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Numerical simulation of turbulent, plane parallel Couette–Poiseuille flow
Published online by Cambridge University Press: 13 January 2023
Abstract
We present numerical simulation and mean-flow modelling of statistically stationary plane Couette–Poiseuille flow in a parameter space 
$(Re,\theta )$ with 
$Re=\sqrt {Re_c^2+Re_M^2}$ and 
$\theta =\arctan (Re_M/Re_c)$, where 
$Re_c,Re_M$ are independent Reynolds numbers based on the plate speed 
$U_c$ 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 
$Re_{\tau,b}$ 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 
$Re_{\tau,b}(Re,\theta )$, 
$Re_{\tau,t}(Re,\theta )$ (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 
$Re$, 
$Re_{\tau,t}$ passes through a peak at approximately 
$\theta = 45^{\circ }$, while 
$Re_{\tau,b}$ 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 
$Re_{\tau,b}=0$ is investigated. Combined WMLES with modelling for these flows enables exploration of the 
$Re\to \infty$ limit, giving 
$\theta _c \to 45^\circ$ as 
$Re\to \infty$.
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- © The Author(s), 2023. Published by Cambridge University Press
Footnotes
‡
Professor Samtaney passed away during the preparation of this paper.
References
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