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Lecturer: Ehsan Saadati
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Eddy-viscosity Models:
• Zero equation model.
• Standard k −ε model.
• RNG k −ε model.
• Standard k −ω model.
• Baseline (BSL) zonal k −ω based model. (This model can only be selected through
Expert Control Parameters.)
• SST zonal k −ω based model.
• (k −ε)1E model.( This model can only be selected through Expert Control Parameters.)
• Curvature correction for two-equation models.
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Reynolds-Stress Models (RSM):
• Launder, Reece and Rodi Isotropization of Production model (LRR Reynolds
Stress).
• Launder, Reece and Rodi Quasi-Isotropic model (QI Reynolds Stress).
• Speziale, Sarkar and Gatski (SSG Reynolds Stress).
• SMC-ω model (Omega Reynolds Stress).
• Baseline (BSL) Reynolds Stress model.
• Explicit Algebraic Reynolds Stress Model (EARSM).
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Laminar flow is governed by the unsteady Navier-Stokes equations. The laminar
option does not apply a turbulence model to the simulation and is only
appropriate if the flow is laminar. This typically applies at low Reynolds number
flows (for example, for pipe flow the laminar flow regime is ~ Re < 1000). Energy
transfer in the fluid is accomplished by molecular interaction (diffusion). In the
case of high speed flows, the work of the viscous stresses can also contribute to
the energy transfer. You should always check in the output file that the maximum
Reynolds number is in the laminar flow regime. If you set up a simulation using
laminar flow, but the real flow is turbulent, convergence is difficult and the
simulation will not reach the correct solution.
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The Zero Equation model implemented in CFX is simple to implement and use,
can produce approximate results very quickly, and provides a good initial guess
for simulations using more advanced turbulence models. In CFX, a constant
turbulent eddy viscosity is calculated for the entire flow domain. If you specify an
eddy viscosity, this value is used instead of the solver calculated value. You should
not use this model to obtain final results.
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The turbulence viscosity is modeled as the product of a turbulent velocity scale,
Ut, and a turbulence length scale, lt, as proposed by Prandtl and Kolmogorov,
μt =ρfμUt lt
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where f μ is a proportionality constant. The velocity scale is taken to be the
maximum velocity in the fluid domain.
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The length scale is derived using the formula:
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where VD is the fluid domain volume. This model has little physical foundation and
is not recommended.
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One of the most prominent turbulence models, the k −ε (k-epsilon) model, has been
implemented in most general purpose CFD codes and is considered the industry
standard model. It has proven to be stable and numerically robust and has a well
established regime of predictive capability. For general purpose simulations, the
k −ε model offers a good compromise in terms of accuracy and robustness. Within
CFX, the k −ε turbulence model uses the scalable wall-function approach to improve
robustness and accuracy when the near-wall mesh is very fine. The scalable wall
functions allow solution on arbitrarily fine near wall grids, which is a significant
improvement over standard wall functions. While standard two-equation models,
such as the k −ε model, provide good predictions for many flows of engineering
interest, there are applications for which these models may not be suitable.
Among these are:
• Flows with boundary layer separation.
• Flows with sudden changes in the mean strain rate.
• Flows in rotating fluids.
• Flows over curved surfaces.
A Reynolds Stress model may be more appropriate for flows with sudden changes in
strain rate or rotating flows, while the SST model may be more appropriate for
separated flows.
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and the momentum equation becomes:
The k-ε model, like the zero equation model, is based on the eddy viscosity concept, so
that:
μeff=μ+μt
where μt is the turbulence viscosity. The k-ε model assumes that the turbulence
viscosity is linked to the turbulence kinetic energy and dissipation via the
relation:
where Cμ is a constant.
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The values of k and ε come directly from the differential transport equations for the
turbulence kinetic energy and turbulence dissipation rate:
where Cε1, Cε2, σk and σε are constants.
Pkb and Pεb represent the influence of the buoyancy forces, which are described
below. Pk is the turbulence production due to viscous forces, which is
modeled using:
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The RNG k −ε model is an alternative to the standard k −ε model. In general it offers
little improvement compared to the standard k −ε model.
 The RNG k − ε model is based on renormalization group analysis of the NavierStokes equations. The transport equations for turbulence generation and
dissipation are the same as those for the standard k − ε model, but the model
constants differ, and the constant Cε1 is replaced by the function Cε1RNG. The
transport equation for turbulence dissipation becomes:
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Currently, the most prominent two-equation models in this area are the k −ω
based models of Menter. The k −ω based Shear-Stress-Transport (SST) model was
designed to give a highly accurate predictions of the onset and the amount of flow
separation under adverse pressure gradients by the inclusion of transport effects
into the formulation of the eddy-viscosity. This results in a major improvement in
terms of flow separation predictions. The superior performance of this model has
been demonstrated in a large number of validation studies (Bardina et al.)
The SST model is recommended for high accuracy boundary layer simulations. To
benefit from this model, a resolution of the boundary layer of more than 10
points is required.
For free shear flows, the SST model is identical to the k −ε model.
The SST model was developed to overcome deficiencies in the k −ω and BSL k −ω
models. Therefore, using the SST model over these models is recommended.
One of the advantages of the k −ω formulation is the near wall treatment for lowReynolds number computations where it is more accurate and more robust.
The convergence behavior of the k −ω model is often similar to that of the k −ε
model.
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One of the advantages of the k-ω formulation is the near wall treatment for lowReynolds number computations. The model does not involve the complex nonlinear damping functions required for the k-ε model and is therefore more
accurate and more robust. A low-Reynolds k-ε model would typically require a
near wall resolution of y+ <0.2, while a low-Reynolds number k-ω model would
require at least y+ <2. In industrial flows, even y+ <2 cannot be guaranteed in most
applications and for this reason, a new near wall treatment was developed for the
k-ω models. It allows for smooth shift from a low-Reynolds number form to a wall
function formulation.
 The k-ω models assumes that the turbulence viscosity is linked to the turbulence
kinetic energy and turbulent frequency via the relation:
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The starting point of the present formulation is the k-ω model developed by
Wilcox .It solves two transport equations, one for the turbulent kinetic energy, k,
and one for the turbulent frequency, ω. The stress tensor is computed from the
eddy-viscosity concept.
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The main problem with the Wilcox model is its well known strong sensitivity to
freestream conditions. Depending on the value specified for ω at the inlet, a
significant variation in the results of the model canbe obtained. This is undesirable
and in order to solve the problem, a blending between the k-ω model near the
surface and the k-ε model in the outer region was developed by Menter. It consists
of a transformation of the k-ε model to a k-ω formulation and a subsequent
addition of the corresponding equations. The Wilcox model is thereby multiplied by
a blending function F1 and the transformed k-ε model by a function 1−F1. F1 is
equal to one near the surface and decreases to a value of zero outside the
boundary layer (that is, a function of the wall distance). At the boundary layer
edge and outside the boundary layer, the standard k-ε model is therefore
recovered.
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The k-ω based SST model accounts for the transport of the turbulent shear stress
and gives highly accurate predictions of the onset and the amount of flow
separation under adverse pressure gradients. The BSL model combines the
advantages of the Wilcox and the k-ε model, but still fails to properly predict the
onset and amount of flow separation from smooth surfaces. The reasons for this
deficiency are given in detail in Menter .The main reason is that both models do
not account for the transport of the turbulent shear stress. This results in an over
prediction of the eddy-viscosity. The proper transport behavior can be obtained
by a limiter to the formulation of the eddy-viscosity:
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Again F2 is a blending function similar to F1, which restricts the limiter to the wall
boundary layer, as the underlying assumptions are not correct for free shear flows.
S is an invariant measure of the strain rate.
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The blending functions are critical to the success of the method. Their
formulation is based on the distance to the nearest surface and on the flow
variables.
where y is the distance to the nearest wall, ν is the kinematic viscosity and:
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A disadvantage of standard two-equation turbulence models is the excessive
generation of turbulence energy, Pk , in the vicinity of stagnation points. In order
to avoid the build-up of turbulent kinetic energy in stagnation regions, a
formulation of limiters for the production term in the turbulence equations is
available.
The coefficient Clim is called Clip Factor and has a value of 10 for ω based models.
This limiter does not affect the shear layer performance of the model, but has
consistently avoided the stagnation point build-up in aerodynamic simulations.
Using the standard Boussinesq-approximation for the Reynolds stress tensor, the
production term Pk can be expressed for incompressible flow as:
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where S denotes the magnitude of the strain rate and Sij the strain rate tensor.
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Kato and Launder noticed that the very high levels of the shear strain rate S in
stagnation regions are responsible for the excessive levels of the turbulence
kinetic energy. Since the deformation near a stagnation point is nearly
irrotational, that is the vorticity rate Ω is very small, they proposed the following
replacement of the production term:
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One weakness of the eddy-viscosity models is that these models are insensitive
to streamline curvature and system rotation. Based on the work of Spalart and
Shur a modification of the production term has been derived which allows to
sensitize the standard two-equation models to these effects .The empirical
function suggested by Spalart and Shur to account for these effects is defined by
It is used as a multiplier of the production term and has been limited in ANSYS CFX in
the following way:
 The specific limiter 1.25 provided a good compromise for different test cases
which have been considered with the SST model (for example, flow through a Uturn, flow in hydro cyclone, and flow over a NACA 0012 wing tip vortex
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A very simple one-equation model has been developed by Menter. It is derived
directly from the k − ε model and is therefore named the (k − ε)1E model
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where ∼v is the kinematic eddy viscosity, ∼v t is the turbulent kinematic eddy
viscosity and σ is a model constant.
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The low Reynolds formulation of the (k − ε)1E model requires a near wall mesh
resolution of y+ <1.
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In flows where the turbulent transport or non-equilibrium effects are important, the
eddy-viscosity assumption is no longer valid and results of eddy-viscosity models
might be inaccurate.
Reynolds Stress (or Second Moment Closure (SMC)) models naturally include the
effects of streamline curvature, sudden changes in the strain rate, secondary flows or
buoyancy compared to turbulence models using the eddy-viscosity approximation.
You may consider using a Reynolds Stress model in the following types of flow:
• Free shear flows with strong anisotropy, like a strong swirl component. This includes flows
in rotating fluids.
• Flows with sudden changes in the mean strain rate.
• Flows where the strain fields are complex, and reproduce the anisotropic nature of
turbulence itself.
• Flows with strong streamline curvature.
• Secondary flow.
• Buoyant flow.
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Reynolds Stress models have shown superior predictive performance compared
to eddy-viscosity models in these cases. This is the major justification for
Reynolds Stress models, which are based on transport equations for the
individual components of the Reynolds stress tensor and the dissipation rate.
These models are characterized by a higher degree of universality. The penalty
for this flexibility is a high degree of complexity in the resulting mathematical
system. The increased number of transport equations leads to reduced numerical
robustness, requires increased computational effort and often prevents their
usage in complex flows. Theoretically, Reynolds Stress models are more suited to
complex flows, however, practice shows that they are often not superior to twoequation models. An example of this is for wall-bounded shear layers, where
despite their (theoretically) higher degree of universality, Reynolds Stress models
often prove inferior to two-equation models. For wall-bounded flows try to use
the SMC-BSL model. It is based on the ω-equation and automatic wall treatment.
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Three varieties of the Reynolds Stress Model are available which use different
model constants:
• Reynolds Stress Model (LRR-IP)
• QI Reynolds Stress Model (LRR-IQ)
• SSG Reynolds Stress Model (SSG)
 In general, the SSG model is more accurate than the LRR versions for most flows.
This is particularly true for swirling flows. The SSG model is therefore
recommended over the other models, which are there for historic reasons and
because they are standard models.
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It is frequently observed that Reynolds Stress models produce unsteady results,
where two-equation models give steady state solutions. This can be correct from a
physical standpoint, but requires the solution of the equations in transient mode.
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Some of the main deficiencies of the Reynolds Stress models for the simulation
