19th Computational Fluid Dynamics Conference, 22-25 June, San Antonio, Texas

Prediction of Separation-induced Transition using a

Correlation-based Transition Model

Roque Corral∗ and Fernando Gisbert

Industria de Turbopropulsores, Madrid, Spain 28830

A correlation-based transition model has been introduced in a RANS solver to improve

the prediction of the transition from laminar to turbulent ow regime in low-pressure

turbine blades. The model has been validated by comparing the numerical results against

a propietary experimental data base . The transition model correctly predicts the transition

process due to the separation of the laminar boundary layer, improving the sensitivity of

the RANS solver to variations in the Reynolds number.

I. Introduction

The design of modern low-pressure turbines tends to reduce the number of blades of each row to reduceboth, the cost and weight of the resulting turbine without incurring in an excessive penalty in performances.As a consequence the loading of each blade is increased. The resulting designs are often characterized bya higher velocity peak in the suction side, followed by a ow deceleration. This in turn causes a strongadverse pressure gradient that may lead to the laminar separation of the boundary layer , especially ifthe Reynolds number is low enough, as it is usually the case during the low-pressure turbine operationin altitude conditions. The presence of the separation bubble strongly inuences the total pressure lossesand the outlet angle downstream the blade row and therefore an accurate prediction of this phenomenon isessential. However, standard turbulence models are not designed taking these phenomena into account, andsome additional extension must be used.

Some authors1 modify the turbulence models to change the damping of the turbulent variables in theviscous sub-layer, hence delaying the onset of the turbulence. However, these modications usually misrep-resent the physics of the transition process for low-pressure turbine blades, that has little to do with thedamping of the turbulent oscillations in the viscous sub-layer, being rather related with the interaction ofthe core ow perturbations (high turbulence levels, potential eects from adjacent rows) with the boundarylayer. Other approaches try to model the way these disturbances aect the laminar boundary layer to appro-priately model the mechanism of bypass transition to turbulence.24 And nally a third approach consistsin using available experimental data to predict the onset of transition.57

In this paper, this latter approach has been chosen to model the transition to turbulence produced bythe separation of the laminar boundary layer in the blade suction side of a low-pressure turbine airfoil at lowReynolds numbers. The model chosen to predict the transition onset is the γ −Reθt framework , developedby Menter et al.5 This model uses two transport equations, one for the intermittency, used to trigger thetransition to turbulence, and another one for the Reθ at the transition point, Reθt. The latter equationis used to diuse into the boundary layer the values of Reθt obtained from an experimental correlation,that usually links the onset of the transition with some outer ow variables, such as the pressure gradientor the freestream turbulence intensity. Unlike other correlation-based transition models, that use non-localvariables such as the shape factor along the boundary layer to determine when the transition is produced,8

the model by Menter et al. only uses local information to determine the onset of transition. This is especiallyconvenient when unstructured grids are used or when three-dimensional geometries are considered, since thedetermination of the boundary layer integral quantities is not always obvious.

∗Head of Technology and Methods Dept., also associate proesor at the School of Aeronautics, UPM, MadridCopyright c© 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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American Institute of Aeronautics and Astronautics Paper 2009-3666

19th AIAA Computational Fluid Dynamics22 - 25 June 2009, San Antonio, Texas

AIAA 2009-3666

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

This model has been implemented in a RANS solver, known as Mu2s2T .9 It uses hybrid unstructuredgrids to discretize the spatial domain and an edge-based data structure to compute the uxes. A second-order MUSCL numerical scheme10 conforms the spatial discretization scheme, which is marched in time withan explicit ve-stage Runge-Kutta.11 Low-Mach number block Jacobi preconditioning12 and multigrid13 areused to accelerate steady state convergence. The dual time stepping technique is used to further improvethe convergence of the solver. Turbulence eects are modeled using the k − ω model.1 The code has beenparallelized using MPI.

The RANS solver together with the γ − Reθt model has been used to enable the prediction of theseparation-induced transition in low-pressure turbine blades. It will be shown that the comparisons betweenthe simulations and the experimental data are satisfactory for a wide range of Reynolds numbers once thecorrelation systems is properly calibrated.

