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May 18, 2007, 01:22 
coordinate transformation

#1 
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When doing calculation in the computational domain for the conservation laws (dU/dt + dF/dx + dG/dy = 0), what are the equations for transforming the flux vectors back to the physical domain(x,y)?. Also how to I attach images?


May 18, 2007, 08:55 
Re: coordinate transformation

#2 
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Just compute the flux vectors using their definitions  you have p, u, and rho. Or are you talking about the derivatives of the flux vectors? In that case compute the derivatives in computational space and use the metrics to transform to physical space, i.e.
d()/dx = d()/dxi*(dxi/dx) Chain rule where xi is the appropriate metric term(s). 

May 20, 2007, 21:09 
Re: coordinate transformation

#3 
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I mean the equation for converting flux vector F from physical to computational domain is:
F_comp = JF_phy*d(zeta)dx + JG_phy*d(zeta)/dy So how what is the equation for vice versa? F_comp to F_phy? 

May 21, 2007, 08:54 
Re: coordinate transformation

#4 
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If you can compute the fluxes directly in the physical domain given the qvariables then why do you need it? To derive it you will need to write the flux transformation for each of the computational fluxes and then solve simultaneously. This entails inverting the the matrix composed of the metric terms. The math is not particularly difficult but is tedious. I don't know of a good reference right off the top of my head, because the inverse transformation is rarely (never) used. The physical flux vectors are directly available once the qvector is computed.


May 21, 2007, 21:02 
Re: coordinate transformation

#5 
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You mean just do all calculations in the physical domain with varying grid spaces rather than converting the fluxes onto a rectangular computational grid and then transforming them back into physical space?.


May 22, 2007, 00:26 
Re: coordinate transformation

#6 
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I'm assuming you are using structured grids.If not discard this note.
You can solve the system of equations in orthogonal curvilinear form or curvilinear form.In the first case the velocity components in the computational space are the transformed velocity components.But in the case of curvilinear form the velocity components are cartesian components and you transform only the derivatives df/dx=df/dx1*dx1/dx+df/dx2*dx2/dx Anderson et. al (computational fluid mechanics and heat transfer)discusses this point in chapter 5. If your grid is orthogonal,use the orthogonal curvilinear form (measure the deviation of g12),if not use the general curvilinear form. 

May 22, 2007, 01:33 
Re: coordinate transformation

#7 
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So the values of the flux vectors when using the general coordinate system dQ/d(tau) + dF/d(zeta) + dG/d(eta) are the same in both computational and Cartesian space?


May 22, 2007, 08:16 
Re: coordinate transformation

#8 
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In a generalized coordinate,you do not look at them as vectors but like each is a scalar ( Ex, FX etc.)


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