of boundary layers are an inheritance from the underlying ε-equation. Particularly
the accurate prediction of flow separation is problematic when the ε-equation is
used. Furthermore, low-Reynolds number formulations for the ε-equation are
usually complex and difficult to integrate, a deficiency which is exaggerated in
combination with a Reynolds Stress model formulation. In order to avoid these
issues, a Reynolds Stress model has been implemented which uses the ω
equation instead of the ε-equation as the scale-determining equation.
There are two types of SMC-ω model in CFX:
• Omega Reynolds Stress
• Baseline (BSL) Reynolds Stress
As the free stream sensitivity of the standard k −ω model does carry over to the
Reynolds Stress model, the BSL Reynolds Stress Model was developed which is
based on the ω-equation used in the BSL model, and is preferred over the Omega
Reynolds Stress Model. This is the formulation recommended when using the ωReynolds Stress model.
 All variants of the ω-Reynolds Stress model are combined with automatic wall
treatment, which allows a flexible grid resolution near the wall.
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Explicit algebraic Reynolds stress models (EARSM) represent an extension of the
standard two-equation models. They are derived from the Reynolds stress
transport equations and give a nonlinear relation between the Reynolds stresses
and the mean strain-rate and vorticity tensors. Due to the higher order terms
many flow phenomena are included in the model without the need to solve
transport equations. The EARSM is an extension of the current k −ε and BSL
model to capture effects of secondary flows and of flows with streamline curvature
and system rotation.
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Engineering transition predictions are based mainly on two modeling concepts. The first is the use
of low-Reynolds number turbulence models, where the wall damping functions of the underlying
turbulence model trigger the transition onset. This concept is attractive, as it is based on transport
equations and can therefore be implemented without much effort. However, experience has
shown that this approach is not capable of reliably capturing the influence of the many different
factors that affect transition, such as free-stream turbulence, pressure gradients and separation.
The second approach is the use of experimental correlations. The correlations usually relate the
turbulence intensity,Tu , in the free-stream to the momentum-thickness Reynolds number, Reθt, at
transition onset. ANSYS has developed a locally formulated transport equation for intermittency,
which can be used to trigger transition. The full model is based on two transport equations, one
for the intermittency and one for the transition onset criteria in terms of momentum thickness
Reynolds number. It is called the ‘Gamma Theta Model’ and is the recommended transition model
for general-purpose applications. It uses a new empirical correlation (Langtry and Menter) that
has been developed to cover standard bypass transition as well as flows in low free-stream
turbulence environments. This built-in correlation has been extensively validated together with
the SST turbulence model for a wide range of transitional flows. The transition model can also be
used with the BSL or SAS-SST turbulence models as well. A description of the full model can be
found in the solver theory chapter.
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Because the transition model requires the solution of two extra transport
equations, there are additional CPU costs associated with using it. A rough
estimate is that for the same grid the transition model solution requires
approximately 18 percent additional CPU time compared to a fully turbulent
solution. As well, the transition model requires somewhat finer grids than are
typically used for routine design purposes. This is because the max grid y+ must be
approximately equal to one (that is, wall-function grids cannot be used because
they cannot properly resolve the laminar boundary layer) and sufficient grid
points in the streamwise direction are needed to resolve the transitional region.
For this reason, it is important to be able to estimate when the additional cost of
using the transition model in terms of CPU and grid generation time is justified.
The relative percentage of laminar flow on a device can be estimated using the
following formula, which is based on the Mayle empirical correlation for
transition onset.
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Where Rext is the transition Reynolds number, Rex is the device Reynolds number,
LDevice is the length of the device, V is a representative velocity, and Tu is the
freestream turbulence intensity, which can be calculated, as follows:
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where k is the turbulent kinetic energy.
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For y+ values between 0.001 and 1, there is almost no effect on the solution. Once the
maximum y+ increases above 8, the transition onset location begins to move upstream. At a
maximum y+ of 25, the boundary layer is almost completely turbulent. For y+ values below
0.001, the transition location appears to move downstream. This is presumably caused by the
large surface value of the specific turbulence frequency ω, which scales with the first grid
point height. Additional simulations on a compressor test case have indicated that at very
small y+ values the SST blending functions switch to k −ε in the boundary layer. For these
reasons, very small (below 0.001) y+ values should be avoided at all costs.
Effect of increasing y+ for the flat plate T3A test case
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Effect of decreasing y+ for the flat plate T3A test case
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Effect of wall normal expansion ratio for the flat plate T3A test case
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The solution differed significantly from the grid independent one only for the case of 25
streamwise nodes where there was only one cell in the transitional region. Nevertheless,
the grid independent solution appears to occur when there is approximately 75 - 100
streamwise grid points.
In CFX, when the transition model is active and the high resolution scheme is selected for
the hydrodynamic equations, the default advection scheme for the turbulence and
transition equations is automatically set to high resolution.
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Based on the grid sensitivity study the recommended best practice mesh
guidelines are a max y+ of 1, a wall normal expansion ratio of 1.1 and about 75 100 grid nodes in the streamwise direction. Note that if separation induced
transition is present, additional grid points in the streamwise direction are most
likely needed. For a typical 2D blade assuming an H-O-H type grid, the above
guide lines result in an inlet H-grid of 15x30, an O-grid around the blade of 200x80
and an outlet H-grid of 100x140 for a total of approximately 30 000 nodes, which
has been found to be grid independent for most turbomachinery cases. It should
also be noted that for the surfaces in the out of plane z-direction, symmetry
planes should always be used, not slip walls. The use of slip walls has been found
to result in an incorrect calculation of the wall distance, which is critical for
calculating the transition onset location accurately. Another point to note is that
all the validation cases for the transition model have been performed on
hexahedral meshes. At this point, the accuracy of the transition model on
tetrahedral meshes has not been investigated.
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It has been observed that the turbulence intensity specified at an inlet can decay
quite rapidly depending on the inlet viscosity ratio (μt ⁄ μ) (and hence turbulence
eddy frequency). As a result, the local turbulence intensity downstream of the inlet
can be much smaller than the inlet value .Typically, the larger the inlet viscosity
ratio, the smaller the turbulent decay rate. However, if too large a viscosity ratio
is specified (that is, >100), the skin friction can deviate significantly from the
laminar value. There is experimental evidence that suggests that this effect
occurs physically; however, at this point it is not clear how accurately the
transition model reproduces this behavior. For this reason, if possible, it is
desirable to have a relatively low (that is ≈1 - 10) inlet viscosity ratio and to
estimate the inlet value of turbulence intensity such that at the leading edge of
the blade/airfoil, the turbulence intensity has decayed to the desired value.
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The decay of turbulent kinetic energy can be calculated with the following
analytical solution:
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Turbulent flows contain a wide range of length and time scales, with large scale
motions being generally more energetic than the small scale ones. The prediction
of industrially important fluctuating flow problems can be performed using the
technique of LES. LES is an approach which solves for large-scale fluctuating
motions and uses "sub-grid" scale turbulence models for the small-scale motion.
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Before starting an LES simulation, you should consider if it is the most appropriate
solution approach. For low Reynolds numbers (Re < 5000) or when it is important to be
able to resolve all scales (for example, for transition to turbulent flow), consider DNS if
you have a large computer available. For Higher Reynolds numbers, LES might be a
suitable option for cases where:
The flow is likely to be unstable, with large scale flapping of a shear layer or vortex
shedding.
The flow is likely to be unsteady with coherent structures (cyclone, flasher).
The flow is buoyant, with large unstable regions created by heating from below, or by
lighter fluid below heavier fluid (multiphase flows in inclined pipelines).
Conventional RANS approach are known to fail (for example due to highly anisotropic
turbulence).
A good representation of the turbulent structure is required for small-scale processes
such as micro-mixing o chemical reaction.
The noise from the flow is to be calculated, and especially when the broadband
contribution is significant.
Other fluctuating information is required (such as fluctuating forces, gusts of winds).
You can afford to wait for up to a week for results, using an 8 to 16 processor system.
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Geometry for LES
Even though there might be symmetries in the geometry and flow, the geometrical
model should include the full region, since, while the time averaged flows may be
symmetric, the instantaneous flows are not, and restricting the domain could
constrain the turbulence. Two-dimensional approximations are particularly poor.
Meshing
The mesh and timesteps are an inherent part of the model. LES models make use of
the grid scale for filtering out the turbulence. If the mesh is anisotropic to resolve
a jet, for example, the longer length scale in the flow direction may also have an
undue effect on the cross-stream turbulence. For this reason, consider the use of
isotropic grids, perhaps using tetrahedral rather than hexahedral elements.
Boundary Layers
If the boundary layer structure is important, you should resolve it with at least 15
points across the boundary layer and with the first grid point at a position of
approximately y+ = 1. Note however, that large aspect rations in the mesh are not
suitable for LES. This often results in excessive resolution requirements for
boundary layer flows.
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Three LES models are available: it is recommended to use the wall-adapted local
eddy-viscosity model by Nicoud and Ducros (LES WALE model) as a first choice.
This is an algebraic model like the Smagorinsky model, but overcomes some
known deficiencies of the Smagorinsky model: the WALE model produces almost
no eddy-viscosity in wall-bounded laminar flows and is therefore capable to
reproduce the laminar to turbulent transition. Furthermore, the WALE model has
been designed to return the correct wall-asymptotic y +3-variation of the subgridscale viscosity and needs no damping functions. In addition to the WALE model,
the Smagorinsky model and the Dynamic Smagorinsky-Lilly model (Germano et
al. Lilly are available. The Dynamic Smagorinsky-Lilly model is based on the
Germano-identity and uses information contained in the resolved turbulent
velocity field to evaluate themodel coefficient, in order to overcome the
deficiencies of the Smagorinsky model. The model coefficient is no longer a
constant value and adjusts automatically to the flow type. However this method
needs explicit (secondary) filtering and is therefore more time consuming than an
algebraic model.
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The representation of the turbulent structures at inlets can be a difficult part of the
setup. The detailed properties of the incoming flow can have a strong effect on
the development of the downstream solution. This has been observed in
turbulent jets, as well as developing boundary layer cases. For other cases,
however, the turbulent conditions at the inlet can have relatively little impact on
the flow in the device. For example, in a cyclone body, the flow can be essentially
determined by very strong anisotropy effects. If the inlet turbulence is felt to play
an important role, you should consider:
1. Moving the inlet far upstream of the geometry of interest. This allows the correct
turbulent structures to establish themselves before the flow reaches that
geometry. Additional obstructions can be placed upstream of geometry to affect
the turbulent structures. This procedure, however, will require the simulation of
the transition process.
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2. Using unsteady values computed from a separate LES simulation (pipe flow and
channel flows).
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
With the transport of turbulent eddies outside of the computational domain,
some recirculation can occur at outlets. Experience shows that if the code is
allowed to bring some fluid back into the domain at outlets, it can destabilize the
solution. Therefore, you should use outlet boundary conditions rather than
openings when using LES. Opening types of boundary conditions allow the flow
to come back in, whereas with outlet boundary conditions, the code builds
artificial walls at the boundary, when the flow tries to come back in. These
artificial walls are later taken away when the flow goes out again. The use of
artificial walls at outlets might introduce some un-physical behavior locally, but it
increases the robustness of the calculation.
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
For simulations initialized with values (rather than from an initial guess field), it is
possible to perturb the initial guess by setting an RMS Velocity Fluctuation. This
has the effect of kickstarting the solution process. For a velocity distribution Vi,
the fluctuation is the quantity:
The behavior of the velocity fluctuation is applied according to the following rules
when you select Automatic or Automatic with Value as the Initialization option,
as well as specify a velocity fluctuation:
• If an initial values file is not found, the fluctuation is applied to the initial velocity
field.
• When an initial values file is used, it is assumed that you are restarting from a
previous set of results, and the fluctuation is not applied.

