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

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.