Hello all,

This one is about the boundary conditions for pressure in the fractional step method. Quite simply, How do you treat pressure at the boundaries, so that the desired velocity is obtained at the boundary?

I can't seem to get the velocity *parallel* to the boundary to approach its prescribed value (e.g. zero at a no-slip wall).

Zang et al. (1994) use their "corrector" (projection) equation in reverse: pressure grad = U_* - U_prescribed to devise a BC for pressure, such that the velocity *across* the boundary yields the appropriate value, e.g. zero across a solid boundary. This is fairly straightforward to apply.

But what about the velocity *parallel* to the boundary? How do you force the pressure gradient parallel to the boundary to be what it should be? Does anyone have ideas or experience with this?

A quick overview of my setup: Steady-state RANS, with pseudo-time derivative Non-staggered, orthogonal grid Fractional step method / LU factorization

Thanks in advance!

Jean-François Corbett

P.S. These are the main references I have tried to get inspiration from. Zang, Street, Koseff, J Comput Phys 1994) Perot 1993 - An Analysis of the Fractional Step Method, and Perot 1995 - Comments on the Fractional Step Method, Journal of Computational Physics

Jean-Francois,

The easiest thing to do with the BC for your pressure, when you have a no-slip wall, is to impose a Dp/Dn=0 (the gradient of the pressure in the direction normal to the no-slip surface = 0). This simply implies that P(i,j=1)=P(i,j=2).

Hope this helps. Sincerely,

Fred

Thanks for your reply. We have indeed tried Dp/Dn=0. However, our problem is rather with Dp/Dt, where by "t" I mean "tangential to the boundary" (not "time"!).

Let u and v be the velocity components tangential and normal to the boundary, respectively, and u* and v* the intermediate velocity components.

The u* and v* equations are set up such that u* and v* obey the BCs for u and v, respectively, e.g. in our no-slip case, u*=0 and v*=0.

Then we calculate p and correct the intermediate velocity. The condition Dp/Dn=0 makes sure that v=v*=0 at the boundary, as desired. However, our pressure solver does *not* yield Dp/Dt=0 at the boundary! Hence u is not equal to u*, i.e. u is not equal to zero and does not obey no-slip.

Neither I nor my colleagues have yet found anything about this in the literature.

A colleague has speculated that our problem may be caused by the fact that we are developing a perturbation solver. The problem arises in the first-order equation, from which higher-order terms are neglected. But it's only speculation and I'm not satisfied with that explanation.

Jean-François

Jean-Francois,

Since you are using a collocated (non-staggered) scheme, then you actually have u,v & p at the cell center correct? If so, then you don't really impose u=0 & v=0 on the no-slip boundary, in fact you impose a zero flux trough that boundary and when you calculate the viscous stresses, then you make sure that you take care of the no-slip.

Now, I don't really understand why you want to impose Dp/Dt=0, because if you really wanted to have Dp/Dt=0, it would mean that P(i,j)=P(i+1,j). If so, for example in the case of an airfoil surface, it is not possible to have P(i,j)=P(i+1,j) since the pressure varies along the airfoil surface.

Ultimatly, if you are solving a pressure poisson equation, and use orthogonal grids at the wall, then imposing Dp/Dn=0 is all you should need.

Good luck to you. Sincerely,

Fred