Analytic approach to nonlinear hydrodynamic instabilities driven by time-dependent accelerations

Abstract

We extend our earlier model for Rayleigh-Taylor and Richtmyer-Meshkov instabilities to the more general class of hydrodynamic instabilities driven by a time-dependent acceleration g(t). Explicit analytic solutions for linear as well as nonlinear amplitudes are obtained for several g(t)s by solving a Schrödinger-like equation d(2)eta/dt(2)-g(t)kAeta=0, where A is the Atwood number and k is the wave number of the perturbation amplitude eta(t). In our model a simple transformation k-->k(L) and A-->A(L) connects the linear to the nonlinear amplitudes: eta(nonlinear)(k,A) approximately (1/k(L))ln eta(linear)(k(L),A(L)). The model is found to be in very good agreement with direct numerical simulations. Bubble amplitudes for a variety of accelerations are seen to scale with s defined by s=integral(square root of [g(t)]dt), while spike amplitudes prefer scaling with displacement Deltax=integral[integralg(t)dt]dt.