Nonlinear effects in buoyancy-driven variable density turbulence

We consider the time-dependence of a hierarchy of scaled $L^{2m}$-norms $D_{m,ω}$ and $D_{m,θ}$ of the vorticity $\boldsymbol ω = \boldsymbol{\nabla} \times {\mathbf u}$ and the density gradient $\boldsymbol{\nabla} θ$, where $θ=\log (ρ^*/ρ^*_0)$, in a buoyancy-driven turbulent flow as simulated by \cite{LR2007}. $ρ^*({\mathbf x},\,t)$ is the composition density of a mixture of two incompressible miscible fluids with fluid densities $ρ^*_2 > ρ^*_1$ and $ρ^*_{0}$ is a reference normalisation density. Using data from the publicly available Johns Hopkins Turbulence Database we present evidence that the $L^{2}$-spatial average of the density gradient $\boldsymbol{\nabla} θ$ can reach extremely large values, even in flows with low Atwood number $At = (ρ^*_{2} - ρ^*_{1})/(ρ^*_{2} + ρ^*_{1}) = 0.05$, implying that very strong mixing of the density field at small scales can arise in buoyancy-driven turbulence. This large growth raises the possibility that the density gradient $\boldsymbol{\nabla} θ$ might blow up in a finite time.

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