Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-20T08:27:38.807Z Has data issue: false hasContentIssue false
  • English
  • Français

Energy balances for axisymmetric gravity currents in homogeneous and linearly stratified ambients

Published online by Cambridge University Press:  10 December 2008

MARIUS UNGARISH  and
HERBERT E. HUPPERT
MARIUS UNGARISH
Affiliation:
Department of Computer Science, Technion, Haifa 32000, Israel
HERBERT E. HUPPERT
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Rd, Cambridge CB3 0WA, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

We analyse the exchange of energy for an axisymmetric gravity current, released instantaneously from a lock, propagating over a horizontal boundary at high Reynolds number. The study is relevant to flow in either a wedge or a full circular geometry. Attention is focused on effects due to a linear stratification in the ambient. The investigation uses both a one-layer shallow-water model and Navier–Stokes finite-difference simulations. There is fair agreement between these two approaches for the energy changes of the dense fluid (the current). The stratification enhances the accumulation of potential energy in the ambient and reduces the energy decay (dissipation) of the two-fluid system. The total energy of the axisymmetric current decays considerably faster with distance of propagation than for the two-dimensional counterpart.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 616 , 10 December 2008 , pp. 303 - 326
Copyright
Copyright © Cambridge University Press 2008

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

Amen, R. & Maxworthy, T. 1980 The gravitational collapse of a mixed region into a linearly stratified fluid. J. Fluid Mech. 96, 65–80.CrossRefGoogle Scholar
Baines, P. G. 1995 Topographic Effects in Stratified Flows. Cambridge University Press.Google Scholar
Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209–248.CrossRefGoogle Scholar
Bonometti, T. & Balachandar, S. 2008 Effect of Schmidt number on the structure and propagation of density currents. Theor. Comput. Fluid Dyn. 22, 341–361.CrossRefGoogle Scholar
Cantero, M., Balachandar, S. & Garcia, M. H. 2007 High resolution simulations of cylindrical density currents. J. Fluid Mech. 590, 437–469.CrossRefGoogle Scholar
Faust, K. M. & Plate, E. J. 1984 Experimental investigation of intrusive gravity currents entering stably stratified fluids. J. Hydraulic Res. 22, 315–325.CrossRefGoogle Scholar
Flynn, M. R. & Sutherland, B. R. 2004 Intrusive gravity currents and internal gravity wave generation in stratified fluid. J. Fluid Mech. 514, 355–383.CrossRefGoogle Scholar
Grundy, R. E. & Rottman, J. 1985 The approach to self-similarity of the solutions of the shallow-water equations representing gravity current releases. J. Fluid Mech. 156, 39–53.CrossRefGoogle Scholar
Hallworth, M. A., Huppert, H. E. & Ungarish, M. 2001 Axisymmetric gravity currents in a rotating system: experimental and numerical investigations. J. Fluid Mech. 447, 1–29.CrossRefGoogle Scholar
Huppert, H. E. 2000 Geological fluid mechanics. In Perspectives in Fluid Dynamics: A Collective Introduction to Current Research (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G.), pp. 447–506. Cambridge University Press.Google Scholar
Huppert, H. E. 2006 Gravity currents: a personal perspective. J. Fluid Mech. 554, 299–322.CrossRefGoogle Scholar
Huppert, H. & Dade, W. 1998 Natural disasters: Explosive volcanic eruptions and gigantic landslides. Theor. Comput. Fluid Dyn. 10, 201–212.CrossRefGoogle Scholar
Huppert, H. E. & Simpson, J. E. 1980 The slumping of gravity currents. J. Fluid Mech. 99, 785–799.CrossRefGoogle Scholar
Kao, T. W. 1976 Principal stage of wake collapse in a stratified fluid: Two-dimensional theory. Phys. Fluids 19, 1071–1074.CrossRefGoogle Scholar
Manins, P. C. 1976 Mixed region collapse in a stratified fluid. J. Fluid Mech. 77, 177–183.CrossRefGoogle Scholar
Maxworthy, T. 1980 On the formation of nonlinear internal waves from the gravitational collapse of mixed regions in two and three dmensions. J. Fluid Mech. 96, 47–64.CrossRefGoogle Scholar
