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Old   April 25, 2017, 19:31
Default Elliptic Grid Generation
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I'm working on generating a grid between a rectangular domain and the surface of an airfoil using Laplace's Equations. I'm trying to convert the physical domain into the computational domain, but I'm struggling with the co-ordinate transformation - I understand how it works but I'm unable to convert the Boundary Conditions. Could anyone walk me through this? How do Xi and Eta from the computational domain relate to x and y in the physical space? Is there a way I can identify the mapping between Xi(x,y), Eta(x,y) and X(Xi,Eta), Y(Xi,Eta)?

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Old   April 30, 2017, 13:22
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Tom-Robin Teschner
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Du you have access to the book "Computational Fluid Dynamics for Engineers, Vol 1" by Hoffmann and Chiang? I could try to give an answer as excellent as they provided in Chapter 9. Alternatively, the book of Anderson (Computational Fluid Dynamics: The Basics with Applications) is a bit less explanatory but equally good on the topic of mesh transformation from physical to computational space with a walk through example for the Prandtl-Meyer expansion wave.

Basically you transform your governing equations into curvelinear form which will feature your metric coefficients (dXi/dx, dEta/dx, dXi/dy, dEta/dy etc.) or the other way around (dx/dXi, dx/dEta, dy/dXi, dy/dEta etc.) depending on your transformation. The latter case will usually have the Jacobian in your governing system and it is this system that you want to aim form, as you can evaluate your metric coefficients here directly on your computational mesh. I.e. dx/dXi, for example, becomes (x_i+1,j - x_i-1,j)/dXi. Since dXi is a constant which you can freely choose, it is usually set to 1 so that dx/dXi = (x_i+1,j - x_i-1,j). Now plug in your x coordinate in for x, di the same for the other metric coefficients and you can solve your governing equations in the computational mesh (rectangular mesh). Your boundary conditions are then imposed at the same boundaries as the physical mesh.
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