7. Turbulence Flow Models
7.1 Introduction
Turbulent flows are characterized by fluctuating velocity fields. These
fluctuations mix transported quantities such as momentum, energy, and
species concentration, and cause the transported quantities to fluctuate as
well. Since these fluctuations can be of a small scale and high frequency,
they are too computationally expensive to simulate directly in practical
engineering calculations. Instead, the instantaneous (exact) governing
equations can be time-averaged, ensemble-averaged, or otherwise
manipulated to remove the small scales, resulting in a modified set of
equations that are computationally less expensive to solve. However, the
modified equations contain additional unknown variables, and turbulence
models are needed to determine these variables in terms of known
quantities. FLUENT provides the following choices of turbulence models:
 Spalart-Allmaras model
 k-ε models: Standard k-ε model, RNG k-ε model, Realizable k-ε
model
 k-ω models: Standard k-ω model, Shear-stress transport (SST) k-ω
model
 Reynolds stress model (RSM)
 Large eddy simulation (LES) model
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7.2 Choosing a Turbulence Model
It is an unfortunate fact that no single turbulence model is universally
accepted as being superior for all classes of problems. The choice of
turbulence model will depend on considerations such as the physics
encompassed in the flow, the established practice for a specific class of
problem, the level of accuracy required, the available computational
resources, and the amount of time available for the simulation. To make
the most appropriate choice of model for your application, you need to
understand the capabilities and limitations of the various options.
The purpose of this section is to give an overview of issues related to the
turbulence models provided in FLUENT. The computational effort and
cost in terms of CPU time and memory of the individual models are
discussed. While it is impossible to state categorically which model is best
for a specific application, general guidelines are presented to help you
choose the appropriate turbulence model for the flow you want to model.
7.2.1 Reynolds-Averaged Approach vs. LES
A complete time-dependent solution of the exact Navier-Stokes equations
for high-Reynolds-number turbulent flows in complex geometries is
unlikely to be attainable for some time to come. Two alternative methods
can be employed to transform the Navier-Stokes equations in such a way
that the small-scale turbulent fluctuations do not have to be directly
simulated: Reynolds averaging and filtering. Both methods introduce
additional terms in the governing equations that need to be modeled in
order to achieve "closure". (Closure implies that there are a sufficient
number of equations for all the unknowns.)
The Reynolds-averaged Navier-Stokes (RANS) equations represent
transport equations for the mean flow quantities only, with all the scales of
the turbulence being modeled. The approach of permitting a solution for
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the mean flow variables greatly reduces the computational effort. If the
mean flow is steady, the governing equations will not contain time
derivatives and a steady-state solution can be obtained economically. A
computational advantage is seen even in transient situations since the time
step will be determined by the global unsteadiness in the mean flow rather
than by the turbulence. The Reynolds-averaged approach is generally
adopted for practical engineering calculations and uses models such as
Spalart-Allmaras, k-ε and its variants, k-ω and its variants, and the RSM.
LES provides an alternative approach in which the large eddies are
computed in a time-dependent simulation that uses a set of "filtered"
equations. Filtering is essentially a manipulation of the exact NavierStokes equations to remove only the eddies that are smaller than the size
of the filter, which is usually taken as the mesh size. Like Reynolds
averaging, the filtering process creates additional unknown terms that must
be modeled in order to achieve closure. Statistics of the mean flow
quantities, which are generally of most engineering interest, are gathered
during the time-dependent simulation. The attraction of LES is that, by
modeling less of the turbulence (and solving more), the error induced by
the turbulence model will be reduced. One might also argue that it ought
to be easier to find a "universal" model for the small scales, which tend to
be more isotropic and less affected by the macroscopic flow features than
the large eddies. It should, however, be stressed that the application of LES
to industrial fluid simulations is in its infancy. As highlighted in a recent
review publication, typical applications to date have been for simple
geometries. This is mainly because of the large computer resources
required to resolve the energy-containing turbulent eddies. Most successful
LES has been done using high-order spatial discretization, with great care
being taken to resolve all scales larger than the inertial subrange. The
degradation of accuracy in the mean flow quantities with poorly resolved
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LES is not well documented. In addition, the use of wall functions with
LES is an approximation that requires further validation.
