We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We consider the time dependence of a hierarchy of scaled 
$L^{2m}$-norms 
$D_{m,\unicode[STIX]{x1D714}}$ and 
$D_{m,\unicode[STIX]{x1D703}}$ of the vorticity 
$\unicode[STIX]{x1D74E}=\unicode[STIX]{x1D735}\times \boldsymbol{u}$ and the density gradient 
$\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}$, where 
$\unicode[STIX]{x1D703}=\log (\unicode[STIX]{x1D70C}^{\ast }/\unicode[STIX]{x1D70C}_{0}^{\ast })$, in a buoyancy-driven turbulent flow as simulated by Livescu & Ristorcelli (J. Fluid Mech., vol. 591, 2007, pp. 43–71). Here, 
$\unicode[STIX]{x1D70C}^{\ast }(\boldsymbol{x},t)$ is the composition density of a mixture of two incompressible miscible fluids with fluid densities 
$\unicode[STIX]{x1D70C}_{2}^{\ast }>\unicode[STIX]{x1D70C}_{1}^{\ast }$, and 
$\unicode[STIX]{x1D70C}_{0}^{\ast }$ is a reference normalization density. Using data from the publicly available Johns Hopkins turbulence database, we present evidence that the 
$L^{2}$-spatial average of the density gradient 
$\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}$ can reach extremely large values at intermediate times, even in flows with low Atwood number 
$At=(\unicode[STIX]{x1D70C}_{2}^{\ast }-\unicode[STIX]{x1D70C}_{1}^{\ast })/(\unicode[STIX]{x1D70C}_{2}^{\ast }+\unicode[STIX]{x1D70C}_{1}^{\ast })=0.05$, 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 
$\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}$ might blow up in a finite time.
The issue of intermittency in numerical solutions of the 3D Navier–Stokes equations on a periodic box 
${[0, L] }^{3} $ is addressed through four sets of numerical simulations that calculate a new set of variables defined by 
${D}_{m} (t)= {({ \varpi }_{0}^{- 1} {\Omega }_{m} )}^{{\alpha }_{m} } $ for 
$1\leq m\leq \infty $ where 
${\alpha }_{m} = 2m/ (4m- 3)$ and 
${[{\Omega }_{m} (t)] }^{2m} = {L}^{- 3} \int \nolimits _{\mathscr{V}} {\vert \boldsymbol{\omega} \vert }^{2m} \hspace{0.167em} \mathrm{d} V$ with 
${\varpi }_{0} = \nu {L}^{- 2} $. All four simulations unexpectedly show that the 
${D}_{m} $ are ordered for 
$m= 1, \ldots , 9$ such that 
${D}_{m+ 1} \lt {D}_{m} $. Moreover, the 
${D}_{m} $ squeeze together such that 
${D}_{m+ 1} / {D}_{m} \nearrow 1$ as 
$m$ increases. The values of 
${D}_{1} $ lie far above the values of the rest of the 
${D}_{m} $, giving rise to a suggestion that a depletion of nonlinearity is occurring which could be the cause of Navier–Stokes regularity. The first simulation is of very anisotropic decaying turbulence; the second and third are of decaying isotropic turbulence from random initial conditions and forced isotropic turbulence at fixed Grashof number respectively; the fourth is of very-high-Reynolds-number forced, stationary, isotropic turbulence at up to resolutions of 
$409{6}^{3} $.
