I am working on Large Eddy Simulation (LES) for compressible flows and using the Smagorinsky model to compute the turbulent viscosity $\nu_t$.
In the filtered Navier-Stokes equations for compressible flows, the viscous stress term involves the divergence of the effective stress tensor, which can be written as:
\begin{equation} \frac{\partial}{\partial x_j} \left[ \rho (\nu + \nu_t) \left( \frac{\partial \overline{u}_i}{\partial x_j} + \frac{\partial \overline{u}_j}{\partial x_i} - \frac{2}{3} \delta_{ij} \frac{\partial \overline{u}_k}{\partial x_k} \right) \right], \end{equation}
where:
$\nu$ is the molecular viscosity;
$\nu_t = (C_s \Delta)^2 |\widetilde{S}|$, according to the Smagorinsky model;
$C_s$ is the Smagorinsky constant;
$\Delta$ is the filter width;
$\overline{u}_i$ are the filtered velocity components.
Since $\nu_t$ depends on local velocity gradients and varies in space, it seems that it should remain inside the derivatives. However, I have seen simplified treatments (particularly for incompressible flows) where $\nu_t$ is treated as locally constant and pulled out of the derivative. Is this appropriate?
For compressible flows, where the stress tensor depends on both the velocity gradients and the divergence of the velocity field, does this simplification also hold?
Any insights or references would be appreciated!