If you want to apply a velocity fluctuation on a restart (for example, if the initial guess
is a RANS solution), you can override this behavior by setting the expert parameter
apply ic fluctuations for les to t.
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
On the Solver Control form, it is recommend that you use the Central Difference
advection scheme rather than the High Resolution scheme because it is less
dissipative and it has provided good answers in CFX. Select the transient scheme
as 2nd Order Backward Euler.
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First order fully implicit methods in time are usually too diffusive, and the turbulence is damped out.
For highly unstable problems, such as cyclones, lower order methods may work, but the results
will be very damped, unless very small timesteps are used.
• For accuracy, the average Courant (or CFL) number should be in the range of 0.5-1. Larger values
can giv stable results, but the turbulence may be damped. For compressible flows where the
acoustic behavior I being modeled (e.g., for noise calculations), this conclusion still holds, but for
the CFL number based on the acoustic velocity as well as the convective velocity.
• 1,000 - 10,000 timesteps are typically required for getting converged statistics. More steps are
required fo second order quantities (for example, variances) than for means. Check the
convergence of the statistics. For vortex-shedding problem, several cycles of the vortex
shedding are required.
• The implicit coupled solver used in CFX requires the equations to be converged within each
timestep to guarantee conservation. The number of coefficient loops required to achieve this is a
function of the timestep size. Wit CFL numbers of order 0.5-1, convergence within each timestep
should be achieved quickly. It is advisable t test the sensitivity of the solution to the number of
coefficient loops, to avoid using more coefficient loops (an hence longer run times) than
necessary. LES tests involving incompressible flow past circular cylinders indicate that one
coefficient loop per timestep is sufficient if the average CFL number is about 0.75. If the physics
o geometry is more complicated, additional coefficient loops (3-5) may be required. Timesteps
size which require more coefficient loops than this will tend to degrade solution accuracy.
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The following values can be written to results files as the solution progresses
using the Solver Output tab in CFX-Pre.
 Although the statistics outlined in the Statistical Reynolds Stresses are useful for
any simulation, they are particularly important in the context of LES. Of
particular interest, are the Reynolds Stress (RS) components:




