Rayleigh Sliding Plate Problem

We verified the ALE code using the Rayleigh problem. The domain is a box with a sliding plate at the lower side. The plate slides horizontally with velocity $cos(\omega t)$. The plate has a periodic length 2. Exact boundary conditions are imposed at y=-0.5,1.5.

The exact solution to the Navier-Stokes equations is:

 

Parameter Value

Dimension 2d

Re 40

$$\Delta t$$ 0.01 and 0.001

N-Range 2 to 9

K_Tri 50

Method ALE Galerkin

First we tested convergence to the exact solution using zero velocity initial conditions. Figure 5.14 shows that at N=5, $\Delta t=0.01$ the simulation converges to 10^-5 accuracy. Also we show that with $\Delta t=0.001$, starting from the exact solution and checking errors at t=2, we obtain exponential convergence with increasing N. A summary of the simulation parameters is given in table 5.5.

 

Figure 5.13: Top: Spectral element mesh KTri=50. Lower: Time convergence history for zero initial conditions N=5. Lower Right: Convergence to exact solution with increasing N.

  

Figure 5.14: Lower Right: Convergence to exact solution with increasing N.