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A shallow-water model for high-Reynolds-number gravity currents for a wide range of density differences and fractional depths

Published online by Cambridge University Press:  02 May 2007

MARIUS UNGARISH
MARIUS UNGARISH*
Affiliation:
Department of Computer Science, Technion, Haifa 32000, Israel
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Abstract

We consider the propagation of a gravity current of density ρc from a lock of length x0 and height h0 into an ambient fluid of density ρa in a channel of height H. When the Reynolds number is large, the resulting flow is governed by the parameters ρc/ρa and H* = H/h0. We show that the shallow-water one-layer model, combined with a Benjamin-type front condition, provides a versatile formulation for the thickness and speed of the current, without any adjustable constants. The results cover in a continuous manner the range of light ρc/ρa ≪ 1, Boussinesq ρc/ρa ≈ 1, and heavy ρc/ρa ≫ 1 currents in a fairly wide range of depth ratio. We obtain analytical solutions for the initial dam-break or slumping stage of propagation with constant speed, and derive explicitly the trends for small and large values of the governing parameters. This reveals the main features: (a) the heavy current propagates faster and its front is thinner than for the light counterpart; (b) the speed of the heavy current depends little on H*, while that of the light current increases with H*; and (c) in the shallow ambient case (H* close to 1) the light current is choked to move with the thickness of half-channel, while the heavy current typically moves with an unrestricted smaller thickness. These qualitative predictions are in accord with previous observations, and some quantitative comparisons with available experimental and numerical simulations data also show fair agreement. However, given the paucity of the available data, the main deficiency of the model is the unknown practical limit of applicability. For large time, t, a self-similar propagation with t2/3 is feasible for both the heavy and light currents, but the thickness profiles display differences.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 579 , 25 May 2007 , pp. 373 - 382
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209–248.CrossRefGoogle Scholar
Birman, V. K., Martin, J. E. & Meiburg, E. 2005 The non-Boussinesq lock exchange problem. Part 2. High-resolution simulations. J. Fluid Mech. 537, 101–124.CrossRefGoogle Scholar
Bonnecaze, R. T., Huppert, H. E. & Lister, J. R. 1993 Particle-driven gravity currents. J. Fluid Mech. 250, 339–369.CrossRefGoogle Scholar
Etienne, J., Hopfinger, E. J. & Saramito, P. 2005 Numerical simulations of high density ratio lock-exchange flows. Phys. Fluids 17, 036601-12.CrossRefGoogle Scholar
Fannel, T. K. & Jacobsen, O. 1984 Gravity spreading of heavy clouds instantaneously released. Z. Angew. Math. Phys. 35, 559–584.CrossRefGoogle Scholar
Gratton, J. & Vigo, C. 1994 Self-similar gravity currents with variable inflow revisited: plane currents. J. Fluid Mech. 258, 77–104.CrossRefGoogle Scholar
Gröbelbauer, H. P., Fannelöp, T. K. & Britter, R. E. 1993 The propagation of intrusion fronts of high density ratios. J. Fluid Mech. 250, 669–687.CrossRefGoogle Scholar
Hogg, A. J. & Pritchard, D. 2004 The effects of hydraulic resistance on dam-break and other shallow inertial flows. J. Fluid Mech. 501, 179–212.CrossRefGoogle Scholar
Keller, J. J. & Chyou, Y.-P. 1991 On the hydraulic lock-exchange problem. Z. Angew. Math. Phys. 42, 874–910.CrossRefGoogle Scholar
Klemp, J. B., Rotunno, R. & Skamarock, W. C. 1994 On the dynamics of gravity currents in a channel. J. Fluid Mech. 269, 169–198.CrossRefGoogle 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.CrossRefGoogle Scholar
Stansby, P. K., Chegini, A. & Barnes, T. C. D. 1998 The initial stages of dam-break flow. J. Fluid Mech. 374, 407–424.CrossRefGoogle Scholar
Ungarish, M. & Zemach, T. 2005 On the slumping of high Reynolds number gravity currents in two-dimensional and axisymmetric configurations. Euro. J. Mech. B/Fluids 24, 71–90.CrossRefGoogle Scholar