where ui ′ is the fluctuation of the of the ith velocity component.
Statistical Reynolds Stresses
In LES runs, Reynolds Stress components are automatically generated using
running statistics of the instantaneous, transient velocity field. As outlined above
for the calculation of the standard deviation, a Reynolds Stress component can
be evaluated using the difference between the running arithmetic average of the
instantaneous velocity correlation and the running arithmetic average of the
instantaneous velocities as:

The noted running averages are also automatically generated for LES runs.
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Data sampling for the Statistical Reynolds Stresses begins on the first timestep
by default. This is because the transient statistics used in the stress evaluation
(that is, the arithmetic averages of velocity and its correlation) start accumulating
on the first timestep by default. Sampling for the Statistical Reynolds Stresses is
deferred, indirectly, by explicitly creating one of the required transient statistics
(that is, the arithmetic averages of velocity or its correlation) and setting the start
iteration (the starting timestep index) to a value larger than unity. If different
start iterations are set for each of the velocity or velocity correlation averages,
then the maximum start iteration set is used for both averages to ensure that the
Statistical Reynolds Stresses are correctly evaluated. The Statistical Reynolds
Stress variable is evaluated using Equation 4.10 (p. 110) during every timestep,
regardless of the start iteration specified for the required velocity-based
statistics. This means that if the current timestep is less than the start iteration
specified for the velocity-based statistics, then those statistics will be initialized
as outlined in Working with Transient Statistics
Solver Memory
 You may find, especially if running averages are to be calculated, that for some
cases you will need to increase the amount of memory needed for the calculation
by using a memory factor above 1 (typically 1.2).

