Kovasznay Flow

  

Figure 5.2: Steady state solution for the Kovasznay flow at Reynolds number Re=40 using a discretization using K=24 elements. Left: The spectral element mesh used, Right: Steady state streamlines.

The first Navier-Stokes solution we shall consider is the Kovasznay flow. This is a laminar flow behind a two-dimensional grid, the exact solution of which is due to Kovasznay [42]. This solution can be written as a function of Reynolds number Re in the form:

where

$$\lambda = \frac{Re}{2} - \left ( \frac{Re^2}{4} + 4 \pi^2 \right )^\frac{1}{2}.$$

Using the exact solution as Dirichlet boundary conditions, a steady state solution was obtained using the discretization shown in figure 5.2. Also shown in this figure are the steady state streamlines and iso-contours of the x-component velocity at a Reynolds number of Re =40 using an expansion basis of N=7.

Using the exact solution allows us to calculate the error, in the $L_\infty$ and H₁ norms, with expansion order as is shown in figure 5.3. A summary of the simulation parameters is given in 5.1.

 

Parameter Value

Dimension 2d

Re 40

$$\Delta t$$ 0.0001

N-Range 5 to 11

K_Tri 12

K_Quad 12

Method Galerkin

  

Figure 5.3: Convergence in the L infty and H1 norms as a function of expansion order for the steady state Kovasznay flow at a Reynolds number Re=40.

We repeated this experiment with a three-dimensional mesh. Again in 5.5 we see that as we increase the expansion order we obtain exponentially increasing accuracy.

 

Figure 5.4: Top Left: Three-dimensional spectral element mesh used to solve Kovasznay flow on, Top Right: Iso-contours of streamwise component of velocity

  

Figure 5.5: Bottom: Convergence to exact solution with increasing order.