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.

Boundary Conditions


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


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 $$ \vec{u} = u_x = 1 $$


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.

  • 2
    $\begingroup$ How, specifically, are you implementing your boundary conditions? This can make all the difference in a simulation like this. $\endgroup$ Commented Mar 6, 2013 at 2:07
  • $\begingroup$ 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. $\endgroup$
    – Hui Zhang
    Commented Mar 7, 2013 at 8:33
  • 2
    $\begingroup$ What is the method that you use? FEM? With stabilization? Did you try to lower the Reynold number? $\endgroup$
    – Dr_Sam
    Commented Mar 7, 2013 at 12:43

1 Answer 1


I have figured out the problem. I had to increase the size of the domain further to remove boundary effects. Moreover, I had to reduce CFL number to around 0.5-1.0

I think CFL number need to be reduced further for higher reynolds number.

Initially, I thought that I had reduced step size enough, but it was not the case.

  • 2
    $\begingroup$ 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 $\nabla u$? $\endgroup$
    – Hui Zhang
    Commented Mar 6, 2013 at 12:16
  • 3
    $\begingroup$ Instead of “answering” your own question, you should edit the original question to include the additional information. This makes it easier to have all the information in one location and thus to answer your question. $\endgroup$ Commented Mar 6, 2013 at 17:03
  • 2
    $\begingroup$ A comment on your thought - the CFL number probably does need to be reduced for higher Reynolds numbers. Max Gunzberger in his FEM book for Viscous Incomp Flows noted that the radius of convergence for the Newton method shrunk with increasing Reynolds number, and decreasing CFL restricts the timestep, which can be interpreted (for implicit timestepping) as adding an increasing amount of regularization to the pure Newton iteration. $\endgroup$
    – Jesse Chan
    Commented Aug 21, 2013 at 5:14
  • $\begingroup$ Will not a Neumann boundary for velocity on the two horizontal boundaries be more appropriate? My guess is as you are imposing a Dirichlet, the boundary is still not far away. $\endgroup$ Commented Jan 24, 2017 at 11:33

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