chapter 10. modeling turbulence - enea · chapter 10. modeling turbulence this chapter provides...

Chapter 10. Modeling Turbulence

This chapter provides details about the turbulence models available inFLUENT.

Information is presented in the following sections:

• Section 10.1: Introduction

• Section 10.2: Choosing a Turbulence Model

• Section 10.3: The Spalart-Allmaras Model

• Section 10.4: The Standard, RNG, and Realizable k-ε Models

• Section 10.5: The Standard and Shear-Stress Transport (SST) k-ωModels

• Section 10.6: The Reynolds Stress Model (RSM)

• Section 10.7: The Large Eddy Simulation (LES) Model

• Section 10.8: Near-Wall Treatments for Wall-Bounded TurbulentFlows

• Section 10.9: Grid Considerations for Turbulent Flow Simulations

• Section 10.10: Problem Setup for Turbulent Flows

• Section 10.11: Solution Strategies for Turbulent Flow Simulations

• Section 10.12: Postprocessing for Turbulent Flows

c© Fluent Inc. November 28, 2001 10-1

Modeling Turbulence

10.1 Introduction

Turbulent flows are characterized by fluctuating velocity fields. Thesefluctuations mix transported quantities such as momentum, energy, andspecies concentration, and cause the transported quantities to fluctuateas well. Since these fluctuations can be of small scale and high frequency,they are too computationally expensive to simulate directly in practicalengineering calculations. Instead, the instantaneous (exact) governingequations can be time-averaged, ensemble-averaged, or otherwise manip-ulated to remove the small scales, resulting in a modified set of equationsthat are computationally less expensive to solve. However, the modifiedequations contain additional unknown variables, and turbulence modelsare needed to determine these variables in terms of known quantities.

FLUENT provides the following choices of turbulence models:

• Spalart-Allmaras model

• k-ε models

– Standard k-ε model

– Renormalization-group (RNG) k-ε model

– Realizable k-ε model

• k-ω models

– Standard k-ω model

– Shear-stress transport (SST) k-ω model

• Reynolds stress model (RSM)

• Large eddy simulation (LES) model

10-2 c© Fluent Inc. November 28, 2001

10.2 Choosing a Turbulence Model


It is an unfortunate fact that no single turbulence model is universallyaccepted as being superior for all classes of problems. The choice ofturbulence model will depend on considerations such as the physics en-compassed in the flow, the established practice for a specific class ofproblem, the level of accuracy required, the available computational re-sources, and the amount of time available for the simulation. To makethe most appropriate choice of model for your application, you need tounderstand the capabilities and limitations of the various options.

The purpose of this section is to give an overview of issues related tothe turbulence models provided in FLUENT. The computational effortand cost in terms of CPU time and memory of the individual models isdiscussed. While it is impossible to state categorically which model isbest for a specific application, general guidelines are presented to helpyou choose the appropriate turbulence model for the flow you want tomodel.

10.2.1 Reynolds-Averaged Approach vs. LES

A complete time-dependent solution of the exact Navier-Stokes equa-tions for high-Reynolds-number turbulent flows in complex geometriesis unlikely to be attainable for some time to come. Two alternativemethods can be employed to transform the Navier-Stokes equations insuch a way that the small-scale turbulent fluctuations do not have tobe directly simulated: Reynolds averaging and filtering. Both methodsintroduce additional terms in the governing equations that need to bemodeled in order to achieve “closure”. (Closure implies that there are asufficient number of equations for all the unknowns.)

The Reynolds-averaged Navier-Stokes (RANS) equations represent trans-port equations for the mean flow quantities only, with all the scales ofthe turbulence being modeled. The approach of permitting a solutionfor the mean flow variables greatly reduces the computational effort. Ifthe mean flow is steady, the governing equations will not contain timederivatives and a steady-state solution can be obtained economically. Acomputational advantage is seen even in transient situations, since thetime step will be determined by the global unsteadiness in the mean


Modeling Turbulence

flow rather than by the turbulence. The Reynolds-averaged approach isgenerally adopted for practical engineering calculations, and uses modelssuch as Spalart-Allmaras, k-ε and its variants, k-ω and its variants, andthe RSM.

LES provides an alternative approach in which the large eddies are com-puted in a time-dependent simulation that uses a set of “filtered” equa-tions. Filtering is essentially a manipulation of the exact Navier-Stokesequations to remove only the eddies that are smaller than the size of thefilter, which is usually taken as the mesh size. Like Reynolds averaging,the filtering process creates additional unknown terms that must be mod-eled in order to achieve closure. Statistics of the mean flow quantities,which are generally of most engineering interest, are gathered during thetime-dependent simulation. The attraction of LES is that, by modelingless of the turbulence (and solving more), the error induced by the tur-bulence model will be reduced. One might also argue that it ought to beeasier to find a “universal” model for the small scales, which tend to bemore isotropic and less affected by the macroscopic flow features thanthe large eddies.

It should, however, be stressed that the application of LES to industrialfluid simulations is in its infancy. As highlighted in a recent review publi-cation [72], typical applications to date have been for simple geometries.This is mainly because of the large computer resources required to re-solve the energy-containing turbulent eddies. Most successful LES hasbeen done using high-order spatial discretization, with great care beingtaken to resolve all scales larger than the inertial subrange. The degra-dation of accuracy in the mean flow quantities with poorly resolved LESis not well documented. In addition, the use of wall functions with LESis an approximation that requires further validation.

As a general guideline, therefore, it is recommended that the conven-tional turbulence models employing the Reynolds-averaged approach beused for practical calculations. The LES approach, described further inSection 10.7, has been made available for you to try if you have the com-putational resources and are willing to invest the effort. The rest of thissection will deal with the choice of models using the Reynolds-averagedapproach.



10.2.2 Reynolds (Ensemble) Averaging

In Reynolds averaging, the solution variables in the instantaneous (ex-act) Navier-Stokes equations are decomposed into the mean (ensemble-averaged or time-averaged) and fluctuating components. For the velocitycomponents:

ui = ui + u′i (10.2-1)

where ui and u′i are the mean and fluctuating velocity components (i =1, 2, 3).

Likewise, for pressure and other scalar quantities:

φ = φ+ φ′ (10.2-2)

where φ denotes a scalar such as pressure, energy, or species concentra-tion.

Substituting expressions of this form for the flow variables into the in-stantaneous continuity and momentum equations and taking a time (orensemble) average (and dropping the overbar on the mean velocity, u)yields the ensemble-averaged momentum equations. They can be writtenin Cartesian tensor form as:

∂ρ

∂t+

∂

∂xi(ρui) = 0 (10.2-3)

∂

∂t(ρui) +

∂

∂xj(ρuiuj) =

− ∂p

∂xi+

∂

∂xj

[µ

(∂ui

∂xj+∂uj

∂xi− 2

3δij∂ul

∂xl

)]+

∂

∂xj(−ρu′iu′j) (10.2-4)

Equations 10.2-3 and 10.2-4 are called Reynolds-averaged Navier-Stokes(RANS) equations. They have the same general form as the instan-taneous Navier-Stokes equations, with the velocities and other solutionvariables now representing ensemble-averaged (or time-averaged) values.


Modeling Turbulence

Additional terms now appear that represent the effects of turbulence.These Reynolds stresses, −ρu′iu′j , must be modeled in order to closeEquation 10.2-4.

For variable-density flows, Equations 10.2-3 and 10.2-4 can be interpretedas Favre-averaged Navier-Stokes equations [91], with the velocities rep-resenting mass-averaged values. As such, Equations 10.2-3 and 10.2-4can be applied to density-varying flows.

10.2.3 Boussinesq Approach vs. Reynolds Stress TransportModels

The Reynolds-averaged approach to turbulence modeling requires thatthe Reynolds stresses in Equation 10.2-4 be appropriately modeled. Acommon method employs the Boussinesq hypothesis [91] to relate theReynolds stresses to the mean velocity gradients:

−ρu′iu′j = µt

(∂ui

∂xj+∂uj

∂xi

)− 2

3

(ρk + µt

∂ui

∂xi

)δij (10.2-5)

The Boussinesq hypothesis is used in the Spalart-Allmaras model, thek-ε models, and the k-ω models. The advantage of this approach is therelatively low computational cost associated with the computation of theturbulent viscosity, µt. In the case of the Spalart-Allmaras model, onlyone additional transport equation (representing turbulent viscosity) issolved. In the case of the k-ε and k-ω models, two additional transportequations (for the turbulence kinetic energy, k, and either the turbulencedissipation rate, ε, or the specific dissipation rate, ω) are solved, and µt

is computed as a function of k and ε. The disadvantage of the Boussi-nesq hypothesis as presented is that it assumes µt is an isotropic scalarquantity, which is not strictly true.

The alternative approach, embodied in the RSM, is to solve transportequations for each of the terms in the Reynolds stress tensor. An addi-tional scale-determining equation (normally for ε) is also required. Thismeans that five additional transport equations are required in 2D flowsand seven additional transport equations must be solved in 3D.



In many cases, models based on the Boussinesq hypothesis perform verywell, and the additional computational expense of the Reynolds stressmodel is not justified. However, the RSM is clearly superior for situationsin which the anisotropy of turbulence has a dominant effect on the meanflow. Such cases include highly swirling flows and stress-driven secondaryflows.

10.2.4 The Spalart-Allmaras Model

The Spalart-Allmaras model is a relatively simple one-equation modelthat solves a modeled transport equation for the kinematic eddy (tur-bulent) viscosity. This embodies a relatively new class of one-equationmodels in which it is not necessary to calculate a length scale related tothe local shear layer thickness. The Spalart-Allmaras model was designedspecifically for aerospace applications involving wall-bounded flows andhas been shown to give good results for boundary layers subjected to ad-verse pressure gradients. It is also gaining popularity for turbomachineryapplications.

In its original form, the Spalart-Allmaras model is effectively a low-Reynolds-number model, requiring the viscous-affected region of theboundary layer to be properly resolved. In FLUENT, however, the Spalart-Allmaras model has been implemented to use wall functions when themesh resolution is not sufficiently fine. This might make it the bestchoice for relatively crude simulations on coarse meshes where accurateturbulent flow computations are not critical. Furthermore, the near-wallgradients of the transported variable in the model are much smaller thanthe gradients of the transported variables in the k-ε or k-ω models. Thismight make the model less sensitive to numerical error when non-layeredmeshes are used near walls. See Section 5.1.2 for further discussion ofnumerical error.

On a cautionary note, however, the Spalart-Allmaras model is still rel-atively new, and no claim is made regarding its suitability to all typesof complex engineering flows. For instance, it cannot be relied on topredict the decay of homogeneous, isotropic turbulence. Furthermore,one-equation models are often criticized for their inability to rapidly ac-commodate changes in length scale, such as might be necessary when


Modeling Turbulence

the flow changes abruptly from a wall-bounded to a free shear flow.

10.2.5 The Standard k-ε Model

The simplest “complete models” of turbulence are two-equation modelsin which the solution of two separate transport equations allows the tur-bulent velocity and length scales to be independently determined. Thestandard k-ε model in FLUENT falls within this class of turbulence modeland has become the workhorse of practical engineering flow calculationsin the time since it was proposed by Launder and Spalding [128]. Ro-bustness, economy, and reasonable accuracy for a wide range of turbulentflows explain its popularity in industrial flow and heat transfer simula-tions. It is a semi-empirical model, and the derivation of the modelequations relies on phenomenological considerations and empiricism.

As the strengths and weaknesses of the standard k-ε model have becomeknown, improvements have been made to the model to improve its per-formance. Two of these variants are available in FLUENT: the RNG k-εmodel [272] and the realizable k-ε model [209].

10.2.6 The RNG k-ε Model

The RNG k-ε model was derived using a rigorous statistical technique(called renormalization group theory). It is similar in form to the stan-dard k-ε model, but includes the following refinements:

• The RNG model has an additional term in its ε equation thatsignificantly improves the accuracy for rapidly strained flows.

• The effect of swirl on turbulence is included in the RNG model,enhancing accuracy for swirling flows.

• The RNG theory provides an analytical formula for turbulentPrandtl numbers, while the standard k-ε model uses user-specified,constant values.

• While the standard k-ε model is a high-Reynolds-number model,the RNG theory provides an analytically-derived differential for-mula for effective viscosity that accounts for low-Reynolds-number



effects. Effective use of this feature does, however, depend on anappropriate treatment of the near-wall region.

These features make the RNG k-ε model more accurate and reliable fora wider class of flows than the standard k-ε model.

10.2.7 The Realizable k-ε Model

The realizable k-ε model is a relatively recent development and differsfrom the standard k-ε model in two important ways:

• The realizable k-ε model contains a new formulation for the tur-bulent viscosity.

• A new transport equation for the dissipation rate, ε, has been de-rived from an exact equation for the transport of the mean-squarevorticity fluctuation.

The term “realizable” means that the model satisfies certain mathemat-ical constraints on the Reynolds stresses, consistent with the physics ofturbulent flows. Neither the standard k-ε model nor the RNG k-ε modelis realizable.

An immediate benefit of the realizable k-ε model is that it more accu-rately predicts the spreading rate of both planar and round jets. It isalso likely to provide superior performance for flows involving rotation,boundary layers under strong adverse pressure gradients, separation, andrecirculation.

Both the realizable and RNG k-ε models have shown substantial im-provements over the standard k-ε model where the flow features includestrong streamline curvature, vortices, and rotation. Since the model isstill relatively new, it is not clear in exactly which instances the real-izable k-ε model consistently outperforms the RNG model. However,initial studies have shown that the realizable model provides the bestperformance of all the k-ε model versions for several validations of sep-arated flows and flows with complex secondary flow features.


Modeling Turbulence

One limitation of the realizable k-εmodel is that it produces non-physicalturbulent viscosities in situations when the computational domain con-tains both rotating and stationary fluid zones (e.g., multiple referenceframes, rotating sliding meshes). This is due to the fact that the realiz-able k-ε model includes the effects of mean rotation in the definition ofthe turbulent viscosity (see Equations 10.4-17–10.4-19). This extra ro-tation effect has been tested on single rotating reference frame systemsand showed superior behavior over the standard k-εmodel. However, dueto the nature of this modification, its application to multiple referenceframe systems should be taken with some caution.

10.2.8 The Standard k-ω Model

The standard k-ω model in FLUENT is based on the Wilcox k-ω model [267],which incorporates modifications for low-Reynolds-number effects, com-pressibility, and shear flow spreading. The Wilcox model predicts freeshear flow spreading rates that are in close agreement with measure-ments for far wakes, mixing layers, and plane, round, and radial jets,and is thus applicable to wall-bounded flows and free shear flows. Avariation of the standard k-ω model called the SST k-ω model is alsoavailable in FLUENT, and is described in Section 10.2.9.

10.2.9 The Shear-Stress Transport (SST) k-ω Model

The shear-stress transport (SST) k-ω model was developed by Menter [153]to effectively blend the robust and accurate formulation of the k-ω modelin the near-wall region with the free-stream independence of the k-εmodel in the far field. To achieve this, the k-ε model is converted intoa k-ω formulation. The SST k-ω model is similar to the standard k-ωmodel, but includes the following refinements:

• The standard k-ω model and the transformed k-ε model are bothmultiplied by a blending function and both models are added to-gether. The blending function is designed to be one in the near-wall region, which activates the standard k-ω model, and zero awayfrom the surface, which activates the transformed k-ε model.



• The SST model incorporates a damped cross-diffusion derivativeterm in the ω equation.

• The definition of the turbulent viscosity is modified to account forthe transport of the turbulent shear stress.

• The modeling constants are different.

These features make the SST k-ω model more accurate and reliable fora wider class of flows (e.g., adverse pressure gradient flows, airfoils, tran-sonic shock waves) than the standard k-ω model.

10.2.10 The Reynolds Stress Model (RSM)

The Reynolds stress model (RSM) is the most elaborate turbulencemodel that FLUENT provides. Abandoning the isotropic eddy-viscosityhypothesis, the RSM closes the Reynolds-averaged Navier-Stokes equa-tions by solving transport equations for the Reynolds stresses, togetherwith an equation for the dissipation rate. This means that four addi-tional transport equations are required in 2D flows and seven additionaltransport equations must be solved in 3D.

Since the RSM accounts for the effects of streamline curvature, swirl,rotation, and rapid changes in strain rate in a more rigorous mannerthan one-equation and two-equation models, it has greater potential togive accurate predictions for complex flows. However, the fidelity ofRSM predictions is still limited by the closure assumptions employed tomodel various terms in the exact transport equations for the Reynoldsstresses. The modeling of the pressure-strain and dissipation-rate termsis particularly challenging, and often considered to be responsible forcompromising the accuracy of RSM predictions.

The RSM might not always yield results that are clearly superior to thesimpler models in all classes of flows to warrant the additional compu-tational expense. However, use of the RSM is a must when the flowfeatures of interest are the result of anisotropy in the Reynolds stresses.Among the examples are cyclone flows, highly swirling flows in combus-tors, rotating flow passages, and the stress-induced secondary flows inducts.


