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Unconfined turbulent entrainment across density interfaces

Published online by Cambridge University Press:  23 September 2014

Ajay B. Shrinivas  and
Gary R. Hunt
Ajay B. Shrinivas
Affiliation:
Department of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, UK
Gary R. Hunt*
Affiliation:
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
*
†Email address for correspondence: gary.hunt@eng.cam.ac.uk
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Abstract

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We present theoretical models describing the quasi-steady downward transport of buoyant fluid across a gravitationally stable density interface separating two unbounded quiescent fluid masses. The primary transport mechanism is turbulent entrainment resulting from the localised impingement of a vertically forced high-Reynolds-number axisymmetric jet with steady source conditions. The entrainment across the interface is examined in the large-time asymptotic state, wherein the interfacial gravity current, formed by the fluid entrained from the upper layer and the jet, becomes infinitesimally thin and a two-layer stratification persists. Characterising flows with small interfacial Froude numbers $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}({{\mathrm{Fr}}}_i)$ as an axisymmetric semi-ellipsoidal impingement dome, we combine conservation equations with a mechanistic model of entrainment and reveal that, in this regime, the dimensionless entrainment flux $E_i$ across the interface follows the power law $E_i = 0.24{{\mathrm{Fr}}}_i^2$. For large-${{\mathrm{Fr}}}_i$ impingements, modelled as a fully penetrating turbulent fountain, we show that $E_i$ no longer scales with ${{\mathrm{Fr}}}_i^2$, but linearly on ${{\mathrm{Fr}}}_i$, following $E_i = 0.42{{\mathrm{Fr}}}_i$. We establish the intermediate range of ${{\mathrm{Fr}}}_i$ over which there is a transition between these quadratic and linear power laws, thus enabling us to classify the dynamics of entrainment across the interface into three distinct regimes. Finally, the close agreement of our solutions with existing experimental results is illustrated.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Convection: Plumes/thermals Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 757 , 25 October 2014 , pp. 573 - 598
Copyright
© 2014 Cambridge University Press 

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