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

We 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. Under a certain stability condition, we prove that this so-called $h$-model is both locally and globally well-posed. 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 mixing layer with experimental data in the "rocket rig" experiment of Read of Youngs. We then derive an RT interface model with a general parameterization $z(\alpha,t)$ such that 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) + 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.