Hello, everyone:

I use a projection method together with finite differences to solve the NaStEq for a turbulent channel flow with time variable boundary conditions (moving walls for example).

My question is: What are the correct (or best) boundary conditions for the intermediate velocity (u*) of the predictor step (with pressure correction).

The time discretization look like this:

a) (u*-u(n))/dt + (conv.)(n+1/2) + grad p(n-1/2) = 1/(2*Re)(laplace)(u*+u(n))

b) (u(n+1)-u*)/dt = grad (p(n+1/2)-p(n-1/2))

How must I choose u* at the walls for step a) ?

Thanks in advance.

Best wishes

Michael Hartmann

In projection methods you solve a Poisson equation for the pressure. This also requires boundary conditions. The important thing is to make sure that the boundary conditions for the pressure and the intermediate velocity are consistant with the boundary condition for the actual velocity, i.e. equation b) must be satisfied at the boundary. As long as this is true it doesn't matter what boundary condition you choose since u* is not real. However, some choices are more convenient than others. I am using a similar method and I apply the same boundary conditions to u* and u. This tells me what my boundary condition for the pressure must be.

Hope this helps. I can't think of any references off the top of my head, but I would think this would be described in more detail in Hirsch or Fletcher.

(Please excuse my english, I'm from germany)

My problem are not the boundary conditions for u* due to the implicit treatment of the pressure (or pressure correction) in the time discret momentum equation. As you have described they can be choosen arbitrarily. I am using a staggered grid so that no boundary conditions for the poisson equation are required (only pseudo b.c. for the implementation).

The implicit treatment of the _velocity_ in the time discret momentum equation leads to an implicit equation for u* (equation a)). Boundary conditions for u* are needed for this first step (predictor step). I think that b.c. for u* in this step can not be choosen arbitrarily. But how ? Currently I use u* = u(n+1), but I am not quite sure about that.

Hi Michael,

I think your approach is correct. I have used projection method alongwith the boundary conditions you have mentioned. That seems to work for all the problems I have solved. However, Kim and Moin (Journal of Computational Physics Vol. 59 pp. 308-323 (1985)) suggested a second order accurate B.C. for u*. They used second order accurate Adams-Bashforth temporal scheme for first equation. Basically, they were following R.J. Leveque's work ( PhD (1982) Stanford University : Time-split methods for partial differential equations)... I would suggest these references if you want to use a consistent BC for your case.

Hope this helps... Take care Mayank