Heat transfer characteristics of fan-shaped hole effusion cooling for a constant hole exit width – Numerical simulation and experimental validation

Applied Thermal Engineering 160 (2019) 113978

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Research Paper

Heat transfer characteristics of fan-shaped hole effusion cooling for a constant hole exit width – Numerical simulation and experimental validation

T

⁎

H. Weia, J.L. Aia, Y.Q. Zua, , L. Dingb a b

Department of Aeronautics and Astronautics, Fudan University, 220 Handan Road, Shanghai 200433, China AECC Commercial Aircraft Engine Co., LTD, 3998 Lianhua South Road, Shanghai 200241, China

H I GH L IG H T S

cooling of fan-shaped holes is studied numerically and experimentally. • Effusion fan-shaped holes with a constant exit width are considered. • The effects of hole geometry and flow condition are evaluated and analyzed. • The • A correlation for area-averaged effusion cooling effectiveness is developed.

A R T I C LE I N FO

A B S T R A C T

Keywords: Adiabatic film cooling effectiveness Fan-shaped hole Effusion cooling Constant hole exit width Correlation

In order to study the effusion cooling mechanism of fan-shaped holes with a constant exit width, numerical simulations and the corresponding experimental verifications are carried out on a flat plate with ten rows of fanshaped film cooling holes. The entrance diameter and the exits width of the cooling holes are kept at D and 3D, respectively. While the thickness of the flat plate is kept at 10D/3. The holes are arranged in a staggered pattern with the constant stream-wise and span-wise spacing of 8.5D and 6.5D, respectively. The hole inclination angles of 20°, 25° and 30°, the hole expansion angles of 10° and 13°, and the blowing ratios ranged from 0.5 to 7.5 are considered. Shear stress transport k - ω turbulence model is adopted for the numerical simulation. The numerical results are validated experimentally using thermo-chromic liquid crystal technology, showing good agreements. Through the simulation, the effects of the inclination angle and the expansion angle of the holes as well as the blowing ratio on the film cooling effectiveness are analyzed and discussed in detail. A correlation for the areaaveraged effectiveness is developed for the prediction of row-by-row superposition effects of effusion cooling with fan-shaped holes.

1. Introduction For the purpose of improving the output power and propulsive efficiency of aero-engines, the turbine inlet temperature increases continuously. As a consequence, the temperature of hot gas is significantly higher than the melting point of high-temperature components. Designers have to explore more proper and effective cooling structures to protect the high-temperature components from the thermal impact of hot gas. The film cooling is a remarkable external cooling strategy, which has been well-accepted and widely used for the cooling of combustor and turbine. The flow and heat transfer mechanism of film cooling has been investigated exhaustively since the 1960s [1].

⁎

The previous studies indicated that the film cooling effectiveness of fan-shaped holes was significantly higher than that of cylindrical holes, especially at high blowing ratios. And, the cooling performance of the fan-shaped holes depends sensitively on the flow conditions at both the entrance and the exit of the holes, which are dominated by the interplay of hole shaping, orientation, spacing pitch of holes, length to diameter ratio, surface roughness, curvature, blowing ratio, density ratio, and turbulence intensity etc. The development of film cooling technology with fan-shaped holes was summarized by Bunker [2]. In the past decades, many experimental and numerical investigations on the flow and heat transfer characteristics of a single fan-shaped film cooling hole or a single-row of the holes on a flat plate have been

Corresponding author. E-mail address: [email protected] (Y.Q. Zu).

https://doi.org/10.1016/j.applthermaleng.2019.113978 Received 8 October 2018; Received in revised form 26 April 2019; Accepted 16 June 2019 Available online 17 June 2019 1359-4311/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature A AR BR Cp D DR H k P Prt R2 T x X y y+ Y z u u u′ W α

β ρ λ η μ μt ω

cross-sectional area area ratio blowing ratio specific heat (J/(kg·K)) the inlet diameter of a hole (mm) density ratio thickness of flat plate (mm) turbulent kinetic energy (m2 s2 ) pressure (Pa) turbulent Prandtl number coefficient of determination temperature (K) coordinate along the stream-wise (mm) stream-wise spacing of the cooling holes (mm) coordinate along the span-wise (mm) non-dimensional distance normal to a wall span-wise spacing of the cooling holes (mm) coordinate normal to wall surface (mm) velocity magnitude (m s ) velocity vector (m s ) fluctuating velocity in turbulent flow (m s ) width of the hole exit (mm) inclination angle

expansion angle density of ideal air (kg/m3) diffusivity coefficient (W/(m·K)) film cooling effectiveness dynamic viscosity (kg/(m·s)) turbulent viscosity (kg/(m·s)) turbulent specific dissipation rate (1 s )

Subscript c start ∞ w

coolant flow starting location mainstream wall

Superscript

¯· ¯· ∼ ·

line-averaged value area-averaged value non-dimensional value

Correlation Constants

a0 , a1, b0 , b1, b2 , ξ0 ,ξ1, ξ2 , ζ 0 , ζ1,ζ2 α∗, α1,Ω –

–

plate. The cooling performance of five different types of holes was evaluated. The laterally diffused hole with a compound angle generated the best cooling performance over the widest ranges of blowing ratios, momentum flux and stream-wise locations. Li et al. [10] carried out a set of experimental studies on the heat transfer coefficient, film cooling effectiveness and heat loads from a row of film cooling holes with cylindrical shape, fan shape and 3-in-1 shape, respectively. The results indicated that the fan-shaped holes provided the best film cooling effectiveness, whereas the cylindrical holes were the worst. Bianchini et al. [11] conducted a numerical analysis of a single laidback fanshaped film cooling hole on a flat plate at the blowing ratios of 1.25 and 2.5 for the capability assessment of four different turbulence models. It was illustrated that the flow and heat transfer predicted by either the v2 - f or the shear stress transport (SST) k − ω model were in good agreements with those of experimental measurements. Saumweber and Schulz [12,13] compared the film cooling performance of cylindrical holes with that of fan-shaped holes under a wide range of free-stream conditions. It was observed that the flow conditions at the entrance and the exit of the hole affected the cooling performance of cylindrical and fan-shaped holes in different ways. Furthermore, the study confirmed the complexity of fan-shaped hole film cooling and so that it was hard to derive a correlation for the combined effects of varying flow conditions and hole geometries. Sun et al. [14] investigated the effect of hole geometries on the film cooling effectiveness experimentally and numerically at the blowing ratios ranged from 0.3 to 1.5. Four kinds of holes with a constant exit area (i.e. the cylindrical hole, the fan-shaped hole, the double jet holes and the hole accompanied with sister holes) were considered. It was noticed that the three kinds of shaped holes had an obvious advantage compared to the cylindrical hole. Colban et al. [15] derived a semi-empirical correlation to predict the laterally averaged film cooling effectiveness of a single row of 30°- inclined fanshaped holes on a flat plate surface. It was indicated that the film cooling effectiveness was strongly affected by the blowing ratio, the coverage ratio, and the outlet-to-inlet area ratio of the holes. Since, as reviewed above, the film cooling effectiveness of the fanshaped hole depends sensitively on the geometrical parameters, optimizing the hole geometry is an important issue for achieving the most favorable film cooling performance. Some research works [16–20] have

