Turbulent entrainment in buoyant releases from horizontal gravity current to vertical planar wall plume

This paper examines the dynamics of a supercritical, steady, miscible, two-dimensional gravity current flowing along an inclined boundary, with a particular focus on the entrainment, which refers to the mixing between the gravity current and the surrounding fluid. Specifically, the study investigates the combined effects of the Richardson number $\text{Ri}$ and slope $\theta$ on the entrainment coefficient $E$. To address these objectives, a theoretical study and large-eddy simulations (LES) were conducted, varying the slope angle from 0° to 90°, while maintaining constant injection conditions.

The theoretical investigation of gravity currents resulted in the extension of the theoretical model of Ellison and Turner (1959) in the general non-Boussinesq case as well as the identification of two specific angles: the critical angle ${\theta }_c$, beyond which no transition between supercritical to subcritical behaviour is observed, and the supercritical angle ${\theta }_{sc}$, above which the flow acquires inertia immediately after injection.

On the other hand, the simulations revealed three distinct flow regimes for our source conditions. The first regime, observed at low slopes, exhibits a non-monotonic behaviour, characterized by a transition from a supercritical to a subcritical regime. The second regime, related to the critical angle ${\theta }_c$, occurs at intermediate slopes, where the flow remains inertial throughout. In the third regime, related to steeper slopes, the supercritical angle ${\theta }_{sc}$ has been exceeded and the flow immediately gains inertia upon injection. The simulations also enabled an analysis of $E$, which was found to increase with slope and reach a constant value at $\theta =90^{\circ }$.

Moreover, a new entrainment law was developed, incorporating both the effects of the Richardson number and the slope: $E=0.002cos\left(\theta \right)/\text{Ri}+0.09{sin}^{1/2}\left(\theta \right)$. This law provides a description of the entrainment behaviour across the full range of slope angles. Comparisons between the LES results and the theoretical model demonstrate that the proposed entrainment law offers improved accuracy over existing models for all slope configurations, including the extreme cases of gravity currents $(\theta =0^{\circ })$ and planar wall plumes $(\theta =90^{\circ })$.