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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
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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
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. 

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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.

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