carried out. The experimental study of Goldstein et al. [3] is normally regarded as the seminal research of shaped-hole film cooling to demonstrate and quantify the film cooling effects of the fan-shaped holes. The results indicated that either a single hole or a row of holes with an expansion angle of 10° significantly enhanced the film cooling effectiveness as compared to cylindrical holes. Kohli and Bogard [4] conducted an experimental study on the film cooling performance of a single row of discrete holes with an inclination angle of 55°. Three different hole shapes were investigated at a density ratio (DR) of 1.6. The obtained results demonstrated that the shaped holes generated higher spatially averaged adiabatic cooling effectiveness than the round hole over the whole range of momentum flux ratios, particularly at higher blowing ratios. Gritsch et al. [5] evaluated the influence of hole geometrical parameters on the film cooling performance of the fanshaped holes in a single row, when the blowing ratios were ranged from 0.5 to 2.5 at an engine-representative density ratio of DR = 1.7. Within the range of parameters considered, the laterally averaged film cooling effectiveness showed only limited sensitive to the variation of the hole geometries. Utilizing a steady state thermo-chromic liquid crystals technique, Bonanni et al. [6] measured the heat transfer coefficient and adiabatic effectiveness downstream of a single laidback fan-shaped hole on a flat plate, when the blowing ratios varied from 0.5 to 2.5. The experimental results showed that the lateral distribution of cooling effectiveness was primarily influenced by the outlet-to-inlet area ratio (AR) of the holes. Gritsh et al. [7] investigated experimentally the local film cooling effectiveness and the heat transfer coefficient in the nearfield of three different film cooling holes (i.e. a cylindrical hole, a fan-shaped hole and a laidback fan-shaped holes) at an engine like coolant-to-mainstream density ratio of DR = 1.85 over a range of blowing ratios of BR = 0.25–1.75. It was observed that both fan-shaped holes provided significantly improvement on overall film cooling performance as compared to the cylindrical hole. In the study of Yu et al. [8], the overall film cooling performance of three different but closely related hole shapes were tested using a transient liquid crystal technique at the blowing ratios of 0.5 and 1.0. The hole with both forward and lateral diffusion showed the highest film cooling effectiveness. Bell et al. [9] carried out an experimental study on the heat transfer behavior of a single row of shaped film cooling holes on a large scale flat 2

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Pressure outlet

Mainstream inlet z

Adiabatic wall

500D/3

focused on the shape optimization of a fan-shaped hole for a single objective enhancing the film cooling effectiveness or multiple objectives considering the film cooling effectiveness and aerodynamic loss. Three-dimensional Reynolds-averaged Navier-Stokes (RANS) analysis combined with different optimization algorithms were applied for the investigations. Some geometric design factors (such as the inclination angle, the lateral expansion angle, the forward expansion angle, and the length-to-diameter ratio of the hole, etc.) were taken into consideration. In the RANS analysis of Lee et al. [16–19], the SST k − ω turbulence model was used as a turbulence closure. The obtained numerical results agreed well with the corresponding experimental data not only for the averaged values but also for the local distributions of film cooling effectiveness. In the study of Wang et al. [20], a renormalized group (RNG) k-ε model was applied instead. The CFD data showed good agreements with the experimental results in terms of the laterally averaged film cooling effectiveness. In addition, several researchers have devoted to analyzing cooling performance of fan-shaped holes on the turbine blades [21–23] and endwalls [24–28]. Colban et al. [21] reported an experimental and computational study of fan-shaped film cooling holes on a gas turbine vane to evaluate the row-by-row interaction of fan-shaped holes. It was found that the multi-row cooling provided the overall higher effectiveness than the single-row cooling. Colban et al. [22] presented an experimental investigation of heat transfer coefficients and adiabatic effectiveness of fan-shaped film cooling holes at the leading edge, the pressure side and the suction side of a turbine guide vane. The obtained results indicated that the behavior of boundary layer transition along the suction side of the vane was sensitive to the location of coolant jet injection. Barigozzi et al. [24] conducted an experimental investigation to compare the film cooling performance of conically expanded fanshaped holes with that of cylindrical holes on a contoured nozzle vane endwall. The results showed that the fan-shaped holes provided a better thermal protection all over the endwall surface, particularly at a higher coolant injection condition. Most previous researches focused primarily on the flow and/or heat transfer characteristics of a single fan-shaped film cooling hole or a single-row of the holes. Very few works have paid attention to the characteristics related to the fan-shaped hole effusion cooling, although the combustor walls and turbine components are more likely to utilize the effusion cooling to achieve high cooling efficiency. It should be emphasized that “effusion cooling is the result of many subsequent rows of cooling holes” [29]. In other words, the film cooling with only two or three rows of holes is not the true effusion cooling [29,30]. The investigation of Harrington et al. [31] indicated that as many as eight rows of holes were required to reach an asymptotic fully developed adiabatic cooling effectiveness level. Krewinkel [29] summarized that seven rows of cooling holes are the minimum for a model to encapsulate all relevant effects of effusion cooling. Besides, previous studies on the influences of geometrical parameters of fan-shaped holes have mainly used the holes with a constant length of the cylindrical/diffuser part and the varying expansion angle, causing the difference in the hole exit width. Note that a wider exit of the fan-shaped hole normally results in a higher cooling performance due to the better coverage of the coolant. Thus, it is unfair to compare the hole geometries with different exit widths. In the present study, the flow and heat transfer characteristics of the fan-shaped hole effusion cooling for a constant hole exit width are studied numerically and experimentally. The studies are carried out on the flat plates, neglecting the surface curvature effect, as an idealization of the actual effusion cooling component. Ten rows of fan-shapes holes are used to establish the effusion cooling configuration. The effects of blowing ratio, hole inclination angle and expansion angle on the effusion cooling performance of the fan-shaped holes are examined and discussed.