II. Model

Even though the γ − Reθt model has been previously reported by Menter et al.5 we reproduce herethe equations that conform it to clarify the discussion of the relevant parameters. As mentioned in theintroduction, the model has two additional transport-diusion equations, one for the intermittency, γ, andanother for the transition Reynolds number based on the momentum thickness, Reθt. These are:

∂

∂t(ργ) +

∂

∂xi(ρUiγ) =

∂

∂xi

[(µ+

µtσf

)∂γ

∂xi

]+

+ (1− ce1γ)FlengthρS (γFonset)ca1 +

+ (1− ce2γ) ca2ρΩγFturb (1)

∂

∂t

(ρReθt

)+

∂

∂xi

(ρUiReθt

)=

∂

∂xi

[σθt (µ+ µt)

∂Reθt∂xi

]+

+ cθt(ρU)2

500µ

(Reθt − Reθt

)(1− Fθt) (2)

The boundary conditions for the model are

γ = 1Reθt = Reθt

for inlet boundaries and

∂γ

∂n= 0

∂Reθt∂n

= 0

at walls.In the γ equation the following functions are dened:

Fturb = e−

“RT4

”4

Fonset = max (0, Fonset2 − Fonset3)

being

RT =ρk

µω

Fonset3 = max

[0, 1−

(RT2.5

)3]

Fonset2 = min[2,max

(Fonset1, F

4onset1

)]Fonset1 =

ReS2.193Reθc

ReS =ρy2S

µ

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American Institute of Aeronautics and Astronautics Paper 2009-3666

where y expresses the distance to the wall. The value S of the ReS expression stands for the absolute valueof the strain rate tensor, S =

√2SijSij . The coecients used to tune the intermittency equation are:

σf = 1ce1 = 1ce2 = 50ca1 = 0.5ca2 = 0.03

In the Reθt equation the function Fθt is dened as follows:

Fθt = min

max

[e−(Reω105 )2−( yδ )4

, 1−(γ − 1/ce21− 1/ce2

)2], 1

(3)

where

Reω =ρy2ω

µ

δ =375ΩyU

θBL

θBL =µReθtρU

(4)

Ω is the absolute value of the vorticity tensor, Ω =√

2ΩijΩij , and the coecients of this equation are

σθt = 10cθt = 0.03

There are two parameters in the γ equation, Flength and Reθc, that control the length and position of the

transition, that are functions of Reθt. These functions are not specied in the paper by Menter et al., andmust be dened to couple the γ and Reθt equations.

These equations provide an appropriate framework to model the boundary layer transition in attachedows. However, when the laminar boundary layer separates, it is observed that the model overpredicts theseparation length, especially for low freestream turbulence levels. To predict the separation length correctly,the model increases the production of turbulent kinetic energy k, hence moving the reattachment pointforward. The parameter used to control the overproduction of k is Fonset1, since it has a value signicantlylarger than 1 near a laminar boundary layer separation.5 Thus, an intermittency value at separation γsep isdened as

γsep = Fθt ·min[5, s1 · e

−“RT15

”4

·max(

0,ReS

2.193Reθc− 1)]

(5)

The parameter s1 controls the length of the separation.The model is easily coupled with the k − ω turbulence model equations. Thus, if the equations are

schematically written as

∂

∂t(ρk) +

∂

∂xi(ρUik) = Pk −Dk +

∂

∂xi

[(µ+ σ∗

ρk

ω

)∂k

∂xi

]∂

∂t(ρω) +

∂

∂xi(ρUiω) = Pω −Dω +

∂

∂xi

[(µ+ σ

ρk

ω

)∂ω

∂xi

]where the terms Pk, Dk, Pω and Dω represent the production and destruction terms of the k and ω equationsrespectively, the modications of the γ −Reθt model consist in modifying Pk and Dk,

Pk = γeffPk (6)

Dk = min [1,max (0.1, γeff )]Dk (7)

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American Institute of Aeronautics and Astronautics Paper 2009-3666

being γeff = max (γ, γsep). Thus, when γeff < 1 the turbulent kinetic energy production term is inhibited,hence avoiding the boundary layer transition. When γeff > 1 the modied production term exceeds theoriginal, but the modied destruction term does not. This overproduction of k is the responsible of theearlier reattachment in separated-induced transition.

