Introduction

Lecture Theme: Part 1.

Turbulence
The problem definition for all CFD simulationsModeling
includes boundary conditions,
cell zone conditions and material properties. The accuracy of the simulation
results depends on defining these properly.

Learning Aims:
You will learn:
• How to define material properties.
• The different boundary condition types in FLUENT and how to use them.
• How to define cell zone conditions in FLUENT including solid zones and
porous media.
• How to specify well–posed boundary conditions.

Learning Objectives:
You will know how to perform these essential steps in setting up a CFD
analysis.

Introduction Material Properties Cell Zone Conditions Boundary Conditions Summary

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 Introduction
Lecture Theme:
The majority of engineering flows are turbulent. Successfully simulating
such flows requires understanding a few basic concepts of turbulence
theory and modeling. This allows one to make the best choice from the
available turbulence models and near –wall options for any given problem.
Learning Aims:
You will learn:
•Basic turbulent flow and turbulence modeling theory .
•Turbulence models and near–wall options available in Fluent.
•How to choose an appropriate turbulence model for a given problem.
•How to specify turbulence boundary conditions at inlets.
Learning Objectives:
You will understand the challenges inherent in turbulent flow simulation
and be able to identify the most suitable model and near–wall treatment
for a given problem.

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 Observation by Osborne Reynolds [1]
• Flows can be classified as either :

Laminar
(Low Reynolds Number)

Transition
(Increasing Reynolds Number)

Turbulent
(Higher Reynolds Number)

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 Observation by Osborne Reynolds [2]
• The Reynolds number is the criterion used to determine whether the flow is
laminar or turbulent.

 .U .L
Re L 

• The Reynolds number is based on the length scale of the flow:
L  x, d, d hyd, etc.

• Transition to turbulence varies depending on the type of flow:

• External flow:
• Along a surface : ReX > 500000
• Around on obstacle : ReL > 20000
• Internal flow: : ReD > 2300

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 Turbulent Flow Structures [1]
• A turbulent flow contains a wide range of turbulent eddy sizes.

• Turbulent flow characteristics:

• Unsteady, three–dimensional, irregular, stochastic motion in which transported
quantities (mass, momentum, scalar species) fluctuate in time and space.
• Enhanced mixing of these quantities results from the fluctuations.
• Unpredictability in detail.
• Large scale coherent structures are different in each flow, whereas small
eddies are more universal.

Small Large
structures structures
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 Turbulent Flow Structures [2]
• Energy is transferred from larger eddies to smaller eddies.
(Kolmogorov Cascade).
• Large scale contains most of the energy.
• In the smallest eddies, turbulent energy is converted to internal energy by viscous
dissipation.

Energy Cascade
Richardson (1922),
Kolmogorov (1941)

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 Backward Facing Step
• As engineers, in most cases we do not actually need to see an exact snapshot of
the velocity at a particular instant.
• Instead for most problems, knowing the time–averaged velocity (and intensity of
the turbulent fluctuations) is all we need to know. This gives us a useful way to
approach modelling turbulence.
Instantaneous velocity contours.

Time–averaged velocity contours.

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 Mean and Instantaneous Velocities
• If we recorded the velocity at a particular point in the real (turbulent) fluid flow,
the instantaneous velocity (U) would look like this:
u  Fluctuating velocity.
Velocity

U Time–average of velocity.

U Instantaneous velocity.

Time

• At any point in time: U  U  u

• The time–average of the fluctuating velocity u must be zero: u  0
• BUT, the RMS of u' is not necessarily zero: u2  0
• Note you will hear reference to the turbulence energy, k. This is the sum of the
three normal fluctuating velocity components: k  1  u2  v2  w2  .
2 
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 Overview of Computational Approaches
• Different approaches to make turbulence computationally tractable.

