The rapid acceleration model and the growth rate of a turbulent mixing zone induced by Rayleigh-Taylor instability

Abstract

A nonlinear model giving the short-time dynamics of turbulent mixing layers of two incompressible miscible fluids submitted to strong accelerations is proposed. This model encompasses both the linear rapid distortion theory applied to unstably stratified flows and an equation of evolution for the mixing zone width L. The nonlinear mechanism coming from the interaction between the turbulent quantities and the mean concentration field leads to a self-similar regime. The convergence to this state is analyzed in depth using dynamical system techniques. In this framework, the existence of a central manifold is established and allows a reduction of dimension of the problem. This is associated with a Lagrangian formulation depending only on \documentclass[12pt]{minimal}\begin{document}$L, \dot{L}$\end{document}L,L̇ so that the dynamics of L degenerates into a buoyancy-drag equation. Here, the expression for the buoyancy coefficient is explicit. It depends only on the global mixing parameter and a quantity called the dimensionality parameter characterizing the form of density turbulent structures inside the mixing zone. An extension of the rapid acceleration model to the classical self-similar Rayleigh-Taylor problem is presented leading to an analytical expression for the growth parameter α, which is compared to existing numerical simulations and experiments.