(a) Contours of Reθt (b) Contours of γeff

Figure 1. Details of the γ −Reθt model solution for a linear cascade

The model presented serves as a general framework to introduce any correlation-based transition model.We have used the correlation proposed by Langtry,14 which agrees well with the available experimental data.The correlation provides a relationship of the type Reθt = F (Tu, λθ), being Tu the turbulence intensity andλθ the Thwaites pressure gradient parameter. By using this correlation we compute a value of Reθt for eachgrid point. Then the freestream values are diused into the boundary layer by the convection and diusionterms of equation (2). The bounds of the boundary layer are delimited by the Fθt function written inequation (3), which has a value of 1 inside the boundary layer and 0 in the freestream. Thus, the production

term of equation (2) is enabled just in the freestream, hence making Reθt = Reθt there. Inside the boundarylayer, the production term is disabled, and the freestream values of Reθt are diused and lagged due to theconvection-diusion terms. Once solved, the Reθt equation assigns each local grid point a value Reθt, as it isdepicted in gure 1a. These values are then used in equations (1) and (5) to predict a γeff value (gure 1b)that will modify the production and destruction terms of the turbulent kinetic energy equation according to(6) and (7). The coupling between the equations is based upon the dependence of Reθc, Flength and s1 on

Reθt. In the next section we detail these dependencies and describe , its sensitivity to the predicted ow-eld.

III. Model Validation

The inuence of Reθc, Flength and s1 in the predictions is discussed by comparing them to the measure-ments performed in a linear cascade geometry representative of a low-pressure turbine. The main character-istics of this cascade, whose geometry has been depicted in gure 1, are specied in table 1. For the range ofReynolds numbers where the measurements are performed, the cascade has a suction side laminar boundarylayer that separates, being that separation the responsible of the transition to turbulence. Hence the correctprediction of the separation has much to do with the ne adjustment of the Reθt dependencies in Eq. (5).

The cascade has been meshed with a very ne two-dimensional unstructured grid of 78000 points, depictedin gure 2. By using such a ne grid we intended to correctly solve the blade boundary layer (ensuringy+ ' 0.1) and to avoid undesired eects of the γ − Reθt model caused by a lack of grid resolution, such asthe upstream displacement of the reattachment point when the mesh is not ne enough, which have beenreported by Menter et al.5 From the numerical point of view, it is mandatory to use the articial viscosity

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American Institute of Aeronautics and Astronautics Paper 2009-3666

Figure 2. Detail of the linear cascade grid

low Mach number modications15 to avoid an excess of damping in the entropy and vorticity waves thatlead to numerical errors that prevent us from obtaining the correct predictions.

Pitch to Chord ratio 0.8447

Stagger Angle 29

Aspect ratio 3

Inlet Angle 31.5

Design Re2is 3 · 105

Mach number Incompressible

Table 1. Main Characteristics of the linear cascade of gure 1

Inuence of Reθc and s1

Piotrowski et al.16 propose a relation between Reθc and Reθt, which is obtained when trying to reproducethe results of a series of experiments in at plate steady state congurations. Even though they match quitewell their simulation results with the experiments, the proposed correlation is not entirely local, since itmakes use of the Reθtmax along the blade wall. This value is easily obtained, even for unstructured grids,but it poses some problems especially in three-dimensional congurations where the ow variables changesubstantially in spanwise direction. Therefore a new correlation has been used, which is based only in localvalues of Reθt. Besides, it has been observed, especially for large bubbles, that just by varying the value ofReθc the transition length is underpredicted if s1 is kept constant, as it is done in the paper by Menter etal.5 Therefore, we have also established a dependence of s1 upon Reθt.

To obtain the previous correlations, a number of factors have been taken into account, given that matchingall the ow variables is rather dicult. First, we want to match the position of the separation bubble inthe experiment and its length in order to obtain the correct pressure distribution along the blade. Specialattention is also paid to the predicted averaged values downstream the blade, since the γ − Reθt model isintended to improve the prediction capabilities of the multistage simulations performed with Mu2s2T . Asthe averaged values are required to compute the boundary conditions at the interfaces between adjacentrows, there is great interest in predicting these values correctly.

