A model for Rayleigh-Taylor mixing and interface turn-over

International audienceWe first develop a new mathematical model for two-fluid interface motion, subjected to the Rayleigh-Taylor (RT) instability in two-dimensional fluid flow, which in its simplest form, is given by$ h_{tt}( \alpha ,t) = A g\, \Lambda h - \frac{ \sigma }{ \rho^++\rho^-} \Lambda^3 h - A \partial_\alpha(H h_t h_t) $, where $\Lambda = H \partial_ \alpha $ and $H$ denotes the Hilbert transform. In this so-called $h$-model, $A$ is the Atwood number, $g$ is the acceleration, $ \sigma $ is surface tension, and $\rho^\pm$ denotes the densities of the two fluids. We derive our $h$-model using asymptotic expansions in theBirkhoff-Rottintegral-kernel formulation for the evolution of an interface separating two incompressible and irrotational fluids. The resulting $h$-model equation is shown to be locally and globallywell-posed in Sobolev spaces when a certain {\it stability condition} is satisfied; this stability condition requires that the product of the Atwood number and the initial velocity field be positive. The asymptotic behavior of these global solutions, as $t \to \infty $, is also described.The $h$-model equation is shown to have interesting balance laws, which distinguish the stable dynamics from the unstable dynamics.Numerical simulations of the $h$-model show that the interface can quickly grow due to nonlinearity, and then stabilize when the lighter fluid is on top of the heavier fluid and acceleration is directed downward. In the unstable case of a heavier fluid being supported by the lighter fluid, we find good agreement for the growth of the mixinglayer with experimental data in the ``rocket rig'' experiment of Read of Youngs.We then derive an interface model for RT instability, with a general parameterization $z( \alpha ,t) = \left( z_1(\alpha ,t), z_2 (\alpha ,t)\right)$such that $z$ satisfies$z_{tt}= \Lambda\bigg{[}\frac{A}{|\partial_\alpha z|^2}H\left(z_t\cdot (\partial_\alpha z)^\perp H(z_t\cdot (\partial_\alpha z)^\perp)\right) + \frac{[p]}{\rho^++\rho^-} + A g z_2 \bigg{]} \frac{(\partial_\alpha z)^\perp}{|\partial_\alpha z|^2} +z_t\cdot (\partial_\alpha z)^\perp\left(\frac{(\partial_\alpha z_t)^\perp}{|\partial_\alpha z|^2}-\frac{(\partial_\alpha z)^\perp 2(\partial_\alpha z\cdot \partial_\alpha z_t)}{|\partial_\alpha z|^4}\right)$. This more general RT $z$-model allows for interface turn-over. Numerical simulations of the $z$-model show an even better agreement with the predicted mixing layer growth for the ``rocket rig'' experiment