Numerical simulation of turbulent, plane parallel Couette-Poiseuille flow

Abstract

We present numerical simulation and mean-flow modelling of statistically stationary plane Couette–Poiseuille flow in a parameter space (Re,θ) with Re = √[Re꜀² + Re²_M] and θ = arctan (Re_M/Re꜀, where Re꜀, Re_M are independent Reynolds numbers based on the plate speed U꜀ 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 = 10000, 20000, 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 Re_(τ,b), changes sign at θ = θ꜀(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 Re_(τ,b)(Re,θ), Re_(τ,t)(Re,θ) (the skin-friction Reynolds number on the top wall) and θ꜀ show general agreement with both DNS and large-eddy simulations. Both model and simulation indicate that, as θ is increased at fixed Re, Re_(τ,t) passes through a peak at approximately θ = 45°, while Re_(τ,b) increases monotonically. Near the bottom wall, the flow laminarizes as θ passes through θ_(c) and then re-transitions to turbulence. As Re increases, θ꜀ 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_(τ,b) = 0 is investigated. Combined WMLES with modelling for these flows enables exploration of the Re → ∞ limit, giving θ꜀ → 45° as Re → ∞.

Additional details