CFD Online Discussion Forums - Non-orthogonality for FV method

It depends on:

(1) what coordinates system you use to express the velocity/tensor components of your solution variables. Cartesian components are common but grid orientated components can have advantages.

(2) what coordinate system you use to express the spatial derivatives. For a structured grid this would usually be a coordinate system based on the grid because it is easy to evaluate.

(3) which independent coordinate directions you choose to decompose the vector/tensor components of the solution variables. This usually follows from (1) but with the occasional exception.

The most common choice is: (1) Cartesian, (2) grid orientated, (3) Cartesian. This is also perhaps the easiest since one can derive all the terms in the Navier-Stokes equations without needing to get to grips with coordinate systems and tensor analysis. All you need is the chain rule of differentation to expand the Cartesian spatial derivatives in terms of the grid coordinates and Cramers rule (or equivalent) to replace terms like di/dx with a combination of terms like dx/di which can be evaluated on the grid.

If I recall correctly, the Introduction to CFD VKI course notes of Anderson have a worked tutorial of the approach. This course ran for many years and the notes are widespread but I do not if they are on the internet.

Tom Pulliam's CFD course notes have the Jacobian flux matrices in the Appendices which you should find useful. These are on the internet somewhere. His 2D CFD code is freely available. I think his 3D code is only freely available to Americans but I could be wrong. Whatever, I am sure it is on his home page or somebodies homepage somewhere. He also wrote quite a good CFD book along with a couple of others which is available on the internet somewhere.

Can anybody firm up my wobbly references?