Reynolds-Stress Equation

Tobi · May 21, 2016, 13:41

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.

kingeorge · May 22, 2016, 03:52

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.

Tobi · May 22, 2016, 04:40

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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.

Tobi · May 22, 2016, 09:31

Quote:

Originally Posted by kingeorge

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.

Tobi · May 22, 2016, 16:18

Okay. Got it (:

May 22, 2016, 04:40		#3
Tobi Super Moderator Tobias Holzmann Join Date: Oct 2010 Location: Bad Wörishofen Posts: 2,711 Blog Entries: 6 Rep Power: 52	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. __________________ Keep foaming, Tobias Holzmann

May 22, 2016, 16:18		#5
Tobi Super Moderator Tobias Holzmann Join Date: Oct 2010 Location: Bad Wörishofen Posts: 2,711 Blog Entries: 6 Rep Power: 52	Okay. Got it (: Dipsomaniac and kingeorge like this. __________________ Keep foaming, Tobias Holzmann

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May 22, 2016, 03:52		#2
kingeorge New Member VladJ Join Date: Feb 2012 Location: Belgium Posts: 18 Rep Power: 14	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.