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In an attempt to improve the predictive capabilities of turbulence models in
highly separated regions, Spalart proposed a hybrid approach, which combines
features of classical RANS formulations with elements of Large Eddy Simulations
(LES) methods. The concept has been termed Detached Eddy Simulation (DES)
and is based on the idea of covering the boundary layer by a RANS model and
switching the model to a LES mode in detached regions. Ideally, DES would
predict the separation line from the underlying RANS model, but capture the
unsteady dynamics of the separated shear layer by resolution of the developing
turbulent structures. Compared to classical LES methods, DES saves orders of
magnitude of computing power for high Reynolds number flows.
 Though this is due to the moderate costs of the RANS model in the boundary
layer region, DES still offers some of the advantages of an LES method in
separated regions.

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It is well known that RANS models does not accurately predict all flow details in
massively separated flow regions. In addition, the RANS formulation does not
provide any information on turbulent flow structures and spectral distribution,
which might be of importance to predict flow-induced noise or vibrations. In
these cases, DES can provide valuable details far exceeding RANS simulations.
Typical examples of DES simulations are:
• Flow around non-aerodynamic obstacles (buildings, bridges, etc.)
• Flow around ground transport vehicles with massively separated regions (cars,
trains, trucks)
• Flow around noise generating obstacles (car side-mirror, etc.)
• Massively separated flow around stalled wings

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
On the other hand, DES is a computer intensive method because large
(detached) turbulent structures need to be resolved in space and time. While the
grid resolution requirements are not significantly higher than RANS for
simulations, the time resolution imposes high CPU demands. In CFX, steadystate solutions can typically be obtained within 100-200 iterations. A typical DES
simulation requires several thousands timesteps using three coefficient loops
each. DES is therefore at least one order of magnitude more computer intensive
than RANS models. DES does not allow the use of symmetry conditions, as
turbulent structures are usually not symmetric. Two-dimensional and
axisymmetric simulations are not feasible, as turbulent structures are always
three-dimensional. Hence, you should carefully weigh the need for additional
information against the required resources.
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
The main issue with geometry is that 2D or axisymmetric simulations are not
visible. In addition, in most cases, symmetry or periodicity conditions are not
allowed. As an example, RANS simulations allow the use of a half model for a car
simulation. As the turbulent structures are asymmetric, this is not possible in
DES.
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There are many requirements in terms of grid generation, which depend on the
selection of the specific formulation of the DES model. The grid resolution
requirement in the attached boundary layers is the same as that of a regular
RANS solution (10-15 nodes in the boundary layer). In the DES region, the grid
has to be designed in all three space dimensions to allow at least the resolution of
the largest turbulent structures. As the largest grid spacing is used in the switch
for the DES limiter, it is essential to have sufficient resolution in the spanwise
direction (if it exists). One of the main issues of DES models is the danger to
produce grid-induced separation due to an activation of the DES limiter in the
RANS region. The zonal formulation used in CFX helps to avoid the severity of
this potential limitation.
 For a cylinder in cross-flow, 50 nodes were used in the spanwise direction for an
extension of 2 diameters. For the Ahmed car body, 70 nodes were used in the
cross-flow direction.

