The structure of a separating turbulent boundary layer. Part 5. Frequency effects on periodic unsteady free-stream flows

Abstract

Measurements of a steady free-stream, nominally two-dimensional, separating turbulent boundary layer have been reported in earlier parts of this work. Here measurements are reported that show the effects of frequency on sinusoidal unsteadiness of the free-stream velocity on this separating turbulent boundary layer at reduced frequencies of 0.61 and 0.90. The ratio of oscillation amplitude to mean velocity is about 1/3 for each flow.

Upstream of flow detachment, hot-wire anemometer measurements were obtained. A surface hot-wire anemometer was used to measure the phase-averaged skin friction. Measurements in the detached-flow zone of phase-averaged velocities and turbulence quantities were obtained with a directionally sensitive laser anemometer. The fraction of time that the flow moves downstream was measured by the LDV and by a thermal flow-direction probe.

Upstream of any flow reversal or backflow, each flow behaves in a quasisteady manner, i.e. the phase-averaged flow is described by the steady free-stream flow structure. The semilogarithmic law-of-the-wall velocity profiles applies at each phase of the cycle. The Perry & Schofield (1973) velocity-profile correlations fit the mean and ensemble-averaged velocity profiles near detachment.

After the beginning of detachment, large amplitude and phase variations develop through each flow. Unsteady effects produce hysteresis in relationships between flow parameters. As the free-stream velocity during a cycle begins to increase, the detached shear layer decreases in thickness, and the fraction of time $\hat{\gamma}_{{\rm p}u} $ that the flow moves downstream increases as backflow fluid is washed downstream. As the free-stream velocity nears the maximum value in a cycle, the increasingly adverse pressure gradient causes progressively greater near-wall backflow at downstream locations while $\hat{\gamma}_{{\rm p}u}$ remains high at the upstream part of the detached flow. After the free-stream velocity begins to decelerate, the detached shear layer grows in thickness, and the location where flow reversal begins moves upstream. This cycle is repeated as the free-stream velocity again increases.

In both unsteady flows, the ensemble-averaged detached-flow velocity profiles agree with steady free-stream profiles for the same $\hat{\gamma}_{{\rm p}u\min} $ value near the wall when $\partial\hat{\gamma}_{{\rm p}u\min}/\partial\hat{t} < 0 $. However, the reduced-frequency k = 0.90 flow has much larger hysteresis in ensemble-averaged velocity profile shapes when $\partial\hat{\gamma}_{{\rm p}u\min}/\partial{t} \geqslant 0 $. Larger and negative values of the profile shape factor $\hat{H}$ occur for this flow during phases when the non-dimensional backflow is greater and $\hat{\gamma}_{{\rm p}u\min}\rightarrow 0.01$.