The subgrid-scale (SGS) stresses resulting from the filtering operation are unknown, and require modeling. The subgrid-scale turbulence models in Ansys Fluent employ the Boussinesq hypothesis [247] as in the RANS models, computing subgrid-scale turbulent stresses from
 | (4–298) |
where 
is the subgrid-scale turbulent viscosity. The isotropic part of the
subgrid-scale stresses 
is not modeled, but added to the filtered static pressure term. 
is the rate-of-strain tensor for the resolved scale defined by
 | (4–299) |
For compressible flows, it is convenient to introduce the density-weighted (or Favre) filtering operator:
 | (4–300) |
The Favre Filtered Navier-Stokes equation takes the same form as Equation 4–9. The compressible form of the subgrid stress tensor is defined as:
 | (4–301) |
This term is split into its isotropic and deviatoric parts
 | (4–302) |
The deviatoric part of the subgrid-scale stress tensor is modeled using the compressible form of the Smagorinsky model:
 | (4–303) |
As for incompressible flows, the term involving 
can be added to the filtered pressure or simply neglected [165]. Indeed, this term can be re-written as 
where 
is the subgrid Mach number. This subgrid Mach number can be expected to be
small when the turbulent Mach number of the flow is small.
Ansys Fluent offers four models for 
: the Smagorinsky-Lilly model, the dynamic Smagorinsky-Lilly model, the WALE
model, and the dynamic kinetic energy subgrid-scale model.
The subgrid-scale turbulent flux of a scalar, 
, is modeled using a subgrid-scale turbulent Prandtl number by
 | (4–304) |
where 
is the subgrid-scale flux.
The compressible subgrid enthalpy flux term is modeled in the same manner:
 | (4–305) |
where 
is the sensible enthalpy, 
is a subgrid viscosity, and 
is a subgrid Prandtl number equal to 0.85.
In the dynamic models, the subgrid-scale turbulent Prandtl number or Schmidt number is obtained by applying the dynamic procedure originally proposed by Germano [199] to the subgrid-scale flux.
This simple model was first proposed by Smagorinsky [603]. In the Smagorinsky-Lilly model, the eddy-viscosity is modeled by
 | (4–306) |
where 
is the mixing length for subgrid scales and 
. In Ansys Fluent, 
is computed using
 | (4–307) |
where 
is the von Kármán constant, 
is the distance to the closest wall, 
is the Smagorinsky constant, and 
is the local grid scale. In Ansys Fluent, 
is computed according to the volume of the computational cell using
 | (4–308) |
Lilly derived a value of 0.23 for 
for homogeneous isotropic turbulence in the inertial subrange. However, this
value was found to cause excessive damping of large-scale fluctuations in the presence of mean
shear and in transitional flows as near solid boundary, and has to be reduced in such regions.
In short, 
is not a universal constant, which is the most serious shortcoming of this
simple model. Nonetheless, a 
value of around 0.1 has been found to yield the best results for a wide range
of flows, and is the default value in Ansys Fluent.
Germano et al. [199] and subsequently Lilly [376] conceived a procedure in which the Smagorinsky model constant,
, is dynamically computed based on the information provided by the resolved
scales of motion. The dynamic procedure therefore obviates the need for users to specify the
model constant 
in advance.
The concept of the dynamic procedure is to apply a second filter (called the test filter)
to the equations of motion. The new filter width 
is equal to twice the grid filter width 
. Both filters produce a resolved flow field. The difference between the two
resolved fields is the contribution of the small scales whose size is in between the grid filter
and the test filter. The information related to these scales is used to compute the model
constant. In Ansys Fluent, the variable density formulation of the model is considered as explained
below.
