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Chapter 4.

Turbulence

This chapter provides theoretical background about the turbulence models available in ANSYS FLUENT. Information is presented in the following sections: • Section 4.1: Introduction • Section 4.2: Choosing a Turbulence Model • Section 4.3: Spalart-Allmaras Model • Section 4.4: Standard, RNG, and Realizable k-ǫ Models • Section 4.5: Standard and SST k-ω Models • Section 4.6: k-kl-ω Transition Model • Section 4.7: Transition SST Model • Section 4.8: The v 2 -f Model • Section 4.9: Reynolds Stress Model (RSM) • Section 4.10: Detached Eddy Simulation (DES) • Section 4.11: Large Eddy Simulation (LES) Model • Section 4.12: Near-Wall Treatments for Wall-Bounded Turbulent Flows For more information about using these turbulence models in ANSYS FLUENT, see Chapter 12: Modeling Turbulence in the separate User’s Guide.

4.1 Introduction Turbulent flows are characterized by fluctuating velocity fields. These fluctuations mix transported quantities such as momentum, energy, and species concentration, and cause the transported quantities to fluctuate as well. Since these fluctuations can be of small scale and high frequency, they are too computationally expensive to simulate directly in practical engineering calculations. Instead, the instantaneous (exact) governing equations can be time-averaged, ensemble-averaged, or otherwise manipulated to remove the resolution of small scales, resulting in a modified set of equations that are computationally less expensive to solve. However, the modified equations contain additional unknown

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Turbulence variables, and turbulence models are needed to determine these variables in terms of known quantities. ANSYS FLUENT provides the following choices of turbulence models: • Spalart-Allmaras model • k-ǫ models

– Standard k-ǫ model

– Renormalization-group (RNG) k-ǫ model – Realizable k-ǫ model • k-ω models

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

• Transition k-kl-ω model • Transition SST model • v 2 -f model (add-on) • Reynolds stress models (RSM)

– Linear pressure-strain RSM model – Quadratic pressure-strain RSM model – Low-Re stress-omega RSM model

• Detached eddy simulation (DES) model, which includes one of the following RANS models. – Spalart-Allmaras RANS model – Realizable k-ǫ RANS model – SST k-ω RANS model • Large eddy simulation (LES) model, which includes one of the following sub-scale models. – Smagorinsky-Lilly subgrid-scale model – WALE subgrid-scale model – Dynamic Smagorinsky model – Kinetic-energy transport subgrid-scale model

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4.2 Choosing a Turbulence Model

4.2 Choosing a Turbulence Model It is an unfortunate fact that no single turbulence model is universally accepted as being superior for all classes of problems. The choice of turbulence model will depend on considerations such as the physics encompassed in the flow, the established practice for a specific class of problem, the level of accuracy required, the available computational resources, and the amount of time available for the simulation. To make the most appropriate choice of model for your application, you need to understand the capabilities and limitations of the various options. The purpose of this section is to give an overview of issues related to the turbulence models provided in ANSYS FLUENT. The computational effort and cost in terms of CPU time and memory of the individual models is discussed. While it is impossible to state categorically which model is best for a specific application, general guidelines are presented to help you choose the appropriate turbulence model for the flow you want to model. Information is presented in the following sections: • Section 4.2.1: Reynolds-Averaged Approach vs. LES • Section 4.2.2: Reynolds (Ensemble) Averaging • Section 4.2.3: Boussinesq Approach vs. Reynolds Stress Transport Models

4.2.1 Reynolds-Averaged Approach vs. LES Time-dependent solutions of the Navier-Stokes equations for high Reynolds-number turbulent flows in complex geometries which set out to resolve all the way down to the smallest scales of the motions are unlikely to be attainable for some time to come. Two alternative methods can be employed to render the Navier-Stokes equations tractable so that the small-scale turbulent fluctuations do not have to be directly simulated: Reynolds-averaging (or ensemble-averaging) and filtering. Both methods introduce additional 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 averaged flow quantities, with the whole range of the scales of turbulence being modeled. The RANS-based modeling approach therefore greatly reduces the required computational effort and resources, and is widely adopted for practical engineering applications. An entire hierarchy of closure models are available in ANSYS FLUENT including SpalartAllmaras, k-ǫ and its variants, k-ω and its variants, and the RSM. The RANS equations are often used 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).

