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 therefore govern the dynamics of large eddies.

A filtered variable (denoted by an overbar) is defined by Equation 4–5:

(4–5)

where 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:

(4–6)

where is the volume of a computational cell. The filter function, , implied here is then

(4–7)

The LES capability in Ansys Fluent is applicable to compressible and incompressible flows. For the sake of concise notation, however, the theory that follows is limited to a discussion of incompressible flows.

Filtering the continuity and momentum equations, one obtains

(4–8)

and

(4–9)

where is the stress tensor due to molecular viscosity defined by

(4–10)

and is the subgrid-scale stress defined by

(4–11)

Filtering the energy equation, one obtains:

(4–12)

where and are the sensible enthalpy and thermal conductivity, respectively.

The subgrid enthalpy flux term in the Equation 4–12 is approximated using the gradient hypothesis:

(4–13)

where is a subgrid viscosity, and is a subgrid Prandtl number equal to 0.85.