The filter operation is a concept to remove small-scale motion.
In the figure below (Rodi et al.), the smooth curves comes from filtering the original one, with the filter width \(\Delta_{1}\) or \(\Delta_{2}\).
In practice, the removal of the small-scale motion and hence the averaging is performed mostly by the numerical grid, as on a given grid only motions with scales larger than the mesh size can be resolved.
The others fall through the mesh.
Filtered Navier-Stokes Equations
By applying a filter on the continuity and Navier-Stokes equations (NSE), the filtered balance equations are obtained, shown here for incompressible flow:
(2)\[\begin{split}\frac{\partial \bar{u}_{\alpha}}{\partial x_{\alpha}}&=0 \\
\frac{\partial \bar{u}_{\alpha}}{\partial t} + \frac{\partial \bar{u}_{\alpha}\bar{u}_{\beta}}{\partial x_{\alpha}}&=-\frac{1}{\rho_{r}}\frac{\partial \bar{p}}{\partial x_{\alpha}}+\frac{\partial}{\partial x_{\alpha}}\left(\nu\frac{\partial \bar{u}_{\alpha}}{\partial x_{\alpha}}\right)-\frac{\partial \tau_{\alpha\beta}^{\mathrm{SGS}}}{\partial x_{\alpha}}+g_{\alpha}\frac{\bar{\rho}-\rho_{r}}{\rho_{r}}\\\end{split}\]
where (\(\bar{\phi}\)) is called resolved variable, \(\rho_{r}\) is the reference density and \(\tau^{\mathrm{SGS}}\) is called subgrid stress, defined by
(3)\[\tau_{\alpha\beta}^{\mathrm{SGS}}=\bar{u_{\alpha}u_{\beta}}-\bar{u}_{\alpha}\bar{u}_{\beta}\]
The subgrid stress represents the effect of the unresolved fluctuations on the resolved motion.
They’re analogous to the Reynolds stresses in the RANS approach, but while the latter represent the effect of the entire turbulent fluctutations on the mean motion, \(\tau_{\alpha\beta}^{\mathrm{SGS}}\) only accounts for the effect of the small-scale motion.
The effect of the subgrid stress is mainly dissipative, i.e, to withdraw energy from the resolved motion.
One way to account for this effect is through an explicit SGS model for \(\tau_{\alpha\beta}^{\mathrm{SGS}}\).
Note
Another approach is to account for it through numerical dissipation introduced by the solution procedure, what’s known as Implicit Large Eddy Simulation (ILES) methods, but this option isn’t pursued here.
Generally, the subgrid scale stress tensor \(\tau_{\alpha\beta}^{\mathrm{SGS}}\) is split into an isotropic and an anisotropic component as
(4)\[\tau_{\alpha\beta}^{\mathrm{SGS}}=\underbrace{\tau_{\alpha\beta}^{\mathrm{SGS}}}_\text{anisotropic}+\underbrace{\frac{1}{3}\tau_{\gamma\gamma}^{\mathrm{SGS}}\delta_{\alpha\beta}}_\text{isotropic}\]
where the isotropic part of \(\tau_{\alpha\beta}^{\mathrm{SGS}}\) is twice the kinetic energy of the SGS fluctuations.
This is absorbed into the pressure term, as a modified pressure \(p^{*}=p+(2/3)k_{\mathrm{SGS}}\).
The anisotropic part is usually approached by an eddy viscosity model:
(5)\[\tau^{\mathrm{SGS}}_{\alpha\beta}=\nu_{SGS}\dot{\gamma}_{\alpha\beta}\]
where \(\nu_{SGS}\) is the subgrid viscosity.
Hence, the resolved Navier-Stokes equations for quasi-incompressible flow is
(6)\[\frac{\partial\bar{u}_{\alpha}}{\partial t}+\frac{\partial}{\partial x_{\beta}}\left(\bar{u}_{\alpha}\bar{u}_{\beta}\right)=-\frac{1}{\rho}\frac{\partial \bar{p}^{*}}{\partial x_{\alpha}}+2\frac{\partial}{\partial x_{\beta}}\left[\left(\nu+\nu_{\mathrm{SGS}}\right)\bar{S}_{\alpha\beta}\right]\]
