Nonlinear effects in buoyancy-driven variable density turbulence

TL;DR

This paper investigates the nonlinear effects in buoyancy-driven turbulence, revealing that density gradients can become extremely large and potentially blow up in finite time, even at low Atwood numbers.

Contribution

It provides evidence that density gradients in buoyancy-driven turbulence can grow significantly, highlighting the importance of nonlinear effects in such flows.

Findings

Density gradients can reach very large values in buoyancy-driven turbulence.

The density gradient may blow up in finite time, indicating possible singularities.

Strong small-scale mixing of density occurs even at low Atwood numbers.

Abstract

We consider the time-dependence of a hierarchy of scaled $L^{2m}$-norms $D_{m,\omega}$ and $D_{m,\theta}$ of the vorticity $\boldsymbol {\omega} = \boldsymbol{\nabla} \times {\mathbf u}$ and the density gradient $\boldsymbol{\nabla} \theta$, where $\theta=\log (\rho^*/\rho^*_0)$, in a buoyancy-driven turbulent flow as simulated by \cite{LR2007}. $\rho^*({\mathbf 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. Using data from the publicly available Johns Hopkins Turbulence Database we present evidence that the $L^{2}$-spatial average of the density gradient $\boldsymbol{\nabla} \theta$ can reach extremely large values, even in flows with low Atwood number $At = (\rho^*_{2} - \rho^*_{1})/(\rho^*_{2} + \rho^*_{1}) = 0.05$, 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 $\boldsymbol{\nabla} \theta$ might blow up in a finite time.