Stokes's First Problem (Impulsively Started Wall)

Overview

Stokes's first problem, also called the Rayleigh problem, is the textbook unsteady viscous flow that develops when an infinite plane wall bounding a fluid at rest is impulsively set into steady motion in its own plane. Momentum diffuses away from the wall into the fluid, and a shear layer of thickness $\delta \approx 4\sqrt{\nu t}$ grows in time. Because there is no pressure gradient and the flow is independent of the streamwise coordinate, the problem has a closed-form self-similar solution: the velocity profile is the complementary error function of a single similarity variable, so profiles taken at different instants collapse onto one curve. The case exercises the unsteady (implicit) time integration coupled to the wall-normal diffusion operator, with no convective or pressure-gradient terms. The reference solution is the one given by Schlichting & Gersten.

Problem Setup

The domain is a tall, thin slab above the moving wall: a streamwise length $L=1$ m and a wall-normal height $H=2.5$ m, meshed as a single spanwise layer of cells so the simulation is quasi-two-dimensional. The fluid is initially at rest everywhere. At $t=0$ the lower wall begins translating in its own plane at the constant speed $U_0=1$ m/s, imposed through the solver's moving-frame mechanism so the no-slip wall carries the fluid with it without any normal motion. The streamwise faces are a translational periodic pair, which removes any dependence on $x$; the top boundary is a symmetry plane placed far above the wall so the diffusing shear layer never reaches it over the simulated interval, matching the quiescent far field; and the spanwise faces are symmetry planes. The fluid has constant density $\rho =1$ kg/m³ and constant viscosity $\mu =1$ Pa·s, giving a kinematic viscosity $\nu =1$ m²/s. The second-order implicit time scheme advances a fixed step $dt={10}^{-3}$ s to a final time $t_{max}=0.2$ s, at which the shear-layer thickness $\delta \approx 1.8$ m is still well inside the domain.

Quantities of Interest

The verification quantity is the streamwise velocity profile $u(y,t)$. With no pressure gradient the streamwise momentum balance reduces to the one-dimensional diffusion equation $$\frac{\partial u}{\partial t}=\nu \frac{{\partial }^{2}u}{\partial y^2},$$ subject to $u=U_0$ at the wall and $u\to 0$ far from it, starting from rest. Its exact self-similar solution is $$\frac{u(y,t)}{U_0}=\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}\left(\eta \right),\eta =\frac{y}{2\sqrt{\nu t}},$$ where $\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}$ is the complementary error function. The computed profile is sampled at three physical times ($t=0.05,0.1,0.2$ s) and plotted as $u/U_0$ against the similarity variable $\eta =y/(2\sqrt{\nu t})$, collapsing onto the single $\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}(\eta )$ curve. The computed $u/U_0$ at the similarity stations $\eta =0.5$ and $\eta =1$ is reported beside its exact $\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}(\eta )$ values; the two agree within 0.1% of the analytic $\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}$ solution.