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December 13, 2021, 14:13 
Grid refinement or Higher order scheme

#1 
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Yusuf Elbadry
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Hello,
I hope you are all fine, I am solving the heaving airfoil problem while using the Eulerian grid (Level Set Method). The 2D Incompressible NS equations are solved using GLS FEM. I am simulating the flow at Re=800, Angle of Attack =20 Deg. I am using bilinear shape function for geometry and solution and I am using an explicit scheme. I have a problem with the accuracy of the result, I am comparing my result (Cl, CD) with the literature and the error is not small, and I was thinking to use a secondorder shape function for both the geometry and solution to get a better result and better shape for the airfoil instead of stairs. Would this be better or should I use a more refined grid with a very small time step? 

December 13, 2021, 15:49 

#2  
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Filippo Maria Denaro
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Quote:
Before introducing a higher order scheme, you must verify that your code works well for via via refined grids. If your solution does not improve under grid refinement, your problem is not the low accuracy order but some different issues. 

December 13, 2021, 16:04 

#3  
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Yusuf Elbadry
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Quote:
Thank you for your reply. 

December 13, 2021, 16:19 

#4  
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Filippo Maria Denaro
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Quote:
For an arbitrary fixed grid of size h, you cannot be sure that the solution obtained with a second order scheme is better than a first order scheme. You need to have a sufficient grid resolution to get a real improving. That is a classical topic in CFD. 

December 13, 2021, 16:58 

#5  
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Yusuf Elbadry
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Quote:
For example, if I have an airfoil with a chord of 1 (nondimensional length), I will need elements with a side length (in ydir) of 3e3 or less to have a smooth airfoil shape, If the airfoil will move in the ydirection with an amplitude of 0.5c (from 0.5c to 0.5c), then I need at least 333 divisions in this direction, and let us say I will have onethird of this number in the xdirection, this may result in around 350,000 elements near the airfoil where it moves, not mentioning the rest of the domain. This may result in a grid of more than 600,000 elements. That is why I thought of using high order shape function, to reduce the number of elements and nodes. I really appreciate your help and guidance and thank you for your patience. 

December 14, 2021, 04:24 

#6  
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Filippo Maria Denaro
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Quote:
I don't understand, the shape functions defines a certain functional relation for the unknown variables, what do you mean now for the highlighted statement? The better description of the airfoil geometry is a different issue. However, grid size and accuracy order defines different topics and different limits.The former fix the range of resolved wavenumber in the solution, the latter is the theoretical and asymptotical velocity rate of the error decreasing. You could have also a spectral accuracy but if you have fixed the grid size you cannot resolve nothing greater than the Nyquist frequency. 

December 14, 2021, 13:59 

#7  
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Yusuf Elbadry
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Quote:
Please correct me if I am wrong or if there is something I am missing. 

December 14, 2021, 15:35 

#8  
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Filippo Maria Denaro
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Quote:
But this is a further topic. Think about a BEM formulation (panel method), you can describe the airfoilf by using straight segment or curved step. Then you prescribe a functional relation for the distribution of singularity. In you case you want to use elements that are not linear triangles but curved triangles and then use high order shape functions. Again, on the same and arbitrary mesh size h you cannot say apriori that the error is for sure lesser for the quadratic shape function. You need to assess that by a grid refinement study. Only at a certain value h you get a monotonically expected rate for the error. 

December 15, 2021, 14:02 

#9  
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Yusuf Elbadry
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Quote:
I really appreciate your patience and your reply. 

December 15, 2021, 14:59 

#10  
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Filippo Maria Denaro
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December 16, 2021, 13:26 

#11  
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Yusuf Elbadry
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Quote:
I will try using a higherorder shape function for both and solution and geometry. Thank you 

December 16, 2021, 13:33 

#12  
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Filippo Maria Denaro
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Quote:
You could search for suitable isoparametric elements. On the other hand, the airfoil is better described by a triangular tesselation. 

December 16, 2021, 14:15 

#13 
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Yusuf Elbadry
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I know that there is a Lagrangian Element(9 nodes in case of biquadratic shape function) and Serendipity Element(8 nodes in case of biquadratic shape function), can I find a different element than these two types of the element?


December 16, 2021, 14:25 

#14  
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Filippo Maria Denaro
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Quote:
Lagrangian simplex, that is triangular elements (or tetrahedron in 3D), having 3, 6, 10, ... nodes 

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fem, high order scheme., lsm 
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