The RNG

model was derived using a rigorous statistical technique (called
renormalization group theory). It is similar in form to the standard

model,
but includes the following refinements:
The RNG model has an additional term in its
equation that
significantly improves the accuracy for rapidly strained flows.
The effect of swirl on turbulence
is included in the RNG model, enhancing accuracy for swirling flows.
The RNG theory provides an analytical formula for turbulent Prandtl
numbers, while the standard

model uses userspecified, constant values.
While the standard

model is a highReynoldsnumber model, the RNG
theory provides an analyticallyderived differential formula for effective
viscosity that accounts for lowReynoldsnumber effects. Effective use of this
feature does, however, depend on an appropriate treatment of the nearwall
region.
These features make the RNG

model more accurate and reliable for a wider
class of flows than the standard

model.
The RNGbased

turbulence model is derived from the instantaneous
NavierStokes equations, using a mathematical technique called "renormalization
group'' (RNG) methods
. The analytical
derivation results in a model with constants different from those in the
standard

model, and additional terms and functions in the transport
equations for
and
. A more comprehensive description of RNG
theory and its application to turbulence can be found in [
59].
Transport Equations for the RNG

Model
The RNG

model has a similar form to the standard

model:
(12.44)
and
(12.45)
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. The quantities
and
are
the inverse effective Prandtl numbers for
and
, respectively.
and
are userdefined source terms.
Modeling the Effective Viscosity
The scale elimination procedure in RNG theory results in a differential equation
for turbulent viscosity:
(12.46)
where
Equation
12.46 is integrated to obtain an accurate description of how
the effective turbulent transport varies with the effective Reynolds number (or
eddy scale), allowing the model to better handle
lowReynoldsnumber
and nearwall
flows
.
In the highReynoldsnumber limit, Equation
12.46 gives
(12.47)
with
, derived using RNG theory. It is interesting to note
that this value of
is very close to the empiricallydetermined value
of 0.09 used in the standard

model.
In
FLUENT, by default, the effective viscosity is computed using the
highReynoldsnumber form in Equation
12.47. However, there is an
option available that allows you to use the differential relation given in
Equation
12.46 when you need to include lowReynoldsnumber effects.
RNG Swirl Modification
Turbulence, in general, is affected by rotation or swirl in the mean flow. The
RNG model in
FLUENT provides an option to account for the effects of swirl
or rotation by modifying the turbulent viscosity appropriately. The
modification takes the following functional form:
(12.48)
where
is the value of turbulent viscosity calculated without the
swirl modification using either Equation
12.46 or
Equation
12.47.
is a characteristic swirl number evaluated
within
FLUENT, and
is a swirl constant that assumes different
values depending on whether the flow is swirldominated or only mildly swirling.
This swirl modification always takes effect for axisymmetric, swirling flows and
threedimensional flows when the RNG model is selected. For mildly swirling
flows (the default in
FLUENT),
is set to 0.07. For strongly swirling flows, however, a higher value of
can be used.
Calculating the Inverse Effective Prandtl Numbers
The inverse effective Prandtl numbers,
and
, are
computed using the following formula derived analytically by the RNG theory:
(12.49)
where
. In the highReynoldsnumber limit
(
),
.
The
Term in the
Equation
The main difference between the RNG and standard

models lies in the
additional term in the
equation given by
(12.410)
where
,
,
.
The effects of this term in the RNG
equation can be seen more clearly
by rearranging Equation
12.45. Using Equation
12.410, the third and fourth
terms on the righthand side of Equation
12.45 can be merged, and the
resulting
equation can be rewritten as
(12.411)
where
is given by
(12.412)
In regions where
, the
term makes a positive contribution,
and
becomes larger than
. In the logarithmic
layer, for instance, it can be shown that
, giving
, which is close in magnitude to the value of
in the standard

model (1.92). As a result, for weakly to
moderately strained flows, the RNG model tends to give results largely
comparable to the standard

model.
In regions of large strain rate (
), however, the
term makes a
negative contribution, making the value of
less than
. In comparison with the standard

model, the smaller
destruction of
augments
, reducing
and, eventually, the
effective viscosity. As a result, in rapidly strained flows, the RNG model
yields a lower turbulent viscosity than the standard

model.
Thus, the RNG model is more responsive to the effects of rapid strain and
streamline curvature than the standard

model, which explains the superior
performance of the RNG model for certain classes of flows.
Model Constants
The model constants
and
in Equation
12.45
have values derived analytically by the RNG theory. These values, used
by default in
FLUENT, are