CFD Online Discussion Forums - Mesh independence studies

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Mesh independence studies

Hey

I am trying to understand the mesh convergence while using interFOAM. So what is have seen are two methods where tthe interface shape is compared after a certain time and the other option is to use Grid Convergence Index (GCI) like described in
http://www.grc.nasa.gov/WWW/wind/val.../spatconv.html

I am trying to figure out GCI method I got confused after looking at how it is used in literature. So I was hoping someone can help it me out :)
What I understood about GCI is that:

Order of grid convergence

$p = \text{ln}\frac{f_3 - f_2}{f_2 - f_1}/Big{\text{ln}r}$

Mesh independence studies

Sorry.. I couldn't edit the post earlier due to some network problem.

So what I understood about GCI is that

First we calcualte the order of convergence as

[math] p = \frac{ln \frac{f_3 - f_2}{f_2 - f_1} }{ln r} [math]

where r is the ratio of the mesh size between the meshes used (greater than 1). This is dependent on the solution methods and is usually lower than 2 because of non-linearities in the governing equations.

Then we calculate relative error as

[math] \epsilon = \frac{f_2 - f_1}{f_1} [math]

Then we calculate GCI as

[math] GCI = F\frac{\epsilon}{r^p - 1} [math]
This GCI is for 'fine' mesh. And F is factor of safety.

Then to show that mesh has actually converged, the GCIs should satisfy :

[math] GCI_{23} = r^p GCI_{12}

Please let me know if I misunderstood something so far :)

Now my questions are:

When I was going through literature, most of the papers usually stop after calcualting GCIs for different meshes and say that the errors are small so it is converged. As GCI is proportional to relative error (whose magnitude of the variable considered for the mesh convergency study). Like relative error in pressure drop would be very small as the denominator is in the order of 1atm.
Then in the work by Roache, GCI is also calcualted for a coarse grids too differently from the previous expression. Typically if we use 3 grids: coarse, medium and fine. When is the GCI expression for coarse grids be used ?

Thank you for your time! :)

Yes, GCI is a relative error, you can interpret it like an error bar.

Say you have 3 grids, coarse, medium, fine. In Roache's proposed method, your working grid is the coarse grid. The purpose of running a medium and fine grid is purely for calculating the GCI and you never see these grids again. Now this sounds silly to a lot of people. Because why would I not utilize the superior results of the finest available grid? Why take results from the coarsest and not finest grid? More technically, it has to do with the way the Richardson extrapolation. You don't need to extrapolate per se when you are using a coarse grid (because you have a medium grid). But if you are using a fine grid, then you have to extrapolate because you don't have a finer grid.

Hence, you will find two different methods for GCI calculation. One with the coarse grid as a reference and another with the fine grid as a reference. You can pick the approach that you prefer, just use the right formulas.

Quote:

Originally Posted by LuckyTran (Post 750670)

Thank you for the quick reply!
Can you recommend a place I can learn more about for calculating GCI and mesh sensitivity tests in literature apart from Roache's work?

Thanks again in advance:)

Well, Patrick Roache is the one that came up with the GCI method. I can't think of a better place to learn specifically about this technique. Roache also write several (pretty thick) books on this topic.

A brief description is also in the Peric & Ferziger textbook

Thank you for the suggestions! :)

Hey again

As you suggested I was going through the work by Roache and I came across : Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications, Journal of Fluids engineering (ASME). They have described each steps.

I have few additional questions:

In this paper, the authors recommend calcualting p using a iterative method. But quite a lot of studies in literature calculate p as ln(eps31/eps21)/ln(r21) like done in https://www.grc.nasa.gov/WWW/wind/va.../spatconv.html, why is the iterative approach not used by everyone?
Also one stupid question what is this sgn function used in GCI? I looked around and I understood it as the sign function (https://en.wikipedia.org/wiki/Sign_function). Then it is the same as writing s = eps32/eps21. Did i misunderstand this?
And one last thing, what value of GCI_fine calculated is 'mesh converged' as GCI_fine provides error graph on the finest gird used in the simulation ?

Sorry to post so many questions and thank you for taking the time to answer them in advance :)