Fractional Step Method with Dirichlet Pressure Boundary Conditions ?
 

Please please please
Could any one tell me how to apply Dirichlet Pressure Boundary Conditions while using fractional step method.
All the paper says that the Poisson equation has to be solved with Neumann BC, what about Dirichlet B.C!!!
Please help me it is urgent

Starting from the Hodge decomposition you have on a boundary:

n.v* = n.v^n+1 + n.Grad Phi

therefore if you fix the pressure at the outlet you have to discretize the normal derivative and use the pressure value. Furthermore, the velocities must be somehow computed.

I remember a post that already treated this topic

So, you mean that the Poisson eq will be solved with Neumann B.C, and the Dirichlet Pressure B.C will be included implicitly in the intermediate velocity B.C ???
If my understanding is true , the question arises will be How can i now v^n+1 ???

You can use some approximation on the derivatives...Some constraint must be fulfilled to ensure the existence of the solution.
Just as example, see http://www.aeromech.usyd.edu.au/~kir...eld-ctac08.pdf

Hello,
I am using a pseudospectral code that uses fractional step algorithm with mixed RK3/CN . It works fine . But I have to modify my velocity field with newton method for some of my problems. After that I would like to recompute the pressure using that velocity field. To verify my poisson solver I used this method :

1. Do a long dns save velocity and pressure(P)

2. Use this velocity to compute pressure using laplacian(P)= -div(convetive terms)
with dp/dy =1/Re(d2v/dy2) as my B.c's . Here I get P'

3. L2norm(grad(P'-P))/L2norm(grad(P)) ~ 1e-2.

4. But to my surprise original pressure P from DNS does not satisfy this laplacian(P)= -div(convetive terms) . I mean this should be staisfied atleast upto machine precision .

In short the gradient is same for P and P' but P does not satisfy pressure poisson equation . Can anyone explain this to me what is the reason behind this

I think your are not ensuring the continuity constraint at all...

I suggest starting from the Hodge decomposition v* = vn+1 + Grad phi, then substitute in the continuity equation to get the elliptic problem

Div (v* = vn+1 + Grad phi) => Div Grad phi = Div v*

To be well posed the problem, you must prescribe the normal component of the Hodge decomposition on the boundary.
When that is correctly done, the divergence-free constraint is ensured to machine precision in case of exact projection method (as happens in the MAC method) or up to the local truncation error magnitude for the approximate projection method (as it happens using colocated variable).

Hello ,
Thanks for your reply . I checked my calculation . The (0,0) harmonic of pressure is not calculated properly by poisson's equation . Here from Poisson I mean not the one in the fractional step , but a seprate subroutine that uses the velocity field computed from the DNS and attempt to recompute the pressure. The problem now is :

1. For the zero zero harmonic the poisson is d2p/dy2 = -RHS(u) . To solve this I need 2 BC's . On the lower wall I apply Drichlet and homogeneous Neuman on the upper wall .

2. When I see the result close to the upperwall my pressure mathches the one from the DNS. But close to the lower wall its wrong . If you apply Drichlet on uuper wall and homogeneous Neuman on lower wall . The results are opposite.

3. I can't apply Neumann on both the walls the matrix becomes singular.

How can I solve this issue .

Thank you

- The pressure equation produces a singular matrix, this is congruent to the fact that the pressure solution is defined apart a constant.
- The system is solvable due to the fact that the compatibility constraint is satisfied. Therefore, your BC must be such as to fulfill this relation, otherwise you do not have a solution.