3 - The Rayleigh–Taylor instability in buoyancy-driven variable density turbulence

Summary

This paper reviews and summarizes two recent pieces of work on the Rayleigh-Taylor instability. The first concerns the 3D Cahn-Hilliard-Navier-Stokes (CHNS) equations and the BKM-type theorem proved by Gibbon, Pal, Gupta, & Pandit (2016). The second and more substantial topic concerns the variable density model, which is a buoyancy-driven turbulent flow considered by Cook & Dimotakis (2001) and Livescu & Ristorcelli (2007, 2008). In this model $\rho^* (x, t)$ is the composition density of a mixture of two incompressible miscible fluids with fluid densities $$\rho^*_2 > \rho^*_1$$ and $$\rho^*_0$$ is a reference normalisation density. Following the work of a previous paper (Rao, Caulfield, & Gibbon, 2017), which used the variable $$\theta = \ln \rho^*/\rho^*_0$$, data from the publicly available Johns Hopkins Turbulence Database suggests that the L2-spatial average of the density gradient $$\nabla \theta$$ can reach extremely large values at intermediate times, even in flows with low Atwood number At = $$(\rho^*_2 - \rho^*_1)/(\rho^*_2 + \rho^*_1) = 0.05$$. This implies that very strong mixing of the density field at small scales can potentially arise in buoyancy-driven turbulence thus raising the possibility that the density gradient $$\nabla \theta$$ might blow up in a finite time.