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.

A subgrid model is used to capture the effects of small scale turbulent structures. The spatial filtering process is used to filter out the turbulent structures from the instantaneous flow field which are smaller than a given filter size. This filtering process is based on the decomposition of instantaneous variables (velocity, pressure) into filtered (resolved) and sub filter (unresolved or residual) variables. Here, velocity is used as an example.

u_{i} = \tilde{u_{i}} + u "_{i}

where

$u_{i}$ : 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 $\tilde{u_{i}} = \int \int \int G (x, x^{'}, Δ) u_{i} (x^{'}, t) d x'_{1} d x'_{2} d x'_{3}$ .

The weighted filter includes

$G (x, x^{'}) = \frac{1}{π} (\frac{sin {(\frac{π (x - x^{'})}{Δ})}^{2}}{(x - x')})$ : a cut-off filter,
$G (x, x^{'}) = \sqrt{\frac{6}{π Δ^{2}}} e x p (\frac{- 6 {(x - x')}^{2}}{Δ^{2}})$ : a Gaussian filter,
$G (x, x^{'}) = {\begin{matrix} \frac{1}{Δ} \forall | x - x^{'} | \leq \frac{1}{2} Δ \\ 0 \forall | x - x^{'} | > \frac{1}{2} Δ \end{matrix}$ : a top-hat filter,

where $Δ = \sqrt[3]{Δ x Δ y Δ z}$ 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. $\tilde{(\tilde{u_{i}})} \neq \tilde{u_{i}}$ , $\tilde{u "_{i}} \neq 0$

Once this concept is substituted into the instantaneous Navier-Stokes equations, and then the spatial-averaging (or filtering) is made, the filtered Navier-Stokes equations are obtained. The filtered Navier-Stokes equations include the equations for the filtered continuity and filtered momentum equations, which are given below.

\frac{\partial (ρ \tilde{u_{i}})}{\partial x_{i}} = 0

\frac{\partial (ρ \tilde{u_{i}})}{\partial t} + \frac{\partial (ρ \tilde{u_{i} u_{j}})}{\partial x_{j}} = - \frac{\partial (\tilde{p} δ_{i j})}{\partial x_{j}} + 2 μ \frac{\partial \tilde{S_{i j}}}{\partial x_{j}}

where

$\tilde{u_{i} u_{j}} = \tilde{(\tilde{u_{i}} \tilde{u_{j}})} + \tilde{(u "_{i} \tilde{u_{j}}) +} \tilde{(\tilde{u_{i}} u "_{j})} + \tilde{(u "_{i} u "_{j})}$ is due to the decomposition of the nonlinear convective term in the momentum equation.
$\tilde{S_{i j}} = \frac{1}{2} (\frac{\partial \tilde{u_{i}}}{\partial x_{j}} + \frac{\partial \tilde{u_{j}}}{\partial x_{i}})$ is the filtered strain rate tensor.

The filtered momentum equation is rearranged as

\frac{\partial (ρ \tilde{u_{i}})}{\partial t} + \frac{\partial (ρ \tilde{\tilde{u_{i}} \tilde{u_{j}}})}{\partial x_{j}} = - \frac{\partial \tilde{p}}{\partial x_{j}} + 2 μ \frac{\partial \tilde{S_{i j}}}{\partial x_{j}} + \frac{\partial τ_{i j}^{*}}{\partial x_{j}}

where

$τ_{i j}^{*} = - ρ (C_{i j} + R_{i j})$ is a double decomposition stress tensor.
$C_{i j} = \tilde{(u "_{i} \tilde{u_{j}}) +} \tilde{(\tilde{u_{i}} u "_{j})}$ is the cross stress tensor, representing the interactions between large and small turbulence eddies.
$R_{i j} = \tilde{(u "_{i} u "_{j})}$ is the Reynolds subgrid stress tensor.

Since $\tilde{(\tilde{u_{i}} \tilde{u_{j}})}$ in the convection term of the filtered momentum equation needs a secondary filtering process, it is rewritten as shown below (Leonard, 1974),

\tilde{(\tilde{u_{i}} \tilde{u_{j}})} = L_{i j} + \tilde{u_{i}} \tilde{u_{j}}

where $L_{i j} = \tilde{(\tilde{u_{i}} \tilde{u_{j}})} - \tilde{u_{i}} \tilde{u_{j}}$ is the Leonard stress tensor, representing interactions among large turbulent eddies.

Utilizing the decomposition process shown above, the filtered momentum equations can be rewritten as

\frac{\partial (ρ \tilde{u_{i}})}{\partial t} + \frac{\partial (ρ \tilde{u_{i}} \tilde{u_{j}})}{\partial x_{j}} = - \frac{\partial \tilde{p}}{\partial x_{j}} + 2 μ \frac{\partial \tilde{S_{i j}}}{\partial x_{j}} + \frac{\partial τ_{i j}^{'}}{\partial x_{j}}

where

$\tilde{S_{i j}} = \frac{1}{2} (\frac{\partial \tilde{u_{i}}}{\partial x_{j}} + \frac{\partial \tilde{u_{j}}}{\partial x_{i}})$ is the filtered strain rate tensor,
$τ_{i j}^{'} = - ρ (C_{i j} + R_{i j} + L_{i j}) = ρ \tilde{u_{i} u_{j}} - ρ \tilde{u_{i}} \tilde{u_{j}}$ is the subgrid stress tensor.

Similar to RANS, the subgrid stress tensor is an unknown and must be computed by a subgrid model. However, the subgrid stress tensor differs from the Reynolds stress tensor. The table below summarizes the differences between time averaging for RANS and spatial averaging for LES. Common subgrid scale (SGS) models available to solve the filtered Navier-Stokes equations include:

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

Table 1. Time-Averaging (RANS) vs. Spatial-Averaging (LES)
	RANS	LES
Double averaging	$\bar{(\bar{u_{i}})} = \bar{u_{i}}$	$\tilde{(\tilde{u_{i}})} \neq \tilde{u_{i}}$
Turbulence averaging	$\bar{u'_{i}} = 0$	$\tilde{u "_{i}} \neq 0$
Averaging (or filtering) of convective term of Navier Stokes	$\bar{u_{i} u_{j}} = (\bar{\bar{u_{i}} \bar{u_{j}}}) + (\bar{u'_{i} \bar{u_{j}}}) +$ $(\bar{\bar{u_{i}} u'_{j}}) + \bar{u'_{i} u'_{j}}$	$\tilde{u_{i} u_{j}} = \tilde{(\tilde{u_{i}} \tilde{u_{j}})} + \tilde{(u "_{i} \tilde{u_{j}}) +}$ $\tilde{(\tilde{u_{i}} u "_{j})} + \tilde{(u "_{i} u "_{j})}$
Turbulent stress	$τ_{i j}^{R} = - ρ \bar{u_{i}^{'} u_{j}^{'}}$	$τ_{i j}^{'} = ρ \tilde{u_{i} u_{j}} - ρ \tilde{u_{i}} \tilde{u_{j}}$