Skip to main content Accessibility help

Inertial gravity currents produced by fluid drainage from an edge

  • Mostafa Momen (a1) (a2), Zhong Zheng (a3) (a4), Elie Bou-Zeid (a2) and Howard A. Stone (a4)


We present theoretical, numerical and experimental studies of the release of a finite volume of fluid instantaneously from an edge of a rectangular domain for high Reynolds number flows. For the cases we considered, the results indicate that approximately half of the initial volume exits during an early adjustment period. Then, the inertial gravity current reaches a self-similar phase during which approximately 40 % of its volume drains and its height decreases as $\unicode[STIX]{x1D70F}^{-2}$ , where $\unicode[STIX]{x1D70F}$ is a dimensionless time that is derived with the typical gravity wave speed and the horizontal length of the domain. Based on scaling arguments, we reduce the shallow-water partial differential equations into two nonlinear ordinary differential equations (representing the continuity and momentum equations), which are solved analytically by imposing a zero velocity boundary condition at the closed end wall and a critical Froude number condition at the open edge. The solutions are in good agreement with the performed experiments and direct numerical simulations for various geometries, densities and viscosities. This study provides new insights into the dynamical behaviour of a fluid draining from an edge in the inertial regime. The solutions may be useful for environmental, geophysical and engineering applications such as open channel flows, ventilations and dam-break problems.


