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 = α_RT Agt² 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 = α R M t θ and θ is rigorously determined from a detailed solution of the model equation, while α_RM retains knowledge of the initial spectrum. The value of θ_RM in two dimensions is θ_{2 D} = 2/5, consistent with the Alon-Shvarts bubble-merger model and numerical simulations, and in three dimensions, it is θ_{3 D} = 1/3. We find that the smaller value θ_{3 D} ∼ 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.