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-   -   Simple vector notation help. uiuj in non index notation (http://www.cfd-online.com/Forums/main/107715-simple-vector-notation-help-uiuj-non-index-notation.html)

meangreen October 4, 2012 12:39

Simple vector notation help. uiuj in non index notation
 
Hey Everyone,

For my paper I have about 50 equations all in the "standard" vector notation
\vec{u}, but there are 2 or 3 turbulence equations with the Reynolds stress therm in index notation that I don't know how to put in the "standard form". The term is below:

u_iu_j = ??? in standard form

I think \vec{u}^2 is not correct or at least will make the reader think I am talking about the dot product of u with u.

meangreen October 4, 2012 12:48

My friend just said he thinks it is

u_iu_j=\vec{u}\vec{u}

agd October 4, 2012 13:01

If you are referring to the terms in the Reynolds stress tensor, your friend is correct. The term for such a representation is dyadic.

meangreen October 4, 2012 13:11

I just had another friend say it is the outer product, so


u_iu_j=\vec{u}\otimes\vec{u}

is it basically the same thing?

\vec{u}\vec{u}=\vec{u}\otimes\vec{u} ?

agd October 4, 2012 16:34

They are not the same thing. The outer product of two vectors is another vector. The inner product of two vectors is, of course, a scalar. The dyadic product of two vectors is a tensor of order 2 (since the vectors are each tensors of order 1). When you write the index notation (ui)(uj) without any caveats, it generally implies the dyadic product.

meangreen October 5, 2012 10:23

I just looked and according to wiki (i know, i know) the outer product does produce a tensor. Are you sure about your answer?

cdegroot October 7, 2012 11:05

Quote:

Originally Posted by meangreen (Post 384956)
I just had another friend say it is the outer product, so


u_iu_j=\vec{u}\otimes\vec{u}

is it basically the same thing?

\vec{u}\vec{u}=\vec{u}\otimes\vec{u} ?

First equation is correct. Second one should be \vec{u}\vec{u}^T=\vec{u}\otimes\vec{u} assuming column vectors.

agd October 7, 2012 11:07

Sorry - I saw the symbol and my brain interpreted it as the cross product. Yes, the outer product results in the same thing as the dyadic product. It is not the same thing as the cross product, which produces a tensor of the same order as the original entries.


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