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Interaction of viscous free-surface flows with topography

Published online by Cambridge University Press:  08 August 2019

Edward M. Hinton Open the ORCID record for Edward M. Hinton [Opens in a new window] ,
Andrew J. Hogg Open the ORCID record for Andrew J. Hogg [Opens in a new window]  and
Herbert E. Huppert
Edward M. Hinton*
Affiliation:
BP Institute for Multiphase Flow, University of Cambridge, Madingley Road, Cambridge CB3 0EZ, UK Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK
Andrew J. Hogg
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
Herbert E. Huppert
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK
*
†Email address for correspondence: edward@bpi.cam.ac.uk
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Abstract

The interaction of gravitationally driven, free-surface flows of viscous fluid with topographic features is investigated theoretically. The motion is studied in the regime where the depth of the flow is much smaller than the streamwise extent of the topography. A lubrication model of the motion is developed, integrated numerically and analysed asymptotically. For small mounds, it is shown that the flow surmounts the obstacles, but for larger mounds the flow is deflected around it and can form dry zones in its wake into which fluid does not flow, as well as forming deeper ponded regions upstream. Which of these phenomena prevails is shown to depend upon the amplitude of the mound height and the thickness of the oncoming flow relative to the streamwise length scale over which the topography varies. By using numerical and asymptotic results, we demonstrate that relatively wide mounds lead to the development of deep ponds of material upstream, which may lead to flow overtopping if the mound is not sufficiently high. These insights can be used to inform the design of barriers that defend built infrastructures from lava flows, and it is shown how this model can also provide an upper bound on the force exerted by the flow on them.

JFM classification

Geophysical and Geological Flows: Magma and lava flow Geophysical and Geological Flows: Topographic effects
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 876 , 10 October 2019 , pp. 912 - 938
Copyright
© 2019 Cambridge University Press 

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