# Dirichlet-Neumann boundary condition solution becomes unstable - Pressure Correction Method

I am simulating incompressible flow over a cylinder at Reynold number of 500. I am solving navier stokes equation using pressure correction method. My solution becomes unstable after certain time (approximately 5s).

I have tried refining my mesh, stepsize (0.05) (making sure my CFL < 1, even though I am using implicit methods)

My boundary conditions, mesh and unstable results are shown in the attached figures. The domain is about 25 times larger than the cylinder diameter.

I have tried simulating this problem O grid (which became unstable almost immediately).

The following link contains the pictures of the boundary conditions and results.

I would be grateful if anyone can share their thoughts/experiences on this problem. Many thanks.

editted:

Apologies for the typing mistake:

I am using the following boundary conditions: Neumann boundary $$\frac{\partial \vec{u}} {\partial n} - \vec{n} p = 0;$$

on Dirichlet Boundary $$\vec{u} = u_x = 1$$

editted:

i have applied velocity boundary conditions on the nodes around the dirichlet boundary. Also, top-right and bottom-right corner node is dirichlet boundary with velocity 1.

After, I looked more deeply into the simulation results, I notice that instability starts to creep in at inflow/outflow junction.

• How, specifically, are you implementing your boundary conditions? This can make all the difference in a simulation like this. Commented Mar 6, 2013 at 2:07
• Mathematically, I do not think N-S in 2-D can behave like this, Navier-Stokes. At the corner node, do you leave the 'Neumann' condition to $0-\mathbf{n}p=0$ because $\partial_{\mathbf{n}} \mathbf{u}=\partial_x (u_x,0,0)=0$ at the corner node along the normal direction of the 'Neumann' boundary. Commented Mar 7, 2013 at 8:33
• What is the method that you use? FEM? With stabilization? Did you try to lower the Reynold number? Commented Mar 7, 2013 at 12:43

• On the Neumann boundary, do you mean normal derivative $\nabla u\cdot \mathbf{n}$ for outer normal unit vector $\mathbf{n}$. Or you really mean gradient $\nabla u$? Commented Mar 6, 2013 at 12:16