Revised mixing coefficient scaling for sheared stably stratified turbulence

Abstract

We revisit and extend the turbulent Froude number ( $Fr_k$ ) scaling for the mixing coefficient ( $\varGamma$ ) introduced by Garanaik & Venayagamoorthy (GV) (J. Fluid Mech., vol. 867, 2019, pp. 323–333) by directly incorporating the effects of mean shear through the non-dimensional shear parameter $S_{\ast } = S k/\epsilon _k$ . For flows where the effects of mean shear are stronger than the background vertical stratification, we find $\varGamma \sim Fr_k^{-2} S_\ast ^{-1}$ for weakly stratified sheared turbulence and $\varGamma \sim Fr_k^{-1}S_\ast ^{-1}$ for moderately stratified sheared turbulence. The scaling procedure is inconclusive for strongly stratified sheared turbulence, but using two independent datasets of homogeneous, sheared, stably stratified turbulence, we empirically observe $\varGamma \sim Fr_k^{-0.5} S_\ast ^{-1}$ . Our revised scaling better collapses both datasets compared with the original GV scaling, and we note that the moderately stratified sheared regime is extremely narrow (or maybe even non-existent). We also apply our scaling to the time-varying open channel simulations of Issaev et al. (J. Fluid Mech., vol. 935, 2022) and observe $\varGamma \sim Fr_k^{-2}S_\ast ^{-1}$ for weakly stratified sheared turbulence, but we observe deviations from our revised scaling for moderate and strong stratifications due to time-varying mean shear and vertical transport. Finally, we apply our revised scaling to field measurements of Conry, Kit & Fernando (Environ. Fluid Mech., vol. 20, 2020, pp. 1177–1197) and observe $\varGamma \sim Fr_k^{-2} S_\ast ^{-1}$ . We emphasize that our revised scaling is applicable only for stably stratified, vertically sheared turbulence with weak spatio-temporal variations of the mean shear and stratification, and we expect different scaling to apply when additional effects such as depth-varying radiative heating/cooling are present or when the orientation of the mean shear relative to the gravity vector is modified (e.g. horizontal shear).