I have a problem unsolved till now. I study the flow around a square cylinder by using SIMPLER method. I had to deal with the pressure boundary condition on the wall of the square cylinder because I define the grid node on the wall. As you know, the four corners of the cylinder are special nodes in this problem. Does anybody know how to give the pressure condition on the wall especially on the corners? please give me your advice! Thank you for any help you can give me!

You can use normal momentum equation

dp/dn = ρ n.u.∇u

where n is unit normal vector. But at a corner there is no unique normal. Mathematically pde does not make sense at such corners. The correct equations to use would be integral equations which are statements of conservation but have to be applied to a control volume.

In your case you will have to use some tricks which cannot be fully justified in terms of the governing equations. The simplest option is to just interpolate the pressure from the surrounding nodes. Or you could apply the normal momentum equation using an average normal taken from the two sides of the corner. Generally such tricks work.

If I remember correctly the pressure in the corner is singular and so your finite difference approximation will be badly in error here - I think it was D.W. Moore did some work on this. What he did was subtract the singularity off and then solve for the smooth bit. (It's the singularity which creates the Moffat eddies near the corner).

Tom.

if you use MAC staggered grid, how can you get a velocity nodes at corners? and how can you get a pressure nodes on wall boundaries?

the correct boundary conditions for pressure poisson equation is not the one believed by many, which is derived from normal component of momentum equation. it is simply "partial p partial n = 0"

essnetially, application of momentum equation on wall boundaries is an illegal but sometimes helpful operation.

I have to make some corrections to my previous post.

1. There should be a negative sign on the right hand side.
2. The equation I wrote is for inviscid flows.
3. For viscous flows, we have dp/dn = n.∇.τ where τ is the shear stress tensor.

For some references see

Roger Peyret, Handbook of Computational Fluid Mechanics, Academic Press.

Check chapter 3.

In boundary layer theory we study that at high Reynolds numbers, when boundary layers are thin, the pressure gradient normal to the wall is zero. Obviously, this is not true in general.

One may wish to look at applied mechanics reviews aug 2003 :"On boundary conditions for incompressible Navier-stokes equation" by Dietmer Rempfer in this regard...this paper tells the truth that the bc to pressure poisson equation cannot be specified with local relationships at the boundary...

a guy with time energy money and proper training can implement Kleiser and Schumann's influence matrix method (proceedings of GAMM conference 1980) in a preprocessing step for BC...

in my opinion : "partial p partial n = 0" is fair (if we trust the schemes we are using)....

"That derived from from normal component of momentum equation" leads to 'ill-posed differential equation problem'

thanks...the idea about normals clarified a problem with boundary condition to electric field on the corners of a liquid-solid interface...

boundary conditions for pressre poisson equations have been resolved. I would like to look at the paper on applied mechanics review, but just keep track of Journal of Computational Physics from 2005 papers.

basically, if you need boundary conditions for pressure poisson, then I always can find an alternative but similar method which does not need any numerical BC at all. tentatively, I name these methods as: exact factorization, approximate factorization, and variable splitting for indefinite system.

I am submitting the paper on ef, and will submit the af next month, and then variable splitting two months later; all to JCP.

I was unable to locate

applied mechanics reviews aug 2003 :"On boundary conditions for incompressible Navier-stokes equation" by Dietmer Rempfer

There is no issue in Aug 2003.

In my case, the primitive variable method (SIMPLER scheme)and the stagged grid is used. The main nodes are set to the wall of the square cylinder. So on the wall, there are only pressure nodes.

In my opinion, the boundary conditions for pressure on the wall,which is derived from normal component of momentum equation, may not be "partial p partial n = 0" .Because another item of the equation--the second normal derivative of velocity is not equal to zero.

by the way, can you tell me how can I DOWNLOAD the paper you've said about the BC for PPE?

Thank you very much and Merry Christmas!

sorry...I have the manuscript of the paper the author submitted..... I can mail you the same(1103k)...

If you can mail it to me then that would be great. My id is

praveen[at]aero.iisc.ernet.in

can you mail the manuscript at :

apurvas@aero.iitb.ac.in

Thanks

Apurva