Hi, What is the so-called "mesh independent issue"? It sounds like that a CFD solution shouldn't depend on grids. So, you compute solutions for some different grids, and make sure the solutions look the same. Is this what you do?

And, What exactly the purpose of it? Is it to check the numerical schemes? So, if the solution depends on grids, you throw that schme away, and try another, looking for a mesh independent scheme? Or something else, like mesh independent boundary conditions or mesh independent turbulence modelling? Or all of them?

Any comments are welcome! Thank you for your attention.

Usually when you refer to mesh-independent you do not refer to the scheme itself, but rather to the application you make of it. I mean, suppose you're studing the flux around an aerodynamic profile with a certain numerical scheme then you use as first step a certain number N of nodes ... now, can you say if your computation is good enough ? Of course not. The correct procedure is solving another time (at least) your problem with twice (it depends) the nodes you've just used and evaluate the percentual error ... when you reach a low value (depending from a lot of factors) you can say you've reached a mesh independent (at least within a certain approximation) solution of your problem : increasing the number of nodes won't give any improvement on the accuracy of your calculations. This procedure, which I've tried to explain just in few words (and NOT at all exhaustive), is called VALIDATION and currently is the main issue for a CFD calculation. I may advice you a site where you'll get some infos about the validation and verification process (although is for NPARC code) : http://www.grc.nasa.gov/www/wind/valid/validation.html Bye.

When reading a book you tend to hold the book just close enough so that you can make out all of the words. If you hold the book too far away then all the letters are blurred and you can't read the text and understand what the book is about.

This analogy holds (somewhat)true for CFD and grid. Sufficient grid is required so that physics of the model can be resolved so that you may understand what is going on.

If the book is held too close then you can make out each and every serif on each and every letter. This is OK but you don't get any more understanding out of the book just because you can see the text VERY clearly.

Sid,

The purpose of finding the mesh independent solution is to minimize the truncation error one obtains due to approximating a differential equation with a difference equation. One does this by refining the grid in a particular location of interest until the solution does not change anymore. There are numerous books on numerical analysis that explain this in detail.

Chuck

Hello, In summary, make a fine mesh for your problem, solve it and get a solution. make another finer mesh. If the two solutions are identical, then you have an independent mesh solution. That is it.

Thank you for your comments! So, it looks like that mesh independent solutions are those cannot be improved "significantly" by further grid refinement.

In a sense, it means that you're looking for the coarsest possible mesh for the accuracy required for your application.

Yet in other words, it is to check if the solution is affected by the choice of the mesh. That is, if you found something strange (i.e. solutions that do not make sense) on the grid of mesh independence, it would be due to something other than the mesh, scheme or BCs etc.

Anyway, I think I begin to see what it is. Thank you. Any further comments, if any, are of course appreciated!

A very good paper on this issue is

Quantification of uncertainty in CFD, P. J. Roache, Annual Review of Fluid Dynamics, Vol. 29, pp. 123-160, 1997.

I feel none of the responses have addressed all the issues regarding the utility and necessity of grid-refinement studies very well. This paper is quite comprehensive and available in the internet.

regards, chidu...

A word of caution on mesh independence :

Mesh independent solutions are not always attainable for unsteady flows. Large eddy simulations of turbulence are good examples.

In large eddy simulations (LES), refinement would continue to produce a better solution and in the limit of very fine grids, a direct numerical simulation solution is reached (which may or may not match the experiments depending on the characteristics of the scheme like artificial dissipation and/or dispersion).

The key issue that determines whether mesh-indepedence applies is whether the flow has multiscale behavior like turbulence. Turbulent flows have eddies of many sizes. The smallest eddies are approximately the size of the so called the Kolmogorov lengths scale. If you use a grid much larger than the Kol. length scale (as one does in a LES), then you are not accounting for some of the small scale turbulence. The smallest eddy resolved depends on your grid size and as you keep refining the grid, more and more small scale turbulence effects are included in the computation. The grid independece is reached only if the mesh spacing is smaller than the Kol. scale. In this limit the large eddy simulation becomes a direct numerical simulation.

It is safe to say that mesh independence does not apply to simulations that use subgrid/sub-filter models.

Thank you for the comments! I got that paper by Roache. It is a 38-page article concerning verification & validation including discussion on turbulence models too. It really is a good article!

Thank you, everybody!