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
Timestep selection is a sensitive issue, as it determines the quality of the
resolution of the turbulent structures. In most cases, a typical shedding frequency
can be estimated for the largest turbulent structures. An example is a cylinder in
cross-flow. For a cylinder, the Strouhal number is of the order of:

In the CFX simulation the free stream velocity is U=46 m/s and the diameter is
D=0.37 m. This results in a frequency f=24.8 1/s. A timestep of Dt=10−4 s is
sufficient to resolve the turbulent structures on the given grid.
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
In almost all cases, the same boundary conditions as in RANS simulation can be
applied. Exceptions are symmetry or periodicity conditions, which may not be
satisfied by the turbulent structures. In most cases, RANS inlet conditions can be
applied, particularly for flows without profile distributions at the inlet, as mostly
used in industrial flows.
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
It is recommended to start the DES simulation from a converged RANS solution.
For comparison purposes, it is useful to use the same RANS model as in the DES
formulation (in CFX this means the SST model). The RANS solution supplies to
the DES formulation important elliptic information on the pressure field, which
might take a long time to build up in an unsteady simulation.
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
Due to high computing costs, it is essential to monitor a DES run to ensure that it
approaches an appropriate solution. It is recommended to check the blending
function for DES model during the simulation (every 50-100 timesteps). In
regions where the function is zero, the LES model is used, and the region where
its value is one, the RANS model is activated. Figure 4.6, “Blending Function for
DES Model for Flow Around Cube.” shows a typical example for the flow around a
cube. The blue region behind the cube is governed by the LES part, and the red
region by the RANS model. This is desirable for this case as only the structures
behind the obstacle are of interest.
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
A second variable to monitor is the invariant:

where S is the absolute value of the strain rate (S = Shear Strain Rate in CFD-Post)
and Ω is the absolute value of the vorticity. Note that the vorticity has to be
activated in order to be available in CFD-Post. The following lines are to be added
to the solver CCL:
LIBRARY:
VARIABLE: vorticity
Output to Postprocessor = Yes
END
END
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
In CFD-Post, you have to define a user variable based on the following
expression:
LIBRARY:
CEL:
EXPRESSIONS:
Invariant = Vorticity^2 - Shear Strain Rate^2
END
END
END
USER SCALAR VARIABLE:VelGradInvariant
Boundary Values = Conservative
Expression = Invariant
Recipe = Expression
END
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Strain Rate Invariant for Flow Around Cube
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Unsteady flow from DES over backward facing step using
insufficient spanwise grid resolution
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The Scale-Adaptive Simulation (SAS)
The Detached Eddy Simulation is a combination of RANS and LES methods. In an
alternative approach, a new class of URANS models has been developed which
can provide a LES-like behavior in detached flow regions, Menter and Egorov,
Egorov and Menter. The Scale-Adaptive Simulation (SAS) concept is based on the
introduction of the von Karman length-scale into the turbulence scale equation.
The information provided by the von Karman length-scale allows SAS models to
dynamically adjust to resolved structures in a URANS simulation, which results in
a LES-like behavior in unsteady regions of the flowfield. At the same time, the
model provides standard RANS capabilities in stable flow regions.
 isosurfaces where S2−Ω2=105[s^−2] for a cylinder in cross flow (Re=3.6x106),
calculated using the SST model (URANS) and the SAS-SST model. The URANS
simulation produces only the large-scale unsteadyness, whereas the SAS-SST
model adjusts to the already resolved scales in a dynamic way and allows the
development of a turbulent spectrum in the detached regions.

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Resolved structures for cylinder in cross flow
(top: URANS; bottom: SAS-SST)
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
The original version of the SST-SAS model has undergone continued evolution.
This model is still available in ANSYS CFX, but the latest version of the SAS model
is now the default model.

The SAS method is an improved URANS formulation, with the ability to adapt
the length scale to resolved turbulent structures. In order for the SAS method to
produce a resolved turbulent spectrum, the underlying turbulence model has to
go unsteady. This is typically the case for flows with a global instability, as
observed for flows with large separation zones, or vortex interactions. There are
many technical flows, where steady state solutions cannot be obtained or where
the use of smaller timesteps results in unsteady solutions. One could argue that
the occurrence of unsteady regions in a RANS simulation is the result of a nonlocal interaction, which cannot be covered by the single-point RANS closure. All
these flows will benefit from the SAS methodology.
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On the other hand, there are classes of flows where the SAS model will return a
steady solution. The two main classes are attached or mildly separated boundary
layers and undisturbed channel/pipe flows. These flows (or areas in a complex
flow domains) will be covered by the RANS formulation. This behavior of SAS is
ideal for many technical applications, as flows for which the model produces
steady solutions are typically those, where a well-calibrated RANS model can
produce accurate results.
 Contrary to DES, SAS cannot be forced to go unsteady by grid refinement.
However, it has been shown in the backstep simulation above that enforcing
unsteadiness by grid refinement does require an in-depth understanding of the
flow, the turbulence model and its interaction with the grid to produce reliable
results.

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
Nevertheless, there will be cases where the grid sensitivity of DES will be
advantageous, as it allows unsteady simulations where SAS might produce a
steady state flow-field. This leads to the following hierarchy of models of
increased complexity, which can be used as a guideline in order to estimate the
needed effort:
1. URANS
2. SAS
3. DES
4. LES
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
Due to the similar functionality of the SAS-SST and DES model, most of the
guidelines which have been given for the DES model regarding timestepping,
boundary conditions and initialization are also valid for the SAS-SST model.
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
Buoyancy Turbulence is an option in CFX-Pre that is available for some turbulence
models when buoyancy is being modeled. When the option is set to Production, a
buoyancy production term is included in the equation for k. When the option is set
to Production and Dissipation, a buoyancy production term is included in the
equations for k, ε, ω (as applicable). See Buoyancy Turbulence and the
accompanying text, which describe the treatment for the k −ε and k −ω models.
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
Near a no-slip wall, there are strong gradients in the dependent variables. In
addition, viscous effects on the transport processes are large. The representation
of these processes within a numerical simulation raises the following problems:
• How to account for viscous effects at the wall.
• How to resolve the rapid variation of flow variables which occurs within the boundary
layer region.

