Hostname: page-component-5c6d5d7d68-wp2c8 Total loading time: 0 Render date: 2024-08-07T23:08:12.089Z Has data issue: false hasContentIssue false
  • English
  • Français

Lock-exchange flows in sloping channels

Published online by Cambridge University Press:  19 April 2007

V. K. BIRMAN ,
B. A. BATTANDIER ,
E. MEIBURG  and
P. F. LINDEN
V. K. BIRMAN
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
B. A. BATTANDIER
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA Ecole Polytechnique, 91128 Palaiseau Cedex, France
E. MEIBURG*
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
P. F. LINDEN
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California, San Diego, CA 92093, USA
*
†Author to whom correspondence should be addressed.
Get access Rights & Permissions [Opens in a new window]

Abstract

Two-dimensional variable-density Navier–Stokes simulations have been conducted in order to investigate the effects of a slope on the classical lock-exchange flow. Simulations of full lock releases show that the flow goes through an initial quasi-steady phase that is characterized by a constant front velocity. This quasi-steady front velocity has a maximum for slope angles around 40°, and it persists up to a dimensionless time of the order of 10. The flow subsequently undergoes a transition to a second phase with a larger, unsteady, front velocity. These computational findings were confirmed by experimental observations of lock-exchange flows in a tube of circular cross-section.

The reason for the observed transition from a quasi-steady front velocity to a larger, unsteady, value is found in the continuous acceleration of the stratified fluid layers connecting the two fronts by the streamwise component of the gravity vector. This acceleration leads to a situation where the fluid layers behind the current front move faster than the front itself. Initially the resulting addition of fluid to the current front from behind affects only the size of the front, while its velocity remains unchanged. Eventually, the current front is unable to absorb more fluid from behind and its velocity has to increase, thereby triggering the transition to the second, unsteady, phase. The transition time is determined as a function of the slope and the density ratio of the two fluids. For increasing density contrast, the transition is seen to occur earlier for the denser current.

Conceptually simple models based on the analysis by Thorpe (1968) are compared with simulation results for the flow in the region connecting the fronts. For the early stages of the flow a two-layer stratification model is found to be appropriate, while the later stages require a three-layer stratification model, owing to the intense mixing in the central part of the channel cross-section. These models are employed to estimate the time after which the accelerating stratified fluid layers will affect the velocities of the current fronts. They provide upper and lower estimates for the transition time which are in good agreement with the simulation results.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 577 , 25 April 2007 , pp. 53 - 77
Copyright
Copyright © Cambridge University Press 2007