Modeling Turbulence

10.2.11 Computational Effort: CPU Time and SolutionBehavior

In terms of computation, the Spalart-Allmaras model is the least expen-sive turbulence model of the options provided in FLUENT, since only oneturbulence transport equation is solved.

The standard k-ε model clearly requires more computational effort thanthe Spalart-Allmaras model since an additional transport equation issolved. The realizable k-ε model requires only slightly more computa-tional effort than the standard k-ε model. However, due to the extraterms and functions in the governing equations and a greater degree ofnon-linearity, computations with the RNG k-ε model tend to take 10–15% more CPU time than with the standard k-ε model. Like the k-εmodels, the k-ω models are also two-equation models, and thus requireabout the same computational effort.

Compared with the k-ε and k-ω models, the RSM requires additionalmemory and CPU time due to the increased number of the transportequations for Reynolds stresses. However, efficient programming in FLU-ENT has reduced the CPU time per iteration significantly. On average,the RSM in FLUENT requires 50–60% more CPU time per iteration com-pared to the k-ε and k-ω models. Furthermore, 15–20% more memory isneeded.

Aside from the time per iteration, the choice of turbulence model canaffect the ability of FLUENT to obtain a converged solution. For example,the standard k-ε model is known to be slightly over-diffusive in certainsituations, while the RNG k-ε model is designed such that the turbulentviscosity is reduced in response to high rates of strain. Since diffusionhas a stabilizing effect on the numerics, the RNG model is more likelyto be susceptible to instability in steady-state solutions. However, thisshould not necessarily be seen as a disadvantage of the RNG model,since these characteristics make it more responsive to important physicalinstabilities such as time-dependent turbulent vortex shedding.

Similarly, the RSM may take more iterations to converge than the k-εand k-ω models due to the strong coupling between the Reynolds stressesand the mean flow.


10.3 The Spalart-Allmaras Model


In turbulence models that employ the Boussinesq approach, the centralissue is how the eddy viscosity is computed. The model proposed bySpalart and Allmaras [226] solves a transport equation for a quantitythat is a modified form of the turbulent kinematic viscosity.

10.3.1 Transport Equation for the Spalart-Allmaras Model

The transported variable in the Spalart-Allmaras model, ν, is identicalto the turbulent kinematic viscosity except in the near-wall (viscous-affected) region. The transport equation for ν is

∂

∂t(ρν) +

∂

∂xi(ρνui) =

Gν +1σν

∂

∂xj

(µ+ ρν)

∂ν

∂xj

+ Cb2ρ

(∂ν

∂xj

)2− Yν + Sν (10.3-1)

where Gν is the production of turbulent viscosity and Yν is the destruc-tion of turbulent viscosity that occurs in the near-wall region due towall blocking and viscous damping. σν and Cb2 are constants and νis the molecular kinematic viscosity. Sν is a user-defined source term.Note that since the turbulence kinetic energy k is not calculated in theSpalart-Allmaras model, it is not taken into account when estimatingthe Reynolds stresses in Equation 10.2-5.

10.3.2 Modeling the Turbulent Viscosity

The turbulent viscosity, µt, is computed from

µt = ρνfv1 (10.3-2)

where the viscous damping function, fv1, is given by

fv1 =χ3

χ3 +C3v1

(10.3-3)

and


Modeling Turbulence

χ ≡ ν

ν(10.3-4)

10.3.3 Modeling the Turbulent Production

The production term, Gν , is modeled as

Gν = Cb1ρSν (10.3-5)

where

S ≡ S +ν

κ2d2fv2 (10.3-6)

and

fv2 = 1 − χ

1 + χfv1(10.3-7)

Cb1 and κ are constants, d is the distance from the wall, and S is ascalar measure of the deformation tensor. By default in FLUENT, as inthe original model proposed by Spalart and Allmaras, S is based on themagnitude of the vorticity:

S ≡√

2ΩijΩij (10.3-8)

where Ωij is the mean rate-of-rotation tensor and is defined by

Ωij =12

(∂ui

∂xj− ∂uj

∂xi

)(10.3-9)

The justification for the default expression for S is that, for the wall-bounded flows that were of most interest when the model was formu-lated, turbulence is found only where vorticity is generated near walls.However, it has since been acknowledged that one should also take intoaccount the effect of mean strain on the turbulence production, and a



modification to the model has been proposed [46] and incorporated intoFLUENT.

This modification combines measures of both rotation and strain tensorsin the definition of S:

S ≡ |Ωij| + Cprod min (0, |Sij | − |Ωij|) (10.3-10)

where

Cprod = 2.0, |Ωij | ≡√

2ΩijΩij, |Sij | ≡√

2SijSij

with the mean strain rate, Sij, defined as

Sij =12

(∂uj

∂xi+∂ui

∂xj

)(10.3-11)

Including both the rotation and strain tensors reduces the productionof eddy viscosity and consequently reduces the eddy viscosity itself inregions where the measure of vorticity exceeds that of strain rate. Onesuch example can be found in vortical flows, i.e., flow near the core of avortex subjected to a pure rotation where turbulence is known to be sup-pressed. Including both the rotation and strain tensors more correctlyaccounts for the effects of rotation on turbulence. The default option (in-cluding the rotation tensor only) tends to overpredict the production ofeddy viscosity and hence overpredicts the eddy viscosity itself in certaincircumstances.

You can select the modified form for calculating production in the ViscousModel panel.

10.3.4 Modeling the Turbulent Destruction

The destruction term is modeled as

Yν = Cw1ρfw

(ν

d

)2

(10.3-12)


Modeling Turbulence

where

fw = g

[1 + C6

w3

g6 + C6w3

]1/6

(10.3-13)

g = r + Cw2

(r6 − r

)(10.3-14)

r ≡ ν

Sκ2d2(10.3-15)

Cw1, Cw2, and Cw3 are constants, and S is given by Equation 10.3-6.Note that the modification described above to include the effects of meanstrain on S will also affect the value of S used to compute r.

10.3.5 Model Constants

The model constants Cb1, Cb2, σν , Cv1, Cw1, Cw2, Cw3, and κ have the fol-lowing default values [226]:

Cb1 = 0.1335, Cb2 = 0.622, σν =23, Cv1 = 7.1

Cw1 =Cb1

κ2+

(1 + Cb2)σν

, Cw2 = 0.3, Cw3 = 2.0, κ = 0.4187

10.3.6 Wall Boundary Conditions

At walls, the modified turbulent kinematic viscosity, ν, is set to zero.

When the mesh is fine enough to resolve the laminar sublayer, the wallshear stress is obtained from the laminar stress-strain relationship:

u

uτ=ρuτy

µ(10.3-16)



If the mesh is too coarse to resolve the laminar sublayer, it is assumedthat the centroid of the wall-adjacent cell falls within the logarithmicregion of the boundary layer, and the law-of-the-wall is employed:

u

uτ=

1κ

lnE(ρuτy

µ

)(10.3-17)

where u is the velocity parallel to the wall, uτ is the shear velocity, y isthe distance from the wall, κ is the von Karman constant (0.4187), andE = 9.793.

10.3.7 Convective Heat and Mass Transfer Modeling

In FLUENT, turbulent heat transport is modeled using the concept ofReynolds’ analogy to turbulent momentum transfer. The “modeled”energy equation is thus given by the following:

∂

∂t(ρE) +

∂

∂xi[ui(ρE + p)] =

∂

∂xj

[(k +

cpµt

Prt

)∂T

∂xj+ ui(τij)eff

]+ Sh

(10.3-18)

where k, in this case, is the thermal conductivity, E is the total energy,and (τij)eff is the deviatoric stress tensor, defined as

(τij)eff = µeff

(∂uj

∂xi+∂ui

∂xj

)− 2

3µeff

∂ui

∂xiδij

The term involving (τij)eff represents the viscous heating, and is alwayscomputed in the coupled solvers. It is not computed by default in thesegregated solver, but it can be enabled in the Viscous Model panel. Thedefault value of the turbulent Prandtl number is 0.85. You can changethe value of Prt in the Viscous Model panel.

Turbulent mass transfer is treated similarly, with a default turbulentSchmidt number of 0.7. This default value can be changed in the ViscousModel panel.


Modeling Turbulence

Wall boundary conditions for scalar transport are handled analogouslyto momentum, using the appropriate “law-of-the-wall”.

10.4 The Standard, RNG, and Realizable k-ε Models

This section presents the standard, RNG, and realizable k-ε models. Allthree models have similar forms, with transport equations for k and ε.The major differences in the models are as follows:

• the method of calculating turbulent viscosity

• the turbulent Prandtl numbers governing the turbulent diffusionof k and ε

• the generation and destruction terms in the ε equation

The transport equations, methods of calculating turbulent viscosity, andmodel constants are presented separately for each model. The featuresthat are essentially common to all models follow, including turbulentproduction, generation due to buoyancy, accounting for the effects ofcompressibility, and modeling heat and mass transfer.

10.4.1 The Standard k-ε Model

The standard k-ε model [128] is a semi-empirical model based on modeltransport equations for the turbulence kinetic energy (k) and its dissi-pation rate (ε). The model transport equation for k is derived from theexact equation, while the model transport equation for ε was obtainedusing physical reasoning and bears little resemblance to its mathemati-cally exact counterpart.

In the derivation of the k-ε model, it was assumed that the flow is fullyturbulent, and the effects of molecular viscosity are negligible. The stan-dard k-ε model is therefore valid only for fully turbulent flows.

Transport Equations for the Standard k-ε Model

The turbulence kinetic energy, k, and its rate of dissipation, ε, are ob-tained from the following transport equations:



∂

∂t(ρk) +

∂

∂xi(ρkui) =

∂

∂xj

[(µ+

µt

σk

)∂k

∂xj

]+Gk +Gb − ρε− YM + Sk

(10.4-1)

and

∂

∂t(ρε) +

∂

∂xi(ρεui) =

∂

∂xj

[(µ+

µt

σε

)∂ε

∂xj

]+ C1ε

ε

k(Gk + C3εGb) −

C2ερε2

k+ Sε (10.4-2)

In these equations, Gk represents the generation of turbulence kineticenergy due to the mean velocity gradients, calculated as described inSection 10.4.4. Gb is the generation of turbulence kinetic energy dueto buoyancy, calculated as described in Section 10.4.5. YM representsthe contribution of the fluctuating dilatation in compressible turbulenceto the overall dissipation rate, calculated as described in Section 10.4.6.C1ε, C2ε, and C3ε are constants. σk and σε are the turbulent Prandtlnumbers for k and ε, respectively. Sk and Sε are user-defined sourceterms.

Modeling the Turbulent Viscosity

The turbulent (or eddy) viscosity, µt, is computed by combining k andε as follows:

µt = ρCµk2

ε(10.4-3)

where Cµ is a constant.

Model Constants

The model constants C1ε, C2ε, Cµ, σk, and σε have the following defaultvalues [128]:

C1ε = 1.44, C2ε = 1.92, Cµ = 0.09, σk = 1.0, σε = 1.3


Modeling Turbulence

These default values have been determined from experiments with airand water for fundamental turbulent shear flows including homogeneousshear flows and decaying isotropic grid turbulence. They have been foundto work fairly well for a wide range of wall-bounded and free shear flows.

Although the default values of the model constants are the standard onesmost widely accepted, you can change them (if needed) in the ViscousModel panel.

10.4.2 The RNG k-ε Model

The RNG-based k-ε turbulence model is derived from the instantaneousNavier-Stokes equations, using a mathematical technique called “renor-malization group” (RNG) methods. The analytical derivation results ina model with constants different from those in the standard k-ε model,and additional terms and functions in the transport equations for k andε. A more comprehensive description of RNG theory and its applicationto turbulence can be found in [36].

Transport Equations for the RNG k-ε Model

The RNG k-ε model has a similar form to the standard k-ε model:

∂

∂t(ρk) +

∂

∂xi(ρkui) =

∂

∂xj

(αkµeff

∂k

∂xj

)+Gk +Gb − ρε− YM + Sk

(10.4-4)

and

∂

∂t(ρε) +

∂

∂xi(ρεui) =

∂

∂xj

(αεµeff

∂ε

∂xj

)+

C1εε

k(Gk + C3εGb) − C2ερ

ε2

k−Rε + Sε (10.4-5)

In these equations, Gk represents the generation of turbulence kineticenergy due to the mean velocity gradients, calculated as described in



Section 10.4.4. Gb is the generation of turbulence kinetic energy dueto buoyancy, calculated as described in Section 10.4.5. YM representsthe contribution of the fluctuating dilatation in compressible turbulenceto the overall dissipation rate, calculated as described in Section 10.4.6.The quantities αk and αε are the inverse effective Prandtl numbers fork and ε, respectively. Sk and Sε are user-defined source terms.

Modeling the Effective Viscosity

The scale elimination procedure in RNG theory results in a differentialequation for turbulent viscosity:

d

(ρ2k√εµ

)= 1.72

ν√ν3 − 1 + Cν

dν (10.4-6)

where

ν = µeff/µ

Cν ≈ 100

Equation 10.4-6 is integrated to obtain an accurate description of howthe effective turbulent transport varies with the effective Reynolds num-ber (or eddy scale), allowing the model to better handle low-Reynolds-number and near-wall flows.

In the high-Reynolds-number limit, Equation 10.4-6 gives

µt = ρCµk2

ε(10.4-7)

with Cµ = 0.0845, derived using RNG theory. It is interesting to notethat this value of Cµ is very close to the empirically-determined value of0.09 used in the standard k-ε model.

In FLUENT, by default, the effective viscosity is computed using thehigh-Reynolds-number form in Equation 10.4-7. However, there is anoption available that allows you to use the differential relation given inEquation 10.4-6 when you need to include low-Reynolds-number effects.


Modeling Turbulence

RNG Swirl Modification

Turbulence, in general, is affected by rotation or swirl in the mean flow.The RNG model in FLUENT provides an option to account for the effectsof swirl or rotation by modifying the turbulent viscosity appropriately.The modification takes the following functional form:

µt = µt0f

(αs,Ω,

k

ε

)(10.4-8)

where µt0 is the value of turbulent viscosity calculated without the swirlmodification using either Equation 10.4-6 or Equation 10.4-7. Ω is acharacteristic swirl number evaluated within FLUENT, and αs is a swirlconstant that assumes different values depending on whether the flow isswirl-dominated or only mildly swirling. This swirl modification alwaystakes effect for axisymmetric, swirling flows and three-dimensional flowswhen the RNG model is selected. For mildly swirling flows (the defaultin FLUENT), αs is set to 0.05 and cannot be modified. For stronglyswirling flows, however, a higher value of αs can be used.

Calculating the Inverse Effective Prandtl Numbers

The inverse effective Prandtl numbers, αk and αε, are computed usingthe following formula derived analytically by the RNG theory:

∣∣∣∣ α− 1.3929α0 − 1.3929

∣∣∣∣0.6321 ∣∣∣∣ α+ 2.3929α0 + 2.3929

∣∣∣∣0.3679

=µmol

µeff(10.4-9)

where α0 = 1.0. In the high-Reynolds-number limit (µmol/µeff 1),αk = αε ≈ 1.393.

The Rε Term in the ε Equation

The main difference between the RNG and standard k-ε models lies inthe additional term in the ε equation given by

Rε =Cµρη

3(1 − η/η0)1 + βη3

ε2

k(10.4-10)



where η ≡ Sk/ε, η0 = 4.38, β = 0.012.

The effects of this term in the RNG ε equation can be seen more clearlyby rearranging Equation 10.4-5. Using Equation 10.4-10, the third andfourth terms on the right-hand side of Equation 10.4-5 can be merged,and the resulting ε equation can be rewritten as

∂

∂t(ρε) +

∂

∂xi(ρεui) =

∂

∂xj

(αεµeff

∂ε

∂xj

)+ C1ε

ε

k(Gk + C3εGb) − C∗

2ερε2

k

(10.4-11)

where C∗2ε is given by

C∗2ε ≡ C2ε +

Cµρη3(1 − η/η0)

1 + βη3(10.4-12)

In regions where η < η0, the R term makes a positive contribution, andC∗

2ε becomes larger than C2ε. In the logarithmic layer, for instance, it canbe shown that η ≈ 3.0, giving C∗

2ε ≈ 2.0, which is close in magnitude tothe value of C2ε in the standard k-ε model (1.92). As a result, for weaklyto moderately strained flows, the RNG model tends to give results largelycomparable to the standard k-ε model.

In regions of large strain rate (η > η0), however, the R term makes a neg-ative contribution, making the value of C∗

2ε less than C2ε. In comparisonwith the standard k-ε model, the smaller destruction of ε augments ε,reducing k and, eventually, the effective viscosity. As a result, in rapidlystrained flows, the RNG model yields a lower turbulent viscosity thanthe standard k-ε model.

Thus, the RNG model is more responsive to the effects of rapid strainand streamline curvature than the standard k-ε model, which explainsthe superior performance of the RNG model for certain classes of flows.