DNS LES RANS

(Direct Numerical Simulation) (Large Eddy Simulation) (Reynolds Averaged Navier–
Stokes Simulation)

• Numerically solving the full • Solves the spatially averaged N–S • Solve time–averaged Navier–Stokes
unsteady Navier–Stokes equations. equations. equations.
• Resolves the whole spectrum of • Large eddies are directly • All turbulent length scales are
scales. resolved, but eddies smaller than modeled in RANS.
the mesh are modeled.
• No modeling is required. • Various different models are
• Less expensive than DNS, but the available.
amount of computational
• But the cost is too prohibitive! resources and efforts are still too • This is the most widely used approach
large for most practical for industrial flows.
Not practical for industrial flows!
applications.

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 RANS Modeling: Averaging
• Thus, the instantaneous Navier–Stokes equations may be rewritten as
Reynolds–Averaged equations (RANS):

 u u  p   ui   Rij Rij  uiuj

 i  uk i      
 x  x
 t xk  xi x j (Reynolds stress tensor)
 j  j

• The Reynolds stresses are additional unknowns introduced by the averaging

procedure, hence they must be modeled (related to the averaged flow quantities) in
order to close the system of governing equations.

   u '2   u ' v '   u ' w ' 

  xx  xy  xz   
 R    uu  
τ   yx ij yy  yzi j    u ' v '   v '2   v ' w ' 
   
 zx  zy  zz     u ' w '   v ' w '   w ' 
2
 
Symmetric tensor  6 unknowns …

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 RANS Modeling: The Closure Problem
• The Reynolds Stress tensor Rij  uiuj must be solved.
• The RANS models can be closed in two ways:
Reynolds–Stress Models (RSM) Eddy Viscosity Models
• Rij is directly solved via transport equations • Boussinesq hypothesis
(modeling is still required for many terms in the Reynolds stresses are modeled using an eddy (or
transport equations). turbulent) viscosity, μT.


t

uiuj  
xk
 
 uk uiuj  Pij  Fij  DijT   ij  ij
 u u j  2 uk
Rij   uiuj   T  i    T
 x x  3 x
2
ij   k ij
 j i  k 3

• RSM is more advantageous in complex 3D • The hypothesis is reasonable for simple turbulent
turbulent flows with large streamline curvature shear flows: boundary layers, round jets, mixing
and swirl, layers, channel flows, etc.
• but the model is more complex, computationally
intensive, more difficult to converge than eddy
viscosity models.
• Note: All turbulence models contain empiricism.
• Equations cannot be derived from fundamental principles.
• Some calibrating to observed solutions and 'intelligent guessing' is contained in the models.

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 Turbulence Models Available in Fluent
One–Equation Model
Spalart–Allmaras
Two–Equation Models
Standard k–ε
RNG k–ε
Increase in
RANS based Realizable k–ε* Computational
models
Standard k–ω Cost
Per Iteration
SST k–ω*
Reynolds Stress Model
k–kl–ω Transition Model
SST Transition Model
Detached Eddy Simulation
Large Eddy Simulation
* Recommended choice for standard cases.

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 Two–Equation Models
• Two transport equations are solved, giving two independent
scales for calculating t.
– Virtually all use the transport equation for the turbulent kinetic energy, k.
Dk   t  k 
        P   ; P  t S 2 ( ske) S  2Sij Sij
Dt x j  k  x j 
production dissipation
– Several transport variables have been proposed, based on dimensional arguments,
and used for second equation. The eddy viscosity t is then formulated from the two
transport variables.

• Kolmogorov, w: t  k / w, l  k1/2 / w, k   / w
– w is specific dissipation rate.
– Defined in terms of large eddy scales that define supply rate of k.
• Chou, : t  k2 / , l  k3/2 / 
• Rotta, l: t  k1/2l,   k3/2 / l

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 RANS:EVM: Standard k–ε (SKE) Model
• The Standard k–ε model (SKE) is the most widely–used engineering turbulence
model for industrial applications.
– Model parameters are calibrated by using data from a number of benchmark
experiments such as pipe flow, flat plate, etc.
– Robust and reasonably accurate for a wide range of applications.
– Contains submodels for compressibility, buoyancy, combustion, etc.

• Known limitations of the SKE model:

– Performs poorly for flows with larger pressure gradient, strong separation, high
swirling component and large streamline curvature.
– Inaccurate prediction of the spreading rate of round jets.
– Production of k is excessive (unphysical) in regions with large strain rate (for example,
near a stagnation point), resulting in very inaccurate model predictions.

