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 inertial–buoyancy or viscous–buoyancy regime is preserved in a uniform-density gravity current (which propagates over a horizontal plane) of variable volume V = qt^\deltain a power-law cross-section (with 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 maximum stratification cases respectively. Earlier theoretical and experimental results valid for a rectangular cross-section (\alpha = 0) and uniform ambient fluid are generalized here to a power-law cross-section and stratified ambient. Novel time scalings, obtained for inertial and viscous regimes, allow a derivation of the critical flow parameter c and the corresponding propagation rate as K^\beta_c as a function of the problem parameters. Estimates of the transition length between the inertial and viscous regimes 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 and 1. The ratio between the 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 2-D (rectangular cross-section channel) gravity currents to power-law cross-sections is available.