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Michail June 24, 2005 10:03

Please help.

I'm writing code for FVM on a non-orthogonal grids.

Main difficulty - discretization of the convective, diffusion and source terms of the transport equation on a non-orthogonal mesh.

I found some description in Joe F. Thomson Grid Book, in some theses and articles, but it's not enough.

Please help. I've spent three whole days and sleepless nights in Google, AltaVista, Yandex and without result.

I'm desperate. I promise, that in my CFD thesis I'll describe this question in smallest details and will place my thesis in Internet, in order that fellows like me could work without such difficulties.


Jonno June 24, 2005 10:27

Re: FINITE VOLUME METHODS for NON-Orthogonal Grids
My dear friend,

There is plenty of literature out ther on this subject. Start with the book of Peric and Muzaferija. Have look through the thesis of Hrvoje Jasak (the link is somehwere in these threads) Just search for Jasak. He and mr Peric are the GURUS of this field.

Let me just add that you are now involving yourself with the cutting edge of CFD! This is the way forward for CFD. Let there be no doubt about it!!!

Happy modelling!

andy June 24, 2005 11:01

Re: FINITE VOLUME METHODS for NON-Orthogonal Grids
I have answered this in your earlier thread.

If you really have spent 3 days and got nowhere I would strongly suggest you find someway to get some supervision. It would take any supervisor familiar with numerical methods about a minute to point you in the right direction.

Failing that, I would recommend reading cover-to-cover a reasonable CFD book before attempting to write any code. There are some on the internet.

I would also strongly recommend downloading one of the available CFD codes and reading it "cover-to-cover". Run the code to check it does what your expect. Then start your coding by modifying it for your purposes. No rational supervisor would ask one of their new student to write a code from scratch. (And this is a supervised student!) They would give them a code to learn from. Later, if necessary, a fresh code might be written but it is rarely necessary.

Michail June 24, 2005 16:05

Re: FINITE VOLUME METHODS for NON-Orthogonal Grids
Dear Jonno, thank You,

I send an E-mail to Mr. Hrvoje Jasak, asking him to allow me to copy of his Ph.D. Thesis.

If he will be kind to me, I will get it.

I am too shy to disturb people, all I can do - to search in Internet and send question at CFD-online.

I haven't connections with any of powerful people in CFD and still work by myself.

Do You know more issues about non-orthogonality which can be downloaded for free? I found a lot, but it's not that suit me.

Below I listed manuals of CFD codes, where I could find description of the problem, but ....


1) CALC-PVM: A Parallel SIMPLEC Multiblock Solver for Turbulent Flow in Complex Domains by Hakan Nilsson and Lars Davidson

2) CALC-BFC: A Finite-Volume Employing Collocated Variable Arrangement and Cartesian Velocity Components for Computation of Fluid Flow and Heat Transfer in Complex Three-Dimensional Geometries by Lars Davidson and Bijan Farhanienh

3) ISAAC version 4.2. by Joseph H. Morrison

4) ISNaS - the Deft incompressible flow solver by Guus Segal, Marcel Zijlema, Ronald van Nooyen, Charles Moulinec, version 1.1

5) STREAM Version 4.5.2

6) caffa3d.MB A fully implicit, finite volume, flow solver for the incompressible 3D Navier-Stokes equations with generic scalar transport, in block structured, non-orthogonal, body fitted grids by Gabriel Usera

7) RELAX3D - program for solving 2- and 3-dimensional Laplace and Poisson equations

8) SPARC Structured PArallel Research Code by Franko Magagnato

9) Fluent descriptiom

10) PHOENICS description

11) MFIX by Madhava Syamlal

12) NaSt3DGP description

Also I can list links to courses of CFD which I found, but there is not description which I need.

Michail June 24, 2005 16:38

Re: FINITE VOLUME METHODS for NON-Orthogonal Grids
Andy thank You very much for Your advice

I also think that it would be better if I had supervisor.

There are only three men who helped me:

- E.M.Smirnov from St.-Petersburg Technical University, he wrote a recommendation letter to Mr. Gunnar Johansson,

- Jonas Larsson who encourage me to apply to Chalmers

- Gunnar Johansson, he send me an invitation to participate in International Masters Progarm in Turbulence at Chalmers.

They are too busy to help me in such questions.

I suppose it's difficult to find supervisor, who would agreed to spend his time for unknown and remote student, besides who doesn't pay money.

I hope that finally I will resolve all contradictions, and finally aplly for Master's degree somewhere either to Chalmers or to KTH.

Here I have a lot to do: basic knowledge of fluid and gas dynamics, fundamentals of CFD, Maple, MATLAB, TEX

It will take at least an year.

I can list links to a lot of CFD courses (course notes, handouts, lectures) wich I found. And I can say that it's more than sufficient.

Buying books is too expensive for me, and thanks to God there is an Internet. When I finish my thesis I'll write free CFD book which I place in Internet, in order that no one suffer like I

zxaar June 24, 2005 18:45

Re: FINITE VOLUME METHODS for NON-Orthogonal Grids
drop me an email, i will try to help you with this. (i am very busy now a days but i think in this regard i could be help)

Harish June 24, 2005 19:59

Re: FINITE VOLUME METHODS for NON-Orthogonal Grids
This paper might be useful.

