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February 12, 2001, 18:31 
Nonstaggered Grids and Momentum Interpolations

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Dear all,
How are you doing. I have some quick questions for you guys who had good experience with Finite Volume Method on nonstaggered grids. I would appreciate it very much if you can drop some words on it. Let's say the center of each control volume is nodal point. 1. To ensure the mass balance in the boundary cell, we need to make a bigger control volume to include half boundary cell (here the domain boundary becomes the boundary of the control volume). The question is: is this kind of control volume necessary for momentum and energy equations? I did this but found that the code was not working well especially for simple driven cavity problem. I thought the problem could be that the coefficient of momentum equation for boundary cell is zero and the momentum of drive lid can not be transfered into the next cell. Am I right? 2. To overcome the checkboard pressure distribution on nonstaggered system, the momentum interpolation should be applied in the calculation of mass fluxes used for continuity equation. Some author said it's not necessary for the fluxes used for momentum equation (just linear interpolation needed), but some author said it would be good to use momentum interpolation for both. The question is which way is better? Also this question is related to the first one. If bigger boundary cell used for momentum equations, then the coefficient of momentum equation for boundary cell is quite different from interior cells and would affect the calculation of fluxes for continuity equation or pressure correction equation. Am I right? 3. If it is compressible flow in a closed domain, the pressure distribution would be shifting since no reference pressure provided (of course the initial pressure was known). For incompressible flow there would be no problem. But there should be a problem for compressible flow (for example the phase separation of siturated steam/water mixture in a closed container) since we need to calculate densiy and temperature etc from updated pressure. I tried floating reference pressure but it seemed not satisfied. Any good suggestion would be very appreciated! Thanks a lot in advance and have a good day. S. Wang 

February 12, 2001, 20:09 
Re: Nonstaggered Grids and Momentum Interpolations

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1. Do you define ghost boundary cells on the other side of the lid. If so, the momenta ghost cell should be prescribed in such a way that when you interpolate (which ever way you are doing it), the momentum at the interface of the boundary cell and ghost cell (i.e. the physical boundary) should equal the velocity of the lid. May be I did not understand your statement about the coefficient of the boundary cell becoming zero. You solve for the boundary cell values just like in the other cells. Only difference is that the boundary cell has a ghost cell adjacent to it. However, if apply the boundary conditions in the ghost cell in the boundary condition subroutine, the solver routine treats the boundary cells much like any other interior cells and updates the variables.
2. Just a linear interpolation would give you second order central scheme and hence very little artificial dissipation (good for DNS or LES). If you use the biased interpolation, there would be some damping. From what I have seen, the linear interpolation would lead to some oscillation at corners (or edges) if the grid is highly nonuniform. If you want to solve some complex problems it is always good to have a code that can handle highly stretched grids. So, I would not use a linear interpolation for such cases. 

February 12, 2001, 21:40 
Re: Nonstaggered Grids and Momentum Interpolations

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Dear Kalyan,
Thank you very much for your kind quick reply. My statement seemed not clear, I am sorry about. I didn't define ghost cells like what I did in the calculation of shock tube problem (pure compressible flow). Actually what I talked about is SIMPLElike finite volume method. I just tried to write a code for transient, 3D, multiphase flow and heat transfer problems with all speeds by using SIMPLElike approach that I thought could be better than methods for pure compressible flow problems. Acutally the code was working fine in incompressible case. Now back to the questions 1. I didn't define ghost cells. I just defined the nodal location as the center of each control volume. From this you can see, in the half of boundary cell we couldn't ensure the mass balance and momentum balance (you can define the center of control volume as the center of surrounding nodal points but there would be another kind of difficulty appeared in the numerical treatment). So I enforced the interface of near boundary cell as the physical boundary. If I did so, the coeffient of the discretized equation (for both momentum and mass balance equations) for boundary cell would be zero for solid wall (like lid) (I didn't mean main coefficient, I meant the coefficient linked to boundary grid, let's say Lid here). That would lead behavior like what I mentioned in previous message. I thought this kind of treatment is necessary for mass balance equation. So my question is: is it necessary for momentum equation? The problem we can see if we did this for momentum equation is that the main coefficient for momentum euqtion of boundary control volume would be much different (the main coefficient would be used to determine the coefficient of pressure correction euqtion and interpolate the mass fluxes for mass residual) that could affect the calculation of pressure correction and further density and temperature. As to question 2, I am sorry I didn't talk about the determination of fluxes in pure compressible flow calculation. I just talked about the momentum interpolation approach in SIMPLElike method when we use nonstaggered system. There are a few altenatives currently used. My question is: we got to use momentum interpolation to determine the mass fluxes which would be used for the calculation of pressure correction, is it necessary to use the fluxes obtained by momentum interpolation in the calculation of momentum euqations? I don't know if my statement is clear now. I really appreciate your kind assistance and your time in this matter. Regards, S. Wang 

March 2, 2001, 14:44 
Re: Nonstaggered Grids and Momentum Interpolations

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