x Ten rows of film cooling holes

1000D/3

Coolant inlet 100D

(a) Side view of the computational domain

Computational domain

(b) Top view of physical domain and computational domain

(c) Three-dimensional view of the computational domain Fig. 1. Computational domain and the corresponding physical domain.

2. Research methodology 2.1. Problem description The physical domain and the corresponding computational domain considered in the present study are shown in Fig. 1. Both of them consist of mainstream duct, coolant plenum, and fan-shaped holes. As shown in Fig. 1(a), hot air flows into the mainstream duct, meanwhile secondary flow enters the coolant plenum and injects through ten rows of fan-shaped film cooling holes into the mainstream. The holes are arranged in staggered pattern and distributed periodically in the lateral direction as shown in Fig. 1(b). Numerical simulations and the corresponding experimental validations on the fan-shaped film cooling with different hole geometries and flow conditions, as presented in Table 1, are conducted. The definitions of geometrical parameters, including hole inclination angle (α ), lateral expansion angle (β ), hole outlet width (W) and the film cooling plate thickness (H) can be found in Fig. 2. The entrance diameter (D) and the exit width of the cooling holes are kept at D (=3mm) and 3D (=9mm), 3

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Table 1 The numerical simulation conditions (* validated with experiments). D (mm)

H (mm)

W (mm)

X D

Y D

α (°)

β (°)

BR

3

10

9

8.5

6.5

20*

10* 13 10 13* 10 13

0.5–7.5 (0.5, 1.0, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5)*

25* 30

BR =

ρc uc ρ∞ u∞

=

4ṁ c πD 2N ρ∞ u∞

(1)

where ρc and uc are the density and the velocity of the coolant respectively, which can be determined on the basis of the hole entrance diameter D, total mass flow rate of the coolant ṁ c and total number of the holes N; ρ∞ and u∞ are the density and the velocity of the mainstream respectively. The adiabatic film cooling effectiveness (η ) can be defined as [32]

η (x , y ) =

T∞ − Tw (x , y ) T∞ − Tc

(2)

where T∞ is the temperature of the mainstream; Tw is the temperature at the hot side surface of the adiabatic film cooling plate; Tc is the temperature of the coolant; the coordinates x and y are taken to be measured along the stream-wise and the span-wise directions respectively, with an origin at the exit center of the film cooling hole in the first row. Then, the laterally averaged film cooling effectiveness, η , can be calculated as

η (x ) =

Y ∫0 η (x , y ) dy

Y

(3)

2.2. Numerical method 2.2.1. Computational domain and boundary conditions Due to the periodicity and symmetry of the configuration, in order to reduce the computational expense, a half span-wise period is taken as the computational domain where exist ten half-fan-shaped film cooling holes, as shown in Fig. 1(b). In the simulation, ideal gas (air) is used as working fluid. At the inlet of the coolant chamber, a constant mass flow rate of 0.002 kg/s and a constant static temperature of 303.15 K are specified. At the inlet of mainstream, the static temperature of the flow is kept at 323.15 K, while the mass flow rate is adjusted to satisfy the required blowing ratio condition. Adiabatic and non-slip boundary condition is specified at all of the walls. In addition, at the mainstream outlet, the averaged gauge pressure is set as 0 Pa.

Fig. 2. Definitions of geometrical parameters.

while H is kept at 10D/3 (=10 mm). Ten rows of cooling holes are arranged in a staggered pattern with stream-wise and span-wise spacing ratios of X/D = 8.5 and Y/D = 6.5, respectively. As summarized by Bunker [2], the most commonly used expansion angles of fan-shaped holes fall in the 10–15° range. Therefore, moderate expansion angles (β ) of 10° and 13° are chosen for the present study. Note that, under the constraints of fixed wall thickness and hole exit width, much smaller expansion angle may result in a non-cylindrical entrance of the hole. The hole inclination angles (α ) of 20°, 25° and 30°, and blowing ratio (BR) ranged from 0.5 to 7.5 (0.5, 1.0, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, and 7.5) are considered in the in the numerical study. Selected cases (marked with * in Table 1) are validated experimentally. In order to investigate the film cooling effectiveness of fan-shaped holes under different blowing ratios, the blowing ratio (BR) is defined as

2.2.2. Turbulence model The three-dimensional flow and convective heat transfer characteristics of fan-shaped hole effusion cooling are analyzed based on CFD commercial code ANSYS CFX. The finite volume method is employed to discretize the RANS equations, and the SST k - ω turbulence model is chosen to close the equations. The steady-state RANS equations in a Cartesian tensor form can be written as

∇ ·(ρ u ) = 0

(a) Mesh of computational domain

(b) Locally enlarged view

Fig. 3. Grid used in current research. 4

(4a)

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(a) α =20°

(b) α =25°

(c) α =30° Fig. 4. Change with grid of the span-wise averaged film cooling effectiveness (β = 10°, BR = 3.5).

Change rate of

(%)

the dynamic viscosity μ , the temperature T , the specific heat Cp and the diffusivity coefficient λ in the equations now represent ensembleaveraged (or time-averaged) values. Prt in the energy equation is the turbulent Prandtl number which is set as 0.9 in the present simulation. Comparing the RANS momentum equation Eq. (4b) to the NS momentum equation, additional terms appear that represent the effect of turbulence. The Reynolds stresses, − ρu′u′, must be modelled in order to close the equations. A common method is to relate the Reynolds stresses to the mean velocity gradients, based on the Boussinesq hypothesis [33], as

− ρ u′u′ = μt [∇u + (∇u)T ] −

2 (ρk + μt ∇ ·u) I 3

(5)

where k is the turbulence kinetic energy and μt the turbulent viscosity. In the SST k - ω model [34], the transport equations for the turbulence kinetic energy k and the specific dissipation rateω are solved.

