evaluate effective viscosity from U and p already known

If you work out the brackets in the term on the right hand side, can't you just solve it like any other equation in OpenFOAM? So you use fvc instead of fvm on the terms with p and U and solve for nuEff. I can imagine that this will work, but I'll give you now guarantee.

Yes, but in that case, I will have :
T1 = fvc::laplacian(U) which is a vector T2 = fvc::grad(p)/rho + fvc::div(phi,U) which is a vector as well. How can I deduce the value of nuEff which should be a scalar (or may be a tensor) ?? 
For the right hand side, don't you get something like
fvm::grad(nuEff) & fvc::grad(U) + nuEff * fvc::laplacian(U) or something like that, which you can solve if you supply boundary conditions? But you mean this is an overdetermined system or something? Maybe you can work componentwise? 
Why don't you just write nut directly then? Seems like a unnecessary exercise to me...

Because the RANS approach are just a particular approach of what I need. It makes sense when LES is considered for example.

Why? Then you still have an effective viscosity in the way it is implemented in OpenFOAM.

Yes, but what I look for is the effective viscosity that corresponds to timeaverage velocity and pressure fields. It is not the case of the nuSgs of the LES..

Why don't you use the timeaveraged value of nuSgs/nut? I don't think this will make a very big difference compared to what you want to do. I still don't get why you want to do it this complex...

Because I do not have it and the simulations are too long to be start one more time...


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