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Mixing and entrainment are suppressed in inclined gravity currents

Published online by Cambridge University Press:  28 June 2019

Maarten van Reeuwijk
Affiliation:
Department of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, UK
Markus Holzner
Affiliation:
Institute of Environmental Engineering, ETH Zürich, CH-8039 Zürich, Switzerland
C. P. Caulfield
Affiliation:
BP Institute, University of Cambridge, Madingley Rise, Madingley Road, Cambridge CB3 0EZ, UK Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
Corresponding

Abstract

We explore the dynamics of inclined temporal gravity currents using direct numerical simulation, and find that the current creates an environment in which the flux Richardson number $\mathit{Ri}_{f}$ , gradient Richardson number $\mathit{Ri}_{g}$ and turbulent flux coefficient $\unicode[STIX]{x1D6E4}$ are constant across a large portion of the depth. Changing the slope angle $\unicode[STIX]{x1D6FC}$ modifies these mixing parameters, and the flow approaches a maximum Richardson number $\mathit{Ri}_{max}\approx 0.15$ as $\unicode[STIX]{x1D6FC}\rightarrow 0$ at which the entrainment coefficient $E\rightarrow 0$ . The turbulent Prandtl number remains $O(1)$ for all slope angles, demonstrating that $E\rightarrow 0$ is not caused by a switch-off of the turbulent buoyancy flux as conjectured by Ellison (J. Fluid Mech., vol. 2, 1957, pp. 456–466). Instead, $E\rightarrow 0$ occurs as the result of the turbulence intensity going to zero as $\unicode[STIX]{x1D6FC}\rightarrow 0$ , due to the flow requiring larger and larger shear to maintain the same level of turbulence. We develop an approximate model valid for small $\unicode[STIX]{x1D6FC}$ which is able to predict accurately $\mathit{Ri}_{f}$ , $\mathit{Ri}_{g}$ and $\unicode[STIX]{x1D6E4}$ as a function of $\unicode[STIX]{x1D6FC}$ and their maximum attainable values. The model predicts an entrainment law of the form $E=0.31(\mathit{Ri}_{max}-\mathit{Ri})$ , which is in good agreement with the simulation data. The simulations and model presented here contribute to a growing body of evidence that an approach to a marginally or critically stable, relatively weakly stratified equilibrium for stratified shear flows may well be a generic property of turbulent stratified flows.

Type
JFM Papers
Copyright
© 2019 Cambridge University Press 

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