Corresponding author

Email addresses for correspondence:,


Hide All
Armi, L. & Farmer, D. M. 1986 Maximal two-layer exchange through a contraction with barotropic net flow. J. Fluid Mech. 164, 2751.
Baines, W. D., Rottman, J. W. & Simpson, J. E. 1985 The motion of constant-volume air cavities in long horizontal tubes. J. Fluid Mech. 161, 313327.
Balmforth, N. J., Hardenberg, J. V. & Zammett, R. J. 2009 Dam-breaking seiches. J. Fluid Mech. 628, 121.
Beghin, P., Hopfinger, E. & Britter, R. 1981 Gravitational convection from instantaneous sources on inclined planes. J. Fluid Mech. 107, 407422.
Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209248.
Berberovic, E., van Hinsberg, N. P., Jakirlic, S., Roisman, I. V. & Tropea, C. 2008 Drop impact onto a liquid layer of finite thickness: dynamics of the cavity evolution. Phys. Rev. E 79, 036306.
Borden, Z. & Meiburg, E. 2013 Circulation based models for Boussinesq gravity currents. Phys. Fluids 25, 101301.
Britter, R. & Linden, P. 1980 The motion of the front of a gravity current travelling down an incline. J. Fluid Mech. 99, 531543.
Clarke, N. S. 1964 On two-dimensional inviscid flow in a waterfall. J. Fluid Mech. 22, 359369.
Dai, A. 2014 Non-Boussinesq gravity currents propagating on different bottom slopes. J. Fluid Mech. 741, 658680.
Deshpande, S. S., Anumolu, L. & Trujillo, M. F. 2012a Evaluating the performance of the two-phase flow solver interFoam. Comput. Sci. Disc. 5, 014016.
Deshpande, S. S., Trujillo, M. F., Wu, X. & Chahine, G. L. 2012b Computational and experimental characterization of a liquid jet plunging into a quiescent pool at shallow inclination. Intl J. Heat Fluid Flow 34, 114.
Dias, F. & Tuck, E. O. 1991 Weir flows and waterfalls. J. Fluid Mech. 230, 525539.
Ellison, T. & Turner, J. 1959 Turbulent entrainment in stratified flow. J. Fluid Mech. 6, 423448.
Gladstone, C., Ritchie, L. J., Sparks, R. S. J. & Woods, A. W. 2004 An experimental investigation of density-stratified inertial gravity currents. Sedimentology 51, 767789.
Gratton, J. & Vigo, C. 1994 Self-similar gravity currents with variable inflow revisited: plane currents. J. Fluid Mech. 258, 77104.
Grobelbauer, H. P., Fannelop, T. K. & Britter, R. E. 1993 The propagation of intrusion fronts of high density ratios. J. Fluid Mech. 250, 669687.
Grundy, R. E. & Rottman, J. W. 1985 The approach to self-similarity of the solutions of the shallow-water equations representing gravity-current releases. J. Fluid Mech. 51, 3953.
Hallworth, M., Huppert, H. H. & Ungarish, M. 2003 On inwardly propagating high-Reynolds-number axisymmetric gravity currents. J. Fluid Mech. 494, 225274.
Hogg, A. J. & Pritchard, D. 2003 The effects of hydraulic resistence on dam-break and other shallow inertial flows. J. Fluid Mech. 501, 179212.
Hoult, D. 1972 Oil spreading in the sea. Annu. Rev. Fluid Mech. 4, 341368.
Huppert, H. E. & Simpson, J. E. 1980 The slumping of gravity currents. J. Fluid Mech. 99, 785799.
Huppert, E. H. & Woods, A. W. 1995 Gravity-driven flows in porous layers. J. Fluid Mech. 292, 5569.
Issa, R. I. 1986 Solution of the implicitly discretised fluid flow equations by operator splitting. J. Comput. Phys. 62, 4065.
Liu, X. & Garcia, M. H. 2008 Three-dimensional numerical model with free water surface and mesh deformation for local sediment scour. ASCE J. Waterway Port Coastal Ocean Engng 134, 203217.
Lowe, R. J., Rottman, J. W. & Linden, P. F. 2005 The non-Boussinesq lock-exchange problem. Part 1. Theory and experiments. J. Fluid Mech. 537, 101124.
Marino, B. M., Thomas, L. P. & Linden, P. F. 2005 The front condition for gravity currents. J. Fluid Mech. 536, 4978.
Maxworthy, T., Leilich, J., Simpson, J. E. & Meiburg, E. H. 2002 The propagation of a gravity current into a linearly stratified fluid. J. Fluid Mech. 453, 371394.
Meiburg, E., Radhakrishnan, S. & Nasr-Azadani, M. 2015 Modeling gravity and turbidity currents: computational approaches and challenges. Appl. Mech. Rev. 67, 0408021.
Momen, M. & Bou-Zeid, E. 2017 Mean and turbulence dynamics in unsteady Ekman boundary layers. J. Fluid Mech. 816, 209242.
Monaghan, J. J., Meriaux, C. A., Huppert, H. E. & Monaghan, J. M. 2009 High Reynolds number gravity currents along V-shaped valleys. Eur. J. Mech. (B/Fluids) 135, 95110.
Moukalled, F., Mangani, L. & Darwish, M. 2015 The Finite Volume Method in Computational Fluid Dynamics: An Advanced Introduction with OpenFOAM and Matlab. Springer.
Naghdi, P. M. & Rubin, M. B. 1981 On inviscid flow in a waterfall. J. Fluid Mech. 103, 375387.
Rhie, C. M. & Chow, W. L. 1983 Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA J. 21, 15251532.
Rooney, G. G. & Linden, P. F. 1996 Similarity considerations for non-Boussinesq plumes in an unstratified environment. J. Fluid Mech. 318, 237250.
Ross, A. N., Linden, P. F. & Dalziel, S. B. 2002 A study of three-dimensional gravity currents on a uniform slope. J. Fluid Mech. 453, 239261.
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, 95110.
Rotunno, R., Klemp, J. B., Bryan, G. H. & Muraki, D. J. 2011 Models of non-Boussinesq lock-exchange flow. J. Fluid Mech. 675, 126.
Sher, D. & Woods, A. W. 2015 Gravity currents: entrainment, stratification and self-similarity. J. Fluid Mech. 521, 134.
Simpson, J. E. 1997 Gravity Currents in the Environment and the Laboratory. Cambridge University Press.
Thomas, L. P., Marino, B. M. & Linden, P. F. 2004 Lock-release inertial gravity currents over a thick porous layer. J. Fluid Mech. 503, 299319.
Tokyay, T., Constantinescu, G. & Meiburg, E. 2011 Lock-exchange gravity currents with a high volume of release propagating over a periodic array of obstacles. J. Fluid Mech. 672, 570605.
Ungarish, M. 2007 A shallow water model for high-Reynolds gravity currents for a wide range of density differences and fractional depths. J. Fluid Mech. 579, 373382.
Ungarish, M. 2009 An Introduction to Gravity Currents and Intrusions. Taylor and Francis Group.
Ungarish, M. 2010 The propagation of high-Reynolds-number non-Boussinesq gravity currents in axisymmetric geometry. J. Fluid Mech. 643, 267277.
Ungarish, M. 2011 Two-layer shallow-water dam-break solutions for non-Boussinesq gravity currents in a wide range of fractional depth. J. Fluid Mech. 675, 2759.
Ungarish, M. 2013a Gravity Currents and Intrusions: Handbook of Environmental Fluid Dynamics (ed. Fernando, H. J. S.), Chapman and Hall/CRC Press.
Ungarish, M. 2013b Two-layer shallow-water dam-break solutions for gravity currents in non- rectangular cross-area channels. J. Fluid Mech. 732, 537570.
Ungarish, M. & Huppert, H. E. 2000 High-Reynolds-number gravity currents over a porous boundary: shallow-water solutions and box-model approximations. J. Fluid Mech. 418, 123.
White, B. L. & Helfrich, K. L. 2008 Gravity currents and internal waves in a stratified fluid. J. Fluid Mech. 616, 327356.
Wilkinson, D. L. 1982 Motion of air cavities in long horizontal ducts. J. Fluid Mech. 118, 109122.
Zemach, T. & Ungarish, M. 2013 Gravity currents in non-rectangular cross-section channels: analytical and numerical solutions of the one-layer shallow-water model for high-Reynolds-number propagation. Phys. Fluids 25, 124.
Zheng, Z., Guo, B., Christov, I. C., Celia, M. A. & Stone, H. A. 2015 Flow regimes for fluid injection into a confined porous medium. J. Fluid Mech. 767, 881909.
Zheng, Z., Soh, B., Huppert, H. E. & Stone, H. A. 2013 Fluid drainage from the edge of a porous reservoir. J. Fluid Mech. 718, 558568.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO
Type Description Title
Supplementary materials

Momen et al supplementary material 1
Momen et al supplementary material

 Unknown (17.7 MB)
17.7 MB

Inertial gravity currents produced by fluid drainage from an edge

  • Mostafa Momen (a1) (a2), Zhong Zheng (a3) (a4), Elie Bou-Zeid (a2) and Howard A. Stone (a4)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.