[karahan]

Hello Everybody,

I am reading the Phd-Thesis of Jasak about the implementation of High resolution schemes for arbitrarily unstructured mesh.

There is one thing I don't understand. For the calculation of normalized variable there is no need for the far upwind value Phi_u. Furthermore it says there is also no need for opposite upstream face f^-. If this opposite upstream face is not used, how is it possible to calculate the divergence (DeltaPhi)_C using the Theorem of Gauss?

How is (DeltaPhi)_C is computed?

The related article is here: http://citeseerx.ist.psu.edu/viewdoc...=rep1&type=pdf
See page 438.

Thanks a lot

Karahan

[mprinkey]

Hrv is often around these forums, so he may give you a more definitive answer, but TVD-type schemes on unstructured meshes almost all work with the values of the phi and the (limited) values of grad(phi) = gradPhi...a vector field computed from Gauss theorem or Least Squares. Simple second-order upwinding can be done via phi_f = phi_upwind + (r_upwind - r_face).gradPhiLimited_upwind.

Here is how Fluent does it. I believe OpenFOAM (and Star and almost every other unstructured FV code) does it the same way.

https://www.sharcnet.ca/Software/Flu...ug/node994.htm

The key difference here between structured algorithms (like QUICK) that use East-of-East and West-of-West cells in the stencil...instead, it uses only the phi and gradPhi(limited) values in the cells on either side of the shared face. That has hugely important ramifications for mesh referencing, parallel layer updated, etc.

This paper has an exhaustive explanation of unstructured TVD treatments, discussing the raw gradient assembly, limiting approaches, and the final face interpolation.

https://www.nas.nasa.gov/assets/pdf/...nas-05-007.pdf

[arjun]

https://feaweb.aub.edu.lb/research/c...ethodology.pdf

[karahan]

Hey thanks a lot. All of those links were very helpful. The calculation of divergence at the cell center makes reference to the far upwind value or the opposite upwind face value in those articles.

My question regarding the elimination of far upwind center and face values in Jasak's paper still persists.

[mprinkey]

Perhaps I am misunderstanding your question. The "far upwind" cell *is* accessed along with all of the other neighbor cells when assembling grad(phi) in each cell. That is step 1...build all of the gradients by using Gauss theorem or Least Squares. Step 2 is generally limiting those gradients in some way...see the Berger paper I referenced earlier. This occurs cell-by-cell. And then Step 3 is the face interpolation/face flux calculation. That step does NOT access the far upwind cell data, but only the upwind cell phi value and the upwind cell (limited) gradPhi value. That is the point...it eliminates the need to identify a single far upwind cell for a given face/flux interpolation and instead references only the derived gradient data based on all of the upwind cell's neighbors.