Effects of pressure gradients on DNS of an oscillating turbulent boundary layer

DNS is used in Stokes’ 2nd Problem to examine the effects of adverse and favorable pressure gradients (APG and FPG) on oscillating turbulent boundary layers (TBLs) for the inner-normalized pressure gradient p⁺ ≈ -0.01∼0.01. Particular attention is given to the resilience of the velocity law of the wall. Three values of the Reynolds number (Re_δs=1000, 1400 and 1800) are used, where Re_δs is based on the amplitude of the freestream velocity and the laminar boundary-layer thickness. The simulations are performed with doubly-periodic boundary conditions in which the mean pressure gradient varies sinusoidally in time (one half-cycle consisting of the phase-angle φ = 0∼π contains all the information). In the cycle, an APG is first imposed (φ = 0∼π/2), causing a reversal of the wall shear stress. This becomes a FPG (φ = π/2∼π) since the flow which has reversed direction is accelerating, and finally approaches a log-law velocity profile. Its inspection reveals a similarity in the U⁺ profile between the present flow and other wall-bounded flows (i.e. channel and spatially-evolving TBLs) for a matched p⁺. At y⁺=50, U⁺ decreases linearly with increasing p⁺, although the behavior is less clear in a FPG. At this y⁺ location, the Reynolds stresses associated with the active motions (〈v⁺v⁺〉 and -〈u⁺v⁺〉) also exhibit linear p⁺ variations but with a positive slope versus p⁺; this is a new result. The results highlight a functional dependence on p⁺ for the mean velocity and the active second-order moments in TBLs with pressure gradients.