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March 18, 2013, 13:40 
Understanding usual mesh constraints

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Malik
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I am currently dealing with meshing issues on a cylinder case. In fact, I saw that in my mesh I have to maintain a cell aspect ratio close to 1, and to make sure that the size a cell over the size of another cell in the vicinity of the first one, should not exceed 1.2. However, I do not understand why we have to respect these ratios, numerically I mean. I need to know the quantification of the numerical error induced by a poor mesh, and this is why I need to know exactly from where does an error come from. If you have any title of books, article or website pages I could read for that, I would be very delighted. Thank you for your help 

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March 19, 2013, 07:27 

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andy
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Quote:
A strong requirement for a constant spacing in all directions can follow from some triangular/tetrahedral based discretizations although not all. Grid generators can be made to automatically enforce it making it a minor practical problem although one can end up using a lot of elements. If the number of elements is becoming a problem then using some hexahedral elements aligned with the flow gradients should bring the element count down a lot. 

March 19, 2013, 08:41 

#3 
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Malik
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Thanks for replying.
Actually I am using Open FOAM for a cylinder case in 2D. i made the mesh without using a mesh generator. I use the blockMesh utility. The cells are hexahedron. I don't see why the way derivation are discretized has an influence on the solution. What matters is the density of points where we make calculation and not the homogeneity of the space between those points, isn't it ? I know I am wrong because I saw with some experiences that the density of mesh is not the only thing important (skewness, aspect ratio and sudden changes in size of cells are very important). But I would like to understand clearly what is going on numerically. Thank you in advance ! 

March 19, 2013, 10:08 

#4 
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Philipp
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Hi Malaboss,
This is what you need http://powerlab.fsb.hr/ped/kturbo/Op...JureticPhD.pdf Also this one is nice: http://www.sciencedirect.com/science...45793012000667 Some of the questions you have are pretty basic, look in some good cfd book: http://www.cfdonline.com/Books/show_book.php?book_id=4 or http://www.cfdonline.com/Books/show...php?book_id=37 Cheers!
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March 19, 2013, 10:50 

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Hi, the effect of size ratio between adjacent cells on the accuracy of the numerical method is clearly seen when performing a finite difference discretization on a nonuniform grid.
You can show with pen and paper that if the cell size ratio is large, the order of the discretization will be lower than that with a uniform grid. The same principle holds for different discretization approaches (i.e. finite volumes). You can check the references you were given above. I can also suggest having a look at C. Hirsch "Numerical computation of internal and external flows". Cheers, Michujo. 

March 27, 2013, 07:12 

#6 
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Malik
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Hi everyone,
I come back to you to give you a feedback about what I just read. The thesis of Franjo Juretic gives clear understanding of the numerical issues. It explains well the effect of skewness and non orhtogonality. There isn't much detail about aspect ratio, but we can deduce it from the expression of the truncation errors. In the paper "High aspect ratio grid effects on the accuracy of Navier–Stokes solutions on unstructured meshes" I did not find much numerical explanations about the effect of aspect ratio. It rather shows numerical experiments of different aspect ratios. It still is very interesting, but less for my problem. Just to be sure, here is what I understood from the links about cell aspect ratio For a first order accuracy, you have something like TruncationError ~ delta(x)*dU/dx + delta(y)*dU/dy + delta(z)*dU/dz For a second order accuracy you have : TruncationError ~ delta(x)²*d²U/dx² + delta(y)²*d²U/dy² + delta(z)²*d²U/dz² With delta(x) delta(y) delta(z), the size of the cell in each direction. Then, to know the what extent I can use cells with a high aspect ratio, I have to know if I have a first or second order accuracy and then compare the first or second order derivative terms of U in each direction. Thanks for your answers, and thank you again for the links ! 

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