Model Constants

The model constants C1ε and C2ε in Equation 10.4-5 have values de-rived analytically by the RNG theory. These values, used by default inFLUENT, are


Modeling Turbulence

C1ε = 1.42, C2ε = 1.68

10.4.3 The Realizable k-ε Model

In addition to the standard and RNG-based k-ε models described inSections 10.4.1 and 10.4.2, FLUENT also provides the so-called realizablek-ε model [209]. The term “realizable” means that the model satisfiescertain mathematical constraints on the normal stresses, consistent withthe physics of turbulent flows. To understand this, consider combiningthe Boussinesq relationship (Equation 10.2-5) and the eddy viscositydefinition (Equation 10.4-3) to obtain the following expression for thenormal Reynolds stress in an incompressible strained mean flow:

u2 =23k − 2 νt

∂U

∂x(10.4-13)

Using Equation 10.4-3 for νt ≡ µt/ρ, one obtains the result that thenormal stress, u2, which by definition is a positive quantity, becomesnegative, i.e., “non-realizable”, when the strain is large enough to satisfy

k

ε

∂U

∂x>

13Cµ

≈ 3.7 (10.4-14)

Similarly, it can also be shown that the Schwarz inequality for shearstresses (uαuβ

2 ≤ u2αu

2β ; no summation over α and β) can be violated

when the mean strain rate is large. The most straightforward way toensure the realizability (positivity of normal stresses and Schwarz in-equality for shear stresses) is to make Cµ variable by sensitizing it to themean flow (mean deformation) and the turbulence (k, ε). The notionof variable Cµ is suggested by many modelers including Reynolds [191],and is well substantiated by experimental evidence. For example, Cµ isfound to be around 0.09 in the inertial sublayer of equilibrium boundarylayers, and 0.05 in a strong homogeneous shear flow.

Another weakness of the standard k-ε model or other traditional k-εmodels lies with the modeled equation for the dissipation rate (ε). Thewell-known round-jet anomaly (named based on the finding that the



spreading rate in planar jets is predicted reasonably well, but predic-tion of the spreading rate for axisymmetric jets is unexpectedly poor) isconsidered to be mainly due to the modeled dissipation equation.

The realizable k-ε model proposed by Shih et al. [209] was intendedto address these deficiencies of traditional k-ε models by adopting thefollowing:

• a new eddy-viscosity formula involving a variable Cµ originallyproposed by Reynolds [191]

• a new model equation for dissipation (ε) based on the dynamicequation of the mean-square vorticity fluctuation

Transport Equations for the Realizable k-ε Model

The modeled transport equations for k and ε in the realizable k-ε modelare

∂

∂t(ρk) +

∂

∂xi(ρkuj) =

∂

∂xi

[(µ+

µt

σk

)∂k

∂xj

]+Gk +Gb − ρε− YM + Sk

(10.4-15)

and

∂

∂t(ρε) +

∂

∂xj(ρεuj) =

∂

∂xj

[(µ+

µt

σε

)∂ε

∂xj

]+ ρC1Sε−

ρC2ε2

k +√νε

+ C1εε

kC3εGb + Sε (10.4-16)

where

C1 = max[0.43,

η

η + 5

]

and


Modeling Turbulence

η = Sk

ε

In these equations, Gk represents the generation of turbulence kineticenergy due to the mean velocity gradients, calculated as described inSection 10.4.4. Gb is the generation of turbulence kinetic energy due tobuoyancy, calculated as described in Section 10.4.5. YM represents thecontribution of the fluctuating dilatation in compressible turbulence tothe overall dissipation rate, calculated as described in Section 10.4.6. C2

and C1ε are constants. σk and σε are the turbulent Prandtl numbers fork and ε, respectively. Sk and Sε are user-defined source terms.

Note that the k equation (Equation 10.4-15) is the same as that in thestandard k-ε model (Equation 10.4-1) and the RNG k-ε model (Equa-tion 10.4-4), except for the model constants. However, the form of the εequation is quite different from those in the standard and RNG-based k-εmodels (Equations 10.4-2 and 10.4-5). One of the noteworthy featuresis that the production term in the ε equation (the second term on theright-hand side of Equation 10.4-16) does not involve the production ofk; i.e., it does not contain the same Gk term as the other k-ε models. Itis believed that the present form better represents the spectral energytransfer. Another desirable feature is that the destruction term (the nextto last term on the right-hand side of Equation 10.4-16) does not haveany singularity; i.e., its denominator never vanishes, even if k vanishesor becomes smaller than zero. This feature is contrasted with traditionalk-ε models, which have a singularity due to k in the denominator.

This model has been extensively validated for a wide range of flows [116,209], including rotating homogeneous shear flows, free flows includingjets and mixing layers, channel and boundary layer flows, and separatedflows. For all these cases, the performance of the model has been found tobe substantially better than that of the standard k-ε model. Especiallynoteworthy is the fact that the realizable k-ε model resolves the round-jet anomaly; i.e., it predicts the spreading rate for axisymmetric jets aswell as that for planar jets.



Modeling the Turbulent Viscosity

As in other k-ε models, the eddy viscosity is computed from

µt = ρCµk2

ε(10.4-17)

The difference between the realizable k-ε model and the standard andRNG k-ε models is that Cµ is no longer constant. It is computed from

Cµ =1

A0 +AskU∗

ε

(10.4-18)

whereU∗ ≡

√SijSij + ΩijΩij (10.4-19)

and

Ωij = Ωij − 2εijkωk

Ωij = Ωij − εijkωk

where Ωij is the mean rate-of-rotation tensor viewed in a rotating refer-ence frame with the angular velocity ωk. The model constants A0 andAs are given by

A0 = 4.04, As =√

6 cosφ

where

φ =13

cos−1(√

6W ), W =SijSjkSki

S, S =

√SijSij

Sij =12

(∂uj

∂xi+∂ui

∂xj

)

It can be seen that Cµ is a function of the mean strain and rotation rates,the angular velocity of the system rotation, and the turbulence fields (k


Modeling Turbulence

and ε). Cµ in Equation 10.4-17 can be shown to recover the standardvalue of 0.09 for an inertial sublayer in an equilibrium boundary layer.

Model Constants

The model constants C2, σk, and σε have been established to ensure thatthe model performs well for certain canonical flows. The model constantsare

C1ε = 1.44, C2 = 1.9, σk = 1.0, σε = 1.2

10.4.4 Modeling Turbulent Production in the k-ε Models

The term Gk, representing the production of turbulence kinetic energy,is modeled identically for the standard, RNG, and realizable k-ε models.From the exact equation for the transport of k, this term may be definedas

Gk = −ρu′iu′j∂uj

∂xi(10.4-20)

To evaluate Gk in a manner consistent with the Boussinesq hypothesis,

Gk = µtS2 (10.4-21)

where S is the modulus of the mean rate-of-strain tensor, defined as

S ≡√

2SijSij (10.4-22)

10.4.5 Effects of Buoyancy on Turbulence in the k-ε Models

When a non-zero gravity field and temperature gradient are present si-multaneously, the k-ε models in FLUENT account for the generation of kdue to buoyancy (Gb in Equations 10.4-1, 10.4-4, and 10.4-15), and thecorresponding contribution to the production of ε in Equations 10.4-2,10.4-5, and 10.4-16.



The generation of turbulence due to buoyancy is given by

Gb = βgiµt

Prt∂T

∂xi(10.4-23)

where Prt is the turbulent Prandtl number for energy and gi is the com-ponent of the gravitational vector in the ith direction. For the standardand realizable k-ε models, the default value of Prt is 0.85. In the caseof the RNG k-ε model, Prt = 1/α, where α is given by Equation 10.4-9,but with α0 = 1/Pr = k/µcp. The coefficient of thermal expansion, β, isdefined as

β = −1ρ

(∂ρ

∂T

)p

(10.4-24)

For ideal gases, Equation 10.4-23 reduces to

Gb = −giµt

ρPrt

∂ρ

∂xi(10.4-25)

It can be seen from the transport equations for k (Equations 10.4-1,10.4-4, and 10.4-15) that turbulence kinetic energy tends to be aug-mented (Gb > 0) in unstable stratification. For stable stratification,buoyancy tends to suppress the turbulence (Gb < 0). In FLUENT, theeffects of buoyancy on the generation of k are always included whenyou have both a non-zero gravity field and a non-zero temperature (ordensity) gradient.

While the buoyancy effects on the generation of k are relatively wellunderstood, the effect on ε is less clear. In FLUENT, by default, thebuoyancy effects on ε are neglected simply by setting Gb to zero in thetransport equation for ε (Equation 10.4-2, 10.4-5, or 10.4-16).

However, you can include the buoyancy effects on ε in the Viscous Modelpanel. In this case, the value of Gb given by Equation 10.4-25 is used inthe transport equation for ε (Equation 10.4-2, 10.4-5, or 10.4-16).

The degree to which ε is affected by the buoyancy is determined by theconstant C3ε. In FLUENT, C3ε is not specified, but is instead calculatedaccording to the following relation [90]:


Modeling Turbulence

C3ε = tanh∣∣∣∣vu∣∣∣∣ (10.4-26)

where v is the component of the flow velocity parallel to the gravita-tional vector and u is the component of the flow velocity perpendicularto the gravitational vector. In this way, C3ε will become 1 for buoyantshear layers for which the main flow direction is aligned with the direc-tion of gravity. For buoyant shear layers that are perpendicular to thegravitational vector, C3ε will become zero.

10.4.6 Effects of Compressibility on Turbulence in the k-εModels

For high-Mach-number flows, compressibility affects turbulence throughso-called “dilatation dissipation”, which is normally neglected in themodeling of incompressible flows [267]. Neglecting the dilatation dis-sipation fails to predict the observed decrease in spreading rate withincreasing Mach number for compressible mixing and other free shearlayers. To account for these effects in the k-ε models in FLUENT, thedilatation dissipation term, YM , is included in the k equation. This termis modeled according to a proposal by Sarkar [197]:

YM = 2ρεM2t (10.4-27)

where Mt is the turbulent Mach number, defined as

Mt =

√k

a2(10.4-28)

where a (≡ √γRT ) is the speed of sound.

This compressibility modification always takes effect when the compress-ible form of the ideal gas law is used.



10.4.7 Convective Heat and Mass Transfer Modeling in the k-εModels

In FLUENT, turbulent heat transport is modeled using the concept ofReynolds’ analogy to turbulent momentum transfer. The “modeled”energy equation is thus given by the following:

∂

∂t(ρE) +

∂


∂

∂xj

(keff

∂T

∂xj+ ui(τij)eff

)+ Sh (10.4-29)

where E is the total energy, keff is the effective thermal conductivity,and (τij)eff is the deviatoric stress tensor, defined as

(τij)eff = µeff

(∂uj

∂xi+∂ui

∂xj

)− 2

3µeff

∂ui

∂xiδij

The term involving (τij)eff represents the viscous heating, and is alwayscomputed in the coupled solvers. It is not computed by default in thesegregated solver, but it can be enabled in the Viscous Model panel.

Additional terms may appear in the energy equation, depending on thephysical models you are using. See Section 11.2.1 for more details.

For the standard and realizable k-ε models, the effective thermal con-ductivity is given by

keff = k +cpµt

Prt

where k, in this case, is the thermal conductivity. The default value ofthe turbulent Prandtl number is 0.85. You can change the value of theturbulent Prandtl number in the Viscous Model panel.

For the RNG k-ε model, the effective thermal conductivity is

keff = αcpµeff


Modeling Turbulence

where α is calculated from Equation 10.4-9, but with α0 = 1/Pr = k/µcp.

The fact that α varies with µmol/µeff , as in Equation 10.4-9, is an advan-tage of the RNG k-ε model. It is consistent with experimental evidenceindicating that the turbulent Prandtl number varies with the molecularPrandtl number and turbulence [111]. Equation 10.4-9 works well acrossa very broad range of molecular Prandtl numbers, from liquid metals(Pr ≈ 10−2) to paraffin oils (Pr ≈ 103), which allows heat transfer to becalculated in low-Reynolds-number regions. Equation 10.4-9 smoothlypredicts the variation of effective Prandtl number from the molecularvalue (α = 1/Pr) in the viscosity-dominated region to the fully turbu-lent value (α = 1.393) in the fully turbulent regions of the flow.

Turbulent mass transfer is treated similarly. For the standard and re-alizable k-ε models, the default turbulent Schmidt number is 0.7. Thisdefault value can be changed in the Viscous Model panel. For the RNGmodel, the effective turbulent diffusivity for mass transfer is calculatedin a manner that is analogous to the method used for the heat trans-port. The value of α0 in Equation 10.4-9 is α0 = 1/Sc, where Sc is themolecular Schmidt number.


10.5 The Standard and SST k-ω Models


This section presents the standard and shear-stress transport (SST) k-ω models. Both models have similar forms, with transport equationsfor k and ω. The major ways in which the SST model differs from thestandard model are as follows:

• gradual change from the standard k-ω model in the inner region ofthe boundary layer to a high-Reynolds-number version of the k-εmodel in the outer part of the boundary layer

• modified turbulent viscosity formulation to account for the trans-port effects of the principal turbulent shear stress

The transport equations, methods of calculating turbulent viscosity, andmethods of calculating model constants and other terms are presentedseparately for each model.

10.5.1 The Standard k-ω Model

The standard k-ω model is an empirical model based on model transportequations for the turbulence kinetic energy (k) and the specific dissipa-tion rate (ω), which can also be thought of as the ratio of ε to k [267].

As the k-ω model has been modified over the years, production termshave been added to both the k and ω equations, which have improvedthe accuracy of the model for predicting free shear flows.

Transport Equations for the Standard k-ω Model

The turbulence kinetic energy, k, and the specific dissipation rate, ω, areobtained from the following transport equations:

∂

∂t(ρk) +

∂

∂xi(ρkui) =

∂

∂xj

(Γk

∂k

∂xj

)+Gk − Yk + Sk (10.5-1)

and


Modeling Turbulence

∂

∂t(ρω) +

∂

∂xi(ρωui) =

∂

∂xj

(Γω

∂ω

∂xj

)+Gω − Yω + Sω (10.5-2)

In these equations, Gk represents the generation of turbulence kineticenergy due to mean velocity gradients. Gω represents the generation ofω. Γk and Γω represent the effective diffusivity of k and ω, respectively.Yk and Yω represent the dissipation of k and ω due to turbulence. Allof the above terms are calculated as described below. Sk and Sω areuser-defined source terms.

Modeling the Effective Diffusivity

The effective diffusivities for the k-ω model are given by

Γk = µ+µt

σk(10.5-3)

Γω = µ+µt

σω(10.5-4)

where σk and σω are the turbulent Prandtl numbers for k and ω, respec-tively. The turbulent viscosity, µt, is computed by combining k and ωas follows:

µt = α∗ ρkω

(10.5-5)

Low-Reynolds-Number Correction

The coefficient α∗ damps the turbulent viscosity causing a low-Reynolds-number correction. It is given by

α∗ = α∗∞(α∗

0 + Ret/Rk

1 + Ret/Rk

)(10.5-6)

where

Ret =ρk

µω(10.5-7)



Rk = 6 (10.5-8)

α∗0 =

βi

3(10.5-9)

βi = 0.072 (10.5-10)

Note that, in the high-Reynolds-number form of the k-ω model, α∗ =α∗∞ = 1.

Modeling the Turbulence Production

Production of k

The term Gk represents the production of turbulence kinetic energy.From the exact equation for the transport of k, this term may be definedas

Gk = −ρu′iu′j∂uj

∂xi(10.5-11)

To evaluate Gk in a manner consistent with the Boussinesq hypothesis,

Gk = µt S2 (10.5-12)

where S is the modulus of the mean rate-of-strain tensor, defined in thesame way as for the k-ε model (see Equation 10.4-22).

Production of ω

The production of ω is given by

Gω = αω

kGk (10.5-13)

where Gk is given by Equation 10.5-11.

The coefficient α is given by

α =α∞α∗

(α0 + Ret/Rω

1 + Ret/Rω

)(10.5-14)


Modeling Turbulence

where Rω = 2.95. α∗ and Ret are given by Equations 10.5-6 and 10.5-7,respectively.

Note that, in the high-Reynolds-number form of the k-ω model, α =α∞ = 1.

Modeling the Turbulence Dissipation

Dissipation of k

The dissipation of k is given by

Yk = ρ β∗fβ∗ k ω (10.5-15)

where

fβ∗ =

1 χk ≤ 01+680χ2

k

1+400χ2k

χk > 0(10.5-16)

whereχk ≡ 1

ω3

∂k

∂xj

∂ω

∂xj(10.5-17)

and

β∗ = β∗i [1 + ζ∗F (Mt)] (10.5-18)

β∗i = β∗∞

(4/15 + (Ret/Rβ)4

1 + (Ret/Rβ)4

)(10.5-19)

ζ∗ = 1.5 (10.5-20)Rβ = 8 (10.5-21)β∗∞ = 0.09 (10.5-22)

where Ret is given by Equation 10.5-7.

Dissipation of ω

The dissipation of ω is given by



Yω = ρ β fβ ω2 (10.5-23)

where

fβ =1 + 70χω

1 + 80χω(10.5-24)

χω =∣∣∣∣ΩijΩjkSki

(β∗∞ω)3

∣∣∣∣ (10.5-25)

Ωij =12

(∂ui

∂xj− ∂uj

∂xi

)(10.5-26)

The strain rate tensor, Sij is defined in Equation 10.3-11. Also,

β = βi

[1 − β∗i

βiζ∗F (Mt)

](10.5-27)

β∗i and F (Mt) are defined by Equations 10.5-19 and 10.5-28, respectively.