As a general guideline, therefore, it is recommended that the conventional
turbulence models employing the Reynolds-averaged approach be used for
practical calculations. The LES approach described further in Section 7,
has been made available for you to try if you have the computational
resources and are willing to invest the effort. The rest of this section will
deal with the choice of models using the Reynolds-averaged approach.
7.2.2 Reynolds (Ensemble) Averaging
In Reynolds averaging, the solution variables in the instantaneous (exact)
Navier-Stokes equations are decomposed into the mean (ensemble
averaged or time-averaged) and fluctuating components. For the velocity
components:
𝑢𝑖 = 𝑢̅ 𝑖 + 𝑢́ 𝑖
⋯ (7.1)
where 𝑢̅i and 𝑢́ i are the mean and fluctuating velocity components (i =1;
2; 3). Likewise, for pressure and other scalar quantities:
̅ + 𝜙́
𝜙 = 𝜙
⋯ (7.2)
where ϕ denotes a scalar such as pressure, energy, or species concentration.
Substituting expressions of this form for the flow variables into the
instantaneous continuity and momentum equations and taking a time (or
ensemble) average (and dropping the overbar on the mean velocity, 𝑢̅)
yields the ensemble-averaged momentum equations. They can be written
in Cartesian tensor form as:
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𝜕𝜌
𝜕
(𝜌𝑢𝑖 ) = 0
+
𝜕𝑡
𝜕𝑥𝑖
𝜕
𝜕𝑡
(𝜌𝑢𝑖 ) +
𝜕
𝜕𝑥𝑗
𝜕
𝜕𝑥𝑖
⋯ (7.3)
𝜕𝑝
𝜕
𝜕𝑢
𝜕𝑢𝑗
2
𝜕𝑢
(𝜌𝑢𝑖 𝑢𝑗 ) = − 𝜕𝑥 + 𝜕𝑥 [𝜇 (𝜕𝑥 𝑖 + 𝜕𝑥 − 3 𝛿𝑖𝑗 𝜕𝑥𝑙 )] +
𝑖
𝑗
𝑗
𝑖
̅̅̅̅̅
́𝑖 𝑢𝑗́ )
(−𝜌𝑢
𝑙
⋯ (7.4)
Equations (3 and 4) are called Reynolds-averaged Navier-Stokes (RANS)
equations. They have the same general form as the instantaneous NavierStokes equations, with the velocities and other solution variables now
representing ensemble-averaged (or time-averaged) values. Additional
terms now appear that represent the effects of turbulence. This Reynolds
̅̅̅̅̅
stresses, (−𝜌𝑢
́𝑖 𝑢𝑗́ ), must be modeled in order to close Equation 4. For
variable-density flows, Equations 3 and 4 can be interpreted as Favreaveraged Navier-Stokes equations, with the velocities representing massaveraged values. As such, Equations 3 and 4 can be applied to densityvarying flows.
7.2.3 Boussinesq Approach vs. Reynolds Stress Transport Models
The Reynolds-averaged approach to turbulence modeling requires that the
Reynolds stresses in Equation 4 be appropriately modeled. A common
method employs the Boussinesq hypothesis to relate the Reynolds stresses
to the mean velocity gradients:
̅̅̅̅̅
−𝜌𝑢
́𝑖 𝑢𝑗́ = 𝜇𝑡 (
𝜕𝑢𝑗
𝜕𝑢𝑖 𝜕𝑢𝑗
2
+
) − (𝜌𝑘 + 𝜇𝑡
) 𝛿𝑖𝑗
𝜕𝑥𝑗 𝜕𝑥𝑖
3
𝜕𝑥𝑖
5
⋯ (7.5)
The Boussinesq hypothesis is used in the Spalart-Allmaras model, the k-ε
models, and the k-ω models. The advantage of this approach is the
relatively low computational cost associated with the computation of the
turbulent viscosity, µt. In the case of the Spalart-Allmaras model, only one
additional transport equation (representing turbulent viscosity) is solved.
