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Reynolds-Stress Equation

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Old   May 21, 2016, 12:41
Default Reynolds-Stress Equation
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Hi all,

I have a problem deriving the Reynolds-Stress equation. I am trying to get the same results as Wilcox but - no chance. I always get one more term in the convective term and I don't know what they actually do to get rid of the term that I get. The derivation is in the attachment. The problem is the term:

-\bar u_k \frac{\partial \tau_{ji}}{\partial x_k}

that is not included in the derivation of Wilcox.

A few hints:
  • Linear terms in fluctuations are zero
  • Derivation in \frac{\partial u_k'}{\partial x_k} = 0
  • Mass conservation is clear (:
All hints are appreciated. I think I make a simple mistake or do something wrong in a mathematic point of view.

I also can achieve this equation:

\underline{-\bar u_k \frac{\partial \tau_{ij}}{\partial x_k}} - \bar u_k \frac{\partial \tau_{ji}}{\partial x_k} -  \bar u_j \frac{\partial \tau_{ik}}{\partial x_k} - \bar u_i  \frac{\partial \tau_{jk}}{\partial x_k} - \underline{ \tau_{ik} \frac{\partial \bar u_j}{\partial x_k} - \tau_{jk} \frac{\partial \bar u_i}{\partial x_k} + \frac{\partial}{\partial x_k}  \rho \overline{u_i' u_k' u_j'}} + \frac{\partial}{\partial x_k} \rho \overline{u_j' u_k' u_i'}

But in Wilcox only the underlines terms are mentioned.
Attached Files
File Type: pdf ReynoldsStressEquation.pdf (28.1 KB, 19 views)
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Old   May 22, 2016, 02:52
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Tobias,
Have you possibly taken a look at the lecture notes of prof. Lars Davidson form Chalmers. He has done a section on transport equation for Reynolds stresses and it might be helpful.

http://www.tfd.chalmers.se/~lada/pos...ing.pdf#page76

Cheers and looking forward to see your derivation.
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Old   May 22, 2016, 03:40
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Please click one of the Quick Reply icons in the posts above to activate Quick Reply.

Hi Vladimir,

thanks for that hint. I will check it out. Yesterday I switched from Einsteins summation convention to the real derivatives. I think in this way it is clearer but for example, in the convective term for the first of 9 parts I get 20 terms

Concentration is the thing you need.
I will see if this will work but I will definitely check out your link. Maybe I am doing something wrong in mathematics or even its only due to the symmetry of the tensor.
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Old   May 22, 2016, 08:31
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Quote:
Originally Posted by kingeorge View Post
Tobias,
Have you possibly taken a look at the lecture notes of prof. Lars Davidson form Chalmers. He has done a section on transport equation for Reynolds stresses and it might be helpful.

http://www.tfd.chalmers.se/~lada/pos...ing.pdf#page76

Cheers and looking forward to see your derivation.

Nice book,

so why I am doing it ... no its good for my understanding. Unfortunately he only showed the Reynolds-Stress equation without the derivation.
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Old   May 22, 2016, 15:18
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Okay. Got it (:
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