Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-02-01T04:09:59.326Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  25 November 2015

Qiang Zhang  and
Wenxuan Guo
Qiang Zhang*
Affiliation:
Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
Wenxuan Guo
Affiliation:
Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong
*
†Email address for correspondence: mazq@cityu.edu.hk
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

JFM classification

Interfacial Flows (free surface): Fingering instability Interfacial Flows (free surface): Interfacial Flows (free surface) Multiphase and Particle-laden Flows: Multiphase flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 786 , 10 January 2016 , pp. 47 - 61
Check for updates
Copyright
© 2015 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.)

References

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), 2190–2197.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
Brouillette, M. 2002 The Richtmyer–Meshkov instability. Annu. Rev. Fluid Mech. 34 (1), 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
Dimonte, G. & Ramaprabhu, P. 2010 Simulations and model of the nonlinear Richtmyer–Meshkov instability. Phys. Fluids 22 (1), 014104.Google Scholar
Dimonte, G. & Schneider, M. 2000 Density ratio dependence of Rayleigh–Taylor mixing for sustained and impulsive acceleration histories. Phys. Fluids 12 (2), 304–321.Google Scholar
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), 821–855.Google Scholar
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), 1–8.Google Scholar
Goncharov, V. 2002 Analytical model of nonlinear, single-mode, classical Rayleigh–Taylor instability at arbitrary Atwood numbers. Phys. Rev. Lett. 88 (13), 134502.Google Scholar
Hecht, J., Alon, U. & Shvarts, D. 1994 Potential flow models of Rayleigh–Taylor and Richtmyer–Meshkov bubble fronts. Phys. Fluids 6, 4019–4030.CrossRefGoogle Scholar
Layzer, D. 1955 On the instability of superposed fluids in a gravitational field. Astrophys. J. 122, 1–12.CrossRefGoogle Scholar
Menikoff, R. & Zemach, C. 1983 Rayleigh–Taylor instability and the 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 wave. Fluid Dyn. 4, 101–104.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.Google Scholar
Mikaelian, K. O. 2008 Limitations and failures of the Layzer model for hydrodynamic instabilities. Phys. Rev. E 78 (1), 015303.Google Scholar
Poujade, O. & Peybernes, M. 2010 Growth rate of Rayleigh–Taylor turbulent mixing layers with the foliation approach. Phys. Rev. E 81 (1), 016316.Google Scholar
Rayleigh, Lord 1883 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. 14, 170–177.Google Scholar
Richtmyer, R. D. 1960 Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Maths 13 (2), 297–319.Google Scholar
Sharp, D. H. 1984 An overview of Rayleigh–Taylor instability. Physica D 12 (1), 3–18.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 Vortex model and simulations for Rayleigh–Taylor and Richtmyer–Meshkov instabilities. Phys. Rev. E 69 (3), 036703.Google Scholar
Sohn, S.-I. 2011 Late-time vortex dynamics of Rayleigh–Taylor instability. J. Phys. Soc. Japan 80 (8), 084401.CrossRefGoogle Scholar
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), 192–196.Google Scholar
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, 99–139.Google Scholar
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, 429–462.Google Scholar
Tryggvason, G. 1988 Numerical simulations of the Rayleigh–Taylor instability. J. Comput. Phys. 75 (2), 253–282.CrossRefGoogle Scholar
Zhang, Q. 1998 Analytical solutions of Layzer-type approach to unstable interfacial fluid mixing. Phys. Rev. Lett. 81 (16), 3391–3394.Google Scholar