In the case of the k-ε and k-ω models, two additional transport equations
(for the turbulence kinetic energy, k, and either the turbulence dissipation
rate, ε, or the specific dissipation rate, ω) are solved, and µt is computed as
a function of k and ε. The disadvantage of the Boussinesq hypothesis as
presented is that it assumes µt is an isotropic scalar quantity, which is not
strictly true. The alternative approach, embodied in the RSM, is to solve
transport equations for each of the terms in the Reynolds stress tensor. An
additional scale-determining equation (normally for ε) is also required.
This means that five additional transport equations are required in 2D flows
and seven additional transport equations must be solved in 3D.
In many cases, models based on the Boussinesq hypothesis perform very
well, and the additional computational expense of the Reynolds stress
model is not justified. However, the RSM is clearly superior for situations
in which the anisotropy of turbulence has a dominant effect on the mean
flow. Such cases include highly swirling flows and stress-driven secondary
flows.
7.3 The Spalart-Allmaras Model
The Spalart-Allmaras model is a relatively simple one-equation model that
solves a modeled transport equation for the kinematic eddy (turbulent)
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viscosity. This embodies a relatively new class of one-equation models in
which it is not necessary to calculate a length scale related to the local shear
layer thickness. The Spalart-Allmaras model was designed specifically for
aerospace applications involving wall-bounded flows and has been shown
to give good results for boundary layers subjected to adverse pressure
gradients. It is also gaining popularity for turbomachinery applications.
In its original form, the Spalart-Allmaras model is effectively a lowReynolds number model, requiring the viscous- affected region of the
boundary layer to be properly resolved. In FLUENT, however, the SpalartAllmaras model has been implemented to use wall functions when the
mesh resolution is not sufficiently fine. This might make it the best choice
for relatively crude simulations on coarse meshes where accurate turbulent
flow computations are not critical. Furthermore, the near-wall gradients of
the transported variable in the model are much smaller than the gradients
of the transported variables in the k-ε or k-ω models. This might make the
model less sensitive to numerical error when non-layered meshes are used
near walls.
On a cautionary note, however, the Spalart-Allmaras model is still
relatively new, and no claim is made regarding its suitability to all types of
complex engineering flows. For instance, it cannot be relied on to predict
the decay of homogeneous, isotropic turbulence. Furthermore, oneequation models are often criticized for their inability to rapidly
accommodate changes in length scale, such as might be necessary when
the flow changes abruptly from a wall-bounded to free shear flow.
Turbulence quantities that can be reported for the Spalart-Allmaras model
are as follows: Modified Turbulent Viscosity, Turbulent Viscosity,
Effective Viscosity, Turbulent Viscosity Ratio, Effective Thermal
Conductivity, Effective Prandtl Number, Wall Yplus
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7.4 k-ε Model
7.4.1 The Standard k-ε Model
The simplest "complete models" of turbulence are two-equation models in
which the solution of two separate transport equations allows the turbulent
velocity and length scales to be independently determined. The standard kε model in FLUENT falls within this class of turbulence model and has
become the workhorse of practical engineering flow calculations.
Robustness, economy, and reasonable accuracy for a wide range of
turbulent flows explain its popularity in industrial flow and heat transfer
simulations.
It is a semi-empirical model, and the derivation of the model equations
relies on phenomenological considerations and empiricism. As the
strengths and weaknesses of the standard k-ε model have become known,
improvements have been made to the model to improve its performance.
Two of these variants are available in FLUENT: the RNG k-ε model and
the realizable k-ε model.
7.4.2 The RNG k-ε Model
The RNG k-ε model was derived using a rigorous statistical technique
(called renormalization group theory). It is similar in form to the standard
k-ε model, but includes the following refinements:
 The RNG model has an additional term in its ε 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.
