[lucamirtanini]

Hi all,
I am trying to understand the mixing length theory from the tenneke book, but I have difficulties to understand all the passages.
I resume below all my difficulties:
1) In figure "mixing length 2" it is written: "the average momentum flux at x2 =0 may be written as..." and then the author use the tau symbol that in the book is used for the notation of the Reynolds stress.
Why he did that? Is the average momentum flux at x2 =0 a Reynolds stress?
Why the average momentum flux at x2 =0 has that formula?

2)In figure "mixing length 2" it is introduced the dispersion rate, this term was never used before in the book. Why that formula can be called dispersion rate?

I hope you can clarify it and answer to my question. I tried to read other explanation to that theory but I want to understand this explanation.

[LuckyTran]

1) Look at 2.3.8. Once you introduce the concept of eddy viscosity, it should be more obvious that it is the Reynolds stress. But even without the mixing length model, the Reynolds stresses are the parts of the momentum fluxes caused by turbulence. If you know that you will one day call these the Reynolds stresses, then it makes sense to label it here. Remember, the Navier-Stokes equation is transport equation for the momentum flux. The very first line of "mixing length 1" says we are trying to estimate the turbulent momentum flux. This should be a surprise only if you are unfamiliar with Reynolds stresses and have never seen them labeled as tau. But since you already know that tau is the symbol, you should have seen this coming.
2) Because if it did not exchange any momentum, that particular term would stay constant. But we know phenomenologically that turbulence tends to decorrelate the two and that term decreases further downstream, which is statistical dispersion.

[lucamirtanini]

Originally Posted by LuckyTran

1) Look at 2.3.8. Once you introduce the concept of eddy viscosity, it should be more obvious that it is the Reynolds stress. But even without the mixing length model, the Reynolds stresses are the parts of the momentum fluxes caused by turbulence. If you know that you will one day call these the Reynolds stresses, then it makes sense to label it here. Remember, the Navier-Stokes equation is transport equation for the momentum flux. The very first line of "mixing length 1" says we are trying to estimate the turbulent momentum flux. This should be a surprise only if you are unfamiliar with Reynolds stresses and have never seen them labeled as tau. But since you already know that tau is the symbol, you should have seen this coming.
2) Because if it did not exchange any momentum, that particular term would stay constant. But we know phenomenologically that turbulence tends to decorrelate the two and that term decreases further downstream, which is statistical dispersion.

Your answers are perfect. I have understand everything. THANK YOU