Experiments and mathematical analysis have shown that the near-wall region
can be subdivided into two layers. In the innermost layer, the so-called viscous
sublayer, the flow is almost laminar-like, and the (molecular) viscosity plays a
dominant role in momentum and heat transfer. Further away from the wall, in
the logarithmic layer, turbulence dominates the mixing process. Finally, there is a
region between the viscous sublayer and the logarithmic layer called the buffer
layer, where the effects of molecular viscosity and turbulence are of equal
importance. The figure below illustrates these subdivisions of the near-wall
region.
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Assuming that the logarithmic profile reasonably approximates the velocity distribution
near the wall, it provides a means to numerically compute the fluid shear stress as a
function of the velocity at a given distance from the wall. This is known as a ‘wall
function' and the logarithmic nature gives rise to the well known ‘log law of the wall.‘
Two approaches are commonly used to model the flow in the near-wall region:
• The wall function method uses empirical formulas that impose suitable conditions near
to the wall without resolving the boundary layer, thus saving computational
resources. All turbulence models in CFX are suitable for a wall function method. The
major advantages of the wall function approach is that the high gradient shear layers
near walls can be modeled with relatively coarse meshes, yielding substantial savings
in CPU time and storage. It also avoids the need to account for viscous effects in the
turbulence model.
• The Low-Reynolds-Number method resolves the details of the boundary layer profile by
using very small mesh length scales in the direction normal to the wall (very thin
inflation layers). Turbulence models based on the ω-equation, such as the SST or
SMC-ω models, are suitable for a low-Re method. Note that the low-Re method does
not refer to the device Reynolds number, but to the turbulent Reynolds number, which is
low in the viscous sublayer. This method can therefore be used even in simulations
with very high device Reynolds numbers, as long as the viscous sublayer has been
resolved.
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
The computations are extended through the viscosity-affected sublayer close to
the wall. The low-Re approach requires a very fine mesh in the near-wall zone and
correspondingly large number of nodes. Computer-storage and runtime
requirements are higher than those of the wall-function approach and care must
be taken to ensure good numerical resolution in the near-wall region to capture
the rapid variation in variables. To reduce the resolution requirements, an
automatic wall treatment was developed by CFX, which allows a gradual switch
between wall functions and low-Reynolds number grids, without a loss in
accuracy.

Wall functions are the most popular way to account for wall effects. In CFX,
Scalable Wall Functions are used for all turbulence models based on the εequation. For k −ω based models (including the SST model), an Automatic nearwall treatment method is applied.
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
In CFX-5.4.1 and earlier, standard wall functions were used. These have been
replaced by scalable wall functions as the default but standard wall functions are
still available for backward compatibility. You should not use standard wall
functions, as they offer no advantage over the scalable wall functions.
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
Scalable wall functions overcome one of the major drawbacks of the standard
wall function approach in that they can be applied on arbitrarily fine meshes. If
the boundary layer is not fully resolved, you will be relying on the logarithmic wall
function approximation to model the boundary layer without affecting the
validity of the scalable wall function approach. If you are not interested in the
details of the boundary layer, then it may not be worth fully resolving it.
However, if you have produced a very fine near-wall mesh to examine details of
the boundary layer, then you should use the SST model with Automatic near-wall
treatment to take advantage of the additional effect in the viscous sublayer.
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
Automatic near-wall treatment automatically switches from wall-functions to a
low-Re near wall formulation as the mesh is refined. One of the well known
deficiencies of the k −ε model is its inability to handle low turbulent Reynolds
number computations. Complex damping functions can be added to the k −ε
model, as well as the requirement of highly refined near-wall grid resolution (y+ <
0.2) in an attempt to model low turbulent Reynolds number flows. This approach
often leads to numerical instability. Some of these difficulties may be avoided by
using the k −ω model, making it more appropriate than the k −ε model for flows
requiring high near-wall resolution (for example, high wall heat transfer,
transition). However, a strict low-Reynolds number implementation of the model
would also require a near wall grid resolution of at least y+ < 2. This condition
cannot be guaranteed in most applications at all walls.
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For this reason, a new near wall treatment was developed by CFX for the k −ω
based models that allows for a smooth shift from a low-Reynolds number form to
a wall function formulation. This near wall boundary condition, named automatic
near wall treatment in CFX, is used as the default in all models based on the ωequation (standard k −ω, Baseline k −ω, SST, ω-Reynolds Stress).
 To take advantage of the reduction in errors offered by the automatic switch to a
low-Re near wall formulation, you should attempt to resolve the boundary layer
using at least 10 nodes when using these models.

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
A user-defined wall function transfer coefficient may be applied to any wall
boundary condition. This is accomplished by adding the following CCL data to the
wall boundary condition, in the HEAT TRANSFER block or ADDITIONAL
VARIABLES block, as appropriate:
TURBULENT WALL FUNCTIONS:
Option = User Defined
Wall Function Transfer Coefficient = [expression goes here]
END

Note that the parameter Wall Function Transfer Coefficient must be a normalized
value having dimensions of [L T^-1] (that is, the units of kinematic diffusivity
divided by length). The definition of the two different types of wall function
transfer coefficient and their normalized values are presented next, followed by
examples of CCL implementations.
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Heat Flux
heat flux = C* (Vertex Temperature - Wall Temperature)
where C represents the wall heat flux transfer coefficient before normalization. The
Wall Function Transfer Coefficient parameter is then calculated as Wall Function
Transfer Coefficient = C / (density * Cp)
Additional Variable Flux
additional variable flux = C * (Vertex AV - Wall AV)
where C represents the wall scalar flux transfer coefficient before normalization.
The Wall Function Transfer Coefficient parameter is then calculated as:
Wall Function Transfer Coefficient = C / density
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Example of User-Defined Wall Function for Heat Flux

