Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-12T02:05:22.041Z Has data issue: false hasContentIssue false
  • English
  • Français

Self-similar viscous gravity currents: phase-plane formalism

Published online by Cambridge University Press:  26 April 2006

Julio Gratton  and
Fernando Minotti
Julio Gratton
Affiliation:
LFP, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, 1428 Buenos Aires, Argentina
Fernando Minotti
Affiliation:
LFP, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, 1428 Buenos Aires, Argentina
Get access Rights & Permissions [Opens in a new window]

Abstract

A theoretical model for the spreading of viscous gravity currents over a rigid horizontal surface is derived, based on a lubrication theory approximation. The complete family of self-similar solutions of the governing equations is investigated by means of a phase-plane formalism developed in analogy to that of gas dynamics. The currents are represented by integral curves in the plane of two phase variables, Z and V, which are related to the depth and the average horizontal velocity of the fluid. Each integral curve corresponds to a certain self-similar viscous gravity current satisfying a particular set of initial and/or boundary conditions, and is obtained by solving a first-order ordinary differential equation of the form dV/dZ = f(Z, V), where f is a rational function. All conceivable self-similar currents can thus be obtained. A detailed analysis of the properties of the integral curves is presented, and asymptotic formulae describing the behaviour of the physical quantities near the singularities of the phase plane corresponding to sources, sinks, and current fronts are given. The derivation of self-similar solutions from the formalism is illustrated by several examples which include, in addition to the similarity flows studied by other authors, many other novel ones such as the extension to viscous flows of the classical problem of the breaking of a dam, the flows over plates with borders, as well as others. A self-similar solution of the second kind describing the axisymmetric collapse of a current towards the origin is obtained. The scaling laws for these flows are derived. Steady flows and progressive wave solutions are also studied and their connection to self-similar flows is discussed. The mathematical analogy between viscous gravity currents and other physical phenomena such as nonlinear heat conduction, nonlinear diffusion, and ground water motion is commented on.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 210 , January 1990 , pp. 155 - 182
Copyright
© 1990 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

Barenblatt, G. I. 1954 On the approximate solution of problems of uniform unsteady filtration in a porous medium, Prikl. Mat. i Mekh. 18, 351–370.Google Scholar
Barenblatt, G. I. 1979 Similarity, Self-similarity, and Intermediate Asymptotics. New York: Consultants Bureau.
Barenblatt, G. I. & Zel'dovich, Ya. B. 1957 On the dipole-type solution in problems of unsteady gas filtration in the polytropic regime. Prilk. Mat. i Mekh. 21, 718–720.Google Scholar
Barenblatt, G. I. & Zel'dovich, Ya. B. 1972 Self-similar solutions as intermediate asymptotics. Ann. Rev. Fluid Mech. 4, 285–312.Google Scholar
Bear, J. 1972 Dynamics of Fluids in Porous Media. Elsevier.
Boyer, R. H. 1962 On some solutions of a nonlinear diffusion equation. J. Maths and Phys. 40, 41.Google Scholar
Britter, R. E. 1979 The spread of a negatively buoyant plume in a calm environment. Atmos. Environ. 13, 1241–1247.Google Scholar
Buckmaster, J. 1977 Viscous sheets advancing over dry beds. J. Fluid Mech. 81, 735–756.Google Scholar
Courant, R. & Friedrichs, K. O. 1948 Supersonic Flow and Shock Waves. Interscience.
Didden, N. & Maxworthy, T. 1982 The viscous spreading of plane and axisymmetric gravity currents. J. Fluid Mech. 121, 27–42.Google Scholar
Fay, J. A. 1969 The spread of oil slicks on a calm sea. In Oil on the Sea (ed. D. P. Hoult), pp. 33–63. Plenum.
Grundy, R. E. 1979 Similarity solutions of the non linear diffusion equation. Q. Appl. Math., October 259–280.Google Scholar
Grundy, R. E. & Rottman, J. W. 1985 The approach to self-similarity of solutions of the shallow-water equations representing gravity current releases. J. Fluid Mech. 156, 39–53.Google Scholar
Grundy, R. E. & Rottman, J. W. 1986 Self-similar solutions of the shallow-water equations representing gravity currents with variable inflow. J. Fluid Mech. 169, 337–351.Google Scholar
Guderley, G. 1942 Starke Kugelige und Zylindrische Verdichtungstösse in der Nähe des Kugelmittelpunktes bzw. der Zylinderachse. Luftfahrtforschung 19, 302–312.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 viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58.Google Scholar
Huppert, H. E. 1986 The intrusion of fluid mechanics into geology. J. Fluid Mech. 173, 557–594.Google Scholar
Kath, W. L. & Cohen, D. S. 1982 Waiting-time behavior in a nonlinear diffusion equation. Stud. Appl. Maths 67, 79–105.Google Scholar
Kerr, R. C. & Lister, J. R. 1987 The spread of subducted lithospheric material along midmantle boundary. Earth Planet. Sci. Lett. 85, 241–247.Google Scholar
Lacey, A. A., Ockendon, J. R. & Tayler, A. B. 1982 ‘Waiting-time’ solutions of a nonlinear diffusion equation. J. Appl. Maths 42, 1252–1264.Google Scholar
Lopez, J., Miller, C. A. & Ruckenstein, E. 1976 Spreading kinetics of liquid drops on solids. J. Colloid and Interface Sci. 53, 460–468.Google Scholar
Maxworthy, T. 1983 Gravity currents with variable inflow. J. Fluid Mech. 128, 247–257.Google Scholar
Nakaya, C. 1974 Spread of fluid drops over a horizontal plane. J. Phys. Soc. Japan 37, 539–543.Google Scholar
Pattle, R. E. 1959 Diffusion from an instantaneous point source with a concentration coefficient. Q. J. Mech. Appl. Maths 12, 407–409.Google Scholar
Peletier, L. A. 1981 Applications of Nonlinear Analysis in the Physical Sciences. Boston: Pitman.
Pert, G. J. 1977 A class of similar solutions of the non-linear diffusion equation. J. Phys. A: Math. Gen. 10, 583–593.Google Scholar
Sedov, L. I. 1959 Similarity and Dimensional Methods in Mechanics. Academic.
Seshadri, R. & Na, T. Y. 1985 Group Invariance in Engineering Boundary Value Problems. Springer.
Simpson, J. E. 1982 Gravity currents in the laboratory, atmosphere, and ocean. Ann. Rev. Fluid Mech. 14, 341–368.Google Scholar
Smith, S. H. 1969 On initial value problems for the flow in a thin sheet of viscous liquid. Z. Angew. Math. Phys. 20, 556–560.Google Scholar
Smyth, N. F. & Hill, J. M. 1988 High-order nonlinear diffusion. IMA J. Appl. Maths 40, 73–86.Google Scholar
Tappert, F. D. 1977 Laser fusion. Proc. Symp. Appl. Maths 21, 55–74.Google Scholar
Vazquez, J. L. 1983 Free Boundary Problems: Theory and Applications, Vol. I. Boston: Pitman.
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
Zel'Dovich, Ya. B. & Raizer, Yu. P. 1967 Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena. Academic.