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 RANS:EVM: Realizable k–epsilon
• Realizable k–ε (RKE) model (Shih):
– Dissipation rate (ε) equation is derived from the mean–square
vorticity fluctuation, which is fundamentally different from the SKE.
– Several realizability conditions are enforced for Reynolds stresses.

– Benefits:
• Accurately predicts the spreading rate of both planar and round jets.
• Also likely to provide superior performance for flows involving rotation,
boundary layers under strong adverse pressure gradients, separation, and
recirculation.

OFTEN PREFERRED TO STANDARD k–ε

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 RANS: EVM: Spalart–Allmaras (S–A) Model
• Spalart–Allmaras is a low–cost RANS model solving a single
transport equation for a modified eddy viscosity.

• Designed specifically for aerospace applications involving wall–

bounded flows .
– Has been shown to give good results for boundary layers subjected to adverse
pressure gradients.
– Used mainly for aerospace and turbomachinery applications.

• Limitations:
– The model was designed for wall bounded flows and flows with mild separation
and recirculation.
– No claim is made regarding its applicability to all types of complex engineering
flows.

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 k–omega Models
• In k–w models, the transport equation for the turbulent dissipation rate, , is
replaced with an equation for the specific dissipation rate, w.
– The turbulent kinetic energy transport equation is still solved.
– See Appendix for details of w equation.
• k–w models have gained popularity in recent years mainly because:
– Much better performance than k– models for boundary layer flows.
• For separation, transition, low –Re effects, and impingement, k–w models are more
accurate than k– models.
– Accurate and robust for a wide range of boundary layer flows with pressure gradient.
• Two variations of the k–w model are available in Fluent.
– Standard k–w model (Wilcox, 1998).
– SST k–w model (Menter).

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 SST Model
• Shear Stress Transport (SST) Model.
– The SST model is an hybrid two–equation model that combines the advantages of
both k– and k–w models.
• k–w model performs much better than k– models for boundary layer flows.
• Wilcox’ original k–w model is overly sensitive to the freestream value (BC) of w,
while k– model is not prone to such problem.

k–

k–w
Wall

– The k–e and k–w models are blended such that the SST model functions like the k–ω
close to the wall and the k–ε model in the freestream.

SST is a good compromise between k– and k–w models

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 RANS: Other Models in Fluent
• RNG k– model.
– Model constants are derived from renormalization group (RNG) theory instead of
empiricism.
– Advantages over the standard k– model are very similar to those of the RKE
model.
• Reynolds Stress model (RSM).
– Instead of using eddy viscosity to close the RANS equations, RSM solves transport
equations for the individual Reynolds stresses.
• 7 additional equations in 3D, compared to 2 additional equations with EVM.
– More computationally expensive than EVM and generally difficult to converge.
• As a result, RSM is used primarily in flows where eddy viscosity models are
known to fail.
• These are mainly flows where strong swirl is the predominant flow feature, for
instance a cyclone (see Appendix).

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 Turbulence Near a Wall [1]
• The Structure of Near–Wall Flows.

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 Turbulence Near a Wall [2]
• Near to a wall, the velocity changes rapidly.
Velocity, U

Distance from Wall, y

• If we plot the same graph again, where:

– Log scale axes are used.

– The velocity is made dimensionless, from U/U ( ,, is friction velocity)

– The wall distance vector is made dimensionless.

• Then we arrive at the graph on the next page. The shape of this is generally the same for all flows:

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 Turbulence Near a Wall [3]
• By scaling the variables near the wall the velocity profile data takes on a
predictable (universal) form (transitioning from linear to logarithmic behavior).

Scaling the non–dimensional

velocity and non–dimensional
distance from the wall results in a Linear
predictable boundary layer profile Logarithmic
for a wide range of flows.

• Since near wall conditions are often predictable, functions can be used to
determine the near wall profiles rather than using a fine mesh to actually
resolve the profile.
– These functions are called wall functions.