Compressibility Correction

The compressibility function, F (Mt), is given by

F (Mt) =

0 Mt ≤ Mt0

M2t − M2

t0 Mt > Mt0(10.5-28)

where

M2t ≡ 2k

a2(10.5-29)

Mt0 = 0.25 (10.5-30)a =

√γRT (10.5-31)

Note that, in the high-Reynolds-number form of the k-ω model, β∗i = β∗∞.In the incompressible form, β∗ = β∗i .


Modeling Turbulence

Model Constants

α∗∞ = 1, α∞ = 0.52, α0 =

19, β∗∞ = 0.09, βi = 0.072, Rβ = 8

Rk = 6, Rω = 2.95, ζ∗ = 1.5, Mt0 = 0.25, σk = 2.0, σω = 2.0

Wall Boundary Conditions

The wall boundary conditions for the k equation in the k-ω models aretreated in the same way as the k equation is treated when enhanced walltreatments are used with the k-ε models. This means that all boundaryconditions for wall-function meshes will correspond to the wall func-tion approach, while for the fine meshes, the appropriate low-Reynolds-number boundary conditions will be applied.

In FLUENT the value of ω at the wall is specified as

ωw =ρ (u∗)2

µω+ (10.5-32)

The asymptotic value of ω+ in the laminar sublayer is given by

ω+ = min(ω+

w ,6

β∗∞(y+)2

)(10.5-33)

where

ω+w =

(

50k+

s

)2k+

s < 25

100k+

sk+

s ≥ 25(10.5-34)

wherek+

s = max(

1.0,ρksu

∗

µ

)(10.5-35)

and ks is the roughness height.

In the logarithmic (or turbulent) region, the value of ω+ is

ω+ =1√β∗∞

du+turb

dy+(10.5-36)



which leads to the value of ω in the wall cell as

ω =u∗√β∗∞κy

(10.5-37)

Note that in the case of a wall cell being placed in the buffer region,FLUENT will blend ω+ between the logarithmic and laminar sublayervalues.

10.5.2 The Shear-Stress Transport (SST) k-ω Model

In addition to the standard k-ω model described in Section 10.5.1, FLU-ENT also provides a variation called the shear-stress transport (SST)k-ω model, so named because the definition of the turbulent viscosityis modified to account for the transport of the principal turbulent shearstress. It is this feature that gives the SST k-ω model an advantage interms of performance over both the standard k-ω model and the standardk-ε model. Other modifications include the addition of a cross-diffusionterm in the ω equation and a blending function to ensure that the modelequations behave appropriately in both the near-wall and far-field zones.

Transport Equations for the SST k-ω Model

The SST k-ω model has a similar form to the standard k-ω model:

∂

∂t(ρk) +

∂

∂xi(ρkui) =

∂

∂xj

(Γk

∂k

∂xj

)+Gk − Yk + Sk (10.5-38)

and

∂

∂t(ρω) +

∂

∂xi(ρωui) =

∂

∂xj

(Γω

∂ω

∂xj

)+Gω − Yω +Dω + Sω (10.5-39)

In these equations, Gk represents the generation of turbulence kineticenergy due to mean velocity gradients, calculated as described in Sec-tion 10.5.1. Gω represents the generation of ω, calculated as described in


Modeling Turbulence

Section 10.5.1. Γk and Γω represent the effective diffusivity of k and ω,respectively, which are calculated as described below. Yk and Yω repre-sent the dissipation of k and ω due to turbulence, calculated as describedin Section 10.5.1. Dω represents the cross-diffusion term, calculated asdescribed below. Sk and Sω are user-defined source terms.

Modeling the Effective Diffusivity

The effective diffusivities for the SST k-ω model are given by

Γk = µ+µt

σk(10.5-40)

Γω = µ+µt

σω(10.5-41)

where σk and σω are the turbulent Prandtl numbers for k and ω, respec-tively. The turbulent viscosity, µt, is computed as follows:

µt =ρk

ω

1

max[

1α∗ ,

ΩF2a1ω

] (10.5-42)

where

Ω ≡√

2ΩijΩij (10.5-43)

σk =1

F1/σk,1 + (1 − F1)/σk,2(10.5-44)

σω =1

F1/σω,1 + (1 − F1)/σω,2(10.5-45)

Ωij is the mean rate-of-rotation tensor and α∗ is defined in Equation 10.5-6.The blending functions, F1 and F2, are given by

F1 = tanh(Φ4

1

)(10.5-46)



Φ1 = min

[max

( √k

0.09ωy,500µρy2ω

),

4ρkσω,2D

+ω y2

](10.5-47)

D+ω = max

[2ρ

1σω,2

1ω

∂k

∂xj

∂ω

∂xj, 10−20

](10.5-48)

F2 = tanh(Φ2

2

)(10.5-49)

Φ2 = max

[2

√k

0.09ωy,500µρy2ω

](10.5-50)

where y is the distance to the next surface and D+ω is the positive portion

of the cross-diffusion term (see Equation 10.5-60).

Modeling the Turbulence Production

Production of k

The term Gk represents the production of turbulence kinetic energy,and is defined in the same manner as in the standard k-ω model. SeeSection 10.5.1 for details.

Production of ω

The term Gω represents the production of ω and is given by

Gω =α

νtGk (10.5-51)

Note that this formulation differs from the standard k-ω model. Thedifference between the two models also exists in the way the term α∞is evaluated. In the standard k-ω model, α∞ is defined as a constant(0.52). For the SST k-ω model, α∞ is given by

α∞ = F1α∞,1 + (1 − F1)α∞,2 (10.5-52)

where


Modeling Turbulence

α∞,1 =βi,1

β∗∞− κ2

σw,1

√β∗∞

(10.5-53)

α∞,2 =βi,2

β∗∞− κ2

σw,2

√β∗∞

(10.5-54)

where κ is 0.41. βi,1 and βi,2 are given by Equations 10.5-58 and 10.5-59,respectively.

Modeling the Turbulence Dissipation

Dissipation of k

The term Yk represents the dissipation of turbulence kinetic energy, andis defined in a similar manner as in the standard k-ω model (see Sec-tion 10.5.1). The difference is in the way the term fβ∗ is evaluated. Inthe standard k-ω model, fβ∗ is defined as a piecewise function. For theSST k-ω model, fβ∗ is a constant equal to 1. Thus,

Yk = ρβ∗kω (10.5-55)

Dissipation of ω

The term Yω represents the dissipation of ω, and is defined in a similarmanner as in the standard k-ω model (see Section 10.5.1). The differ-ence is in the way the terms βi and fβ are evaluated. In the standardk-ω model, βi is defined as a constant (0.072) and fβ is defined in Equa-tion 10.5-24. For the SST k-ω model, fβ is a constant equal to 1. Thus,

Yk = ρβω2 (10.5-56)

Instead of a having a constant value, βi is given by

βi = F1βi,1 + (1 − F1)βi,2 (10.5-57)

where


10.6 The Reynolds Stress Model (RSM)

βi,1 = 0.075 (10.5-58)βi,2 = 0.0828 (10.5-59)

and F1 is obtained from Equation 10.5-46.

Cross-Diffusion Modification

The SST k-ω model is based on both the standard k-ω model and thestandard k-ε model. To blend these two models together, the stan-dard k-ε model has been transformed into equations based on k and ω,which leads to the introduction of a cross-diffusion term (Dω in Equa-tion 10.5-39). Dω is defined as

Dω = 2 (1 − F1) ρσω,21ω

∂k

∂xj

∂ω

∂xj(10.5-60)

For details about the various k-ε models, see Section 10.4.

Model Constants

σk,1 = 1.176, σω,1 = 2.0, σk,2 = 1.0, σω,2 = 1.168

a1 = 0.31, βi,1 = 0.075 βi,2 = 0.0828

All additional model constants (α∗∞, α∞, α0, β∗∞, Rβ, Rk, Rω, ζ∗, andMt0) have the same values as for the standard k-ω model (see Sec-tion 10.5.1).


The Reynolds stress model [75, 125, 126] involves calculation of the in-dividual Reynolds stresses, u′iu′j , using differential transport equations.The individual Reynolds stresses are then used to obtain closure of theReynolds-averaged momentum equation (Equation 10.2-4).


Modeling Turbulence

The exact form of the Reynolds stress transport equations may be de-rived by taking moments of the exact momentum equation. This isa process wherein the exact momentum equations are multiplied by afluctuating property, the product then being Reynolds-averaged. Unfor-tunately, several of the terms in the exact equation are unknown andmodeling assumptions are required in order to close the equations.

In this section, the Reynolds stress transport equations are presentedtogether with the modeling assumptions required to attain closure.

10.6.1 The Reynolds Stress Transport Equations

The exact transport equations for the transport of the Reynolds stresses,ρu′iu′j, may be written as follows:

∂

∂t(ρ u′iu′j)︸︷︷︸

Local Time Derivative

+∂

∂xk(ρuku

′iu

′j)︸︷︷︸

Cij ≡ Convection

=

− ∂

∂xk

[ρ u′iu′ju′k + p

(δkju

′i + δiku

′j

)]︸︷︷︸

DT,ij ≡ Turbulent Diffusion

+∂

∂xk

[µ∂

∂xk(u′iu′j)

]︸︷︷︸

DL,ij ≡ Molecular Diffusion

− ρ

(u′iu′k

∂uj

∂xk+ u′ju′k

∂ui

∂xk

)︸︷︷︸Pij ≡ Stress Production

− ρβ(giu′jθ + gju

′iθ)︸︷︷︸

Gij ≡ Buoyancy Production

+ p

(∂u′i∂xj

+∂u′j∂xi

)︸︷︷︸

φij ≡ Pressure Strain

− 2µ∂u′i∂xk

∂u′j∂xk︸︷︷︸

εij ≡ Dissipation

−2ρΩk

(u′ju′mεikm + u′iu′mεjkm

)︸︷︷︸

Fij ≡ Production by System Rotation

+ Suser︸︷︷︸User-Defined Source Term

(10.6-1)



Of the various terms in these exact equations, Cij , DL,ij, Pij, and Fij

do not require any modeling. However, DT,ij , Gij , φij, and εij need tobe modeled to close the equations. The following sections describe themodeling assumptions required to close the equation set.

10.6.2 Modeling Turbulent Diffusive Transport

DT,ij can be modeled by the generalized gradient-diffusion model of Dalyand Harlow [48]:

DT,ij = Cs∂

∂xk

(ρku′ku

′`

ε

∂u′iu′j∂x`

)(10.6-2)

However, this equation can result in numerical instabilities, so it has beensimplified in FLUENT to use a scalar turbulent diffusivity as follows [138]:

DT,ij =∂

∂xk

(µt

σk

∂u′iu′j∂xk

)(10.6-3)

The turbulent viscosity, µt, is computed using Equation 10.6-27.

Lien and Leschziner [138] derived a value of σk = 0.82 by applying thegeneralized gradient-diffusion model, Equation 10.6-2, to the case of aplanar homogeneous shear flow. Note that this value of σk is differentfrom that in the standard and realizable k-ε models, in which σk = 1.0.

10.6.3 Modeling the Pressure-Strain Term

Linear Pressure-Strain Model

By default in FLUENT, the pressure-strain term, φij, in Equation 10.6-1is modeled according to the proposals by Gibson and Launder [75], Fuet al. [71], and Launder [124, 125].

The classical approach to modeling φij uses the following decomposition:

φij = φij,1 + φij,2 + φij,w (10.6-4)


Modeling Turbulence

where φij,1 is the slow pressure-strain term, also known as the return-to-isotropy term, φij,2 is called the rapid pressure-strain term, and φij,w isthe wall-reflection term.

The slow pressure-strain term, φij,1, is modeled as

φij,1 ≡ −C1ρε

k

[u′iu′j −

23δijk

](10.6-5)

with C1 = 1.8.

The rapid pressure-strain term, φij,2, is modeled as

φij,2 ≡ −C2

[(Pij + Fij +Gij − Cij) − 2

3δij(P +G− C)

](10.6-6)

where C2 = 0.60, Pij , Fij , Gij , and Cij are defined as in Equation 10.6-1,P = 1

2Pkk, G = 12Gkk, and C = 1

2Ckk.

The wall-reflection term, φij,w, is responsible for the redistribution ofnormal stresses near the wall. It tends to damp the normal stress per-pendicular to the wall, while enhancing the stresses parallel to the wall.This term is modeled as

φij,w ≡ C ′1

ε

k

(u′ku′mnknmδij − 3

2u′iu′knjnk − 3

2u′ju′knink

)k3/2

C`εd

+ C ′2

(φkm,2nknmδij − 3

2φik,2njnk − 3

2φjk,2nink

)k3/2

C`εd

(10.6-7)

where C ′1 = 0.5, C ′

2 = 0.3, nk is the xk component of the unit normal tothe wall, d is the normal distance to the wall, and C` = C

3/4µ /κ, where

Cµ = 0.09 and κ is the von Karman constant (= 0.4187).

φij,w is included by default in the Reynolds stress model.



Low-Re Modifications to the Linear Pressure-Strain Model

When the RSM is applied to near-wall flows using the enhanced walltreatment described in Section 10.8.3, the pressure-strain model needsto be modified. The modification used in FLUENT specifies the valuesof C1, C2, C ′

1, and C ′2 as functions of the Reynolds stress invariants and

the turbulent Reynolds number, according to the suggestion of Launderand Shima [127]:

C1 = 1 + 2.58A√A2

1 − exp

[−(0.0067Ret)2

](10.6-8)

C2 = 0.75√A (10.6-9)

C ′1 = −2

3C1 + 1.67 (10.6-10)

C ′2 = max

[23C2 − 1

6

C2, 0

](10.6-11)

with the turbulent Reynolds number defined as Ret = (ρk2/µε). Theparameter A and tensor invariants, A2 and A3, are defined as

A ≡[1 − 9

8(A2 −A3)

](10.6-12)

A2 ≡ aikaki (10.6-13)A3 ≡ aikakjaji (10.6-14)

aij is the Reynolds-stress anisotropy tensor, defined as

aij = −(−ρu′iu′j + 2

3ρkδij

ρk

)(10.6-15)

The modifications detailed above are employed only when the enhancedwall treatment is selected in the Viscous Model panel.


Modeling Turbulence

Quadratic Pressure-Strain Model

An optional pressure-strain model proposed by Speziale, Sarkar, andGatski [228] is provided in FLUENT. This model has been demonstratedto give superior performance in a range of basic shear flows, includingplane strain, rotating plane shear, and axisymmetric expansion/contrac-tion. This improved accuracy should be beneficial for a wider class ofcomplex engineering flows, particularly those with streamline curvature.The quadratic pressure-strain model can be selected as an option in theViscous Model panel.

This model is written as follows:

φij = − (C1ρε+ C∗1P ) bij + C2ρε

(bikbkj − 1

3bmnbmnδij

)+(C3 − C∗

3

√bijbij

)ρkSij

+C4ρk

(bikSjk + bjkSik − 2

3bmnSmnδij

)+C5ρk (bikΩjk + bjkΩik) (10.6-16)

where bij is the Reynolds-stress anisotropy tensor defined as

bij = −(−ρu′iu′j + 2

3ρkδij

2ρk

)(10.6-17)

The mean strain rate, Sij , is defined as

Sij =12

(∂uj

∂xi+∂ui

∂xj

)(10.6-18)

The mean rate-of-rotation tensor, Ωij, is defined by

Ωij =12

(∂ui

∂xj− ∂uj

∂xi

)(10.6-19)



The constants are

C1 = 3.4, C∗1 = 1.8, C2 = 4.2, C3 = 0.8, C∗

3 = 1.3, C4 = 1.25, C5 = 0.4

The quadratic pressure-strain model does not require a correction toaccount for the wall-reflection effect in order to obtain a satisfactorysolution in the logarithmic region of a turbulent boundary layer. Itshould be noted, however, that the quadratic pressure-strain model isnot available when the enhanced wall treatment is selected in the ViscousModel panel.

10.6.4 Effects of Buoyancy on Turbulence

The production terms due to buoyancy are modeled as

Gij = βµt

Prt

(gi∂T

∂xj+ gj

∂T

∂xi

)(10.6-20)

where Prt is the turbulent Prandtl number for energy, with a defaultvalue of 0.85.

Using the definition of the coefficient of thermal expansion, β, given byEquation 10.4-24, the following expression is obtained for Gij for idealgases:

Gij = − µt

ρPrt

(gi∂ρ

∂xj+ gj

∂ρ

∂xi

)(10.6-21)

10.6.5 Modeling the Turbulence Kinetic Energy

In general, when the turbulence kinetic energy is needed for modeling aspecific term, it is obtained by taking the trace of the Reynolds stresstensor:

k =12u′iu′i (10.6-22)


Modeling Turbulence

As described in Section 10.6.8, an option is available in FLUENT to solvea transport equation for the turbulence kinetic energy in order to obtainboundary conditions for the Reynolds stresses. In this case, the followingmodel equation is used:

∂

∂t(ρk) +

∂

∂xi(ρkui) =

∂

∂xj

[(µ+

µt

σk

)∂k

∂xj

]+

12

(Pii +Gii) − ρε(1 + 2M2t ) + Sk (10.6-23)

where σk = 0.82 and Sk is a user-defined source term. Equation 10.6-23 isobtainable by contracting the modeled equation for the Reynolds stresses(Equation 10.6-1). As one might expect, it is essentially identical toEquation 10.4-1 used in the standard k-ε model.

