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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 Open the ORCID record for Maarten van Reeuwijk [Opens in a new window] ,
Markus Holzner  and
C. P. Caulfield Open the ORCID record for C. P. Caulfield [Opens in a new window]
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
*
†Email address for correspondence: m.vanreeuwijk@imperial.ac.uk
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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.

JFM classification

Geophysical and Geological Flows: Gravity currents Geophysical and Geological Flows: Stratified flows Mixing: Turbulent mixing

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Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 873 , 25 August 2019 , pp. 786 - 815
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© 2019 Cambridge University Press 

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