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Z. Zeng May 16, 2000 21:07

Would you please suggest me some references on the finite volume methods(FVM) in body fitting coordinate(BFC). (method and application)

Thanks in advance.


CONSULTANT May 18, 2000 00:24

Re: FVM and BFC
I can suggest the following JOURNALS/NOTES as general reference material for both theory and application. A more specific statement of your requirements may get a better response !

1. Journal of Computational Physics.

2. VKI Lecture Notes (past 5-6 years).

3. Computers and Fluids.

4. AIAA Journal.

5. International Journal of Numerical Methods in Fluids.

6. International Journal of Numerical Methods in Engineering.

You can also refer to books by the following authors:

1. HIRSCH (vol I & II)

2. Dale Anderson, Tannehill, Pletcher.

3. C.A.J. Fletcher.

Hope this helps,


Zeng May 18, 2000 22:49

Re: FVM and BFC
Thanks for your suggestions. In fact, I have experience on the FVM in Cartesian & cylindrical coordinate systems. Now, I would like to get same basic ideas on the FVM in BFC in following aspect:

(1) Comparing with FVM in Cartesian & cylindrical coordinate system, what is the difference for the expressions of governing equations in BFC?

(2) What is the main difference in programming for FVM between BFC and Cartesian & cylindrical coordinate systems? ( I am considering to modify a code of FVM in Cartesian & cylindrical coordinate systems to BFC).

Thanks for your response in advance.

CONSULTANT May 19, 2000 01:27

Re: FVM and BFC

The main difference is in the definition of flux normal to an interface.

In the case of 2-D Cartesian grid, the grid normals to interfaces perpandicular to the co-ordinate axes x & y coincide with x & y respectively.

But, in the case of body fitted grid, this may not be true. The grid normals may not coincide with the local co-ordinate directions say, i & j, in the x-y plane.

So, you have to transform the equations to local co-ordinates.

Consider a 2-D problem in which the two components of fluxes are f and g, for example.

Then the flux normal to an interface (F) can be obtained by taking the scalar product with the local grid notmal vector n {= (dy/ds) i - (dx/ds) j }.


F = f (dy/ds) - g (dx/ds) ..................(1)

For example, in the case of 2-D Cartesian grids dx = dy = ds where ds = SQRT(dx*dx+dy*dy). The above expression can be simplified for an interface that is perpandicular to x-axis (dx=0 & ds=dy) as

F = f

and for an interface that is perpandicular to y-axis (dy=0 & ds=dx)

F = g.

But, you have to retain the form [ Eqn. (1) ] for the body fitted grid because, dx & dy may be non-zero and non-uniform in general.

As for the programming is concerned, it should be clear from the above discussions that both the components (f & g) contribute to the computation of fluxes for ALL the interfaces and that involve normal and tangent vectors.

It is a better idea to write the expression for flux as in Eqn. (1) and compute dx, dy and ds for all interfaces and store them in a one-dimensional array because it can be used for both Cartesian as well as body fitted grid methods.

There are many papers in the references I had posted earlier that give the required forms of equations. Otherwise, you can derive them yourself.

Good Luck,


John Law May 19, 2000 08:20

Re: FVM and BFC
You may find the following paper useful since it does not use equation transfer (and this methodology is actually used in many leading FV based commercial CFD codes):

I. Demirdzic and M Peric 'Finite volume method for prediction of fluid flow in arbitrary shaped domains with moving boundary', Int. J. for numerical methods in fluids, Vol. 10, pp771-790, 1990.

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