Fig. 5.

The relative changing rate of span-wise averaged film cooling effectiveness caused by mesh

∇ ·(ρku ) = ∇ ·(Γk ∇k ) + Gk − Yk

(6a)

∇ ·(ρωu ) = ∇ ·(Γω ∇ω) + Gω − Yω + Dω

(6b)

refinement.

∇ ·(ρuu ) = −∇p + ∇ ·[μ (∇u + (∇u )T )] + ∇ ·(−ρu′u′)

(4b)

μ ∇ ·(ρuCp T ) = ∇ ·⎡ ⎛λ + t Cp ⎞ ∇T⎤ ⎢⎝ Pr t ⎠ ⎥ ⎣ ⎦

(4c)

⎜

where Gk and Gω represent the generation of k and ω due to the mean velocity gradients, respectively. Γk and Γω represent the effective diffusivity of k and ω , respectively. Yk and Yω represent the dissipation of k and ω due to turbulence, respectively. Dω is the cross-diffusion term. To take the transport of the principal turbulent shear stress into consideration, μt is given as [35,36]

⎟

The above equations have the similar general form as the steady state Navier-Stokes (NS) equations. However, the velocity u , the density ρ ,

μt = 5

ρ k 1 ω max[1 α∗, ΩF2 α1 ω]

(7)

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Fig. 6. The distribution of y+ on the hot-side surface of the plate and the hole surfaces (α = 20°, β = 10°, BR = 3.5).

million hexahedral elements is regarded as the best compromise between the computational cost and accuracy and is used for all of the following simulations. The solution is assumed to be converged when the normalized residuals of the continuity, momentum and energy equations are lower than 10−6. The grids in the vicinity of the walls of the cooling holes and the plate have been fined, as shown in Fig. 3, by specifying the thickness of the first layer of elements adjacent to the walls as 10−6 m and the grid expansion ratio as 1.1 so as to obtain the accurate velocity and thermal boundary layers. Fig. 6 shows the distribution of y+ on the hot-side surface of the plate and the hole surfaces for α = 20°, β = 10° and BR = 3.5. It is found that the values of y+ are always less than 1.1. 2.3. Experimental validation method 2.3.1. Experimental parameters For (α = 20°, β = 10°) and (α = 25°, β = 13°) , the numerical results are validated with experiments to examine the reliability and accuracy of the chosen numerical method. The geometry and distribution of the holes considered in the experiments are exactly the same as those of numerical simulations. To ensure the periodicity of flow and heat transfer in the span-wise direction, 145 effusion cooling holes are arranged in ten rows, each with 14 or 15 holes as shown in Fig. 7. For the measurements of adiabatic film cooling effectiveness, the test plates are made of polyether-ether-ketone (PEEK) with a low thermal conductivity (< 0.2 W/(m·K)), so that the convective heat transfer effect on the opposite surface of the effusion cooling plate as well as the conduction effect in the plate are very weak and thus ignorable [37]. The detailed experimental parameters considered in the present study are shown in Table 1.

Fig. 7. The effusion cooling test plate.

where α∗ is a function of Reynolds number, Ω is a function of the mean rate-of-rotation tensor, F2 is the blending function, and α1 is a constant. 2.2.3. Mesh generation and grid independence analysis Structured multi-block grids, as shown in Fig. 3, are generated in the computational domain using ICEM CFD. C-type grids are applied to the fan-shaped holes to increase the orthogonality of the mesh. In order to ensure the grid independence of the numerical results, the initially generated coarse mesh is successively refined in the hole region and in the near wall region of the plate downstream of the hole for test. The comparisons of calculation results based on three different grid density conditions (3.8 million, 6.9 million and 9.5 million elements) have been performed. For BR = 3.5, β = 10° and three different inclination angles, Fig. 4 shows the change with grid of the span-wise averaged film cooling effectiveness, η . It is found that η changes significantly when element number increases from 3.8 million to 6.9 million, but changes a little when element number increases further to 9.5 million. Furthermore, Fig. 5 plots the relative changing rates of η when the element number increases from 6.9 million to 9.5 million. It can be noticed from the figure that, for all of three cases, the changing rates are less than 5% over a wide range of x. The high changing rates obtained in x < 1.5D are not representative since η approximates zero there. Based on the above grid-independent tests, the mesh with 6.9

2.3.2. Experimental setup Fig. 8 presents a schematic view of the experimental system, which consists of a re-circulating wind tunnel for mainstream, an open-circuit wind tunnel for secondary stream supply, test control units and data acquisition units. In wind tunnel for the mainstream, a pair of special designed 60 KW guide-vane heaters [38] are located at two left corners of the tunnel, as shown in Fig. 8, to heat the air in the mainstream channel. Multi layers of damping screens are arranged upstream the test section to obtain uniform temperature and velocity distributions. The test section is made up of 15 mm thickness transparent Plexiglas plates, the specific schematic diagram is shown in Fig. 9. One of the sidewalls of the test section is manufactured with an appropriate designed square-shaped 6

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Centrifugal blower

Guide-vane heater 1

CCD camera LED headlight 1

Guide vane 2

LED headlight 2

Damping Screen Experimental section Guide vane 1

Guide-vane heater 2

Coolant plenum Roots blower

Air intake

Filte

Data acquisition

Orifice plate flowmeter

Computer

Data Transmission line Fig. 8. Sketch of the experimental system.

