March 8, 2007: Intrusive gravity currents in a stratified ambient - shallow-water theory, numerical results and new interpretations of previous experiments

Marius Ungarish, Technion-Israel Institute of Technology

The intrusion of a fixed volume of fluid which is released from rest and then propagates horizontally at the neutral buoyancy level in a vertically-stratified ambient fluid is investigated. The density change is linear, in a restricted layer or over the full-depth of the container, and locks of both rectangular and cylindrical shapes are considered. The investigation was motivated by the fact that previous theoretical models ([4, 5, 8]) were in conflict with experimental observations (see [21). The present theory employs a closed one-layer shallow-water inviscid formulation. The propagation of the intrusion, in realistic configurations, is obtained from the solutions of a hyperbolic well-posed system. A similarity solution for the large-time developed motion, and an approximate box-model are also presented. The results are corroborated by numerical solutions of the full two-dimensional Navier-Stokes equations and comparisons with previously published experiments ( [1, 2, 6, 8]). It is shown that the model is a versatile prediction too] which clarifies essential features of the flow-field. Accurate insights are provided concerning: (1) the governing dimensionless parameters; (2) the fact that the initial propagation is with constant speed for intrusions released from a rectangular lock but time-dependent for the cylindrical lock used in Wu's experiments; (3) the spread with time at some power; (4) the sub-critical (compared to the mode 2 linear waves) speed in a full-depth stratified container configuration; and (5) the first interaction of the head with the internal gravity waves. This rehabilitates the validity of the inviscid shallow-water modeling for these problems. It is shown that the previous theoretical models can now be dismissed (with due credit) because the present approach has a much broader range of validity and relevance. Some connections with other recent results [3] and extensions [7] will be briefly discussed.

References

[1] R. Amen and T. Maxworthy. The gravitational collapse of a mixed region into a linearly stratified fluid. J. Fluid Mech., 96:65-80, 1980.

[2] K. M. Faust and E. J. Plate. Experimental investigation of intrusive gravity currents entering stably stratified fluids. J. Hydraulic Res., 22:315-325, 1984.

[3] M. R. Flynn and B. R Sutherland. Intrusive gravity currents and internal gravity wave generation in stratified fluid. J. Fluid Mech., 514:355 383, 2004.

[4] T. W. Kao. Principal stage of wake collapse in a stratified fluid: Two-dimensional theory. Phys. Fluids, 19:1071-1074, 1976.

[5] P. C. Manins. Mixed region collapse in a stratified fluid. J. Fluid Mech., 77:177-183, 1976.

[6] F. de Rooij. Sedirmenting particle-laden flows in confined geometries. PhD thesis, DAMTP, University of Cambridge , 1999.