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Shock formation in two-layer equal-density viscous gravity currents

Published online by Cambridge University Press:  25 January 2019

Tim-Frederik Dauck Open the ORCID record for Tim-Frederik Dauck [Opens in a new window] ,
Finn Box Open the ORCID record for Finn Box [Opens in a new window] ,
Laura Gell ,
Jerome A. Neufeld Open the ORCID record for Jerome A. Neufeld [Opens in a new window]  and
John R. Lister Open the ORCID record for John R. Lister [Opens in a new window]
Tim-Frederik Dauck*
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK
Finn Box
Affiliation:
BP Institute, University of Cambridge, Cambridge CB3 0EZ, UK Bullard Laboratories, Department of Earth Sciences, University of Cambridge, Cambridge CB3 0WA, UK
Laura Gell
Affiliation:
BP Institute, University of Cambridge, Cambridge CB3 0EZ, UK
Jerome A. Neufeld
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK BP Institute, University of Cambridge, Cambridge CB3 0EZ, UK Bullard Laboratories, Department of Earth Sciences, University of Cambridge, Cambridge CB3 0WA, UK
John R. Lister
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK
*
†Email address for correspondence: tfd23@cam.ac.uk
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Abstract

The flow of a viscous gravity current over a lubricating layer of fluid is modelled using lubrication theory. We study the case of an axisymmetric current with constant influx which allows for a similarity solution, which depends on three parameters: a non-dimensional influx rate ${\mathcal{Q}}$; a viscosity ratio $m$ between the lower and upper layer fluid; and a relative density difference $\unicode[STIX]{x1D700}$. The limit of equal densities $\unicode[STIX]{x1D700}=0$ is singular, as the interfacial evolution equation changes nature from parabolic to hyperbolic. Theoretical analysis of this limit reveals that a discontinuity, or shock, in the interfacial height forms above a critical viscosity ratio $m_{crit}=3/2$, i.e. for a sufficiently less viscous upper-layer fluid. The physical mechanism for shock formation is described, which is based on advective steepening of the interface between the two fluids and relies on the lack of a contribution to the pressure gradient from the interfacial slope for equal-density fluids. In the limit of small but non-zero density differences, local travelling-wave solutions are found which regularise the singular structure of a potential shock and lead to a constraint on the possible shock heights in the form of an Oleinik entropy condition. Calculation of a simplified time-dependent system reveals the appropriate boundary conditions for the late-time similarity solution, which includes a shock at the nose of the current for $m>3/2$. The numerically calculated similarity solutions compare well to experimental measurements with respect to the predictions of self-similarity, the radial extent and the self-similar top-surface shapes of the current.

JFM classification

Geophysical and Geological Flows: Gravity currents Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 863 , 25 March 2019 , pp. 730 - 756
Copyright
© 2019 Cambridge University Press 

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