Contravariant and covariant velocity decomposition
Currently, it has taken me lots of times in understanding contravarient and covariant velocity decompsition in CFD. I have checked some books. However, I still think that I do not understand very well regarding them. Why some researchers chose to use contravariant or covariant velocity decomposition in CFD, but not Cartisian one. Are there any advantages in CFD with contranvariant or covariant velocity decomposition?
Tanks in advance. cheers Ben 
Re: Contravariant and covariant velocity decomposition
Hi. When you deal with a simple (regular) computational domain, everything is fine in terms of mesh, grid, and grid (nodal) points. Say, you utilize finite difference approach, and discretize your governing equations (partial differential equations) into algebraic equations. You can see that boundary nodes are easily taken of care as long as your grid is regular. Now, you pick more complex geometry and try to write difference equations for boundary nodes. Since the boundary geometry is irregular, you encounter tremendeous difficulty in formulating difference equations for the boundary nodes. Basically, you have two options if you decide to stick to finite differences. First option is: you write out difference (algebraic) equations manually for EVERY grid point that lies on irregular boundary, as there is actually no way to implement it automatically unless you involve some intellegence there. The reason for that is you need to perform interpolation for every irregular grid point, and eventually you may lose accuracy even compared to internal (regular) grid points. The second option is, you apply some "magic" transformation, so that your initial irregular domain is mapped onto regular domain. Of course, now you know how easy is to write difference equations and to apply boundary conditions. However, nothing is for free! Here it comes all transformation jacobians that you actually used to do a magic! They show up in nothing but in "modified" governing equations and involve covariant and contrvariant tensors, as you said. As you already noticed, they are quite involved. Another not too bright side of the mapping is that you can do it for some cases (not too complex, say) only, while there is no generality there. A transformation that has been helpful for one geometry, may be of no use for the others. Moreover, it involves a lot of cumbersome analytical work to get a proper mapping. Note, that when people talk about structured and/or unstructured meshes, the former has to do with the mapping as people generate a nice looking mesh first, and then map it. Some people who routinely run cfdcodes do not even realize it, though. The latter, i.e. unstructured mesh, is more or less similar (formally, though) to the finite element mesh. Generally, you do not use a transformation for the governing equations, rather you work with the cartesian coordinate system, however, discretize PDEs into algebraical equations directly within, say, finite element framework, or, utilize a sophisticated control(finite) volume approach that involves intrinsic interpolation functions to represent temperature (velocity) variations within control volume. hope that helps.

Re: Contravariant and covariant velocity decomposition
Usually covariant or contravariant velocity components (more choices are available) are adopted in cases when the staggered grid is used with curvilinear (e.g. bodyfitted) meshes. At a price of more complexity in the governing equations (the curvature terms) one can minimize the need for interpolation (surface mass flow rates), minimize storage requirements and minimize the effects of grid nonorthogonality.
I would say, that if one accepts the cellcentered scheme, there is no need to take anything else than the Cartesian velocity components. This is my preferred choice. If you need more information on the usage of various velocity components, have a look at the book by Aris (Vectors, tensors and the basic equations of fluid mechanics). If you want something with more numerical details, see e.g. the Ph.D.Thesis of S.Parameswaran (Finite Volume Equations for Fluid Flow Based on NonOrthogonal Velocity Projections, Imperial Coll.Sci.Tech., CFD/86/12, Univ. London 1985. regards DML 
Re: Contravariant and covariant velocity decomposition
A brief addition to the previous posting.
A covariant representation will eliminate the cross derivatives terms in the pressure equation. This a major practical advantage for a solution procedure but the number of terms is excessive for most applications. In general, the advantages of grid staggering and efficient solution procedures win out only when one can use an orthogonal grid to bring the number of terms to a sensible number and when strict numerical conservation of vector and tensor quantities is not essential (you can conserve scalars like mass). Unfortunately, curvilinear orthogonal coordinate systems do not exist in general in 3D. 
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