Figure 3 shows the comparisons between the predictions made by the k−ω model alone or in conjunctionwith the γ−Reθt model and the measurements for the linear cascade presented at table 1 for various Reynoldsnumbers. Three curves are shown for each Reynolds number: the Cp distribution along the blade and thetotal pressure loss and swirl angle distributions at a distance x/Cax = 1.5 measured from the blade leading

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American Institute of Aeronautics and Astronautics Paper 2009-3666

0 0.2 0.4 0.6 0.8 1x/C

ax

0

0.5

1

1.5

Cp Experiment

κ−ω + γ − Reθκ−ω

0 0.5 1 1.5 2 2.5 3ζ/ζ

ref

-50

-25

0

25

50

%pitch

Experimentκ−ω + γ−Reθtκ−ω

(a) Re2is = 3 · 105

-64 -63.5 -63 -62.5 -62α

-50

-25

0

25

50

%pitch

Experimentκ−ω + γ−Reθtκ−ω

0 0.2 0.4 0.6 0.8 1x/C

ax

0

0.5

1

1.5

Cp Experiment

κ−ω + γ − Reθκ−ω

0 0.5 1 1.5 2 2.5 3ζ/ζ

ref

-50

-25

0

25

50

%pitch

Experimentκ−ω + γ−Reθtκ−ω

(b) Re2is = 2.6 · 105

-64 -63.5 -63 -62.5 -62α

-50

-25

0

25

50

%pitch

Experimentκ−ω + γ−Reθtκ−ω

0 0.2 0.4 0.6 0.8 1x/C

ax

0

0.5

1

1.5

Cp Experiment

κ−ω + γ − Reθκ−ω

0 0.5 1 1.5 2 2.5 3ζ/ζ

ref

-50

-25

0

25

50

%pitch

Experimentκ−ω + γ−Reθtκ−ω

(c) Re2is = 2 · 105

-64 -63.5 -63 -62.5 -62α

-50

-25

0

25

50

%pitch

Experimentκ−ω + γ−Reθtκ−ω

0 0.2 0.4 0.6 0.8 1x/C

ax

0

0.5

1

1.5

Cp Experiment

κ−ω + γ − Reθκ−ω

0 0.5 1 1.5 2 2.5 3ζ/ζ

ref

-50

-25

0

25

50

%pitch

Experimentκ−ω + γ−Reθtκ−ω

(d) Re2is = 1.6 · 105

-64 -63.5 -63 -62.5 -62α

-50

-25

0

25

50

%pitch

Experimentκ−ω + γ−Reθtκ−ω

0 0.2 0.4 0.6 0.8 1x/C

ax

0

0.5

1

1.5

Cp Experiment

κ−ω + γ − Reθκ−ω

0 0.5 1 1.5 2 2.5 3ζ/ζ

ref

-50

-25

0

25

50

%pitch

Experimentκ−ω + γ−Reθtκ−ω

(e) Re2is = 1.24 · 105

-64 -63.5 -63 -62.5 -62α

-50

-25

0

25

50

%pitch

Experimentκ−ω + γ−Reθtκ−ω

Figure 3. Comparison of the predicted and measured Cp distributions (left), losses distribution (middle) andswirl angle distribution (right) at x/Cax = 1.5 for the linear cascade at dierent Reynolds numbers

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American Institute of Aeronautics and Astronautics Paper 2009-3666

edge. It is observed that the γ−Reθt model appropriately captures the changes in the Cp distributions due tothe separation bubbles, that are not predicted by the k−ω model alone. The length of the separation bubblesis slightly underpredicted for all cases, but it is especially noticeable for the lowest Reynolds numbers. Theimprovement of the Cp distribution is directly translated into an improvement in the downstream swirl angledistribution. Thus, the matching between the experiments and the γ−Reθt model is very good for the caseswith highest Reynolds numbers (Re2is = 3 · 105, 2.6 · 105 and 2 · 105), while the dierences are larger for thecases Re2is = 1.6 · 105 and 1.24 · 105. Nevertheless, the dierences are less than 1 at the peak value. Whenthe γ −Reθt model is not used, the discrepancies in the swirl angle predictions are much larger. The largestdierences between the γ −Reθt model and the experiments are seen in the downstream total pressure lossdistribution. For the highest Reynolds number, the matching is satisfactory, but as the Reynolds numberdecreases, the γ − Reθt model does not capture the total pressure loss distribution appropriately. Whilethe experiments show that the amplitude of the peak of losses diminishes and the distribution widens, theγ − Reθt model predicts an increase of the peak value, and just a weak increase of the wake width, that isunderpredicted for the whole range of Reynolds numbers. This mismatch has also been reported by otherauthors7 and it is believed to be produced due to the incorrect modeling of the pretransitional behavior ofthe laminar boundary layer. Other models, such as the k − kl − ω model of Mayle and Schulz,2 comparebetter with the experimental losses distribution.17 However, the averaged value is quite well approximated,as it is shown in gure 4, where it is seen that the dependence of the mass-averaged total pressure losseswith the Reynolds number is well captured by the γ − Reθt model. When it is not used, the total pressureloss peak is much deeper for all Reynolds numbers, and the averaged distribution hardly reproduces thevariation with this parameter.

