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Self-similar gravity currents with variable inflow revisited: plane currents

Published online by Cambridge University Press:  26 April 2006

Julio Gratton  and
Claudio Vigo
Julio Gratton
Affiliation:
INFIP-CONICET, Departamento de Fisica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, 1428 Buenos Aires, Argentina
Claudio Vigo
Affiliation:
INFIP-CONICET, Departamento de Fisica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, 1428 Buenos Aires, Argentina
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Abstract

We use shallow-water theory to study the self-similar gravity currents that describe the intrusion of a heavy fluid below a lighter ambient fluid. We consider in detail the case of currents with planar symmetry produced by a source with variable inflow, such that the volume of the intruding fluid varies in time according to a power law of the type tα. The resistance of the ambient fluid is taken into account by a boundary condition of the von Kármán type, that depends on a parameter β that is a function of the density ratio of the fluids. The flow is characterized by β, α, and the Froude number [Fscr ]0 near the source. We find four kinds of self-similar solutions: subcritical continuous solutions (Type I), continuous solutions with a supercritical-subcritical transition (Type II), discontinuous solutions (Type III) that have a hydraulic jump, and discontinuous solutions having hydraulic jumps and a subcritical-supercritical transition (Type IV). The current is always subcritical near the front, but near the source it is subcritical ([Fscr ]0 < 1) for Type I currents, and supercritical ([Fscr ]0 > 1) for Types II, III, and IV. Type I solutions have already been found by other authors, but Type II, III, and IV currents are novel. We find the intervals of parameters for which these solutions exist, and discuss their properties. For constant-volume currents one obtains Type I solutions for any β that, when β > 2, have a ‘dry’ region near the origin. For steady inflow one finds Type I currents for O < β < ∞ and Type II and III Currents for and β, if [Fscr ]0 is sufficiently large.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 258 , 10 January 1994 , pp. 77 - 104
Copyright
© 1994 Cambridge University Press

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References

Barenblatt, G. I. 1979 Similarity, Self-Similarity, and Intermediate Asymptotics. Consultants Bureau.
Britter, R. E. 1979 The spread of a negatively buoyant plume in a calm environment. Atmos. Environ. 13, 1241–1247.Google Scholar
Courant, R. & Friedrichs, K. O. 1948 Supersonic Flow and Shock Waves. Interscience.
Fannelop, T. K. & Waldman, G. D. 1972 Dynamics of oil slicks. AIAA J. 10, 506–510.Google Scholar
Gratton, J. 1988 Corrientes de gravedad autosimilares. Unpublished report.
Gratton, J. 1991 Similarity and self similarity in fluid dynamics. Found. Cosmic Phys. 15, 1–106.Google Scholar
Gratton, J. & Minotti, F. 1990 Self-similar viscous gravity currents: phase-plane formalism. J. Fluid Mech. 210, 155–182.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 (referred to herein as GR).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 (referred to herein as GR).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
Kármán, von T. 1940 The engineer grapples with nonlinear problems. Bull. Am. Math. Soc. 46, 615–683.Google Scholar
Lamb, H. 1945 Hydrodynamics. Dover.
Landau, L. D. & Lifschitz, E. M. 1959 Fluid Mechanics. Pergamon.
Maxworthy, T. 1983 Gravity currents with variable inflow. J. Fluid Mech. 128, 247–257.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 release 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 fixedvolume release 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. Academic.
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
Stanyukovich, K. P. 1960 Unsteady Motion of Continuous Media. Pergamon.
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.
Yih, C. S. 1965 Dynamics of Nonhomogeneous Fluids. Macmillan.
Yih, C. S. & Guha, C. R. 1955 Hydraulic jumps in a fluid system of two layers. Tellus 7, 358–366.Google Scholar
Zel'dovich, Ya. B. & Raizer, Yu. P. 1968 Physics of Shock Waves and High Temperature Hydrodynamics Phenomena. Academic.