Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T07:13:06.286Z Has data issue: false hasContentIssue false
  • English
  • Français

Viscous gravity currents over flat inclined surfaces

Published online by Cambridge University Press:  23 November 2021

Herbert E. Huppert Open the ORCID record for Herbert E. Huppert [Opens in a new window] ,
Vitaly A. Kuzkin Open the ORCID record for Vitaly A. Kuzkin [Opens in a new window]  and
Svetlana O. Kraeva Open the ORCID record for Svetlana O. Kraeva [Opens in a new window]
Herbert E. Huppert
Affiliation:
Institute of Theoretical Geophysics, King's College, Cambridge CB2 1ST, UK
Vitaly A. Kuzkin
Affiliation:
Department of Theoretical Mechanics, Peter the Great St Petersburg Polytechnic University, St Petersburg 195251, Russia Laboratory for Discrete Models in Mechanics, Institute for Problems in Mechanical Engineering RAS, St Petersburg 199178, Russia
Svetlana O. Kraeva*
Affiliation:
Department of Theoretical Mechanics, Peter the Great St Petersburg Polytechnic University, St Petersburg 195251, Russia
*
†Email address for correspondence: kraeva.so96@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Previous analyses of the flow of low-Reynolds-number, viscous gravity currents down inclined planes are investigated further and extended. Particular emphasis is on the motion of the fluid front and tail, which previous analyses treated somewhat cavalierly. We obtain reliable, approximate, analytic solutions in these regions, the accuracies of which are satisfactorily tested against our numerical evaluations. The solutions show that the flow has several time scales determined by the inclination angle, $\alpha$. At short times, the influence of initial and boundary conditions is important and the flow is governed by both the pressure gradient and the direct action of gravity due to inclination. Later on, the areas where the boundary conditions are important shrink. This fact explains why previous solutions, being inaccurate near the front and the tail, described experimental data with high accuracy. At larger times, of the order of $\alpha ^{-5/2}$, the influence of the pressure gradient may be neglected and the fluid profile converges to the square-root shape predicted in previous works. Important extensions of our approach are also outlined.

JFM classification

Geophysical and Geological Flows: Gravity currents Mathematical Foundations: General fluid mechanics Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 931 , 25 January 2022 , A12
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Fay, J.A. 1969 The spread of oil slicks on a calm sea. In Oil on the Sea (ed. D.P. Hoult), pp. 53–63. Plenum.CrossRefGoogle Scholar
Head, J.W., Campbell, D.B., Elachi, C., Guest, J.E., McKenzie, D.P., Saunders, R.S., Schaber, G.G. & Schubert, G. 1991 Venus volcanism: initial analysis from Magellan data. Science 252, 276–288.CrossRefGoogle ScholarPubMed
Hoult, D.P. 1972 Oil spreading on the sea. Annu. Rev. Fluid Mech. 4, 341–368.CrossRefGoogle Scholar
Huppert, H.E. 1982 Flow and instability of a viscous current down a slope. Nature 300 (5891), 427–429.CrossRefGoogle Scholar
Huppert, H.E. 1982 The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58.CrossRefGoogle Scholar
Huppert, H.E., Shepherd, J.B., Sigurdsson, H. & Sparks, R.S.J. 1982 On lava dome growth, with application to the 1979 lava extrusion of the soufriere of St. Vincent. J. Volcanol. Geotherm. Res. 14, 199–222.CrossRefGoogle Scholar
Huppert, H.E. & Simpson, J.E. 1980 The slumping of gravity currents. J. Fluid Mech. 99, 785–799.CrossRefGoogle Scholar
von Karman, T. 1940 The engineer grapples with nonlinear problems. Bull. Am. Math. Soc. 46, 615–683.CrossRefGoogle Scholar
Linkov, A.M. 2015 The particle velocity, speed equation and universal asymptotics for the efficient modelling of hydraulic fractures. J. Appl. Math. Mech. 79 (1), 54–63.CrossRefGoogle Scholar
Ungarish, M. 2020 Gravity Currents and Intrusions: Analysis and Predictions. World Scientific.CrossRefGoogle Scholar