Regime transitions in stratified shear flows: the link between horizontal and inclined ducts

TL;DR

This paper derives an analytical solution for laminar stratified inclined duct flows under specific approximations, linking flow regimes to a non-dimensional Froude number and validating findings with experiments.

Contribution

It introduces a new analytical solution for SID flows under the HGV-A approximation, connecting flow regimes with a non-dimensional parameter and validating with experimental data.

Findings

Constant Froude number characterizes flow regime transitions.

Analytical solution matches laboratory experiment results.

Flow regimes progress from laminar to turbulence with increasing Froude number.

Abstract

We present the analytical solution for the two-dimensional velocity and density fields within an approximation for laminar stratified inclined duct (SID) flows where diffusion dominates over inertia in the along-channel momentum equation but it is negligible in the density transport equation. We refer to this approximation as the hydrostatic/gravitational/viscous in momentum and advective in density (HGV-A) approximation due to the leading balances in the governing equations. The analytical solution is valid for laminar flows in a two-layer configuration in the limit of long ducts. Under such conditions, the non-dimensional volume flux is given by the Froude number $Fr^* =Re_g/(A\,K)$ with $Re_g$ the gravitational Reynolds number, $A$ the aspect ratio of the duct, and $K$ a geometrical parameter that depends on the tilt of the duct and is obtained from the analytical solution. The analytical solution in the HGV-A approximation is validated against results from laboratory experiments, and allows us to gain new insight into the dynamics and properties of SID flows. Most importantly, constant values of $Fr^*$ describe, in both horizontal and inclined ducts, the transitions between increasingly turbulent flow regimes: from laminar flow, to interfacial waves, to intermittent turbulence and sustained turbulence.