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 Choice of Wall Modeling Strategy.
• In the near–wall region, the solution gradients are very high, but accurate calculations
in the near–wall region are paramount to the success of the simulation.

The choice is between:

Resolving the Viscous Sublayer.
– First grid cell needs to be at about y+ = 1.
– This will add significantly to the mesh count.
– Use a low–Reynolds number turbulence model (like k–omega) .
– Generally speaking, if the forces on the wall are key to your simulation (aerodynamic drag,
turbomachinery blade performance) this is the approach you will take.
Using a Wall Function.
– First grid cell needs to be 30<y+<300.
(Too low, and model is invalid. Too high and the wall is not properly resolved).
– Use a wall function, and a high Re turbulence model (SKE, RKE, RNG).
– Generally speaking, this is the approach if you are more interested in the mixing in the middle of
the domain, rather than the forces on the wall.

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 Limitations of Wall Functions
• In some situations, such as boundary layer separation, logarithmic–based
wall functions do not correctly predict the boundary layer profile.

Wall functions applicable. Wall functions not applicable.

Non–Equilibrium Wall Functions have been developed
in Fluent to address this situation but they are very
empirical. A more rigorous approach is recommended if
affordable.

• In these cases logarithmic–based wall functions should not be used.

• Instead, directly resolving the boundary layer can provide accurate results.
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 Choosing a Near Wall Treatment
• Standard Wall Functions.
– The Standard Wall Function options
is designed for high Re attached flows.
– The near–wall region is not resolved.
– Near–wall mesh is relatively coarse.

• Non–Equilibrium Wall Functions.

– For better prediction of adverse pressure gradient flows and
separation.
– Near–wall mesh is relatively coarse.

• Enhanced Wall Treatment*

– Used for low–Re flows or flows with complex
near–wall phenomena.
– Generally requires a very fine near–wall mesh capable of
resolving the near–wall region.
– Can also handle coarse near–wall mesh.

• User–Defined Wall Functions.

– Can host user specific solutions.
* Recommended choice for standard cases.
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 Inlet Boundary Conditions
• When turbulent flow enters a domain at inlets or outlets (backflow), boundary
conditions for k, ε, ω and/or u'u' must be specified, depending on which
i j
turbulence model has been selected.

• Four methods for directly or indirectly specifying turbulence parameters:

1) Explicitly input k, ε, ω, or Reynolds stress components (this is the only method that
allows for profile definition).
• Note by default, the Fluent GUI enters k=1 [m²/s²] and  =1 [m²/s³]. These values
MUST be changed, they are unlikely to be correct for your simulation.

2) Turbulence intensity and length scale.

• Length scale is related to size of large eddies that contain most of energy.
– For boundary layer flows: l  0.4δ99.
– For flows downstream of grid: l  opening size.
3) Turbulence intensity and hydraulic diameter (primarily for internal flows).

4) Turbulence intensity and viscosity ratio (primarily for external flows).

The default setting is turbulent intensity=5% and turbulent viscosity ratio=10. This
should be reasonable for many flows if more precise information not available.

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 Inlet Turbulence Conditions
• If you have absolutely no idea of the turbulence levels in your simulation, you
could use following values of turbulence intensities and viscosity ratios:

– Usual turbulence intensity ranges from 1% to 5%.

– The default turbulence intensity value of 0.037 (that is, 3.7%) is sufficient for nominal
turbulence through a circular inlet, and is a good estimate in the absence of
experimental data.
– For external flows, turbulent viscosity ratio of 1÷10 is typically a good value.
– For internal flows, turbulent viscosity ratio of 10÷100 it typically a good value.
• For fully developed pipe flow at Re=50,000, the turbulent viscosity ratio is around
100.

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 RANS Turbulence Model Usage
Model Behavior and Usage
Spalart–Allmaras Economical for large meshes. Performs poorly for 3D flows, free shear flows, flows with strong
separation. Suitable for mildly complex (quasi–2D) external/internal flows and boundary layer flows
under pressure gradient (e.g. airfoils, wings, airplane fuselages, missiles, ship hulls).

