[bumper]

Hello,

i'm trying to understand the theory behind the eddy viscosity turbulence modelling.

First i'm doing the RANS approach on the conservation equations and i get the Reynolds Stress tensor like in the picture (Reynold_Stress_Tensor) below. Each line characterize the additonal terms that i get in the three momentum equation. By choosing a eddy viscosity turbulence model, i use the approach of Boussinesq for modelling the expressions of the Reynolds Stress Tensor. There is an unknown variable, the turbulent viscosity. The definition of the turbulent viscosity depends on the choosen turbulence model. Let's choose the k-epsilon model for example.

https://www.cfd-online.com/Wiki/Stan...-epsilon_model

In the equations of k and epsilon, you can find the indices i and j which go from 1 to 3 each (direction x,y,z). If i=1, j could be 1,2,3, so i have 3 equations in total for k in x-direction ? All in all i have 9 equations for k and 9 equations for epsilon? Than i have lots of opportunities to calculate the turbulent viscosity and the entries of the Reynolds Stress Tensor.

Or do i have just 9 additonal equations (one for each entrie of the Reynolds Stress Tensor). For example i want to calculate the first entry. Than, i and j are 1 and it "goes down" to the equation of the turbulence model?

So the main question is, how many equations do i get in total. I hope that i describe my problem in a good way. It would be great if someone could explain it to me.

Thanks a lot,

bumper