Mean-velocity scaling in and around a mild, turbulent separation bubble

Abstract

This paper examines the question of the scaling of mean-velocity profiles in adverse-pressure-gradient flows. In these flows, the mean velocity scaling must be different than in zero-pressure-gradient flows, because the friction velocity used in the latter case can become vanishingly small in the former. Two decades ago, Perry and Schofield [Phys. Fluids 16, 2068 (1973)] proposed a new outer-region scaling law to be used when the boundary layer approaches separation. Since that time, a number of sets of experimental data close to separation have been shown to fall on a universal curve when the profiles are plotted in Perry–Schofield coordinates, and the profile shape was given by Dengel and Fernholz [J. Fluid Mech. 212, 615 (1990)]. Recently, however, a new set of scaling laws has been proposed by Durbin and Belcher [J. Fluid Mech. 238, 699 (1992)] as a result of their asymptotic analysis, in which they assumed the appropriate near-wall velocity scale to be based on the local strength of the pressure gradient. The resulting scaling laws are different than Perry and Schofield’s scaling and, in fact, predict a three-layered rather than a two-layered boundary-layer structure. Here, experimental results are shown for an adverse-pressure-gradient boundary layer which separates from and then reattaches to a smooth surface. These data provide a wide range of flow conditions for comparing the conflicting scaling laws mentioned above, under conditions of both decreasing and increasing skin friction, with and without instantaneous reverse flow. It is found that the Perry–Schofield coordinates provide better collapse, over a wider range of streamwise positions and over a larger fraction of the boundary layer, than the scaling laws of Durbin and Belcher. Other proposed scaling laws are also evaluated. Yaglom’s half-power law is shown to hold for a subset of the profiles which fall on Dengel and Fernholz’s universal profile. And the data provide a test of the range of validity of the (zero-pressure-gradient) logarithmic law of the wall. The law is violated here when instantaneous reverse flow exists in the boundary layer and/or when the local pressure gradient is strong enough, as is consistent with earlier work. However, after reattachment these criteria are insufficient to indicate the return to the log law, and several bubble lengths are required after reattachment before the universal log law is satisfied. The wake region responds to reattachment more slowly and does not appear fully recovered six bubble lengths (twenty boundary-layer thicknesses) after reattachment.