Standard k–ε Robust. Widely used despite the known limitations of the model. Performs poorly for complex flows
involving severe pressure gradient, separation, strong streamline curvature. Suitable for initial
iterations, initial screening of alternative designs, and parametric studies.

Realizable k–ε* Suitable for complex shear flows involving rapid strain, moderate swirl, vortices, and locally transitional
flows (e.g. boundary layer separation, massive separation, and vortex shedding behind bluff bodies, stall
in wide–angle diffusers, room ventilation).

RNG k–ε Offers largely the same benefits and has similar applications as Realizable. Possibly harder to converge
than Realizable.

Standard k–ω Superior performance for wall–bounded boundary layer, free shear, and low Reynolds number flows.
Suitable for complex boundary layer flows under adverse pressure gradient and separation (external
aerodynamics and turbomachinery). Can be used for transitional flows (though tends to predict early
transition). Separation is typically predicted to be excessive and early.

SST k–ω* Offers similar benefits as standard k–ω. Dependency on wall distance makes this less suitable for free
shear flows.

RSM Physically the most sound RANS model. Avoids isotropic eddy viscosity assumption. More CPU time
and memory required. Tougher to converge due to close coupling of equations. Suitable for complex
3D flows with strong streamline curvature, strong swirl/rotation (e.g. curved duct, rotating flow
passages, swirl combustors with very large inlet swirl, cyclones).

* Recommended choice for standard cases

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 RANS Turbulence Model Descriptions
Model Description
Spalart – A single transport equation model solving directly for a modified turbulent viscosity. Designed specifically
for aerospace applications involving wall–bounded flows on a fine near–wall mesh. Fluent’s implementation
Allmaras allows the use of coarser meshes. Option to include strain rate in k production term improves predictions of
vortical flows.

Standard k–ε The baseline two–transport–equation model solving for k and ε. This is the default k–ε model. Coefficients
are empirically derived; valid for fully turbulent flows only. Options to account for viscous heating,
buoyancy, and compressibility are shared with other k–ε models.

RNG k–ε A variant of the standard k–ε model. Equations and coefficients are analytically derived. Significant changes
in the ε equation improves the ability to model highly strained flows. Additional options aid in predicting
swirling and low Reynolds number flows.

Realizable k–ε A variant of the standard k–ε model. Its "realizability" stems from changes that allow certain mathematical
constraints to be obeyed which ultimately improves the performance of this model.

Standard k–ω A two–transport–equation model solving for k and ω, the specific dissipation rate (ε / k) based on Wilcox
(1998). This is the default k–ω model. Demonstrates superior performance for wall–bounded and low
Reynolds number flows. Shows potential for predicting transition. Options account for transitional, free
shear, and compressible flows.

SST k–ω A variant of the standard k–ω model. Combines the original Wilcox model for use near walls and the
standard k–ε model away from walls using a blending function. Also limits turbulent viscosity to guarantee
that τT ~ k. The transition and shearing options are borrowed from standard k–ω. No option to include
compressibility.

RSM Reynolds stresses are solved directly using transport equations, avoiding isotropic viscosity assumption of
other models. Use for highly swirling flows. Quadratic pressure–strain option improves performance for
many basic shear flows.

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 Summary – Turbulence Modeling Guidelines
• Successful turbulence modeling requires engineering judgment of:
– Flow physics.
– Computer resources available.
– Project requirements.
• Accuracy.
• Turnaround time.
– Choice of Near–wall treatment.
• Modeling procedure
1. Calculate characteristic Reynolds number and determine whether flow is turbulent.
2. If the flow is in the transition (from laminar to turbulent) range, consider the use of one
of the turbulence transition models (not covered in this training).
3. Estimate wall–adjacent cell centroid y+ before generating the mesh.
4. Prepare your mesh to use wall functions except for low–Re flows and/or flows with
complex near–wall physics (non–equilibrium boundary layers).
5. Begin with RKE (Realizable k–ε) and change to S–A, RNG, SKW, or SST if needed. Check
the tables on previous slides as a guide for your choice.
6. Use RSM for highly swirling, 3–D, rotating flows.
7. Remember that there is no single, superior turbulence model for all flows!

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