Application of γ-Reθ Transition Model for Internal CoolingSimulation

Lin Li, Ren Jing, and Jiang HongdeDepartment of Thermal Engineering, Tsinghua University, Beijing, China

The aero thermodynamics of the flow field in a cascade with an internal coolingsystem plays a big role in the whole efficiency of a gas turbine. For an accuratenumerical analysis, this transition is one of the main obstacles. In the present work,the γ-Reθ transition model combined with the SST turbulence model is applied toanalyze the heat transfer of the Mark II blade. The prediction of the onset of theturbulence layer is proven to be satisfied, while the distribution of the blade surfacetemperature before the transition is also captured through the model. Meanwhile, it ispointed out that the turbulent kinetic energy might be over-suppressed in the laminarlayer before the bypass transition due to the combination of the transition model andthe turbulence model. © 2010 Wiley Periodicals, Inc. Heat Trans Asian Res, 40(1),69–77, 2011; Published online 3 December 2010 in Wiley Online Library (wileyon-linelibrary.com). DOI 10.1002/htj.20331

Key words: transition model, conjugate heat transfer, gas turbine, internalcooling

1. Introduction

With the development of the gas turbines, the turbine inlet temperature has exceeded the limitof the material so that the appropriate cooling of the blade is necessary. However, the estimations ofthe heat transfer coefficients and the temperature distribution of the blade surface are not accurateenough using the current numerical algorithm. Therefore, an “over-cooled” technique is normallyapplied to protect the safety of the blades [1]. The waste of the cooling air leads to a decrease in thetotal efficiency. Therefore, predicting the heat transfer coefficients and the temperature distributionof the blade surface accurately is an important issue.

Generally, both the laminar boundary layer and the turbulent boundary layer exist on the bladesurface, while these two kinds of boundary layers have quite different characters with respect to theheat transfer. Thus, how to predict the position of the transition accurately is important to the heattransfer design. This is also a main reason for the discrepancy between the conjugate heat transfersimulations and the experiments in much current literature [2, 7]. Although there are many difficultiesin simulating the transition, Menter [4] pointed out, “Transition should not be viewed as outside the

© 2010 Wiley Periodicals, Inc.

Heat Transfer—Asian Research, 40 (1), 2011

Contract grant sponsor: National Basic Research Program of China (2007CB210100).

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range of the RANS methods. Even relatively simple models can capture these effects with sufficientengineering accuracy. The challenge to a proper engineering model is therefore mainly the formula-tion of models, which are suitable for implementation into a general RANS environment.”

Although the simulation of transition did not develop as quickly as the turbulence model, therehas been some important progress. The NASA report [5] in 1983 put forward an algebraic turbulencemodel concerning bypass transition. The diffusion of turbulence intensity in the free stream is takeninto account through the effective viscosity derived from the experiment data to capture the bypasstransition. Mayle [6] presented a theory for calculating the fluctuations in a laminar boundary layerwhen the free stream is turbulent and derived the kinetic energy equations for these fluctuations in1996. Menter [4] introduced a new concept in 2004 for transition modeling, on which he raised theγ-Reθ transition model. This transition model not only takes into account the non-local effects butalso settles the compatibility with the modern CFD methods.

In the present work, the γ-Reθ transition model is validated and evaluated. The conjugate heattransfer method will be applied to study a benchmark test case: the NASA Mark II turbine vane (testrun number 5411) measured by Hylton et al. [5]. The transition position, the temperature distributionover the blade surface, and the simulation of the boundary layer before transition are the three mainaspects studied in this paper.

Nomenclature

k: turbulent kinetic energy, m2/s2

Nu: Nusselt numberPr: Prandtl numberp: pressure, PaReθ: transition momentum thickness Reynolds numberReθt: momentum thickness Reynolds number of transition onsetT: temperature, KTu: turbulence intensity defined by Eq. (3)U: velocity, m/s γ: intermittencyρ: density, kg/m3

Subscripts and Superscripts

inlet: inlet valuet: stagnation value1: inlet value2: outlet value

2. Geometric Model and Numerical Model

2.1 Geometric model

The transonic Mark II turbine vane test has detailed measurement of the external and internalconvection and the metal temperature. It is often used to validate CFD codes. A linear cascade of three

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vanes is contained in the experiment, while the center vane was cooled by air flowing radially throughten round pipes from the hub to the shroud.

The computations in this paper correspond to the number 5411 test run. The geometric modeland the boundary condition are shown in Fig. 1. The heat transfer coefficients, derived from Eq. (1),and the coolant temperature are given as boundary conditions for the inner surfaces of the coolingchannels. The material of the vane is stainless steel (ASTM Type 310), which has a relatively lowthermal conductivity. The details about the experiment can be found in Ref. 5.