Flange

Mainstream channel

Test plate Coolant plenum Fig. 9. 3D view of the test section.

cut-out where the effusion film cooling test flat plate can be inserted and tighten so that the hot-side surface of the test plate is flush with the inner surface of the mainstream channel. The wind tunnel for the secondary stream consists of air intake, filter, Roots blower, orifice plate flowmeter and coolant plenum. In the wind tunnel, the air is drawn into the air intake and filtered. It then passes through a calibrated orifice flow-meter, progresses through the coolant plenum, and finally ejects through the fan-shaped holes to form a cooling film over the hot side surface of the test plate. A frame structure is used to connect the coolant chamber with the mainstream channel. Black paint backed thermo-chromic liquid crystal (TLC) (R30C20W, Hallcrest) is sprayed on the hot-side surface of the test plate for the measurement of Tw . The color display of the TLC is recorded using a CCD camera mounted perpendicular to the test plate. A pair of 70 W LED lamps are fixed on both sides of the camera as the light sources during the test. Before Tw measurements, the TLC has been calibrated with a Hue based technique [39] to relate its color response with the temperature quantitatively. In order to minimize the differences in

lighting conditions between the calibration procedure and Tw measurements, the calibration apparatus (a calibration plate) was mounted at the same location as that of the effusion cooling test plates. The lighting arrangements, the optical path and the camera setting are fixed for all of the test runs. The temperatures of mainstream air (T∞) and secondary flow (Tc ) are both monitored by Pt100 RTDs. The velocity of the mainstream is measured utilizing a Pitot-static tube, while the mass flow rate of the secondary flow is measured using the orifice flowmeter. A Keysight 34980A system is applied to acquire data. In the experiments, both T∞ and Tc have the uncertainty of ± 0.15 K, while Tw captured with TLC has an uncertainty of ± 0.5 K. Following the procedure described by Moffat [40], the relative uncertainty of η, is estimated, based on a 95% confidence level, as 3.5% at a typical value of η = 0.5. However, it rises with the decrease of η , changing to 5.8% at η = 0.3 and 17.4% at η = 0.1. The experimental results, which can be found in the next section, indicate that the relative uncertainty of η is less than 5.8% in most of the test region (x/D ≥ 8.5, where η > 0.3), and goes down to 3.5% or even much lower in the wake of the jets (η > 0.5). The high relative uncertainties of η obtained in-between the 7

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(a) Experimental results

(b) Numerical results

Fig. 10. Local distribution of adiabatic film cooling effectiveness (α = 20°, β = 10°).

(a) Experimental results

(b) Numerical results

Fig. 11. Local distribution of adiabatic film cooling effectiveness (α = 25°, β = 13°).

first-row of cooling holes are not representative as η goes to zero in these regions.

(α = 20°, β = 10°) and (α = 25°, β = 13°) are compared with the corresponding experimental data obtained by the present validation experiments. Figs. 10 and 11 present the comparisons of the spatiallyresolved distribution of adiabatic film cooling effectiveness, η, obtained by CFD simulations with those obtained by experiments when (α = 20°, β = 10°) and (α = 25°, β = 13°) , respectively. Since the CFD simulations are carried out on the basis of a half span-wise period of real physical domain, the results are symmetrically extended for the

3. Results and analysis 3.1. Validation of CFD model In order to validate the CFD model, the numerical results for 8

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effectiveness. As a result, the development tendency of film cooling performance at this stage depends on the competition between the above two factors. Finally, at the third stage, the coolant blows off totally as BR increases further, reducing the cooling performance in the upstream region of the hot side surfaces. On the other hand, at the second and the third stages mentioned above, the mainstream in the near wall area is cooled significantly due to the mixing with the blowoff coolant, improving the heat transfer performance downstream. Thus, the gradient of η (x ) increases with BR. For examples, when (α = 20°, β = 10°), as shown in Fig. 10, the coolant of the higher speed is concentrated in the central position of the hole exits, accelerated with the increasing BR and finally penetrates into the mainstream. Meanwhile, in both lateral side regions of the holes exits, the coolant of relatively lower speed can still form cooling film and improve the cooling performance until blowing off from the hot side surface with the increase of BR. While, compared to the cases of (α = 20°, β = 10°), the lateral distributions of the coolant velocity at the hole exits are relatively more uniform for the cases of (α = 25°, β = 13°), when the coolant of the higher speed is distributed at the central position and double lateral borders of the hole exits, as shown in Fig. 11. With the increasing BR, higher speed part of coolant permeates into the mainstream, reducing the local cooling performance.

(a) α =20°, β = 10°

3.2.1. Effects of inclination angle Fig. 13(a) and (b) show the compilations of the local adiabatic film cooling effectiveness distributions downstream the fan-shaped holes with various combinations of inclination angle and expansion angle when BR = 1.5 and 3.5, respectively. The cooling effectiveness in a region of x/D < 40 is taken for comparisons. In each column, the expansion angle is kept constant. While, in each row, the inclination angle is the same. It is known that, under the same blowing ratio condition, an increase in the inclination angle tends to magnify the vertical component of the coolant jet momentum, which can enhance the coolant jet blowoff at high blowing ratios without the consideration of the uniformity of the coolant velocity. However, as mentioned above, the velocity distribution of the coolant at the hole outlet is generally non-uniform, which causes an obvious change in the pattern of the local film cooling effectiveness with the variation of the inclination angle as shown in Fig. 13. A diagrammatic sketch of the evolution of local cooling effectiveness patterns downstream the hole exits is presented in Fig. 14. When α = 20°, the local film cooling effectiveness presents a threeforked pattern, as shown in Fig. 14(a), one tine related to the centerline and the other two corresponding to the lateral borders of the hole. A similar pattern has been reported in the study of Lee et al. [19]. Similar to the cases of α = 20°, the local film cooling effectiveness for α = 25° is also distributed in the three-forked pattern as shown in Fig. 14(b). It, however, compared to α = 20° cases, becomes relatively more uniform, as shown in Fig. 13, thereby enhancing the lateral spreading of the coolant also contributing to the increase of spatially averaged film cooling effectiveness. Then, the fan-shaped holes with α = 30° yield a typical bimodal pattern of the local film cooling effectiveness like Fig. 14(c) due to the separation of the coolant inside the diffuser, which confirm the findings of Saumweber and Schulz [13]. The effects of the inclination angle on the footprint of the coolant observed above should be caused by the helical motion of the coolant inside the holes and hence the velocity distribution at the hole exits. For the blowing ratio of 1.5 and the expansion angle of β = 13°, Fig. 15 present the distributions of vertical component of the coolant jet velocity at the outlet of first-row of holes with varying inclination angles. When α = 20° and 25°, as shown in Fig. 15(a) and (b), the coolant preferably exits from the downstream part of the hole along the centerline region. Nevertheless, the distribution of the positive normal velocity at the outlet of the film cooling hole with α = 25° is relatively more uniform then that of α = 20°. Unlike that in the holes with α = 20° and 25°, the coolant in the hole with α = 30°, preferably ejects from the