In the following example, a user-defined wall function, in the form of a wall function transfer
coefficient for heat flux, is defined and applied to a wall boundary condition.
LIBRARY :
CEL:
EXPRESSIONS:
sn = 1.E-20
prandtl = viscosity * Cp / cond
pryplus = prandtl * yplus
beta1 = (3.85 * prandtl^1/3 -1.3) ^ 2
gamma = 0.01 * prandtl * yplus ^ 3 / \
( 1./(yplus*prandtl^3+sn) + 5.0)
tplussub = pryplus
tpluslog = max(2.12*loge(pryplus+sn)+beta1,0.)
tplus = tplussub*exp(-gamma) + tpluslog*exp(-1./(gamma+sn))
twftfc1 = ustar / (tplus + sn)
END
END
END
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FLOW
DOMAIN: flatplate
BOUNDARY : Wall
Boundary Type = WALL
Coord Frame = Coord 0
BOUNDARY CONDITIONS :
MASS AND MOMENTUM :
Option = No Slip Wall
END
WALL ROUGHNESS :
Option = Rough Wall
Roughness Height = 0.019 [in]
END
HEAT TRANSFER :
Option = Fixed Temperature
Fixed Temperature = 100[C]
TURBULENT WALL FUNCTIONS:
Option = User Defined
Wall Function Transfer Coefficient = twftfc1
END
END
END
END
END
MAPNAEND
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
The CCL implementation of a user-defined wall function for an Additional
Variable is similar to that shown in the example for heat flux. The CCL block in the
boundary condition is (using a specified quantity as an example):
FLOW
DOMAIN: flatplate
BOUNDARY : Wall
BOUNDARY CONDITIONS :
ADDITIONAL VARIABLE: my_av
Wall Function Transfer Coefficient = 10 [m s^-1]
END
END
END
END
END
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
The near wall treatment which has been discussed above (Scalable Wall
Functions, Automatic Wall Treatment) is appropriate when walls can be
considered as hydraulically smooth. For rough walls, the logarithmic profile
exists, but moves closer to the wall. The near wall treatment becomes more
complex, since it depends now on two variables: the dimensionless wall distance
y+ and the roughness height. The roughness height which must be specified in the
wall boundary section ,is the equivalent sand grain roughness height. Note that
the sand-grain roughness is not equal to the geometric roughness height of the
surface under consideration. Wall friction depends not only on roughness height
but also on the type of roughness (shape, distribution, etc.). Therefore, one must
determine the appropriate equivalent sand-grain roughness height. Details on
the formulation of the rough wall treatment for turbulence models based on the
ε and ω-equation can be found in Treatment of Rough Walls in the ANSYS CFXSolver Theory Guide. If the ANSYS-CFX two-equation transition model is used
together with rough walls, then the roughness correlation must be turned on in
the GUI. This correlation requires the geometric roughness height as input
parameter.
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Yplus (y+) is the dimensionless distance from the wall. It is used to check the location of
the first node away from a wall. This was more important when standard wall
functions were used, as you had to avoid y+ values smaller than approximately
20. With the scalable wall functions and the automatic wall treatment, these
values are only provided for information on the near wall resolution. Two
variables related to y+ are output to the results file, Yplus and Solver Yplus.
• The variable Yplus is based on the distance from the wall to the first node and the
wall shear stress. This is the usual definition in the CFD community.
• The variable Solver Yplus is used internally by the CFX-Solver. It uses different
definitions for the wall distance of the first wall point. This variable is available for
backwards consistency, but is of little use to you.
There is usually no need for you to work with the y+ used internally (Solver Yplus). The
information required to judge the near wall mesh spacing should be based on the
standard y+ (Yplus) definition.
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One of the most essential issues for the optimal performance of turbulence models
is the proper resolution of the boundary layer. In this section, two criteria are
suggested for judging the quality of a mesh:
• Minimum spacing between nodes in the boundary layer
• Minimum number of nodes in the boundary layer
These are simple guidelines for the generation of meshes which satisfy the minimal
requirements for accurate boundary layer computations.
Minimum Node Spacing
The goal is to determine the required near wall mesh spacing, Δy, in terms of
Reynolds number, running length, and a Δ y+ target value. This target value
depends on the flow type and the turbulence model in use, or in other words the
near wall treatment in use.
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Determination of the Near Wall Spacing
The estimates will be based on correlations for a flat plate with a Reynolds number
of:
with characteristic velocity U∞ and length of the plate L. The correlation for the wall
shear stress coefficient, cf , is given by White
where x is the distance along the plate from the leading edge. The definition of Δ y+ for
this estimate is:
with Δy being the mesh spacing between the wall and the first node away from the
wall.
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Using the definition
uτ can be eliminated in Equation 4.15 to yield:
cf can be eliminated using Equation 4.14 to yield:
Further simplification can be made by assuming that:
where C is some fraction.
Assuming that C1⁄14≈1 , then, except for very small Rex, the result is:
This equation allows us to set the target Δ y+ value at a given x location and obtain
the mesh
spacing,
ΔyCourse-Turbulence-Rev-01-Januray2012
for nodes in the boundary layer.
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Minimum Number of Nodes
Goal
A good mesh should have a minimum number of mesh points inside the boundary
layer in order for the turbulence model to work properly. As a general guideline, a
boundary layer should be resolved with at least:
where Nnormal, min is the minimum number of nodes that should be placed in the
boundary layer in the direction normal to the wall.
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Formulation
The boundary layer thickness δ can then be computed from the correlation:
to be:
The boundary layer for a blunt body does not start with zero thickness at the
stagnation point for Rex. It is, therefore, safe to assume that Reδ is some fraction of
ReL, say 25%. With this assumption, you get:
You would, therefore, select a point, say the fifteenth off the surface (for a low-Re
model, or 10th for a wall function model) and check to make sure that:
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The near wall treatment which has been discussed above (Scalable Wall
Functions, Automatic Wall Treatment) is appropriate when walls can be
considered as hydraulically smooth. Surface roughness can have a significant
effect on flows of engineering interest. Surface roughness typically leads to an
increase in turbulence production near the wall. This in turn can result in
significant increases in the wall shear stress and the wall heat transfer
coefficients. For the accurate prediction of near wall flows, particularly with heat
transfer, the proper modeling of surface roughness effects is essential for a good
agreement with experimental data.
 Wall roughness increases the wall shear stress and breaks up the viscous sublayer
in turbulent flows.


where B=5.2. The shift ΔB is a function of the dimensionless roughness height, h +,
defined as h+ =huτ / υ.
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
For sand-grain roughness, the downward shift can be expressed as:
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
It has been shown that a technical roughness, which has peaks and valleys of
different shape and size, can be described by an equivalent sand-grain roughness

Depending on the dimensionless sand-grain roughness hs+, three roughness
regimes can be distinguished:
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
The viscous sublayer is fully established near hydraulically smooth walls. In the
transitional roughness regime the roughness elements are slightly thicker than the
viscous sublayer and start to disturb it, so that in fully rough flows the sublayer is
destroyed and viscous effects become negligible.

Rough Wall Treatment for Turbulence Models Based on the Dissipation Equation:

The Scalable Wall Function approach neglects the viscous sublayer and limits the
value used in the logarithmic formulation by a lower value of y+ =max(y* ,11.06) . This
limiter must be combined with the formulation Equation 2.238 (p. 92) for the rough wall
treatment and reads:
y+=max(y* , hS+/ 2, 11.06)

Since the automatic wall treatment works together with the transition model, it is the
default rough wall treatment. This default Automatic setting for the Wall Function
option can be set on the Fluid Models tab of the Domain details view in CFX-Pre. The
CFX-Solver checks internally if a roughness height has been specified for a wall and
then uses the proper wall treatment: when a roughness height exists, then the
automatic rough wall treatment is used, otherwise the automatic smooth wall
treatment.
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