Exact, approximate, and hybrid treatments of viscous Rayleigh-Taylor and Richtmyer-Meshkov instabilities
We consider Rayleigh-Taylor and Richtmyer-Meshkov instabilities at the interface between two fluids, one or both of which may be viscous. We derive exact analytic expressions for the amplitude η(t) in the linear regime when only one of the fluids is viscous. The more general case is solved numerically using Laplace transforms. We compare the exact solutions of the initial-value problem with the approximate solutions of the eigenvalue problem used in a simple expression for η(t) in terms of two growth rates, γ+ and γ-. We then propose a hybrid model as an improvement on the approximate model. The hybrid model is based on the same expression for η(t) in terms of γ± but uses exact eigenvalues for γ+, the larger growth rate, and a relationship between γ- and γ+. We also discuss two concepts: isogrowth wave number pairs and asymptotic decay. The first relies on viscosity in one or both fluids to identify perturbations of two different wavelengths having the same γ+. The second concept, which is more general, can be found in viscous as well as inviscid fluids and requires only a specific initial growth rate $\dot{η}$$critical\atop{0}$ to force η(t) → 0 as t → ∞. We present several examples illustrating these two concepts and comparing exact, approximate, and hybrid treatments.