Chp12 RNG K-Epsilon

Chapter 12. Modeling Turbulence

This chapter provides details about the turbulence models available in FLUENT.

Information is presented in the following sections:

• Section 12.1: Introduction

• Section 12.2: Choosing a Turbulence Model

• Section 12.3: Spalart-Allmaras Model Theory

• Section 12.4: Standard, RNG, and Realizable k-ε Models Theory

• Section 12.5: Standard and SST k-ω Models Theory

• Section 12.6: The v2-f Model Theory

• Section 12.7: Reynolds Stress Model (RSM) Theory

• Section 12.8: Detached Eddy Simulation (DES) Model Theory

• Section 12.9: Large Eddy Simulation (LES) Model Theory

• Section 12.10: Near-Wall Treatments for Wall-Bounded Turbulent Flows

• Section 12.11: Grid Considerations for Turbulent Flow Simulations

• Section 12.12: Steps in Using a Turbulence Model

• Section 12.13: Setting Up the Spalart-Allmaras Model

• Section 12.14: Setting Up the k-ε Model

• Section 12.15: Setting Up the k-ω Model

• Section 12.16: Setting Up the Reynolds Stress Model

• Section 12.17: Setting Up the Detached Eddy Simulation Model

• Section 12.18: Setting Up the Large Eddy Simulation Model

• Section 12.19: Setup Options for all Turbulence Modeling

• Section 12.20: Defining Turbulence Boundary Conditions

c© Fluent Inc. September 29, 2006 12-1

Modeling Turbulence

• Section 12.21: Providing an Initial Guess for k and ε (or k and ω)

• Section 12.22: Solution Strategies for Turbulent Flow Simulations

• Section 12.23: Postprocessing for Turbulent Flows

12.1 Introduction

Turbulent flows are characterized by fluctuating velocity fields. These fluctuations mixtransported quantities such as momentum, energy, and species concentration, and causethe transported quantities to fluctuate as well. Since these fluctuations can be of smallscale and high frequency, they are too computationally expensive to simulate directly inpractical engineering calculations. Instead, the instantaneous (exact) governing equationscan be time-averaged, ensemble-averaged, or otherwise manipulated to remove the smallscales, resulting in a modified set of equations that are computationally less expensiveto solve. However, the modified equations contain additional unknown variables, andturbulence models are 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

• v2-f model (addon)

• Reynolds stress model (RSM)

– Linear pressure-strain RSM model

– Quadratic pressure-strain RSM model

– Low-Re stress-omega RSM model

12-2 c© Fluent Inc. September 29, 2006

12.2 Choosing a Turbulence Model

• Detached eddy simulation (DES) model

– Spalart-Allmaras RANS model

– Realizable k-ε RANS model

– SST k-ω RANS model

• Large eddy simulation (LES) model

– Smagorinsky-Lilly subgrid-scale model

– WALE subgrid-scale model

– Kinetic-energy transport subgrid-scale model


It is an unfortunate fact that no single turbulence model is universally accepted as be-ing superior for all classes of problems. The choice of turbulence model will depend onconsiderations such as the physics encompassed in the flow, the established practice fora specific class of problem, the level of accuracy required, the available computationalresources, and the amount of time available for the simulation. To make the most ap-propriate choice of model for your application, you need to understand the capabilitiesand limitations of the various options.

The purpose of this section is to give an overview of issues related to the turbulencemodels provided in FLUENT. The computational effort and cost in terms of CPU time andmemory of the individual models is discussed. While it is impossible to state categoricallywhich model is best for a specific application, general guidelines are presented to helpyou choose the appropriate turbulence model for the flow you want to model.

12.2.1 Reynolds-Averaged Approach of the DES Model vs. LES

Time-dependent solutions of the Navier-Stokes equations for high Reynolds-number tur-bulent flows in complex geometries which set out to resolve all the way down to thesmallest scales of the motions are unlikely to be attainable for some time to come. Twoalternative methods can be employed to render the Navier-Stokes equations tractableso that the small-scale turbulent fluctuations do not have to be directly simulated:Reynolds-averaging (or ensemble-averaging) and filtering. Both methods introduce ad-ditional terms in the governing equations that need to be modeled in order to achieve a“closure” for the unknowns.

The Reynolds-averaged Navier-Stokes (RANS) equations govern the transport of the aver-aged flow quantities, with the whole range of the scales of turbulence being modeled. TheRANS-based modeling approach therefore greatly reduces the required computational ef-fort and resources, and is widely adopted for practical engineering applications. An entirehierarchy of closure models are available in FLUENT including Spalart-Allmaras, k-ε andits variants, k-ω and its variants, and the RSM. The RANS equations are often used


Modeling Turbulence

to compute time-dependent flows, whose unsteadiness may be externally imposed (e.g.,time-dependent boundary conditions or sources) or self-sustained (e.g., vortex-shedding,flow instabilities).

LES provides an alternative approach in which large eddies are explicitly computed (re-solved) in a time-dependent simulation using the “filtered” Navier-Stokes equations. Therationale behind LES is that by modeling less of turbulence (and resolving more), theerror introduced by turbulence modeling can be reduced. It is also believed to be easierto find a “universal” model for the small scales, since they tend to be more isotropic andless affected by the macroscopic features like boundary conditions, than the large eddies.Filtering is essentially a mathematical manipulation of the exact Navier-Stokes equationsto remove the eddies that are smaller than the size of the filter, which is usually taken asthe mesh size when spatial filtering is employed as in FLUENT. Like Reynolds-averaging,the filtering process creates additional unknown terms that must be modeled to achieveclosure. Statistics of the time-varying flow-fields such as time-averages and r.m.s. valuesof the solution variables, which are generally of most engineering interest, can be col-lected during the time-dependent simulation. LES for high Reynolds number industrialflows requires a significant amount of compute resources. This is mainly because of theneed to accurately resolve the energy-containing turbulent eddies in both space and timedomains, which becomes most acute in near-wall regions where the scales to be resolvedbecome increasingly smaller. Wall functions in combination with a coarse near wall meshcan be employed, often with some success, to reduce the cost of LES for wall-boundedflows. However, one needs to carefully consider the ramification of using wall functionsfor the flow in question. For the same reason (to accurately resolve the eddies), LES alsorequires highly accurate spatial and temporal discretizations.

12.2.2 Reynolds (Ensemble) Averaging

In Reynolds averaging, the solution variables in the instantaneous (exact) Navier-Stokesequations are decomposed into the mean (ensemble-averaged or time-averaged) and fluc-tuating components. For the velocity components:

ui = ui + u′i (12.2-1)

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′ (12.2-2)

where φ denotes a scalar such as pressure, energy, or species concentration.

Substituting expressions of this form for the flow variables into the instantaneous conti-nuity and momentum equations and taking a time (or ensemble) average (and dropping



the overbar on the mean velocity, u) yields the ensemble-averaged momentum equations.They can be written in Cartesian tensor form as:

∂ρ

∂t+

∂

∂xi(ρui) = 0 (12.2-3)

∂

∂t(ρui) +

∂

∂xj(ρuiuj) = − ∂p

∂xi+

∂

∂xj

[µ

(∂ui∂xj

+∂uj∂xi− 2

3δij∂ul∂xl

)]+

∂

∂xj(−ρu′iu′j)

(12.2-4)

Equations 12.2-3 and 12.2-4 are called Reynolds-averaged Navier-Stokes (RANS) equa-tions. They have the same general form as the instantaneous Navier-Stokes equations,with the velocities and other solution variables now representing ensemble-averaged (ortime-averaged) values. Additional terms now appear that represent the effects of tur-bulence. These Reynolds stresses, −ρu′iu′j, must be modeled in order to close Equa-tion 12.2-4.

For variable-density flows, Equations 12.2-3 and 12.2-4 can be interpreted as Favre-averaged Navier-Stokes equations [141], with the velocities representing mass-averagedvalues. As such, Equations 12.2-3 and 12.2-4 can be applied to density-varying flows.

12.2.3 Boussinesq Approach vs. Reynolds Stress Transport Models

The Reynolds-averaged approach to turbulence modeling requires that the Reynoldsstresses in Equation 12.2-4 be appropriately modeled. A common method employs theBoussinesq hypothesis [141] to relate the Reynolds stresses to the mean velocity gradients:

− ρu′iu′j = µt

(∂ui∂xj

+∂uj∂xi

)− 2

3

(ρk + µt

∂uk∂xk

)δij (12.2-5)

The Boussinesq hypothesis is used in the Spalart-Allmaras model, the k-ε models, andthe k-ω models. The advantage of this approach is the relatively low computationalcost associated with the computation of the turbulent viscosity, µt. In the case of theSpalart-Allmaras model, only one additional transport equation (representing turbulentviscosity) is solved. In the case of the k-ε and k-ω models, two additional transportequations (for the turbulence kinetic energy, k, and either the turbulence dissipationrate, ε, or the specific dissipation rate, ω) are solved, and µt is computed as a function ofk and ε. The disadvantage of the Boussinesq hypothesis as presented is that it assumesµt is an isotropic scalar quantity, which is not strictly true.


Modeling Turbulence

The alternative approach, embodied in the RSM, is to solve transport equations for eachof the terms in the Reynolds stress tensor. An additional scale-determining equation(normally for ε) is also required. This means that five additional transport equations arerequired in 2D flows and seven additional transport equations must be solved in 3D.

In many cases, models based on the Boussinesq hypothesis perform very well, and theadditional computational expense of the Reynolds stress model is not justified. However,the RSM is clearly superior for situations in which the anisotropy of turbulence has adominant effect on the mean flow. Such cases include highly swirling flows and stress-driven secondary flows.

12.2.4 Computational Effort: CPU Time and Solution Behavior

In terms of computation, the Spalart-Allmaras model is the least expensive turbulencemodel of the options provided in FLUENT, since only one turbulence transport equationis solved.

The standard k-ε model clearly requires more computational effort than the Spalart-Allmaras model since an additional transport equation is solved. The realizable k-εmodel requires only slightly more computational effort than the standard k-ε model.However, due to the extra terms and functions in the governing equations and a greaterdegree of non-linearity, computations with the RNG k-ε model tend to take 10–15% moreCPU time than with the standard k-ε model. Like the k-ε models, the k-ω models arealso two-equation models, and thus require about the same computational effort.

Compared with the k-ε and k-ω models, the RSM requires additional memory and CPUtime due to the increased number of the transport equations for Reynolds stresses. How-ever, efficient programming in FLUENT has reduced the CPU time per iteration signifi-cantly. On average, the RSM in FLUENT requires 50–60% more CPU time per iterationcompared to the k-ε and k-ω models. Furthermore, 15–20% more memory is needed.

Aside from the time per iteration, the choice of turbulence model can affect the ability ofFLUENT to obtain a converged solution. For example, the standard k-ε model is knownto be slightly over-diffusive in certain situations, while the RNG k-ε model is designedsuch that the turbulent viscosity is reduced in response to high rates of strain. Sincediffusion has a stabilizing effect on the numerics, the RNG model is more likely to besusceptible to instability in steady-state solutions. However, this should not necessarilybe seen as a disadvantage of the RNG model, since these characteristics make it moreresponsive to important physical instabilities such as time-dependent turbulent vortexshedding.

Similarly, the RSM may take more iterations to converge than the k-ε and k-ω modelsdue to the strong coupling between the Reynolds stresses and the mean flow.


12.3 Spalart-Allmaras Model Theory


12.3.1 Overview

The Spalart-Allmaras model is a relatively simple one-equation model that solves a mod-eled transport equation for the kinematic eddy (turbulent) viscosity. This embodies arelatively new class of one-equation models in which it is not necessary to calculate alength scale related to the local shear layer thickness. The Spalart-Allmaras model wasdesigned specifically for aerospace applications involving wall-bounded flows and has beenshown to give good results for boundary layers subjected to adverse pressure gradients.It is also gaining popularity for turbomachinery applications.

In its original form, the Spalart-Allmaras model is effectively a low-Reynolds-numbermodel, requiring the viscous-affected region of the boundary layer to be properly resolved.In FLUENT, however, the Spalart-Allmaras model has been implemented to use wallfunctions when the mesh resolution is not sufficiently fine. This might make it the bestchoice for relatively crude simulations on coarse meshes where accurate turbulent flowcomputations are not critical. Furthermore, the near-wall gradients of the transportedvariable in the model are much smaller than the gradients of the transported variablesin the k-ε or k-ω models. This might make the model less sensitive to numerical errorwhen non-layered meshes are used near walls. See Section 6.1.3: Numerical Diffusion forfurther discussion of numerical error.

On a cautionary note, however, the Spalart-Allmaras model is still relatively new, andno claim is made regarding its suitability to all types of complex engineering flows. Forinstance, it cannot be relied on to predict the decay of homogeneous, isotropic turbu-lence. Furthermore, one-equation models are often criticized for their inability to rapidlyaccommodate changes in length scale, such as might be necessary when the flow changesabruptly from a wall-bounded to a free shear flow.

In turbulence models that employ the Boussinesq approach, the central issue is how theeddy viscosity is computed. The model proposed by Spalart and Allmaras [348] solvesa transport equation for a quantity that is a modified form of the turbulent kinematicviscosity.

12.3.2 Transport Equation for the Spalart-Allmaras Model

The transported variable in the Spalart-Allmaras model, ν, is identical to the turbu-lent kinematic viscosity except in the near-wall (viscous-affected) region. The transportequation for ν is

∂

∂t(ρν)+

∂

∂xi(ρνui) = Gν+

1

σν

∂

∂xj

(µ+ ρν)

∂ν

∂xj

+ Cb2ρ

(∂ν

∂xj

)2−Yν+Sν (12.3-1)


Modeling Turbulence

where Gν is the production of turbulent viscosity and Yν is the destruction of turbulentviscosity that occurs in the near-wall region due to wall blocking and viscous damping.σν and Cb2 are constants and ν is the molecular kinematic viscosity. Sν is a user-definedsource term. Note that since the turbulence kinetic energy k is not calculated in theSpalart-Allmaras model, the last term in Equation 12.2-5 is ignored when estimating theReynolds stresses.

12.3.3 Modeling the Turbulent Viscosity

The turbulent viscosity, µt, is computed from

µt = ρνfv1 (12.3-2)

where the viscous damping function, fv1, is given by

fv1 =χ3

χ3 + C3v1

(12.3-3)

and

χ ≡ ν

ν(12.3-4)

12.3.4 Modeling the Turbulent Production

The production term, Gν , is modeled as

Gν = Cb1ρSν (12.3-5)

where

S ≡ S +ν

κ2d2fv2 (12.3-6)

and

fv2 = 1− χ

1 + χfv1

(12.3-7)



Cb1 and κ are constants, d is the distance from the wall, and S is a scalar measure of thedeformation tensor. By default in FLUENT, as in the original model proposed by Spalartand Allmaras, S is based on the magnitude of the vorticity:

S ≡√

2ΩijΩij (12.3-8)

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

Ωij =1

2

(∂ui∂xj− ∂uj∂xi

)(12.3-9)

The justification for the default expression for S is that, for the wall-bounded flows thatwere of most interest when the model was formulated, turbulence is found only wherevorticity is generated near walls. However, it has since been acknowledged that oneshould also take into account the effect of mean strain on the turbulence production, anda modification to the model has been proposed [72] and incorporated into FLUENT.

This modification combines measures of both rotation and strain tensors in the definitionof S:

S ≡ |Ωij|+ Cprod min (0, |Sij| − |Ωij|) (12.3-10)

where

Cprod = 2.0, |Ωij| ≡√

2ΩijΩij, |Sij| ≡√

2SijSij

with the mean strain rate, Sij, defined as

Sij =1

2

(∂uj∂xi

+∂ui∂xj

)(12.3-11)

Including both the rotation and strain tensors reduces the production of eddy viscosityand consequently reduces the eddy viscosity itself in regions where the measure of vortic-ity exceeds that of strain rate. One such example can be found in vortical flows, i.e., flownear the core of a vortex subjected to a pure rotation where turbulence is known to besuppressed. Including both the rotation and strain tensors more correctly accounts forthe effects of rotation on turbulence. The default option (including the rotation tensoronly) tends to overpredict the production of eddy viscosity and hence overpredicts theeddy viscosity itself in certain circumstances.

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


Modeling Turbulence

12.3.5 Modeling the Turbulent Destruction

The destruction term is modeled as

Yν = Cw1ρfw

(ν

d

)2

(12.3-12)

where

fw = g

[1 + C6

w3

g6 + C6w3

]1/6

(12.3-13)

g = r + Cw2

(r6 − r

)(12.3-14)

r ≡ ν

Sκ2d2(12.3-15)

Cw1, Cw2, and Cw3 are constants, and S is given by Equation 12.3-6. Note that themodification described above to include the effects of mean strain on S will also affectthe value of S used to compute r.

12.3.6 Model Constants

The model constants Cb1, Cb2, σν , Cv1, Cw1, Cw2, Cw3, and κ have the following defaultvalues [348]:

Cb1 = 0.1355, Cb2 = 0.622, σν =2

3, Cv1 = 7.1

Cw1 =Cb1κ2

+(1 + Cb2)

σν, Cw2 = 0.3, Cw3 = 2.0, κ = 0.4187

12.3.7 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 wall shear stress isobtained from the laminar stress-strain relationship:

u

uτ=ρuτy

µ(12.3-16)



If the mesh is too coarse to resolve the laminar sublayer, it is assumed that the centroidof the wall-adjacent cell falls within the logarithmic region of the boundary layer, andthe law-of-the-wall is employed:

u

uτ=

1

κlnE

(ρuτy

µ

)(12.3-17)

where u is the velocity parallel to the wall, uτ is the shear velocity, y is the distance fromthe wall, κ is the von Karman constant (0.4187), and E = 9.793.

12.3.8 Convective Heat and Mass Transfer Modeling

In FLUENT, turbulent heat transport is modeled using the concept of Reynolds’ analogyto turbulent momentum transfer. The “modeled” energy equation is thus given by thefollowing:

∂

∂t(ρE) +

∂

∂xi[ui(ρE + p)] =

∂

∂xj

[(k +

cpµtPrt

)∂T

∂xj+ ui(τij)eff

]+ Sh (12.3-18)

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

(τij)eff = µeff

(∂uj∂xi

+∂ui∂xj

)− 2

3µeff

∂uk∂xk

δij

The term involving (τij)eff represents the viscous heating, and is always computed in thedensity-based solvers. It is not computed by default in the pressure-based solver, but itcan be enabled in the Viscous Model panel. The default value of the turbulent Prandtlnumber is 0.85. You can change the value of Prt in the Viscous Model panel.

Turbulent mass transfer is treated similarly, with a default turbulent Schmidt number of0.7. This default value can be changed in the Viscous Model panel.

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


Modeling Turbulence

12.4 Standard, RNG, and Realizable k-ε Models Theory

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

• the method of calculating turbulent viscosity

• the turbulent Prandtl numbers governing the turbulent diffusion of k and ε

• the generation and destruction terms in the ε equation

The transport equations, methods of calculating turbulent viscosity, and model constantsare presented separately for each model. The features that are essentially common to allmodels follow, including turbulent production, generation due to buoyancy, accountingfor the effects of compressibility, and modeling heat and mass transfer.

12.4.1 Standard k-ε Model

Overview

The simplest “complete models” of turbulence are two-equation models in which the so-lution of two separate transport equations allows the turbulent velocity and length scalesto be independently determined. The standard k-ε model in FLUENT falls within thisclass of turbulence model and has become the workhorse of practical engineering flowcalculations in the time since it was proposed by Launder and Spalding [195]. Robust-ness, economy, and reasonable accuracy for a wide range of turbulent flows explain itspopularity in industrial flow and heat transfer simulations. It is a semi-empirical model,and the derivation of the model equations relies on phenomenological considerations andempiricism.

As the strengths and weaknesses of the standard k-ε model have become known, improve-ments have been made to the model to improve its performance. Two of these variantsare available in FLUENT: the RNG k-ε model [407] and the realizable k-ε model [329].

The standard k-ε model [195] is a semi-empirical model based on model transport equa-tions for the turbulence kinetic energy (k) and its dissipation rate (ε). The model trans-port equation for k is derived from the exact equation, while the model transport equationfor ε was obtained using physical reasoning and bears little resemblance to its mathe-matically exact counterpart.

In the derivation of the k-ε model, the assumption is that the flow is fully turbulent, andthe effects of molecular viscosity are negligible. The standard k-ε model is therefore validonly for fully turbulent flows.



Transport Equations for the Standard k-ε Model

The turbulence kinetic energy, k, and its rate of dissipation, ε, are obtained from thefollowing transport equations:

∂

∂t(ρk) +

∂

∂xi(ρkui) =

∂

∂xj

[(µ+

µtσk

)∂k

∂xj

]+Gk +Gb − ρε− YM + Sk (12.4-1)

and

∂

∂t(ρε) +

∂

∂xi(ρεui) =

∂

∂xj

[(µ+

µtσε

)∂ε

∂xj

]+C1ε

ε

k(Gk + C3εGb)−C2ερ

ε2

k+ Sε (12.4-2)

In these equations, Gk represents the generation of turbulence kinetic energy due to themean velocity gradients, calculated as described in Section 12.4.4: Modeling TurbulentProduction in the k-ε Models. Gb is the generation of turbulence kinetic energy dueto buoyancy, calculated as described in Section 12.4.5: Effects of Buoyancy on Turbu-lence in the k-ε Models. YM represents the contribution of the fluctuating dilatation incompressible turbulence to the overall dissipation rate, calculated as described in Sec-tion 12.4.6: Effects of Compressibility on Turbulence in the k-ε Models. C1ε, C2ε, and C3ε

are constants. σk and σε are the turbulent Prandtl numbers for k and ε, respectively. Skand Sε are user-defined source terms.

Modeling the Turbulent Viscosity

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

µt = ρCµk2

ε(12.4-3)

where Cµ is a constant.

Model Constants

The model constants C1ε, C2ε, Cµ, σk, and σε have the following default values [195]:

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 air and water for funda-mental turbulent shear flows including homogeneous shear flows and decaying isotropicgrid turbulence. They have been found to 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 ones most widelyaccepted, you can change them (if needed) in the Viscous Model panel.

12.4.2 RNG k-ε Model

Overview

The RNG k-ε model was derived using a rigorous statistical technique (called renormal-ization group theory). It is similar in form to the standard k-ε model, but includes thefollowing refinements:

• The RNG model has an additional term in its ε equation that significantly improvesthe accuracy for rapidly strained flows.

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

• The RNG theory provides an analytical formula for turbulent Prandtl 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 theoryprovides an analytically-derived differential formula for effective viscosity that ac-counts for low-Reynolds-number effects. Effective use of this feature does, however,depend on an appropriate treatment of the near-wall region.

These features make the RNG k-ε model more accurate and reliable for a wider class offlows than the standard k-ε model.

The RNG-based k-ε turbulence model is derived from the instantaneous Navier-Stokesequations, using a mathematical technique called “renormalization group” (RNG) meth-ods. The analytical derivation results in a model with constants different from those inthe standard k-ε model, and additional terms and functions in the transport equationsfor k and ε. A more comprehensive description of RNG theory and its application toturbulence can be found in [58].



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 (12.4-4)

and

∂

∂t(ρε)+

∂

∂xi(ρεui) =

∂

∂xj

(αεµeff

∂ε

∂xj

)+C1ε

ε

k(Gk + C3εGb)−C2ερ

ε2

k−Rε+Sε (12.4-5)

In these equations, Gk represents the generation of turbulence kinetic energy due to themean velocity gradients, calculated as described in Section 12.4.4: Modeling TurbulentProduction in the k-ε Models. Gb is the generation of turbulence kinetic energy dueto buoyancy, calculated as described in Section 12.4.5: Effects of Buoyancy on Turbu-lence in the k-ε Models. YM represents the contribution of the fluctuating dilatation incompressible turbulence to the overall dissipation rate, calculated as described in Sec-tion 12.4.6: Effects of Compressibility on Turbulence in the k-ε Models. The quantitiesαk and αε are the inverse effective Prandtl numbers for k and ε, respectively. Sk and Sεare user-defined source terms.

Modeling the Effective Viscosity

The scale elimination procedure in RNG theory results in a differential equation forturbulent viscosity:

d

(ρ2k√εµ

)= 1.72

ν√ν3 − 1 + Cν

dν (12.4-6)

where

ν = µeff/µ

Cν ≈ 100

Equation 12.4-6 is integrated to obtain an accurate description of how the effective tur-bulent transport varies with the effective Reynolds number (or eddy scale), allowing themodel to better handle low-Reynolds-number and near-wall flows.


Modeling Turbulence

In the high-Reynolds-number limit, Equation 12.4-6 gives

µt = ρCµk2

ε(12.4-7)

with Cµ = 0.0845, derived using RNG theory. It is interesting to note that this valueof Cµ is very close to the empirically-determined value of 0.09 used in the standard k-εmodel.

In FLUENT, by default, the effective viscosity is computed using the high-Reynolds-number form in Equation 12.4-7. However, there is an option available that allows youto use the differential relation given in Equation 12.4-6 when you need to include low-Reynolds-number effects.

RNG Swirl Modification

Turbulence, in general, is affected by rotation or swirl in the mean flow. The RNG modelin FLUENT provides an option to account for the effects of swirl or rotation by modifyingthe turbulent viscosity appropriately. The modification takes the following functionalform:

µt = µt0 f

(αs,Ω,

k

ε

)(12.4-8)

where µt0 is the value of turbulent viscosity calculated without the swirl modificationusing either Equation 12.4-6 or Equation 12.4-7. Ω is a characteristic swirl number eval-uated within FLUENT, and αs is a swirl constant that assumes different values dependingon whether the flow is swirl-dominated or only mildly swirling. This swirl modificationalways takes effect for axisymmetric, swirling flows and three-dimensional flows when theRNG model is selected. For mildly swirling flows (the default in FLUENT), αs is set to0.07. For strongly swirling 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 using the followingformula derived analytically by the RNG theory:

∣∣∣∣ α− 1.3929

α0 − 1.3929

∣∣∣∣0.6321 ∣∣∣∣ α + 2.3929

α0 + 2.3929

∣∣∣∣0.3679

=µmol

µeff

(12.4-9)

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 in the additionalterm in the ε equation given by

Rε =Cµρη

3(1− η/η0)

1 + βη3

ε2

k(12.4-10)

where η ≡ Sk/ε, η0 = 4.38, β = 0.012.

The effects of this term in the RNG ε equation can be seen more clearly by rearrangingEquation 12.4-5. Using Equation 12.4-10, the third and fourth terms on the right-handside of Equation 12.4-5 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∗2ερ

ε2

k(12.4-11)

where C∗2ε is given by

C∗2ε ≡ C2ε +Cµη

3(1− η/η0)

1 + βη3(12.4-12)

In regions where η < η0, the R term makes a positive contribution, and C∗2ε becomeslarger than C2ε. In the logarithmic layer, for instance, it can be shown that η ≈ 3.0,giving C∗2ε ≈ 2.0, which is close in magnitude to the value of C2ε in the standard k-εmodel (1.92). As a result, for weakly to moderately strained flows, the RNG model tendsto give results largely comparable to the standard k-ε model.

In regions of large strain rate (η > η0), however, the R term makes a negative contribu-tion, making the value of C∗2ε less than C2ε. In comparison with the standard k-ε model,the smaller destruction of ε augments ε, reducing k and, eventually, the effective viscosity.As a result, in rapidly strained flows, the RNG model yields a lower turbulent viscositythan the standard k-ε model.

Thus, the RNG model is more responsive to the effects of rapid strain and streamlinecurvature than the standard k-ε model, which explains the superior performance of theRNG model for certain classes of flows.

Model Constants

The model constants C1ε and C2ε in Equation 12.4-5 have values derived analytically bythe RNG theory. These values, used by default in FLUENT, are

C1ε = 1.42, C2ε = 1.68


Modeling Turbulence

12.4.3 Realizable k-ε Model

Overview

The realizable k-ε model [329] is a relatively recent development and differs from thestandard k-ε model in two important ways:

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

• A new transport equation for the dissipation rate, ε, has been derived from an exactequation for the transport of the mean-square vorticity fluctuation.

The term “realizable” means that the model satisfies certain mathematical constraintson the Reynolds stresses, consistent with the physics of turbulent flows. Neither thestandard k-ε model nor the RNG k-ε model is realizable.

An immediate benefit of the realizable k-ε model is that it more accurately predictsthe spreading rate of both planar and round jets. It is also likely to provide superiorperformance for flows involving rotation, boundary layers under strong adverse pressuregradients, separation, and recirculation.

To understand the mathematics behind the realizable k-ε model, consider combiningthe Boussinesq relationship (Equation 12.2-5) and the eddy viscosity definition (Equa-tion 12.4-3) to obtain the following expression for the normal Reynolds stress in anincompressible strained mean flow:

u2 =2

3k − 2 νt

∂U

∂x(12.4-13)

Using Equation 12.4-3 for νt ≡ µt/ρ, one obtains the result that the normal stress, u2,which by definition is a positive quantity, becomes negative, i.e., “non-realizable”, whenthe strain is large enough to satisfy

k

ε

∂U

∂x>

1

3Cµ≈ 3.7 (12.4-14)

Similarly, it can also be shown that the Schwarz inequality for shear stresses (uαuβ2 ≤

u2αu

2β; no summation over α and β) can be violated when the mean strain rate is large.

The most straightforward way to ensure the realizability (positivity of normal stressesand Schwarz inequality for shear stresses) is to make Cµ variable by sensitizing it tothe mean flow (mean deformation) and the turbulence (k, ε). The notion of variableCµ is suggested by many modelers including Reynolds [302], and is well substantiatedby experimental evidence. For example, Cµ is found to be around 0.09 in the inertialsublayer of equilibrium boundary layers, and 0.05 in a strong homogeneous shear flow.



Both the realizable and RNG k-ε models have shown substantial improvements over thestandard k-ε model where the flow features include strong streamline curvature, vortices,and rotation. Since the model is still relatively new, it is not clear in exactly whichinstances the realizable k-ε model consistently outperforms the RNG model. However,initial studies have shown that the realizable model provides the best performance of allthe k-ε model versions for several validations of separated flows and flows with complexsecondary flow features.

One of the weaknesses of the standard k-ε model or other traditional k-ε models lies withthe modeled equation for the dissipation rate (ε). The well-known round-jet anomaly(named based on the finding that the spreading rate in planar jets is predicted reasonablywell, but prediction 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. [329] was intended to address thesedeficiencies of traditional k-ε models by adopting the following:

• A new eddy-viscosity formula involving a variable Cµ originally proposed byReynolds [302].

• A new model equation for dissipation (ε) based on the dynamic equation of themean-square vorticity fluctuation.

One limitation of the realizable k-ε model is that it produces non-physical turbulentviscosities in situations when the computational domain contains both rotating and sta-tionary fluid zones (e.g., multiple reference frames, rotating sliding meshes). This is dueto the fact that the realizable k-ε model includes the effects of mean rotation in thedefinition of the turbulent viscosity (see Equations 12.4-17–12.4-19). This extra rotationeffect has been tested on single rotating reference frame systems and showed superior be-havior over the standard k-ε model. However, due to the nature of this modification, itsapplication to multiple reference frame systems should be taken with some caution. SeeSection 12.4.3: Modeling the Turbulent Viscosity for information about how to includeor exclude this term from the model.


Modeling Turbulence

Transport Equations for the Realizable k-ε Model

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

∂

∂t(ρk) +

∂

∂xj(ρkuj) =

∂

∂xj

[(µ+

µtσk

)∂k

∂xj

]+Gk +Gb − ρε− YM + Sk (12.4-15)

and

∂

∂t(ρε) +

∂

∂xj(ρεuj) =

∂

∂xj

[(µ+

µtσε

)∂ε

∂xj

]+ ρC1Sε− ρC2

ε2

k +√νε

+ C1εε

kC3εGb + Sε

(12.4-16)

where

C1 = max

[0.43,

η

η + 5

], η = S

k

ε, S =

√2SijSij

In these equations, Gk represents the generation of turbulence kinetic energy due to themean velocity gradients, calculated as described in Section 12.4.4: Modeling TurbulentProduction in the k-ε Models. Gb is the generation of turbulence kinetic energy dueto buoyancy, calculated as described in Section 12.4.5: Effects of Buoyancy on Turbu-lence in the k-ε Models. YM represents the contribution of the fluctuating dilatation incompressible turbulence to the overall dissipation rate, calculated as described in Sec-tion 12.4.6: Effects of Compressibility on Turbulence in the k-ε Models. C2 and C1ε areconstants. σk and σε are the turbulent Prandtl numbers for k and ε, respectively. Sk andSε are user-defined source terms.

Note that the k equation (Equation 12.4-15) is the same as that in the standard k-ε model (Equation 12.4-1) and the RNG k-ε model (Equation 12.4-4), except for themodel constants. However, the form of the ε equation is quite different from those inthe standard and RNG-based k-ε models (Equations 12.4-2 and 12.4-5). One of thenoteworthy features is that the production term in the ε equation (the second term onthe right-hand side of Equation 12.4-16) does not involve the production of k; i.e., it doesnot contain the same Gk term as the other k-ε models. It is believed that the presentform better represents the spectral energy transfer. Another desirable feature is thatthe destruction term (the next to last term on the right-hand side of Equation 12.4-16)does not have any singularity; i.e., its denominator never vanishes, even if k vanishes orbecomes smaller than zero. This feature is contrasted with traditional k-ε models, whichhave a singularity due to k in the denominator.



This model has been extensively validated for a wide range of flows [182, 329], includingrotating homogeneous shear flows, free flows including jets and mixing layers, channeland boundary layer flows, and separated flows. For all these cases, the performance ofthe model has been found to be substantially better than that of the standard k-ε model.Especially noteworthy is the fact that the realizable k-ε model resolves the round-jetanomaly; i.e., it predicts the spreading rate for axisymmetric jets as well as that forplanar jets.

Modeling the Turbulent Viscosity

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

µt = ρCµk2

ε(12.4-17)

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

Cµ =1

A0 + AskU∗

ε

(12.4-18)

where

U∗ ≡√SijSij + ΩijΩij (12.4-19)

and

Ωij = Ωij − 2εijkωk

Ωij = Ωij − εijkωk

where Ωij is the mean rate-of-rotation tensor viewed in a rotating reference frame withthe angular velocity ωk. The model constants A0 and As are given by

A0 = 4.04, As =√

6 cosφ

where

φ =1

3cos−1(

√6W ), W =

SijSjkSki

S3, S =

√SijSij, Sij =

1

2

(∂uj∂xi

+∂ui∂xj

)


Modeling Turbulence

It can be seen that Cµ is a function of the mean strain and rotation rates, the angular ve-locity of the system rotation, and the turbulence fields (k and ε). Cµ in Equation 12.4-17can be shown to recover the standard value of 0.09 for an inertial sublayer in an equilib-rium boundary layer.

i In FLUENT, the term −2εijkωk is, by default, not included in thecalculation of Ωij. This is an extra rotation term that is not com-patible with cases involving sliding meshes or multiple reference frames.If you want to include this term in the model, you can enable it by using thedefine/models/viscous/turbulence-expert/rke-cmu-rotation-term?

text command and entering yes at the prompt.

Model Constants

The model constants C2, σk, and σε have been established to ensure that the modelperforms well for certain canonical flows. The model constants are

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

12.4.4 Modeling Turbulent Production in the k-ε Models

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

Gk = −ρu′iu′j∂uj∂xi

(12.4-20)

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

Gk = µtS2 (12.4-21)

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

S ≡√

2SijSij (12.4-22)

i When using the high-Reynolds number k-ε versions, µeff is used in lieu ofµt in Equation 12.4-21.



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

When a non-zero gravity field and temperature gradient are present simultaneously, thek-ε models in FLUENT account for the generation of k due to buoyancy (Gb in Equa-tions 12.4-1, 12.4-4, and 12.4-15), and the corresponding contribution to the productionof ε in Equations 12.4-2, 12.4-5, and 12.4-16.

The generation of turbulence due to buoyancy is given by

Gb = βgiµtPrt

∂T

∂xi(12.4-23)

where Prt is the turbulent Prandtl number for energy and gi is the component of thegravitational vector in the ith direction. For the standard and realizable k-ε models, thedefault value of Prt is 0.85. In the case of the RNG k-ε model, Prt = 1/α, where αis given by Equation 12.4-9, but with α0 = 1/Pr = k/µcp. The coefficient of thermalexpansion, β, is defined as

β = −1

ρ

(∂ρ

∂T

)p

(12.4-24)

For ideal gases, Equation 12.4-23 reduces to

Gb = −giµtρPrt

∂ρ

∂xi(12.4-25)

It can be seen from the transport equations for k (Equations 12.4-1, 12.4-4, and 12.4-15)that turbulence kinetic energy tends to be augmented (Gb > 0) in unstable stratification.For stable stratification, buoyancy tends to suppress the turbulence (Gb < 0). In FLU-ENT, the effects of buoyancy on the generation of k are always included when you haveboth a non-zero gravity field and a non-zero temperature (or density) gradient.

While the buoyancy effects on the generation of k are relatively well understood, theeffect on ε is less clear. In FLUENT, by default, the buoyancy effects on ε are neglectedsimply by setting Gb to zero in the transport equation for ε (Equation 12.4-2, 12.4-5, or12.4-16).

However, you can include the buoyancy effects on ε in the Viscous Model panel. In thiscase, the value of Gb given by Equation 12.4-25 is used in the transport equation for ε(Equation 12.4-2, 12.4-5, or 12.4-16).


Modeling Turbulence

The degree to which ε is affected by the buoyancy is determined by the constant C3ε.In FLUENT, C3ε is not specified, but is instead calculated according to the followingrelation [139]:

C3ε = tanh∣∣∣∣vu∣∣∣∣ (12.4-26)

where v is the component of the flow velocity parallel to the gravitational vector andu is the component of the flow velocity perpendicular to the gravitational vector. Inthis way, C3ε will become 1 for buoyant shear layers for which the main flow direction isaligned with the direction of gravity. For buoyant shear layers that are perpendicular tothe gravitational vector, C3ε will become zero.

12.4.6 Effects of Compressibility on Turbulence in the k-ε Models

For high-Mach-number flows, compressibility affects turbulence through so-called “di-latation dissipation”, which is normally neglected in the modeling of incompressibleflows [402]. Neglecting the dilatation dissipation fails to predict the observed decreasein spreading rate with increasing Mach number for compressible mixing and other freeshear layers. To account for these effects in the k-ε models in FLUENT, the dilatationdissipation term, YM , is included in the k equation. This term is modeled according toa proposal by Sarkar [314]:

YM = 2ρεM2t (12.4-27)

where Mt is the turbulent Mach number, defined as

Mt =

√k

a2(12.4-28)

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

This compressibility modification always takes effect when the compressible form of theideal gas law is used.



12.4.7 Convective Heat and Mass Transfer Modeling in the k-ε Models

In FLUENT, turbulent heat transport is modeled using the concept of Reynolds’ analogyto turbulent momentum transfer. The “modeled” energy equation is thus given by thefollowing:

∂

∂t(ρE) +

∂


∂

∂xj

(keff

∂T

∂xj+ ui(τij)eff

)+ Sh (12.4-29)

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

∂uk∂xk

δij

The term involving (τij)eff represents the viscous heating, and is always computed in thedensity-based solvers. It is not computed by default in the pressure-based solver, but itcan be enabled in the Viscous Model panel.

Additional terms may appear in the energy equation, depending on the physical modelsyou are using. See Section 13.2.1: Heat Transfer Theory for more details.

For the standard and realizable k-ε models, the effective thermal conductivity is givenby

keff = k +cpµtPrt

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

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

keff = αcpµeff

where α is calculated from Equation 12.4-9, but with α0 = 1/Pr = k/µcp.

The fact that α varies with µmol/µeff , as in Equation 12.4-9, is an advantage of the RNG k-ε model. It is consistent with experimental evidence indicating that the turbulent Prandtlnumber varies with the molecular Prandtl number and turbulence [174]. Equation 12.4-9works well across a very broad range of molecular Prandtl numbers, from liquid metals(Pr ≈ 10−2) to paraffin oils (Pr ≈ 103), which allows heat transfer to be calculated in low-Reynolds-number regions. Equation 12.4-9 smoothly predicts the variation of effective


Modeling Turbulence

Prandtl number from the molecular value (α = 1/Pr) in the viscosity-dominated regionto the fully turbulent value (α = 1.393) in the fully turbulent regions of the flow.

Turbulent mass transfer is treated similarly. For the standard and realizable k-ε models,the default turbulent Schmidt number is 0.7. This default value can be changed in theViscous Model panel. For the RNG model, the effective turbulent diffusivity for masstransfer is calculated in a manner that is analogous to the method used for the heattransport. The value of α0 in Equation 12.4-9 is α0 = 1/Sc, where Sc is the molecularSchmidt number.

12.5 Standard and SST k-ω Models Theory

This section presents the standard [402] and shear-stress transport (SST) [236] k-ω mod-els. Both models have similar forms, with transport equations for k and ω. The majorways in which the SST model [237] differs from the standard model are as follows:

• gradual change from the standard k-ω model in the inner region of the boundarylayer to a high-Reynolds-number version of the k-ε model in the outer part of theboundary layer

• modified turbulent viscosity formulation to account for the transport effects of theprincipal turbulent shear stress

The transport equations, methods of calculating turbulent viscosity, and methods ofcalculating model constants and other terms are presented separately for each model.

12.5.1 Standard k-ω Model

Overview

The standard k-ω model in FLUENT is based on the Wilcox k-ω model [402], whichincorporates modifications for low-Reynolds-number effects, compressibility, and shearflow spreading. The Wilcox model predicts free shear flow spreading rates that are inclose agreement with measurements for far wakes, mixing layers, and plane, round, andradial jets, and is thus applicable to wall-bounded flows and free shear flows. A variationof the standard k-ω model called the SST k-ω model is also available in FLUENT, and isdescribed in Section 12.5.2: Shear-Stress Transport (SST) k-ω Model.

The standard k-ω model is an empirical model based on model transport equations forthe turbulence kinetic energy (k) and the specific dissipation rate (ω), which can also bethought of as the ratio of ε to k [402].

As the k-ω model has been modified over the years, production terms have been addedto both the k and ω equations, which have improved the accuracy of the model forpredicting free shear flows.



Transport Equations for the Standard k-ω Model

The turbulence kinetic energy, k, and the specific dissipation rate, ω, are obtained fromthe following transport equations:

∂

∂t(ρk) +

∂

∂xi(ρkui) =

∂

∂xj

(Γk

∂k

∂xj

)+Gk − Yk + Sk (12.5-1)

and

∂

∂t(ρω) +

∂

∂xi(ρωui) =

∂

∂xj

(Γω

∂ω

∂xj

)+Gω − Yω + Sω (12.5-2)

In these equations, Gk represents the generation of turbulence kinetic energy due to meanvelocity gradients. Gω represents the generation of ω. Γk and Γω represent the effectivediffusivity of k and ω, respectively. Yk and Yω represent the dissipation of k and ω dueto turbulence. All of 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+µtσk

(12.5-3)

Γω = µ+µtσω

(12.5-4)

where σk and σω are the turbulent Prandtl numbers for k and ω, respectively. Theturbulent viscosity, µt, is computed by combining k and ω as follows:

µt = α∗ρk

ω(12.5-5)


Modeling Turbulence

Low-Reynolds-Number Correction

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

α∗ = α∗∞

(α∗0 + Ret/Rk

1 + Ret/Rk

)(12.5-6)

where

Ret =ρk

µω(12.5-7)

Rk = 6 (12.5-8)

α∗0 =βi3

(12.5-9)

βi = 0.072 (12.5-10)

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 exactequation for the transport of k, this term may be defined as

Gk = −ρu′iu′j∂uj∂xi

(12.5-11)

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

Gk = µt S2 (12.5-12)

where S is the modulus of the mean rate-of-strain tensor, defined in the same way as forthe k-ε model (see Equation 12.4-22).

Production of ω

The production of ω is given by

Gω = αω

kGk (12.5-13)



where Gk is given by Equation 12.5-11.

The coefficient α is given by

α =α∞α∗

(α0 + Ret/Rω

1 + Ret/Rω

)(12.5-14)

where Rω = 2.95. α∗ and Ret are given by Equations 12.5-6 and 12.5-7, 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 (12.5-15)

where

fβ∗ =

1 χk ≤ 01+680χ2

k

1+400χ2k

χk > 0(12.5-16)

where

χk ≡1

ω3

∂k

∂xj

∂ω

∂xj(12.5-17)

and

β∗ = β∗i [1 + ζ∗F (Mt)] (12.5-18)

β∗i = β∗∞

(4/15 + (Ret/Rβ)4

1 + (Ret/Rβ)4

)(12.5-19)

ζ∗ = 1.5 (12.5-20)

Rβ = 8 (12.5-21)

β∗∞ = 0.09 (12.5-22)

where Ret is given by Equation 12.5-7.


Modeling Turbulence

Dissipation of ω

The dissipation of ω is given by

Yω = ρ β fβ ω2 (12.5-23)

where

fβ =1 + 70χω1 + 80χω

(12.5-24)

χω =

∣∣∣∣∣ΩijΩjkSki(β∗∞ω)3

∣∣∣∣∣ (12.5-25)

Ωij =1

2


)(12.5-26)

The strain rate tensor, Sij is defined in Equation 12.3-11. Also,

β = βi

[1− β∗i

βiζ∗F (Mt)

](12.5-27)

β∗i and F (Mt) are defined by Equations 12.5-19 and 12.5-28, respectively.

Compressibility Correction

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

F (Mt) =

0 Mt ≤ Mt0

M2t −M2

t0 Mt > Mt0(12.5-28)

where

M2t ≡

2k

a2(12.5-29)

Mt0 = 0.25 (12.5-30)

a =√γRT (12.5-31)

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



Model Constants

α∗∞ = 1, α∞ = 0.52, α0 =1

9, 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

12.5.2 Shear-Stress Transport (SST) k-ω Model

Overview

The shear-stress transport (SST) k-ω model was developed by Menter [236] to effectivelyblend the robust and accurate formulation of the k-ω model in the near-wall region withthe free-stream independence of the k-ε model in the far field. To achieve this, the k-εmodel is converted into a k-ω formulation. The SST k-ω model is similar to the standardk-ω model, but includes the following refinements:

• The standard k-ω model and the transformed k-ε model are both multiplied by ablending function and both models are added together. The blending function isdesigned to be one in the near-wall region, which activates the standard k-ω model,and zero away from the surface, which activates the transformed k-ε model.

• The SST model incorporates a damped cross-diffusion derivative term in the ωequation.

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

• The modeling constants are different.

These features make the SST k-ω model more accurate and reliable for a wider classof flows (e.g., adverse pressure gradient flows, airfoils, transonic shock waves) than thestandard k-ω model. Other modifications include the addition of a cross-diffusion termin the ω equation and a blending function to ensure that the model equations behaveappropriately in both the near-wall and far-field zones.


Modeling Turbulence

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 (12.5-32)

and

∂

∂t(ρω) +

∂

∂xi(ρωui) =

∂

∂xj

(Γω

∂ω

∂xj

)+Gω − Yω +Dω + Sω (12.5-33)

In these equations, Gk represents the generation of turbulence kinetic energy due tomean velocity gradients, calculated as described in Section 12.5.1: Modeling the Tur-bulence Production. Gω represents the generation of ω, calculated as described in Sec-tion 12.5.1: Modeling the Turbulence Production. Γk and Γω represent the effectivediffusivity of k and ω, respectively, which are calculated as described below. Yk andYω represent the dissipation of k and ω due to turbulence, calculated as described inSection 12.5.1: Modeling the Turbulence Dissipation. Dω represents the cross-diffusionterm, calculated as described 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+µtσk

(12.5-34)

Γω = µ+µtσω

(12.5-35)

where σk and σω are the turbulent Prandtl numbers for k and ω, respectively. Theturbulent viscosity, µt, is computed as follows:

µt =ρk

ω

1

max[

1α∗, SF2

a1ω

] (12.5-36)

where S is the strain rate magnitude and

σk =1

F1/σk,1 + (1− F1)/σk,2(12.5-37)

σω =1

F1/σω,1 + (1− F1)/σω,2(12.5-38)



α∗ is defined in Equation 12.5-6. The blending functions, F1 and F2, are given by

F1 = tanh(Φ4

1

)(12.5-39)

Φ1 = min

[max

( √k

0.09ωy,500µ

ρy2ω

),

4ρk

σω,2D+ω y

2

](12.5-40)

D+ω = max

[2ρ

1

σω,2

1

ω

∂k

∂xj

∂ω

∂xj, 10−10

](12.5-41)

F2 = tanh(Φ2

2

)(12.5-42)

Φ2 = max

[2

√k

0.09ωy,500µ

ρy2ω

](12.5-43)

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

diffusion term (see Equation 12.5-52).

Modeling the Turbulence Production

Production of k

The term Gk represents the production of turbulence kinetic energy, and is defined as:

Gk = min(Gk, 10ρβ∗kω) (12.5-44)

where Gk is defined in the same manner as in the standard k-ω model. See Sec-tion 12.5.1: Modeling the Turbulence Production for details.

Production of ω

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

Gω =α

νtGk (12.5-45)

Note that this formulation differs from the standard k-ω model. The difference betweenthe two models also exists in the way the term α∞ is evaluated. In the standard k-ωmodel, 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 (12.5-46)


Modeling Turbulence

where

α∞,1 =βi,1β∗∞− κ2

σw,1√β∗∞

(12.5-47)

α∞,2 =βi,2β∗∞− κ2

σw,2√β∗∞

(12.5-48)

where κ is 0.41.

Modeling the Turbulence Dissipation

Dissipation of k

The term Yk represents the dissipation of turbulence kinetic energy, and is defined in asimilar manner as in the standard k-ω model (see Section 12.5.1: Modeling the TurbulenceDissipation). The difference is in the way the term fβ∗ is evaluated. In the standard k-ωmodel, fβ∗ is defined as a piecewise function. For the SST k-ω model, fβ∗ is a constantequal to 1. Thus,

Yk = ρβ∗kω (12.5-49)

Dissipation of ω

The term Yω represents the dissipation of ω, and is defined in a similar manner as inthe standard k-ω model (see Section 12.5.1: Modeling the Turbulence Dissipation). Thedifference is in the way the terms βi and fβ are evaluated. In the standard k-ω model, βiis defined as a constant (0.072) and fβ is defined in Equation 12.5-24. For the SST k-ωmodel, fβ is a constant equal to 1. Thus,

Yk = ρβω2 (12.5-50)

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

βi = F1βi,1 + (1− F1)βi,2 (12.5-51)

and F1 is obtained from Equation 12.5-39.



Cross-Diffusion Modification

The SST k-ω model is based on both the standard k-ω model and the standard k-ε model.To blend these two models together, the standard k-ε model has been transformed intoequations based on k and ω, which leads to the introduction of a cross-diffusion term(Dω in Equation 12.5-33). Dω is defined as

Dω = 2 (1− F1) ρσω,21

ω

∂k

∂xj

∂ω

∂xj(12.5-52)

For details about the various k-ε models, see Section 12.4: Standard, RNG, and Realizablek-ε Models Theory.

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∗, and Mt0) have the samevalues as for the standard k-ω model (see Section 12.5.1: Model Constants).


The wall boundary conditions for the k equation in the k-ω models are treated in thesame way as the k equation is treated when enhanced wall treatments are used withthe k-ε models. This means that all boundary conditions for wall-function meshes willcorrespond to the wall function approach, while for the fine meshes, the appropriatelow-Reynolds-number boundary conditions will be applied.

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

ωw =ρ (u∗)2

µω+ (12.5-53)

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

ω+ = min

(ω+w ,

6

βi(y+)2

)(12.5-54)


Modeling Turbulence

where

ω+w =

(

50k+s

)2k+s < 25

100k+s

k+s ≥ 25

(12.5-55)

where

k+s = max

(1.0,

ρksu∗

µ

)(12.5-56)

and ks is the roughness height.

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

ω+ =1√β∗∞

du+turb

dy+(12.5-57)

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

ω =u∗√β∗∞κy

(12.5-58)

Note that in the case of a wall cell being placed in the buffer region, FLUENT will blendω+ between the logarithmic and laminar sublayer values.

12.6 The v2-f Model Theory

The v2-f model is similar to the standard k-ε model, but incorporates near-wall turbu-lence anisotropy and non-local pressure-strain effects. A limitation of the v2-f model isthat it cannot be used to solve Eulerian multiphase problems, whereas the k-ε model istypically used in such applications. The v2-f model is a general low-Reynolds-numberturbulence model that is valid all the way up to solid walls, and therefore does not needto make use of wall functions. Although the model was originally developed for attachedor mildly separated boundary layers [90], it also accurately simulates flows dominated byseparation [30].

The distinguishing feature of the v2-f model is its use of the velocity scale, v2, insteadof the turbulent kinetic energy, k, for evaluating the eddy viscosity. v2, which can bethought of as the velocity fluctuation normal to the streamlines, has shown to providethe right scaling in representing the damping of turbulent transport close to the wall, afeature that k does not provide.


12.7 Reynolds Stress Model (RSM) Theory

For more information about the theoretical background and usage of the v2-f model,please visit the Fluent User Services Center (www.fluentusers.com).


12.7.1 Overview

The Reynolds stress model (RSM) [117, 192, 193] is the most elaborate turbulence modelthat FLUENT provides. Abandoning the isotropic eddy-viscosity hypothesis, the RSMcloses the Reynolds-averaged Navier-Stokes equations by solving transport equations forthe Reynolds stresses, together with an equation for the dissipation rate. This meansthat five additional 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 rapidchanges in strain rate in a more rigorous manner than one-equation and two-equationmodels, it has greater potential to give accurate predictions for complex flows. However,the fidelity of RSM predictions is still limited by the closure assumptions employed tomodel various terms in the exact transport equations for the Reynolds stresses. Themodeling of the pressure-strain and dissipation-rate terms is particularly challenging, andoften considered to be responsible for compromising the accuracy of RSM predictions.

The RSM might not always yield results that are clearly superior to the simpler modelsin all classes of flows to warrant the additional computational expense. However, useof the RSM is a must when the flow features of interest are the result of anisotropy inthe Reynolds stresses. Among the examples are cyclone flows, highly swirling flows incombustors, rotating flow passages, and the stress-induced secondary flows in ducts.

The exact form of the Reynolds stress transport equations may be derived by taking mo-ments of the exact momentum equation. This is a process wherein the exact momentumequations are multiplied by a fluctuating property, the product then being Reynolds-averaged. Unfortunately, several of the terms in the exact equation are unknown andmodeling assumptions are required in order to close the equations.


Modeling Turbulence

12.7.2 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

(12.7-1)

Of the various terms in these exact equations, Cij, DL,ij, Pij, and Fij do not require anymodeling. However, DT,ij, Gij, φij, and εij need to be modeled to close the equations.The following sections describe the modeling assumptions required to close the equationset.

12.7.3 Modeling Turbulent Diffusive Transport

DT,ij can be modeled by the generalized gradient-diffusion model of Daly and Harlow [74]:

DT,ij = Cs∂

∂xk

(ρku′ku

′`

ε

∂u′iu′j

∂x`

)(12.7-2)

However, this equation can result in numerical instabilities, so it has been simplified inFLUENT to use a scalar turbulent diffusivity as follows [207]:

DT,ij =∂

∂xk

(µtσk

∂u′iu′j

∂xk

)(12.7-3)

The turbulent viscosity, µt, is computed using Equation 12.7-33.



Lien and Leschziner [207] derived a value of σk = 0.82 by applying the generalizedgradient-diffusion model, Equation 12.7-2, to the case of a planar homogeneous shearflow. Note that this value of σk is different from that in the standard and realizable k-εmodels, in which σk = 1.0.

12.7.4 Modeling the Pressure-Strain Term

Linear Pressure-Strain Model

By default in FLUENT, the pressure-strain term, φij, in Equation 12.7-1 is modeledaccording to the proposals by Gibson and Launder [117], Fu et al. [112], and Launder [191,192].

The classical approach to modeling φij uses the following decomposition:

φij = φij,1 + φij,2 + φij,w (12.7-4)

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 is the wall-reflection term.

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

φij,1 ≡ −C1ρε

k

[u′iu′j −

2

3δijk

](12.7-5)

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)

](12.7-6)

where C2 = 0.60, Pij, Fij, Gij, and Cij are defined as in Equation 12.7-1, P = 12Pkk,

G = 12Gkk, and C = 1

2Ckk.

The wall-reflection term, φij,w, is responsible for the redistribution of normal stresses nearthe wall. It tends to damp the normal stress perpendicular to the wall, while enhancingthe stresses parallel to the wall. This term is modeled as

φij,w ≡ C ′1ε

k

(u′ku

′mnknmδij −

3

2u′iu′knjnk −

3

2u′ju

′knink

)C`k

3/2

εd

+ C ′2

(φkm,2nknmδij −

3

2φik,2njnk −

3

2φjk,2nink

)C`k

3/2

εd(12.7-7)


Modeling Turbulence

where C ′1 = 0.5, C ′2 = 0.3, nk is the xk component of the unit normal to the wall, d isthe normal distance to the wall, and C` = C3/4

µ /κ, where Cµ = 0.09 and κ is the vonKarman 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 wall treatment describedin Section 12.10.4: Two-Layer Model for Enhanced Wall Treatment, the pressure-strainmodel needs to be modified. The modification used in FLUENT specifies the values of C1,C2, C ′1, and C ′2 as functions of the Reynolds stress invariants and the turbulent Reynoldsnumber, according to the suggestion of Launder and Shima [194]:

C1 = 1 + 2.58A√A2

1− exp

[−(0.0067Ret)

2]

(12.7-8)

C2 = 0.75√A (12.7-9)

C ′1 = −2

3C1 + 1.67 (12.7-10)

C ′2 = max

[23C2 − 1

6

C2

, 0

](12.7-11)

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

A ≡[1− 9

8(A2 − A3)

](12.7-12)

A2 ≡ aikaki (12.7-13)

A3 ≡ aikakjaji (12.7-14)

aij is the Reynolds-stress anisotropy tensor, defined as

aij = −(−ρu′iu′j + 2

3ρkδij

ρk

)(12.7-15)

The modifications detailed above are employed only when the enhanced wall treatmentis selected in the Viscous Model panel.



Quadratic Pressure-Strain Model

An optional pressure-strain model proposed by Speziale, Sarkar, and Gatski [351] isprovided in FLUENT. This model has been demonstrated to give superior performance in arange of basic shear flows, including plane strain, rotating plane shear, and axisymmetricexpansion/contraction. This improved accuracy should be beneficial for a wider class ofcomplex engineering flows, particularly those with streamline curvature. The quadraticpressure-strain model can be selected as an option in the Viscous 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) (12.7-16)

where bij is the Reynolds-stress anisotropy tensor defined as

bij = −(−ρu′iu′j + 2

3ρkδij

2ρk

)(12.7-17)

The mean strain rate, Sij, is defined as

Sij =1

2

(∂uj∂xi

+∂ui∂xj

)(12.7-18)

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

Ωij =1

2


)(12.7-19)

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 to account for thewall-reflection effect in order to obtain a satisfactory solution in the logarithmic regionof a turbulent boundary layer. It should be noted, however, that the quadratic pressure-strain model is not available when the enhanced wall treatment is selected in the ViscousModel panel.


Modeling Turbulence

Low-Re Stress-Omega Model

The low-Re stress-omega model is a stress-transport model that is based on the omegaequations and LRR model [402]. This model is ideal for modeling flows over curvedsurfaces and swirling flows. The low-Re stress-omega model can be selected in the Vis-cous Model panel and requires no treatments of wall reflections. The closure coefficientsare identical to the k-ω model (Section 12.5.1: Model Constants), however, there areadditional closure coefficients, C1 and C2, defined in Section 12.7.4: Model Constants.

The low-Re stress-omega model resembles the k-ω model due to its excellent predictionsfor a wide range of turbulent flows. Furthermore, low Reynolds number modificationsand surface boundary conditions for rough surfaces are similar to the k-ω model.

Equation 12.7-4 can be re-written for the low-Re stress-omega model such that wallreflections are excluded:

φij = φij,1 + φij,2 (12.7-20)

Hence,

φij = − (C1ρε+ C∗1P ) bij + C2ρε(bikbkj −

1

3bmnbmnδij

)+(C3 − C∗3

√bijbij

)ρkSij

+ C4ρk(bikSjk + bjkSik −

2

3bmnSmnδij

)+ C5ρk (bikΩjk + bjkΩik) (12.7-21)

where bij is the Reynolds-stress anisotropy tensor defined as

bij = −(−ρu′iu′j + 2

3ρkδij

2ρk

)(12.7-22)

The mean strain rate, Sij, is defined in Equation 12.7-18 and the mean rate-of-rotationtensor, Ωij, is defined by Equation 12.7-19.

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

Near-wall treatment options in the Viscous Model panel are not available with the low-Restress-omega model.



Model Constants

C1 = 1.8, C2 = 0.52

α∗∞ = 1, α∞ = 0.52, α0 =1

9, 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 low-Re stress-omega equation in the RSM modelsare treated in the same way as the k equation in the k-ω models.

FLUENT defines the value of ω at the wall as

ωw =ρ (u∗)2

µω+ (12.7-23)

where ω+ is dimensionless and is defined as

ω+w =

(

50k+s

)2k+s < 25

500k+s

k+s ≥ 25

(12.7-24)

where

k+s =

ρksu∗

µ(12.7-25)

and ks is the roughness height.


Modeling Turbulence

12.7.5 Effects of Buoyancy on Turbulence

The production terms due to buoyancy are modeled as

Gij = βµtPrt

(gi∂T

∂xj+ gj

∂T

∂xi

)(12.7-26)

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

Using the definition of the coefficient of thermal expansion, β, given by Equation 12.4-24,the following expression is obtained for Gij for ideal gases:

Gij = − µtρPrt

(gi∂ρ

∂xj+ gj

∂ρ

∂xi

)(12.7-27)

12.7.6 Modeling the Turbulence Kinetic Energy

In general, when the turbulence kinetic energy is needed for modeling a specific term, itis obtained by taking the trace of the Reynolds stress tensor:

k =1

2u′iu′i (12.7-28)

As described in Section 12.7.9: Wall Boundary Conditions, an option is available inFLUENT to solve a transport equation for the turbulence kinetic energy in order toobtain boundary conditions for the Reynolds stresses. In this case, the following modelequation is used:

∂

∂t(ρk)+

∂

∂xi(ρkui) =

∂

∂xj

[(µ+

µtσk

)∂k

∂xj

]+

1

2(Pii +Gii)−ρε(1+2M2

t )+Sk (12.7-29)

where σk = 0.82 and Sk is a user-defined source term. Equation 12.7-29 is obtainableby contracting the modeled equation for the Reynolds stresses (Equation 12.7-1). Asone might expect, it is essentially identical to Equation 12.4-1 used in the standard k-εmodel.

Although Equation 12.7-29 is solved globally throughout the flow domain, the values ofk obtained are used only for boundary conditions. In every other case, k is obtained fromEquation 12.7-28. This is a minor point, however, since the values of k obtained witheither method should be very similar.



12.7.7 Modeling the Dissipation Rate

The dissipation tensor, εij, is modeled as

εij =2

3δij(ρε+ YM) (12.7-30)

where YM = 2ρεM2t is an additional “dilatation dissipation” term according to the model

by Sarkar [314]. The turbulent Mach number in this term is defined as

Mt =

√k

a2(12.7-31)

where a (≡√γRT ) is the speed of sound. This compressibility modification always takes

effect when the compressible form of the ideal gas law is used.

The scalar dissipation rate, ε, is computed with a model transport equation similar tothat used in the standard k-ε model:

∂

∂t(ρε)+

∂

∂xi(ρεui) =

∂

∂xj

[(µ+

µtσε

)∂ε

∂xj

]Cε1

1

2[Pii + Cε3Gii]

ε

k−Cε2ρ

ε2

k+Sε (12.7-32)

where σε = 1.0, Cε1 = 1.44, Cε2 = 1.92, Cε3 is evaluated as a function of the local flowdirection relative to the gravitational vector, as described in Section 12.4.5: Effects ofBuoyancy on Turbulence in the k-ε Models, and Sε is a user-defined source term.

12.7.8 Modeling the Turbulent Viscosity

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

µt = ρCµk2

ε(12.7-33)

where Cµ = 0.09.


Modeling Turbulence


The RSM model in FLUENT requires boundary conditions for individual Reynolds stresses,u′iu′j, and for the turbulence dissipation rate, ε (or ω if the low-Re stress-omega model

is used). These quantities can be input directly or derived from the turbulence intensityand characteristic length, as described in Section 12.20.3: Reynolds Stress Model.

At walls, FLUENT computes the near-wall values of the Reynolds stresses and ε fromwall functions (see Section 12.10.2: Standard Wall Functions, Section 12.10.3: Non-Equilibrium Wall Functions, and Section 12.10.4: Enhanced Wall Functions). FLUENTapplies explicit wall boundary conditions for the Reynolds stresses by using the log-lawand the assumption of equilibrium, disregarding convection and diffusion in the transportequations for the stresses (Equation 12.7-1). Using a local coordinate system, where τis the tangential coordinate, η is the normal coordinate, and λ is the binormal coordi-nate, the Reynolds stresses at the wall-adjacent cells (assuming standard wall functionsor non-equilibrium wall functions) are computed from

u′2τk

= 1.098,u′2ηk

= 0.247,u′2λ

k= 0.655, −

u′τu′η

k= 0.255 (12.7-34)

To obtain k, FLUENT solves the transport equation of Equation 12.7-29. For reasons ofcomputational convenience, the equation is solved globally, even though the values of kthus computed are needed only near the wall; in the far field k is obtained directly from thenormal Reynolds stresses using Equation 12.7-28. By default, the values of the Reynoldsstresses near the wall are fixed using the values computed from Equation 12.7-34, andthe transport equations in Equation 12.7-1 are solved only in the bulk flow region.

Alternatively, the Reynolds stresses can be explicitly specified in terms of wall-shearstress, instead of k:

u′2τu2τ

= 5.1,u′2ηu2τ

= 1.0,u′2λ

u2τ

= 2.3, −u′τu

′η

u2τ

= 1.0 (12.7-35)

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 equation is not solved.

When using enhanced wall treatments as the near-wall treatment, FLUENT applies zeroflux wall boundary conditions to the Reynolds stress equations.


12.8 Detached Eddy Simulation (DES) Model Theory

12.7.10 Convective Heat and Mass Transfer Modeling

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

∂

∂t(ρE) +

∂


∂

∂xj

[(k +

cpµtPrt

)∂T

∂xj+ ui(τij)eff

]+ Sh (12.7-36)

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

∂uk∂xk

δij

The term involving (τij)eff represents the viscous heating, and is always computed in thedensity-based solvers. It is not computed by default in the pressure-based solver, but itcan be enabled in the Viscous Model panel. The default value of the turbulent Prandtlnumber is 0.85. You can change the value of Prt in the Viscous Model panel.

Turbulent mass transfer is treated similarly, with a default turbulent Schmidt number of0.7. This default value can be changed in the Viscous Model panel.

12.8 Detached Eddy Simulation (DES) Model Theory

Overview

FLUENT offers three different models for the detached eddy simulation: the Spalart-Allmaras model, the realizable k-ε model, and the SST k-ω model.

In the DES approach, the unsteady RANS models are employed in the near-wall regions,while the filtered versions of the same models are used in the regions away from thenear-wall. The LES region is normally associated with the core turbulent region wherelarge turbulence scales play a dominant role. In this region, the DES models recoverthe respective subgrid models. In the near-wall region, the respective RANS models arerecovered.

The application of DES, however, may still require significant CPU resources and there-fore, as a general guideline, it is recommended that the conventional turbulence modelsemploying the Reynolds-averaged approach be used for practical calculations.

The DES models, often referred to as the hybrid LES/RANS models combine RANSmodeling with LES for applications such as high-Re external aerodynamics simulations.In FLUENT, the DES model is based on the one-equation Spalart-Allmaras model, therealizable k-ε model, and the SST k-ω model. The computational costs, when using theDES models, is less than LES computational costs, but greater than RANS.


Modeling Turbulence

12.8.1 Spalart-Allmaras RANS Model

The standard Spalart-Allmaras model uses the distance to the closest wall as the defini-tion for the length scale d, which plays a major role in determining the level of productionand destruction of turbulent viscosity (Equations 12.3-6, 12.3-12, and 12.3-15). The DESmodel, as proposed by Shur et al. [330] replaces d everywhere with a new length scale d,defined as

d = min(d, Cdes∆) (12.8-1)

where the grid spacing, ∆, is based on the largest grid space in the x, y, or z directionsforming the computational cell. The empirical constant Cdes has a value of 0.65.

12.8.2 Realizable k-ε RANS Model

This RANS model is similar to the Realizable k-ε model discussed in Section 12.4.3: Re-alizable k-ε Model, with the exception of the dissipation term in the k equation. In theDES model, the Realizable k-ε RANS dissipation term is modified such that:

Yk =ρk

32

ldes(12.8-2)

where

ldes = min(lrke, lles) (12.8-3)

lrke =k

32

ε(12.8-4)

lles = Cdes∆ (12.8-5)

where Cdes is a calibration constant used in the DES model and has a value of 0.61 and∆ is the maximum local grid spacing (∆x,∆y,∆z).

For the case where ldes = lrke, you will obtain an expression for the dissipation of the kformulation for the Realizable k-ε model (Section 12.4.3: Realizable k-ε Model):Yk = ρε


12.9 Large Eddy Simulation (LES) Model Theory

12.8.3 SST k-ω RANS Model

The dissipation term of the turbulent kinetic energy (see Section 12.5.1: Modeling the Tur-bulence Dissipation) is modified for the DES turbulence model as described in Menter’swork [237] such that

Yk = ρβ∗kωfβ∗ (12.8-6)

where fβ∗ is no longer a constant equal to 1 as in the SST k-ω model (see Section 12.5.1: Mod-eling the Turbulence Dissipation), but is now expressed as

fβ∗ = max(

LtCdes∆

, 1)

(12.8-7)

where Cdes is a calibration constant used in the DES model and has a value of 0.61, ∆ isthe maximum local grid spacing (∆x,∆y,∆z) and fβ∗ is defined in Equation 12.5-16.

The turbulent length scale is the parameter that defines this RANS model:

Lt =

√k

β∗ω(12.8-8)


12.9.1 Overview

Turbulent flows are characterized by eddies with a wide range of length and time scales.The largest eddies are typically comparable in size to the characteristic length of themean flow. The smallest scales are responsible for the dissipation of turbulence kineticenergy.

It is possible, in theory, to directly resolve the whole spectrum of turbulent scales usingan approach known as direct numerical simulation (DNS). No modeling is required inDNS. However, DNS is not feasible for practical engineering problems involving highReynolds number flows. The cost required for DNS to resolve the entire range of scalesis proportional to Re3

t , where Ret is the turbulent Reynolds number. Clearly, for highReynolds numbers, the cost becomes prohibitive.


Modeling Turbulence

In LES, large eddies are resolved directly, while small eddies are modeled. Large eddysimulation (LES) thus falls between DNS and RANS in terms of the fraction of theresolved scales. The rationale behind LES can be summarized as follows:

• Momentum, mass, energy, and other passive scalars are transported mostly by largeeddies.

• Large eddies are more problem-dependent. They are dictated by the geometriesand boundary conditions of the flow involved.

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

• The chance of finding a universal turbulence model is much higher for small eddies.

Resolving only the large eddies allows one to use much coarser mesh and larger times-step sizes in LES than in DNS. However, LES still requires substantially finer meshesthan those typically used for RANS calculations. In addition, LES has to be run fora sufficiently long flow-time to obtain stable statistics of the flow being modeled. Asa result, the computational cost involved with LES is normally orders of magnitudeshigher than that for steady RANS calculations in terms of memory (RAM) and CPUtime. Therefore, high-performance computing (e.g., parallel computing) is a necessity forLES, especially for industrial applications.

The following sections give details of the governing equations for LES, the subgrid-scaleturbulence models, and the boundary conditions.

12.9.2 Filtered Navier-Stokes Equations

The governing equations employed for LES are obtained by filtering the time-dependentNavier-Stokes equations in either Fourier (wave-number) space or configuration (physical)space. The filtering process effectively filters out the eddies whose scales are smaller thanthe filter width or grid spacing used in the computations. The resulting equations thusgovern the dynamics of large eddies.

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

φ(x) =∫Dφ(x′)G(x,x′)dx′ (12.9-1)

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



In FLUENT, the finite-volume discretization itself implicitly provides the filtering opera-tion:

φ(x) =1

V

∫Vφ(x′) dx′, x′ ∈ V (12.9-2)

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

G(x,x′)

1/V, x′ ∈ V0, x′ otherwise

(12.9-3)

The LES capability in FLUENT is applicable to compressible flows. For the sake of concisenotation, however, the theory is presented here for incompressible flows.

Filtering the Navier-Stokes equations, one obtains

∂ρ

∂t+

∂

∂xi(ρui) = 0 (12.9-4)

and

∂

∂t(ρui) +

∂

∂xj(ρuiuj) =

∂

∂xj

(µ∂σij∂xj

)− ∂p

∂xi− ∂τij∂xj

(12.9-5)

where σij is the stress tensor due to molecular viscosity defined by

σij ≡[µ

(∂ui∂xj

+∂uj∂xi

)]− 2

3µ∂ul∂xl

δij (12.9-6)

and τij is the subgrid-scale stress defined by

τij ≡ ρuiuj − ρuiuj (12.9-7)


Modeling Turbulence

12.9.3 Subgrid-Scale Models

The subgrid-scale stresses resulting from the filtering operation are unknown, and re-quire modeling. The subgrid-scale turbulence models in FLUENT employ the Boussinesqhypothesis [141] as in the RANS models, computing subgrid-scale turbulent stresses from

τij −1

3τkkδij = −2µtSij (12.9-8)

where µt is the subgrid-scale turbulent viscosity. The isotropic part of the subgrid-scalestresses τkk is not modeled, but added to the filtered static pressure term. Sij is therate-of-strain tensor for the resolved scale defined by

Sij ≡1

2

(∂ui∂xj

+∂uj∂xi

)(12.9-9)

For compressible flows, it is convenient to introduce the density-weighted (or Favre)filtering operator:

φ =ρφ

ρ(12.9-10)

The Favre Filtered Navier-Stokes equation takes the same form as Equation 12.9-5. Thecompressible form of the subgrid stress tensor is defined as:

Tij = −ρuiuj + ρuiuj (12.9-11)

This term is split into its isotropic and deviatoric parts

Tij = Tij −1

3Tllδij︸︷︷︸

deviatoric

+1

3Tllδij︸︷︷︸

isotropic

(12.9-12)

The deviatoric part of the subgrid-scale stress tensor is modeled using the compressibleform of the Smagorinsky model:

Tij −1

3Tllδij = 2µt(δij −

1

3δiiδij) (12.9-13)

As for incompressible flows, the term involving Tll can be added to the filtered pressureor simply neglected [97]. Indeed, this term can be re-written as Tll = γM2

sgsp whereMsgs is the subgrid Mach number. This subgrid Mach number can be expected to besmall when the turbulent Mach number of the flow is small.



FLUENT offers four models for µt: the Smagorinsky-Lilly model, the dynamic Smagorinsky-Lilly model, the WALE model, and the dynamic kinetic energy subgrid-scale model.

Subgrid-scale turbulent flux of a scalar, φ, is modeled using s subgrid-scale turbulentPrandtl number by

qj = −µtσt

∂φ

∂xj(12.9-14)

where qj is the subgrid-scale flux.

In the dynamic models, the subgrid-scale turbulent Prandtl number or Schmidt numberis obtained by applying the dynamic procedure originally proposed by Germano [114] tothe subgrid-scale flux.

Smagorinsky-Lilly Model

This simple model was first proposed by Smagorinsky [336]. In the Smagorinsky-Lillymodel, the eddy-viscosity is modeled by

µt = ρL2s

∣∣∣S∣∣∣ (12.9-15)

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

2SijSij. In FLUENT, Ls iscomputed using

Ls = min(κd, CsV

1/3)

(12.9-16)

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

Lilly derived a value of 0.17 for Cs for homogeneous isotropic turbulence in the inertialsubrange. However, this value was found to cause excessive damping of large-scale fluc-tuations in the presence of mean shear and in transitional flows as near solid boundary,and has to be reduced in such regions. In short, Cs is not an universal constant, whichis the most serious shortcoming of this simple model. Nonetheless, Cs value of around0.1 has been found to yield the best results for a wide range of flows, and is the defaultvalue in FLUENT.


Modeling Turbulence

Dynamic Smagorinsky-Lilly Model

Germano et al. [114] and subsequently Lilly [210] conceived a procedure in which theSmagorinsky model constant, Cs, is dynamically computed based on the informationprovided by the resolved scales of motion. The dynamic procedure thus obviates theneed for users to specify the model constant Cs in advance. The details of the modelimplementation in FLUENT and its validation can be found in [180].

The Cs obtained using the dynamic Smagorinsky-Lilly model varies in time and spaceover a fairly wide range. To avoid numerical instability, in FLUENT, Cs is clipped at zeroand 0.23 by default.

Wall-Adapting Local Eddy-Viscosity (WALE) Model

In the WALE model [261], the eddy viscosity is modeled by

µt = ρL2s

(SdijSdij)

3/2

(SijSij)5/2 + (SdijSdij)

5/4(12.9-17)

where Ls and Sdij in the WALE model are defined, respectively, as

Ls = min(κd, CwV

1/3)

(12.9-18)

Sdij =1

2

(g2ij + g2

ji

)− 1

3δijg

2kk , gij =

∂ui∂xj

(12.9-19)

In FLUENT, the default value of the WALE constant, Cw, is 0.325 and has been foundto yield satisfactory results for a wide range of flow. The rest of the notation is thesame as for the Smagorinsky-Lilly model. With this spatial operator, the WALE modelis designed to return the correct wall asymptotic (y3) behavior for wall bounded flows.



Dynamic Kinetic Energy Subgrid-Scale Model

The original and dynamic Smagorinsky-Lilly models, discussed previously, are essentiallyalgebraic models in which subgrid-scale stresses are parameterized using the resolved ve-locity scales. The underlying assumption is the local equilibrium between the transferredenergy through the grid-filter scale and the dissipation of kinetic energy at small sub-grid scales. The subgrid-scale turbulence can be better modeled by accounting for thetransport of the subgrid-scale turbulence kinetic energy.

The dynamic subgrid-scale kinetic energy model in FLUENT replicates the model pro-posed by Kim and Menon [183].

The subgrid-scale kinetic energy is defined as

ksgs =1

2

(u2k − u2

k

)(12.9-20)

which is obtained by contracting the subgrid-scale stress in Equation 12.9-7.

The subgrid-scale eddy viscosity, µt, is computed using ksgs as

µt = Ckk1/2sgs ∆f (12.9-21)

where ∆f is the filter-size computed from ∆f ≡ V 1/3.

The subgrid-scale stress can then be written as

τij −2

3ksgsδij = −2Ckk

1/2sgs ∆fSij (12.9-22)

ksgs is obtained by solving its transport equation

∂ksgs

∂t+∂ujksgs∂xj

= −τij∂ui∂xj− Cε

k3/2sgs

∆f

+∂

∂xj

(µtσk

∂ksgs

∂xj

)(12.9-23)

In the above equations, the model constants, Ck and Cε, are determined dynamically [183].σk is hardwired to 1.0. The details of the implementation of this model in FLUENT andits validation is given by Kim [180].


Modeling Turbulence

12.9.4 Inlet Boundary Conditions for the LES Model

This section describes the three algorithms available in FLUENT to model the fluctuatingvelocity at velocity inlet boundaries.

No Perturbations

The stochastic components of the flow at the velocity-specified inlet boundaries are ne-glected if the No Perturbations option is used. In such cases, individual instantaneousvelocity components are simply set equal to their mean velocity counterparts. This op-tion is suitable only when the level of turbulence at the inflow boundaries is negligible ordoes not play a major role in the accuracy of the overall solution.

Vortex Method

To generate a time-dependent inlet condition, a random 2D vortex method is considered.With this approach, a perturbation is added on a specified mean velocity profile via afluctuating vorticity field (i.e. two-dimensional in the plane normal to the streamwisedirection). The vortex method is based on the Lagrangian form of the 2D evolutionequation of the vorticity and the Biot-Savart law. A particle discretization is used tosolve this equation. These particles, or “vortex points” are convected randomly andcarry information about the vorticity field. If N is the number of vortex points and Ais the area of the inlet section, the amount of vorticity carried by a given particle i isrepresented by the circulation Γi and an assumed spatial distribution η:

Γi(x, y) = 4

√√√√ πAk(x, y)

3N [2 ln(3)− 3 ln(2)](12.9-24)

η(~x) =1

2πσ2

(2e−|x|

2/2σ2 − 1)

2e−|x|2/2σ2

(12.9-25)

where k is the turbulence kinetic energy. The parameter σ provides control over the sizeof a vortex particle. The resulting discretization for the velocity field is given by

~u(~x) =1

2π

N∑i=1

Γi((~xi − ~x)× ~z)(1− e|~x−~x′|2/2σ2

)

|~x− ~x′i|2(12.9-26)



Where ~z is the unit vector in the streamwise direction. Originally [326], the size ofthe vortex was fixed by an ad hoc value of σ. To make the vortex method generallyapplicable, a local vortex size is specified through a turbulent mixing length hypothesis. σis calculated from a known profile of mean turbulence kinetic energy and mean dissipationrate at the inlet according to the following:

σ =ck3/2

2ε(12.9-27)

where c = 0.16. To ensure that the vortex will always belong to resolved scales, theminimum value of σ in Equation 12.9-27 is bounded by the local grid size. The signof the circulation of each vortex is changed randomly each characteristic time scale τ .In the general implementation of the vortex method, this time scale represents the timenecessary for a 2D vortex convected by the bulk velocity in the boundary normal directionto travel along n times its mean characteristic 2D size (σm), where n is fixed equal to100 from numerical testing. The vortex method considers only velocity fluctuations inthe plane normal to the streamwise direction.

In FLUENT however, a simplified linear kinematic model (LKM) for the streamwisevelocity fluctuations is used [230]. It is derived from a linear model that mimics theinfluence of the two-dimensional vortex in the streamwise mean velocity field. If themean streamwise velocity U is considered as a passive scalar, the fluctuation u′ resultingfrom the transport of U by the planar fluctuating velocity field v′ is modeled by

u′ = −~v′ · ~g (12.9-28)

where ~g is the unit vector aligned with the mean velocity gradient ~∇U . When this meanvelocity gradient is equal to zero, a random perturbation can be considered instead.

i Since the vortex method theory is based on the modification of the velocityfield normal to the streamwise direction, it is imperative that the usercreates an inlet plane normal (or as close as possible) to the streamwisevelocity direction.

Spectral Synthesizer

The spectral synthesizer provides an alternative method of generating fluctuating velocitycomponents. It is based on the random flow generation technique originally proposed byKraichnan [185] and modified by Smirnov et al. [337]. In this method, fluctuating velocitycomponents are computed by synthesizing a divergence-free velocity-vector field fromthe summation of Fourier harmonics. In the implementation in FLUENT, the number ofFourier harmonics is fixed to 100.


Modeling Turbulence

12.10 Near-Wall Treatments for Wall-Bounded Turbulent Flows

12.10.1 Overview

Turbulent flows are significantly affected by the presence of walls. Obviously, the meanvelocity field is affected through the no-slip condition that has to be satisfied at the wall.However, the turbulence is also changed by the presence of the wall in non-trivial ways.Very close to the wall, viscous damping reduces the tangential velocity fluctuations, whilekinematic blocking reduces the normal fluctuations. Toward the outer part of the near-wall region, however, the turbulence is rapidly augmented by the production of turbulencekinetic energy due to the large gradients in mean velocity.

The near-wall modeling significantly impacts the fidelity of numerical solutions, inasmuchas walls are the main source of mean vorticity and turbulence. After all, it is in the near-wall region that the solution variables have large gradients, and the momentum and otherscalar transports occur most vigorously. Therefore, accurate representation of the flow inthe near-wall region determines successful predictions of wall-bounded turbulent flows.

The k-ε models, the RSM, and the LES model are primarily valid for turbulent coreflows (i.e., the flow in the regions somewhat far from walls). Consideration thereforeneeds to be given as to how to make these models suitable for wall-bounded flows. TheSpalart-Allmaras and k-ω models were designed to be applied throughout the boundarylayer, provided that the near-wall mesh resolution is sufficient.

Numerous experiments have shown that the near-wall region can be largely subdividedinto three layers. In the innermost layer, called the “viscous sublayer”, the flow is almostlaminar, and the (molecular) viscosity plays a dominant role in momentum and heator mass transfer. In the outer layer, called the fully-turbulent layer, turbulence playsa major role. Finally, there is an interim region between the viscous sublayer and thefully turbulent layer where the effects of molecular viscosity and turbulence are equallyimportant. Figure 12.10.1 illustrates these subdivisions of the near-wall region, plottedin semi-log coordinates.



Figure 12.10.1: Subdivisions of the Near-Wall Region

In Figure 12.10.1, y+ ≡ ρuτy/µ, where uτ is the friction velocity, defined as√

τwρ

.

Wall Functions vs. Near-Wall Model

Traditionally, there are two approaches to modeling the near-wall region. In one ap-proach, the viscosity-affected inner region (viscous sublayer and buffer layer) is not re-solved. Instead, semi-empirical formulas called “wall functions” are used to bridge theviscosity-affected region between the wall and the fully-turbulent region. The use of wallfunctions obviates the need to modify the turbulence models to account for the presenceof the wall.

In another approach, the turbulence models are modified to enable the viscosity-affectedregion to be resolved with a mesh all the way to the wall, including the viscous sublayer.For purposes of discussion, this will be termed the “near-wall modeling” approach. Thesetwo approaches are depicted schematically in Figure 12.10.2.

In most high-Reynolds-number flows, the wall function approach substantially saves com-putational resources, because the viscosity-affected near-wall region, in which the solutionvariables change most rapidly, does not need to be resolved. The wall function approachis popular because 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 where the low-Reynolds-number effects are pervasive in the flow domain in question, and the hypotheses under-lying the wall functions cease to be valid. Such situations require near-wall models that


Modeling Turbulence

Figure 12.10.2: Near-Wall Treatments in FLUENT

are valid in the viscosity-affected region and accordingly integrable all the way to thewall.

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

Wall Functions

Wall functions are a collection of semi-empirical formulas and functions that in effect“bridge” or “link” the solution variables at the near-wall cells and the correspondingquantities on the wall. The wall functions comprise

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

• formulas for near-wall turbulent quantities

Depending on the turbulent model you choose, FLUENT offers three to four choices ofwall function approaches:

• Standard Wall Functions

• Non-Equilibrium Wall Functions

• Enhanced Wall Treatment

• User-Defined Wall Functions



12.10.2 Standard Wall Functions

The standard wall functions in FLUENT are based on the proposal of Launder and Spald-ing [196], and have been most widely used for industrial flows. They are provided as adefault option in FLUENT.

Momentum

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

U∗ =1

κln(Ey∗) (12.10-1)

where

U∗ ≡UPC

1/4µ k

1/2P

τw/ρ(12.10-2)

y∗ ≡ρC1/4

µ k1/2P yP

µ(12.10-3)

and κ = von Karman constant (= 0.4187)E = empirical constant (= 9.793)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 30 < y∗ < 300. InFLUENT, the log-law is employed when y∗ > 11.225.

When the mesh is such that y∗ < 11.225 at the wall-adjacent cells, FLUENT applies thelaminar stress-strain relationship that can be written as

U∗ = y∗ (12.10-4)

It should be noted that, in FLUENT, the laws-of-the-wall for mean velocity and temper-ature are based on the wall unit, y∗, rather than y+ (≡ ρuτy/µ). These quantities areapproximately equal in equilibrium turbulent boundary layers.


Modeling Turbulence

Energy

Reynolds’ analogy between momentum and energy transport gives a similar logarithmiclaw for mean temperature. As in the law-of-the-wall for mean velocity, the law-of-the-wallfor temperature employed in FLUENT comprises the following two different laws:

• linear law for the thermal conduction sublayer where conduction is important

• logarithmic law for the turbulent region where effects of turbulence dominate con-duction

The thickness of the thermal conduction layer is, in general, different from the thicknessof the (momentum) viscous sublayer, and changes from fluid to fluid. For example, thethickness of the thermal sublayer for a high-Prandtl-number fluid (e.g., oil) is much lessthan its momentum sublayer thickness. For fluids of low Prandtl numbers (e.g., liquidmetal), on the contrary, it is much larger than the momentum sublayer thickness.

In highly compressible flows, the temperature distribution in the near-wall region canbe significantly different from that of low subsonic flows, due to the heating by viscousdissipation. In FLUENT, the temperature wall functions include the contribution fromthe viscous heating [381].

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

T ∗ ≡(Tw − TP ) ρcpC

1/4µ k

1/2P

q=

Pr y∗ + 12ρPr

C1/4µ k

1/2P

qU2P (y∗ < y∗T )

Prt[

1κ

ln(Ey∗) + P]

+

12ρC

1/4µ k

1/2P

qPrtU

2P + (Pr− Prt)U

2c (y∗ > y∗T )

(12.10-5)

where P is computed by using the formula given by Jayatilleke [164]:

P = 9.24

[(Pr

Prt

)3/4

− 1

] [1 + 0.28e−0.007Pr/Prt

](12.10-6)



and

kP = turbulent kinetic energy at point Pρ = 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 = Van Driest constant (= 26)Uc = mean velocity magnitude at y∗ = y∗T

Note that, for the pressure-based solver, the terms

1

2ρPr

C1/4µ k

1/2P

qU2P

and1

2ρC1/4µ k

1/2P

q

PrtU

2P + (Pr− Prt)U

2c

will be included in Equation 12.10-5 only for compressible flow calculations.

The non-dimensional thermal sublayer thickness, y∗T , in Equation 12.10-5 is computed asthe y∗ value at which the linear law and the logarithmic law intersect, given the molecularPrandtl number of the fluid being modeled.

The procedure of applying the law-of-the-wall for temperature is as follows. Once thephysical properties of the fluid being modeled are specified, its molecular Prandtl numberis computed. Then, given the molecular Prandtl number, the thermal sublayer thickness,y∗T , is computed from 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 orthe logarithmic profile in Equation 12.10-5 is applied to compute the wall temperatureTw or heat flux q (depending on the type of the thermal boundary conditions).

The function for P given by equation Equation 12.10-6 is relevant for the smooth walls.For the rough walls, however, this function is modified as follows:

Prough = 3.15Pr0.695(

1

E ′− 1

E

)0.359

+

(E ′

E

)0.6

P (12.10-7)

where E ′ is the wall function constant modified for the rough walls, defined by E ′ = E/fr.To find a description of the roughness function fr, you may refer to Equation 7.13-3 inSection 7.13.1: Wall Roughness Effects in Turbulent Wall-Bounded Flows.


Modeling Turbulence

Species

When using wall functions for species transport, FLUENT assumes that species transportbehaves analogously to heat transfer. Similarly to Equation 12.10-5, the law-of-the-wallfor species can be expressed for constant property flow with no viscous dissipation as

Y ∗ ≡(Yi,w − Yi) ρC1/4

µ k1/2P

Ji,w=

Sc y∗ (y∗ < y∗c )

Sct[

1κ

ln(Ey∗) + Pc]

(y∗ > y∗c )(12.10-8)

where Yi is the local species mass fraction, Sc and Sct are molecular and turbulentSchmidt numbers, and Ji,w is the diffusion flux of species i at the wall. Note that Pc andy∗c are calculated in a similar way as P and y∗T , with the difference being that the Prandtlnumbers are always replaced by the corresponding Schmidt numbers.

Turbulence

In the k-ε models and in the RSM (if the option to obtain wall boundary conditions fromthe k equation is enabled), the k equation is solved in the whole domain including thewall-adjacent cells. The boundary condition for k imposed at the wall is

∂k

∂n= 0 (12.10-9)

where n is the local coordinate normal to the wall.

The production of kinetic energy, Gk, and its dissipation rate, ε, at the wall-adjacentcells, which are the source terms in the k equation, are computed on the basis of the localequilibrium hypothesis. Under this assumption, the production of k and its dissipationrate are assumed to be 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

(12.10-10)

and ε is computed from

εP =C3/4µ k

3/2P

κyP(12.10-11)

The ε equation is not solved at the wall-adjacent cells, but instead is computed us-ing Equation 12.10-11. ω and Reynolds stress equations are solved as detailed in Sec-tions 12.5.3 and 12.7.9, respectively.



Note that, as shown here, the wall boundary conditions for the solution variables, in-cluding mean velocity, temperature, species concentration, k, and ε, are all taken care ofby the wall functions. Therefore, you do not need to be concerned about the boundaryconditions at the walls.

The standard wall functions described so far are provided as a default option in FLUENT.The standard wall functions work reasonably well for a broad range of wall-bounded flows.However, they tend to become less reliable when the flow situations depart too much fromthe ideal conditions that are assumed in their derivation. Among others, the constant-shear and local equilibrium hypotheses are the ones that most restrict the universalityof the standard wall functions. Accordingly, when the near-wall flows are subjected tosevere pressure gradients, and when the flows are in strong non-equilibrium, the qualityof the predictions is likely to be compromised.

The non-equilibrium wall functions offered as an additional option can improve the resultsin such situations.

i Standard wall functions are available with the following viscous models:

• K-epsilon

• Reynolds Stress

12.10.3 Non-Equilibrium Wall Functions

In addition to the standard wall function described above (which is the default near-walltreatment) a two-layer-based, non-equilibrium wall function [181] is also available. Thekey elements in the non-equilibrium wall functions are as follows:

• Launder and Spalding’s log-law for mean velocity is sensitized to pressure-gradienteffects.

• The two-layer-based concept is adopted to compute the budget of turbulence kineticenergy (Gk,ε) in the wall-neighboring cells.

The law-of-the-wall for mean temperature or species mass fraction remains the same asin the standard wall function described above.

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

UC1/4µ k1/2

τw/ρ=

1

κln

(EρC1/4

µ k1/2y

µ

)(12.10-12)


Modeling Turbulence

where

U = U − 1

2

dp

dx

[yv

ρκ√k

ln

(y

yv

)+y − yvρκ√k

+y2v

µ

](12.10-13)

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

yv ≡µy∗v

ρC1/4µ k

1/2P

(12.10-14)

where y∗v = 11.225.

The non-equilibrium wall function employs the two-layer concept in computing the bud-get of turbulence kinetic energy at the wall-adjacent cells, which is needed to solve the kequation at the wall-neighboring cells. The wall-neighboring cells are assumed to consistof a viscous sublayer and a fully turbulent layer. The following profile assumptions forturbulence quantities are made:

τt =

0, y < yvτw, y > yv

k =

(yyv

)2kP , y < yv

kP , y > yvε =

2νky2 , y < yvk3/2

C`∗y, y > yv

(12.10-15)

where C`∗ = κC−3/4

µ , and yv is the dimensional thickness of the viscous sublayer, definedin Equation 12.10-14.

Using these profiles, the cell-averaged production of k, Gk, and the cell-averaged dissipa-tion rate, ε, can be computed from the volume average of Gk and ε of the wall-adjacentcells. For quadrilateral and hexahedral cells for which the volume average can be ap-proximated with a depth-average,

Gk ≡1

yn

∫ yn

0τt∂U

∂ydy =

1

κyn

τ 2w

ρC1/4µ k

1/2P

ln

(ynyv

)(12.10-16)

and

ε ≡ 1

yn

∫ yn

0ε dy =

1

yn

2ν

yv+k

1/2P

C`∗ ln

(ynyv

) kP (12.10-17)

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

In Equations 12.10-16 and 12.10-17, the turbulence kinetic energy budget for the wall-neighboring cells is effectively sensitized to the proportions of the viscous sublayer andthe fully turbulent layer, which varies widely from cell to cell in highly non-equilibrium



flows. It effectively relaxes the local equilibrium assumption (production = dissipation)that is adopted by the standard wall function in computing the budget of the turbulencekinetic energy at wall-neighboring cells. Thus, the non-equilibrium wall functions, ineffect, partly account for non-equilibrium effects neglected in the standard wall function.

Limitations of the Wall Function Approach

The standard wall functions give reasonably accurate predictions for the majority ofhigh-Reynolds-number, wall-bounded flows. The non-equilibrium wall functions furtherextend the applicability of the wall function approach by including the effects of pressuregradient and strong non-equilibrium. However, the wall function approach becomes lessreliable when the flow conditions depart too much from the ideal conditions underlyingthe wall functions. Examples are as follows:

• Pervasive low-Reynolds-number or near-wall effects (e.g., flow through a small gapor 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-driven flows).

• High three-dimensionality in the near-wall region (e.g., Ekman spiral flow, stronglyskewed 3D boundary layers).

If any of the items listed above is a prevailing feature of the flow you are modeling, andif it is considered critically important to capture that feature for the success of yoursimulation, you must employ the near-wall modeling approach combined with adequatemesh resolution in the near-wall region. FLUENT provides the enhanced wall treatmentfor such situations. This approach can be used with the three k-ε models, the k-ω models,and the RSM.


Modeling Turbulence

Standard Wall Functions vs. Non-Equilibrium Wall Functions

Because of the capability to partly account for the effects of pressure gradients anddeparture from equilibrium, the non-equilibrium wall functions are recommended for usein complex flows involving separation, reattachment, and impingement where the meanflow and turbulence are subjected to severe pressure gradients and change rapidly. Insuch flows, improvements can be obtained, particularly in the prediction of wall shear(skin-friction coefficient) and heat transfer (Nusselt or Stanton number).

i Non-equilibrium wall functions are available with the following viscousmodels:

• K-epsilon

• Reynolds Stress

12.10.4 Enhanced Wall Treatment

Enhanced wall treatment is a near-wall modeling method that combines a two-layermodel with enhanced wall functions. If the near-wall mesh is fine enough to be able toresolve the laminar sublayer (typically y+ ≈ 1), then the enhanced wall treatment willbe identical to the traditional two-layer zonal model (see below for details). However,the restriction that the near-wall mesh must be sufficiently fine everywhere might imposetoo large a computational requirement. Ideally, then, one would like to have a near-wallformulation that can be used with coarse meshes (usually referred to as wall-functionmeshes) as well as fine meshes (low-Reynolds-number meshes). In addition, excessiveerror should not be incurred for intermediate meshes that are too fine for the near-wallcell centroid to lie in the fully turbulent region, but also too coarse to properly resolvethe sublayer.

To achieve the goal of having a near-wall modeling approach that will possess the accuracyof the standard two-layer approach for fine near-wall meshes and that, at the same time,will not significantly reduce accuracy for wall-function meshes, FLUENT can combine thetwo-layer model with enhanced 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 is completely re-solved all the way to the viscous sublayer. The two-layer approach is an integral partof the enhanced wall treatment and is used to specify both ε and the turbulent vis-cosity in the near-wall cells. In this approach, the whole domain is subdivided into aviscosity-affected region and a fully-turbulent region. The demarcation of the two re-gions is determined by a wall-distance-based, turbulent Reynolds number, Rey, definedas

Rey ≡ρy√k

µ(12.10-18)

where y is the normal distance from the wall at the cell centers. In FLUENT, y isinterpreted as the distance to the nearest wall:

y ≡ min~rw∈Γw

‖~r − ~rw‖ (12.10-19)

where ~r is the position vector at the field point, and ~rw is the position vector on thewall boundary. Γw is the union of all the wall boundaries involved. This interpretationallows y to be uniquely defined in flow domains of complex shape involving multiplewalls. Furthermore, y defined in this way is independent of the mesh topology used, andis definable even on unstructured meshes.

In the fully turbulent region (Rey > Re∗y; Re∗y = 200), the k-ε models or the RSM(described in Sections 12.4 and 12.7) are employed.

In the viscosity-affected near-wall region (Rey < Re∗y), the one-equation model of Wolf-stein [405] is employed. In the one-equation model, the momentum equations and thek equation are retained as described in Sections 12.4 and 12.7. However, the turbulentviscosity, µt, is computed from

µt,2layer = ρ Cµ`µ√k (12.10-20)

where the length scale that appears in Equation 12.10-20 is computed from [53]

`µ = yC`∗(1− e−Rey/Aµ

)(12.10-21)

The two-layer formulation for turbulent viscosity described above is used as a part of theenhanced wall treatment, in which the two-layer definition is smoothly blended with thehigh-Reynolds-number µt definition from the outer region, as proposed by Jongen [167]:

µt,enh = λεµt + (1− λε)µt,2layer (12.10-22)


Modeling Turbulence

where µt is the high-Reynolds-number definition as described in Section 12.4: Standard,RNG, and Realizable k-ε Models Theory or 12.7 for the k-ε models or the RSM. Ablending 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 =1

2

[1 + tanh

(Rey − Re∗y

A

)](12.10-23)

The constant A determines the width of the blending function. By defining a width suchthat the value of λε will be within 1% of its far-field value given a variation of ∆Rey, theresult is

A =|∆Rey|

tanh(0.98)(12.10-24)

Typically, ∆Rey would be assigned a value that is between 5% and 20% of Re∗y. Themain purpose of the blending function λε is to prevent solution convergence from beingimpeded when the k-ε solution in the outer layer does not match with the two-layerformulation.

The ε field is computed from

ε =k3/2

`ε(12.10-25)

The length scales that appear in Equation 12.10-25 are again computed from Chen andPatel [53]:

`ε = yC`∗(1− e−Rey/Aε

)(12.10-26)

If the whole flow domain is inside the viscosity-affected region (Rey < 200), ε is notobtained by solving the transport equation; it is instead obtained algebraically fromEquation 12.10-25. FLUENT uses a procedure for the ε specification that is similar tothe µt blending in order to ensure a smooth transition between the algebraically-specifiedε in the inner region and the ε obtained from solution of the transport equation in theouter region.

The constants in the length scale formulas, Equations 12.10-21 and 12.10-26, are takenfrom [53]:

C`∗ = κC−3/4

µ , Aµ = 70, Aε = 2C`∗ (12.10-27)



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-turbulent outer region) it is necessary toformulate the law-of-the wall as a single wall law for the entire wall region. FLUENTachieves this by blending linear (laminar) and logarithmic (turbulent) laws-of-the-wallusing a function suggested by Kader [169]:

u+ = eΓu+lam + e

1Γu+

turb (12.10-28)

where the blending function is given by:

Γ = − a(y+)4

1 + by+(12.10-29)

where a = 0.01 and b = 5.

Similarly, the general equation for the derivative du+

dy+ is

du+

dy+= eΓ du

+lam

dy++ e

1Γdu+

turb

dy+(12.10-30)

This approach allows the fully turbulent law to be easily modified and extended to takeinto account other effects such as pressure gradients or variable properties. This formulaalso guarantees the correct asymptotic behavior for large and small values of y+ andreasonable representation of velocity profiles in the cases where y+ falls inside the wallbuffer region (3 < y+ < 10).

The enhanced wall functions were developed by smoothly blending an enhanced turbulentwall law with the laminar wall law. The enhanced turbulent law-of-the-wall for compress-ible flow with heat transfer and pressure gradients has been derived by combining theapproaches of White and Cristoph [401] and Huang et al. [148]:

du+turb

dy+=

1

κy+

[S ′(1− βu+ − γ(u+)2)

]1/2(12.10-31)

where

S ′ =

1 + αy+ for y+ < y+

s

1 + αy+s for y+ ≥ y+

s

(12.10-32)


Modeling Turbulence

and

α ≡ νwτwu∗

dp

dx=

µ

ρ2(u∗)3

dp

dx(12.10-33)

β ≡ σtqwu∗

cpτwTw=

σtqwρcpu∗Tw

(12.10-34)

γ ≡ σt(u∗)2

2cpTw(12.10-35)

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 12.10-31 represents the influences of pressure gradientswhile the coefficients β and γ represent thermal effects. Equation 12.10-31 is an ordinarydifferential equation and FLUENT will provide an appropriate analytical solution. If α, β,and γ all equal 0, an analytical solution would lead to the classical turbulent logarithmiclaw-of-the-wall.

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

du+lam

dy+= 1 + αy+ (12.10-36)

Note that the above expression only includes effects of pressure gradients through α,while the effects of variable properties due to heat transfer and compressibility on thelaminar wall law are neglected. These effects are neglected because they are thought to beof minor importance when they occur close to the wall. Integration of Equation 12.10-36results in

u+lam = y+

(1 +

α

2y+)

(12.10-37)

Enhanced thermal wall functions follow the same approach developed for the profile ofu+. The unified wall thermal formulation blends the laminar and logarithmic profilesaccording to the method of Kader [169]:

T+ ≡ (Tw − TP ) ρcpu∗

q= eΓT+

lam + e1ΓT+

turb (12.10-38)

where the notation for TP and q is the same as for standard thermal wall functions (seeEquation 12.10-5). Furthermore, the blending factor Γ is defined as

Γ = − a(Pr y+)4

1 + bPr3 y+(12.10-39)



where Pr is the molecular Prandtl number, and the coefficients a and b are defined as inEquation 12.10-29.

Apart from the formulation for T+ in Equation 12.10-38, enhanced thermal wall functionsfollow the same logic as for standard thermal wall functions (see Section 12.10.2: Energy),resulting in the following definition for turbulent and laminar thermal wall functions:

T+lam = Pr

(u+

lam +ρu∗

2qu2

)(12.10-40)

T+turb = Prt

u+

turb + P +ρu∗

2q

[u2 −

(Pr

Prt

− 1)

(u+c )2(u∗)2

](12.10-41)

where the quantity u+c is the value of u+ at the fictitious “crossover” between the laminar

and turbulent region. The function P is defined in the same way as for standard wallfunctions.

A similar procedure is also used for species wall functions when the enhanced wall treat-ment is used. In this case, the Prandtl numbers in Equations 12.10-40 and 12.10-41 arereplaced by adequate Schmidt numbers. See Section 12.10.2: Species for details aboutspecies wall functions.

The boundary condition for turbulence kinetic energy is the same as for standard wallfunctions (Equation 12.10-9). However, the production of turbulence kinetic energy Gk iscomputed using the velocity gradients that are consistent with the enhanced law-of-the-wall (Equations 12.10-28 and 12.10-30), ensuring a formulation that is valid throughoutthe near-wall region.

i The enhanced wall treatment is available with the following viscous models:

• K-epsilon

• Reynolds Stress

Enhanced wall functions are available with the following viscous models:

• Spalart-Allmaras

• K-omega

• Large Eddy Simulation


Modeling Turbulence

12.10.5 User-Defined Wall Functions

This option is only available when the k-ε model is enabled. Selecting the User-DefinedWall Functions under Near-wall Treatment allows you to hook a Law-of-the-Wall UDF.See Section 2.3.23: DEFINE WALL FUNCTIONS of the separate UDF Manual for details onuser-defined wall functions.

i User-defined wall functions are available with the following viscous model:

• K-epsilon

12.10.6 LES Near-Wall Treatment

When the mesh is fine enough to resolve the laminar sublayer, the wall shear stress isobtained from the laminar stress-strain relationship:

u

uτ=ρuτy

µ(12.10-42)

If the mesh is too coarse to resolve the laminar sublayer, it is assumed that the centroidof the wall-adjacent cell falls within the logarithmic region of the boundary layer, andthe law-of-the-wall is employed:

u

uτ=

1

κlnE

(ρuτy

µ

)(12.10-43)

where κ is the von Karman constant and E = 9.793. If the mesh is a such that the firstnear wall point is within the buffer region, then two above laws are blended in accordancewith equation Equation 12.10-28.

For the LES simulations in FLUENT, there is an alternative near wall approach based onthe work of Werner and Wengle [397], who proposed analytical integration of power-lawnear-wall velocity distribution resulting in the following expressions for the wall shearstress:

|τw| =

2µ|up|

∆zfor |up| ≤ µ

2ρ∆zA

21−B

ρ[

1−B2A

1+B1−B

(µρ∆z

)1+B+ 1+B

A

(µρ∆z

)B|up|

] 21+B

for |up| > µ2ρ∆z

A2

1−B

(12.10-44)

where up is velocity parallel to the wall, A = 8.3, B = 1/7 are the constants, and ∆z isthe near-wall control volume length scale.


12.11 Grid Considerations for Turbulent Flow Simulations

The Werner-Wengle wall functions can be enabled using the define/models/viscous/

near-wall-treatment/werner-wengle-wall-fn? text command.

12.11 Grid Considerations for Turbulent Flow Simulations

Successful computations of turbulent flows require some consideration during the meshgeneration. Since turbulence (through the spatially-varying effective viscosity) plays adominant role in the transport of mean momentum and other parameters, you mustascertain that turbulence quantities in complex turbulent flows are properly resolved ifhigh accuracy is required. Due to the strong interaction of the mean flow and turbulence,the numerical results for turbulent flows tend to be more susceptible to grid dependencythan those for laminar flows.

It is therefore recommended that you resolve, with sufficiently fine meshes, the regionswhere the mean flow changes rapidly and there are shear layers with a large mean rateof strain.

You can check the near-wall mesh by displaying or plotting the values of y+, y∗, andRey, which are all available in the postprocessing panels. It should be remembered thaty+, y∗, and Rey are not fixed, geometrical quantities. They are all solution-dependent.For example, when you double the mesh (thereby halving the wall distance), the new y+

does not necessarily become half of the y+ for the original mesh.

For the mesh in the near-wall region, different strategies must be used depending on whichnear-wall option you are using. In Sections 12.11.1 and 12.11.1 are general guidelines forthe near-wall mesh.

12.11.1 Near-Wall Mesh Guidelines

Wall Functions

The log-law, which is valid for equilibrium boundary layers and fully developed flows,provides upper and lower bounds on the acceptable distance between the cell centroidand the wall for wall-adjacent cells. The distance is usually measured in the wall unit,y+ (≡ ρuτy/µ), or y∗. Note that y+ and y∗ have comparable values when the first cell isplaced in the log-layer.

• For standard or non-equilibrium wall functions, each wall-adjacent cell’s centroidshould be located within the log-law layer, 30 < y+ < 300. A y+ value close to thelower bound (y+ ≈ 30) is most desirable.

• Although FLUENT employs the linear (laminar) law when y∗ < 11.225, using anexcessively fine mesh near the walls should be avoided, because the wall functionscease to be valid in the viscous sublayer.


Modeling Turbulence

• As much as possible, the mesh should be made either coarse or fine enough toprevent the wall-adjacent cells from being placed in the buffer layer (y+ = 5 ∼ 30).

• The upper bound of the log-layer depends on, among others, pressure gradientsand Reynolds number. As the Reynolds number increases, the upper bound tendsto also increase. y+ values that are too large are not desirable, because the wakecomponent becomes substantially large above the log-layer.

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

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

Enhanced Wall Treatment

Although the enhanced wall treatment is designed to extend the validity of near-wallmodeling beyond the viscous sublayer, it is still recommended that you construct a meshthat will fully resolve the viscosity-affected near-wall region. In such a case, the two-layercomponent of the enhanced wall treatment will be dominant and the following meshrequirements are recommended (note that, here, the mesh requirements are in terms ofy+, not y∗):

• When the enhanced wall treatment is employed with the intention of resolving thelaminar sublayer, y+ at the wall-adjacent cell should be on the order of y+ = 1.However, a higher y+ is acceptable as long as it is well inside the viscous sublayer(y+ < 4 to 5).

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

Spalart-Allmaras Model

The Spalart-Allmaras model in its complete implementation is a low-Reynolds-numbermodel. This means that it is designed to be used with meshes that properly resolve theviscous-affected region, and damping functions have been built into the model in order toproperly attenuate the turbulent viscosity in the viscous sublayer. Therefore, to obtainthe full benefit of the Spalart-Allmaras model, the near-wall mesh spacing should be asdescribed in Section 12.11.1: Enhanced Wall Treatment for the enhanced wall treatment.

However, as discussed in Section 12.3.7: Wall Boundary Conditions, the boundary condi-tions for the Spalart-Allmaras model have been implemented so that the model will workon coarser meshes, such as would be appropriate for the wall function approach. If you areusing a coarse mesh, you should follow the guidelines described in Section 12.11.1: WallFunctions.


12.12 Steps in Using a Turbulence Model

In summary, for best results with the Spalart-Allmaras model, you should use either avery fine near-wall mesh spacing (on the order of y+ = 1) or a mesh spacing such thaty+ ≥ 30.

k-ω Models

Both k-ω models available in FLUENT are available as low-Reynolds-number models aswell as high-Reynolds-number models. If the Transitional Flows option is enabled in theViscous Model panel, low-Reynolds-number variants will be used, and, in that case, meshguidelines should be 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 wall functions.

Large Eddy Simulation

For the LES implementation in FLUENT, the wall boundary conditions have been imple-mented using a law-of-the-wall approach as described in Section 12.9.4: Inlet BoundaryConditions for the LES Model. This means that there are no computational restrictionson the near-wall mesh spacing. However, for best results, it might be necessary to use avery fine near-wall mesh spacing (on the order of y+ = 1).


When your FLUENT model includes turbulence you need to activate the relevant modeland options, and supply turbulent boundary conditions. These inputs are described inthis section.

The procedure for setting up a turbulent flow problem is described below. (Note thatthis procedure includes only those steps necessary for the turbulence model itself; youwill need to set up other models, boundary conditions, etc. as usual.)


Modeling Turbulence

1. To activate the turbulence model, select Spalart-Allmaras, k-epsilon, k-omega, ReynoldsStress, Detached Eddy Simulation (3D), and Large Eddy Simulation (LES) (3D) underModel in the Viscous Model panel (Figure 12.12.1).

Define −→ Models −→Viscous...

Figure 12.12.1: The Viscous Model Panel



If you choose the k-epsilon model, select Standard, RNG, or Realizable under k-epsilonModel. If you choose the k-omega model, select Standard or SST under k-omegaModel.

i The Detached Eddy Simulation and the Large Eddy Simulation (LES) modelsare available only for 3D cases.

2. If the flow involves walls, and you are using one of the k-εmodels or the RSM, chooseone of the following options for the Near-Wall Treatment in the Viscous Model panel:

• Standard Wall Functions

• Non-Equilibrium Wall Functions

• Enhanced Wall Treatment

• User-Defined Wall Functions

These near-wall options are described in detail in Section 12.10: Near-Wall Treat-ments for Wall-Bounded Turbulent Flows. By default, the standard wall functionis enabled.

The near-wall treatment for the Spalart-Allmaras, k-ω, and LES models is definedautomatically, as described in Sections 12.3.7, 12.5.3, and 12.9.4, respectively.

3. Enable the appropriate turbulence modeling options in the Viscous Model panel.See Section 12.19: Setup Options for all Turbulence Modeling for details.

4. Specify the boundary conditions for the solution variables.

Define −→Boundary Conditions...

See Section 12.20: Defining Turbulence Boundary Conditions for details.

5. Specify the initial guess for the solution variables.

Solve −→ Initialize −→Initialize...

See Section 12.21: Providing an Initial Guess for k and ε (or k and ω) for details.Note that Reynolds stresses are automatically initialized using k, and thereforeneed not be initialized.


Modeling Turbulence

12.13 Setting Up the Spalart-Allmaras Model

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

• vorticity-based production (Section 12.19.3: Vorticity- and Strain/Vorticity-BasedProduction)

• strain/vorticity-based production (Section 12.19.3: Vorticity- and Strain/Vorticity-Based Production)

• viscous heating (always activated for the density-based solvers) (Section 12.19.1: In-cluding the Viscous Heating Effects)

Figure 12.13.1: The Viscous Model Panel Displaying the Spalart-AllmarasOptions


12.14 Setting Up the k-ε Model


12.14.1 Setting Up the Standard or Realizable k-ε Model

If you choose the standard k-ε model or the realizable k-ε model, the following optionsare available:


• inclusion of buoyancy effects on ε (Section 12.4.5: Effects of Buoyancy on Turbulencein the k-ε Models)

Figure 12.14.1: The Viscous Model Panel Displaying the Standard k-ε Model


Modeling Turbulence

12.14.2 Setting Up the RNG k-ε Model

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

• differential viscosity model (Section 12.19.5: Differential Viscosity Modification)

• swirl modification (Section 12.19.6: Swirl Modification)



For all k-εmodels, one the following near-wall treatments must be selected (Section 12.10: Near-Wall Treatments for Wall-Bounded Turbulent Flows):

• standard wall functions

• non-equilibrium wall functions

• enhanced wall treatment

• user-defined wall functions

If you choose the enhanced wall treatment, the following options are available:

• pressure gradient effects (Section 12.19.9: Including Pressure Gradient Effects)

• thermal effects (Section 12.19.10: Including Thermal Effects)

If you choose the user-defined wall functions near-wall treatment, hook your UDF underLaw of the Wall, as shown in Figure 12.14.1.



Figure 12.14.2: The Viscous Model Panel Displaying the RNG k-ε Model


Modeling Turbulence

12.15 Setting Up the k-ω Model

12.15.1 Setting Up the Standard k-ω Model

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

• transitional flows (Section 12.19.7: Transitional Flows)

• shear flow corrections (Section 12.19.8: Shear Flow Corrections)


Figure 12.15.1: The Viscous Model Panel Displaying the Standard k-ω Model

The k-ω models use enhanced wall functions, described in Section 12.10.4: EnhancedWall Functions, as the near-wall treatment.


12.15 Setting Up the k-ω Model

12.15.2 Setting Up the Shear-Stress Transport k-ω Model

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



Figure 12.15.2: The Viscous Model Panel Displaying the SST k-ω Model


Modeling Turbulence

12.16 Setting Up the Reynolds Stress Model

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

• Linear pressure-strain model (Section 12.7.4: Linear Pressure-Strain Model)

• Quadratic pressure-strain model (Section 12.19.13: Quadratic Pressure-Strain Model)

• Low-Re Stress-Omega (Section 12.19.14: Low-Re Stress-Omega Pressure-Strain)

Figure 12.16.1: The Viscous Model Panel Displaying the Reynolds StressModel Options


12.16 Setting Up the Reynolds Stress Model

The following Reynolds-stress options are available:

• wall boundary conditions for the Reynolds stresses from the k equation (Sec-tion 12.19.12: Solving the k Equation to Obtain Wall Boundary Conditions) forthe linear and quadratic pressure-strain models

• wall reflection effects on Reynolds stresses (Section 12.19.11: Including the WallReflection Term) for the linear pressure-strain model

Other options that are available based on your case setup include:



For the Reynolds stress model, the following near-wall treatments are available (Sec-tion 12.10: Near-Wall Treatments for Wall-Bounded Turbulent Flows):

• standard wall functions

• non-equilibrium wall functions

• enhanced wall treatment

If wall boundary conditions for the Reynolds stresses from the k equation and/or wallreflection effects on Reynolds stresses are/is selected, then all the above near-wall treat-ments are available for selection.

If you choose the enhanced wall treatment, the following options are available:

• pressure gradient effects (Section 12.19.9: Including Pressure Gradient Effects)

• thermal effects (Section 12.19.10: Including Thermal Effects)

If the quadratic pressure-strain model is selected, then you can set either the standardwall functions or the non-equilibrium wall functions.

If Low-Re Stress-Omega is selected, you cannot select any near-wall treatments. You dohave the option of selecting any or all of the following k − ω options:


• shear flow corrections (Section 12.19.8: Shear Flow Corrections)


Modeling Turbulence

Figure 12.16.2: The Viscous Model Panel Displaying the Low-Re Stress-Omega Model Options


12.17 Setting Up the Detached Eddy Simulation Model

12.17 Setting Up the Detached Eddy Simulation Model

The following submodels are available when selecting the DES model (Section 12.19.4: De-tached Eddy Simulation (DES) Modeling):

• Spalart-Allmaras

• Realizable k-ε

• SST k-ω

Figure 12.17.1: The Viscous Model Panel Displaying the Detached Eddy Sim-ulation Model Options

For the Realizable k-ε submodel, there are no other model-specific options to set. As forthe SST k-ω sub-model, the model-specific option that you can select is the TransitionalFlows k-omega Option (Section 12.19.7: Transitional Flows).


Modeling Turbulence

Additionally, you can perform the following DES-specific functions by using the/define/models/viscous/detached-eddy-simulation? text command:

• Use cell volume-based LES length scale (default is to use maximum cell edge)

• Modify only the length scales that appear in the destruction term in νt equation(the default is to modify all length scales within the νt equation)

12.18 Setting Up the Large Eddy Simulation Model

If you choose the LES model, the following subgrid-scale submodels are available (Sec-tion 12.19.15: Subgrid-Scale Model):

• Smagorinsky-Lilly

• WALE

• Kinetic-Energy Transport

The LES options that are available for the Smagorinsky-Lilly are

• Dynamic Stress

• Dynamic Energy Flux (available only when the Dynamic Stress Model is enabled)

• Dynamic Scalar Flux

The LES option that is available when the Kinetic-Energy Transport submodel is selectedis the Dynamic Energy Flux Model.

It is also possible to modify the Model Constants, but this is not necessary for mostapplications. See Sections 12.3 through 12.9 for details about these constants. Note thatC1-PS and C2-PS are the constants C1 and C2 in the linear pressure-strain approximationof Equations 12.7-5 and 12.7-6, and C1’-PS and C2’-PS are the constants C ′1 and C ′2 inEquation 12.7-7. 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 quadraticpressure-strain approximation of Equation 12.7-16.


12.18 Setting Up the Large Eddy Simulation Model

Figure 12.18.1: The Viscous Model Panel Displaying the Large Eddy Simu-lation Model Options


Modeling Turbulence

12.19 Setup Options for all Turbulence Modeling

The various options available for the turbulence models are described in detail in Sec-tions 12.3 through 12.9. Instructions for activating these options are provided here.

12.19.1 Including the Viscous Heating Effects

See Sections 13.2.1 and 13.2.2 for information on including viscous heating effects in yourmodel.

12.19.2 Including Turbulence Generation Due to Buoyancy

If you specify a non-zero gravity force (in the Operating Conditions panel), and you aremodeling a non-isothermal flow, the generation of turbulent kinetic energy due to buoy-ancy (Gb in Equation 12.4-1) is, by default, always 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 Buoyancy Effects optionunder Options in the Viscous Model panel.

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

12.19.3 Vorticity- and Strain/Vorticity-Based Production

For the Spalart-Allmaras model, you can choose either Vorticity-Based Production orStrain/Vorticity-Based Production under Spalart-Allmaras Options in the Viscous Modelpanel. If you choose Vorticity-Based Production, FLUENT will use Equation 12.3-8 tocompute the value of the deformation tensor S; if you choose Strain/Vorticity-Based Pro-duction, it will use Equation 12.3-10.

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

12.19.4 Detached Eddy Simulation (DES) Modeling

If you enabled DES for the Spalart-Allmaras model as described at the beginning of thissection, FLUENT will use Equation 12.8-1 to compute the value of the length scale d. Bydefault, the empirical constant Cdes is set to 0.65. You can change its value in the Cdesfield under Model Constants. Cdes is set to 0.61 for the Realizable k-ε and SST k-ω RANSmodels (Sections 12.8.2 and 12.8.3).



12.19.5 Differential Viscosity Modification

In the RNG turbulence model in FLUENT, you have an option to use a differential formulafor effective viscosity µeff (Equation 12.4-6) to account for low-Reynolds-number effects.To enable this option, turn on Differential Viscosity Model under RNG Options in theViscous Model panel.

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

12.19.6 Swirl Modification

Once you choose the RNG model, the swirl modification takes effect, by default, for allthree-dimensional flows and axisymmetric flows with swirl. The default swirl constant(αs in Equation 12.4-8) is set to 0.07, which works well for weakly to moderately swirlingflows. However, for strongly swirling flows, you may need to use a larger swirl constant.

To change the value of the swirl constant, you must first turn on the Swirl DominatedFlow option under RNG Options in the Viscous Model panel. (This option will not appearunless you have activated the RNG k-ε model.)

12.19.7 Transitional Flows

If either of the k-ω models are used, you may enable a low-Reynolds-number correction tothe turbulent viscosity by enabling the Transitional Flows option under k-omega Optionsin the Viscous Model panel. By default, this option is not enabled, and the dampingcoefficient (α∗ in Equation 12.5-6) is equal to 1.

12.19.8 Shear Flow Corrections

In the standard k-ω model, you also have the option of including corrections to improvethe accuracy in predicting free shear flows. The Shear Flow Corrections option underk-omega Options is enabled by default in the Viscous Model panel, as these correctionsare included in the standard k-ω model [402]. When this option is enabled, FLUENT willcalculate f ∗β and fβ using Equations 12.5-16 and 12.5-24, respectively. If this option isdisabled, f ∗β and fβ will be set equal to 1.

12.19.9 Including Pressure Gradient Effects

If the enhanced wall treatment is used, you may include the effects of pressure gradientsby enabling the Pressure Gradient Effects option under Enhanced Wall Treatment Options.When this option is enabled, FLUENT will include the coefficient α in Equation 12.10-31.


Modeling Turbulence

12.19.10 Including Thermal Effects

If the enhanced wall treatment is used, you may include thermal effects by enablingthe Thermal Effects option under Enhanced Wall Treatment Options. When this optionis enabled, FLUENT will include the coefficient β in Equation 12.10-31. γ will also beincluded in Equation 12.10-31 when the Thermal Effects option is enabled if the ideal gaslaw is selected for the fluid density in the Materials panel.

12.19.11 Including the Wall Reflection Term

If the RSM is used with the default model for pressure strain, FLUENT will, by default,include the wall-reflection effects in the pressure-strain term. That is, FLUENT willcalculate φij,w using Equation 12.7-7 and include it in Equation 12.7-4. Note that wall-reflection effects are not included if you have selected the quadratic pressure-strain model.

i The empirical constants and the function f used in the calculation of φij,ware calibrated for simple canonical flows such as channel flows and flat-plateboundary layers involving a single wall. If the flow involves multiple wallsand the wall has significant curvature (e.g., an axisymmetric pipe or curvi-linear duct), the inclusion of the wall-reflection term in Equation 12.7-7may not improve the accuracy of the RSM predictions. In such cases,you can disable the wall-reflection effects by turning off the Wall ReflectionEffects under Reynolds-Stress Options in the Viscous Model panel.

12.19.12 Solving the k Equation to Obtain Wall Boundary Conditions

In the RSM, FLUENT, by default, uses the explicit setting of boundary conditions forthe Reynolds stresses near the walls, with the values computed with Equation 12.7-34.k is calculated by solving the k equation obtained by summing Equation 12.7-1 fornormal stresses. To disable this option and use the wall boundary conditions given inEquation 12.7-35, turn off Wall B.C. from k Equation under Reynolds-Stress Options in theViscous Model panel. (This option will not appear unless you have activated the RSM.)

12.19.13 Quadratic Pressure-Strain Model

To use the quadratic pressure-strain model described in Section 12.7.4: Quadratic Pressure-Strain Model, turn on the Quadratic Pressure-Strain Model option under Reynolds-StressOptions in the Viscous Model panel. (This option will not appear unless you have activatedthe RSM.) The following options are not available when the Quadratic Pressure-StrainModel is enabled:

• Wall Reflection Effects under Reynolds-Stress Options

• Enhanced Wall Treatment under Near-Wall Treatment



12.19.14 Low-Re Stress-Omega Pressure-Strain

To use the Low-Re Stress-Omega option described in Section 12.7.4: Low-Re Stress-Omega Model, turn on the Low-Re Stress-Omega option under Reynolds-Stress Optionsin the Viscous Model panel. (This option will not appear unless you have activated theRSM.) The following options are not available when the Low-Re Stress-Omega is enabled:

• Wall BC from k Equation under Reynolds-Stress Options

• Quadratic Pressure-Strain Model under Reynolds-Stress Options

• Wall Reflection Effects under Reynolds-Stress Options

• Standard Wall Functions under Near-Wall Treatment

• Non-Equilibrium Wall Functions under Near-Wall Treatment

• Enhanced Wall Treatment under Near-Wall Treatment

Instead, the following options have to be set:

• Transitional Flows under k-omega Options

• Shear Flow Corrections under k-omega Options

12.19.15 Subgrid-Scale Model

If you have selected the Large Eddy Simulation model, you will be able to choose one of thesubgrid-scale models described in Section 12.9.3: Subgrid-Scale Models. You can choosefrom the Smagorinsky-Lilly, WALE, or Kinetic-Energy Transport subgrid-scale models. Notethat the Dynamic Model is an option with the Smagorinsky-Lilly model, while the Kinetic-Energy Transport model is always run as a dynamic model.

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

12.19.16 Customizing the Turbulent Viscosity

If you are using the Spalart-Allmaras, k-ε, k-ω, or LES model, a user-defined functioncan be used to customize the turbulent viscosity. This option will enable you to modifyµt in the case of the Spalart-Allmaras, k-ε, and k-ω models, and incorporate completelynew subgrid models in the case of the LES model. See the separate UDF Manual forinformation about user-defined functions.

In the Viscous Model panel, under User-Defined Functions, select the appropriate user-defined function in the Turbulent Viscosity drop-down list. For the LES model, select theappropriate UDF in the Subgrid-Scale Turbulent Viscosity drop-down list.


Modeling Turbulence

12.19.17 Customizing the Turbulent Prandtl Numbers

If you are using the standard or realizable k-ε model or the standard k-ω model, a user-defined function can be used to customize the turbulent Prandtl numbers. This optionwill allow you to calculate σk and either σε or σω (depending on if you have enabled theappropriate k-ε or k-ω model) by using a UDF. You will also be able to calculate thevalue of the energy Prandtl number (Prt in Equation 12.4-23) and the Prandtl numberat the wall (Prt in Equation 12.10-5) in this way. See the separate UDF Manual forinformation about user-defined functions.

In the Viscous Model panel, under User-Defined Functions, select the appropriate user-defined function from the drop-down lists under Prandtl Numbers. Options include: TKEPrandtl Number, TDR Prandtl Number (k-ε models only), SDR Prandtl Number (k-ω modelonly), Energy Prandtl Number, and Wall Prandtl Number.

12.19.18 Modeling Turbulence with Non-Newtonian Fluids

If the turbulent flow involves non-Newtonian fluids, you can use the define/models/

viscous/turbulence-expert/turb-non-newtonian? text command to enable the se-lection of non-Newtonian options for the material viscosity. See Section 8.4.5: Viscosityfor Non-Newtonian Fluids for details about these options.

12.20 Defining Turbulence Boundary Conditions

12.20.1 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 in addition to other mean solution vari-ables. The boundary conditions for ν at the walls are internally taken care of by FLUENT,which obviates the need for your inputs. The boundary condition input for ν you mustsupply to FLUENT is the one at inlet boundaries (velocity inlet, pressure inlet, etc.). Inmany situations, it is important to specify correct or realistic boundary conditions at theinlets, because the inlet turbulence can significantly affect the downstream flow.

See Section 7.2.2: Determining Turbulence Parameters for details about specifying theboundary condition for ν at the inlets.

You may want to include the effects of the wall roughness on selected wall boundaries.In such cases, you can specify the roughness parameters (roughness height and roughnessconstant) in the panels for the corresponding wall boundaries (see Section 7.13.1: Settingthe Roughness Parameters).



12.20.2 k-ε Models and k-ω Models

When you are modeling turbulent flows in FLUENT using one of the k-ε models or oneof the k-ω models, you must provide the boundary conditions for k and ε (or k and ω)in addition to other mean solution variables. The boundary conditions for k and ε (or kand ω) at the walls are internally taken care of by FLUENT, which obviates the need foryour inputs. The boundary condition inputs for k and ε (or k and ω) you must supplyto FLUENT are the ones at inlet boundaries (velocity inlet, pressure inlet, etc.). In manysituations, it is important to specify correct or realistic boundary conditions at the inlets,because the inlet turbulence can significantly affect the downstream flow.

See Section 7.2.2: Determining Turbulence Parameters for details about specifying theboundary conditions for k and ε (or k and ω) at the inlets.

You may want to include the effects of the wall roughness on selected wall boundaries.In such cases, you can specify the roughness parameters (roughness height and roughnessconstant) in the panels for the corresponding wall boundaries (see Section 7.13.1: Settingthe Roughness Parameters).

Additionally, you can control whether or not to set the turbulent viscosity to zero withina laminar zone. If the fluid zone in question is laminar, the text command define/

boundary-conditions/fluid will contain an option called Set Turbulent Viscosity

to zero within laminar zone?. By setting this option to yes, FLUENT will set boththe production term in the turbulence transport equation and µt to zero. In contrast,when the Laminar Zone option is turned on in a Fluid boundary condition panel, only theproduction term is set to zero. See Section 7.17.1: Specifying a Laminar Zone for detailsabout laminar zones.

i Note that the laminar zone feature is also available for the Spalart-Allmarasand RSM models.


Modeling Turbulence

12.20.3 Reynolds Stress Model

The specification of turbulent boundary conditions for the RSM is the same as for theother turbulence models for all boundaries except at boundaries where flow enters thedomain. Additional input methods are available for these boundaries and are describedhere.

When you choose to use the RSM, the default inlet boundary condition inputs requiredare identical to those required when the k-ε model is active. You can input the tur-bulence quantities using any of the turbulence specification methods described in Sec-tion 7.2.2: Determining Turbulence Parameters. FLUENT then uses the specified tur-bulence quantities to derive the Reynolds stresses at the inlet from the assumption ofisotropy of turbulence:

u′2i =

2

3k (i = 1, 2, 3) (12.20-1)

u′iu′j = 0.0 (12.20-2)

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

condition for ε is determined in the same manner as for the k-ε turbulence models (seeSection 7.2.2: Determining Turbulence Parameters). To use this method, you will selectK or Turbulence Intensity as the Reynolds-Stress Specification Method in the appropriateboundary condition panel.

Alternately, you can directly specify the Reynolds stresses by selecting Reynolds-StressComponents as the Reynolds-Stress Specification Method in the boundary condition panel.When this option is enabled, you should input the Reynolds stresses directly.

You can set the Reynolds stresses by using constant values, profile functions of coordinates(see Section 7.26: Boundary Profiles), or user-defined functions (see the separate UDFManual).



Figure 12.20.1: Specifying Inlet Boundary Conditions for the ReynoldsStresses


Modeling Turbulence

12.20.4 Large Eddy Simulation Model

It is possible to specify the magnitude of random fluctuations of the velocity componentsat an inlet only if the velocity inlet boundary condition is selected. In this case, you mustspecify a Turbulence Intensity that determines the magnitude of the random perturbationson individual mean velocity components as described in Section 12.9.4: Inlet BoundaryConditions for the LES Model. For all boundary types other than velocity inlets, theboundary conditions for LES remain the same as for laminar flows.

12.21 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 convergedsolutions or (for unsteady calculations) the solutions after a sufficiently long time haselapsed should be independent of the initial values for k and ε (or k and ω). For betterconvergence, however, it is beneficial to use a reasonable initial guess for k and ε (or kand ω).

In general, it is recommended that you start from a fully-developed state of turbulence.When you use the enhanced wall treatment for the k-ε models or the RSM, it is criticallyimportant to specify fully-developed turbulence fields. Guidelines are provided below.

• If you were able to specify reasonable boundary conditions at the inlet, it may bea good idea to compute the initial values for k and ε (or k and ω) in the wholedomain from these boundary values. (See Section 25.14: Initializing the Solutionfor details.)

• For more complex flows (e.g., flows with multiple inlets with different conditions) itmay be better to specify the initial values in terms of turbulence intensity. 5–10%is enough to represent fully-developed turbulence. k can then be computed fromthe turbulence intensity 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 eddy viscosity (Cµk2

ε)

is sufficiently large in comparison to the molecular viscosity. In fully-developedturbulence, the turbulent viscosity is roughly two orders of magnitude larger thanthe molecular viscosity. From this, you can compute ε.

Note that, for the RSM, Reynolds stresses are initialized automatically using Equa-tions 12.20-1 and 12.20-2.


12.22 Solution Strategies for Turbulent Flow Simulations


Compared to laminar flows, simulations of turbulent flows are more challenging in manyways. For the Reynolds-averaged approach, additional equations are solved for the tur-bulence quantities. Since the equations for mean quantities and the turbulent quantities(µt, k, ε, ω, or the Reynolds stresses) are strongly coupled in a highly non-linear fashion,it takes more computational effort to obtain a converged turbulent solution than to ob-tain a converged laminar solution. The LES model, while embodying a simpler, algebraicmodel for the subgrid-scale viscosity, requires a transient solution on a very fine mesh.

The fidelity of the results for turbulent flows is largely determined by the turbulencemodel being used. Here are some guidelines that can enhance the quality of your turbulentflow simulations.

12.22.1 Mesh Generation

The following are suggestions to follow when generating the mesh for use in your turbulentflow simulation:

• Picture in your mind the flow under consideration using your physical intuition orany data for a similar flow situation, and identify the main flow features expectedin the flow you want to model. Generate a mesh that can resolve the major featuresthat you expect.

• If the flow is wall-bounded, and the wall is expected to significantly affect the flow,take additional care when generating the mesh. You should avoid using a meshthat is too fine (for the wall function approach) or too coarse (for the enhancedwall treatment approach). See Section 12.11: Grid Considerations for TurbulentFlow Simulations for details.

12.22.2 Accuracy

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

• Use the turbulence model that is better suited for the salient features you expectto see in the flow (see Section 12.2: Choosing a Turbulence Model).

• Because the mean quantities have larger gradients in turbulent flows than in laminarflows, it is recommended that you use high-order schemes for the convection terms.This is especially true if you employ a triangular or tetrahedral mesh. Note thatexcessive numerical diffusion adversely affects the solution accuracy, even with themost elaborate turbulence model.

• In some flow situations involving inlet boundaries, the flow downstream of the inletis dictated by the boundary conditions at the inlet. In such cases, you shouldexercise care to make sure that reasonably realistic boundary values are specified.


Modeling Turbulence

12.22.3 Convergence

The suggestions below are provided to help you enhance convergence for turbulent flowcalculations:

• Starting with excessively crude initial guesses for mean and turbulence quantitiesmay cause the solution to diverge. A safe approach is to start your calculationusing conservative (small) under-relaxation parameters and (for the density-basedsolvers) a conservative Courant number, and increase them gradually as the itera-tions proceed and the solution begins to settle down.

• It is also helpful for faster convergence to start with reasonable initial guesses forthe k and ε (or k and ω) fields. Particularly when the enhanced wall treatmentis used, it is important to start with a sufficiently developed turbulence field, asrecommended in Section 12.21: Providing an Initial Guess for k and ε (or k and ω),to avoid the need for an excessive number of iterations to develop the turbulencefield.

• When you are using the RNG k-ε model, an approach that might help you achievebetter convergence is to obtain a solution with the standard k-εmodel before switch-ing to the RNG model. Due to the additional non-linearities in the RNG model,lower under-relaxation factors and (for the density-based solvers) a lower Courantnumber might also be necessary.

Note that when you use the enhanced wall treatment, you may sometimes find duringthe calculation that the residual for ε is reported to be zero. This happens when yourflow is such that Rey is less than 200 in the entire flow domain, and ε is obtained fromthe algebraic formula (Equation 12.10-25) instead of from its transport equation.

12.22.4 RSM-Specific Solution Strategies

Using the RSM creates a high degree of coupling between the momentum equations andthe turbulent stresses in the flow, and thus the calculation can be more prone to stabilityand convergence difficulties than with the k-ε models. When you use the RSM, therefore,you may need to adopt special solution strategies in order to obtain a converged solution.The following strategies are generally recommended:

• Begin the calculations using the standard k-ε model. Turn on the RSM and usethe k-ε solution data as a starting point for the RSM calculation.

• Use low under-relaxation factors (0.2 to 0.3) and (for the density-based solvers) alow Courant number for highly swirling flows or highly complex flows. In thesecases, you may need to reduce the under-relaxation factors both for the velocitiesand for all of the stresses.



Instructions for setting these solution parameters are provided below. If you are applyingthe RSM to prediction of a highly swirling flow, you will want to consider the solutionstrategies discussed in Section 9.5: Swirling and Rotating Flows as well.

Under-Relaxation of the Reynolds Stresses

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

Solve −→ Controls −→Solution...

The default settings of 0.5 are recommended for most cases. You may be able to increasethese settings and speed up the convergence when the RSM solution begins to converge.

Disabling Calculation Updates of the Reynolds Stresses

In some instances, you may wish to let the current Reynolds stress field remain fixed,skipping the solution of the Reynolds transport equations while solving the other trans-port equations. You can activate/deactivate all Reynolds stress equations in the SolutionControls panel.


Residual Reporting for the RSM

When you use the RSM for turbulence, FLUENT reports the equation residuals for theindividual Reynolds stress transport equations. You can apply the usual convergencecriteria to the Reynolds stress residuals: normalized residuals in the range of 10−3 usu-ally indicate a practically-converged solution. However, you may need to apply tighterconvergence criteria (below 10−4) to ensure full convergence.

12.22.5 LES-Specific Solution Strategies

Large eddy simulation involves running a transient solution from some initial condition,on an appropriately fine grid, using an appropriate time step size. The solution mustbe run long enough to become independent of the initial condition and to enable thestatistics of the flow field to be determined.


Modeling Turbulence

The following are suggestions to follow when running a large eddy simulation:

1. Start by running a steady state flow simulation using a Reynolds-averaged tur-bulence model such as standard k-ε, k-ω, Spalart-Allmaras, or even RSM. Rununtil the flow field is reasonably converged and then use the solve/initialize/

init-instantaneous-vel text command to generate the instantaneous velocityfield out of the steady-state RANS results. This command must be executed be-fore LES is enabled. This option is available for all RANS-based models and it willcreate a much more realistic initial field for the LES run. Additionally, it will helpin reducing the time needed for the LES simulation to reach a statistically stablemode. This step is optional.

2. When you enable LES, FLUENT will automatically turn on the unsteady solveroption and choose the second-order implicit formulation. You will need to setthe appropriate time step size and all the needed solution parameters. (See Sec-tion 25.17.1: User Inputs for Time-Dependent Problems for guidelines on settingsolution parameters for transient calculations in general.) The bounded central-differencing spatial discretization scheme will be automatically enabled for momen-tum equations. Both the bounded central-differencing and pure central-differencingschemes are available for all equations when running LES simulations.

3. Run LES until the flow becomes statistically steady. The best way to see if the flowis fully developed and statistically steady is to monitor forces and solution variables(e.g., velocity components or pressure) at selected locations in the flow.

4. Zero out the initial statistics using the solve/initialize/init-flow-statisticstext command. Before you restart the solution, enable Data Sampling for TimeStatistics in the Iterate panel, as described in Section 25.17.1: User Inputs for Time-Dependent Problems. With this option enabled, FLUENT will gather data fortime statistics while performing a large eddy simulation. You can set the SamplingInterval such that Data Sampling for Time Statistics can be performed at the specifiedfrequency. When Data Sampling for Time Statistics is enabled, the statistics collectedat each sampling interval can be postprocessed and you can then view both themean and the root-mean-square (RMS) values in FLUENT.

5. Continue until you get statistically stable data. The duration of the simulationcan be determined beforehand by estimating the mean flow residence time in thesolution domain (L/U , where L is the characteristic length of the solution domainand U is a characteristic 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 provided below.


12.23 Postprocessing for Turbulent Flows

Temporal Discretization

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

Define −→ Models −→Solver...

Spatial Discretization

Overly diffusive schemes such as the first-order upwind or power law scheme should beavoided, because they may unduly damp out the energy of the resolved eddies. Thecentral-differencing based schemes are recommended for all equations when you use theLES model. FLUENT provides two central-differencing based schemes: pure central-differencing and bounded central-differencing. The bounded scheme is the default optionwhen you select LES or DES.



FLUENT provides postprocessing options for displaying, plotting, and reporting vari-ous turbulence quantities, which include the main solution variables and other auxiliaryquantities.

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

• 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


Modeling Turbulence

• Wall Ystar

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

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



• Specific Dissipation Rate (Omega)

• Production of k






• Wall Ystar

• Wall Yplus

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

• Modified Turbulent Viscosity






• Wall Yplus



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



• 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 DES model are as follows:

• Modified Turbulent Viscosity






Modeling Turbulence


• Wall Yplus

• Relative Length Scale (DES)

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

• Turbulence Kinetic Energy


• Subgrid Kinetic Energy

• Production of k

• Subgrid Turbulent Viscosity

• Subgrid Effective Viscosity

• Subgrid Turbulent Viscosity Ratio

• Subgrid Filter Length

• Subgrid Test-Filter Length

• Subgrid Dissipation Rate

• Subgrid Dynamic Viscosity Constant

• Subgrid Dynamic Prandtl Number

• Subgrid Dynamic Sc of Species

• Subtest Kinetic Energy



• Wall Ystar

• Wall Yplus

All of these variables can be found in the Turbulence... category of the variable selectiondrop-down list that appears in postprocessing panels. See Chapter 30: Field FunctionDefinitions for their definitions.



12.23.1 Custom Field Functions for Turbulence

In addition to the quantities listed above, you can define your own turbulence quantitiesusing the Custom Field Function Calculator panel.

Define −→Custom Field Functions...

The following functions may be useful:

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

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

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

)

12.23.2 Postprocessing Turbulent Flow Statistics

As described in Section 12.9: Large Eddy Simulation (LES) Model Theory, LES involvesthe solution of a transient flow field, but it is the mean flow quantities that are of interestfrom an engineering standpoint.

For all other turbulent flow, if Data Sampling for Time Statistics is enabled in the Iteratepanel, FLUENT gathers data for time statistics while performing the simulation. Thestatistics that FLUENT collects at each sampling interval (which consists of the meanand the root-mean-square (RMS) values) can be postprocessed by selecting UnsteadyStatistics... in any of the postprocessing panels. You can view several variables thatinclude, but are not limited to, shear stresses (Resolved UV/UW/VW Reynolds Stress), flowheat fluxes (Resolved UT/VT/WT Heat Flux), and species statistics (RMS Mass Fractionof species and Mean Mass Fraction of species). If you select Unsteady Wall Statistics...in any of the postprocessing panels, you can view wall statistics such as Mean PressureCoefficient, Mean Wall Shear Stress, Mean X-Wall Shear Stress, Mean Y-Wall Shear Stress,Mean Z-Wall Shear Stress, Mean Skin Friction Coefficient, Mean Surface Heat Flux, MeanSurface Heat Transfer Coef., Mean Surface Nusselt Number, Mean Surface Stanton Number.See Section 25.17.4: Postprocessing for Time-Dependent Problems for details.

i Note that mean statistics are collected only in interior cells and not onwall surfaces. Therefore, when node or cell values of mean quantities areplotted on the wall surface, you are actually plotting values in nearby cellsattached to the wall.


Modeling Turbulence

Figure 12.23.1: Postprocessing Choices for Unsteady Wall Statistics...

There may be cases when you want to control what set of variables are available forpostprocessing. To enable or disable certain variables, use the following text command:

solve −→ set −→data-sampling

The text command prompts you with a set of questions that you will answer yes or noto depending on whether or not you want statistics collected on certain variables. Thefollowing example demonstrates using the text command when do not want to collectstatistics on species:

Data Sampling for Time Statistics? [yes]Sampling interval [2]Collect statistics for flow shear stresses? [yes]Collect statistics for flow heat fluxes? [yes]Collect wall statistics? [yes]Collect statistics for ch3oh? [yes] no

i When including or excluding statistics on variables, it is recommended thatyou re-initialize the flow statistics.



12.23.3 Troubleshooting

You can use the postprocessing options not only for the purpose of interpreting yourresults but also for investigating any anomalies that may appear in the solution. Forinstance, you may want to plot contours of the k field to check if there are any regionswhere k is erroneously large or small. You should see a high k region in the regionwhere the production of k is large. You may want to display the turbulent viscosityratio field in order to see whether or not turbulence takes full effect. Usually turbulentviscosity is at least two orders of magnitude larger than molecular viscosity for fully-developed turbulent flows modeled using the RANS approach (i.e., not using LES). Youmay also want to see whether you are using a proper near-wall mesh for the enhancedwall treatment. In this case, you can display filled contours of Rey (turbulent Reynoldsnumber) overlaid on the mesh.


Modeling Turbulence


Documents

Chp12 RNG K-Epsilon