Entraining gravity currents

Abstract

Entrainment of ambient fluid into a gravity current, while often negligible in laboratory-scale flows, may become increasingly significant in large-scale natural flows. We present a theoretical study of the effect of this entrainment by augmenting a shallow water model for gravity currents under a deep ambient with a simple empirical model for entrainment, based on experimental measurements of the fluid entrainment rate as a function of the bulk Richardson number. By analysing long-time similarity solutions of the model, we find that the decrease in entrainment coefficient at large Richardson number, due to the suppression of turbulent mixing by stable stratification, qualitatively affects the structure and growth rate of the solutions, compared to currents in which the entrainment is taken to be constant or negligible. In particular, mixing is most significant close to the front of the currents, leading to flows that are more dilute, deeper and slower than their non-entraining counterparts. The long-time solution of an inviscid entraining gravity current generated by a lock-release of dense fluid is a similarity solution of the second kind, in which the current grows as a power of time that is dependent on the form of the entrainment law. With an entrainment law that fits the experimental measurements well, the length of currents in this entraining inviscid regime grows with time approximately as t 0.447 . For currents instigated by a constant buoyancy flux, a different solution structure exists in which the current length grows as t 4/5 . In both cases, entrainment is most significant close to the current front.

Sustained gravity currents in a channel

“…However over the time-and lengthscales of these simulations, the emergence of this mixed zone does not significantly modify the speed of the current, presumably because the vertical integral of the excess density field over the depth of the current, ( h 0ρ dz), remains approximately constant under mixing and this determines the speed of the density-driven spreading. Johnson & Hogg (2013) have demonstrated that entrainment alone modifies the predictions of shallow layer models, but only at time-and length-scales far beyond those computed in these simulations.…”

“…Experiments, numerical simulations and shallow layer models, which are built upon the assumption of hydrostatic balance, reveal that provided the effects of viscosity are negligible the fluid slumps at an initially constant rate of propagation, before progressively slowing as the finite volume of released material is spread out along the boundary underlying the fluid domain (Huppert & Simpson 1980;Rottman & Simpson 1983;Hogg 2006). Subsequently the motion is retarded by the action of drag (Huppert 1982;Hogg & Woods 2001) and mixing processes between the two fluids (Johnson & Hogg 2013). Of particular importance for applications is that it is possible to identify timescales and lengthscales at which these transitions in behaviour occur in terms of the properties of the fluid and its initial conditions (see, for example, Huppert & Simpson (1980)).…”

“…Our model neglects mixing between the two fluid layers and drag from either the underlying boundary or the interfacial processes. Such processes may become important, or even dominant, after sufficiently long times and distances of propagation (see, for example, Hogg & Woods (2001) and Johnson & Hogg (2013)). …”

Key Future Directions For Research On Turbidity Currents and Their Deposits

“…Scaling of laboratory experiments needs to go beyond Froude or Reynolds Number similarity, and also consider the ratio of flow power (shear velocity) to sediment settling velocity, such as that needed to suspend sediment or generate bedload. Furthermore, the scale of laboratory experiments may preclude many other processes such as mixing with the surrounding ambient water, which may significantly alter the dynamics (see, for example, Johnson and Hogg 2013). Field data from the Agadir Basin, offshore NW Africa, was presented by Stevenson et al (2014).…”

A numerical shallow-water model for gravity currents for a wide range of density differences

“…The dynamics of PDCs is complex because the dilute and dense currents are influenced by a number of physical processes such as particle settling (e.g., Bonnecaze et al 1993), entrainment of ambient air (e.g., Johnson and Hogg 2013;Sher and Woods 2015), and basal resistance (e.g., Roche et al 2008). In addition to the effects of these processes, our results suggest that the application of the correct numerical model to the flow front is important if we are to understand the dynamics and sedimentation of PDCs.…”