(b) α =25°, β = 13° Fig. 12. The comparisons between numerical results and experimental results of η .

convenience of comparison. It is noticed that the patterns of η obtained by CFD simulations agree well with experiments for nine different blowing ratios ranged from 0.5 to 7.5. To further check the reliability of the CFD model quantitatively, the comparisons of laterally averaged adiabatic film cooling effectiveness between the numerical simulations and the experiments are carried out at typical blowing ratios (BR = 0.5, 1.0, 1.5, 7.5). Good agreements appear although the CFD results are slightly different from the experimental data at high blowing ratios as shown in Fig. 12. 3.2. Numerical simulation results Generally, the lateral distributions of η just downstream the outlet of film cooling holes are non-uniform, as shown in Figs. 10 and 11, which should be caused by the non-uniform lateral distribution of the coolant velocity at the hole exits. The evolution of the film cooling effectiveness normally experiences three stages with the increase of blowing ratio BR. At the first stage, the coolant does not blow off under lower BR conditions, and the film cooling effectiveness increases with BR. Then, at the second stage, with the further increase of BR, high velocity part of coolant penetrates into the mainstream, reducing the local cooling performance. While other part of coolant with lower speed can still attach on the hot side surface, enhancing the film cooling 9

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(a) BR=1.5

(b) BR=3.5 Fig. 13. Effects of inclination angle and expansion angle on local adiabatic film cooling effectiveness.

upstream part of the hole along two lateral borders, as shown in Fig. 15(c). The velocity distribution at the hole exits mentioned above is decisive for the vortical structure and thus the local film cooling effectiveness downstream of the holes, as shown in Fig. 16. In the figure, the three-dimensional vortex sheet is obtained on the basis of Q-Criterion ∼ and colorized with non-dimensional temperature, T , which is defined as

Three-forked pattern Coolant diffusion (a)

T − Tc ∼ T = T∞ − Tc

Three-forked pattern

(8)

∼ where T is the local physical temperature. Obviously, 0≤T ≤ 1, and ∼ T ≡ 1 − η on the walls. It can be noticed that, for each case shown in Fig. 16, the vortex system downstream of the holes consists of a pair of coherent counterrotating main vortices, namely the so-called kidney-shaped pair of vortices, which are accompanied by some secondary vortices. For α = 20° and 25°, since the coolant jet with higher outflow velocity concentrates in the vicinity of the hole centerlines, as shown in Fig. 15(a) and (b), the flow at the position where two main vortices touch can be characterized by streamlines going away from the wall. At the same time, due to the rotating effect, the flow at the lateral sides of the main vortices carries the mainstream hot gas towards the wall, reducing the cooling effectiveness there, as shown in Fig. 16(a) and (b). This statement can be confirmed quantitatively by the temperature distribution on the main vortex sheet, where the temperature at the medial sides is lower than that at the lateral sides. Moreover, this type of vortical structure should be the main cause for the formation of

Coolant diffusion (b)

Bimodal pattern

Coolant diffusion (c) Fig. 14. Diagrammatic sketch of local effectiveness patterns downstream the hole exits.

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(a) α =20°

(b) α =25°

(c) α =30°

Fig. 15. Vertical component of the coolant jet velocity at the outlet of the first row of film cooling holes, β = 13° and BR = 1.5.

three-forked pattern of the local film cooling effectiveness as shown in Fig. 14(a) and (b). While, the vortical structure downstream of the α = 30° hole is obviously different from that of α = 20° and 25° holes. Compared to the latter two, the former have a reversed rotating direction as shown in Fig. 16(c). More specifically, the vortices carry the hot gas to the wall and mixing with coolant in the centerline region, meanwhile bring the coolant away from the lateral walls. Thus, the temperature at the medial sides of vortex sheet is higher than that at the lateral sides, as shows in Fig. 16(c). Consequently, this vortical structure leads to a bimodal pattern of the local film cooling effectiveness like Fig. 14(c). In order to supply a deeper insight on the characteristics of fanshape hole effusion cooling, two-dimensional vortical structures and the corresponding temperature distribution at the vertical cross-sections 1D downstream of the fifth row of film cooling holes are checked as shown in Fig. 17. Combining the observation from Fig. 16, it can be noticed that the pairs of vortices far away from the centerline of the holes, for all the cases presented in Fig. 17, should be corresponding to the vortices generated from upstream row of holes, and the pair of vortices in the vicinity of the centerline are developed from the nearest holes upstream, i.e. the cross section of the above described “main vortices”. When α = 20° and α = 25°, the pairs of main vortices carry the coolant away from the wall in the centerline region thickening the thermal boundary layer and meanwhile carry the high temperature mainstream hot gas to the lateral sides of the wall, reducing the cooling performance there, as shown in Fig. 17(a) and (b). However, unlike α = 20° case where only main vortices appear, the hole with α = 25° generates some additional smaller and squeezed vertices which are covered by the main vortices. Thus, the distance between the centers of the main vortices downstream of the holes with α = 20° is longer than that of α = 25° case (The former is 1.2D, while the latter is only 0.9D). As a result, the local film cooling efficiency of α = 25° holes shows a more uniform three-forked pattern than that of α = 20° holes, see Figs. 13 and 14. Regarding the holes with α = 30°, it is noticed from Fig. 17(c) that the pairs of main vortices carry the coolant away from the lateral sides of the wall thickening the thermal boundary layer and meanwhile bring the high temperature mainstream hot gas to the centerline region of the wall, thereby reducing the local film cooling performance. Moreover, a pair of secondary vortices are generated from the lateral borders of the holes. As shown in Fig. 17(c), the secondary vortices, combining with the main vortices, develop a relatively stronger outgoing lateral momentum component, contributing to the widening of the coolant footprint. Fig. 18(a) and (b) show the comparisons of the laterally averaged cooling effectiveness of fan-shaped holes with various inclination angles for BR = 1.5 and 3.5, respectively. The expansion angle of the holes are kept at β = 10°. In each case, due to superposition effect of the