Although Equation 10.6-23 is solved globally throughout the flow do-main, the values of k obtained are used only for boundary conditions. Inevery other case, k is obtained from Equation 10.6-22. This is a minorpoint, however, since the values of k obtained with either method shouldbe very similar.

10.6.6 Modeling the Dissipation Rate

The dissipation tensor, εij, is modeled as

εij =23δij(ρε+ YM) (10.6-24)

where YM = 2ρεM2t is an additional “dilatation dissipation” term ac-

cording to the model by Sarkar [197]. The turbulent Mach number inthis term is defined as

Mt =

√k

a2(10.6-25)



where a (≡ √γRT ) is the speed of sound. This compressibility modifi-

cation always takes effect when the compressible form of the ideal gaslaw is used.

The scalar dissipation rate, ε, is computed with a model transport equa-tion similar to that used in the standard k-ε model:

∂

∂t(ρε) +

∂

∂xi(ρεui) =

∂

∂xj

[(µ+

µt

σε

)∂ε

∂xj

]+

Cε112

[Pii + Cε3Gii]ε

k−Cε2ρ

ε2

k+ Sε (10.6-26)

where σε = 1.0, Cε1 = 1.44, Cε2 = 1.92, Cε3 is evaluated as a function ofthe local flow direction relative to the gravitational vector, as describedin Section 10.4.5, and Sε is a user-defined source term.

10.6.7 Modeling the Turbulent Viscosity

The turbulent viscosity, µt, is computed similarly to the k-ε models:

µt = ρCµk2

ε(10.6-27)

where Cµ = 0.09.

10.6.8 Boundary Conditions for the Reynolds Stresses

Whenever flow enters the domain, FLUENT requires values for individualReynolds stresses, u′iu′j, and for the turbulence dissipation rate, ε. Thesequantities can be input directly or derived from the turbulence intensityand characteristic length, as described in Section 10.10.2.

At walls, FLUENT computes the near-wall values of the Reynolds stressesand ε from wall functions (see Section 10.8.2). FLUENT applies explicitwall boundary conditions for the Reynolds stresses by using the log-lawand the assumption of equilibrium, disregarding convection and diffusionin the transport equations for the stresses (Equation 10.6-1). Usinga local coordinate system, where τ is the tangential coordinate, η is


Modeling Turbulence

the normal coordinate, and λ is the binormal coordinate, the Reynoldsstresses at the wall-adjacent cells are computed from

u′2τ

k= 1.098,

u′2η

k= 0.247,

u′2λ

k= 0.655, −u

′τu

′η

k= 0.255 (10.6-28)

To obtain k, FLUENT solves the transport equation of Equation 10.6-23.For reasons of computational convenience, the equation is solved globally,even though the values of k thus computed are needed only near the wall;in the far field k is obtained directly from the normal Reynolds stressesusing Equation 10.6-22. By default, the values of the Reynolds stressesnear the wall are fixed using the values computed from Equation 10.6-28,and the transport equations in Equation 10.6-1 are solved only in thebulk flow region.

Alternatively, the Reynolds stresses can be explicitly specified in termsof wall-shear stress, instead of k:

u′2τ

u2τ

= 5.1,u′2

η

u2τ

= 1.0,u

′2λ

u2τ

= 2.3, −u′τu

′η

u2τ

= 1.0 (10.6-29)

where uτ is the friction velocity defined by uτ ≡ √τw/ρ, where τw is the

wall-shear stress. When this option is chosen, the k transport equationis not solved.

10.6.9 Convective Heat and Mass Transfer Modeling

With the Reynolds stress model in FLUENT, turbulent heat transport ismodeled using the concept of Reynolds’ analogy to turbulent momentumtransfer. The “modeled” energy equation is thus given by the following:

∂

∂t(ρE) +

∂


∂

∂xj

[(k +

cpµt

Prt

)∂T

∂xj+ ui(τij)eff

]+ Sh

(10.6-30)


10.7 The Large Eddy Simulation (LES) Model

where E is the total energy and (τij)eff is the deviatoric stress tensor,defined as

(τij)eff = µeff

(∂uj

∂xi+∂ui

∂xj

)− 2

3µeff

∂ui

∂xiδij

The term involving (τij)eff represents the viscous heating, and is alwayscomputed in the coupled solvers. It is not computed by default in thesegregated solver, but it can be enabled in the Viscous Model panel. Thedefault value of the turbulent Prandtl number is 0.85. You can changethe value of Prt in the Viscous Model panel.

Turbulent mass transfer is treated similarly, with a default turbulentSchmidt number of 0.7. This default value can be changed in the ViscousModel panel.


Turbulent flows are characterized by eddies with a wide range of lengthand time scales. The largest eddies are typically comparable in sizeto the characteristic length of the mean flow. The smallest scales areresponsible for the dissipation of turbulence kinetic energy.

It is theoretically possible to directly resolve the whole spectrum of tur-bulent scales using an approach known as direct numerical simulation(DNS). DNS is not, however, feasible for practical engineering prob-lems. To understand the large computational cost of DNS, consider thatthe ratio of the large (energy-containing) scales to the small (energy-dissipating) scales is proportional to Re3/4

t , where Ret is the turbulentReynolds number. Therefore, to resolve all the scales, the mesh sizein three dimensions will be proportional to Re9/4

t . Simple arithmeticshows that, for high Reynolds numbers, the mesh sizes required for DNSare prohibitive. Adding to the computational cost is the fact that thesimulation will be a transient one with very small time steps, since thetemporal resolution requirements are governed by the dissipating scales,rather than the mean flow or the energy-containing eddies.


Modeling Turbulence

As explained in Section 10.2.1, the conventional approach to flow sim-ulations employs the solution of the Reynolds-averaged Navier-Stokes(RANS) equations. In the RANS approach, all the turbulent motionsare modeled, resulting in a significant savings in computational effort.

Conceptually, large eddy simulation (LES) is situated somewhere be-tween DNS and the RANS approach. Basically large eddies are resolveddirectly in LES, while small eddies are modeled. The rationale behindLES can be summarized as follows:

• Momentum, mass, energy, and other passive scalars are trans-ported mostly by large eddies.

• Large eddies are more problem-dependent. They are dictated bythe geometries and boundary conditions of the flow involved.

• Small eddies are less dependent on the geometry, tend to be moreisotropic, and are consequently more universal.

• The chance of finding a universal model is much higher when onlysmall eddies are modeled.

Solving only for the large eddies and modeling the smaller scales resultsin mesh resolution requirements that are much less restrictive than withDNS. Typically, mesh sizes can be at least one order of magnitude smallerthan with DNS. Furthermore, the time step sizes will be proportional tothe eddy-turnover time, which is much less restrictive than with DNS.In practical terms, however, extremely fine meshes are still required. Itis only due to the explosive increases in computer hardware performancecoupled with the availability of parallel processing that LES can even beconsidered as a possibility for engineering calculations.

The following sections give details of the governing equations for LES,present the two options for modeling the subgrid-scale stresses (necessaryto achieve closure of the governing equations), and discuss the relevantboundary conditions.



10.7.1 Filtered Navier-Stokes Equations

The governing equations employed for LES are obtained by filtering thetime-dependent Navier-Stokes equations in either Fourier (wave-number)space or configuration (physical) space. The filtering process effectivelyfilters out the eddies whose scales are smaller than the filter width or gridspacing used in the computations. The resulting equations thus governthe dynamics of large eddies.

A filtered variable (denoted by an overbar) is defined by

φ(x) =∫Dφ(x′)G(x,x′)dx′ (10.7-1)

where D is the fluid domain, and G is the filter function that determinesthe scale of the resolved eddies.

In FLUENT, the finite-volume discretization itself implicitly provides thefiltering operation:

φ(x) =1V

∫Vφ(x′) dx′, x′ ∈ V (10.7-2)

where V is the volume of a computational cell. The filter function,G(x,x′), implied here is then

G(x,x′)

1/V, x′ ∈ V0, x′ otherwise

(10.7-3)

Since the application of LES to compressible flows is still in its infancy,the theory is presented here for incompressible flows. It is assumed thatthe LES model in FLUENT will be applied to essentially incompressible(but not necessarily constant-density) flows.

Filtering the incompressible Navier-Stokes equations, one obtains

∂ρ

∂t+

∂

∂xi(ρui) = 0 (10.7-4)


Modeling Turbulence

and

∂

∂t(ρui) +

∂

∂xj(ρuiuj) =

∂

∂xj

(µ∂ui

∂xj

)− ∂p

∂xi− ∂τij∂xj

(10.7-5)

where τij is the subgrid-scale stress defined by

τij ≡ ρuiuj − ρuiuj (10.7-6)

The similarity between the filtered equations, 10.7-4 through 10.7-6, andthe incompressible form of the RANS equations, Equations 10.2-3 and10.2-4, is obvious. The major difference is that the dependent variablesare now filtered quantities rather than mean quantities, and the expres-sions for the turbulent stresses differ.

10.7.2 Subgrid-Scale Models

The subgrid-scale stresses resulting from the filtering operation are un-known, and require modeling. The majority of subgrid-scale models inuse today are eddy viscosity models of the following form:

τij − 13τkkδij = −2µtSij (10.7-7)

where µt is the subgrid-scale turbulent viscosity, and Sij is the rate-of-strain tensor for the resolved scale defined by

Sij ≡ 12

(∂ui

∂xj+∂uj

∂xi

)(10.7-8)

FLUENT contains two models for µt: the Smagorinsky-Lilly model andthe RNG-based subgrid-scale model.



Smagorinsky-Lilly Model

The most basic of subgrid-scale models was proposed by Smagorin-sky [214] and further developed by Lilly [139]. In the Smagorinsky-Lillymodel, the eddy viscosity is modeled by

µt = ρL2s

∣∣∣S∣∣∣ (10.7-9)

where Ls is the mixing length for subgrid scales and∣∣∣S∣∣∣ ≡ √

2SijSij. Cs

is the Smagorinsky constant. In FLUENT, Ls is computed using

Ls = min(κd,CsV

1/3)

(10.7-10)

where κ is the von Karman constant, d is the distance to the closest wall,and V is the volume of the computational cell.

Lilly derived a value of 0.23 for Cs from homogeneous isotropic turbu-lence in the inertial subrange. However, this value was found to causeexcessive damping of large-scale fluctuations in the presence of meanshear or in transitional flows. Cs=0.1 has been found to yield the bestresults for a wide range of flows, and is the default value in FLUENT.

RNG-Based Subgrid-Scale Model

Renormalization group (RNG) theory can be used to derive a model forthe subgrid-scale eddy viscosity [271]. The RNG procedure results in aneffective subgrid viscosity, µeff = µ+ µt, given by

µeff = µ [1 +H(x)]1/3 (10.7-11)

H(x) is the Heaviside function:

H(x) =

x, x > 00, x ≤ 0

(10.7-12)

where


Modeling Turbulence

x =µ2

sµeff

µ3− C (10.7-13)

and

µs = (CrngV1/3)2

√2SijSij (10.7-14)

where V is the volume of the computational cell. The theory gives Crng =0.157 and C = 100.

In highly turbulent regions of the flow (µt µ), µeff ≈ µs, and theRNG-based subgrid-scale model reduces to the Smagorinsky-Lilly modelwith a different model constant. In low-Reynolds-number regions ofthe flow, the argument of the ramp function becomes negative and theeffective viscosity recovers molecular viscosity. This enables the RNG-based subgrid-scale eddy viscosity to model the low-Reynolds-numbereffects encountered in transitional flows and near-wall regions.

10.7.3 Boundary Conditions for the LES Model

The stochastic components of the flow at the velocity-specified inletboundaries are accounted for by superposing random perturbations onindividual velocity components as

ui =< ui > +I ψ |u| (10.7-15)

where I is the intensity of the fluctuation, ψ is a Gaussian random num-

ber satisfying ψ = 0, and√ψ′ = 1.

When the mesh is fine enough to resolve the laminar sublayer, the wallshear stress is obtained from the laminar stress-strain relationship:

u

uτ=ρuτy

µ(10.7-16)


10.8 Near-Wall Treatments for Wall-Bounded Turbulent Flows

If the mesh is too coarse to resolve the laminar sublayer, it is assumedthat the centroid of the wall-adjacent cell falls within the logarithmicregion of the boundary layer, and the law-of-the-wall is employed:

u

uτ=

1κ

lnE(ρuτy

µ

)(10.7-17)

where κ is the von Karman constant and E = 9.793.

10.8 Near-Wall Treatments for Wall-Bounded TurbulentFlows

10.8.1 Overview

Turbulent flows are significantly affected by the presence of walls. Ob-viously, the mean velocity field is affected through the no-slip conditionthat has to be satisfied at the wall. However, the turbulence is alsochanged by the presence of the wall in non-trivial ways. Very close tothe wall, viscous damping reduces the tangential velocity fluctuations,while kinematic blocking reduces the normal fluctuations. Toward theouter part of the near-wall region, however, the turbulence is rapidlyaugmented by the production of turbulence kinetic energy due to thelarge gradients in mean velocity.

The near-wall modeling significantly impacts the fidelity of numerical so-lutions, inasmuch as walls are the main source of mean vorticity and tur-bulence. After all, it is in the near-wall region that the solution variableshave large gradients, and the momentum and other scalar transports oc-cur most vigorously. Therefore, accurate representation of the flow inthe near-wall region determines successful predictions of wall-boundedturbulent flows.

The k-ε models, the RSM, and the LES model are primarily valid forturbulent core flows (i.e., the flow in the regions somewhat far fromwalls). Consideration therefore needs to be given as to how to makethese models suitable for wall-bounded flows. The Spalart-Allmaras andk-ω models were designed to be applied throughout the boundary layer,provided that the near-wall mesh resolution is sufficient.


Modeling Turbulence

Numerous experiments have shown that the near-wall region can belargely subdivided into three layers. In the innermost layer, called the“viscous sublayer”, the flow is almost laminar, and the (molecular) vis-cosity plays a dominant role in momentum and heat or mass transfer. Inthe outer layer, called the fully-turbulent layer, turbulence plays a ma-jor role. Finally, there is an interim region between the viscous sublayerand the fully turbulent layer where the effects of molecular viscosity andturbulence are equally important. Figure 10.8.1 illustrates these subdi-visions of the near-wall region, plotted in semi-log coordinates.

U/U = U y/τ τ ν

τ ν

U/U

τ

2.5 ln(U y/ ) + 5.45U/U =τ τ ν

viscous sublayer

buffer layeror

blendingregion

fully turbulent regionor

log-law region

y+y+ 5 60∼ −∼− ln U y/

outer layer

Upper limitdepends onReynolds no.

inner layer

Figure 10.8.1: Subdivisions of the Near-Wall Region

Wall Functions vs. Near-Wall Model

Traditionally, there are two approaches to modeling the near-wall region.In one approach, the viscosity-affected inner region (viscous sublayerand buffer layer) is not resolved. Instead, semi-empirical formulas called“wall functions” are used to bridge the viscosity-affected region betweenthe wall and the fully-turbulent region. The use of wall functions obviatesthe need to modify the turbulence models to account for the presence ofthe wall.



In another approach, the turbulence models are modified to enable theviscosity-affected region to be resolved with a mesh all the way to thewall, including the viscous sublayer. For purposes of discussion, this willbe termed the “near-wall modeling” approach. These two approachesare depicted schematically in Figure 10.8.2.

Wall Function Approach Near-Wall Model Approach

buffer &sublayer

turb

ulen

t cor

e

wall functions.

The viscosity-affected region is not The near-wall region is resolved

used. High-Re turbulence models can be

?

resolved, instead is bridged by the all the way down to the wall.

throughout the near-wall region.The turbulence models ought to be valid

Figure 10.8.2: Near-Wall Treatments in FLUENT

In most high-Reynolds-number flows, the wall function approach sub-stantially saves computational resources, because the viscosity-affectednear-wall region, in which the solution variables change most rapidly,does not need to be resolved. The wall function approach is popular be-cause it is economical, robust, and reasonably accurate. It is a practicaloption for the near-wall treatments for industrial flow simulations.

The wall function approach, however, is inadequate in situations wherethe low-Reynolds-number effects are pervasive in the flow domain inquestion, and the hypotheses underlying the wall functions cease to bevalid. Such situations require near-wall models that are valid in theviscosity-affected region and accordingly integrable all the way to thewall.


Modeling Turbulence

FLUENT provides both the wall function approach and the near-wallmodeling approach.

Near-Wall Treatments for the Spalart-Allmaras, k-ω, and LESModels

See Sections 10.3.6, 10.5.1, and 10.7.3, respectively, for a description ofthe near-wall treatments applied by the Spalart-Allmaras, k-ω, and LESmodels.

