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Co-variant and Contra-variant

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Old   March 24, 2003, 07:28
Default Co-variant and Contra-variant
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James.
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Can anyone explain (in plain english please!) what co-variant and contra-variant fluxes/ velocities are? Where are they used, what are their physical interpretations and where can I read about them in a CFD sense? Thanks for any help.
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Old   March 24, 2003, 09:47
Default Re: Co-variant and Contra-variant
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Rami
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James,

You will find detailed explanation in any textbook on vector and tensor analysis. The following link includes such an explanation and illustrations: http://www.mathpages.com/rr/s5-02/5-02.htm

In short, you may cast the conservation laws in tensor form, such that they are valid in any coordinate system (including non-orthogonal systems) if you use appropriate covariant and contravariant tensors and derivatives. This is helpful in many fields, including CFD, where you wish to deal with general coordinate systems, e.g., in skew quadrilateral cells.

I hope this helps, Rami
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Old   March 24, 2003, 09:49
Default Re: Co-variant and Contra-variant
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Nina Shokina
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Do you mean covariant and contravariant components of velocity vector?

Yours,

Nina
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Old   March 24, 2003, 09:55
Default Re: Co-variant and Contra-variant
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franck
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Considering a curvilinear coordinate system (xi_i, i=1,2,3 in 3 dimensional space), you can define two kinds of vector basis. The covariant vector basis e_i (i=1,2,3) is the set of vectors tangential to each coordinates line (xi_i). The contravariant vector basis e^i (i=1,2,3) is the set of vectors orthogonal to the coordinate surfaces (xi_i=const). It is easy to show that e_i and e^j are orthogonal if i and j are not equal.

Define now a vector field (let say the velocity v), then the covariant components v_i are the projection of v on the covariant basis, and the contravariant components v^i are the projection of v on the contravariant basis. It can be shown that the vector v is expressed in the covariant basis with its contravariant components, and vice-versa (which is a bit confusing indeed...) i.e. v = v_i e^i = v^i e_i (with summation over repeated indices)

Because a contravariant component is the projection of a vector onto the normal to a surface, it is naturally associated with a flux. The covariant components are the projection of a vector along a line, and therefore are associated to a circulation.

Take a look at 'Aris, R., Vectors, Tensors and the Basic Equations of Fluid Mechanics, Prentice-Hall, Englewood Cliffs, NJ, 1962' for more details.

Hope this helps...
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Old   March 24, 2003, 10:47
Default Re: Co-variant and Contra-variant
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James.
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Sorry, yes.
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Old   March 24, 2003, 11:01
Default Re: Co-variant and Contra-variant
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Nina Shokina
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While I was waiting other guys dave you good answers.

Try also http://www.mathpages.com/home/kmath398.htm

It give an idea too.

Yours,

Nina
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