The 
-
-
model is considered to be a three-equation eddy-viscosity type, which includes
transport equations for turbulent kinetic energy (
), laminar kinetic energy (
), and the inverse turbulent time scale (
)
 | (4–133) |
 | (4–134) |
 | (4–135) |
The inclusion of the turbulent and laminar fluctuations on the mean flow and energy equations via the eddy viscosity and total thermal diffusivity is as follows:
 | (4–136) |
 | (4–137) |
The effective length is defined as
 | (4–138) |
where 
is the turbulent length scale and is defined by
 | (4–139) |
and the small scale energy is defined by
 | (4–140) |
 | (4–141) |
 | (4–142) |
The large scale energy is given by
 | (4–143) |
Note that the sum of Equation 4–140 and Equation 4–143 yields the turbulent kinetic energy 
.
The turbulence production term generated by turbulent fluctuations is given by
 | (4–144) |
where the small-scale turbulent viscosity is 
 | (4–145) |
and
 | (4–146) |
 | (4–147) |
A damping function defining the turbulent production due to intermittency is given by
 | (4–148) |
 | (4–149) |
In Equation 4–134, 
is the production of laminar kinetic energy by large scale turbulent
fluctuations, such that
 | (4–150) |
The large-scale turbulent viscosity 
is modeled as
 | (4–151) |
where
 | (4–152) |
The limit in Equation 4–151 binds the realizability such that it is
not violated in the two-dimensional developing boundary layer. The time-scale-based damping
function 
is
 | (4–153) |
where 
from Equation 4–152 is
 | (4–154) |
 | (4–155) |
Near-wall dissipation is given by
 | (4–156) |
 | (4–157) |
In Equation 4–133 – Equation 4–135, 
represents the averaged effect of the breakdown of streamwise fluctuations into
turbulence during bypass transition:
 | (4–158) |
, which is the threshold function controls the bypass transition process:
 | (4–159) |
 | (4–160) |
The breakdown to turbulence due to instabilities is considered to be a natural transition production term, given by
 | (4–161) |
 | (4–162) |
 | (4–163) |
The use of 
as the scale-determining variable can lead to a reduced intermittency effect in
the outer region of a turbulent boundary layer, and consequently an elimination of the wake
region in the velocity profile. From Equation 4–135, the following
damping is defined as
 | (4–164) |
The total eddy viscosity and eddy diffusivity included in Equation 4–136 and Equation 4–137 are given by
 | (4–165) |
 | (4–166) |
The turbulent scalar diffusivity in Equation 4–133 – Equation 4–135 is defined as
 | (4–167) |
 | (4–168) |
A compressibility effects option, similar to the one in the 
-
model (Effects of Compressibility on Turbulence in the k-ε Models) is available for the
-
-
model. By default, this compressibility effects option is turned off. For
details see, Model Enhancements in the Fluent User's Guide.
