Physical Intuitions for the k-epsilon Turbulence Model

[Twigg]

Hey all!

I am trying to understand the terms in the source-side of the k-epsilon transport equations. My only reference on this so far has been the wikipedia article (in my defense, it's not in Landau! the gospel has forsaken me ). If you have a reference that goes into gory detail on the transport equations, that'll probably answer my questions.

But my specific questions are:
1) How is the double contraction of the strain rate tensor related to "production of k (or epsilon)"? My gut reaction is that eddy viscosity times strain rate is Reynolds stress, and Reynolds stress contracted with strain rate is somehow like a "turbulent work" thingy, but I'd like to know more. Also that doesn't explain the term in the epsilon transport equation.

2) How is ##\epsilon^2/k## related to the rate of decay of ##\epsilon##? I get the dimensions are right, but is that all there is to it?

3) How is ##\epsilon## different from ##\omega## in the k-omega model?

 

[eljose79]

Hi there,

I can definitely understand your confusion about the terms in the source-side of the k-epsilon transport equations. It can be a bit overwhelming to try and understand all the different terms and their relationships. Fortunately, there are some great resources available that can help you dive deeper into this topic.

One reference that I would highly recommend is the book "Turbulence Modeling for CFD" by David C. Wilcox. It provides a comprehensive and in-depth explanation of the k-epsilon transport equations, including the source terms. It also includes derivations and explanations for each term, which can be very helpful in understanding their physical significance.

To answer your specific questions:

1) The double contraction of the strain rate tensor is related to the "production of k (or epsilon)" in the sense that it represents the transfer of kinetic energy from the mean flow to the turbulence. This is often referred to as "turbulent production" and is represented by the term eddy viscosity times strain rate. The Reynolds stress term, on the other hand, represents the transport of momentum by the turbulent eddies.

2) The term ##\epsilon^2/k## in the epsilon transport equation is related to the rate of decay of ##\epsilon## in the sense that it represents the dissipation of turbulent kinetic energy. This term essentially accounts for the loss of energy due to the dissipation of turbulent eddies, which leads to a decrease in the overall turbulence intensity.

3) In the k-omega model, ##\epsilon## and ##\omega## are both variables used to model the turbulent kinetic energy and dissipation rate, respectively. However, the main difference between them is that ##\epsilon## is based on the k-epsilon model, which uses the eddy viscosity assumption, while ##\omega## is based on the k-omega model, which uses the eddy diffusivity assumption. This can lead to different predictions for the turbulence intensity and length scales in different flow situations.

I hope this helps clarify some of your questions. Again, I would highly recommend checking out the book I mentioned for a more detailed explanation of the k-epsilon transport equations. Best of luck with your research!