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

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.