10.8.2 Wall Functions

Wall functions are a collection of semi-empirical formulas and functionsthat in effect “bridge” or “link” the solution variables at the near-wallcells and the corresponding quantities on the wall. The wall functionscomprise

• laws-of-the-wall for mean velocity and temperature (or other scalars)

• formulas for near-wall turbulent quantities

FLUENT offers two choices of wall function approaches:

• standard wall functions

• non-equilibrium wall functions

Standard Wall Functions

The standard wall functions in FLUENT are based on the proposal ofLaunder and Spalding [129], and have been most widely used for indus-trial flows. They are provided as a default option in FLUENT.

Momentum

The law-of-the-wall for mean velocity yields



U∗ =1κ

ln(Ey∗) (10.8-1)

where

U∗ ≡ UPC1/4µ k

1/2P

τw/ρ(10.8-2)

y∗ ≡ ρC1/4µ k

1/2P yP

µ(10.8-3)

and κ = von Karman constant (= 0.42)E = empirical constant (= 9.81)UP = mean velocity of the fluid at point PkP = turbulence kinetic energy at point PyP = distance from point P to the wallµ = dynamic viscosity of the fluid

The logarithmic law for mean velocity is known to be valid for y∗ >about 30 to 60. In FLUENT, the log-law is employed when y∗ > 11.225.

When the mesh is such that y∗ < 11.225 at the wall-adjacent cells, FLU-ENT applies the laminar stress-strain relationship that can be writtenas

U∗ = y∗ (10.8-4)

It should be noted that, in FLUENT, the laws-of-the-wall for mean ve-locity and temperature are based on the wall unit, y∗, rather than y+

(≡ ρuτy/µ). These quantities are approximately equal in equilibriumturbulent boundary layers.

Energy

Reynolds’ analogy between momentum and energy transport gives a sim-ilar logarithmic law for mean temperature. As in the law-of-the-wall for


Modeling Turbulence

mean velocity, the law-of-the-wall for temperature employed in FLUENTcomprises the following two different laws:

• linear law for the thermal conduction sublayer where conduction isimportant

• logarithmic law for the turbulent region where effects of turbulencedominate conduction

The thickness of the thermal conduction layer is, in general, differentfrom the thickness of the (momentum) viscous sublayer, and changesfrom fluid to fluid. For example, the thickness of the thermal sublayer fora high-Prandtl-number fluid (e.g., oil) is much less than its momentumsublayer thickness. For fluids of low Prandtl numbers (e.g., liquid metal),on the contrary, it is much larger than the momentum sublayer thickness.

In highly compressible flows, the temperature distribution in the near-wall region can be significantly different from that of low subsonic flows,due to the heating by viscous dissipation. In FLUENT, the temperaturewall functions include the contribution from the viscous heating [246].

The law-of-the-wall implemented in FLUENT has the following compositeform:

T ∗ ≡ (Tw − TP ) ρcpC1/4µ k

1/2P

q

=

Pr y∗ + 12ρPrC

1/4µ k

1/2P

q U2P (y∗ < y∗T )

Prt

[1κ ln(Ey∗) + P

]+

12ρ

C1/4µ k

1/2P

q

PrtU

2P + (Pr − Prt)U2

c

(y∗ > y∗T )

(10.8-5)

where P is computed by using the formula given by Jayatilleke [104]:

P = 9.24

[(σ

σt

)3/4

− 1

] [1 + 0.28e−0.007σ/σt

](10.8-6)



and

kf = thermal conductivity of fluidρ = density of fluidcp = specific heat of fluidq = wall heat fluxTP = temperature at the cell adjacent to wallTw = temperature at the wallPr = molecular Prandtl number (µcp/kf )Prt = turbulent Prandtl number (0.85 at the wall)A = 26 (Van Driest constant)κ = 0.4187 (von Karman constant)E = 9.793 (wall function constant)Uc = mean velocity magnitude at y∗ = y∗T

Note that, for the segregated solver, the terms

12ρPr

C1/4µ k

1/2P

qU2

P

and12ρC

1/4µ k

1/2P

q

PrtU2

P + (Pr − Prt)U2c

will be included in Equation 10.8-5 only for compressible flow calcula-tions.

The non-dimensional thermal sublayer thickness, y∗T , in Equation 10.8-5is computed as the y∗ value at which the linear law and the logarithmiclaw intersect, given the molecular Prandtl number of the fluid beingmodeled.

The procedure of applying the law-of-the-wall for temperature is as fol-lows. Once the physical properties of the fluid being modeled are speci-fied, its molecular Prandtl number is computed. Then, given the molec-ular Prandtl number, the thermal sublayer thickness, y∗T , is computedfrom the intersection of the linear and logarithmic profiles, and stored.

During the iteration, depending on the y∗ value at the near-wall cell,either the linear or the logarithmic profile in Equation 10.8-5 is appliedto compute the wall temperature Tw or heat flux q (depending on thetype of the thermal boundary conditions).


Modeling Turbulence

Species

When using wall functions for species transport, FLUENT assumes thatspecies transport behaves analogously to heat transfer. Similarly toEquation 10.8-5, the law-of-the-wall for species can be expressed for con-stant property flow with no viscous dissipation as

Y ∗ ≡ (Yi,w − Yi) ρC1/4µ k

1/2P

Ji,w

=

Sc y∗ (y∗ < y∗c )Sct

[1κ ln(Ey∗) + Pc

](y∗ > y∗c )

(10.8-7)

where Yi is the local species mass fraction, Sc and Sct are molecular andturbulent Schmidt numbers, and Ji,w is the diffusion flux of species iat the wall. Note that Pc and y∗c are calculated in a similar way as Pand y∗T , with the difference being that the Prandtl numbers are alwaysreplaced by the corresponding Schmidt numbers.

Turbulence

In the k-ε models and in the RSM (if the option to obtain wall boundaryconditions from the k equation is enabled), the k equation is solved in thewhole domain including the wall-adjacent cells. The boundary conditionfor k imposed at the wall is

∂k

∂n= 0 (10.8-8)

where n is the local coordinate normal to the wall.

The production of kinetic energy, Gk, and its dissipation rate, ε, at thewall-adjacent cells, which are the source terms in the k equation, arecomputed on the basis of the local equilibrium hypothesis. Under thisassumption, the production of k and its dissipation rate are assumed tobe equal in the wall-adjacent control volume.

Thus, the production of k is computed from



Gk ≈ τw∂U

∂y= τw

τw

κρC1/4µ k

1/2P yP

(10.8-9)

and ε is computed from

εP =C

3/4µ k

3/2P

κyP(10.8-10)

The ε equation is not solved at the wall-adjacent cells, but instead iscomputed using Equation 10.8-10.

Note that, as shown here, the wall boundary conditions for the solutionvariables, including mean velocity, temperature, species concentration,k, and ε, are all taken care of by the wall functions. Therefore, you donot need to be concerned about the boundary conditions at the walls.

The standard wall functions described so far are provided as a defaultoption in FLUENT. The standard wall functions work reasonably well fora broad range of wall-bounded flows. However, they tend to become lessreliable when the flow situations depart too much from the ideal condi-tions that are assumed in their derivation. Among others, the constant-shear and local equilibrium hypotheses are the ones that most restrictthe universality of the standard wall functions. Accordingly, when thenear-wall flows are subjected to severe pressure gradients, and when theflows are in strong non-equilibrium, the quality of the predictions is likelyto be compromised.

The non-equilibrium wall functions offered as an additional option canimprove the results in such situations.

Non-Equilibrium Wall Functions

In addition to the standard wall function described above (which isthe default near-wall treatment) a two-layer-based, non-equilibrium wallfunction [115] is also available. The key elements in the non-equilibriumwall functions are as follows:


Modeling Turbulence

• Launder and Spalding’s log-law for mean velocity is sensitized topressure-gradient effects.

• The two-layer-based concept is adopted to compute the budget ofturbulence kinetic energy (Gk,ε) in the wall-neighboring cells.

The law-of-the-wall for mean temperature or species mass fraction re-mains the same as in the standard wall function described above.

The log-law for mean velocity sensitized to pressure gradients is

UC1/4µ k1/2

τw/ρ=

1κ

ln

(EρC

1/4µ k1/2y

µ

)(10.8-11)

where

U = U − 12dp

dx

[yv

ρκ√k

ln(y

yv

)+y − yv

ρκ√k

+y2

v

µ

](10.8-12)

and yv is the physical viscous sublayer thickness, and is computed from

yv ≡ µy∗vρC

1/4µ k

1/2P

(10.8-13)

where y∗v = 11.225.

The non-equilibrium wall function employs the two-layer concept in com-puting the budget of turbulence kinetic energy at the wall-adjacent cells,which is needed to solve the k equation at the wall-neighboring cells. Thewall-neighboring cells are assumed to consist of a viscous sublayer anda fully turbulent layer. The following profile assumptions for turbulencequantities are made:

τt =

0, y < yv

τw, y > yvk =

(

yyv

)2kP , y < yv

kP , y > yv

ε =

2νky2 , y < yv

k3/2

C`y, y > yv

(10.8-14)



where C` = κC−3/4µ , and yv is the dimensional thickness of the viscous

sublayer, defined in Equation 10.8-13.

Using these profiles, the cell-averaged production of k, Gk, and the cell-averaged dissipation rate, ε, can be computed from the volume averageof Gk and ε of the wall-adjacent cells. For quadrilateral and hexahedralcells for which the volume average can be approximated with a depth-average,

Gk ≡ 1yn

∫ yn

0τt∂U

∂ydy

=1κyn

τ2w

ρC1/4µ k

1/2P

ln(yn

yv

)(10.8-15)

and

ε =1yn

∫ yn

0ε dy

≡ 1yn

[2νyv

+k

1/2P

C`ln(yn

yv

)]kP (10.8-16)

where yn is the height of the cell (yn = 2yP ). For cells with other shapes(e.g., triangular and tetrahedral grids), the appropriate volume averagesare used.

In Equations 10.8-15 and 10.8-16, the turbulence kinetic energy budgetfor the wall-neighboring cells is effectively sensitized to the proportionsof the viscous sublayer and the fully turbulent layer, which varies widelyfrom cell to cell in highly non-equilibrium flows. It effectively relaxes thelocal equilibrium assumption (production = dissipation) that is adoptedby the standard wall function in computing the budget of the turbulencekinetic energy at wall-neighboring cells. Thus, the non-equilibrium wallfunctions, in effect, partly account for non-equilibrium effects neglectedin the standard wall function.


Modeling Turbulence

Standard Wall Functions vs. Non-Equilibrium Wall Functions

Because of the capability to partly account for the effects of pressuregradients and departure from equilibrium, the non-equilibrium wall func-tions are recommended for use in complex flows involving separation,reattachment, and impingement where the mean flow and turbulenceare subjected to severe pressure gradients and change rapidly. In suchflows, improvements can be obtained, particularly in the prediction ofwall shear (skin-friction coefficient) and heat transfer (Nusselt or Stantonnumber).

Limitations of the Wall Function Approach

The standard wall functions give reasonably accurate predictions forthe majority of high-Reynolds-number, wall-bounded flows. The non-equilibrium wall functions further extend the applicability of the wallfunction approach by including the effects of pressure gradient and strongnon-equilibrium. However, the wall function approach becomes less reli-able when the flow conditions depart too much from the ideal conditionsunderlying the wall functions. Examples are as follows:

• Pervasive low-Reynolds-number or near-wall effects (e.g., flowthrough a small gap or highly viscous, low-velocity fluid flow)

• Massive transpiration through the wall (blowing/suction)

• Severe pressure gradients leading to boundary layer separations

• Strong body forces (e.g., flow near rotating disks, buoyancy-drivenflows)

• High three-dimensionality in the near-wall region (e.g., Ekman spi-ral flow, strongly skewed 3D boundary layers)

If any of the items listed above is a prevailing feature of the flow youare modeling, and if it is considered critically important to capture thatfeature for the success of your simulation, you must employ the near-wall modeling approach combined with adequate mesh resolution in the



near-wall region. FLUENT provides the enhanced wall treatment for suchsituations. This approach can be used with the three k-ε models and theRSM.

10.8.3 Enhanced Wall Treatment

Enhanced wall treatment is a near-wall modeling method that com-bines a two-layer model with enhanced wall functions. If the near-wallmesh is fine enough to be able to resolve the laminar sublayer (typicallyy+ ≈ 1), then the enhanced wall treatment will be identical to the tradi-tional two-layer zonal model (see below for details). However, the restric-tion that the near-wall mesh must be sufficiently fine everywhere mightimpose too large a computational requirement. Ideally, then, one wouldlike to have a near-wall formulation that can be used with coarse meshes(usually referred to as wall-function meshes) as well as fine meshes (low-Reynolds-number meshes). In addition, excessive error should not beincurred for intermediate meshes that are too fine for the near-wall cellcentroid to lie in the fully turbulent region, but also too coarse to prop-erly resolve the sublayer.

To achieve the goal of having a near-wall modeling approach that willpossess the accuracy of the standard two-layer approach for fine near-wallmeshes and that, at the same time, will not significantly reduce accuracyfor wall-function meshes, FLUENT can combine the two-layer model withenhanced wall functions, as described in the following sections.

Two-Layer Model for Enhanced Wall Treatment

In FLUENT’s near-wall model, the viscosity-affected near-wall region iscompletely resolved all the way to the viscous sublayer. The two-layerapproach is an integral part of the enhanced wall treatment and is usedto specify both ε and the turbulent viscosity in the near-wall cells. Inthis approach, the whole domain is subdivided into a viscosity-affectedregion and a fully-turbulent region. The demarcation of the two regionsis determined by a wall-distance-based, turbulent Reynolds number, Rey,defined as


Modeling Turbulence

Rey ≡ ρy√k

µ(10.8-17)

where y is the normal distance from the wall at the cell centers. InFLUENT, y is interpreted as the distance to the nearest wall:

y ≡ min~rw∈Γw

‖~r − ~rw‖ (10.8-18)

where ~r is the position vector at the field point, and ~rw is the positionvector on the wall boundary. Γw is the union of all the wall boundariesinvolved. This interpretation allows y to be uniquely defined in flow do-mains of complex shape involving multiple walls. Furthermore, y definedin this way is independent of the mesh topology used, and is definableeven on unstructured meshes.

In the fully turbulent region (Rey > Re∗y; Re∗y = 200), the k-ε models orthe RSM (described in Sections 10.4 and 10.6) are employed.

In the viscosity-affected near-wall region (Rey < Re∗y), the one-equationmodel of Wolfstein [269] is employed. In the one-equation model, themomentum equations and the k equation are retained as described inSections 10.4 and 10.6. However, the turbulent viscosity, µt, is computedfrom

µt,2layer = ρ Cµ`µ√k (10.8-19)

where the length scale that appears in Equation 10.8-19 is computedfrom [34]

`µ = yc`(1 − e−Rey/Aµ

)(10.8-20)

The two-layer formulation for turbulent viscosity described above is usedas a part of the enhanced wall treatment, in which the two-layer definitionis smoothly blended with the high-Reynolds-number µt definition fromthe outer region, as proposed by Jongen [106]:



µt,enh = λεµt + (1 − λε)µt,2layer (10.8-21)

where µt is the high-Reynolds-number definition as described in Sec-tion 10.4 or 10.6 for the k-ε models or the RSM. A blending function,λε, is defined in such a way that it is equal to unity far from walls andis zero very near to walls. The blending function chosen is

λε =12

[1 + tanh

(Rey − Re∗y

A

)](10.8-22)

The constant A determines the width of the blending function. By defin-ing a width such that the value of λε will be within 1% of its far-fieldvalue given a variation of ∆Rey, the result is

A =|∆Rey|

tanh(0.98)(10.8-23)

Typically, ∆Rey would be assigned a value that is between 5% and 20%of Re∗y. The main purpose of the blending function λε is to preventsolution convergence from being impeded when the k-ε solution in theouter layer does not match with the two-layer formulation.

The ε field is computed from

ε =k3/2

`ε(10.8-24)

The length scales that appear in Equation 10.8-24 are again computedfrom Chen and Patel [34]:

`ε = yc`(1 − e−Rey/Aε

)(10.8-25)

If the whole flow domain is inside the viscosity-affected region(Rey < 200), ε is not obtained by solving the transport equation; itis instead obtained algebraically from Equation 10.8-24. FLUENT uses


Modeling Turbulence

a procedure for the ε specification that is similar to the µt blending inorder to ensure a smooth transition between the algebraically-specifiedε in the inner region and the ε obtained from solution of the transportequation in the outer region.

