A mathematical model for calculating the transient volume of gravity currents: An experimental analysis focused on mixing mechanisms and volume growth for accidental releases

We present experiments of two dimensional high-Reynolds ($Re>2400$) turbulent gravity current advancing down on slope $(\theta =5°)$ generated by continuous buoyancy flux. The current research is focused on understanding the flow mixing mechanism and consequently the rate of volume growth for the development of mathematical model to calculate the volume of the current. The gravity currents were obtained pumping saline solution continuously into a channel filled with fresh water. Images of the flow were taken with a ratio of 4 frames per second (fps). The gravity current buoyancy distribution was obtained by using light attenuation technique to calculate the cross-channel average of the density. It was found that the proportional parameter $\lambda$ in $U_F=\lambda {\left(Q_0{g}_0^{\prime }\right)}^{1/3}$ is $\lambda =1.23/F{r}_0^{1/12}$. The head reaches a dynamic equilibrium for $F{r}_F\approx 0.95$, where $F{r}_H=U_F/\sqrt{g_F^{\prime }{h}_F}$. Three mixing zones were observed; near the source, tail and head. The ambient fluid volume fluxes entraining the current into this three zones were modelled in terms of entrainment coefficients ${\varepsilon }_j$, ${\varepsilon }_T$, ${\varepsilon }_H$, respectively. A model of the rate of growth of the volume of the current was developed, it is written as $dV/dt=Q_0(1+{\varepsilon }_J)/(1-{\lambda }^3{\varepsilon }_H)$. One of the simplifications implies that the volume growth rate is constant. Good agreement with experimental data is observed.