Double inner product?

  • Thread starter Smed
  • Start date
  • #1
36
1
Hi, I'm having trouble understanding how to perform the following calculation:

[tex]
u=(u,v,w)
[/tex]

[tex]
(\nabla u + (\nabla u)^T) : \nabla u
[/tex]

I get the following by doing the dot product of the first term and then adding the dot product of the second term, but I'm pretty sure it isn't correct.

[tex]
2\left(\frac{\partial u}{\partial x}\right)^2
+ 2\left(\frac{\partial v}{\partial y}\right)^2
+ 2\left(\frac{\partial w}{\partial z}\right)^2
[/tex]

Could someone please shed some light on how the double inner product should work?
Thanks
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
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Hi, I'm having trouble understanding how to perform the following calculation:

[tex]
u=(u,v,w)
[/tex]

[tex]
(\nabla u + (\nabla u)^T) : \nabla u
[/tex]
Your notation here doesn't make sense to me. If you are using "u" to represent a vector, don't use the same "u" to represent one of its components. If you are writing [itex]\nabla u[/itex] as, say, a row vector, then [itex[(\nabl u)^T[/itex] would be a column vector and you cannot add them. In general, a vector and its transpose are in different vector spaces and cannot be added. Finally, I don't know what ":" means. Was that supposed to be [itex]\cdot[/itex]?

I get the following by doing the dot product of the first term and then adding the dot product of the second term, but I'm pretty sure it isn't correct.

[tex]
2\left(\frac{\partial u}{\partial x}\right)^2
+ 2\left(\frac{\partial v}{\partial y}\right)^2
+ 2\left(\frac{\partial w}{\partial z}\right)^2
[/tex]

Could someone please shed some light on how the double inner product should work?
Thanks
 
  • #3
649
2
I think it is related to the definition in section 1.3.2 found here:
http://www.foamcfd.org/Nabla/guides/ProgrammersGuidese3.html

It is for a pair of rank 2 tensors, and is denoted by a :

The \nabla u's used in the original post are interpreted as second rank tensors, and the double inner product is applied between the terms on each side of the :

Haven't got time to check the calculation myself.

Torquil
 

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