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Turbulence LES provides an alternative approach in which large eddies are explicitly computed (resolved) in a time-dependent simulation using the “filtered” Navier-Stokes equations. The rationale behind LES is that by modeling less of turbulence (and resolving more), the error introduced by turbulence modeling can be reduced. It is also believed to be easier to find a “universal” model for the small scales, since they tend to be more isotropic and less affected by the macroscopic features like boundary conditions, than the large eddies. Filtering is essentially a mathematical manipulation of the exact Navier-Stokes equations to remove the eddies that are smaller than the size of the filter, which is usually taken as the mesh size when spatial filtering is employed as in ANSYS FLUENT. Like Reynoldsaveraging, the filtering process creates additional unknown terms that must be modeled to achieve closure. Statistics of the time-varying flow-fields such as time-averages and r.m.s. values of the solution variables, which are generally of most engineering interest, can be collected during the time-dependent simulation. LES for high Reynolds number industrial flows requires a significant amount of computational resources. This is mainly because of the need to accurately resolve the energycontaining turbulent eddies in both space and time domains, which becomes most acute in near-wall regions where the scales to be resolved become much smaller. Wall functions in combination with a coarse near wall mesh can be employed, often with some success, to reduce the cost of LES for wall-bounded flows. However, one needs to carefully consider the ramification of using wall functions for the flow in question. For the same reason (to accurately resolve the eddies), LES also requires highly accurate spatial and temporal discretizations.

4.2.2 Reynolds (Ensemble) Averaging In Reynolds averaging, the solution variables in the instantaneous (exact) Navier-Stokes equations are decomposed into the mean (ensemble-averaged or time-averaged) and fluctuating components. For the velocity components: ui = u¯i + ui′

(4.2-1)

where u¯i and u′i are the mean and fluctuating velocity components (i = 1, 2, 3). Likewise, for pressure and other scalar quantities: φ = φ¯ + φ′

(4.2-2)

where φ denotes a scalar such as pressure, energy, or species concentration. Substituting expressions of this form for the flow variables into the instantaneous continuity 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:

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4.2 Choosing a Turbulence Model

∂ ∂ρ + (ρui ) = 0 ∂xi ∂t "

∂p ∂ ∂ui ∂uj 2 ∂ul ∂ ∂ (ρui uj ) = − + + µ (ρui )+ − δij ∂xj ∂xi ∂xj ∂xj ∂t ∂xi 3 ∂xl

(4.2-3)

!#

+

∂ (−ρu′i uj′ ) (4.2-4) ∂xj

Equations 4.2-3 and 4.2-4 are called Reynolds-averaged Navier-Stokes (RANS) equations. They have the same general form as the instantaneous Navier-Stokes equations, with the velocities and other solution variables now representing ensemble-averaged (or timeaveraged) values. Additional terms now appear that represent the effects of turbulence. These Reynolds stresses, −ρu′i uj′ , must be modeled in order to close Equation 4.2-4. For variable-density flows, Equations 4.2-3 and 4.2-4 can be interpreted as Favre-averaged Navier-Stokes equations [130], with the velocities representing mass-averaged values. As such, Equations 4.2-3 and 4.2-4 can be applied to density-varying flows.

4.2.3 Boussinesq Approach vs. Reynolds Stress Transport Models The Reynolds-averaged approach to turbulence modeling requires that the Reynolds stresses in Equation 4.2-4 are appropriately modeled. A common method employs the Boussinesq hypothesis [130] to relate the Reynolds stresses to the mean velocity gradients: −

ρu′i uj′

= µt

∂ui ∂uj + ∂xi ∂xj

!

!

