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September 14, 2004, 10:11 
momentum interpolation for collocated grid

#1 
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Dear friends
I am developing a 2D implicit unsteady code using collocated grid. As you know, to avoid the Checkerboard pressure prediction, the momentum interpolation is needed. since I use the underrelaxation factor in solving momentum equation the original Rhie&Chow method can not be used directly. To have the results independent of underrelaxation and time increment a modified Rhie&Chow interpolation should be used. One of them has been explained in the following paper: Lien ,F.S.; Leschziner, M.A. "A General NonOrthogonal Collocated Finite Volume Algorithm for Turbulent flow at all Speeds…..", Comput. Methods Appl. Mech. Engrg, 114(1994) 123148 Now my question is about implementation of such a method. 1 In this type of momentum interpolation the Ap at cell face is needed. should I interpolate the Ap by linear interpolation or 1/Ap should be interpolated? 2 In the momentum interpolation equation there are the cell face velocity at previous iteration and previous time step. So we should store 2 velocity component at each face. So in this case the required memory will be more than what is needed for staggered grid. Is this true? Or there is a trick not to store the velocities at cell faces? 3 Should I use the cell face velocity resulted from momentum interpolation only in pressure correction or it should be used in momentum equation too? Thanks in advance for your replies. Regards, Hadian 

September 19, 2004, 21:05 
Re: momentum interpolation for collocated grid

#2 
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(1) Yes, usually 1/Ap is linearly interpolated. (2) Yes, the cell face velocities at the previous iteration and time step should be stored in order to get solutions which are independent of underrelaxation factor and time. (3) Usually the mass fluxes (density*velocity*area) are stored at the cellfaces instead of the velocities. The cell face mass fluxes are used not only for the source term of pressure correction equation, but also for treating the convection terms of transport equations such as the momentum, energy and various turbulent quantities.
I hope this helps, Halim Choi 

September 20, 2004, 02:08 
Re: momentum interpolation for collocated grid

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Thank you for your reply.
As you stated, we need the velocity of last iteration and last time step to calculate the new cell face velocity. this need to store 2 velocity component (u and v) at each face in nonorthogonal grid. if i store only the mass flux at each face, i can not calculate the velocity components which are needed for the momentum interpolation. how should i treat it? if we assume the same underrelaxation factor for u and v, it is possible to combine the equations and interpolate the contravariant velocity components directly on the cell face. so i will need only one velocity for each face. what is your suggestion for this trick? Thanks again for your attention. Hadian 

September 20, 2004, 05:14 
Re: momentum interpolation for collocated grid

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Dear Hadian;
(1) Store the Cartesian velocity components (u,v) at the cellcenters and store mass flux only at the cellfaces. The velocity components which are needed for the momentum interpolation to get the cellface mass flux (in nonorthogonal grid the cellface mass flux is density*contravariant velocity component) are the cellcentered Cartesian velocity components and the mass flux at the previous time step and iteration. Combining the expression for the cellface velocities from the momentum interpolation, you can calculate the contravariant velocities at the cellfaces. Instead of storing the mass flux at the cellfaces, you can store both Cartesian velocity components at the cellfaces. In this case, you need one more storage and a routine to calculate the cellface mass flux which are needed to treat the convection terms in the transport equation. (2) Yes. You got it. It is generally assumed that the underrelaxation factors for both cellcentered velocity components are same in the momentum interpolation method. If you do not like this, store both Cartesian velocity components at the cellfaces. Then, you do not have any problems, but it is messy and need more computer storage and programing. Could you please read my paper "Use of the momentum interpolation method for numerical solution of incompressible flows in complex geometries : choosing cell face velocities" Numerical Heat Transfer, Part B, vol.23, pp.2141, 1993. Good luck Halim Choi 

December 25, 2009, 08:25 
bc for colocated grid

#5 
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jk
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Dear Halim Choi,
In the case of colocated grid, can you please tell me the boundary condition (right side, consider a one dimensional case) at the right side. Basically i am using cell centered CV. Assume I have 7 points (cell centered). I have 1 and 7 as my ghost point. i have on the west face of point 2 my velocity inlet boundary condition. At 7 i have my pressure boundary condition. I dont know how to give the bc at the exit. I tried with some extrapolations at the exit but nothing worked out well. In the momentum equation, for finding the convective quantity (F) whether i have to use average nodal velocity or directly the face velocity. I am interpolating the ap(i) value at the faces. Please help me in this regard. thanks jyothish 

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