Nonlinear effects in buoyancy-driven variable density turbulence

Abstract

We consider the time-dependence of a hierarchy of scaled $L^{2m}$-norms $D_{m,\omega }$ and $D_{m,\theta }$ of the vorticity $\mathbf{\omega }=\nabla \times \mathbf{u}$ and the density gradient $\nabla \theta$, where $\theta =\log ({\rho }^{\ast }/{\rho }_0^{\ast })$, in a buoyancy-driven turbulent flow as simulated by \cite{LR2007}. ${\rho }^{\ast }(\mathbf{x},t)$ is the composition density of a mixture of two incompressible miscible fluids with fluid densities ${\rho }_2^{\ast }>{\rho }_1^{\ast }$ and ${\rho }_0^{\ast }$ 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 $\nabla \theta$ can reach extremely large values, even in flows with low Atwood number $At=({\rho }_2^{\ast }-{\rho }_1^{\ast })/({\rho }_2^{\ast }+{\rho }_1^{\ast })=0.05$, 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 $\nabla \theta$ might blow up in a finite time.

References in corpus (2)

• Potentially Singular Solutions of the 3D Incompressible Euler Equations
• Regimes of nonlinear depletion and regularity in the 3D Navier-Stokes equations

Cited by in corpus (3)

• Atwood and Reynolds numbers effects on the evolution of buoyancy-driven homogeneous variable-density turbulence
• A regularity criterion for solutions of the three-dimensional Cahn-Hilliard-Navier-Stokes equations and associated computations
• The role of BKM-type theorems in $3D$ Euler, Navier-Stokes and Cahn-Hilliard-Navier-Stokes analysis