An analytical model for a density current from a supercritical to a subcritical state

In the seminal paper by Ellison and Turner [1], the authors develop a theoretical model to calculate the longitudinal evolution of the characteristic quantities (velocity, thickness and density) of a continuous steady density current. These equations introduce the Richardson number Ri, which is a ratio between the buoyancy and the inertia forces of the current (Δρgh/ρ_aU²) and characterizes the stability of the flow. They also introduce two flow regimes related to the Richardson number: the subcritical regime when buoyancy forces dominate (Ri>1) and the supercritical regime when inertia forces dominate (Ri<1). For an initially supercritical flow, the set of equations developed by Ellison and Turner [1] presents a singularity when the flow transitions from the supercritical to the subcritical state through the critical state (Ri=1). In such a case, a mathematical discontinuity, similar to a jump, allows the super- and sub-critical solutions to be matched, enabling a complete computational resolution of the problem (Haddad et al. [2]).

To go further mathematically with this theoretical model, we propose in this study an algebraic development allowing a direct calculation of the evolution of the Richardson number as a function of the length of the studied domain. This method provides direct results, in contrast to those obtained by a computational resolution of coupled ordinary differential equations. This method also gives the variations of the primary variables from the Richardson number. It is even possible, for the limiting cases, to find strict analytical solutions by means of asymptotic developments. We perform numerical simulations using the large-eddy simulation code CALIF³S-Isis to obtain reference results and compare them with the theoretical solutions obtained. The extension proposed is interesting and allows, in a qualitatively acceptable way, the non-monotonic evolution of the density current primary variables to be reproduced.