The constants in the length scale formulas, Equations 10.8-20 and 10.8-25,are taken from [34]:

c` = κC−3/4µ , Aµ = 70, Aε = 2c` (10.8-26)

Enhanced Wall Functions

To have a method that can extend its applicability throughout the near-wall region (i.e., laminar sublayer, buffer region, and fully-turbulentouter region) it is necessary to formulate the law-of-the wall as a sin-gle wall law for the entire wall region. FLUENT achieves this by blend-ing linear (laminar) and logarithmic (turbulent) laws-of-the-wall using afunction suggested by Kader [108]:

u+ = eΓu+lam + e

1Γu+

turb (10.8-27)

where the blending function is given by:

Γ = − a(y+)4

1 + by+(10.8-28)

c = exp(E

E′′ − 1.0)

(10.8-29)

a = 0.01c (10.8-30)

b =5c

(10.8-31)

Similarly, the general equation for the derivative du+

dy+ is

du+

dy+= eΓ

du+lam

dy++ e

1Γdu+

turb

dy+(10.8-32)



This approach allows the fully turbulent law to be easily modified andextended to take into account other effects such as pressure gradients orvariable properties. This formula also guarantees the correct asymptoticbehavior for large and small values of y+ and reasonable representationof velocity profiles in the cases where y+ falls inside the wall buffer region(3 < y+ < 10).

The enhanced wall functions were developed by smoothly blending anenhanced turbulent wall law with the laminar wall law. The enhancedturbulent law-of-the-wall for compressible flow with heat transfer andpressure gradients has been derived by combining the approaches ofWhite and Cristoph [266] and Huang et al. [95]:

du+turb

dy+=

1κy+

[S′(1 − βu+ − γ(u+)2)

]1/2(10.8-33)

where

S′ =

1 + αy+ for y+ < y+

s

1 + αy+s for y+ ≥ y+

s(10.8-34)

and

α ≡ νw

τwu∗dp

dx=

µ

ρ2(u∗)3dp

dx(10.8-35)

β ≡ σtqwu∗

cpτwTw=

σtqwρcpu∗Tw

(10.8-36)

γ ≡ σt(u∗)2

2cpTw(10.8-37)

where y+s is the location at which the log-law slope will remain fixed. By

default, y+s = 60. The coefficient α in Equation 10.8-33 represents the

influences of pressure gradients while the coefficients β and γ representthermal effects. Equation 10.8-33 is an ordinary differential equation andFLUENT will provide an appropriate analytical solution. If α, β, and γall equal 0, an analytical solution would lead to the classical turbulentlogarithmic law-of-the-wall.

The laminar law-of-the-wall is determined from the following expression:


Modeling Turbulence

du+lam

dy+= 1 + αy+ (10.8-38)

Note that the above expression only includes effects of pressure gradientsthrough α, while the effects of variable properties due to heat transferand compressibility on the laminar wall law are neglected. These effectsare neglected because they are thought to be of minor importance whenthey occur close to the wall. Integration of Equation 10.8-38 results in

u+lam = y+

(1 +

α

2y+)

(10.8-39)

Enhanced thermal wall functions follow the same approach developedfor the profile of u+. The unified wall thermal formulation blends thelaminar and logarithmic profiles according to the method of Kader [108]:

T+ = eΓT+lam + e

1ΓT+

turb (10.8-40)

where

Γ = − a(Pr y+)4

1 + bPr3 y+(10.8-41)

where Pr is the molecular Prandtl number, and the coefficients a and bare defined as in Equations 10.8-30 and 10.8-31. Apart from the aboveformulation for T+, enhanced thermal wall functions follow the samelogic as previously described for standard thermal wall functions (seeSection 10.8.2). A similar procedure is also used for species wall functionswhen the enhanced wall treatment is used. See Section 10.8.2 for detailsabout species wall functions.

The boundary condition for turbulence kinetic energy is the same as forstandard wall functions (Equation 10.8-8). However, the production ofturbulence kinetic energy Gk is computed using the velocity gradientsthat are consistent with the enhanced law-of-the-wall (Equations 10.8-27and 10.8-32), ensuring a formulation that is valid throughout the near-wall region.


10.9 Grid Considerations for Turbulent Flow Simulations


Successful computations of turbulent flows require some considerationduring the mesh generation. Since turbulence (through the spatially-varying effective viscosity) plays a dominant role in the transport of meanmomentum and other scalars for the majority of complex turbulent flows,you must ascertain that turbulence quantities are properly resolved, ifhigh accuracy is required. Due to the strong interaction of the meanflow and turbulence, the numerical results for turbulent flows tend to bemore susceptible to grid dependency than those for laminar flows.

It is therefore recommended that you resolve, with sufficiently fine meshes,the regions where the mean flow changes rapidly and there are shear lay-ers with a large mean rate of strain.

You can check the near-wall mesh by displaying or plotting the values ofy+, y∗, and Rey, which are all available in the postprocessing panels. Itshould be remembered that y+, y∗, and Rey are not fixed, geometricalquantities. They are all solution-dependent. For example, when youdouble the mesh (thereby halving the wall distance), the new y+ doesnot necessarily become half of the y+ for the original mesh.

For the mesh in the near-wall region, different strategies must be useddepending on which near-wall option you are using. In Sections 10.9.1and 10.9.2 are general guidelines for the near-wall mesh.

10.9.1 Near-Wall Mesh Guidelines for Wall Functions

The distance from the wall at the wall-adjacent cells must be determinedby considering the range over which the log-law is valid. The distanceis usually measured in the wall unit, y+ (≡ ρuτy/µ), or y∗. Note thaty+ and y∗ have comparable values when the first cell is placed in thelog-layer.

• It is known that the log-law is valid for y+ > 30 to 60.

• Although FLUENT employs the linear (laminar) law when y+ <11.225, using an excessively fine mesh near the walls should beavoided, because the wall functions cease to be valid in the viscoussublayer.


Modeling Turbulence

• The upper bound of the log-layer depends on, among others, pres-sure gradients and Reynolds number. As the Reynolds numberincreases, the upper bound tends to also increase. y+ values thatare too large are not desirable, because the wake component be-comes substantially large above the log-layer.

• A y+ value close to the lower bound (y+ ≈ 30) is most desirable.

• Using excessive stretching in the direction normal to the wall shouldbe avoided.

• It is important to have at least a few cells inside the boundarylayer.

10.9.2 Near-Wall Mesh Guidelines for the Enhanced WallTreatment

Although the enhanced wall treatment is designed to extend the valid-ity of near-wall modeling beyond the viscous sublayer, it is still recom-mended that you construct a mesh that will fully resolve the viscosity-affected near-wall region. In such a case, the two-layer component ofthe enhanced wall treatment will be dominant and the following meshrequirements are recommended (note that, here, the mesh requirementsare in terms of y+, not y∗):

• When the enhanced wall treatment is employed with the inten-tion of resolving the laminar sublayer, y+ at the wall-adjacent cellshould be on the order of y+ = 1. However, a higher y+ is ac-ceptable as long as it is well inside the viscous sublayer (y+ < 4 to5).

• You should have at least 10 cells within the viscosity-affected near-wall region (Rey < 200) to be able to resolve the mean velocityand turbulent quantities in that region.

10.9.3 Near-Wall Mesh Guidelines for the Spalart-AllmarasModel

The Spalart-Allmaras model in its complete implementation is a low-Reynolds-number model. This means that it is designed to be used with



meshes that properly resolve the viscous-affected region, and dampingfunctions have been built into the model in order to properly attenuatethe turbulent viscosity in the viscous sublayer. Therefore, to obtain thefull benefit of the Spalart-Allmaras model, the near-wall mesh spacingshould be as described in Section 10.9.2 for the enhanced wall treatment.

However, as discussed in Section 10.3.6, the boundary conditions forthe Spalart-Allmaras model have been implemented so that the modelwill work on coarser meshes, such as would be appropriate for the wallfunction approach. If you are using a coarse mesh, you should follow theguidelines described in Section 10.9.1.

In summary, for best results with the Spalart-Allmaras model, you shoulduse either a very fine near-wall mesh spacing (on the order of y+ = 1) ora mesh spacing such that y+ ≥ 30.

10.9.4 Near-Wall Mesh Guidelines for the k-ω Models

Both k-ω models available in FLUENT are available as low-Reynolds-number models as well as high-Reynolds-number models. If the Transi-tional Flows option is enabled in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines shouldbe the same as for the enhanced wall treatment. However, if this optionis not active, then the mesh guidelines should be the same as for the wallfunctions.

10.9.5 Near-Wall Mesh Guidelines for Large Eddy Simulation

For the LES implementation in FLUENT, the wall boundary conditionshave been implemented using a law-of-the-wall approach as described inSection 10.7.3. This means that there are no computational restrictionson the near-wall mesh spacing. However, for best results, it might benecessary to use a very fine near-wall mesh spacing (on the order ofy+ = 1).


Modeling Turbulence

10.10 Problem Setup for Turbulent Flows

When your FLUENT model includes turbulence you need to activate therelevant model and options, and supply turbulent boundary conditions.These inputs are described in this section.

The procedure for setting up a turbulent flow problem is described below.(Note that this procedure includes only those steps necessary for theturbulence model itself; you will need to set up other models, boundaryconditions, etc. as usual.)

1. To activate the turbulence model, select Spalart-Allmaras, k-epsilon,k-omega, Reynolds Stress, or (in 3D) Large Eddy Simulation underModel in the Viscous Model panel (Figure 10.10.1).

Define −→ Models −→Viscous...

If you choose the k-epsilon model, select Standard, RNG, or Realiz-able under k-epsilon Model. If you choose the k-omega model, selectStandard or SST under k-omega Model.

The Large Eddy Simulation model is available only for 3D cases.!

2. If the flow involves walls, and you are using one of the k-ε modelsor the RSM, choose one of the following options for the Near-WallTreatment in the Viscous Model panel:

• Standard Wall Functions

• Non-Equilibrium Wall Functions

• Enhanced Wall Treatment

These near-wall options are described in detail in Section 10.8. Bydefault, the standard wall function is enabled.

The near-wall treatment for the Spalart-Allmaras, k-ω, and LESmodels is defined automatically, as described in Sections 10.3.6,10.5.1, and 10.7.3, respectively.

3. Enable the appropriate turbulence modeling options in the ViscousModel panel. See Section 10.10.1 for details.



Figure 10.10.1: The Viscous Model Panel


Modeling Turbulence

4. Specify the boundary conditions for the solution variables.

Define −→Boundary Conditions...

See Section 10.10.2 for details.

5. Specify the initial guess for the solution variables.

Solve −→ Initialize −→Initialize...

See Section 10.10.3 for details. Note that Reynolds stresses areautomatically initialized using k, and therefore need not be initial-ized.

10.10.1 Turbulence Options

The various options available for the turbulence models are describedin detail in Sections 10.3 through 10.7. Instructions for activating theseoptions are provided here.

If you choose the Spalart-Allmaras model, the following options are avail-able:

• Vorticity-based production

• Strain/vorticity-based production

• Viscous heating (always activated for the coupled solvers)

If you choose the standard k-ε model or the realizable k-ε model, thefollowing options are available:


• Inclusion of buoyancy effects on ε

If you choose the RNG k-ε model, the following options are available:

• Differential viscosity model

• Swirl modification





If you choose the standard k-ω model, the following options are available:

• Transitional flows

• Shear flow corrections


If you choose the shear-stress transport k-ω model, the following optionsare available:

• Transitional flows


If you choose the RSM, the following options are available:

• Wall reflection effects on Reynolds stresses

• Wall boundary conditions for the Reynolds stresses from the kequation

• Quadratic pressure-strain model



If you choose the enhanced wall treatment (available for the k-ε modelsand the RSM), the following options are available:

• Pressure gradient effects

• Thermal effects


Modeling Turbulence

If you choose the LES model, the following options are available:

• Smagorinsky-Lilly model for the subgrid-scale viscosity

• RNG model for the subgrid-scale viscosity


It is also possible to modify the Model Constants, but this is not nec-essary for most applications. See Sections 10.3 through 10.7 for detailsabout these constants. Note that C1-PS and C2-PS are the constants C1

and C2 in the linear pressure-strain approximation of Equations 10.6-5and 10.6-6, and C1’-PS and C2’-PS are the constants C ′

1 and C ′2 in

Equation 10.6-7. C1-SSG-PS, C1’-SSG-PS, C2-SSG-PS, C3-SSG-PS, C3’-SSG-PS, C4-SSG-PS, and C5-SSG-PS are the constants C1, C∗

1 , C2, C3,C∗

3 , C4, and C5 in the quadratic pressure-strain approximation of Equa-tion 10.6-16.

Including the Viscous Heating Effects

See Sections 11.2.1 and 11.2.2 for information on including viscous heat-ing effects in your model.

Including Turbulence Generation Due to Buoyancy

If you specify a non-zero gravity force (in the Operating Conditions panel),and you are modeling a non-isothermal flow, the generation of turbulentkinetic energy due to buoyancy (Gb in Equation 10.4-1) is, by default, al-ways included in the k equation. However, FLUENT does not, by default,include the buoyancy effects on ε.

To include the buoyancy effects on ε, you must turn on the Full BuoyancyEffects option under Options in the Viscous Model panel.

This option is available for the three k-ε models and for the RSM.



Vorticity- and Strain/Vorticity-Based Production

For the Spalart-Allmaras model, you can choose either Vorticity-BasedProduction or Strain/Vorticity-Based Production under Spalart-AllmarasOptions in the Viscous Model panel. If you choose Vorticity-Based Pro-duction, FLUENT will use Equation 10.3-8 to compute the value of thedeformation tensor S; if you choose Strain/Vorticity-Based Production, itwill use Equation 10.3-10.

(These options will not appear unless you have activated the Spalart-Allmaras model.)

Differential Viscosity Modification

In the RNG turbulence model in FLUENT, you have an option to usea differential formula for effective viscosity µeff (Equation 10.4-6) toaccount for low-Reynolds-number effects. To enable this option, turnon Differential Viscosity Model under RNG Options in the Viscous Modelpanel.

(This option will not appear unless you have activated the RNG k-εmodel.)

Swirl Modification

Once you choose the RNG model, the swirl modification takes effect,by default, for all three-dimensional flows and axisymmetric flows withswirl. The default swirl constant (αs in Equation 10.4-8) is set to 0.05,which works well for weakly to moderately swirling flows. However, forstrongly swirling flows, you may need to use a larger swirl constant.

In order to change the value of the swirl constant, you must first turnon the Swirl Dominated Flow option under RNG Options in the ViscousModel panel. (This option will not appear unless you have activated theRNG k-ε model.)

Once you turn on the Swirl Dominated Flow option, the swirl constantαs is increased to 0.07. You can change its value in the Swirl Factor fieldunder Model Constants.


Modeling Turbulence

Transitional Flows

If either of the k-ω models are used, you may enable a low-Reynolds-number correction to the turbulent viscosity by enabling the TransitionalFlows option under k-omega Options in the Viscous Model panel. Bydefault, this option is not enabled, and the damping coefficient (α∗ inEquation 10.5-6) is equal to 1.

Shear Flow Corrections

In the standard k-ω model, you also have the option of including correc-tions to improve the accuracy in predicting free shear flows. The ShearFlow Corrections option under k-omega Options is enabled by default inthe Viscous Model panel, as these corrections are included in the standardk-ω model [267]. When this option is enabled, FLUENT will calculate f∗βand fβ using Equations 10.5-16 and 10.5-24, respectively. If this optionis disabled, f∗β and fβ will be set equal to 1.

Including Pressure Gradient Effects

If the enhanced wall treatment is used, you may include the effects ofpressure gradients by enabling the Pressure Gradient Effects option underEnhanced Wall Treatment Options. When this option is enabled, FLUENTwill include the coefficient α in Equation 10.8-33.

Including Thermal Effects

If the enhanced wall treatment is used, you may include thermal effectsby enabling the Thermal Effects option under Enhanced Wall TreatmentOptions. When this option is enabled, FLUENT will include the coeffi-cient β in Equation 10.8-33. γ will also be included in Equation 10.8-33when the Thermal Effects option is enabled if the ideal gas law is selectedfor the fluid density in the Materials panel.

Including the Wall Reflection Term

If the RSM is used with the default model for pressure strain, FLUENTwill, by default, include the wall-reflection effects in the pressure-strainterm. That is, FLUENT will calculate φw

ij using Equation 10.6-7 and



include it in Equation 10.6-4. Note that wall-reflection effects are notincluded if you have selected the quadratic pressure-strain model.

The empirical constants and the function f used in the calculation of!φw

ij are calibrated for simple canonical flows such as channel flows andflat-plate boundary layers involving a single wall. If the flow involvesmultiple walls and the wall has significant curvature (e.g., an axisym-metric pipe or curvilinear duct), the inclusion of the wall-reflection termin Equation 10.6-7 may not improve the accuracy of the RSM predic-tions. In such cases, you can disable the wall-reflection effects by turningoff the Wall Reflection Effects under Reynolds-Stress Options in the ViscousModel panel.

Solving the k Equation to Obtain Wall Boundary Conditions

In the RSM, FLUENT, by default, uses the explicit setting of boundaryconditions for the Reynolds stresses near the walls, with the values com-puted with Equation 10.6-28. k is calculated by solving the k equationobtained by summing Equation 10.6-1 for normal stresses. To disable thisoption and use the wall boundary conditions given in Equation 10.6-29,turn off Wall B.C. from k Equation under Reynolds-Stress Options in theViscous Model panel. (This option will not appear unless you have acti-vated the RSM.)

