On scaling of mass entrainment in separated shear layers: the footprint of the incoming boundary layer

Abstract

We experimentally investigate the effects on scaling of separating/reattaching flows of the ratio $\delta_e/h$, where $\delta_e$ is the thickness at separation of the incoming boundary layer and $h$ the characteristic cross-stream scale of the flow. In the present study, we propose an original approach based on mean mass entrainment, which is the driving mechanism accounting for the growth of the separated shear layer. The focus is on mass transfer at the Turbulent/Non-Turbulent Interface (TNTI). In particular, the scaling of the TNTI, which is well documented in turbulent boundary layers, is used to trace changes in the scaling properties of the flow. To emphasise the influence of the incoming boundary layer, two geometrically similar, descending ramps with sizeably different heights $h$ but fundamentally similar values of $\delta_e$ are compared. The distribution in space of the TNTI highlights a sizeable footprint of the incoming boundary layer on the separated flow, the scaling of which results of the competition between $h$ and $\delta_e$. On the basis of a simple mass budget within the neighbourhood of separation, we propose to model this competition by introducing the scaling factor $C_{h,\delta} = 1 + \delta_e/h$. With this model, we demonstrate that the relationship between shear layer growth and mass entrainment rates established for free shear layers (i.e. $\delta_e = 0$) might be extended to flows where $\delta_e/h > 0$. Since many control systems rely on mass entrainment to modify separation properties, our findings suggest that the parameter $\delta_e/h$ needs to be taken into account when choosing the most relevant strategies for controlling or predicting separating/reattaching flows.