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, theta) with Re = root Re-c(2) + Re-M(2) and theta = arctan(Re-M/Re-c), where Re-c, Re-M 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 = 10(10). 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 Re-tau,Re-b changes sign at theta = theta(c)(Re). The mean flow in the (Re, theta) 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 Re-tau,Re-b(Re, theta), Re-tau,Re-t(Re, theta) (the skin-friction Reynolds number on the top wall) and theta c show general agreement with both DNS and large-eddy simulations. Both model and simulation indicate that, as theta is increased at fixed Re, Re-tau,Re-t passes through a peak at approximately theta = 45 degrees, while Re-tau,Re-b increases monotonically. Near the bottom wall, the flow laminarizes as theta passes through theta(c )and then re-transitions to turbulence. As Re increases, theta(c) 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 Re-tau,Re-b = 0 is investigated. Combined WMLES with modelling for these flows enables exploration of the Re -> infinity limit, giving theta(c) -> 45 degrees as Re -> infinity.