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Inertial gravity currents produced by fluid drainage from an edge

Published online by Cambridge University Press:  29 August 2017

Mostafa Momen Open the ORCID record for Mostafa Momen [Opens in a new window] ,
Zhong Zheng ,
Elie Bou-Zeid  and
Howard A. Stone
Mostafa Momen*
Affiliation:
Department of Earth System Science, Stanford University, Stanford, CA 94305, USA Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544, USA
Zhong Zheng
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Elie Bou-Zeid
Affiliation:
Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544, USA
Howard A. Stone*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

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.

JFM classification

Geophysical and Geological Flows: Gravity currents Waves/Free-surface Flows: Hydraulics Geophysical and Geological Flows: Shallow water flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 827 , 25 September 2017 , pp. 640 - 663
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Copyright
© 2017 Cambridge University Press 

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References

Armi, L. & Farmer, D. M. 1986 Maximal two-layer exchange through a contraction with barotropic net flow. J. Fluid Mech. 164, 27–51.CrossRefGoogle Scholar
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, 313–327.Google Scholar
Balmforth, N. J., Hardenberg, J. V. & Zammett, R. J. 2009 Dam-breaking seiches. J. Fluid Mech. 628, 1–21.Google Scholar
Beghin, P., Hopfinger, E. & Britter, R. 1981 Gravitational convection from instantaneous sources on inclined planes. J. Fluid Mech. 107, 407–422.Google Scholar
Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209–248.Google Scholar
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.Google Scholar
Borden, Z. & Meiburg, E. 2013 Circulation based models for Boussinesq gravity currents. Phys. Fluids 25, 101301.CrossRefGoogle Scholar
Britter, R. & Linden, P. 1980 The motion of the front of a gravity current travelling down an incline. J. Fluid Mech. 99, 531–543.CrossRefGoogle Scholar
Clarke, N. S. 1964 On two-dimensional inviscid flow in a waterfall. J. Fluid Mech. 22, 359–369.Google Scholar
Dai, A. 2014 Non-Boussinesq gravity currents propagating on different bottom slopes. J. Fluid Mech. 741, 658–680.CrossRefGoogle Scholar
Deshpande, S. S., Anumolu, L. & Trujillo, M. F. 2012a Evaluating the performance of the two-phase flow solver interFoam. Comput. Sci. Disc. 5, 014016.CrossRefGoogle Scholar
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, 1–14.CrossRefGoogle Scholar
Dias, F. & Tuck, E. O. 1991 Weir flows and waterfalls. J. Fluid Mech. 230, 525–539.CrossRefGoogle Scholar
Ellison, T. & Turner, J. 1959 Turbulent entrainment in stratified flow. J. Fluid Mech. 6, 423–448.CrossRefGoogle Scholar
Gladstone, C., Ritchie, L. J., Sparks, R. S. J. & Woods, A. W. 2004 An experimental investigation of density-stratified inertial gravity currents. Sedimentology 51, 767–789.Google Scholar
Gratton, J. & Vigo, C. 1994 Self-similar gravity currents with variable inflow revisited: plane currents. J. Fluid Mech. 258, 77–104.Google Scholar
Grobelbauer, H. P., Fannelop, T. K. & Britter, R. E. 1993 The propagation of intrusion fronts of high density ratios. J. Fluid Mech. 250, 669–687.Google Scholar
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, 39–53.CrossRefGoogle Scholar
Hallworth, M., Huppert, H. H. & Ungarish, M. 2003 On inwardly propagating high-Reynolds-number axisymmetric gravity currents. J. Fluid Mech. 494, 225–274.Google Scholar
Hogg, A. J. & Pritchard, D. 2003 The effects of hydraulic resistence on dam-break and other shallow inertial flows. J. Fluid Mech. 501, 179–212.Google Scholar
Hoult, D. 1972 Oil spreading in the sea. Annu. Rev. Fluid Mech. 4, 341–368.CrossRefGoogle Scholar