effusion cooling, laterally averaged effectiveness increases in the stream-wise direction, and the maximum value appears at the position just downstream the tenth row of holes (x/D = 76.5 approx.). Particularly, when α = 25°, the maximal value is almost close to 1.0. Moreover, it can be seen from Fig. 18 that, for both BR = 1.5 and 3.5, the holes with inclination angle of α = 25° provides the higher laterally averaged effectiveness in the region of x/D ≥ 17 than that of α = 20° and α = 30°. While, the laterally averaged adiabatic film cooling effectiveness of the holes with α = 30° is the lowest in the entire region, although its maximum value can approximate 0.7. From the comparisons of laterally averaged effectiveness of the holes with β = 13°, as shown in Fig. 19, similar statements in regard to the effects of the inclination angle can be deduced. In the downstream area (x/D ≥ 17), the laterally averaged effectiveness of the holes with an inclination angle of α = 25° is higher than that of other two inclination angles. While, the effectiveness of α = 30° is still the lowest one, although, in the downstream area, it tends to be closer to the effectiveness of α = 20° at a larger blowing ratio. 3.2.2. Effects of expansion angle It can be seen from Fig. 13 that an increase of the expansion angle (β ), at the same blowing ratio condition, leads to a better lateral spreading of the coolant and a shortening of the intensely cooled area downstream of the holes. These observations should be attributed to the augment of lateral component of the coolant jet momentum with the increasing β . In addition, an improved lateral spreading of the coolant can suppress coolant jet blow-off at high blowing ratios. Figs. 20–22 are plotted to clarify the influence of the expansion angle on the laterally averaged adiabatic film cooling effectiveness for α = 20°, 25°, and 30°, respectively. Where, the expansion angles are 10° and 13°, and the blowing ratios are BR = 1.5 and BR = 3.5. From Fig. 20, it can be noticed that, when α = 20°, the laterally averaged film cooling effectiveness increases slightly with the expansion angle for the whole hot-side surface of the film plate except limited regions. This phenomenon, again, can be attributed to the improved lateral spreading of the coolant jet. For α = 25°, blow-off phenomena takes place partially at BR ≥ 1.5. And, as mentioned above, the coolant jet ejected from the film cooling holes with larger expansion angle tends to attach on the hot-side surface of the film cooling plate more easily. Therefore, with the decrease of the expansion angle, more coolant air tends to detach from the hot-side surface, reducing the local film cooling performance. However, on the other hand, the mainstream temperature in the near wall area is cooled significantly due to the mixing with the “blow-off” coolant, improving the cooling performance downstream as shown in Fig. 21. As a result, compared to the cases of β = 13°, the laterally averaged film cooling effectiveness of β = 10° is lower in the upstream region (0 ≤ x/D ≤ 17 11

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(a) α =20°

(b) α =25°

(c) α =30° Fig. 16. Three dimensional vortical structure and the corresponding distribution of cooling effectiveness, β = 13° and BR = 1.5.

for BR = 1.5; 0 ≤ x/D ≤ 34 for BR = 3.5), but slightly higher in the downstream area. From the viewpoint of the uniformity of film cooling effectiveness in the entire region of the film cooling plate, the cooling performance of the fan-shaped holes with an expansion angle of 13° is

more preferable than that of the expansion angle is 10°. When α = 30°, as shown in Fig. 22, it is quite obvious that the film cooling effectiveness of the holes with the expansion angle of 13° is higher than that of 10°, which becomes much more evident with the 12

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(a) α =20°, β =13°

(b) α =25°, β =13°

(c) α =30°, β =13° Fig. 17. Vortices and the corresponding non-dimensional temperature distribution at the vertical cross-sections (BR = 1.5).

that, for the effusion cooling, the effectiveness changes significantly with the variation of hole inclination angle and expansion angle even under a constant blowing ratio BR. In order to study the influence of the hole geometries as well as the superposition effect of effusion cooling, an area-averaged film cooling effectiveness, η , is introduced. Here, the η is based on the area between the locations of xstart and x, i.e.

increasing blowing ratio, which can be seen from the quantitative analysis in Fig. 22(a) and (b). 3.2.3. Superposition effect of effusion cooling In Ref. [15], the laterally averaged film cooling effectiveness of a single row of fan-shaped holes with α = 30° is correlated as a function of the blowing ratio BR, the coverage ratio Y/W, and the outlet-to-inlet area ratio of the holes AR. Note that the values of Y/W and AR in the present study are basically kept to be constant. However, it is revealed

Y

η (x ) = 13

x

x

∫ 0 ∫ xstart η (x , y ) dxdy x −xstart Y

=

∫ xstart η (x ) dx x −xstart

(9)

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(a) BR=1.5

(b) BR=3.5

Fig. 18. Influence of the inclination angle on the laterally averaged adiabatic film cooling effectiveness when β = 10°.

(a) BR=1.5

(b) BR=3.5

Fig. 19. Influence of the inclination angle on the laterally averaged adiabatic film cooling effectiveness when β = 13°.

(a) BR=1.5

(b) BR=3.5

Fig. 20. Influence of expansion angle on laterally averaged cooling effectiveness, α = 20°.

14

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(a) BR=1.5

(b) BR=3.5

Fig. 21. Influence of expansion angle on laterally averaged cooling effectiveness, α = 25°.

(a) BR=1.5

(b) BR=3.5

Fig. 22. Influence of expansion angle on laterally averaged cooling effectiveness, α = 30°. Table 2 Coefficients in Eq. (10) and the corresponding coefficient of determination.

