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  and extensions  will be briefly discussed.
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Last updated: March 22, 2007