Modal Model Mean Field Self-Similar Solutions To The Asymptotic Evolution Of Rayleigh-Taylor And Richtmyer-Meshkov Instabilities And Its Dependence On The Initial Conditions

The evolution of Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities for incompressible and immiscible fluids and their dependence on the initial perturbation spectrum is evaluated using a new mean field formulation of the Haan-Ofer-Shvarts mode coupling model. The height of the lighter fluid bubbles penetrating into the denser fluid is shown to reach asymptotic, universal, self-similar behavior when the initial spectrum is dominated by short wavelengths and at least 3-4 mode coupling generations have occurred. For RT, the model yields h = alpha(RT)Agt(2) for the bubble front penetration height, in good agreement with experimental data and 3D numerical simulations for various initial conditions. For RM, the lack of a natural length scale leads to a 2nd type self-similar solution h = alpha(RM)t(theta) and theta is rigorously determined from a detailed solution of the model equation, while alpha(RM) retains knowledge of the initial spectrum. The value of theta(RM) in two dimensions is theta(2D) = 2/5, consistent with the Alon-Shvarts bubble-merger model and numerical simulations, and in three dimensions, it is theta(3D) = 1/3. We find that the smaller value theta(3D)similar to 0.25 +/- 0.05 obtained in numerical simulations and experiments [Dimonte and Schneider, Phys. Fluids 12, 304 (2000)] results from the lack of enough mode coupling generations needed to reach the RM asymptotic self-similar stage. The feasibility of a true self-similar RM experiment on NIF is discussed. Published by AIP Publishing.