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 The RNG theory provides an analytical formula for turbulent Prandtl
numbers, while the standard k-ε model uses user-specified, constant
values.
 While the standard k-ε model is a high-Reynolds-number model, the
RNG theory provides an analytically-derived differential formula
for effective viscosity that accounts for low-Reynolds-number
effects. Effective use of this feature does, however, depend on
appropriate treatment of the near-wall region.
These features make the RNG k-ε model more accurate and reliable for a
wider class of flows than the standard k-ε model.
7.4.3 The Realizable k-ε Model
The realizable k-ε model is a relatively recent development and differs
from the standard k-ε model in two important ways:
 The realizable k-ε model contains a new formulation for the
turbulent viscosity.
 A new transport equation for the dissipation rate, ε, has been derived
from an exact equation for the transport of the mean-square 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 k-ε model nor the RNG k-ε model is
realizable.
An immediate benefit of the realizable k-ε 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.
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Both the realizable and RNG k-ε models have shown substantial
improvements over the standard k-ε 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 k-ε
model consistently outperforms the RNG model. However, initial studies
have shown that the realizable model provides the best performance of all
the k-ε model versions for several validations of separated flows and flows
with complex secondary flow features.
One limitation of the realizable k-ε model is that it produces non-physical
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 k-ε model
includes the effects of mean rotation in the definition of the turbulent
viscosity. This extra rotation effect has been tested on single rotating
reference frame systems and showed superior behavior over the standard
k-ε model. However, due to the nature of this modification, its application
to multiple reference frame systems should be taken with some caution.
Turbulence quantities that can be reported for the k-ε models are as follows:
Turbulent Kinetic Energy (k), Turbulence Intensity, Turbulent Dissipation
Rate (Epsilon), Production of k, Turbulent Viscosity, Effective Viscosity,
Turbulent Viscosity Ratio, Effective Thermal Conductivity, Effective
Prandtl Number, Wall Yplus, Wall Ystar
7.5 k-ω Model
7.5.1 The Standard k-ω Model
The standard k-ω model in FLUENT is based on the Wilcox k-ω model,
which incorporates modifications for low-Reynolds-number effects,
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compressibility, and shear flow spreading. The Wilcox model predicts free
shear flow spreading rates that are in close agreement with measurements
for far wakes, mixing layers, and plane, round, and radial jets, and is thus
applicable to wall-bounded flows and free shear flows. A variation of the
standard k-ω model called the SST k-ω model is also available in
FLUENT, and is described in Section 5.2.
7.5.2 The Shear-Stress Transport (SST) k-ω Model
The shear-stress transport (SST) k-ω model was developed to effectively
blend the robust and accurate formulation of the k-ω model in the nearwall region with the free-stream independence of the k-ε model in the far
field. To achieve this, the k-ε model is converted into a k-ω formulation.
The SST k-ω model is similar to the standard k-ω model, but includes the
following refinements:
 The standard k-ω model and the transformed k-ε model are both
multiplied by a blending function and both models are added
together. The blending function is designed to be one in the near wall
region, which activates the standard k-ω model and zero away from
the surface, which activates the transformed k-ε model.
 The SST model incorporates a damped cross-diffusion derivative
the term in the ω equation.
 The definition of the turbulent viscosity is modified to account for
the transport of the turbulent shear stress.
 The modeling constants are different.
These features make the SST k-ω model more accurate and reliable for a
wider class of flows (e.g., adverse pressure gradient flows, airfoils,
transonic shock waves) than the standard k-ω model.
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Turbulence quantities that can be reported for the k-ω models are as
follows:
Turbulent Kinetic Energy (k),
Turbulence Intensity, Specific
Dissipation Rate (Omega), Production of k, Turbulent Viscosity,
Effective Viscosity, Turbulent Viscosity Ratio, Effective Thermal
Conductivity, Effective Prandtl Number, Wall Ystar, Wall Yplus
7.6 The Reynolds Stress Model (RSM)
The Reynolds stress model (RSM) is the most elaborate turbulence model
that FLUENT provides. Abandoning the isotropic eddy-viscosity
hypothesis, the RSM closes the Reynolds-averaged Navier-Stokes
equations by solving transport equations for the Reynolds stresses, together
with an equation for the dissipation rate. This means that four additional
transport equations are required in 2D flows and seven additional transport
equations must be solved in 3D.
