ANSYS FLUENT 12.0 Theory Guide - 4.11.2 Filtered Navier-Stokes Equations

4.11.2 Filtered Navier-Stokes Equations

The governing equations employed for LES are obtained by filtering the time-dependent Navier-Stokes equations in either Fourier (wave-number) space or configuration (physical) space. The filtering process effectively filters out the eddies whose scales are smaller than the filter width or grid spacing used in the computations. The resulting equations thus govern the dynamics of large eddies.

A filtered variable (denoted by an overbar) is defined by

$\overline{\phi}({\bf x}) = \int_{\cal D} \phi({\bf x}') G({\bf x,x'}) d{\bf x}'$

(4.11-1)

where $\cal D$ is the fluid domain, and is the filter function that determines the scale of the resolved eddies.

In ANSYS FLUENT, the finite-volume discretization itself implicitly provides the filtering operation:

$\overline{\phi}({\bf x}) = \frac{1}{V} \int_{\nu} \phi({\bf x}') ~d{\bf x}', \; \; \; {\bf x}' \in {\nu}$

(4.11-2)

where is the volume of a computational cell. The filter function, $G(\bf {x},\bf {x'})$ , implied here is then

$G({\bf x, x'}) = \left\{ \begin{array}{ll} 1/V, & {\bf x'} \in {\nu} \\ 0, & {\bf x'} \; {\rm otherwise} \end{array}\right.$

(4.11-3)

The LES capability in ANSYS FLUENT is applicable to compressible flows. For the sake of concise notation, however, the theory is presented here for incompressible flows.

Filtering the Navier-Stokes equations, one obtains

$\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x_i}(\rho \overline{u}_i) = 0$

(4.11-4)

and

$\frac{\partial}{\partial t} (\rho \overline{u}_{i}) + \frac{... ...}{\partial x_{i}} - \frac{\partial \tau_{ij}} {\partial x_{j}}$

(4.11-5)

where $\sigma_{ij}$ is the stress tensor due to molecular viscosity defined by

$\sigma_{ij} \equiv \left[\mu\left(\frac{\partial \overline{... ...} \mu \frac{\partial \overline{u}_l}{\partial x_l} \delta_{ij}$

(4.11-6)

and $\tau_{ij}$ is the subgrid-scale stress defined by

$\tau_{ij} \equiv \rho {\overline{u_i u_j}} - \rho {\overline{u}_i} {\overline{u}_j}$

(4.11-7)

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