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

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 ${\mathcal{V}}=qt^{\unicode[STIX]{x1D6FF}}$ in a power-law cross-section (with width described by $f(z)=bz^{\unicode[STIX]{x1D6FC}}$, where $z$ is the vertical coordinate, $b$ and $q$ are positive real numbers, and $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FF}$ 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 ($\unicode[STIX]{x1D6FC}=0$) 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 $\unicode[STIX]{x1D6FF}_{c}$ and the corresponding propagation rate as $Kt^{\unicode[STIX]{x1D6FD}_{c}}$ 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 ($\unicode[STIX]{x1D6FC}=1/2$) validate the critical values $\unicode[STIX]{x1D6FF}_{c}=2$ and $\unicode[STIX]{x1D6FF}_{c}=9/4$ 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.

Experimental study on radial gravity currents flowing in a vegetated channel

“…The presence of turbulent drag suggests an analogy between GC flow and flow in the Forchheimer regime in porous media (Hatcher, Hogg & Woods 2000), with a velocity profile that appears uniform in the vertical and with dissipation controlled by the obstacles, with negligible bottom effects. If the GC can spread over a sufficient space, a transition is expected with the current initially in the inertial-buoyancy regime, then in the viscous-buoyancy regime depending on the Reynolds number (with the exception of critical GCs, see Maxworthy 1983;Chiapponi et al 2018). In the presence of turbulent drag, a further transition is expected, with inertia initially dominant, then overtaken by turbulent drag and finally replaced by viscous drag.…”

Effect of stratification on the propagation of a cylindrical gravity current

“…Planar release gravity currents in stratified (Maxworthy et al 2002;Ungarish & Huppert 2002;Birman et al 2007b;Longo et al 2016;Chiapponi et al 2018;Lam et al 2018b;la Forgia et al 2020;Ottolenghi et al 2020;Dai et al 2021;Zahtila et al 2024), and unstratified ambients (Rottman & Simpson 1983;Shin, Dalziel & Linden 2004;La Rocca et al 2008;Dai 2015;Pelmard, Norris & Friedrich 2018;Dai & Huang 2020;De Falco, Adduce & Maggi 2021;Maggi, Adduce & Negretti 2022, 2023aMaggi et al 2023b), as well as cylindrical release gravity currents in unstratified environment have been widely studied experimentally and numerically (Cantero et al 2007a,b;Zgheib, Bonometti & Balachandar 2014, 2015aDai & Huang 2016). However, experimental and numerical studies on stratified gravity currents with the cylindrical release are limited, and therefore we are interested in cylindrical gravity currents propagating into a linearly stratified ambient.…”