100 150 200 250 300Re x 10

-3

ζ/ζref

Experimentκ−ω + γ−Reθtκ−ω

0.25%

Figure 4. Variation of the linear cascade mass-averaged total pressure loss with the Reynolds number atx/Cax = 1.5

Figure 5 shows the changes in the Cp curves when the Reθc dependence upon Reθt is shifted a constantvalue. For both high and low Reynolds numbers, the reattachment is moved downstream when Reθc isincreased and upstream when it is reduced. It is also observed that by reducing the Reθc the matchingbetween the measured and predicted Cp curves is improved. This is better seen in the Re2is = 1.24 ·105 case.This has also an eect in the downstream swirl angle distribution, as showed in gure 6, where it is seen thatthe case with a -100 shift in the Reθc value is the one that best approaches the measurements. Nevertheless,when looking at the averaged downstream total pressure losses (gure 7), we see that the proposed Reθcrelation is the one that best ts the experimental data for the lowest Reynolds numbers.

The trends with s1 are similar to those showed for Reθc. Thus, when s1 is increased, the reattachment ismoved upstream and the total pressure losses decrease. When s1 decreases, the opposite trends are obtained.

Inuence of Flength

The value of Flength has been initially set following the criteria of Piotrowski et al.16 These authors use values

of Flength ranging from 0.5 to 1.75, depending on the value of Reθtmax along the blade wall. However, wehave varied Flength from 0.5 to 50 and the dierences in the predictions are insignicant, as shown in gure8, where the Cp of the linear cascade for the design Reynolds number is depicted. This is an expected result,since there is no dependence on Flength in equation (5), which is the equation that controls the separatedboundary layer transition. A constant value Flength = 1.25 has been nally used throughout this work.

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American Institute of Aeronautics and Astronautics Paper 2009-3666

0.7 0.8 0.9 1x/C

ax

1

1.1

1.2

1.3

1.4

1.5

Cp Experiment

ReθcReθc

- 100Reθc

+ 100

0.7 0.8 0.9 1x/C

ax

1

1.1

1.2

1.3

1.4

1.5

Cp Experiment

ReθcReθc

- 100Reθc

+ 100

Figure 5. Inuence of Reθc in the Cp distribution of the linear cascade. Left: Re2is = 3·105 Right: Re2is = 1.24·105

-64 -63.5 -63 -62.5 -62α

-50

-25

0

25

50

%pitch

ExperimentReθcReθc

- 100Reθc

+ 100

-64 -63.5 -63 -62.5 -62α

-50

-25

0

25

50

%pitch

ExperimentReθcReθc

- 100Reθc

+ 100

Figure 6. Inuence of Reθc in the swirl angle distribution of the linear cascade at x/Cax = 1.5. Left: Re2is = 3·105

Right: Re2is = 1.24 · 105

100 150 200 250 300Re x 10

-3

ζ/ζref

ExperimentReθcReθc

- 100Reθc

+ 100

0.5%

Figure 7. Inuence of Reθc in the variation of the linear cascade mass-averaged total pressure loss with theReynolds number at x/Cax = 1.5

0.7 0.8 0.9 1x/C

ax

1

1.1

1.2

1.3

1.4

1.5

Cp Experiment

Flength

=0.5

Flength

=10

Flength

=50

Figure 8. Inuence of Flength in the Cp distribution of the linear cascade. Re2is = 3 · 105

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American Institute of Aeronautics and Astronautics Paper 2009-3666

IV. Steady three-dimensional results

After the γ − Reθt model has been ne-tuned, it has been applied to predict the behavior of a 3D low-pressure turbine vane when the Reynolds number is changed. The 3D domain has been meshed with asemi-unstructured grid18 with 2.5 million grid points. A detail of the midspan mesh is depicted in gure 9.It is not as ne as the one used in the linear cascade, since using an equivalent grid would result prohibitivein terms of CPU cost, but the resolution is believed to be high enough to obtain accurate predictions. Theturbine vane has an aspect ratio Λ = 3.7, the design Reynolds number is Re2is = 2 · 105 and the exit Machnumber M2is = 0.6.

