hi,all!

In the dynamic smagrinsky procedure we get Cs=0.5*(Lij*Mij)/(Mij*Mij) ^^^^^^^^(1) the equation is from D.K.Lilly ,

But I have also seen that the isotropic part of Lij should be removed from (1) so

Cs=0.5*(Lij-1/3*DELTAij*Lkk)*Mij/(Mij*Mij)^^^^^^^(2)

I can understand (2) but for (1) Lilly explained that the isotropic part of Lij does not appear because in incompressible flow Sii=0, I dont really understand this ,since no direct links between Lij 's isotropic part and Sii exist.

can anyone tell me which equation I should use,maybe their effect doesn't differ much,so either will be ok?

any comment is welcome,thanks.

The isotropic part of S is simply added to the pressure gradient term (since it is a normal stress). It is usually not included in the SGS stress formulation. Whenever people refer to S_ij, they usually mean only the anisotropic part of it.