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

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.

Mechanism and modelling of the secondary baroclinic vorticity in the Richtmyer–Meshkov instability

“…This SBV effect also provides a physical interpretation for the acceleration/deceleration caused by the perturbation parameter in the perturbation analysis of RMI (see Velikovich et al. 2014).

Figure 12.Temporal evolution of the scaled circulation variations, SBV and viscous components of the circulation in the () spike and () bubble regions.

…”

Analytical asymptotic velocities in linear Richtmyer-Meshkov-like flows

“…Then, as the shock travels farther from the piston surface, the velocity field that develops in the whole fluid downstream can be decomposed as the sum of an irrotational component (due to the fluid pressure fluctuations) and a rotational part (due to the vorticity created just behind the shock, that is, the divergence free perturbation mode commented in Refs. [13,14]). As is known, a shock moving inside an ideal gas is stable [23]; this means that pressure perturbations downstream of the shock will be zero for t → ∞.…”

“…Recently, the use of corrugated shock waves has been suggested as an important tool to diagnose material properties [7][8][9] within the domains of high-energy-density physics (HEDP) experiments or within the domain of geophysics or planetary sciences [10]. Therefore, analytical models that reveal the details of the linear phase are extremely important in order to develop consistent nonlinear theories of the perturbation evolution [11][12][13] or helping in the design of experiments [9] and/or assisting in the benchmarking of simulation hydrocodes dealing with RM-like flows. The work shown here is a natural extension of previous work on the subject [15,16] and the objective is very specific: to obtain accurate analytical estimates of the asymptotic velocities in an RM-like environment.…”

Tantalum strength at extreme strain rates from impact-driven Richtmyer-Meshkov instabilities

“…The ability of viscosity to reduce the growth rate of Richtmyer-Meshkov instabilities (RMI) in fluids has been studied for some time [1][2][3][4][5]. The sensitivity of RMI to strength in solids, the analog of viscosity, has received explicit attention more recently [6][7][8][9][10][11][12] with increasing attention also in regard to ejecta [13][14][15][16].…”