The standard 
-
model in Ansys Fluent is based on a 
-
model proposed by Wilcox in [708], which incorporates
modifications for low-Reynolds number effects, compressibility, and shear flow spreading. One of
the weak points of the 1998 Wilcox model is the sensitivity of the solutions to values for k and
outside the shear layer (freestream sensitivity)., which can have a
significant effect on the solution, especially for free shear flows [429].
There is a newer version of the model (Wilcox 2006 k-ω model [709]), which did also not fully resolve the freestream sensitivity as
shown in [429] .
The standard 
-
model is an empirical model based on model transport equations for the
turbulence kinetic energy (
) and the specific dissipation rate (
), which can also be thought of as the ratio of 
to 
[708].
As the 
-
model has been modified over the years, production terms have been added to
both the 
and 
equations, which have improved the accuracy of the model for predicting free
shear flows.
The turbulence kinetic energy, 
, and the specific dissipation rate, 
, are obtained from the following transport equations:
 | (4–71) |
and
 | (4–72) |
In these equations, 
represents the generation of turbulence kinetic energy due to mean velocity
gradients. 
represents the generation of 
. 
and 
represent the effective diffusivity of 
and 
, respectively. 
and 
represent the dissipation of 
and 
due to turbulence. All of the above terms are calculated as described below.
and 
are user-defined source terms. 
and 
account for buoyancy terms as described in Effects of Buoyancy on Turbulence in the k-ω Models.
The effective diffusivities for the 
-
model are given by
 | (4–73) |
where 
and 
are the turbulent Prandtl numbers for 
and 
, respectively. The turbulent viscosity, 
, is computed by combining 
and 
as follows:
 | (4–74) |
The term 
represents the production of turbulence kinetic energy. From the exact
equation for the transport of 
, this term may be defined as
 | (4–80) |
To evaluate 
in a manner consistent with the Boussinesq hypothesis,
 | (4–81) |
where 
is the modulus of the mean rate-of-strain tensor, defined in the same way as
for the 
-
model (see Equation 4–62).
The production of 
is given by
 | (4–82) |
where 
is given by Equation 4–80.
The coefficient 
is given by
 | (4–83) |
where 
= 2.95. 
and 
are given by Equation 4–75 and Equation 4–76, respectively.
Note that in the high-Reynolds number form of the 
-
model, 
.
The dissipation of 
is given by
 | (4–84) |
where
 | (4–85) |
where
 | (4–86) |
and
 | (4–87) |
 | (4–88) |
 | (4–89) |
 | (4–90) |
 | (4–91) |
where 
is given by Equation 4–76.
The dissipation of 
is given by
 | (4–92) |
where
 | (4–93) |
 | (4–94) |
 | (4–95) |
The strain rate tensor, 
is defined in Equation 4–25. Also,
 | (4–96) |
and 
are defined by Equation 4–88 and Equation 4–97, respectively.
The compressibility function, 
, is given by
 | (4–97) |
where
 | (4–98) |
 | (4–99) |
 | (4–100) |
Note that, in the high-Reynolds number form of the 
-
model, 
. In the incompressible form, 
.
Note: The compressibility effects have been calibrated for a very limited number of free shear flow experiments, and it is not recommended for general use. It is disabled by default. For details, see Model Enhancements in the Fluent User's Guide.
