Entrainment into two-dimensional and axisymmetric turbulent gravity currents

Abstract

Entrainment of ambient fluid into both two-dimensional and axisymmetric gravity currents is investigated experimentally using a novel neutralization technique. The technique involves the titrative neutralization of an alkaline gravity current which intrudes into and entrains an acidic ambient, and is visualized using a pH indicator solution. Using this technique, we can determine quantitative results for the amount of dilution in the head of the current. The head of the current is able to entrain ambient fluid both by shear instabilities on the current/ambient interface and by over-riding (relatively light) ambient fluid. Guided by our experimental observations, we present two slightly different theoretical models to determine the entrainment into the head of the current as a function of distance from the source for the instantaneous release of a constant volume of fluid in a two-dimensional geometry. By dimensional analysis, we determine from both models that the dimensionless entrainment or dilution ratio, E , defined as the ratio of the volumes of ambient and original fluid in the head, is independent of the initial reduced gravity of the current; and this result is confirmed by our experiments in Boussinesq situations. Our theoretical evaluation of E in terms of the initial cross-sectional area of the current agrees very well with our experimental measurements on the incorporation of an entrainment coefficient a, evaluated experimentally to be 0.063 f 0.003. We also obtain experimental results for constant-volume gravity currents moving over horizontal surfaces of varying roughness. A particularly surprising result from all the experiments, which is reflected in the theoretical models, is that the head remains essentially unmixedthe entrainment is negligiblein the slumping phase. Thus the heads of gravity currents with identical initial cross-sectional areas but different initial aspect ratios (lock lengths) will begin to be diluted by ambient fluid at different positions and hence propagate at different rates. A range of similar results is determined, both theoretically and experimentally, for the instantaneous release of a fixed volume of (heavy) fluid in an axisymmetric geometry. By contrast, the results of our experiments with gravity currents fed by a constant flux exhibit markedly different entrainment dynamics due to the continual replenishment of the fluid in the head by the constant input of undiluted fluid from the tail.

Dynamics of the head of gravity currents

“…This model is, in its essence, similar to the three-layer model proposed earlier by Britter and Simpson [12]. Regarding entrainment into the head region, Hallworth et al [26] argue that during the slumping phase the head remains undiluted, i.e., no entrainment takes place during this phase, and the head is progressively reduced in length. Thereafter, when the reflected bore reaches the current front, even though ambient fluid is continuously entrained into the head region, the head volume decreases as the current develops towards downstream due to mixed fluid left behind in the tail.…”

“…Unsteady gravity currents experiments are usually performed through the lock-exchange technique [6,26,29,53,63], which consists in the release of a fixed volume of dense fluid into a lower density fluid. The differences in hydrostatic pressure between contacting fluids cause the denser fluid to flow along the bottom of the tank, while the lighter fluid flows along the top boundary, in the opposite direction.…”

“…Based on steady state experiments and using the shadowgraph technique to visualize the flow, [12] presented a model for the structure of the flow in the head region of a stationary current where three distinct layers are identified: a lower layer of undiluted dense fluid, an upper layer of ambient fluid and the layer in between, where mixing takes place. Hallworth et al [26] and Hacker et al [25] investigated unsteady gravity currents over horizontal beds produced by the lock-exchange technique and, through two different visualization techniques, commented the inner structure of the current. Hallworth et al [26] observed that unsteady gravity currents have a distinct inner structure when compared to steady gravity currents, namely in what regards the extent of mixing: steady currents have a shorter mixing layer, near the upper boundary, above an undiluted layer which is continuously replenished by denser fluid from the tail.…”

“…Hallworth et al [26] and Hacker et al [25] investigated unsteady gravity currents over horizontal beds produced by the lock-exchange technique and, through two different visualization techniques, commented the inner structure of the current. Hallworth et al [26] observed that unsteady gravity currents have a distinct inner structure when compared to steady gravity currents, namely in what regards the extent of mixing: steady currents have a shorter mixing layer, near the upper boundary, above an undiluted layer which is continuously replenished by denser fluid from the tail. This model is, in its essence, similar to the three-layer model proposed earlier by Britter and Simpson [12].…”

Particulate gravity currents along V-shaped valleys

“…Although it has become customary to use γ equal to 1 we should expect that γ will vary with the mixing process associated with gravity current and be space and time dependent. In the case of saline gravity currents the mixing and entrainment in the head is known to differ from that along the tail of the current (Hacker et al 1996;Hallworth et al 1996) and to depend on the aspect ratio (height to length) of the lock (Hacker et al 1996). Although these results are for saline currents, they suggest that the mixing in particulate currents may vary along the current, and may depend on the aspect ratio of the lock and in this way affect the deposit.…”