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Spalart-Allmaras and "Non-Conservative Diffusion"

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Old   April 15, 2013, 18:22
Default Spalart-Allmaras and "Non-Conservative Diffusion"
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Peter Doings
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I'm an undergrad student and I'm implementing the S-A model for incompressible flow in a FV solver;

I have access to STAR-CCM+'s documentation and there I found an entry explaining that they did not discretize explicitly one of parcels of the diffusion term but that they combined it with the conservative diffusion one. I tried to get to the formula on my own using the identity:

\nabla\cdot(\phi\nabla\psi)=\phi\nabla^{2}\psi + \nabla\phi\cdot\nabla\psi

And got a very similar result to STAR's:

\frac{1}{\sigma_\nu}[\int_S(\nu+\tilde{\nu})\nabla\nu \cdot dS + \int_V C_{b2} (\nabla \nu \cdot \nabla \nu)dV] = \frac{1+C_{b2}}{\sigma_\nu}  \int(\nu+\tilde{\nu})\nabla\tilde{\nu} \cdot dS -  \int_V\frac{C_{b2}}{\sigma_\nu}(\nu+\tilde{\nu})\nabla^2\tilde{\nu}dV

Except that in the documentation it appears as a \approx and the last term has, instead of the Laplacian, a \nabla\tilde{\nu}^2 which I assume is their mistake (is it?).
My questions are: a) is this common practice? and b) I'll treat the first term as a regular diffusion term with the diffusivity given by last iteration's \tilde{\nu}, but how can I best discretize the last one? Explicitly calculate the laplacian on a cell by cell basis, multiply by \nu+\tilde{\nu} and put the result in the right-hand side?

Any help apreciated.
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