At the test filtered field level, the SGS stress tensor can be expressed as:
 | (4–309) |
Both 
and 
are modeled in the same way with the Smagorinsky-Lilly model, assuming scale
similarity:
 | (4–310) |
 | (4–311) |
In Equation 4–310 and Equation 4–311, the coefficient C is assumed to be the same and
independent of the filtering process (note that per Equation 4–307, 
). The grid filtered SGS and the test-filtered SGS are related by the Germano
identity [199] such that:
 | (4–312) |
Where 
is computable from the resolved large eddy field. Substituting the grid-filter
Smagorinsky-Lilly model and Equation 4–311 into Equation 4–312, the following expressions can be derived to solve for C
with the contraction obtained from the least square analysis of Lilly (1992).
 | (4–313) |
With
 | (4–314) |
More details of the model implementation in Ansys Fluent and its validation can be found in [307].
The 
obtained using the dynamic Smagorinsky-Lilly model varies in time and space
over a fairly wide range. To avoid numerical instability, both the numerator and the denominator
in Equation 4–313 are locally averaged (or filtered) using the
test-filter. In Ansys Fluent, 
is also clipped at zero and 0.23 by default.
In the WALE model [478], the eddy viscosity is modeled by
 | (4–315) |
where 
and 
in the WALE model are defined, respectively, as
 | (4–316) |
 | (4–317) |
where 
is the von Kármán constant. The published value for the WALE
constant, 
, is 0.5; however, intensive validation during a European Union research
project involving the original model developers has shown consistently superior results in
Ansys Fluent with 
= 0.325, and so it is this value that is used as the default setting. The rest
of the notation is the same as for the Smagorinsky-Lilly model. With this spatial operator, the
WALE model is designed to return the correct wall asymptotic (
) behavior for wall bounded flows.
Another advantage of the WALE model is that it returns a zero turbulent viscosity for laminar shear flows. This allows the correct treatment of laminar zones in the domain. In contrast, the Smagorinsky-Lilly model produces nonzero turbulent viscosity. The WALE model is therefore preferable compared to the Smagorinsky-Lilly model.
While widely used in the academic community, LES has had very limited impact on industrial simulations. The reason lies in the excessively high resolution requirements for wall boundary layers. Near the wall, the largest scales in the turbulent spectrum are nevertheless geometrically very small and require a very fine grid and a small time step. In addition, unlike RANS, the grid cannot only be refined in the wall normal direction, but also must resolve turbulence in the wall parallel plane. This can only be achieved for flows at very low Reynolds number and on very small geometric scales (the extent of the LES domain cannot be much larger than 10-100 times the boundary layer thickness parallel to the wall). For this reason, using LES is only recommended for flows where wall boundary layers are not relevant and need not be resolved, or for flows where the boundary layers are laminar due to the low Reynolds number.
However, there are only very few such flows and other approaches need to be employed. A promising approach to overcome the Reynolds number scaling limitations of LES is the algebraic Wall-Modeled LES (WMLES) approach [592]. In WMLES, the RANS portion of the model is only activated in the inner part of the logarithmic layer and the outer part of the boundary layer is covered by a modified LES formulation. Since the inner portion of the boundary layer is responsible for the Reynolds number dependency of the LES model, the WMLES approach can be applied at the same grid resolution to an ever increasing Reynolds number for channel flow simulations.
Note that for wall boundary layers, the Reynolds number scaling is not entirely avoided, as the thickness of the boundary layer declines relative to body dimensions with increasing Reynolds number. Assuming a certain number of grid nodes per ‘boundary layer volume’, the overall grid spacing will decrease and the overall number of cells will increase with the Reynolds number.
Another interesting option, and possibly the best use of the WMLES option, is when it is combined with the Embedded LES (ELES) implementation in Ansys Fluent. ELES limits the LES zone to small regions of the domain and covers the less critical areas with RANS. WMLES extends this functionality to high Reynolds numbers in the LES domain. To apply this option, refer to Setting Up the Large Eddy Simulation Model and Setting Up the Embedded Large Eddy Simulation (ELES) Model in the User’s Guide.
The original Algebraic WMLES formulation was proposed in the works of Shur et.al. [592]. It combines a mixing length model with a modified Smagorinsky model [603] and with the wall-damping function of Piomelli [520].
