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A numerical investigation of horizontal viscous gravity currents

Published online by Cambridge University Press:  10 July 2009

YANNICK HALLEZ  and
JACQUES MAGNAUDET
YANNICK HALLEZ
Affiliation:
Université de Toulouse; INPT, UPS; Institut de Mécanique des Fluides de Toulouse, Allée Camille Soula, F-31400 Toulouse, France CNRS, Institut de Mécanique des Fluides de Toulouse; F-31400 Toulouse, France
JACQUES MAGNAUDET*
Affiliation:
Université de Toulouse; INPT, UPS; Institut de Mécanique des Fluides de Toulouse, Allée Camille Soula, F-31400 Toulouse, France CNRS, Institut de Mécanique des Fluides de Toulouse; F-31400 Toulouse, France
*
†Email address for correspondence: [email protected]
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Abstract

We study numerically the viscous phase of horizontal gravity currents of immiscible fluids in the lock-exchange configuration. A numerical technique capable of dealing with stiff density gradients is used, allowing us to mimic high-Schmidt-number situations similar to those encountered in most laboratory experiments. Plane two-dimensional computations with no-slip boundary conditions are run so as to compare numerical predictions with the ‘short reservoir’ solution of Huppert (J. Fluid Mech., vol. 121, 1982, pp. 43–58), which predicts the front position lf to evolve as t1/5, and the ‘long reservoir’ solution of Gratton & Minotti (J. Fluid Mech., vol. 210, 1990, pp. 155–182) which predicts a diffusive evolution of the distance travelled by the front xf ~ t1/2. In line with dimensional arguments, computations indicate that the self-similar power law governing the front position is selected by the flow Reynolds number and the initial volume of the released heavy fluid. We derive and validate a criterion predicting which type of viscous regime immediately succeeds the slumping phase. The computations also reveal that, under certain conditions, two different viscous regimes may appear successively during the life of a given current. Effects of sidewalls are examined through three-dimensional computations and are found to affect the transition time between the slumping phase and the viscous regime. In the various situations we consider, we make use of a force balance to estimate the transition time at which the viscous regime sets in and show that the corresponding prediction compares well with the computational results.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 630 , 10 July 2009 , pp. 71 - 91
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

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. Theoret. Comput. Fluid Dyn. 22, 341–361.CrossRefGoogle Scholar
Bonometti, T., Balachandar, S. & Magnaudet, J. 2008 Wall effects in non-Boussinesq density currents. J. Fluid Mech. 616, 445–475.CrossRefGoogle Scholar
Bonometti, T. & Magnaudet, J. 2006 Transition from spherical caps to toroidal bubbles. Phys. Fluids 18, 052102.CrossRefGoogle Scholar
Bonometti, T. & Magnaudet, J. 2007 A front-capturing technique for the computation of incompressible two-phase flows: validation and application to bubbly flows. Intl J. Multiphase Flow 33, 109–133.CrossRefGoogle Scholar
Cantero, M. I., Lee, J. R., Balachandar, S. & Garcia, M. H. 2007 On the front velocity of density currents. J. Fluid Mech. 586, 1–39.CrossRefGoogle Scholar
Didden, W. & Maxworthy, T. 1982 The viscous spreading of plane and axisymmetric gravity currents. J. Fluid Mech. 121, 27–42.CrossRefGoogle Scholar
Gratton, J. & Minotti, F. 1990 Self-similar viscous gravity currents: phase-plane formalism. J. Fluid Mech. 210, 155–182.CrossRefGoogle Scholar
Hallez, Y. & Magnaudet, J. 2008 Effects of channel geometry on buoyancy-driven mixing. Phys. Fluids 20, 053306.CrossRefGoogle Scholar
Härtel, C., Meiburg, E. & Necker, F. 2000 Analysis and direct numerical simulation of the flow at a gravity-current head. Part 1. Flow topology and front speed for slip and non-slip boundaries. J. Fluid Mech. 418, 189–212.CrossRefGoogle Scholar
Hinch, E. J., Hulin, J. P., Salin, D., Perrin, B., Séon, T. & Znaien, J. 2007 Inclined to exchange: shocking gravity currents. http://www.damtp.com.ac.uk/user/hinch/talks/nottingham.pdfGoogle Scholar
Hoult, D. P. 1972 Oil spreading on the sea. Annu. Rev. Fluid Mech. 4, 341–368.CrossRefGoogle 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.CrossRefGoogle Scholar
Huppert, H. E. 2000 Geological fluid mechanics. In Perspectives in Fluid Dynamics (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G.), pp. 447–506. Cambridge University Press.Google Scholar
Huppert, H. E. & Simpson, J. E. 1980 The slumping of gravity currents. J. Fluid Mech. 99, 785–799.CrossRefGoogle Scholar
Linden, P. F. 1999 The fluid mechanics of natural ventilation. Annu. Rev. Fluid Mech. 31, 201–238.CrossRefGoogle 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
Necker, F., Härtel, C., Kleiser, L. & Meiburg, E. 2002 High-resolution simulations of particle-driven gravity currents. Intl J. Multiphase Flow 28, 279–300.CrossRefGoogle Scholar
Necker, F., Härtel, C., Kleiser, L. & Meiburg, E. 2005 Mixing and dissipation in particle-driven gravity currents. J. Fluid Mech. 545, 339–372.CrossRefGoogle Scholar
Rottman, J. W. & Simpson, J. E. 1983 Gravity currents produced by insantaneous releases of a heavy fluid in a rectangular channel. J. Fluid Mech. 135, 95–110.CrossRefGoogle Scholar
Séon, T., Znaien, J., Hinch, E. J., Perrin, B., Salin, D. & Hulin, J. P. 2007 Transient buoyancy-driven front dynamics in nearly horizontal tubes. Phys. Fluids 19, 123603.CrossRefGoogle Scholar
Simpson, J. E. 1982 Gravity currents in the laboratory, atmosphere, and ocean. Annu. Rev. Fluid Mech. 14, 213–234.CrossRefGoogle Scholar
Simpson, J. E. 1997 Gravity Currents. Cambridge University Press.Google Scholar
Takagi, D. & Huppert, H. E. 2007 The effect of confining boundaries on viscous gravity currents. J. Fluid Mech. 577, 495–505.CrossRefGoogle Scholar
Zalesak, S. T. 1979 Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys. 31, 335–362CrossRefGoogle Scholar