Smagorinsky model and Sij terms
 

I'm currently updating a code that uses the Smagorinsky model as its sgs model. At present the code calculates Sij in the form:

Sij**2 = S11**2 + S22**2 + S33**2 + 2S12**2 + 2S13**2 + 2S23**2

in order to calculate the eddiy viscostiy.

Now, as the |S| term is SQRT(2SijSij), is is clear that the Sij**2 shown above and SijSij calculated normally do not commute, so which one should I use? Is the above implmentation wrong?

Thanks,

James

Re: Smagorinsky model and Sij terms
 

I think the above implementation is correct. Sij**2 = Sij:Sij = Sij*Sji

Re: Smagorinsky model and Sij terms
 

It all depends on the definitions of S11, S22, S12, etc. in the code. The equation you have written is neither wrong nor right; you have simply not provided sufficient information.

This is a two minute exercise in algebra. You know Sij = {ui,j + uj,i}/2 and Sij*Sij simply uses the repeated index notation. You can then verify by yourself whether your code is right or wrong

Adrin Gharakhani

Re: Smagorinsky model and Sij terms
 

The current code in indeed correct. I should try to remember how tensor notation works!