Couette–Poiseuille Flow (Velocity Profiles)

Overview

Couette–Poiseuille flow is the fully-developed laminar flow between two parallel plates driven by both a moving wall and a streamwise pressure gradient. It generalizes plane Couette flow (pure shear, a linear profile) by superimposing the parabolic Poiseuille profile produced by the pressure gradient. Because the governing equation is linear, the exact solution is simply the sum of the two, giving a one-parameter family of profiles controlled by the dimensionless pressure-gradient parameter $P$. It is a standard textbook verification with a closed-form answer and no turbulence, compressibility, or geometric complication.

A favorable gradient ($P>0$) accelerates the fluid and can raise the peak velocity above the lid speed; an adverse gradient ($P<0$) decelerates the near-wall fluid and, if strong enough, drives a region of reversed flow along the stationary wall. The pure-Couette limit $P=0$ recovers the straight-line profile.

Problem Setup

The domain is the same unit-height channel as the plane-Couette case: a quasi-2D box, periodic in the streamwise direction, with the top wall translating at constant speed $U$ and the bottom wall stationary. The flow is incompressible (constant density), laminar and steady, with $Re=\rho Uh/\mu =1000$ on the channel height $h$.

The streamwise pressure gradient is imposed as a uniform body force through the gravity term, with an x-acceleration $a_x=2\mu UP/(\rho h^2)$ chosen to set the target value of $P$. Three cases are run: an adverse gradient $P=-2$, the pure-Couette case $P=0$, and a favorable gradient $P=+2$.

Quantities of Interest

The verification quantity is the streamwise velocity profile $u(y)/U$ across the channel, sampled on a vertical line and compared point-by-point to the analytic Couette–Poiseuille superposition $u/U=\eta +P\eta (1-\eta )$ with $\eta =y/h$. The three values of $P$ exercise the favorable, zero and adverse pressure-gradient branches of the solution family, including the near-wall reversed flow at $P=-2$ and the above-lid peak velocity at $P=+2$.