Since the RSM accounts for the effects of streamline curvature, swirl,
rotation, and rapid changes in strain rate in a more rigorous manner than
one-equation and two-equation models, it has greater potential to give
accurate predictions for complex flows. However, the fidelity of RSM
predictions is still limited by the closure assumptions employed to model
various terms in the exact transport equations for the Reynolds stresses.
The modeling of the pressure-strain and dissipation-rate terms is
particularly challenging, and often considered to be responsible for
compromising the accuracy of RSM predictions.
The RSM might not always yield results that are clearly superior to the
simpler models in all classes of flows to warrant the additional
computational expense. However, use of the RSM is a must when the flow
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features of interest are the result of anisotropy in the Reynolds stresses.
Among the examples is cyclone flows, highly swirling flows in
combustors, rotating flow passages, and the stress-induced secondary
flows in ducts.
Turbulence quantities that can be reported for the RSM are as follows:
Turbulent Kinetic Energy (k), Turbulence Intensity, UU Reynolds Stress,
VV Reynolds Stress, WW Reynolds Stress, UV Reynolds Stress, VW
Reynolds Stress,
UW Reynolds Stress, Turbulent Dissipation Rate
(Epsilon), Production of k, Turbulent Viscosity, Effective Viscosity,
Turbulent Viscosity Ratio, Effective Thermal Conductivity, Effective
Prandtl Number Wall Yplus, Wall Ystar, Turbulent Reynolds Number (Ret)
7.7 Large eddy simulation (LES) model
Turbulent flows are characterized by eddies with a wide range of length
and time scales. The largest eddies are typically comparable in size to the
characteristic length of the mean flow. The smallest scales are responsible
for the dissipation of turbulence kinetic energy. It is theoretically possible
to directly resolve the whole spectrum of turbulent scales using an
approach known as direct numerical simulation (DNS). DNS is not,
however, feasible for practical engineering problems. To understand the
large computational cost of DNS, consider that the ratio of the large
(energy-containing) scales to the small (energy dissipating) scales are
⁄
proportional to Re3𝑡 4 , where Ret is the turbulent Reynolds number.
Therefore, to resolve all the scales, the mesh size in three dimensions will
⁄
be proportional to Re9𝑡 4 . Simple arithmetic shows that, for high Reynolds
numbers, the mesh sizes required for DNS are prohibitive. Adding to the
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computational cost is the fact that the simulation will be a transient one
with very small-time steps since the temporal resolution requirements are
governed by the dissipating scales, rather than the mean flow or the energycontaining eddies.
As explained in Section 7.2.1, the conventional approach to flow
simulations employs the solution of the Reynolds-averaged Navier-Stokes
(RANS) equations. In the RANS approach, all the turbulent motions are
modeled, resulting in significant savings in computational effort.
Conceptually, large eddy simulation (LES) is situated somewhere between
DNS and the RANS approach. Basically, large eddies are resolved directly
in LES, while small eddies are modeled. The rationale behind LES can be
summarized as follows:

Momentum, mass, energy, and other passive scalars are transported
mostly by large eddies.

Large eddies are more problem-dependent. They are dictated by the
geometries and boundary conditions of the flow involved.

Small eddies are less dependent on the geometry, tend to be more
isotropic, and are consequently more universal.
 The chance of finding a universal model is much higher when only
small eddies are modeled.
Solving only for the large eddies and modeling the smaller scales results in
mesh resolution requirements that are much less restrictive than with DNS.
