Reynolds stresses and second law
 

Hi all,

I wonder if there is a mathematical proof showing that the Reynolds stresses follows the second principle of thermodynamic. Of course, I know that the unsteady Navier-Stokes follows the second pricinciple. But if we (Reynolds) average these equations and that we look directly at the Reynolds stresses, can we state that the conservation of entropy won't be violated ? In fact, this question should arise when you model the Reynolds stresses. In this case, how to be sure that the model is consistent with the second law ?

Thank you for your enlightment.

JF

Re: Reynolds stresses and second law
 

All turbulence models that approximate the Reynolds stress tensor using an eddy viscosity model generally end up defining a positive eddy viscosity. This is simply the proof of second law. It is easy to prove that entropy only increases with a positive diffusion constant and can never decrease. (e.g. the first integral of the heat equation).

E.g.

Constant Eddy visc model: nu_T > 0

Mixing Length models: nu_T = l_m^2 * abs(dU / dy) > 0

One equation models: nu_T = C * k^(0.5) * l_m > 0

k-epsilon model and other two equation models: nu_T = c_mu * k^2 / epsilon > 0

And the list can go on ...

Re: Reynolds stresses and second law
 

Thank you for responding.

All turbulence models that approximate the Reynolds stress tensor using an eddy viscosity model generally end up defining a positive eddy viscosity. This is simply the proof of second law.

I agree with that, the proof is similar to the (laminar) Navier-Stokes equations one's. But what happens if you don't take the eddy viscosity assumption and that you directly look at the Reynolds stresses. What is the condition that must fullfill the Reynolds stresses to follow the conservation of entropy. Do you know a reference dealing with this ?

Re: Reynolds stresses and second law
 

I admit, I don't know any restrictions on the Reynolds stress tensor that are required for second law satisfaction. If you do come across something, please let the forum know!

Re: Reynolds stresses and second law
 

I have measured Reynolds stresses in a flow in the lab, and what I found was related to this question. It seems the Reynolds stresses can take both signs (positive and negative) for some flow configurations (for some components of the Reynolds stress tensor at least)!! That would imply transfer of mom. energy from fluctuations (disordered motion) to mean flow (order). I have read some articles that speak of "negative turbulent energy production" (???). I am also trying to find out more about this issue, so if someone has any ideas, they are wellcome!!

Re: Reynolds stresses and second law
 

I think it is fairly well known that energy transfer can take place in both directions (forward = cascade & backward = backscatter). However, both such energy transfers should result in entropy increase. I think a proof of that is lacking...

Re: Reynolds stresses and second law
 

Dear Sirs,