2 ∂u − ρk + µt k δij 3 ∂xk

(4.2-5)

The Boussinesq hypothesis is used in the Spalart-Allmaras model, the k-ǫ models, and the k-ω models. The advantage of this approach is the relatively low computational cost associated with the computation of the turbulent viscosity, µt . In the case of the Spalart-Allmaras model, only one additional transport equation (representing turbulent viscosity) is solved. In the case of the k-ǫ and k-ω models, two additional transport equations (for the turbulence kinetic energy, k, and either the turbulence dissipation rate, ǫ, or the specific dissipation rate, ω) are solved, and µt is computed as a function of k and ǫ or k and ω. The disadvantage of the Boussinesq hypothesis as presented is that it assumes µt is an isotropic scalar quantity, which is not strictly true. The alternative approach, embodied in the RSM, is to solve transport equations for each of the terms in the Reynolds stress tensor. An additional scale-determining equation (normally for ǫ) is also required. This means that five additional transport equations are required in 2D flows and seven additional transport equations must be solved in 3D.

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

4.3 Spalart-Allmaras Model This section describes the theory behind the Spalart-Allmaras model. Information is presented in the following sections: • Section 4.3.1: Overview • Section 4.3.2: Transport Equation for the Spalart-Allmaras Model • Section 4.3.3: Modeling the Turbulent Viscosity • Section 4.3.4: Modeling the Turbulent Production • Section 4.3.5: Modeling the Turbulent Destruction • Section 4.3.6: Model Constants • Section 4.3.7: Wall Boundary Conditions • Section 4.3.8: Convective Heat and Mass Transfer Modeling For details about using the model in ANSYS FLUENT, see Chapter 12: Modeling Turbulence and Section 12.5: Setting Up the Spalart-Allmaras Model in the separate User’s Guide.

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

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4.3 Spalart-Allmaras Model In its original form, the Spalart-Allmaras model is effectively a low-Reynolds-number model, requiring the viscosity-affected region of the boundary layer to be properly resolved. In ANSYS FLUENT, however, the Spalart-Allmaras model has been implemented to use wall functions when the mesh resolution is not sufficiently fine. This might make it the best choice for relatively crude simulations on coarse meshes where accurate turbulent flow computations are not critical. Furthermore, the near-wall gradients of the transported variable in the model are much smaller than the gradients of the transported variables in the k-ǫ or k-ω models. This might make the model less sensitive to numerical errors when non-layered meshes are used near walls. See Section 6.1.3: Numerical Diffusion in the separate User’s Guide for a further discussion of the numerical errors. On a cautionary note, however, the Spalart-Allmaras model is still relatively new, and no claim is made regarding its suitability to all types of complex engineering flows. For instance, it cannot be relied on to predict the decay of homogeneous, isotropic turbulence. Furthermore, one-equation models are often criticized for their inability to rapidly accommodate changes in length scale, such as might be necessary when the flow changes abruptly from a wall-bounded to a free shear flow. In turbulence models that employ the Boussinesq approach, the central issue is how the eddy viscosity is computed. The model proposed by Spalart and Allmaras [331] solves a transport equation for a quantity that is a modified form of the turbulent kinematic viscosity.

4.3.2 Transport Equation for the Spalart-Allmaras Model e is identical to the turbulent The transported variable in the Spalart-Allmaras model, ν, kinematic viscosity except in the near-wall (viscosity-affected) region. The transport equation for νe is 

∂ ∂ 1 ∂ e + (ρν) (ρνe ui ) = Gν +  ∂t σνe ∂xj ∂xi

(

∂νe e (µ + ρν) ∂xj

)

∂νe + Cb2 ρ ∂xj

!2   − Yν + S

ν e

(4.3-1)

where Gν is the production of turbulent viscosity, and Yν is the destruction of turbulent viscosity that occurs in the near-wall region due to wall blocking and viscous damping. σνe and Cb2 are the constants and ν is the molecular kinematic viscosity. Sνe is a userdefined source term. Note that since the turbulence kinetic energy, k, is not calculated in the Spalart-Allmaras model, while the last term in Equation 4.2-5 is ignored when estimating the Reynolds stresses.