Multigrid acceleration for recirculating laminar and turbulent flows computed with a non-orthogonal, collocated finite-volume scheme

Computer Methods in Applied Mechanics and Engineering (ISSN 0045-7825), vol. 118, no. 3-4, p. 351-371


Michail June 25, 2005 07:46

Re: FINITE VOLUME METHODS for NON-Orthogonal Grids
Andy many thanks, for that, I just discover Your responce. I looked through Thomas H. Pulliam's VKI lectures notes, but unfortunately it's not that I would like to see. I need implementation for Finite-Volume application.

Anyway great thanks for Your help and support.

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?

Michail June 25, 2005 07:52

Re: FINITE VOLUME METHODS for NON-Orthogonal Grids
I still complete fool in such questions. I collected a lot of literature, printed all Numerical Grid generation by Joe F. Thomson & Z.U.A.Warsi (it lies on my table in front of me), but I still can not understand, how to derivate equation for convective and diffusion fluxes through cell face for FV in curvilinear coordinates.

Michail June 25, 2005 08:11

Re: FINITE VOLUME METHODS for NON-Orthogonal Grids
Many thanks Harish, I found it

, but to have an article it's necessary to pay. I'm out of money. If Your have scanner, please send a copy of it in any graphical form .gif, .jpg or any other


Harish June 26, 2005 03:16

Re: FINITE VOLUME METHODS for NON-Orthogonal Grids
I got the paper from scholar search option in google and I do not have access to that paper either :(.


Harish June 27, 2005 01:08

Re: FINITE VOLUME METHODS for NON-Orthogonal Grids
Your email does not accept the file that i send because it seems that the size was huge. Gimme another account where i can send the file


Michail June 27, 2005 01:12

Re: FINITE VOLUME METHODS for NON-Orthogonal Grids
Thank You very much Here is second Email

Michail June 27, 2005 11:26

Re: FINITE VOLUME METHODS for NON-Orthogonal Grids

Thank You very much for Your help and support

But OE (in OutLook Express) removed access to the attachment

If it won't be very difficult to You, please, rename it to the .txt and send again.

Great Thanks Michail

harish June 27, 2005 12:32

Re: FINITE VOLUME METHODS for NON-Orthogonal Grids
go to and login with your id and password and download it.


Michail June 27, 2005 13:09

Re: FINITE VOLUME METHODS for NON-Orthogonal Grids
I try but I can't. After OutLook Express it dissapeared from Yahoo.

Ahmed June 30, 2005 02:35

Re: FINITE VOLUME METHODS for NON-Orthogonal Grids
The finite volume method is designed from the beginning to deal with volumes of any shape, the mesh does not have to be orthogoal, structured or any other limitation, all these limitations need clever coding and the data structure has to cope with this opennness. I am not sure if you are talking about the finite volume method or something else. For information about the finite volume method, read the following book: Computational Fluid Dynamics, Principles and Applications By J. Blazek Published by Elsevier.

Michail June 30, 2005 03:02

Re: FINITE VOLUME METHODS for NON-Orthogonal Grids
Dear Ahmed

Thank You for Your response. I indeed talk abot FVM.

Main difficulty:

how make discretization of general transport equation on a non-orthogonal grid, in general curvilinear boundary fitted coordinates.

how to derive equations for

1) surface area

2) convective fluxes

3) diffusion fluxes

There is some description in Joe Thomson's "Grid generation" book, but it rather complicated, and difficult for understanding. Besides I never use only one issue for getting information.

In vector notation its rather difficult to understand. I have some experience and know that all must be in conservative form.

There is some description in Lars Davidson's work

"Flux2D A Finite-Volume Computer program written in general Non-Orthogonal Coordinates for Calculationof two-dimensional flow"

BUT. There is only for 2D case, but I would like to have generalization for 3D case.

Also I don't like manner of description.

Will be very pleased if You can help

ANy free-downloadable references are very welcome

Ahmed June 30, 2005 15:01

Re: FINITE VOLUME METHODS for NON-Orthogonal Grids
I gave you the name of the only book I know of where all your questions are already answered. As you understand it, I cannot scan any copyrighted material, but the book is well worth the money you pay for it. Good luck with your project, it takes a year to develop that but you will end up with another FLUENT

Michail June 30, 2005 15:37

Re: FINITE VOLUME METHODS for NON-Orthogonal Grids
Dear Ahmed, thank You a lot for Your help.

What is the name of the book?

About project.

I am going to re-write codes by Peric and caffa3D and develop free code for educational/not very serious research purposes in FORTRAN.

It will be a good practice for me and good for CFD community.

I don't mind that it will take an year or may be even more, I have all my life in the future I have nothing to hurry to.

Till I have to be in Lithuania with my mom I will work.

Then I hope to join one of universities, preferably KTH or Chalmers.

So we will see what I will do.

Thank You for Your help. Waiting for the name of book You recommended

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