1. Chapter 9 Turbulence
Introduction to CFX

2. What is Turbulence?
Unsteady, irregular (non-periodic) motion in which transported quantities (mass, momentum, scalar species) fluctuate in time and space
Identifiable swirling patterns characterize turbulent eddies
Enhanced mixing (matter, momentum, energy, etc.) results
Fluid properties and velocity exhibit random variations
Statistical averaging results in accountable, turbulence related transport mechanisms
This characteristic allows for turbulence modeling
Contains a wide range of turbulent eddy sizes (scales spectrum)
The size/velocity of large eddies is on the order of the mean flow
Large eddies derive energy from the mean flow
Energy is transferred from larger eddies to smaller eddies
In the smallest eddies, turbulent energy is converted to internal energy by viscous dissipation
Point out that the governing equations for turbulence are well-known and are the non-linear, unsteady three-dimensional Navier-Stokes equations. Useful to think of the instantaneous velocity in terms of a mean velocity with random fluctuations superimposed. Not only are there fluctuations in velocity but also in pressure, temperature, and scalar variables. The ability to predict the enhanced mixing resulting from turbulence is important in a large number of applications.

3. Is the Flow Turbulent?
Flows can be characterized by the Reynolds Number, Re
External Flows
where
along a surface
around an obstacle
Other factors such as free-stream turbulence, surface conditions, and disturbances may cause transition to turbulence at lower Reynolds numbers
Internal Flows
That first thing to consider is whether or not you need to consider turbulence modeling at all. Basically, the types of flows can be classified as either external, internal, or natural convection, The criteria for transition to turbulent flow is different depending on the type of flow you are considering. For external/internal flows common criteria are based on Reynolds number where the length scale varies depending on the flow. For flows along a surface, the Reynolds number is based on the distance along the surface. For flows about some object the Reynolds number is based on the diameter of the obstruction. Internal flows have the Reynolds number based on the hydraulic diameter. These criteria are not steadfast and can be affected by the other factors listed. Flows involving natural convection have been observed to transition from laminar to turbulent flow over a range of Rayleigh numbers.
Natural Convection
where
is the Rayleigh number
is the Prandtl number

4. Observation by O. Reynolds
Laminar
(Low Reynolds Number)
Transition
(Increasing Reynolds Number)
Turbulent
(Higher Reynolds Number)

5. Turbulent Flow Structures
Small
structures
Large
Energy Cascade Richardson (1922)

6. Conservation Equations
Governing Equations
Conservation Equations
Continuity
Momentum
Energy
These are the basic equations that are solved for fluid flow problems.
More equations must be solved for more complex flows involving combustion, radiation, etc.
where
Note that there is no turbulence equation in the governing Navier-Stokes equations!

7. Overview of Computational Approaches
Direct Numerical Simulation (DNS)
Theoretically, all turbulent (and laminar / transition) flows can be simulated by numerically solving the full Navier-Stokes equations
Resolves the whole spectrum of scales. No modeling is required
But the cost is too prohibitive! Not practical for industrial flows
Large Eddy Simulation (LES) type models
Solves the spatially averaged N-S equations
Large eddies are directly resolved, but eddies smaller than the mesh are modeled
Less expensive than DNS, but the amount of computational resources and efforts are still too large for most practical applications
Reynolds-Averaged Navier-Stokes (RANS) models
Solve time-averaged Navier-Stokes equations
All turbulent length scales are modeled in RANS
Various different models are available
This is the most widely used approach for calculating industrial flows
There is not yet a single, practical turbulence model that can reliably predict all turbulent flows with sufficient accuracy
For DNS, you may mention the scaling argument;
where R is turbulent Reynolds number
Therefore, the minimum number of grid points per integral scale is
Including time discretization, the computational work

8. RANS Modeling – Time Averaging
Ensemble (time) averaging may be used to extract the mean flow properties from the instantaneous ones
The instantaneous velocity, ui, is split into average and fluctuating components
The Reynolds-averaged momentum equations are as follows
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
Example: Fully-Developed
Turbulent Pipe Flow
Velocity Profile
Instantaneous
component
Time-average
component
Fluctuating
component
(Reynolds stress tensor)

9. RANS Modeling – The Closure Problem
Closure problem: Relate the unknown Reynolds Stresses to the known mean flow variables through new equations
The new equations are the turbulence model
Equations can be:
Algebraic
Transport equations
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

10. RANS Modeling – The Closure Problem
The RANS models can be closed in one of the following ways
(1) Eddy Viscosity Models (via the Boussinesq hypothesis)
Boussinesq hypothesis – Reynolds stresses are modeled using an eddy (or turbulent) viscosity, μT. The hypothesis is reasonable for simple turbulent shear flows: boundary layers, round jets, mixing layers, channel flows, etc.
(2) Reynolds-Stress Models (via transport equations for Reynolds stresses)
Modeling is still required for many terms in the transport equations
RSM is more advantageous in complex 3D turbulent flows with large streamline curvature and swirl, but the model is more complex, computationally intensive, more difficult to converge than eddy viscosity models
Boussinesq hypothesis is reasonable for simple turbulent shear flows---see “Turbulent Flows,” S.B. Pope, p

11. Available Turbulence Models
A large number of turbulence models are available in CFX, some have very specific applications while others can be applied to a wider class of flows with a reasonable degree of confidence
RANS Eddy-viscosity Models:
1) Zero Equation model.
2) Standard k-ε model.
3) RNG k-ε model.
4) Standard k-ω model.
5) Baseline (BSL) zonal k-ω based model.
6) SST zonal k-ω based model.
7) (k-ε)1E model.
RANS Reynolds-Stress Models:
1) LRR Reynolds Stress
2) QI Reynolds Stress
3) Speziale, Sarkar and Gatski Reynolds Stress
4) SMC-ω model
5) Baseline (BSL) Reynolds' Stress model
Eddy Simulation Models:
1) Large Eddy Simulation (LES) [transient]
2) Detached Eddy Simulation (DES)* [transient]
3) Scale Adaptive Simulation SST (SAS)* [transient]
* Not available in the ANSYS CFD-Flo product
--A large number of models have been developed that can be used to approximate turbulence
--The choice of turbulence model determines how the unsteady behaviour affects the meanflow.
--Some have very specific applications, while others can be applied to a wider class of flows with a reasonable degree of confidence.
Simulation Models
--The high end models actually simulate the transient nature of large scale eddy fluctuations.
--for certain applications, simulation is needed. E.g. buffet hood of vehicle

12. Turbulence Near the Wall
The velocity profile near the wall is important:
Pressure Drop
Separation
Shear Effects
Recirculation
Turbulence models are generally suited to model the flow outside the boundary layer
Examination of experimental data yields a wide variety of results in the boundary layer
The above graph shows non-dimensional velocity versus non-dimensional distance from the wall. Different flows show different boundary layer profiles.

13. Turbulence Near the Wall
By scaling the variables near the wall the velocity profile data takes on a predictable form (transitioning from linear to logarithmic behavior)
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
Scaling the non-dimensional velocity and non-dimensional distance from the wall results in a predictable boundary layer profile for a wide range of flows
Linear
Logarithmic

14. Turbulence Near the Wall
Fewer nodes are needed normal to the wall when wall functions are used u y
Boundary layer
Wall functions used to resolve boundary layer
Wall functions not used to resolve boundary layer

15. Turbulence Near The Wall
y+ is the non-dimensional distance from the wall
It is used to measure the distance of the first node away from the wall y u
Boundary layer y+
Wall functions are only valid within specific y+ values
If y+ is too high the first node is outside the boundary layer and wall functions will be imposed too far into the domain
If y+ is too low the first node will lie in the laminar (viscous) part of the boundary layer where wall functions are not valid

16. Turbulence Near the Wall
In some situations, such as boundary layer separation, wall functions do not correctly predict the boundary layer profile
In these cases wall functions should not be used
Instead, directly resolving the boundary layer can provide accurate results
Not all turbulence models allow the wall functions to be turned off
Wall functions applicable
Wall functions not applicable

17. k-epsilon Model Standard k- Model
The “industrial CFD” standard since it offer a good compromise between numerical effort and computational accuracy
Wall functions are always used
y+ should typically be < 300 for the wall functions to be valid
There is no lower limit on y+
CFX uses Scalable wall functions
If your mesh results in y+ values below the valid range of the wall functions, the nodes nearest the wall are effectively ignored
This ensures valid results, within the model limitations, but is a waste of mesh
Known limitations:
Separation generally under predicted since wall functions are used
Inaccuracies with swirling flows and flows with strong streamline curvature
The most widely used turbulence model is the k-epsilon turbulence model. It is known as the “Industrial CFD” standard since it is applicable to such a broad range of problems. There are, however, a few known limitations. It is not ideal for separation prediction, swirling flows, and flows with strong streamline curvature.
Let’s take a look at these two cases to better understand what this means. CLICK for the first case we have an airfoil at low angles of attack. CLICK the velocity profile along the blade would be shaped like this…there is no separation.
Comparatively, in the second case with a high angle of attack the fluid does not stay attached to the blade. The fluid separates resulting in a low pressure region. The fluid downstream is drawn to the low pressure region CLICK resulting in recirculation. If we look at the velocity profile in this recirculation zone, we see that it does not have the usual shape. For cases such as this, SST is a much better turbulence model to use.

18. k-omega Model k- Model
One of the advantages of the k- formulation is the near wall treatment for low-Reynolds number computations
Here “low-Reynolds” refers to the turbulent Reynolds number, which is low in the viscous sub-layer, not the device Reynolds number
In other words “low-Reynolds number computations” means the near wall mesh is fine enough to resolve the laminar (viscous) part of the boundary layer which is very close to the wall
A low-Reynolds number k- model only requires y+ <= 2
If a low-Re k-e model were available, it would require a much small y+
In industrial flows, even y+ <= 2 cannot be guaranteed in most applications and for this reason, a new automatic near wall treatment was developed for the k- models
The most widely used turbulence model is the k-epsilon turbulence model. It is known as the “Industrial CFD” standard since it is applicable to such a broad range of problems. There are, however, a few known limitations. It is not ideal for separation prediction, swirling flows, and flows with strong streamline curvature.
Let’s take a look at these two cases to better understand what this means. CLICK for the first case we have an airfoil at low angles of attack. CLICK the velocity profile along the blade would be shaped like this…there is no separation.
Comparatively, in the second case with a high angle of attack the fluid does not stay attached to the blade. The fluid separates resulting in a low pressure region. The fluid downstream is drawn to the low pressure region CLICK resulting in recirculation. If we look at the velocity profile in this recirculation zone, we see that it does not have the usual shape. For cases such as this, SST is a much better turbulence model to use.

19. k-omega Model k- Model (continued)
The Automatic wall treatment for the k- models switches between a low-Reynolds number formulation (i.e. direct resolution of the boundary layer) at low y+ values and a wall function approach at higher y+ values
This lets you take advantage of a fine near-wall mesh when present
Airfoil at Mach 0.5 showing the mesh and y+ values. y+ values are >2. A finer near wall mesh is required to achieve y+ < 2.

20. Experiment Gersten et al.
SST Model
Shear Stress Transport (SST) Model
The SST model is based on the k- model and has the same automatic wall treatment
It accounts for the transport of the turbulent shear stress and gives highly accurate predictions of the onset and the amount of flow separation
This is a good default choice
SST result and experiment
k- fails to predict separation
Experiment Gersten et al.
The velocity profile in the boundary layer cannot be modeled, it must be resolved and SST allows for this. SST offers a great balance of accuracy and stability; however benefits only come with finer near-wall grid spacing.
Looking at this case, we see that k-epsilon fails to predict separation while SST results compare very well with experimental results. What should be of even more concern is the direction that k-epsilon errs on. K-epsilon predicts separation late therefore it over predicts efficiency.

21. y+ for the SST and k-omega Models
When using the SST or k- models y+ should be < 300 so that the wall function approach is valid
This will not take advantage of the low-Reynolds formulation, which is necessary for accurate separation prediction
However, the model can still be used on these coarser near-wall mesh and produce valid results, within the limitations of the wall functions
To take full advantage of the low-Reynolds formulation y+ should be < 2

22. Estimating y+ It is useful to estimate y+ before obtaining a solution
Saves time!
Use the following formula based on flow over a flat plate:
Dy is the actual distance between the wall and first node
L is a flow length scale
y+ is the desired y+ value
ReL is the Reynolds Number based on the length scale L
See the documentation for a derivation of this formula
ANSYS CFX-Solver Modeling Guide >> Turbulence and Near-Wall Modeling >> Modeling Flow Near the Wall >> Guidelines for Mesh Generation

23. Other Turbulence Models
When RANS models are not adequate, Eddy Simulation models can be used
As already mentioned, these are more computationally expensive
Large Eddy Simulation (LES)
Resolves the large eddies, models the small eddies
Problem: Requires a very fine boundary layer mesh, making it impractical for most flows
Detached Eddy Simulation (DES)
Uses a RANS model in the boundary layer, switches over to LES in the bulk flow
A “standard” boundary layer mesh can be used
Problem: the RANS to LES switch depends on the mesh, which can give unphysical results on the “wrong” mesh
Scale-Adaptive Simulation (SAS)
Like DES, but without the mesh dependency problems

24. Inlet Turbulence Conditions
Unless turbulence is being directly simulated, it is accounted for by modeling the transport of turbulence properties, for example k and ε
Similar to mass and momentum, turbulence variables require boundary condition specifications
Several options exist for the specification of turbulence quantities at inlets (details on next slide)
Unless you have absolutely no idea of the turbulence levels in your simulation (in which case, you can use the Medium (Intensity = 5%) option), you should use well chosen values of turbulence intensities and length scales
Nominal turbulence intensities range from 1% to 5% but will depend on your specific application
The default turbulence intensity value of (that is, 3.7%) is sufficient for nominal turbulence through a circular inlet, and is a good estimate in the absence of experimental data

25. Inlet Turbulence Conditions
Default Intensity and Autocompute Length Scale
The default turbulence intensity of (3.7%) is used together with a computed length scale to approximate inlet values of k and . The length scale is calculated to take into account varying levels of turbulence. In general, the autocomputed length scale is not suitable for external flows
Intensity and Autocompute Length Scale
This option allows you to specify a value of turbulence intensity but the length scale is still automatically computed. The allowable range of turbulence intensities is restricted to 0.1%-10.0% to correspond to very low and very high levels of turbulence accordingly. In general, the autocomputed length scale is not suitable for external flows
Intensity and Length Scale
You can specify the turbulence intensity and length scale directly, from which values of k and ε are calculated
Low (Intensity = 1%)
This defines a 1% intensity and a viscosity ratio equal to 1
Medium (Intensity = 5%)
This defines a 5% intensity and a viscosity ratio equal to 10
This is the recommended option if you do not have any information about the inlet turbulence
High (Intensity = 10%)
This defines a 10% intensity and a viscosity ratio equal to 100
Specified Intensity and Eddy Viscosity Ratio
Use this feature if you wish to enter your own values for intensity and viscosity ratio
k and Epsilon
Specify the values of k and ε directly
Zero Gradient
Use this setting for fully developed turbulence conditions

26. Example: Pipe Expansion with Heat Transfer
Reynolds Number ReD= 40750
Fully Developed Turbulent Flow at Inlet
Experiments by Baughn et al. (1984)
q=const .
Outlet
axis H
40 x H
Inlet
q=0 d D

27. Example: Pipe Expansion with Heat Transfer
Plot shows dimensionless distance versus Nusselt Number
Best agreement is with SST and k-omega models which do a better job of capturing flow recirculation zones accurately

28. Summary: Turbulence Modeling Guidelines
Successful turbulence modeling requires engineering judgment of:
Flow physics
Computer resources available
Project requirements
Accuracy
Turnaround time
Near-wall treatments
Modeling procedure
Calculate characteristic Re and determine whether the flow is turbulent
Estimate y+ before generating the mesh
The SST model is good choice for most flows
Use the Reynolds Stress Model or the SST model with Curvature Correction (see documentation) for highly swirling, 3-D, rotating flows
We have described the turbulence models and near-wall treatments available in Fluent CFD software and have tried to show how successful modeling of turbulent flows requires engineering judgement.