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Turbulence

4.3.3 Modeling the Turbulent Viscosity The turbulent viscosity, µt , is computed from e v1 µt = ρνf

(4.3-2)

where the viscous damping function, fv1 , is given by fv1 =

χ3 3 χ3 + Cv1

(4.3-3)

and χ≡

νe ν

(4.3-4)

4.3.4 Modeling the Turbulent Production The production term, Gν , is modeled as

where

and

Gν = Cb1 ρSeνe

(4.3-5)

Se ≡ S +

νe fv2 κ2 d2

(4.3-6)

fv2 = 1 −

χ 1 + χfv1

(4.3-7)

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

q

2Ωij Ωij

(4.3-8)

where Ωij is the mean rate-of-rotation tensor and is defined by ∂uj 1 ∂ui − Ωij = ∂x 2 ∂xi j

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!

(4.3-9)

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4.3 Spalart-Allmaras Model The justification for the default expression for S is that, in the wall-bounded flows that were of most interest when the model was formulated, the turbulence production found only where vorticity is generated near walls. However, it has since been acknowledged that one should also take into account the effect of mean strain on the turbulence production, and a modification to the model has been proposed [65] and incorporated into ANSYS FLUENT. This modification combines the measures of both vorticity and the strain tensors in the definition of S : S ≡ |Ωij | + Cprod min (0, |Sij | − |Ωij |)

(4.3-10)

where Cprod = 2.0, |Ωij | ≡

q

2Ωij Ωij , |Sij | ≡

q

2Sij Sij

with the mean strain rate, Sij , defined as 1 Sij = 2

∂uj ∂ui + ∂xi ∂xj

!

(4.3-11)

Including both the rotation and strain tensors reduces the production of eddy viscosity and consequently reduces the eddy viscosity itself in regions where the measure of vorticity exceeds that of strain rate. One such example can be found in vortical flows, i.e., flow near the core of a vortex subjected to a pure rotation where turbulence is known to be suppressed. Including both the rotation and strain tensors more correctly accounts for the effects of rotation on turbulence. The default option (including the rotation tensor only) tends to overpredict the production of eddy viscosity and hence overpredicts the eddy viscosity itself in certain circumstances. You can select the modified form for calculating production in the Viscous Model dialog box.

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Turbulence

4.3.5 Modeling the Turbulent Destruction The destruction term is modeled as Yν = Cw1 ρfw

 2 νe

(4.3-12)

d

where "

6 1 + Cw3 fw = g 6 g + C 6w3



#1/6

(4.3-13)



(4.3-14)

g = r + Cw2 r 6 − r νe r≡ e 2 2 Sκ d

(4.3-15)

Cw1 , Cw2 , and Cw3 are constants, and Se is given by Equation 4.3-6. Note that the modification described above to include the effects of mean strain on S will also affect the value of Se used to compute r.

4.3.6 Model Constants

The model constants Cb1 , Cb2 , σνe, Cv1 , Cw1 , Cw2 , Cw3 , and κ have the following default values [331]: 2 Cb1 = 0.1355, Cb2 = 0.622, σeν = , Cv1 = 7.1 3 Cw1 =

Cb1 (1 + Cb2 ) + , Cw2 = 0.3, Cw3 = 2.0, κ = 0.4187 σνe κ2

4.3.7 Wall Boundary Conditions

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

When the mesh is fine enough to resolve the viscosity-dominated sublayer, the wall shear stress is obtained from the laminar stress-strain relationship: u ρuτ y = µ uτ

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(4.3-16)

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4.4 Standard, RNG, and Realizable k-ǫ Models If the mesh is too coarse to resolve the viscous sublayer, then it is assumed that the centroid of the wall-adjacent cell falls within the logarithmic region of the boundary layer, and the law-of-the-wall is employed: ρuτ y u 1 = ln E κ µ uτ

!

(4.3-17)

where u is the velocity parallel to the wall, uτ is the shear velocity, y is the distance from the wall, κ is the von K´ arm´ an constant (0.4187), and E = 9.793.

4.3.8 Convective H...