# Filtered Navier-Stokes Equations

While Reynolds-averaged Navier-Stokes (RANS) resolves the mean flow and requires a turbulence model to account for the effect of turbulence on the mean flow, Large Eddy Simulation (LES) computes both the mean flow and the large energy containing eddies.

- ${u}_{i}$ : instantaneous velocity,
- $\tilde{{u}_{i}}$ : filtered velocity,
- $u{"}_{i}$ : sub filtered (unresolved) velocity.

The filtered velocity field is obtained from a low pass filtering operation due to a weighted filter G, defined as $\tilde{{u}_{i}}=\int \int \int G\left(x,{x}^{\prime},\Delta \right){u}_{i}\left({x}^{\prime},t\right)dx{\text{'}}_{1}dx{\text{'}}_{2}dx{\text{'}}_{3}$ .

- $G\left(x,{x}^{\prime}\right)=\frac{1}{\pi}\left(\frac{\text{sin}{\left(\frac{\pi \left(x-{x}^{\prime}\right)}{\Delta}\right)}^{2}}{\left(x-x\text{'}\right)}\right)$ : a cut-off filter,
- $G\left(x,{x}^{\prime}\right)=\sqrt{\frac{6}{\pi {\Delta}^{2}}}exp\left(\frac{-6{\left(x-x\text{'}\right)}^{2}}{{\Delta}^{2}}\right)$ : a Gaussian filter,
- $G\left(x,{x}^{\prime}\right)=\{\begin{array}{c}\frac{1}{\Delta}\forall \left|x-{x}^{\prime}\right|\le \frac{1}{2}\Delta \\ 0\forall \left|x-{x}^{\prime}\right|\frac{1}{2}\Delta \end{array}$ : a top-hat filter,

where $\Delta =\sqrt[3]{\Delta x\Delta y\Delta z}$ is a cutoff width, representing a spatial averaging over a grid element. Among those, a top-hat filter (or similar one) is a popular choice for commercial CFD codes where unstructured meshes are usually adopted.

Although the LES decomposition method resembles the Reynolds method, they have an important difference due to the following. $\tilde{\left(\tilde{{u}_{i}}\right)}\ne \tilde{{u}_{i}}$ , $\tilde{u{"}_{i}}\ne 0$

- $\tilde{{u}_{i}{u}_{j}}=\tilde{\left(\tilde{{u}_{i}}\tilde{{u}_{j}}\right)}+\tilde{\left(u{"}_{i}\tilde{{u}_{j}}\right)+}\tilde{\left(\tilde{{u}_{i}}u{"}_{j}\right)}+\tilde{\left(u{"}_{i}u{"}_{j}\right)}$ is due to the decomposition of the nonlinear convective term in the momentum equation.
- $\tilde{{S}_{ij}}=\frac{1}{2}\left(\frac{\partial \tilde{{u}_{i}}}{\partial {x}_{j}}+\frac{\partial \tilde{{u}_{j}}}{\partial {x}_{i}}\right)$ is the filtered strain rate tensor.

- ${\tau}_{ij}^{*}=-\rho \left({C}_{ij}+{R}_{ij}\right)$ is a double decomposition stress tensor.
- ${C}_{ij}=\tilde{\left(u{"}_{i}\tilde{{u}_{j}}\right)+}\tilde{\left(\tilde{{u}_{i}}u{"}_{j}\right)}$ is the cross stress tensor, representing the interactions between large and small turbulence eddies.
- ${R}_{ij}=\tilde{\left(u{"}_{i}u{"}_{j}\right)}$ is the Reynolds subgrid stress tensor.

where ${L}_{ij}=\tilde{\left(\tilde{{u}_{i}}\tilde{{u}_{j}}\right)}-\tilde{{u}_{i}}\tilde{{u}_{j}}$ is the Leonard stress tensor, representing interactions among large turbulent eddies.

- $\tilde{{S}_{ij}}=\frac{1}{2}\left(\frac{\partial \tilde{{u}_{i}}}{\partial {x}_{j}}+\frac{\partial \tilde{{u}_{j}}}{\partial {x}_{i}}\right)$ is the filtered strain rate tensor,
- ${\tau}_{ij}^{\text{'}}=-\rho \left({C}_{ij}+{R}_{ij}+{L}_{ij}\right)=\rho \tilde{{u}_{i}{u}_{j}}-\rho \tilde{{u}_{i}}\tilde{{u}_{j}}$ is the subgrid stress tensor.

- Smagorinsky-Lilly SGS model
- Germano Dynamic Smagorinsky-Lilly model
- Wall-Adapting Local Eddy-Viscosity (WALE)

RANS | LES | |
---|---|---|

Double averaging | $\overline{\left(\overline{{u}_{i}}\right)}=\overline{{u}_{i}}$ | $\tilde{\left(\tilde{{u}_{i}}\right)}\ne \tilde{{u}_{i}}$ |

Turbulence averaging | $\overline{u{\text{'}}_{i}}=0$ | $\tilde{u{"}_{i}}\ne 0$ |

Averaging (or filtering) of convective term of Navier Stokes | $\overline{{u}_{i}{u}_{j}}=\left(\overline{\overline{{u}_{i}}\overline{{u}_{j}}}\right)+\left(\overline{u{\text{'}}_{i}\overline{{u}_{j}}}\right)+$ $\left(\overline{\overline{{u}_{i}}u{\text{'}}_{j}}\right)+\overline{u{\text{'}}_{i}u{\text{'}}_{j}}$ | $\tilde{{u}_{i}{u}_{j}}=\tilde{\left(\tilde{{u}_{i}}\tilde{{u}_{j}}\right)}+\tilde{\left(u{"}_{i}\tilde{{u}_{j}}\right)+}$ $\tilde{\left(\tilde{{u}_{i}}u{"}_{j}\right)}+\tilde{\left(u{"}_{i}u{"}_{j}\right)}$ |

Turbulent stress | ${\tau}_{ij}^{R}=-\rho \overline{{u}_{i}^{\text{'}}{u}_{j}^{\text{'}}}$ | ${\tau}_{ij}^{\text{'}}=\rho \tilde{{u}_{i}{u}_{j}}-\rho \tilde{{u}_{i}}\tilde{{u}_{j}}$ |