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

Beghin, P., Hopfinger, E. J. & Britter, R. E 1981 Gravitational convection from instantaneous sources on inclined boundaries. J. Fluid Mech. 107, 407.CrossRefGoogle Scholar
Benjamin, T. B. 1968 Gravity currents and related phenomena. J. Fluid Mech. 31, 209.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, 125.CrossRefGoogle Scholar
Bonnecaze, R., Huppert, H. E. & Lister, J. R. 1993 Particle-driven gravity currents. J. Fluid Mech. 250, 339.CrossRefGoogle Scholar
Bonnecaze, R. & Lister, J. 1999 Particle-driven gravity currents down planar slopes. J. Fluid Mech. 390, 75.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.CrossRefGoogle Scholar
Britter, R. & Simpson, J. E. 1978 Experiments on the dynamics of a gravity current head. J. Fluid Mech. 88, 223.CrossRefGoogle Scholar
Cantero, M. I., Lee, J. R., Balachandar, S., Garcia, M. H. & Ha, M. Y. 2006 On the front velocity of gravity currents. J. Fluid Mech. (submitted).CrossRefGoogle Scholar
Ellison, T. & Turner, J. 1959 Turbulent entrainment in stratified flow. J. Fluid Mech. 6, 423.CrossRefGoogle Scholar
Etienne, J., Hopfinger, E. J. & Saramito, P. 2005 Numerical simulations of high density lock-exchange flows. Phys. Fluids 17, 036601.CrossRefGoogle Scholar
Gröbelbauer, H. P., Fannelöp, T. K. & Britter, R. E. 1993 The propagation of intrusion fronts of high density ratio. J. Fluid Mech. 250, 669.CrossRefGoogle Scholar
Hacker, J., Linden, P. F. & Dalziel, S. B. 1995 Mixing in lock-release gravity currents. Dyn. Atmos. Oceans 24, 183.CrossRefGoogle Scholar
Hallworth, M., Huppert, H., Phillips, J. & Sparks, S. 1996 Entrainment into two-dimensional and axisymmetric turbulent gravity currents. J. Fluid Mech. 308, 289.CrossRefGoogle Scholar
Hallworth, M. A., Phillips, J. C., Huppert, H. E. & Sparks, S. J. 1993 Entrainment in turbulent gravity currents. Nature 362, 829.CrossRefGoogle Scholar
Härtel, C., Meiburg, E. & Necker, F. 1999 Vorticity dynamics during the start-up phase of gravity currents. Il Nuovo Cimento C 22, 823.Google 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 no-slip boundaries. J. Fluid Mech. 418, 189.CrossRefGoogle Scholar
Huppert, H. E. & Simpson, J. 1980 The slumping of gravity currents. J. Fluid Mech. 99, 785.CrossRefGoogle Scholar
Joseph, D. & Renardy, Y. 1992 Fundamentals of Two-Fluid Dynamics, Part II. Springer.Google Scholar
Keller, J. J. & Chyou, Y. P. 1991 On the hydraulic lock-exchange problem. Z. Angew. Math. Phys. 42, 874.CrossRefGoogle Scholar
Lele, S. K. 1992 Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16.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.CrossRefGoogle Scholar
Maxworthy, T. & Nokes, R. 2006 Experiments on gravity currents propagating down slopes in open channels. Part 1. The effect of initial conditions on the release of a fixed volume of heavy fluid from an enclosed lock. J. Fluid Mech. (submitted).Google 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.CrossRefGoogle Scholar
Ross, A. N., Linden, P. F. & Dalziel, S. B. 2002 Three-dimensional gravity currents on slopes. J. Fluid Mech. 453, 239.CrossRefGoogle Scholar
Rottman, J. W. & Simpson, J. E. 1983 Gravity currents produced by instantaneous releases of a heavy fluid in a rectangular channel. J. Fluid Mech. 135, 95.CrossRefGoogle Scholar
Séon, T., Hulin, J. P. & Salin, D. 2005 Buoyancy driven miscible front dynamics in tilted tubes. Phys. Fluids 17, 031702.CrossRefGoogle Scholar
Shin, J. O., Dalziel, S. B. & Linden, P. F. 2004 Gravity currents produced by lock exchange. J. Fluid Mech. 521, 1.CrossRefGoogle Scholar
Simpson, J. E. 1997 Gravity Currents in the Environment and the Laboratory. Cambridge University Press.Google Scholar
Simpson, J. E. & Britter, R. E. 1979 The dynamics of the head of a gravity current advancing over a horizontal surface. J. Fluid Mech. 94, 477.CrossRefGoogle Scholar
Staquet, C. 2000 Mixing in a stably stratified shear layer: two- and three-dimensional numerical experiments. Fluid Dyn. Res. 27, 367–404.CrossRefGoogle Scholar
Thorpe, S. A. 1968 A method of producing a shear flow in a stratified fluid. J. Fluid Mech. 32, 693.CrossRefGoogle Scholar
Thorpe, S. A. 1971 Experiments on the instability of stratified shear flows: miscible fluids. J. Fluid Mech. 46, 299–319.CrossRefGoogle Scholar
Tickle, G. 1996 A model of the motion and dilution of a heavy gas cloud released on a uniform slope in calm conditions. J. Haz. Mat. 49, 29.CrossRefGoogle Scholar
Webber, D., Jones, S. & Martin, D. 1993 A model of motion of a heavy gas cloud released on a uniform slope. J. Haz. Mat. 33, 101.CrossRefGoogle Scholar
Zukoski, E. E. 1966 Influence of viscosity, surface tension and inclination angle on motion of long bubbles in closed tubes. J. Fluid Mech. 25, 821.CrossRefGoogle Scholar