The characterization of the dynamics of a continuous gravity current

The present work investigates the dynamical characteristics of a gravity current produced by a continuous and steady release of a dense fluid, with a source density ρ₀, into a lighter ambient fluid, with density ρ_a < ρ₀. A planar geometry is considered in which the ambient fluid is initially at rest inside a rectangular tank and the dense current is continuously released from a source located at the bottom-left corner of the tank.  The released current propagates along the horizontal bottom boundary of the domain displacing the ambient fluid. This configuration has been considered experimentally by Sher & Woods (2017). A numerical study has been carried out using the Direct Numerical Simulations (DNS) of the governing equations via the NEK5000 solver. Instantaneous three-dimensional velocity, pressure and density fields were extracted and two-dimensionalized by width-averaging. The state of the release is characterized by the source Froude number Fr₀ = u₀ / √(g^'₀ h₀ ) with u₀ being the velocity of the release, h₀ being the height at the inlet, and g^'₀ = g (ρ₀ - ρ_a)/ρ_a being the source buoyancy. Throughout the series of simulations, we control the state of the current at the source by only varying the source density ρ₀, resulting in a range of source Froude number between 0.6 < Fr₀ < 2.7, and we seek to record the effects of this variation on the dynamics. The source discharge Q₀ = u₀ h₀ and buoyancy flux B₀ = Q₀ g^'₀ are kept constant over time. The front speed, u_f, was shown to remain steady; a well-known feature of continuous gravity currents. A dimensionless parameter, λ = u_f/B₀^1/3, that characterizes the front speed was computed as a function of Fr₀ and the result shows a good agreement with the range recorded by Sher & Woods (2017). The entrainment of ambient fluid into the current is parametrized with two methods. First, we estimate the rate of change of the volume of the current, dV/dt, and we recorded the range 1.8Q₀ < dV/dt < 2.1Q₀ for the selected Fr₀  range. Secondly, the theory of inclined plumes introduced by Ellison & Turner (1959) was considered to estimate a local entrainment parameter, E, as a function of the local stratification represented by the local Richardson number Ri. The well-known relation, E proportional to Ri^-1, was held when Ri < 0.8; otherwise, the entrainment parameter tends to near-zero values.