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Nonlinear effects in buoyancy-driven variable-density turbulence

Published online by Cambridge University Press:  25 November 2016

P. Rao ,
C. P. Caulfield  and
J. D. Gibbon
P. Rao*
Affiliation:
Department of Applied Mathematics and Statistics, State University of New York, Stony Brook, NY 11790, USA
C. P. Caulfield
Affiliation:
BP Institute, University of Cambridge, Madingley Rise, Madingley Road, Cambridge CB3 0EZ, UK Department of Applied Mathematics & Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
J. D. Gibbon
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

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.

JFM classification

Convection: Buoyancy-driven instability Mathematical Foundations: Mathematical Foundations
Type
Papers
Information
Journal of Fluid Mechanics , Volume 810 , 10 January 2017 , pp. 362 - 377
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Copyright
© 2016 Cambridge University Press 

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