Hi all!!

I am trying to solve the NS equations, and I am trying to implement the method described by Hino in his paper "An unstructured grid method for incompressible viscous flowa with free surface", and I would like to know why the convection terms are evaluated by the second order upwind scheme and the viscous terms are by the central differencing? could I use central scheme in both??

Thanks in advance

Ruben

Hi ruben,

I am pavitran, I am still naive to this forum and also in CFD. So, please dont mind if my answer is not the one your expecting.

I think i know the answer for your query.

In general we have 3 requirements for a differencing scheme

1.Conservativeness 2.Boundedness 3.Transportiveness

In order to satisfy the 3rd condition that is transportiveness 'second order upwind scheme is used'

if you are already aware of what i mentioned above then inform me.

or else i can tell you more about those 3 requirements.

A good reference:

'An introduction to computational fluid dynamics - The finite volume method'; H.K. Versteeg & W. Malalasekera

He provides a reasonable explanation.

diaw...

Thanks Pavitran

I don´t have enough information about this topic, I would like to know more about these requirements.

Thanks diaw

The problem is that There aren´t books about CFD in the libraries of my University, and I can´t buy all the books that I should read, but thanks for your help.

Hi Rueben,

If I could only buy one book, then it would be Computational Methods for Fluid Dynamics by Ferziger and Peric, now in the 3rd edition.

It covers everything from basics right up to the complex topics such as Multigrid, turbulence , LES, DES, etc, free surface problems, and current research topics. Also included are some simple codes in Fortran which are readable and understandable such that you can fully learn the methods.

Regards,

Bak_flow

Maybe it is satisfying enough for you to understand it in a very simplified way (for more rigorous understanding consult a textbook):

The choice of a scheme is related to the physics you would like to model (or the properties of the equations you would like to solve). Now, if you look at convective terms and viscous terms, they both describe two very different types of physical phenomena. Convection (as the word suggests) essentially describes the transport of some quantity (for example mass, momentum, energy) associated with the flow velocity. This transport has a distinct direction (depending on flow velocity) and upwinding schemes are designed to take directions and transport speed into account. (There is also transport of information in all directions due to wave propagation). On the other hand, look at viscous diffusion, and you basically deal with another type of transport which does not have a clear direction, as quantities diffuse in all possible directions. For this type of behavior, a central difference scheme, which does not account for directions, works well.

This is a simplified physical argument just to give you some basic idea. There are numerical, or mathematical, arguments as well, but they are actually related to physics. For example, if you were to try a pure (and explicit) central differencing scheme on the convective terms, you may not be able to make it work. It would be unstable because you are disregarding the physical nature of the problem (or the mathematical nature of the flow equations).

In general we have 3 requirements for a differencing scheme.

1.CONSERVATIVENESS:

The flux out of a cell should be the same as that into the neighbouring cell(ex: flux out of cell 'i' through interface 'east' should be same as that into cell 'i+1' through its face 'west'). This is automatically satisfied for finite volume methods

2. BOUNDEDNESS:

For the east face, for ex: this means that some property (either velocity or temperature) on east face must not be smaller or larger than cell values used to compute the property on the face. if all coefficients are positive this is satisfied.

3.TRANSPORTIVENESS:

The scheme should reflect the way in which the information is transported. The way information is transported is dependant on the ratio

convection/diffusion= peclet number

If the peclet number is small the transport is dominated by diffusion which transports information equally in all directions. If on the other hand peclet number is large, information is transported in the direction of U(velocity).

If peclet number is greater than 2 convection dominates and less than 2 diffusion dominates

So, if we use central difference for convection terms We get negative coefficients and this gives oscillations.

Then no convergent solution can be obtained at all, or if a solution is obtained it contains oscillation?

Oscillation means the property(either velocity or temperature) which we are computing at the centre of the cell may take any value , I mean it will violate 2nd condition(bounded ness)

So, if we use central difference scheme for convection terms the final result will not be good at all. There may be over or under predictions of properties. So we need to use upwind biased discretization schemes.

The central difference scheme works very good for diffusion problems

For more details u can go to this link http://www.tfd.chalmers.se/gr-kurs/MTF071/

You can go to lecture notes in that link and can get more information about different type of schemes.

Out of interest, where does the 'magic' Peclet=2 stability criterion stem from?

This instability point is evident in both FEM approaches with 2-node elements & FVM central discretisation.

diaw...

hi diaw,

In my previous post actually i mentioned about peclet number in a wrong way .

central differencing scheme gives negitive coeffcients when peclet number > 2.

>>In my previous post actually i mentioned about peclet number in a wrong way .

>>central differencing scheme gives negitive coeffcients when peclet number > 2.

That is true.. but, why do these negative coefficients - of the matrix diagonal - cause instability?

diaw...

hi diaw,

If we get negative coefficients, the 2nd requirement that is boundedness is violated.

boundedness is very important for stability reasons.

>>If we get negative coefficients, the 2nd requirement that is boundedness is violated.

>>boundedness is very important for stability reasons.

This is very true & we are on the correct path, but, why does the solution become unbounded if the dominant coefficients of the system matrix become negative?

Actually, the exact same instability problem occurs in the Finite Element Method with 2 node linear elements eg. for a 1D problem.

diaw...

Hi,

There was a very nice topic posted by Michail, with lots of links to CFD courses: http://www.cfd-online.com/Forum/main...cgi?read=37970

As far as I recall, this has been very clearly explained in Patankar's book.

>As far as I recall, this has been very clearly explained in Patankar's book.

Do you have the page numbers in Patankar? I am currently reading through it...

Thanks, diaw...