|
[Sponsors] |
March 25, 2015, 16:44 |
What is even-odd decoupling ?
|
#1 |
Member
Tommy Chen
Join Date: Mar 2011
Location: University of Michigan
Posts: 96
Rep Power: 15 |
Hi everyone
I am a CFD beginner currently reading as many books as I can. A lot of books say that artificial dissipation has to be added to central difference scheme in order to avoid the odd-even decoupling. could anyone explain to me a little bit what exactly the odd-even decoupling is ? For instance , in the finite volume approach, if I use the JST scheme for inviscid flux without adding the artificial dissipation term, what would happen? what is the relationship between this consequence with the odd-even decoupling ? Thanks ! |
|
March 25, 2015, 16:55 |
|
#2 | |
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,773
Rep Power: 71 |
Quote:
You can find this issue treated in many textbooks, odd-even decoupling can be seen also for the incompressible flows formulation. Briefly, here an example: consider the case in which you want to compute on 1D grid a term like d/dx (df/dx). Using central discretization you have (df/dx|i+1 - df/dx|i-1)/2h = (f(i+2) - 2f(i) + f(i-2))/4h^2 Such discretization, compared to the compact counterpart (f(i+1) - 2f(i) + f(i-1))/h^2 show lack of data comunication between adjacent nodes. This aspect can produce spurious modes. Details are explained for example in the Peric & Ferziger textbooks |
||
March 25, 2015, 17:03 |
|
#3 | |
Member
Tommy Chen
Join Date: Mar 2011
Location: University of Michigan
Posts: 96
Rep Power: 15 |
Quote:
However , how to interpret or how to understand the " lack of communication between adjacent nodes " ? Isn't the i+1 and i-1 cell the adjacent cell of the i cell ? |
||
March 25, 2015, 17:09 |
|
#4 |
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,773
Rep Power: 71 |
when you write the equation at node i+1 you get
(f(i+3) - 2f(i+1) + f(i-1))/4h^2 and when you write at node i-1 you get (f(i+1) - 2f(i-1) + f(i-3))/4h^2 Therefore, you formally have 2 indipendent grids over which solutions do not communicate each other. |
|
March 25, 2015, 21:44 |
|
#5 |
Member
Tommy Chen
Join Date: Mar 2011
Location: University of Michigan
Posts: 96
Rep Power: 15 |
But does the concept ‘odd-even decoupling’ exist when using unstructured mesh ?
|
|
March 26, 2015, 11:41 |
|
#6 | |
Senior Member
Michael Prinkey
Join Date: Mar 2009
Location: Pittsburgh PA
Posts: 363
Rep Power: 25 |
Quote:
One variation does happen depending on grid tolopology...where a neighbor of neighbor actually maps back to a first-degree neighbor. But that is not guaranteed and it is, in practice, not enough connectivity to avoid even-odd decoupling |
||
March 26, 2015, 11:47 |
|
#7 | |
Member
Tommy Chen
Join Date: Mar 2011
Location: University of Michigan
Posts: 96
Rep Power: 15 |
Quote:
In finite volume approach, the residual is basically the flux across the interface between to cells, taking unstructured mesh for examples, it is really hard for me to build an analogy from the structured mesh finite referencing to unstructured mesh finite volume on this concept. |
||
March 26, 2015, 12:37 |
|
#8 |
Senior Member
Michael Prinkey
Join Date: Mar 2009
Location: Pittsburgh PA
Posts: 363
Rep Power: 25 |
FV works with the volume integral of each cell. Gradient, Laplacian, and Div terms are converted to surface integrals by Green's/Stoke's/Divergence theorems. But the calculations end up being the same, with Area/Volume generally replacing dx. If you use the surface integrals to compute the vector field gradPhi = grad(phi) in all of the cells. And then use the surface integrals AGAIN to compute the div(gradPhi) to find the Laplacian, you will get the same neighbor-of-neighbor (4*h^2) form.
And, in fact, that is exactly how FV schemes compute the skew portion of face fluxes for diffusion terms. The "straight line" gradient at the face can be built using the cell and its neighbor, but that will not represent the normal flux on meshes were the line connecting the neighboring cell centroids is not parallel to the shared face normal. So, the resulting face gradient ends up being computed as a mixture of the straight line gradient (corrected for direction) and the average of the cell gradient fields in the neighboring cells. There is a good overview of this in the Ansys Fluent documentation if you have access to it. |
|
March 27, 2015, 09:03 |
|
#9 |
Member
shahrooz
Join Date: Aug 2011
Posts: 41
Rep Power: 14 |
When you use simple central difference scheme for a grid like below :
[1]----[2]----[3]----[4]----[5]----[6] all equations on grid points with odd number involve data from grid points from even numbers and vice versa. as you can see it's like having 2 separate grids for a single equation. the solution on odd grid points becomes idependent / decoupled from even grid points and vice versa. |
|
Thread Tools | Search this Thread |
Display Modes | |
|
|
Similar Threads | ||||
Thread | Thread Starter | Forum | Replies | Last Post |
PimpleFoam odd behavior, any suggestions? | NJG | OpenFOAM Running, Solving & CFD | 1 | March 20, 2013 07:52 |
flow out of an odd shaped tank | halfrhovsquared | FLOW-3D | 8 | March 13, 2012 14:48 |
extrudeMesh - odd behaviour | grjmell | OpenFOAM | 0 | September 20, 2011 08:41 |
ODD-EVEN DECOUPLING | Al | Main CFD Forum | 1 | October 25, 2007 10:58 |
Odd Even Decoupling Validation | Sachin | Main CFD Forum | 1 | October 8, 2005 06:58 |