W. H. 1957 Nonuniformities in shock-tube flow due to unsteady-boundary-layer action. NACA TN4021.
Motl, B., Niederhaus, J., Ranjan, D., Oakley, J., Anderson, M. & Bonazza, R. 2007 Experimental studies for ICF related Richtmyer–Meshkov instabilities. Fus. Sci. Tech. 52, 10791083.
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.
Oron, D., Arazi, L., Kartoon, D., Rkanati, A., Alon, U. & Shvarts, D. 2001 Dimensionality dependence of the Rayleigh–Taylor and Richtmyer–Meshkov instability late-time scaling laws. Phys. Plasmas 8, 21082115.
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, 37303744.
Prasad, J. K., Rasheed, A., Kumar, S. & Sturtevant, B. 2000 The late-time development of the Richtmyer–Meshkov instability. Phys. Fluids 12, 37303744.
Raffel, M., Kompenhans, J. & Willert, C. E. 1998 Particle Image Velocimetry: A Practical Guide. Springer.
Rayleigh, Lord 1900 Investigation of the Character of the Equilibrium of an Incompressible Heavy Fluid of Variable Density. Cambridge University Press.
Richtmyer, R. D. 1960 Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Math. 23, 297319.
Roshko, A. 1960 On flow duration in low-pressure shock tubes. Phys. Fluids 3, 835842.
Sadot, O., Erez, L., Alon, U., Oron, D., Levin, L. A., Erez, G., Ben-Dor, B. & Shvarts, D. 1998 Study of nonlinear evolution of single-mode and two-bubble interaction under Richtmyer–Meshkov instability. Phys. Rev. Lett. 80, 16541657.
Schlichting, H. & Gersten, K. 2000 Boundary-Layer Theory, 8th edn. Springer.
Schilling, O., Latini, M. & Don, W. S. 2007 Physics of reshock and mixing in single-mode Richtmyer–Meshkov instability. Phys. Rev. E 76, 026319.
Sharp, R. W. Jr & Barton, R. T. 1981 Hemp advection model. UCID-17809 Rev.1, Lawrence Livermore Laboratory.
Sohn, S.-I. 2011 Inviscid and viscous vortex models for Richtmyer–Meshkov instability. Fluid Dyn. Res. 43, 065506.
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Part 1. Proc. R. Soc. A 201, 192196.
Thornber, B., Drikakis, D., Youngs, D. L. & Williams, R. J. R. 2011 Growth of a Richtmyer–Meshkov turbulent layer after reshock. Phys. Fluids 23, 095107.
Tomkins, C., Kumar, S., Orlicz, G. & Prestridge, K. 2008 An experimental investigation of mixing mechanisms in shock-accelerated flow. J. Fluid Mech. 611, 131150.
von Kármán, T. 1921 Laminar and turbulent friction. Z. Angew. Math. Mech. 1, 233252.
Vandenboomgaerde, M., Gauthier, S. & Mügler, C 2002 Nonlinear regime of a multimode Richtmyer–Meshkov instability: a simplified perturbation theory. Phys. Fluids 14, 11111122.
Vetter, M. & Sturtevant, B. 1995 Experiments on the Richtmyer–Meshkov instability of an air/ ${\mathrm{SF} }_{6} $ interface. Shock Waves 4, 247252.
Westerweel, J. 1993 Digital particle image velocimetry, theory and application. PhD thesis, Technische Universiteit Delft.
Wilkins, M. L. 1963 Calculation of elastic–plastic flow. UCRL-7322, Lawrence Radiation Laboratory.
Wouchuk, J. G. & Nishihara, K. 1997 Asymptotic growth in the linear Richtmyer–Meshkov instability. Phys. Plasmas 4, 10281038.
Yang, J., Kubota, Y. & Zukoski, E. E. 1993 Applications of shock-induced mixing to supersonic combustion. AIAA J. 31, 854862.
Zhang, Q. & Sohn, S.-I. 1997a Nonlinear theory of unstable fluid mixing driven by shock wave. Phys. Fluids 9, 11061124.
Zhang, Q. & Sohn, S.-I. 1997b Padé approximation to an interfacial fluid mixing problem. Appl. Math. Lett. 10, 121127.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed