1. Modeling Turbulent Flows
General introduction that Fluent CFD software contains a variety of state-of-the-art turbulence modeling options; however, successful predictions of turbulent flows will only result from turbulence modeling decisions based on sound engineering judgement. In this lecture we will identify the issues which must be considered, discuss the turbulence modeling options available in Fluent CFD software, and describe how an engineer should decide how to model turbulent flows.

2. What is Turbulence?
Unsteady, irregular (aperiodic) motion in which transported quantities (mass, momentum, scalar species) fluctuate in time and space
Identifiable swirling patterns characterizes turbulent eddies.
Enhanced mixing (matter, momentum, energy, etc.) results
Fluid properties exhibit random variations
Statistical averaging results in accountable, turbulence related transport mechanisms.
This characteristic allows for Turbulence Modeling.
Wide range in size of turbulent eddies (scales spectrum).
Size/velocity of large eddies on order of mean flow.
derive energy from mean flow
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 mixing resulting from turbulence is important in a large number of applications.

3. Is the Flow Turbulent? External Flows Internal Flows
where
along a surface
L = x, D, Dh, etc.
around an obstacle
Other factors such as free-stream turbulence, surface conditions, and disturbances may cause earlier transition to turbulent flow.
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 transiton from laminar to turbulent flow over a range of Rayleigh numbers.
Natural Convection
where

4. Choices to be Made Flow Computational Physics Resources
Turbulence Model
&
Near-Wall Treatment
Computational
Grid
This slide tries to show that the turbulence model selection needs to be coupled with the selection of a near-wall treatment and that those decisions are closely related to development of an appropriate computational grid. The slide also tries to show the factors which the engineer must consider in selecting a turbulence modeling approach what will meet his needs best. The most obvious factor is the physics of the flow and making sure that the turbulence model selected will be able to capture the phenomena identified as important by the engineer. Another issue is what level of accuracy does the engineer require and may help determine whether or not to use a more sophisticated turbulence model. Most of us also have certain hardware restrictions in terms of RAM available, parallel processing capability, and cpu speed. Moreover, don’t forget the constraints on turnaround time imposed by the demands of an urgent project or anxious boss.
Stress the big picture approach.
Turnaround
Time
Constraints
Accuracy
Required

5. Modeling Turbulence
Direct numerical simulation (DNS) is the solution of the time-dependent Navier-Stokes equations without recourse to modeling.
Mesh must be fine enough to resolve smallest eddies, yet sufficiently large to encompass complete model.
Solution is inherently unsteady to capture convecting eddies.
DNS is only practical for simple low-Re flows.
The need to resolve the full spectrum of scales is not necessary for most engineering applications.
Mean flow properties are generally sufficient.
Most turbulence models resolve the mean flow.
Many different turbulence models are available and used.
There is no single, universally reliable engineering turbulence model for wide class of flows.
Certain models contain more physics that may be better capable of predicting more complex flows including separation, swirl, etc.
Although governing equations for turbulence are well-known, and even with todays supercomputers, we are only able to solve simple low-Re turbulent flows with DNS and we must find another approach to solve real engineering problems.

6. Modeling Approaches
‘Mean’ flow can be determined by solving a set of modified equations.
Two modeling approaches:
(1) Governing equations are ensemble or time averaged (RANS-based models).
Transport equations for mean flow quantities are solved.
All scales of turbulence are modeled.
If mean flow is unsteady, Dt is set by global unsteadiness.
(2) Governing equations are spatially averaged (LES).
Transport equations for ‘resolvable scales.’
Resolves larger eddies; models smaller ones.
Inherently unsteady, Dt set by small eddies.
Resulting models requires more CPU time/memory and is not practical for the majority of engineering applications.
Both approaches requires modeling of the scales that are averaged out.
Although governing equations for turbulence are well-known, and even with todays supercomputers, we are only able to solve simple low-Re turbulent flows with DNS and we must find another approach to solve real engineering problems.

7. RANS Modeling - Ensemble Averaging
Imagine how velocity, temperature, pressure, etc. might vary in a turbulent flow field downstream of a valve that has been slightly perturbed:
Ensemble averaging may be used to extract the mean flow properties from the instantaneous properties. U
n identifies the ‘sample’ ID
Ensemble averaging can be used for periodic or quasi-periodic flows.
Suppose the first figure represents a velocity trace at a point downstream of a valve that has been adjusted slightly. Subtle variations in the unsteady mean velocity are observed. The random turbulent fluctuations are superimposed on top of that. Every time the same adjustment is made, another trace can be generated, different from the first with respect to the turbulent fluctuations, but similar on average. The ensemble average is as defined by the equation.
u'i Ui ui t u

8. Deriving RANS Equations
Substitute mean and fluctuating velocities in instantaneous Navier-Stokes equations and average:
Reynolds Averaged Navier-Stokes equations:
where are the Reynolds Stresses.
The transported variables, U, r, p, etc., now represent the mean flow quantities
The Reynolds Stress terms are modeled using functions containing empirical constants and information about the mean flow.
To arrive at the Navier Stokes equations for mean flow quantities, we substitute the mean and fluctuating quantities and time average. The equation here is written using indicial notation.
Since the equation is written to include the unsteady term we assume we use ensemble averaging vs. long term time averaging.
Density falls outside of averaging process since we will assume incompressible flow.
Nonlinear convective term results in products of Uu and uu, uv, etc.
The time average of uv does not disappear.

9. Modeling the Reynolds Stresses
The RANS based turbulence models calculate the Reynolds Stresses by one of two methods:
(1) Using the Boussinesq assumption, the Reynolds stresses are related to the mean flow by a turbulent viscosity, mt:
Strain rate tensor, Sij, described in terms of mean flow.
Isotropic viscosity assumed
(2) Solving individual transport equations for the Reynolds stresses.
Turbulent viscosity is not employed, no assumption of isotropy
Contains more “physics”
More complex and computationally expensive than (1)
2Sij
This is where the term modeling in Turbulence Modeling becomes a reality.
There are two approaches in this case for defining the Reynolds stresses.

10. Calculating mt for Boussinesq Formula
Based on dimensional arguments, mt can be determined from the conservative variables k, e or w.
is the turbulent kinetic energy
is the dissipation rate of k (to thermal energy)
is the specific dissipation rate
mt is calculated differently depending upon the turbulence model.
Spalart-Allmaras
This ‘single equation’ model solves one additional transport equation for a modified viscosity.
Standard k-e, RNG k-e, Realizable k-e
These ‘two equation’ models solve transport equations for k and e.
Standard k-w, SST k-w
These ‘two equation’ models solve transport equations for k and w.
This is where the term modeling in Turbulence Modeling becomes a reality.
There are two approaches in this case for defining the Reynolds stresses.

11. Example: Fluent’s Standard k- Model
Transport equation for k:
Transport equation for e:
Turbulent viscosity:
are empirically defined constants.
Turbulence modeling options account for:
viscous heating (in energy equation)
compressibility effects, YM (activated when ideal gas is used)
buoyancy, Gb
user defined sources, Sk and Se
These transport equations represent a basic bookkeepping approach applied to a differential control volume where convection of the parameter is equal to the sum of the generation, diffusion, and dissipation terms. The simplified equations for steady, incompressible flow without body forces are shown here for clarity. The turbulent kinetic energy equation can be derived directly from the transport equations for the velocity fluctuations. The exact equation for the dissipation rate is very complex and virtually all turbulence models that use the dissipation rate use a transport equation that is “modeled” and has the same types of terms as the TKE transport equation.

12. Example: Fluent’s Reynolds Stress Model
Transport equation for Rij:
Generation:
Pressure-Strain Redistribution:
Dissipation:
Diffusion:
(computed)
(modeled)
Point out each term and note that the generation term does not require modeling. Terms should look familiar since the contraction of this equation results in the exact turbulent kinetic energy equation.
(related to e)
(modeled)
Pressure/velocity
fluctuations
Turbulent
transport
Molecular

13. Turbulence Models in Fluent
Zero-Equation Models
One-Equation Models
Spalart-Allmaras
Two-Equation Models
Standard k-e
RNG k-e
Realizable k-e
Standard k-w
SST k-w
Reynolds-Stress Model
Large-Eddy Simulation
Direct Numerical Simulation
RANS-based
models
Increase
Computational
Cost
Per Iteration
Available
in FLUENT 6
Over the years, many different types of turbulence models have been developed and this slide tries to show you where the turbulence models available in Fluent CFD software fit in the grand scheme of things. The various types of turbulence models are listed in the center in order of increasing sophistication or increasing inclusion of more physics (more physics is supposed to be good !?). Unfortunately, inclusion of more physics usally increases the computational cost. Engineers must find the turbulence modeling approach that satisfies their technical needs at the lowest cost! Fluent Inc. offers several state-of-the-art turbulence modeling options that provide the accuracy that engineers need at reasonable computational expense. These turbulence models include the two-equation and RSM models. We are also working to expand the number of turbulence modeling options available to take advantage of progres in turbulence modeling field that improve the accuracy of one-equation models, develop a k-e model that is fully realizable, and make LES methods more practical.

14. RANS Turbulence Model Descriptions

15. RANS Turbulence Model Behavior and Usage

16. Large Eddy Simulation (LES)
Motivation:
Large eddies:
Mainly responsible for transport of momentum, energy, and other scalars, directly affecting the mean fields.
Anisotropic, subjected to history effects, and flow-dependent, i.e., strongly dependent on flow configuration, boundary conditions, and flow parameters.
Small eddies tend to be more isotropic, less flow-dependent, and hence more amenable to modeling.
Approach:
LES resolves large eddies and models only small eddies.
Equations are similar in form to RANS equations
Dependent variables are now spatially averaged instead of time averaged.
Large computational effort
Number of grid points, NLES 
Unsteady calculation

17. Problem Setup for LES Small eddies defined by grid cell size.
Good starting point for cell size is Taylor length scale, l  (10nk/e)1/2.
You can use a two-equation model on coarse mesh to determine range of k and e.
Time step size can be defined by: t  l/U.
Effective stress requires definition of Subgrid Scale viscosity.
Smagorinsky-Lilly model
constant Cs must be tuned to flow
RNG-based model
Useful for low Re flows where mt  m
Post-processing
Statistically time averaged results are available.

18. Modeling the Near-Wall Region
Accurate near-wall modeling is important for most engineering applications.
Successful prediction of frictional drag, pressure drop, separation, etc., depends on fidelity of local wall shear predictions.
Most k-e and RSM turbulence models will not predict correct near-wall behavior if integrated down to the wall.
Problem is the inability to resolve e.
Special near-wall treatment is required.
Standard Wall Functions
Non-Equilibrium Wall Functions
Enhanced wall treatment
S-A and k-w models are capable of resolving the near-wall flow provided near-wall mesh is sufficient. u+ y+
Make sure to identify the various regions in the figure showing the boundary-layer structure because those regions will be discussed later in the near-wall treatment section. Also, point out the the first grid point should lie in the log-law region if wall functions are used or at y+=1 if the two-layer zonal model is used.

19. Near-Wall Modeling Options
In general, ‘wall functions’ are a collection or set of laws that serve as boundary conditions for momentum, energy, and species as well as for turbulence quantities.
Wall Function Options
The Standard and Non-equilibrium Wall Function options refer to specific ‘sets’ designed for high Re flows.
The viscosity affected, near-wall region is not resolved.
Near-wall mesh is relatively coarse.
Cell center information bridged by empirically-based wall functions.
Enhanced Wall Treatment Option
This near-wall model combines the use of enhanced wall functions and a two-layer model.
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.
Turbulence models are modified for ‘inner’ layer.
inner layer
outer layer
Given the behavior discussed on the previous slide, there are two basic approaches for handling the flow near walls…

20. Standard and Non-Equilibrium Wall Functions
Standard Wall Function
Momentum boundary condition based on Launder-Spaulding law-of-the-wall:
Similar ‘wall laws’ apply for energy and species.
Additional formulas account for k, e, and ruiuj.
Less reliable when flow departs from conditions assumed in their derivation.
Severe p or highly non-equilibrium near-wall flows, high transpiration or body forces, low Re or highly 3D flows
Non-Equilibrium Wall Function
SWF is modified to account for stronger p and non-equilibrium flows.
Useful for mildly separating, reattaching, or impinging flows.
Less reliable for high transpiration or body forces, low Re or highly 3D flows.
The Standard and Non-Equilibrium Wall functions are options for the k-e and RSM turbulence models.
where
for .
The inclusion of k^1/2 as an additional velocity scale in the wall function allows the wall function to be less susceptible to error for slight non-equilibrium boundary layer flows. If k^1/2 over u_tau =1 then the flow is at equilbrium. >1 implies non equilbrium(higher freestream turbulent flows, or wall jets…) and <1 implies developing boundary layer.
In solving for tke at the wall adjacent cell, production does not cancel out dissipation explicitly since they are calculated separately. This allows partial non-equilibrium flows.
The impact of having the wall adjacent cells to be too near the wall is that excessive tke will be generated, increasing turbulent viscosity, turbulent mixing, etc.

21. Enhanced Wall Treatment
Enhanced wall functions
Momentum boundary condition based on blended law-of-the-wall (Kader).
Similar blended ‘wall laws’ apply for energy, species, and w.
Kader’s form for blending allows for incorporation of additional physics.
Pressure gradient effects
Thermal (including compressibility) effects
Two-layer model
A blended two-layer model is used to determine near-wall e field.
Domain is divided into viscosity-affected (near-wall) region and turbulent core region.
Based on ‘wall-distance’ turbulent Reynolds number:
Zoning is dynamic and solution adaptive.
High Re turbulence model used in outer layer.
‘Simple’ turbulence model used in inner layer.
Solutions for e and mt in each region are blended, e.g.,
The Enhanced Wall Treatment near-wall model are options for the k-e and RSM turbulence models.
Slide is self-explanatory.

22. Estimating Placement of First Grid Point
Ability for near-wall treatments to accurately predict near-wall flows depends on placement of wall adjacent cell centroids (cell size).
For SWF and NWF, centroid should be located in log-layer:
For best results using EWT, centroid should be located in laminar sublayer:
This near-wall treatment can accommodate cells placed in the log-layer.
To determine actual size of wall adjacent cells, recall that:
The skin friction coefficient can be estimated from empirical correlations:
Flat Plate-
Pipe Flow-
Use post-processing to confirm near-wall mesh resolution.

23. Near-Wall Modeling Recommended Strategy
Use SWF or NWF for most high Re applications (Re > 106) for which you cannot afford to resolve the viscous sublayer.
There is little gain from resolving viscous sublayer (choice of core turbulence model is more important).
Use NWF for mildly separating, reattaching, or impinging flows.
You may consider using EWT if:
The characteristic Re is low or if near wall characteristics need to be resolved.
The same or similar cases ran successfully previously with the two-layer zonal model (in Fluent v5).
The physics and near-wall mesh of the case is such that y+ is likely to vary significantly over a wide portion of the wall region.
Try to make the mesh either coarse or fine enough, and avoid putting the wall-adjacent cells in the buffer layer (y+ = 5 ~ 30).

24. Setting Boundary Conditions
When turbulent flow enters a domain at inlets or outlets (backflow), boundary values for:
k, , w and/or must be specified
Four methods for directly or indirectly specifying turbulence parameters:
Explicitly input k, , w, or
This is the only method that allows for profile definition.
Turbulence intensity and length scale
Length scale is related to size of large eddies that contain most of energy.
For boundary layer flows: l  0.4d99
For flows downstream of grid: l  opening size
Turbulence intensity and hydraulic diameter
Ideally suited for duct and pipe flows
Turbulence intensity and turbulent viscosity ratio
For external flows: 1 < mt/m < 10
Turbulence intensity depends on upstream conditions:
The code will internally calculate k and e based on turbulence intensity and other parameters input.
Non-uniform profiles would be done using user defined functions.

25. GUI for Turbulence Models
Define  Models  Viscous...
Inviscid, Laminar, or Turbulent
Turbulence Model options
Although model constants may be changed, the need to do this is very rare.
Near Wall Treatments
Additional Turbulence options

26. Example: Ship Hull Flow
Experiments: KRISO’s 300K VLCC (1998)
Complex, high ReL (4.6  106) 3D Flow
Thick 3D boundary layer in moderate pressure gradient
Streamline curvature
Crossflow
Free vortex-sheet formation (“open separation”)
Streamwise vortices embedded in TBL and wake
Simulation
Wall Functions used to manage mesh size.
y+ 
Hex mesh  ~200,000 cells
Experimentally derived contours of axial velocity

27. Comparing Contour Plots of Axial VelocitySA
RKE
RNG
SKE
SKO
RSM
SKO and RSM models capture characteristic shape at propeller plane.

28. Comparing Wake Fraction and Drag
Though SKO (and SST) were able to resolve salient features in propeller plane, not all aspects of flow could be accurately captured.
Eddy viscosity model
RSM models accurately capture all aspects of the flow.
Complex industrial flows provide new challenges to turbulence models.

29. Summary: Turbulence Modeling Guidelines
Successful turbulence modeling requires engineering judgement of:
Flow physics
Computer resources available
Project requirements
Accuracy
Turnaround time
Turbulence models & near-wall treatments that are available
Modeling Procedure
Calculate characteristic Re and determine if Turbulence needs modeling.
Estimate wall-adjacent cell centroid y+ first before generating mesh.
Begin with SKE (standard k-) and change to RNG, RKE, SKO, or SST if needed.
Use RSM for highly swirling flows.
Use wall functions unless low-Re flow and/or complex near-wall physics are present.
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.