In the Shur et al. model [592], the eddy viscosity is calculated with the use of a hybrid length scale:
 | (4–318) |
where 
is the wall distance, 
is the strain rate, 
, and 
are constants, and 
is the normal to the wall inner scaling. The LES model is based on a modified
grid scale to account for the grid anisotropies in wall-modeled flows:
 | (4–319) |
Here, 
is the maximum edge length for a rectilinear hexahedral cell (for other cell
types and/or conditions an extension of this concept is used). 
is the wall-normal grid spacing, and 
is a constant.
The above model has been optimized for use in Ansys Fluent but remains within the spirit of the original model.
The main advantage of the WMLES formulation is the improved Reynolds number scaling. The classical resolution requirements for wall resolved LES is
 | (4–320) |
where
 | (4–321) |
and 
is the number of cells across the boundary layer, h,
(or half channel height h) and ν is the kinematic viscosity. The wall
friction velocity is defined as
 | (4–322) |
where 
is the wall shear stress and 
is the density.
The resolution requirement for WMLES is
 | (4–323) |
Again with 
for a channel (where H is the channel height), or
(where 
is the boundary layer thickness).
Considering a periodic channel (in the span and streamwise direction) with the geometry dimensions:
 | (4–324) |
you obtain the following cell numbers (
total number of cells in channel) for the wall-resolved LES and
WMLES:
Wall-Resolved LES:
WMLES:
Table 4.2: WMLES Grid Size as a Function of Reynolds Number
 | 500 |  |  |  |
 | 5.0e5 | 5.0e5 | 5.0e5 | 5.0e5 |
| Ratio LES/WMLES | 1 | 4 |  |  |
| Ratio CPU LES/WMLES (CFL=0.3) | 1 |  |  |  |
The Reynolds number is based on the friction velocity:
 | (4–325) |
From the above tables, it can be seen that the quadratic Reynolds number dependence of the Wall-Resolved LES has been avoided by WMLES and that the CPU effort is substantially reduced.
However, it is still important to remember that WMLES is much more computationally
expensive than RANS. Considering a wall boundary layer flow, with WMLES one must cover every
boundary layer volume 
with 10 x (30-40) x 20 = 6000-8000 cells in the stream-wise, normal, and
span-wise direction. RANS can be estimated to 1 x (30-40) x 1 =30-40 cells. In addition, RANS
can be computed steady-state whereas WMLES requires an unsteady simulation with CFL~0.3.
One of the deficiencies of the WMLES approach is that the model does not provide zero eddy-viscosity for flows with constant shear when you use a modified Smagorinsky model in the LES zone. For this reason, the WMLES model does not allow the computation of transitional effects, and can produce overly large eddy-viscosities in separating shear layers (like from a backstep).
A way to enhance the WMLES formulation given in Equation 4–318
is to compute the LES portion of the model using the difference 
instead of 
, where 
is the strain rate and 
is the vorticity magnitude. This enhancement is referred to as the WMLES
-
formulation.
The original and dynamic Smagorinsky-Lilly models, discussed previously, are essentially algebraic models in which subgrid-scale stresses are parameterized using the resolved velocity scales. The underlying assumption is the local equilibrium between the transferred energy through the grid-filter scale and the dissipation of kinetic energy at small subgrid scales. The subgrid-scale turbulence can be better modeled by accounting for the transport of the subgrid-scale turbulence kinetic energy.
The dynamic subgrid-scale kinetic energy model in Ansys Fluent replicates the model proposed by Kim and Menon [310].
The subgrid-scale kinetic energy is defined as
 | (4–326) |
which is obtained by contracting the subgrid-scale stress in Equation 4–11.
The subgrid-scale eddy viscosity, 
, is computed using 
as
 | (4–327) |
where 
is the filter-size computed from 
.
The subgrid-scale stress can then be written as
 | (4–328) |
is obtained by solving its transport equation
 | (4–329) |
In the above equations, the model constants, 
and 
, are determined dynamically [310]. 
is hardwired to 1.0. The details of the implementation of this model in
Ansys Fluent and its validation is given by Kim [307].