Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T22:25:52.568Z Has data issue: false hasContentIssue false
  • English
  • Français

Self-similar solutions of the shallow-water equations representing gravity currents with variable inflow

Published online by Cambridge University Press:  21 April 2006

R. E. Grundy  and
James W. Rottmant
R. E. Grundy
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK
James W. Rottmant
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK Present address: Fluid Modelling Facility MD-81, Atmospheric Sciences Research Laboratory, United States Environmental Protection Agency, Research Triangle Park, NC 2711, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

A phase-plane method is used to study the existence of similarity solutions of the two-dimensional and axisymmetric shallow-water equations representing gravity currents with volumes proportional to tα, where α ≥ 0 and t is the time after the flow is initiated. Only currents for which there is a balance between the inertia of the current and the driving buoyancy force are considered. It is found that similarity solutions exist for the two-dimensional problem for all α ≥ 0, with some restrictions on the condition at the current front. It is shown that no similarity solutions satisfying the boundary conditions on the axis of symmetry exist for the axisymmetric problem except when α = 0.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 169 , August 1986 , pp. 337 - 351
Copyright
© 1986 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

Britter, R. E. 1979 The spread of a negatively buoyant plume in a calm environment. Atmos. Environ. 13, 1241–1247.Google Scholar
Chen, J.-C. 1980 Studies on gravitational spreading currents. Ph.D. thesis, California Institute of Technology.
Courant, R. & Friedrichs, K. O. 1948 Supersonic Flow and Shock Waves. New York: Interscience.
Didden, N. & Maxworthy, T. 1982 The viscous spreading of plane and axisymmetric gravity currents. J. Fluid Mech. 121, 27–42.Google Scholar
Fannelop, T. K. & Waldman, G. D. 1972 Dynamics of oil slicks. AIAA J. 10, 506–510.Google Scholar
Garvine, R. W. 1984 Radial spreading of buoyant surface plumes. J. Geophys. Res. 89, 1989–1996.Google Scholar
Grundy, R. E. & Rottman, J. W. 1985 The approach to self-similarity of the solution of the shallow-water equations representing gravity current releases. J. Fluid Mech. 156, 39–53.Google Scholar
Guderley, K. G. 1942 Starke Kugelige und zylindrische Verdichtungsstosse in der Nahe des Kugelmittelpunktes bzw. der Zylinderachse. Luftfahrtforschung 19, No 9.Google Scholar
Hoult, D. P. 1972 Oil spreading on the sea. Ann. Rev. Fluid Mech. 4, 341–368.Google 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.Google Scholar
Huppert, H. E. & Simpson, J. E. 1980 The slumping of gravity currents. J. Fluid Mech. 90, 785–799.Google Scholar
Ivey, G. N. & Blake, S. 1985 Axisymmetric withdrawal and inflow in a density-stratified container. J. Fluid Mech. 161, 115–137.Google Scholar
Maxworthy, T. 1983 Gravity currents with variable inflow. J. Fluid Mech. 128, 247–257.Google Scholar
Mcquaid, J. 1985 Objectives and design of the Phase I heavy gas dispersion trials. J. Hazardous Materials, 11, 1–33.Google Scholar
Penney, W. G. & Thornhill, C. K. 1952 The dispersion, under gravity, of a column of fluid supported by a rigid horizontal plane. Phil. Trans. R. Soc. Lond. A 244, 285–311.Google Scholar
Rottman, J. W. & Simpson, J. E. 1983 Gravity currents produced by instantaneous releases of a heavy fluid in a rectangular channel. J. Fluid Mech. 135, 95–110.Google Scholar
Rottman, J. W. & Simpson, J. E. 1984 The initial development of gravity currents from fixed-volume releases of heavy fluids. Proc. IUTAM Symp. on Atmospheric Dispersion of Heavy Gases and Small Particles. Delft, The Netherlands.
Sedov, L. I. 1959 Similarity and Dimensional Methods in Mechanics. London: Infosearch.
Simpson, J. E. 1982 Gravity currents in the laboratory, atmosphere, and ocean. Ann. Rev. Fluid Mech. 14, 213–234.Google Scholar
Simpson, J. E. & Britter, R. E. 1979 The dynamics of the head of a gravity current advancing over a horizontal surface. J. Fluid Mech. 94, 477–495.Google Scholar