Skip to main content Accessibility help

Universality of finger growth in two-dimensional Rayleigh–Taylor and Richtmyer–Meshkov instabilities with all density ratios

  • Qiang Zhang (a1) and Wenxuan Guo (a1)


Interfacial fluid mixing driven by an external acceleration or a shock wave are common phenomena known as Rayleigh–Taylor instability and Richtmyer–Meshkov instability, respectively. The most significant feature of these instabilities is the penetrations of heavy (light) fluid into light (heavy) fluid known as spikes (bubbles). The study of the growth rate of these fingers is a classical problem in fundamental science and has important applications. Research on this topic has been very active over the past half-century. In contrast to the well-known phenomena that spikes and bubbles can have quantitatively, even qualitatively, different behaviours, we report a surprising result for fingers in a two-dimensional system: in terms of scaled dimensionless variables, all spikes and bubbles at any density ratio closely follow a universal curve, up through a pre-asymptotic stage. Such universality holds not only among bubbles and among spikes of different density ratios, but also between bubbles and spikes of different density ratios. The data from numerical simulations show good agreement with our theoretical predictions.


Corresponding author

Email address for correspondence:


Hide All
Abarzhi, S. I., Glimm, J. & Lin, A.-D. 2003 Dynamics of two-dimensional Rayleigh–Taylor bubbles for fluids with a finite density contrast. Phys. Fluids 15 (8), 21902197.
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), 534537.
Brouillette, M. 2002 The Richtmyer–Meshkov instability. Annu. Rev. Fluid Mech. 34 (1), 445468.
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, 6084.
Dimonte, G. & Ramaprabhu, P. 2010 Simulations and model of the nonlinear Richtmyer–Meshkov instability. Phys. Fluids 22 (1), 014104.
Dimonte, G. & Schneider, M. 2000 Density ratio dependence of Rayleigh–Taylor mixing for sustained and impulsive acceleration histories. Phys. Fluids 12 (2), 304321.
E, W. & Hou, T. Y. 1990 Homogenization and convergence of the vortex method for 2-D Euler equations with oscillatory vorticity fields. Commun. Pure Appl. Maths 43 (7), 821855.
Glimm, J., Li, X.-L. & Lin, A.-D. 2002 Nonuniform approach to terminal velocity for single mode Rayleigh–Taylor instability. Acta Math. Appl. Sinica 18 (1), 18.
Goncharov, V. 2002 Analytical model of nonlinear, single-mode, classical Rayleigh–Taylor instability at arbitrary Atwood numbers. Phys. Rev. Lett. 88 (13), 134502.
Hecht, J., Alon, U. & Shvarts, D. 1994 Potential flow models of Rayleigh–Taylor and Richtmyer–Meshkov bubble fronts. Phys. Fluids 6, 40194030.
Layzer, D. 1955 On the instability of superposed fluids in a gravitational field. Astrophys. J. 122, 112.
Menikoff, R. & Zemach, C. 1983 Rayleigh–Taylor instability and the use of conformal maps for ideal fluid flow. J. Comput. Phys. 51 (1), 2864.
Meshkov, E. E. 1969 Instability of the interface of two gases accelerated by a shock wave. Fluid Dyn. 4, 101104.
Mikaelian, K. O. 1998 Analytic approach to nonlinear Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Rev. Lett. 80 (3), 508511.
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.
Mikaelian, K. O. 2008 Limitations and failures of the Layzer model for hydrodynamic instabilities. Phys. Rev. E 78 (1), 015303.
Poujade, O. & Peybernes, M. 2010 Growth rate of Rayleigh–Taylor turbulent mixing layers with the foliation approach. Phys. Rev. E 81 (1), 016316.
Rayleigh, Lord 1883 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. 14, 170177.
Richtmyer, R. D. 1960 Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Maths 13 (2), 297319.
Sharp, D. H. 1984 An overview of Rayleigh–Taylor instability. Physica D 12 (1), 318.
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.
Sohn, S.-I. 2004 Vortex model and simulations for Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Rev. E 69 (3), 036703.
Sohn, S.-I. 2011 Late-time vortex dynamics of Rayleigh–Taylor instability. J. Phys. Soc. Japan 80 (8), 084401.
Taylor, G. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A 201 (1065), 192196.
Thornber, B., Drikakis, D., Youngs, D. L. & Williams, R. J. R. 2010 The influence of initial conditions on turbulent mixing due to Richtmyer–Meshkov instability. J. Fluid Mech. 654, 99139.
Tritschler, V. K., Olson, B. J., Lele, S. K., Hickel, S., Hu, X. Y. & Adams, N. A. 2014 On the Richtmyer–Meshkov instability evolving from a deterministic multimode planar interface. J. Fluid Mech. 755, 429462.
Tryggvason, G. 1988 Numerical simulations of the Rayleigh–Taylor instability. J. Comput. Phys. 75 (2), 253282.
Zhang, Q. 1998 Analytical solutions of Layzer-type approach to unstable interfacial fluid mixing. Phys. Rev. Lett. 81 (16), 33913394.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

Universality of finger growth in two-dimensional Rayleigh–Taylor and Richtmyer–Meshkov instabilities with all density ratios

  • Qiang Zhang (a1) and Wenxuan Guo (a1)


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.