Figure 9. Detail of the low-pressure turbine vane semi-unstructured grid at midspan

Figure 10 shows the comparisons between the measurements and the γ −Reθt model for three Reynoldsnumbers at a measurement plane located at x/Cax = 1.6 from the midspan leading edge. On one hand,all results show a good agreement between the measured and predicted eciency at midspan, while theresults with the k − ω model alone do not capture the variation of the eciency with the Reynolds numberappropriately. The secondary ow zone is not reproduced with the same degree of accuracy. The correctshape of the secondary losses is not captured by the simulations, especially for the cases with highest Reynoldsnumbers, that have a larger two-dimensional ow zone. This mismatch is believed to be produced due togeometric features that have not been appropriately taken into account. Besides, the hub and tip peaks oflosses are deeper for both k − ω and k − ω + γ − Reθt simulations. On the other hand, the predicted swirlangle shows a very good agreement between the experimental data and the simulations for all Reynoldsnumbers The ow underturning due to the secondary vortices is well captured by the solver, even thoughthere is a slight mismatch in the position of the peaks. The dierences between the measured and predictedswirl angle at midspan are not larger than 1 in any case.

V. Conclusions

The γ−Reθt model has been used to predict the transition to turbulent of a separated laminar boundarylayer. The model has been integrated in an in-house RANS solver for unstructured grids. The link betweenthe γ and Reθt equations is performed by specifying the dependence of Reθc and s1 upon the local value ofReθt. It has been shown that the proposed relations yield a good agreement between the model predictionsand the linear cascade measurements that served as an initial test case. If the γ − Reθt model is not used,the k − ω equations cannot reproduce correctly the variation of the total pressure losses with the Reynoldsnumber, and the dierences between the measured and predicted swirl angle are larger.

The model has been also used to predict the behavior of a low-pressure turbine blade. The agreementbetween the experimental data and the simulations is also good when comparing both the eciency and theswirl angle distributions.

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American Institute of Aeronautics and Astronautics Paper 2009-3666

Efficiency0

20

40

60

80

100

Span

(%)

Experimentκ−ω + γ−Reθtκ−ω

1.25%

-70 -65 -60 -55 -50α (deg)

0

20

40

60

80

100

Span

(%)

Experimentκ−ω + γ−Reθtκ−ω

(a) Re2is = 3 · 105

Efficiency0

20

40

60

80

100

Span

(%)

Experimentκ−ω + γ−Reθtκ−ω

1.25%

-70 -65 -60 -55 -50α (deg)

0

20

40

60

80

100

Span

(%)

Experimentκ−ω + γ−Reθtκ−ω

(b) Re2is = 2 · 105

Efficiency0

20

40

60

80

100

Span

(%)

Experimentκ−ω + γ−Reθtκ−ω

1.25%

-70 -65 -60 -55 -50α (deg)

0

20

40

60

80

100

Span

(%)

Experimentκ−ω + γ−Reθtκ−ω

(c) Re2is = 1.5 · 105

Figure 10. Comparison between the measured and predicted eciency (left) and swirl angle (right) at x/Cax =1.6 for the 3D low-pressure turbine vane

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American Institute of Aeronautics and Astronautics Paper 2009-3666

It is believed that the γ −Reθt model is a good tool to enhance the predicting capabilities of an existingRANS solver. The next steps will consist in using the model to perform steady multistage and rotor-statorsimulations and analyze how the γ−Reθt model improves the resulting solution when compared with existingexperimental data.

Acknowledgments

The authors wish to thank ITP for the permission to publish this paper and for its support during theproject. This work has been partially funded by the Spanish Ministry of Science and Technology underOPENAER Cenit framework.

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11Martinelli, L., Calculations of viscous ow with a multigrid method , Ph.D. thesis, Princeton University, 1987.12Pierce, N., Preconditioned Multigrid Methods for Compressible Flow Calculations on Stretched Meshes, Ph.D. thesis,

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