Dirichlet condition incremental pressure correction scheme
 

Hello,

I was wondering to what extent it is / if it is physically corerct to prescribe a pressure dirichlet BC in an IPCS method. Due to the fact that I seem to remember that there is a slight discrepancy because you are prescribing the perssure, however the equation solves for the pressure correction.
So my question is, is it actually correct to prescirbe the pressure in an IPCS method.

Cheers,
Jack

Mathematically, the Poisson problem for the pressure is elliptic and the Dirichlet BC is a correct choice. However the problem is more general, you can prescribe physically a pressure value? That depends on the assumption of your problem. For example an outlet in a constant-pressure enviroment.

Do not forget that there is no physical thermodynamic pressure for incompressible flows.

Hi,

You can. There's just one think one should be careful about. What is important in a pressure correction scheme in this regard is that you cannot prescribe pressure and velocity boundary condition (BC) independently:
- If you prescribe a Dirichlet velocity BC, you want that the pressure correction step does not compromise this constrain on the velocity. So the pressure gradient normal to the boundary has to be zero (Neumann BC).
- Similarly, you can prescribe a zero pressure boundary condition (e.g. outflow), but then you should make sure that the velocity BC normal to the boundary is consistent with this condition. For that you can employ the pressure correction step also at the boundary.

Let me highlight that, the normal derivative of the pressure is not zero but it is expressed according to the Hodge decomposition, that is expressed by the momentum equation projected along the normal direction to the boundary



dp/dn = n.(a* -a )


The homogenous Neumann condition is an approximation that can generate errors if n.a* is not added to the source term of the Poisson equation.

Perhaps we are talking about different things, so let me clarify. The boundary condition for the correction pressure has to be homogeneous Neumann, because if you discretise your pressure projection step at the boundary, it is the only boundary condition that will not disturb the velocity BC.

Are we talking about the closure for the pressure equation? If yes, the Poisson problem has the closure defined by the non homogeneous Neuman condition that satisfy the compatibility condition.
At a boundary, the pressure derivative must be never discretized by it is substituted into the Div Grad () operator.
That issue is general and does not depend on the specific incremental pressure, pressur-free or gauge method.

If I am wrong to understand you comment, please write down the equation you are talking about.

I understand that perhaps I was a bit confusing. Let's not call this variable pressure, but the scalar potential \psi that projects the prediction velocity u* into a divergence-free space u^{n+1}.



When we solve the Poisson equation for this scalar \psi, often denoted correction pressure, we need to prescribe BCs for it. The question is: which BC shall we prescribe when we impose no-slip/no-penetration?


If we look at the correction step, we that:


u^{n+1} = u^* - dt grad \psi



(dt is the time step)



since at the boundary we want to impose a no-penetration BC: u^{n+1} = u* = u_b, we do not want this gradient to change the flow at the wall, so the gradient of this potential normal to the wall should to be zero at the boundary.

Yes, that is what I am talking, too. According to your notation, project along the normal to the boundary (a wall but also possible to be inflow/outflow) your equation





un^{n+1} = un^* - dt d(psi)/dn



As you see, this is the projection of the Hodge decomposition. You have to use it as it is, just substitute the derivative into the Div Grad psi at the boundary. Then, you will see that un^* is exactly as same as in the source term, but with opposite sign. Thus, the source term has no longer un^* on the boundary but directly the physical velocity un^n+1.
Apparently, that could be seen as the application of the homogeneous Neumann condition but you have, correspondigly to modify also the source term close to the boundary. That will satisfy the compatibility condition and a solution exists apart a function of time.

I worked for some years on the theory of projection methods, if you are interested to the details of what I am talking you can read here in Remark 1

https://www.researchgate.net/publica...ary_conditions

Thanks for your comment and reference. I agree with your point. I guess that the difficulty is that in a staggered grid the source term is not defined at the boundary, and since it will cancel out in the pressure correction step, it is simpler to impose a homogeneous Neumann BC for solving the Poisson equation.

The key is that you can indeed set the homogenoeus Neumann condition (see Remark 1 in the paper) but when you look at the source term it has the Div v* expression so that on a staggered grid the component n.v* appears. But actually you do not need at all to use it as the Hodge decomposition is used and you simply cancel it. But doing that the source term must be computed using n.v^n+1 on the boundary. In conclusion, if you prescribe dpsi/dn= 0 you must also change the source term accordingly. This way the similarity between the twoi BCs is mathematically correct.

This point is often not fully understood and the consequence is that the compatibility condition is not satisfied and people are forced to set an arbitrary value for the pressure to get a convergent solution.