Huppert, H. E. & Simpson, J. E. 1980 The slumping of gravity currents. J. Fluid Mech. 99, 785–799.CrossRefGoogle Scholar
Huppert, E. H. & Woods, A. W. 1995 Gravity-driven flows in porous layers. J. Fluid Mech. 292, 55–69.Google Scholar
Issa, R. I. 1986 Solution of the implicitly discretised fluid flow equations by operator splitting. J. Comput. Phys. 62, 40–65.Google Scholar
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, 203–217.Google Scholar
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, 101–124.Google Scholar
Marino, B. M., Thomas, L. P. & Linden, P. F. 2005 The front condition for gravity currents. J. Fluid Mech. 536, 49–78.CrossRefGoogle Scholar
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, 371–394.Google Scholar
Meiburg, E., Radhakrishnan, S. & Nasr-Azadani, M. 2015 Modeling gravity and turbidity currents: computational approaches and challenges. Appl. Mech. Rev. 67, 0408021.Google Scholar
Momen, M. & Bou-Zeid, E. 2017 Mean and turbulence dynamics in unsteady Ekman boundary layers. J. Fluid Mech. 816, 209–242.CrossRefGoogle Scholar
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, 95–110.Google Scholar
Moukalled, F., Mangani, L. & Darwish, M. 2015 The Finite Volume Method in Computational Fluid Dynamics: An Advanced Introduction with OpenFOAM and Matlab. Springer.Google Scholar
Naghdi, P. M. & Rubin, M. B. 1981 On inviscid flow in a waterfall. J. Fluid Mech. 103, 375–387.Google Scholar
Rhie, C. M. & Chow, W. L. 1983 Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA J. 21, 1525–1532.CrossRefGoogle Scholar
Rooney, G. G. & Linden, P. F. 1996 Similarity considerations for non-Boussinesq plumes in an unstratified environment. J. Fluid Mech. 318, 237–250.Google Scholar
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, 239–261.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
Rotunno, R., Klemp, J. B., Bryan, G. H. & Muraki, D. J. 2011 Models of non-Boussinesq lock-exchange flow. J. Fluid Mech. 675, 1–26.CrossRefGoogle Scholar
Sher, D. & Woods, A. W. 2015 Gravity currents: entrainment, stratification and self-similarity. J. Fluid Mech. 521, 1–34.Google Scholar
Simpson, J. E. 1997 Gravity Currents in the Environment and the Laboratory. Cambridge University Press.Google Scholar
Thomas, L. P., Marino, B. M. & Linden, P. F. 2004 Lock-release inertial gravity currents over a thick porous layer. J. Fluid Mech. 503, 299–319.Google Scholar
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, 570–605.Google Scholar
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, 373–382.Google Scholar
Ungarish, M. 2009 An Introduction to Gravity Currents and Intrusions. Taylor and Francis Group.Google Scholar
Ungarish, M. 2010 The propagation of high-Reynolds-number non-Boussinesq gravity currents in axisymmetric geometry. J. Fluid Mech. 643, 267–277.Google Scholar
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, 27–59.CrossRefGoogle Scholar
Ungarish, M. 2013a Gravity Currents and Intrusions: Handbook of Environmental Fluid Dynamics (ed. Fernando, H. J. S.), Chapman and Hall/CRC Press.Google Scholar
Ungarish, M. 2013b Two-layer shallow-water dam-break solutions for gravity currents in non- rectangular cross-area channels. J. Fluid Mech. 732, 537–570.Google Scholar
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, 1–23.Google Scholar
White, B. L. & Helfrich, K. L. 2008 Gravity currents and internal waves in a stratified fluid. J. Fluid Mech. 616, 327–356.Google Scholar
Wilkinson, D. L. 1982 Motion of air cavities in long horizontal ducts. J. Fluid Mech. 118, 109–122.CrossRefGoogle Scholar
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, 1–24.Google Scholar
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, 881–909.Google Scholar
Zheng, Z., Soh, B., Huppert, H. E. & Stone, H. A. 2013 Fluid drainage from the edge of a porous reservoir. J. Fluid Mech. 718, 558–568.Google Scholar
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