Maxworthy, T. 1983 Experiments on solitary internal Kelvin waves. J. Fluid Mech. 128, 365–383.CrossRefGoogle Scholar
Maxworthy, T., Leilich, J., Simpson, J. E. & Meiburg, E. H. 2002 The propagation of gravity currents in a linearly stratified fluid. J. Fluid Mech. 453, 371–394.CrossRefGoogle Scholar
Morton, K. W. & Mayers, D. F. 1994 Numerical Solutions of Partial Differential Equations. Cambridge University Press.Google Scholar
Necker, F., Härtel, C., Kleiser, L. & Meiburg, E. 2005 Mixing and disipation in particle-driven gravity currents. J. Fluid Mech. 545, 339–372.CrossRefGoogle Scholar
Patterson, M. D., Simpson, J. E., Dalziel, S. B. & van Heijst, G. J. F. 2006 Vortical motion in the head of an axisymmetric gravity current. Phys. Fluids 18, 046601.CrossRefGoogle Scholar
Press, W. H., Teukolski, S. A., Vetterling, W. T. & Flannery, B. P. 1992 Numerical Recipes in Fortran. Cambridge University Press.Google Scholar
de Rooij, F. 1999 Sedimenting particle-laden flows in confined geometries. PhD thesis, DAMTP, University of Cambridge.Google Scholar
Schooley, A. H. & Hughes, B. A. 1972 An experimental and theoretical study of internal waves generated by the collapse of a two-dimensional mixed region in a density gradient. J. Fluid Mech. 51, 159–175.CrossRefGoogle Scholar
Shin, J. O., Dalziel, S. B. & Linden, P. F. 2004 Gravity currents produced by lock exchange. J. Fluid Mech. 521, 1–34.CrossRefGoogle Scholar
Simpson, J. E. 1997 Gravity Currents in the Environment and the Laboratory, 2nd EdnCambridge University Press.Google Scholar
Stegner, A., Bouruet-Aubertot, P. & Pichon, T. 2004 Nonlinear adjustment of density fronts. Part 1. The Rossby scenario and the experimental reality. J. Fluid Mech. 502, 335–360.CrossRefGoogle Scholar
Ungarish, M. 2005 a Dam-break release of a gravity current in a stratified ambient. Eur. J. Mech. B/Fluids 24, 642–658.CrossRefGoogle Scholar
Ungarish, M. 2005 b Intrusive gravity currents in a stratified ambient-shallow-water theory and numerical results. J. Fluid Mech. 535, 287–323.CrossRefGoogle Scholar
Ungarish, M. 2006 On gravity currents in a linearly stratified ambient: a generalization of Benjamin's steady-state propagation results. J. Fluid Mech. 548, 49–68.CrossRefGoogle Scholar
Ungarish, M. 2007 Axisymmetric gravity currents at high Reynolds number – on the quality of shallow-water modeling of experimental observations. Phys. Fluids 19, 036602/7.CrossRefGoogle Scholar
Ungarish, M. 2008 Energy balances and front speed conditions of two-layer models for gravity currents produced by lock release. Acta Mechanica, (in press).CrossRefGoogle Scholar
Ungarish, M. & Huppert, H. E. 1999 Simple models of Coriolis-influenced axisymmetric particle-driven gravity currents. Int. J. Multiphase Flow 25, 715–737.CrossRefGoogle Scholar
Ungarish, M. & Huppert, H. E. 2002 On gravity currents propagating at the base of a stratified ambient. J. Fluid Mech. 458, 283–301.CrossRefGoogle Scholar
Ungarish, M. & Huppert, H. E. 2004 On gravity currents propagating at the base of a stratified ambient: effects of geometrical constraints and rotation. J. Fluid Mech. 521, 69–104.CrossRefGoogle Scholar
Ungarish, M. & Huppert, H. E. 2006 Energy balances for propagating gravity currents: homogeneous and stratified ambients. J. Fluid Mech. 565, 363–380.CrossRefGoogle Scholar
Ungarish, M. & Zemach, T. 2007 On axisymmetric intrusive gravity currents in a stratified ambient - shallow-water theory and numerical results. Eur. J. Mech. B/Fluids 26, 220–235.CrossRefGoogle Scholar
Winters, K. B., Lombard, P. N., Riley, J. J. & D'Asaro, E. A. 1995 Available potential energy and mixing in density-stratified fluids. J. Fluid Mech. 289, 115–128.CrossRefGoogle Scholar
Wu, J. 1969 Mixed region collapse with internal wave generation in a density-stratified medium. J. Fluid Mech. 35, 531–544.CrossRefGoogle Scholar
Zemach, T. & Ungarish, M. 2007 On axisymmetric intrusive gravity currents in a deep fully-linearly stratified ambient: the approach to self-similarity solutions of the shallow-water equations. Proc. R. Soc. Lond. 463, 2165–2183.Google Scholar