Numerical simulation of turbulent, plane parallel Couette–Poiseuille flow. (25th January 2023) - Cheng, W.; Pullin, D.I.; Samtaney, R.; Luo, X.
Abstract: Abstract : We present numerical simulation and mean-flow modelling of statistically stationary plane Couette–Poiseuille flow in a parameter space $(Re, θ )$ with $Re=\sqrt {Re_c^2+Re_M^2}$ and $θ =\arctan (Re_M/Re_c)$, where $Re_c, Re_M$ are independent Reynolds numbers based on the plate speed $U_c$ 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 $Re_{τ, b}$ changes sign at $θ =θ _c(Re)$ . 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 $θ _c$ 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, $θ _c$ 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 $Re_{τ, b}=0$ is investigated. Combined WMLES with modelling for these flows enables exploration of the $Re→ ∞$ limit, giving $θ _c → 45^°$ as $Re→ ∞$ .