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Chuck Leakeas July 13, 2000 14:29

Grid Independent Solution
 
Hello all! I was just wondering about finding the grid independent solution for a chemically reacting flow. I have been performing many mesh refinements near a reacting surface and so far the deposition rate there keeps changing a great deal. Is it possible for there not to be a grid independent solution for certain problems? If so, how would one determine when a solution is 'good enough'? Thanks, Chuck

John C. Chien July 14, 2000 00:39

Re: Grid Independent Solution
 
(1). If you are tired of cfd, you can try the control volume approach. (2). Just create one control volume, then model the flow and reaction, then you should have a simple mesh independent solution. (3). It is easier to understand the need of grid independent solution based on the finite-difference approach. This is because as the mesh size approach zero, the algebraic equations will approximate the real governing equations. (4). So, if the mesh is of finite size, it is likely that you are not solving the real governing equations, thus, not solving the real problem. (5). The number of mesh points needed depend on the solution profile distribution, the algorithm used. So, lower-order algorithms most of the time require more mesh points. Higher Reynolds number flows, wall boundary layer flows, flows with chemical reactions, requires both the mesh points and local mesh refinement.

Chuck Leakeas July 14, 2000 10:18

Re: Grid Independent Solution
 
John,

Does this mean a grid independent solution always exists no matter what method one uses? I am already using the FV method and I move the first grid point next to the wafer surface closer by one-half the distance for each new grid refinement. Is this a reasonable way to find the grid independent solution?

Thanks, Chuck

John C. Chien July 14, 2000 11:16

Re: Grid Independent Solution
 
(1). Yes, unless the model or the boundary conditions are explicit function of the mesh spacing (such as the wall function), you should be able to get the grid independent solution. (2). Run a calculation, plot the result, identify the high gradient areas, put more mesh points in those high gradient areas, re-run the case, check the result and compare the difference. When you no longer see the difference, you can stop. (that depends on the requirement, the difference between the acceptable solution and the real solution. It is one way to guarantee the solution is repeatable , within certain limit.)

Duane Baker July 15, 2000 03:27

Re: Grid Independent Solution
 
Hi,

what are you using for the advection term discretization?

If you are using a really bad scheme such as first order UDS, the error reduction with grid refinement is so slow for some problems that it does seem to never get to a grid-independant solution. With Richardson extrapolatinon you can estimate the discretization error.

Say for example, a 3D problem with UDS and reasonable but coarse grid of 50 cells in each direction gives a discretization error of 20%. Then doubling the grid will take the following sequence (if we are in the assymptotic convergence region):

Grid Cells Total Cells Error

1 50x50x50 125000 20%

2 100x100x100 1000000 10%

3 200x200x200 8000000 5%

etc.

So, with most people's resources you will never see a grid independant solution before you are tapped out. As Ferziger and Peric state: UDS is inaccurate and should not be used.

May I suggest that you read up on this issue in Ferziger and Peric's text: Computational Methods for Fluid Dynamcis.

Regards..........Duane


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