NuD = Cr(0.022Pr0.5ReD0.8) (1)

The solution domain is divided into two parts: the flow field outside the vane and the vaneitself. Both parts are unstructured mesh, and prisms are applied in the near wall region in the outsideflow field. The y+ is smaller than 1 over the whole vane surface. There are 2.03 million cells for theoutside part and 1 million cells for the vane part.

Fig. 1. Geometry model and boundary conditions of the internal cooling channels.

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2.2 Numerical model and transition model

The computation is carried on the commercial CFD package CFX. The SST model isimplemented, which handles the free stream and boundary layer with k–ε equations and k–ω equationsrespectively. The transport of viscous stress is also taken into account within the SST model [9].

The transition model used in this paper is the γ-Reθ model raised by Menter. The transportequations of intermittency γ and transition momentum thickness Reynolds number Reθ can be foundin Ref. 4. The experimental transition correlations relate the momentum thickness Reynolds numberof transition onset, Reθt, to the turbulence intensity, Tu, and other quantities in free stream. Then theReθt computed in the free stream is allowed to diffuse into the boundary layer through transportequations. The coupling of the transition model with the turbulence model is realized by usingintermittency γ to limit the source terms in the k-equation. The transport equation of turbulence kineticenergy k is shown as Eq. (2).

∂∂t

(ρk) + ∂

∂xj(ρujk) = γPk − D

~k +

∂∂xj

(µ + σkµk)

∂k

∂xj

(2)

Instead of using the transition model to predict transition, one can set the distribution of theintermittency γ, if the transition position is known from the experiment. This is an easier way to treatthe transition, and is also tested in this paper.

3. Results and Discussion

The original SST model was tested first, which is shown in Fig. 2 as the dashed line. As thesecond step, the SST turbulence model corrected with intermittency γ was tested, and the predictedprofiles of suction side temperature are shown in Fig. 3. Finally, the SST model with transition modelwas tested, and the result is also shown in Fig. 2 as the solid line. The experimental data are shownas discrete points in the two figures.

3.1 Original SST model

The dashed line in Fig. 2 shows the predicted results using the SST turbulence model withouttransition correction. At the leading edge and the suction side, the predicted temperatures are higherthan the experiment results, with the largest discrepancy about 8%, while the tendencies are alsodifferent. This is because the boundary layer on the suction side is laminar before the shock wave,while the overall boundary layers are treated as the turbulent boundary layers in the computation withthe original SST model. The temperature profile over the suction side downstream of the shock waveis lower than the test data by about 5% to 8%, and the reason is unclear. Probably it is due to theCauchy boundary condition setting on the inner face of the cooling channel when taking the CHTmethod.

The predicted temperature profile over the pressure side agrees well with the test data.However, the NASA report says that the boundary layer over the pressure side is at the transitionalstage between the laminar state and the turbulent state. That means the original SST model, whichdealt with the pressure side boundary layer that is not totally turbulent as a turbulent boundary layer,

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leads to a rational result. There are two possible reasons: one is that the diffusion of the free streamturbulence intensity into the boundary layer makes the boundary layer more like a turbulent one ratherthan a laminar one; the other is that the k–ω equations used in the SST model in the boundary layerscan capture the transition process to some extent [11].

3.2 SST model corrected with intermittency

As the predicted result with the original SST model agrees well with the experiment data onthe pressure side, the focus below will be set on comparison over the suction side. According to theposition of the transition onset determined from the experiment, approximately, the boundary layerupstream of the transition point can be treated as laminar boundary layer, and the boundary layerdownstream can be treated as turbulent. The intermittency γ equals 1 in the turbulent boundary layer,while it can take any value in 0.1 in the laminar layer. Such a setting is due to the following tworeasons. First, the physical quantities in the laminar boundary layer before bypass transition havefluctuations. Second, given the distribution of γ in CFX instead of using the transition model willcause the free stream upstream of the transition to have the same intermittency as the local boundarylayer, which will weaken the effect of the diffusion of turbulence intensity from free stream toboundary layer.

For convenience, the intermittency for the boundary layer before transition is set uniform.After some trials, the predicted results are summarized in Fig. 3. It is shown that the experiment datalies in a band, and the predicted results in which the prescribed γ equals 0.6 and 1 form the bounds.The results with prescribed γ equalling 0.9 agree well with the experiment data. So it is inferred thatif an appropriate value of γ is chosen for the laminar boundary layer, the predicted results will besatisfactory. However, an appropriate value of γ is related with the turbulence intensity in free stream,

Fig. 2. Profile of blade surface temperature.

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the local streamline curvature, and the upstream influence, and the authors have not found anyquantitative regularity among them. Thus setting the distribution of γ needs much trial and error andexperience, and is not a common way to treat transition.

