Hi,

the pressure Poisson equation is derived from the momentum equation. Be-

cause it is derived by taking the divergence of the momentum equation, it requires

that the solution be sufficiently smooth up to the boundary; that is, the solution

must be more smooth than would otherwise be required. The pressure Poisson equation is, for sufficiently smooth solutions, equivalent to the continuum Navier-Stokes eq. in the non-steady case. For the steady case, notes that equivalence is achieved when the pressure Poisson equation is written without the viscosity term and a divergence free boundary condition is enforced for the velocity.

The boundary conditions exist for the velocity. What we need are boundary conditions for both the flow velocity and the pressure that allow us to develop the velocity field in time for a given pressure and then solve for the pressure at each fixed time given the velocity.

For an uncompressible fluid you should choose the normal (Neumann) and tangent (Dirichlet) component on the boundary: The two conditions are equivalent and you calculate pressure up to an arbitrary additive constant.

The discretization of the pressure equation, being a derived one, requires special care in order to control the accumulation of numerical errors that might invalidate the continuity equation. This means there is a need to introduce corrective terms into the pressure equation and you may use a Neumann condition to develop the solution.

@Pablo Montalvo, In fact I don't expect the boundary term to vanish, what I'm looking for is a physical information about the pressure gradient on the boundary that I can understand why it's imposed this way and not otherwise, and consequently solve numerically my Poisson's equation for the pressure which eventually verify the Navier-Stokes system.

regards

As far as I understand, natural way to proceed would be to use Navier-Stokes equation to deduce the boundary condition. Since you already used that equation to deduce the equation for pressure Poisson equation, I assume that you don't have anything against using it for boundary condition.

Since you know the velocity field, you can use it to find pressure gradient at the boundaries (directly from Navier-Stokes equations), be it steady or time dependent. After that your Poisson equation (with natural boundary conditions over all of the surface) would be determined up to an additive constant, which you can fix at one point in the domain (or at boundary).

Good luck!

Bonjour,

I am not sure that followings are what you would like to find...

All the boundary points should be assigned as Dirichlet (pressure) or Neumann (pressure gradient) B.C., while at least one Dirichlet condition is required to avoid the singularity of Poisson equation.

Laplacian of pressure, div ( grad p ), can be expressed in central differential scheme as, (assuming as 1D case for simplifying the equation...)

 Eq. 1   :   ( grad p(i+0.5) - grad p (i-0.5) ) / dx                 

or

Eq.   2  :    ( ( p(i+1) - p(i) ) / dx - ( p(i) - p(i-1) ) / dx ) / dx

(here ( p(i+1) - p(i) ) / dx = grad p(i+0.5),  p(i) - p(i-1) ) / dx = grad p(i-0.5) )

You can find here how you can apply Dirichlet (Eq. 2) or Neumann (Eq.1) boundary condition. For a general example, Neumann condition at the left side, the above equation can be expressed as

div ( grad p ) = ( ( p(i+1) - p(i) ) / dx - [ pressure gradient at the left side] ) / dx 

Best,

@Uģis Lācis,

if I do what you suggest I would take the following possible B.C _{H^{-1/2},H^{1/2}} = 0 for all \phi in H^1, so I get the equivalence between the weak and strong form of Poisson's pressure equation. what I I'm not sure of, is if the pressure field that I will obtain using this B.C will approximate the solution up to a constant !

hi Mourad Oulghelou, for incompressible fluid, the question is still not clearly understood from theoretical aspect. Unlike compressible fluid, the pressure difference makes sense in incompressible fluid. So commonly, Raffaele Salerno's opinion is a classic point of view, and many commercial CFD utilize this kind idea. Obviously, the detailed pressure exerted on the wall boundary is not known and not possible to find. So the boundary pressure value is commonly assigned the inner grid point value, which means the Neumann B.C. is applied.