Critical regime of gravity currents flowing in non--rectangular channels with density stratification

We present theoretical and experimental analyses of the critical condition where the regime is preserved inertial-buoyancy or viscous-buoyancy in uniform-density gravity current (which propagates over a horizontal plane) of variable volume $\V = qt^\delta$ in a power-law cross-section (width described by $f(z)=bz^{\alpha}$, where $z$ is the vertical coordinate), occupied by homogeneous or linearly stratified ambient fluid. The magnitude of the ambient stratification is represented by the parameter $S$, with $S=0$ and $S=1$ describing the homogeneous and the maximum stratification case, respectively. Earlier theoretical and experimental results valid for a rectangular cross-section ($\alpha=0$) and uniform ambient fluid are generalized here to power-law cross-section and stratified ambient. Novel time scalings, obtained for inertial and viscous regimes, allow a derivation of the critical flow parameter $\delta_c$ and the corresponding propagation rate like $Kt^{\beta_c}$ as a function of problem parameters. Estimates of the transition length between inertial and viscous regime are also derived. A series of experiments conducted in a circular cross-section ($\alpha=1/2$) validate the critical values $\delta_c=2$ and $\delta_c=9/4$ for the two cases $S=0,\,1$. The ratio between inertial and viscous forces is determined by an effective Reynolds number proportional to $q$ at some power; the threshold value of this number, which enables a determination of the regime of the current (inertial-buoyancy or viscous-buoyancy) in critical conditions, is determined experimentally for both $S=0$ and $S=1$. We conclude that a very significant generalization of the insights and results from the 2-D (rectangular cross--section channel) gravity currents to power-law cross-sections is available.