Typically, mesh sizes can be at least one order of magnitude smaller than
with DNS. Furthermore, the time step sizes will be proportional to the
eddy-turnover time, which is much less restrictive than with DNS. In
practical terms, however, extremely fine meshes are still required. It is only
due to the explosive increases in computer hardware performance coupled
with the availability of parallel processing that LES can even be considered
as a possibility for engineering calculations.
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The following sections give details of the governing equations for LES,
present the two options for modeling the subgrid-scale stresses (necessary
to achieve closure of the governing equations), and discuss the relevant
boundary conditions.
Turbulence quantities that can be reported for the LES model are as
follows: Subgrid Turbulent Kinetic Energy, Subgrid Turbulent Viscosity,
Subgrid Effective Viscosity, Subgrid Turbulent Viscosity Ratio, Effective
Thermal Conductivity, Effective Prandtl Number, Wall Yplus
7.8 Wall Functions vs. Near-Wall Model
Turbulent flows are significantly affected by the presence of walls.
Obviously, the mean velocity field is affected through the no-slip condition
that has to be satisfied at the wall. However, the turbulence is also changed
by the presence of the wall in non-trivial ways. Very close to the wall,
viscous damping reduces the tangential velocity fluctuations, while
kinematic blocking reduces the normal fluctuations. Toward the outer part
of the near-wall region, however, the turbulence is rapidly augmented by
the production of turbulence kinetic energy due to the large gradients in
mean velocity.
The near-wall modeling significantly impacts the fidelity of numerical
solutions, inasmuch as walls are the main source of mean vorticity and
turbulence. After all, it is in the near-wall region that the solution variables
have large gradients, and the momentum and other scalar transports occur
most vigorously. Therefore, accurate representation of the flow in the nearwall region determines successful predictions of wall-bounded turbulent
flows.
The k-ε models, the RSM, and the LES model are primarily valid for
turbulent core flows (i.e., the flow in the regions somewhat far from walls).
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Consideration therefore needs to be given as to how to make these models
suitable for wall-bounded flows. The Spalart-Allmaras and k-ω models
were designed to be applied throughout the boundary layer, provided that
the near-wall mesh resolution is sufficient.
Numerous experiments have shown that the near-wall region can be largely
subdivided into three layers. In the innermost layer, called the "viscous
sublayer", the flow is almost laminar, and the (molecular) viscosity plays
a dominant role in momentum and heat or mass transfer. In the outer layer,
called the fully-turbulent layer, turbulence plays a major role. Finally, there
is an interim region between the viscous sublayer and the fully turbulent
layer where the effects of molecular viscosity and turbulence are equally
important. Figure 7.1 illustrates these subdivisions of the near-wall region,
plotted in semi-log coordinates.
Figure 7.1 Subdivisions of the Near-Wall Region
Traditionally, there are two approaches to modeling the near-wall region.
In one approach, the viscosity-affected inner region (viscous sublayer and
buffer layer) is not resolved. Instead, semi-empirical formulas called \wall
16
functions" are used to bridge the viscosity-affected region between the wall
and the fully-turbulent region. The use of wall functions obviates the need
to modify the turbulence models to account for the presence of the wall.
In another approach, the turbulence models are modified to enable the
viscosity-affected region to be resolved with a mesh all the way to the wall,
including the viscous sublayer. For purposes of discussion, this will be
termed the "near-wall modeling" approach. These two approaches are
depicted in Figure 7.2
Figure 7.2 Near-Wall Treatments in FLUENT
In most high-Reynolds-number flows, the wall function approach
substantially saves computational resources, because the viscosity-affected
17
near-wall region, in which the solution variables change most rapidly, does
not need to be resolved. The wall function approach is popular because it
is economical, robust, and reasonably accurate. It is a practical option for
the near-wall treatments for industrial flow simulations. The wall function
approach, however, is inadequate in situations where the low-Reynoldsnumber effects are pervasive in the flow domain in question, and the
hypotheses underlying the wall functions cease to be valid. Such situations
require near-wall models that are valid in the viscosity-affected region and
accordingly integrable all the way to the wall. FLUENT provides both the
wall function approach and the near-wall modeling approach.
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