Numerical simulation of turbulent, plane parallel Couette-Poiseuille flow

We present numerical simulation and mean-flow modelling of statistically stationary plane Couette–Poiseuille flow in a parameter space (Re,θ) with Re = Re2 c + Re2 M and θ = arctan(ReM/Rec), where Rec, ReM are independent Reynolds numbers based on the plate speed Uc and the volume flow rate per unit span, respectively. The database comprises direct numerical simulations (DNS) at Re = 4000, 6000, wall-resolved large-eddy simulations at Re = 10 000, 20 000, and some wall-modelled large-eddy simulations (WMLES) up to Re = 1010. Attention is focused on the transition (from Couette-type to Poiseuille-type flow), defined as where the mean skin-friction Reynolds number on the bottom wall Reτ,b changes sign at θ = θc(Re). The mean flow in the (Re,θ) plane is modelled with combinations of patched classical log-wake profiles. Several model versions with different structures are constructed in both the Couette-type and Poiseuille-type flow regions. Model calculations of Reτ,b(Re,θ), Reτ,t(Re,θ) (the skin-friction Reynolds number on the top wall) and θc show general agreement with both DNS and large-eddy simulations. Both model and simulation indicate that, as θ is increased at fixed Re, Reτ,t passes through a peak at approximately θ = 45◦, while Reτ,b increases monotonically. Near the bottom wall, the flow laminarizes as θ passes through θc and then re-transitions to turbulence. As Re increases, θc increases monotonically. The transition from Couette-type to Poiseuille-type flow is accompanied by the rapid attenuation of streamwise rolls observed in pure Couette flow. A subclass of flows with Reτ,b = 0 is investigated. Combined WMLES with modelling for these flows enables exploration of the Re → ∞ limit, giving θc → 45◦ as Re → ∞.