π 9

≤α≤

α=

5π 36

π 6

R2

a0

a1

b0

b1

b2

ξ0

ξ1

0.5 ≤ BR ≤ 7.5

0.2083

1.4670

1.3282

0.0235

0.0832

0.1304

−0.7503

0.0946

1.2942

0.2459

0.7021

97.1%

BR ≤ 1.5 BR > 1.5

0.4282 0.7480

1.1918 1.5798

1.2323 1.0971

0.7745 0.9782

0.0664 0.0058

0 1.4832

0 0

−1.9949 −11.7272

2.3114 0.4783

0 0

2.4722 −2.6150

98.4% 99.4%

ξ2

ζ0

ζ1

ζ2

continuous variation of η in the stream-wise direction. The comparisons of ηi with the corresponding fitting curves have been presented in Fig. 23, showing good agreements.

where the value of xstart is set at zero in the present study. Then, the CFD data are correlated for the prediction of η (x i ) = η (i·X ) , i = 1, 2, …8. For the staggered pattern of fan-shaped holes, η (x i ) could be correlated as

4. Conclusions

η (x , BR, α, β ) −ξ0 + ξ1 α + ξ2 β x ⎤ = (−a0 + a1 BR) BR−ζ0+ ζ1 α + ζ2 β ⎡b0 − ⎛b1 + b2⎞ ⎢ ⎥ ⎝ D ⎠ ⎣ ⎦

The effusion cooling characteristics of ten rows of staggered arranged fan-shaped holes with various combinations of inclination angle and expansion angle have been investigated numerically. Selected cases are validated with experiments using thermo-chromic liquid crystal technology. The obtained numerical results agreed well with the corresponding experimental data. Through the simulation, the effects of blowing ratio, inclination angle and expansion angle on the local and spatially averaged film cooling effectiveness are analyzed. The results clearly indicated that the interplay of the hole geometry and the flow condition dominate the helical motion of the coolant inside the holes and hence the velocity distribution at the hole exits, which is then

(10)

π 9 ⩽ α ⩽ π 6, for X/D = 8.5, Y/D = 6.5, 0.5 ≤ BR ≤ 7.5, 10π 180 ⩽ β ⩽ 13π 180 . The coefficients in Eq. (10) and the corresponding coefficient of determination R2 quantifying the wellness-of-fit are given by Table 2. The above correlation can be used to predict the stream-wise row-byrow superposition of cooling effectiveness as multiple rows of fanshaped holes exist in the cooling system. Note that the correlation equation is of a relatively simple form, and that it can’t well fit the 15

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(b) α = 20°, β = 13°

(a) α = 20°, β = 10°

(c) α = 25°, β = 10°

(d) α = 25°, β = 13°

(e) α = 30°, β = 10°

(f) α = 30°, β = 13°

Fig. 23. Comparisons of correlation curves with CFD data.

decisive for the vortical structure and thus the local film cooling effectiveness downstream of the holes. The main conclusions are summarized as follows.

performance of the downstream region. (2) For smaller inclination angles of α = 20° and 25°, the local film cooling performance present the three-forked pattern, one tine related to the centerline and the other two corresponding to the lateral borders of the hole, while the holes with α = 30° yield a typical bimodal pattern of the local film cooling performance. (3) The variation of the expansion angle β from 10° to 13° does not show such evident influence as that of the inclination angle. As

(1) The coolant tends to blow off at higher blowing ratio, reducing the local film cooling performance. Meanwhile, the mainstream temperature in the near wall area is cooled significantly due to the mixing with the blow-off coolant, improving the heat transfer 16

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compared to the hole with β = 10°, the hole with a 13° expansion forms a slightly wider cooling film. (4) From the viewpoints of the effectiveness and the uniformity of film cooling performance, the fan-shaped hole with α = 25° and β = 13° is more preferable for the conditions considered in the present study. (5) For all the cases considered, the film cooling effectiveness increased in the stream-wise direction due to the accumulation of the coolant from multiple rows of film cooling holes. A correlation for the areaaveraged cooling effectiveness is developed, which could be useful for the prediction of row-by-row superposition effects of effusion cooling with fan-shaped holes.

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It should be pointed out that the current study mainly focuses on the flow and heat transfer characteristics of effusion cooling from the fanshaped holes with limited geometric parameters. Whereas, it seems that there exists an optimal combination of the inclination angle and the diffusion angle under the constraint of constant hole exit width. To achieve the most favorable film cooling performance, an optimization of hole geometrical parameters should be carried out with a proper algorithm, which will be considered in the further work. Acknowledgements This work is supported by United Innovation Program of Shanghai Commercial Aircraft Engine under grant AR908 (The program was founded by Shanghai Municipal Commission of Economy and Informatization, China; Shanghai Municipal Education Commission, China; and AECC Commercial Aircraft Engine Co., LTD, China); Natural Science Foundation of Shanghai, China; Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, China; and the Doctoral Students Research Funding of Fudan University, China. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.applthermaleng.2019.113978. References [1] R.J. Goldstein, E.R.G. Eckert, J.W.R. Rhine, Film cooling with injection through holes: adiabatic wall temperatures downstream of a circular hole, J. Eng. Power 90 (1968) 384–395. [2] R.S. Bunker, A review of shaped hole turbine film cooling technology, J. heat transf. 127 (2005) 441–453. [3] R.J. Goldstein, E.R.G. Eckert, F. Burggraf, Effect of hole geometry and density on three -dimensional film cooling, Int. J. Heat Mass Transf. 17 (1974) 595–607. [4] A. Kohli, D. Bogard, Effects of hole shape on film cooling with large angle injection, in: Proceedings of ASME Turbo Expo 1999, 99-GT-165, 1999. [5] M. Gritsch, W. Colban, H. Schar, K. Dobbeling, Effect of hole geometry on the thermal performance of fan-shaped film cooling holes, ASME J. Turbomach. 127 (2005) 718–725. [6] L. Bonanni, B. Facchini, L. Tarchi, Heat transfer performance of fan shaped film cooling holes, Part I: Experimental analysis, in: Proceedings of ASME Turbo Expo 2010, GT2010-22808, 2010. [7] M. Gritsch, A. Schulz, S. Wittig, Heat transfer coefficients measurements of film cooling holes with expanded exits, in: Proceedings of ASME Turbo Expo 1998, 98GT-28, 1998. [8] Y. Yu, C.-H. Yen, T.I.-P. Shih, M.K. Chyu, S. Gogineni, Film cooling effectiveness and heat transfer coefficient distributions around diffusion shaped holes, ASME J. Turbomach. 124 (2002) 820–827. [9] C.M. Bell, H. Hamakawa, P.M. Ligrani, Film cooling from shaped holes, J. Heat Transf. 122 (2000) 224–232. [10] G. Li, H. Zhu, H. Fan, Influences of hole shape on film cooling characteristics with

17