Quadratic Pressure-Strain Model

To use the quadratic pressure-strain model described in Section 10.6.3,turnon the Quadratic Pressure-Strain Model option under Reynolds-Stress Op-tions in the Viscous Model panel. (This option will not appear unless youhave activated the RSM.) The following options are not available whenthe Quadratic Pressure-Strain Model is enabled:

• Wall Reflection Effects under Reynolds-Stress Options

• Enhanced Wall Treatment under Near-Wall Treatment


Modeling Turbulence

Subgrid-Scale Model

If you have selected the Large Eddy Simulation model, you will be able tochoose which of the two subgrid-scale models described in Section 10.7.2is to be used. You can choose either the Smagorinsky-Lilly or the RNGsubgrid-scale model.

(These options will not appear unless you have activated the LES model.)

Customizing the Turbulent Viscosity

If you are using the Spalart-Allmaras, k-ε, k-ω, or LES model, a user-defined function can be used to customize the turbulent viscosity. Thisoption will enable you to modify µt in the case of the Spalart-Allmaras,k-ε, and k-ω models, and incorporate completely new subgrid models inthe case of the LES model. See the separate UDF Manual for informationabout user-defined functions.

In the Viscous Model panel, under User-Defined Functions, select the ap-propriate user-defined function in the Turbulent Viscosity drop-down list.

10.10.2 Defining Turbulence Boundary Conditions

k-ε Models and k-ω Models

When you are modeling turbulent flows in FLUENT using one of thek-ε models or one of the k-ω models, you must provide the boundaryconditions for k and ε (or k and ω) in addition to other mean solutionvariables. The boundary conditions for k and ε (or k and ω) at thewalls are internally taken care of by FLUENT, which obviates the needfor your inputs. The boundary condition inputs for k and ε (or k and ω)you must supply to FLUENT are the ones at inlet boundaries (velocityinlet, pressure inlet, etc.). In many situations, it is important to specifycorrect or realistic boundary conditions at the inlets, because the inletturbulence can significantly affect the downstream flow.

See Section 6.2.2 for details about specifying the boundary conditionsfor k and ε (or k and ω) at the inlets.



You may want to include the effects of the wall roughness on selectedwall boundaries. In such cases, you can specify the roughness param-eters (roughness height and roughness constant) in the panels for thecorresponding wall boundaries (see Section 6.13.1).

The Spalart-Allmaras Model

When you are modeling turbulent flows in FLUENT using the Spalart-Allmaras model, you must provide the boundary conditions for ν inaddition to other mean solution variables. The boundary conditions forν at the walls are internally taken care of by FLUENT, which obviatesthe need for your inputs. The boundary condition input for ν you mustsupply to FLUENT is the one at inlet boundaries (velocity inlet, pressureinlet, etc.). In many situations, it is important to specify correct orrealistic boundary conditions at the inlets, because the inlet turbulencecan significantly affect the downstream flow.

See Section 6.2.2 for details about specifying the boundary condition forν at the inlets.

You may want to include the effects of the wall roughness on selectedwall boundaries. In such cases, you can specify the roughness param-eters (roughness height and roughness constant) in the panels for thecorresponding wall boundaries (see Section 6.13.1).

Reynolds Stress Model

The specification of turbulent boundary conditions for the RSM is thesame as for the other turbulence models for all boundaries except atboundaries where flow enters the domain. Additional input methods areavailable for these boundaries and are described here.

When you choose to use the RSM, the default inlet boundary conditioninputs required are identical to those required when the k-ε model isactive. You can input the turbulence quantities using any of the tur-bulence specification methods described in Section 6.2.2. FLUENT thenuses the specified turbulence quantities to derive the Reynolds stressesat the inlet from the assumption of isotropy of turbulence:


Modeling Turbulence

u′2i =

23k (i = 1, 2, 3) (10.10-1)

u′iu′j = 0.0 (10.10-2)

where u′2i is the normal Reynolds stress component in each direction.

The boundary condition for ε is determined in the same manner as forthe k-ε turbulence models (see Section 6.2.2). To use this method, youwill select K or Turbulence Intensity as the Reynolds-Stress SpecificationMethod in the appropriate boundary condition panel.

Alternately, you can directly specify the Reynolds stresses by selectingReynolds-Stress Components as the Reynolds-Stress Specification Methodin the boundary condition panel. When this option is enabled, youshould input the Reynolds stresses directly.

You can set the Reynolds stresses by using constant values, profile func-tions of coordinates (see Section 6.25), or user-defined functions (see theseparate UDF Manual).

Large Eddy Simulation Model

It is possible to specify the magnitude of random fluctuations of thevelocity components at an inlet only if the velocity inlet boundary con-dition is selected. In this case, you must specify a Turbulence Intensitythat determines the magnitude of the random perturbations on individ-ual mean velocity components as described in Section 10.7.3. For allboundary types other than velocity inlets, the boundary conditions forLES remain the same as for laminar flows.

10.10.3 Providing an Initial Guess for k and ε (or k and ω)

For flows using one of the k-ε models, one of the k-ω models, or the RSM,the converged solutions or (for unsteady calculations) the solutions aftera sufficiently long time has elapsed should be independent of the initialvalues for k and ε (or k and ω). For better convergence, however, it isbeneficial to use a reasonable initial guess for k and ε (or k and ω).



Figure 10.10.2: Specifying Inlet Boundary Conditions for the ReynoldsStresses


Modeling Turbulence

In general, it is recommended that you start from a fully-developed stateof turbulence. When you use the enhanced wall treatment for the k-εmodels or the RSM, it is critically important to specify fully-developedturbulence fields. Guidelines are provided below.

• If you were able to specify reasonable boundary conditions at theinlet, it may be a good idea to compute the initial values for k andε (or k and ω) in the whole domain from these boundary values.(See Section 22.13 for details.)

• For more complex flows (e.g., flows with multiple inlets with dif-ferent conditions) it may be better to specify the initial values interms of turbulence intensity. 5–10% is enough to represent fully-developed turbulence. k can then be computed from the turbulenceintensity and the characteristic mean velocity magnitude of yourproblem (k = 1.5(Iuavg)2).

You should specify an initial guess for ε so that the resulting eddyviscosity (Cµ

k2

ε ) is sufficiently large in comparison to the molecularviscosity. In fully-developed turbulence, the turbulent viscosity isroughly two orders of magnitude larger than the molecular viscos-ity. From this, you can compute ε.

Note that, for the RSM, Reynolds stresses are initialized automaticallyusing Equations 10.10-1 and 10.10-2.


10.11 Solution Strategies for Turbulent Flow Simulations


Compared to laminar flows, simulations of turbulent flows are more chal-lenging in many ways. For the Reynolds-averaged approach, additionalequations are solved for the turbulence quantities. Since the equationsfor mean quantities and the turbulent quantities (µt, k, ε, ω, or theReynolds stresses) are strongly coupled in a highly non-linear fashion,it takes more computational effort to obtain a converged turbulent solu-tion than to obtain a converged laminar solution. The LES model, whileembodying a simpler, algebraic model for the subgrid-scale viscosity, re-quires a transient solution on a very fine mesh.

The fidelity of the results for turbulent flows is largely determined by theturbulence model being used. Here are some guidelines that can enhancethe quality of your turbulent flow simulations.

10.11.1 Mesh Generation

The following are suggestions to follow when generating the mesh for usein your turbulent flow simulation:

• Picture in your mind the flow under consideration using your phys-ical intuition or any data for a similar flow situation, and identifythe main flow features expected in the flow you want to model.Generate a mesh that can resolve the major features that you ex-pect.

• If the flow is wall-bounded, and the wall is expected to significantlyaffect the flow, take additional care when generating the mesh. Youshould avoid using a mesh that is too fine (for the wall function ap-proach) or too coarse (for the enhanced wall treatment approach).See Section 10.9 for details.

10.11.2 Accuracy

The suggestions below are provided to help you obtain better accuracyin your results:


Modeling Turbulence

• Use the turbulence model that is better suited for the salient fea-tures you expect to see in the flow (see Section 10.2).

• Because the mean quantities have larger gradients in turbulentflows than in laminar flows, it is recommended that you use high-order schemes for the convection terms. This is especially true ifyou employ a triangular or tetrahedral mesh. Note that excessivenumerical diffusion adversely affects the solution accuracy, evenwith the most elaborate turbulence model.

• In some flow situations involving inlet boundaries, the flow down-stream of the inlet is dictated by the boundary conditions at theinlet. In such cases, you should exercise care to make sure thatreasonably realistic boundary values are specified.

10.11.3 Convergence

The suggestions below are provided to help you enhance convergence forturbulent flow calculations:

• Starting with excessively crude initial guesses for mean and tur-bulence quantities may cause the solution to diverge. A safe ap-proach is to start your calculation using conservative (small) under-relaxation parameters and (for the coupled solvers) a conservativeCourant number, and increase them gradually as the iterationsproceed and the solution begins to settle down.

• It is also helpful for faster convergence to start with reasonable ini-tial guesses for the k and ε (or k and ω) fields. Particularly whenthe enhanced wall treatment is used, it is important to start witha sufficiently developed turbulence field, as recommended in Sec-tion 10.10.3, to avoid the need for an excessive number of iterationsto develop the turbulence field.

• When you are using the RNG k-ε model, an approach that mighthelp you achieve better convergence is to obtain a solution withthe standard k-ε model before switching to the RNG model. Due



to the additional non-linearities in the RNG model, lower under-relaxation factors and (for the coupled solvers) a lower Courantnumber might also be necessary.

Note that when you use the enhanced wall treatment, you may some-times find during the calculation that the residual for ε is reported tobe zero. This happens when your flow is such that Rey is less than 200in the entire flow domain, and ε is obtained from the algebraic formula(Equation 10.8-24) instead of from its transport equation.

10.11.4 RSM-Specific Solution Strategies

Using the RSM creates a high degree of coupling between the momentumequations and the turbulent stresses in the flow, and thus the calculationcan be more prone to stability and convergence difficulties than with thek-ε models. When you use the RSM, therefore, you may need to adoptspecial solution strategies in order to obtain a converged solution. Thefollowing strategies are generally recommended:

• Begin the calculations using the standard k-ε model. Turn on theRSM and use the k-ε solution data as a starting point for the RSMcalculation.

• Use low under-relaxation factors (0.2 to 0.3) and (for the coupledsolvers) a low Courant number for highly swirling flows or highlycomplex flows. In these cases, you may need to reduce the under-relaxation factors both for the velocities and for all of the stresses.

Instructions for setting these solution parameters are provided below. Ifyou are applying the RSM to prediction of a highly swirling flow, youwill want to consider the solution strategies discussed in Section 8.4 aswell.

Under-Relaxation of the Reynolds Stresses

FLUENT applies under-relaxation to the Reynolds stresses. You can setunder-relaxation factors using the Solution Controls panel.


Modeling Turbulence

Solve −→ Controls −→Solution...

The default settings of 0.5 are recommended for most cases. You may beable to increase these settings and speed up the convergence when theRSM solution begins to converge.

Disabling Calculation Updates of the Reynolds Stresses

In some instances, you may wish to let the current Reynolds stress fieldremain fixed, skipping the solution of the Reynolds transport equationswhile solving the other transport equations. You can activate/deactivateall Reynolds stress equations in the Solution Controls panel.


Residual Reporting for the RSM

When you use the RSM for turbulence, FLUENT reports the equationresiduals for the individual Reynolds stress transport equations. Youcan apply the usual convergence criteria to the Reynolds stress residuals:normalized residuals in the range of 10−3 usually indicate a practically-converged solution. However, you may need to apply tighter convergencecriteria (below 10−4) to ensure full convergence.

10.11.5 LES-Specific Solution Strategies

Large eddy simulation involves running a transient solution from someinitial condition, on an appropriately fine grid, using an appropriate timestep size. The solution must be run long enough to become independentof the initial condition and to enable the statistics of the flow field to bedetermined.

The following are suggestions to follow when running a large eddy sim-ulation:

1. Start by running a flow simulation assuming laminar flow or usinga simple Reynolds-averaged turbulence model such as standard k-ε or Spalart-Allmaras. Since this is only an initial condition, youneed run only until the flow field is somewhat converged. This stepis optional.



2. When you enable LES, FLUENT will automatically turn on theunsteady solver option and choose the second-order implicit for-mulation. You will need to set the appropriate time step size andall the needed solution parameters. (See Section 22.15.1 for guide-lines on setting solution parameters for transient calculations ingeneral.) Use the central-differencing spatial discretization schemefor all equations.

3. Run LES until the flow becomes statistically steady. The best wayto see if the flow is fully developed and statistically steady is tomonitor forces and solution variables (e.g., velocity components orpressure) at selected locations in the flow.

4. Zero out the initial statistics using the solve/initialize/init-flow-statistics text command. Before you restart the so-lution, enable Data Sampling for Time Statistics in the Iterate panel,as described in Section 22.15.1.

5. Continue until you get statistically stable data. The duration ofthe simulation can be determined beforehand by estimating themean flow residence time in the solution domain (L/U , where L isthe characteristic length of the solution domain and U is a charac-teristic mean flow velocity). The simulation should be run for atleast a few mean flow residence times.

Instructions for setting the solution parameters for LES are providedbelow.

Temporal Discretization

FLUENT provides both first-order and second-order temporal discretiza-tions. For LES, the second-order discretization is recommended.

Define −→ Models −→Solver...

Spatial Discretization

Overly diffusive schemes such as the first-order upwind or power lawscheme should be avoided, because they may unduly damp out the energy


Modeling Turbulence

of the resolved eddies. The central-differencing scheme is recommendedfor all equations when you use the LES model.


10.12 Postprocessing for Turbulent Flows

FLUENT provides postprocessing options for displaying, plotting, andreporting various turbulence quantities, which include the main solutionvariables and other auxiliary quantities.

Turbulence quantities that can be reported for the k-ε models are asfollows:

• Turbulent Kinetic Energy (k)

• Turbulence Intensity

• Turbulent Dissipation Rate (Epsilon)

• Production of k

• Turbulent Viscosity

• Effective Viscosity

• Turbulent Viscosity Ratio

• Effective Thermal Conductivity

• Effective Prandtl Number

• Wall Yplus

• Wall Ystar

• Turbulent Reynolds Number (Re y) (only when the enhanced walltreatment is used for the near-wall treatment)

Turbulence quantities that can be reported for the k-ω models are asfollows:





• Specific Dissipation Rate (Omega)

• Production of k






• Wall Ystar

• Wall Yplus

Turbulence quantities that can be reported for the Spalart-Allmarasmodel are as follows:

• Modified Turbulent Viscosity






• Wall Yplus

Turbulence quantities that can be reported for the RSM are as follows:



Modeling Turbulence


• UU Reynolds Stress

• VV Reynolds Stress

• WW Reynolds Stress

• UV Reynolds Stress

• VW Reynolds Stress

• UW Reynolds Stress

• Turbulent Dissipation Rate (Epsilon)

• Production of k






• Wall Yplus

• Wall Ystar

• Turbulent Reynolds Number (Re y)

Turbulence quantities that can be reported for the LES model are asfollows:

• Subgrid Turbulent Kinetic Energy

• Subgrid Turbulent Viscosity

• Subgrid Effective Viscosity



• Subgrid Turbulent Viscosity Ratio



• Wall Yplus

All of these variables can be found in the Turbulence... category of thevariable selection drop-down list that appears in postprocessing panels.See Chapter 27 for their definitions.

10.12.1 Custom Field Functions for Turbulence

In addition to the quantities listed above, you can define your own tur-bulence quantities using the Custom Field Function Calculator panel.

Define −→Custom Field Functions...

The following functions may be useful:

• Ratio of production of k to its dissipation (Gk/ρε)

• Ratio of the mean flow to turbulent time scale, η (≡ Sk/ε)

• Reynolds stresses derived from the Boussinesq formula (e.g., −uv =νt

∂u∂y )

10.12.2 Postprocessing LES Statistics

As described in Section 10.7, LES involves the solution of a transientflow field, but it is the mean flow quantities that are of most engineeringinterest. If you turn on the Data Sampling for Time Statistics optionin the Iterate panel, FLUENT will gather data for time statistics whileperforming a large eddy simulation. You can then view both the meanand the root-mean-square (RMS) values in FLUENT. See Section 22.15.3for details.


Modeling Turbulence

10.12.3 Troubleshooting

You can use the postprocessing options not only for the purpose of in-terpreting your results but also for investigating any anomalies that mayappear in the solution. For instance, you may want to plot contours ofthe k field to check if there are any regions where k is erroneously largeor small. You should see a high k region in the region where the produc-tion of k is large. You may want to display the turbulent viscosity ratiofield in order to see whether or not turbulence takes full effect. Usu-ally turbulent viscosity is at least two orders of magnitude larger thanmolecular viscosity for fully-developed turbulent flows modeled using theRANS approach (i.e., not using LES). You may also want to see whetheryou are using a proper near-wall mesh for the enhanced wall treatment.In this case, you can display filled contours of Rey (turbulent Reynoldsnumber) overlaid on the mesh.


chapter 10. modeling turbulence - enea · chapter 10. modeling turbulence this chapter provides...

Documents