3.3 SST model with transition model

3.3.1 Numerical result

The solid line in Fig. 2 shows the predicted results using the SST turbulence model with theγ-Reθ transition model. The numerical result is lower than the test data by about 5% to 8%, but capturesthe shape of the profile of the test data, especially at the leading edge of the suction side and the wholepressure side. The following discussion is focused on the comparison between the two numericalresults.

A discrepancy between the two numerical results is noticeable. The predicted temperaturedistribution with transition is lower than with the original SST model. This is because the boundarylayer over the suction side before the shock wave is treated as a laminar boundary layer by thetransition model. The distribution of the intermittency is shown in Fig. 3. The position where the onsetof transition takes place agrees with test data.

Second, in the result without the transition model, the temperature profile of the suction sidebefore the shock wave is a convex curve, which is also the case in Refs. 2 and 8. But in the result withthe transition model, the temperature profile of that segment agrees well with the test data, which isuncommon in the literature about the computation of the Mark II vane.

Third, some information can be found when looking into the distribution of turbulenceintensity Tu, defined by Eq. (3). It is shown in Fig. 4 that the turbulence intensity in the boundarylayer over the pressure side in the results with the transition model is smaller than that without thetransition model. Recall the distribution of γ in Fig. 3 and the transport equation of the turbulencekinetic energy k, Eq. (2), the discrepancy between the two numerical results conforms to the

Fig. 3. Predicted suction side temperature profile with different prescribed γ before transition.

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distribution of γ. As a result of the decrease of Tu, the local heat transfer coefficient decreases, as doesthe temperature over the surface.

Tu = √2 / 3k / Uinlet × 100 (3)

3.3.2 Discussions on the transition model’s behavior in boundary layer of pressure side

As shown in Fig. 2, the temperature distribution over the pressure side predicted with theoriginal SST model agrees well with the test data, but the result with the transition model is lowerthan the test data. A possible explanation is as follows:

First, Menter [4, 10] pointed out that one of the basic requirements of the underlying turbulencemodel is that it must produce full turbulent flow from the location where the model is first activated(i.e., low Reynolds number turbulence models which often predict some amount of laminar flow ontheir own could potentially affect the accuracy of the present transition model).

Fig. 4. Distribution of γ in the cascade.

Fig. 5. Distribution of turbulence intensity.

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The SST turbulence model is implemented in the work, which uses the k–ω model in the nearwall region and accounts for the effect of the transport of the principal turbulent shear stress. Theauthors have not found any evidence to show that this kind of near wall treatment may capture somelaminar characters. However, Wilcox showed that the capacity of the k–ω model on modelingtransition is not as good as supposed [11]. It means that the k–ω model is somehow able to model thelaminar boundary layer before transition to a certain extent. Besides, it is shown in Ref. 8 that the heattransfer coefficients predicted with the V2F model and the LRN nonlinear quadratic k–ε model overthe pressure side agree well with the C3X vane test data, and the boundary layer over the pressureside of the C3X vane is similar to that of the Mark II vane according to the NASA report [5]. It meansthat some turbulence models can capture the characters of laminar boundary layer although treatingthe whole computational domain as a turbulent flow.

The physical quantities in the laminar boundary layer before bypass transition have fluctua-tions as a result of the diffusion from the free stream [3], so this kind of laminar boundary layer canbe treated as a kind of “turbulent” boundary layer, in which the average effect of fluctuations is muchweaker. The results in Section 3.2 also support this viewpoint. It is reasonable that some turbulencemodels can produce satisfactory results when dealing with this kind of laminar boundary layer.However, the turbulence model is corrected for the laminar boundary layer to reduce the turbulentkinetic energy while applying the transition model. It may cause the turbulent kinetic energy in thelaminar boundary layer before bypass transition to be smaller than in the real case, which may accountfor the discrepancy between the predicted temperature in the suction side before the shock wave andthe test data.

4. Conclusions

The γ-Reθ transition model is validated and valued with the SST model as the underlyingturbulence model in the Mark II vane test case. The result is that the position where the onset oftransition takes place is in accordance with the test data. The trend of the blade surface temperaturedistribution before the transition is also captured with the transition model. The turbulent kineticenergy might be over-suppressed in the laminar boundary layer before the bypass transition whenusing SST as the underlying turbulence model for the γ-Reθ transition model.

Further research would be valuable to make the over-suppress effect clear. Meanwhile, moreexperiments concerning the boundary layer over the pressure side are necessary to improve thetransition model.

Acknowledgments

The authors would like to acknowledge the financial support from the National Basic ResearchProgram of China (2007CB210100).

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Originally published in J Eng Thermophys 30, 2009, 1849–1852.Translated by Lin Li, Department of Thermal Engineering, Tsinghua University, Beijing 100084,

China.

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