Critical regime of gravity currents flowing in non-rectangular channels with density stratification. (14th February 2018) - Chiapponi, L.; Ungarish, M.; Longo, S.; Di Federico, V.; Addona, F.
Abstract : 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 time-variable volume V=qt^[STIX]x1D6FF in a power-law cross-section (with width described by f(z)=bz^[STIX]x1D6FC, where z is the vertical coordinate, b and q are positive real numbers, and [STIX]x1D6FC and [STIX]x1D6FF are non-negative real numbers) 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 ( [STIX]x1D6FC=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 [STIX]x1D6FF_c and the corresponding propagation rate as Kt^[STIX]x1D6FD_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 semicircular cross-section ( [STIX]x1D6FC=1/2 ) validate the critical values [STIX]x1D6FF_c=2 and [STIX]x1D6FF_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 two-dimensional (rectangular cross-section channel) gravity currents to power-law cross-sections is available.