Wannier Flow

The first test we consider is Stokes flow past a rotating circular cylinder next to a moving wall. The exact solution due to Wannier [41] allows us to evaluate the error in a domain involving curvilinear elements. The exact solution can be written:

where we define:

$$A = -\frac{Ud}{\ln (\Gamma)} + \frac{d\omega}{2s}, \hspace{3... ...ac{2U(d-s)}{\ln (\Gamma)} + \frac{(d-s)\omega}{s}, \hspace{3mm}$$

$$D=-U, \hspace{3mm} F = \frac{U}{\ln(\Gamma)},$$

$$K_1 = x^2 + (s+y_1)^2, \hspace{3mm} k_2 = x^2 + (s-y_1)^2, \... ...{3mm} s^2 = d^2 + R^2, \hspace{3mm} \Gamma = \frac{d+s}{d-s}.$$

Here we have used a cylinder of radius R=0.25 which is a distance d=0.5 from the moving wall. The wall is moving with a velocity of U=1 and the cylinder is rotating in a counter clockwise sense with an angular velocity of $\omega = 2$. The domain was split into 65 elements and the discretized domain is shown in figure 5.1.

  

Figure 5.1: Discretized solution domain for the Wannier-Stokes flow using 68 elements. Left: Graph of exponential spatial convergence, Top Right: Hybrid spectral element mesh, Bottom Right: Streamlines of the steady state Stokes flow.

The steady state solution with Dirichlet boundary conditions using an expansion basis of L=11 is shown in figure 5.1. This figure shows iso-contours of velocity, streamlines of the steady flow and the exponential convergence to the exact solution with increasing order.