pressure poisson equation on staggerred grid
 

Hello!

I have got the pressure poission equation "DEL2_p = d". I am using cylindrical co-ordinate system on staggerred grid. Off course the R.H.S. "d = func(v, DEL_v, DEL2_v)".

I am a little confused about discritization of this Poisson Equation regarding whether I should discretize it on velocity points(face of the cell) or at the pressure point(center of the cell).

This Poisson Equation is obtained after taking divergence of the momentum equation which are discretized at velocity points and so this poisson's equation should also. But I went through a document and there the term DEL2_p was discretized on the 'pressure point'. Nothing was mentioned about how to discretize the R.H.S. term.

Please help me out.

thanks!!

chandra

Re: pressure poisson equation on staggerred grid
 

The purpose of the pressure correction is to guide velocity corrections that balance the mass conservation equation (at least for incompressible flows).

So the pressure correction is derived by taking the divergence of the velocity vector around a cell (with velocities on the edges). At least this is the way it was done for MAC and SMAC techniques at Los Alamos.

MAC/SMAC employ a two-step process that splits the velocity advancement into two parts, essentially a predictor-corrector. The predictor is explicit. The velocity correction - based on the Poisson solution - is of course implicit.

The MAC/SMAC techniques are available in JCP. The SIMPLE technique (and descendents) are totally implicit but the pressure correction still serves to satisfy mass conservation.

I'm sure others will have comments on those techniques.

Re: pressure poisson equation on staggerred grid
 

Thanks Jim for your suggestions. I will have a look at the paper. But could you suggest me whether I should discretize the obtained Poisson equation at the 'face' or at the center of the 'cell'.....or 'DEL2_p' term at center of the 'cell' and velocity gradients (on right hand side of the equation) at the 'face'?

thanks

chandra

Re: pressure poisson equation on staggerred grid
 

Pressure correction at cell centers to balance mass (or volume) flow into the cell, thus satisfying the mass conservation law over the cell. The pressure changes at the cell center are used to correct the velocities/fluxes across the faces.

Re: pressure poisson equation on staggerred grid
 

Discretize over the pressure points (centre of the cell), the velocity gradients will then become central derivatives and hence second order.

Re: pressure poisson equation on staggerred grid
 

Thanks a lot. But I think all the linear terms should be discretized at the cell-centers while the non-linear convective term can't be discretized at cell-center. It is because we are taking divergence of the momentum equation and "DEL.DEL(n)_V (at right face) - DEL.DEL(n)_V (left face) = DEL(n+1)_V (at the center)". But this relation doesn't hold if we replace "V" with a non-linear term, say "V.DEL_V". So, in my opinion, all the non-linear terms (here V.DEL_V) should be evaluated at the 'face' centers first and then divergence of it should be obtained after subtacting the value at the left face from the value at the right face.

e.g. if DEL2_P = DEL.Q + DEL.R, here P = pressure, Q = non-linear term (V.DEL_V), and R = linear terms, then first evaluate Q = V.DEL_V at face centers. and then, evaluate div(Q) = DEL.Q = [Q(i+1) - Q(i)]/[x(i+1) - x(i)] + [Q(j+1) - Q(j)]/[y(j+1) - y(j)].

DEL2_P and DEL.R can be discretized directly on the face centers because these are linear terms.

Please suggest if something went wrong in my understanding.

Thanks again,

chandra

Re: pressure poisson equation on staggerred grid
 

I do not see what you mean with V DEL V, this non-linear term does not appear in the pressure equation. The momentum equation does not appear in the pressure equation, is the continuity eq.

Re: pressure poisson equation on staggerred grid
 

Heyy, sorry...this is (V.DEL)V

Here DEL is the operator "(del/del_x)ex + (del/del_y)ey + (del/del_z)ez". ex, ey and ez represents unit vectors along x, y and z directions respectively.

V is the velocity vector and so "(V.DEL)V" is the convective term. Off course, V = u*ex + v*ey + w*ez. Momentum equation: dV/dt + (V.DEL)V = -(1/DENSITY)*DEL_P + KIN_VISC*(DEL2)V + F.

The pressure poisson equation is obtained after taking divergence of this momentum equation. i.e. DEL.(dV/dt) + DEL.((V.DEL)V) = -(1/DENSITY)*DEL.DEL_P + KIN_VISC*DEL.((DEL2)V) + DEL.F.

=> dD/dt + DEL.((V.DEL)V) = -(1/DENSITY)*DEL2_P + KIN_VISC*(DEL2)D + DEL.F. Here, D = DEL.V = divergence.

Since, our requirement is D = 0 (from continuity), the equation becomes: DEL.((V.DEL)V) = -(1/DENSITY)*DEL2_P + DEL.F => DEL2_P = DENSITY*(DEL.F - DEL.((V.DEL)V)). This is the pressure-poisson equation and the term "DEL.((V.DEL)V)" is the non-linear term I mentioned in my previous posting. Here, we can discretize DEL2_P on the pressure point since it is linear term.

Please correct me if there is something wrong.

Thanks,

chandra

Re: pressure poisson equation on staggerred grid
 

Sorry, my mistake, I got confused

Two things here, assuming constant density and viscoisty

1) Pressure equation

DEL^2 p = - DEL (V DEL V)

2) Pressure CORRECTION equation

C DEL^2 p'= DEL V'

which C= 2/3 dt and V' is the velocity obtaine from the momentum equation.

So which method are you using, if you solve eq 1) you obvoulsy got a non-linear term. In eq 2, you obtain the pressure correction which you have to add to the pressure and as well as correct the velocities

V(n+1)= V' - C DEL p '

p(n+1)=p(n) +p'

Schem 2) is better for unsteady flows and do not have the problem to discretize. Eq 2 is the basics of SIMPLE type methods, and I think is predominant in current incompressible CFD (see books Patankar, Ferziger). Eq 1 is mostly used if you work with thw stream function and also in steady flows (book Tanehill), or the MAC type methods, which sincerely are a little bit outdated (see Tanehill book, last reference late 70's early 80's)

Re: pressure poisson equation on staggerred grid
 

Hey, thanks for your response. I am using the first procedure, i.e. i m solving the Pressure Poisson Equation and so I got the non-linear term and as did what I mentioned in my previous posting. To check the correctness of my code, I applied the boundary condition for simple pipe flow. I am in process to check whether the code is diverging after what Reynolds Number.

It would be a great help to me if you could please suggest me what time step should I implement in my step for a flow well within Leminar limits...say for Re = 1000. I am using 1.0E-3 now.

Yes, I am using the MAC algorithm to implement the solution. Actually I found the iterative method u suggested a little complicated in comparision with the current one I am using.

Thanks a lot in anticipation,

chandra