The realizable

model [
330] is a relatively recent development
and differs from the standard

model in two important ways:
The realizable

model contains a new formulation for the turbulent
viscosity.
A new transport equation for the dissipation rate,
, has been
derived from an exact equation for the transport of the meansquare vorticity
fluctuation.
The term "realizable'' means that the model satisfies certain mathematical
constraints on the Reynolds stresses, consistent with the physics of turbulent
flows. Neither the standard

model nor the RNG

model is realizable.
An immediate benefit of the realizable

model is that it more accurately
predicts the spreading rate of both planar and round jets. It is also likely to
provide superior performance for flows involving rotation, boundary layers
under strong adverse pressure gradients, separation, and recirculation.
To understand the mathematics behind the realizable

model,
consider combining the Boussinesq relationship (Equation
12.25) and the eddy
viscosity definition (Equation
12.43) to obtain the following expression for
the normal Reynolds stress in an incompressible strained mean flow:
(12.413)
Using Equation
12.43 for
, one obtains the result
that the normal stress,
, which by definition is a positive
quantity, becomes negative, i.e., "nonrealizable'', when the strain is large
enough to satisfy
(12.414)
Similarly, it can also be shown that the Schwarz inequality for shear stresses
(
;
no summation over
and
) can be violated when the mean strain
rate is large. The most straightforward way to ensure the realizability
(positivity of normal stresses and Schwarz inequality for shear stresses) is to
make
variable by sensitizing it to the mean flow (mean deformation)
and the turbulence (
,
). The notion of variable
is
suggested by many modelers including Reynolds [
303], and is well
substantiated by experimental evidence. For example,
is found to be
around 0.09 in the inertial sublayer of equilibrium boundary layers, and 0.05 in
a strong homogeneous shear flow.
Both the realizable and RNG

models have shown substantial improvements
over the standard

model where the flow features include strong streamline
curvature, vortices, and rotation.
Since the model is still
relatively new, it is not clear in exactly which instances the realizable

model consistently outperforms the RNG model. However, initial studies have
shown that the realizable model provides the best performance of all the

model versions for several validations of separated flows and flows with
complex secondary flow features.
One of the weaknesses of the standard

model or other traditional

models
lies with the modeled equation for the dissipation rate (
). The
wellknown roundjet anomaly (named based on the finding that the spreading rate
in planar jets is predicted reasonably well, but prediction of the spreading
rate for axisymmetric jets is unexpectedly poor) is considered to be mainly due
to the modeled dissipation equation.
The realizable

model proposed by Shih et al. [
330] was intended to
address these deficiencies of traditional

models by adopting the following:
A new eddyviscosity formula involving a variable
originally
proposed by Reynolds [
303].
A new model equation for dissipation (
) based on
the dynamic equation of the meansquare vorticity fluctuation.
One limitation of the realizable

model is that it produces nonphysical
turbulent viscosities in situations when the computational domain contains
both rotating and stationary fluid zones (e.g., multiple reference frames,
rotating sliding meshes). This is due to the fact that the realizable

model includes the effects of mean rotation in the definition of the turbulent
viscosity (see Equations
12.417
12.419). This extra
rotation effect has been tested on single rotating reference frame systems and
showed superior behavior over the standard

model. However, due to the
nature of this modification, its application to multiple reference frame
systems should be taken with some caution. See Section
12.4.3 for
information about how to include or exclude this term from the model.
Transport Equations for the Realizable

Model
The modeled transport equations for
and
in the
realizable

model are
(12.415)
and
(12.416)
where
In these equations,
represents the generation of turbulence kinetic energy
due to the mean velocity gradients, calculated as described in
Section
12.4.4.
is the generation of turbulence kinetic energy due
to buoyancy, calculated as described in Section
12.4.5.
represents the contribution of the fluctuating dilatation in compressible
turbulence to the overall dissipation rate, calculated as described in
Section
12.4.6.
and
are constants.
and
are the turbulent Prandtl numbers for
and
, respectively.
and
are userdefined source
terms.
Note that the
equation (Equation
12.415) is the same as
that in the standard

model (Equation
12.41) and the RNG

model (Equation
12.44), except for the model constants. However, the
form of the
equation is quite different from those in the standard
and RNGbased

models (Equations
12.42 and
12.45). One of
the noteworthy features is that the production term in the
equation
(the second term on the righthand side of Equation
12.416) does
not involve the production of
; i.e., it does not contain the same
term
as the other

models. It is believed that the present form better
represents the spectral energy transfer. Another desirable feature is that the
destruction term (the next to last term on the righthand side of
Equation
12.416) does not have any singularity; i.e., its
denominator never vanishes, even if
vanishes or becomes smaller than zero.
This feature is contrasted with traditional

models, which have a
singularity due to
in the denominator.
This model has been extensively validated for a wide range of
flows [
183,
330], including rotating homogeneous shear flows,
free flows including jets and mixing layers, channel and boundary layer flows,
and separated flows. For all these cases, the performance of the model has been
found to be substantially better than that of the standard

model.
Especially noteworthy is the fact that the realizable

model resolves the
roundjet anomaly; i.e., it predicts the spreading rate for axisymmetric jets as
well as that for planar jets.
Modeling the Turbulent Viscosity
As in other

models, the eddy viscosity is computed from
(12.417)
The difference between the realizable

model and the standard and RNG

models is that
is no longer constant. It is computed from
(12.418)
where
(12.419)
and
where
is the mean rateofrotation tensor viewed in a
rotating reference frame with the angular velocity
. The model
constants
and
are given by
where
It can be seen that
is a function of the mean strain and rotation
rates, the angular velocity of the system rotation, and the turbulence fields
(
and
).
in Equation
12.417 can be shown to
recover the standard value of 0.09 for an inertial sublayer in an equilibrium
boundary layer.
In
FLUENT, the term
is, by default, not included
in the calculation of
. This is an extra rotation term
that is not compatible with cases involving sliding meshes or multiple reference
frames. If you want to include this term in the model, you can enable it by
using the
define/models/viscous/turbulenceexpert/rkecmurotationterm?
text command and entering
yes at the
prompt.
Model Constants
The model constants
,
, and
have been
established to ensure that the model performs well for certain canonical flows.
The model constants are