Experimental research on the thermal performance of converging slot holes with different divergence angles

Experimental Thermal and Fluid Science 33 (2009) 808–817

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Experimental Thermal and Fluid Science journal homepage: www.elsevier.com/locate/etfs

Experimental research on the thermal performance of converging slot holes with different divergence angles Cun-liang Liu *, Hui-ren Zhu, Jiang-tao Bai, Du-chun Xu School of Engine and Energy, Northwestern Polytechnical University, 127 You Yi Xi Lu, Xi’an 710072, China

a r t i c l e

i n f o

Article history: Received 27 November 2008 Received in revised form 22 February 2009 Accepted 23 February 2009

Keywords: Film cooling Converging slot-hole Thermal performance Divergence angle Transient measurement

a b s t r a c t Thermal performances of two kinds of converging slot-hole (console) with different divergence angles have been measured using transient liquid crystal measurement technique which can process the nonuniform initial wall temperature. Four momentum ratios are tested. Consoles with different divergence angles produce different cooling effectiveness distributions in the upstream region. However, the cooling effectiveness distributions of the two consoles are similar in the downstream. The laterally averaged cooling effectiveness results show that the differences between the two consoles are very small and the best momentum ratio for both consoles’ cooling effectiveness distribution is around two. With the momentum ratio increasing, the normalized heat transfer coefficient h/h0 of both consoles increases, but the h/h0 value of small divergence case is higher and becomes progressively higher than that of large divergence case. Moreover, the effect of the couple vortices on the heat transfer coefficient distributions is more significant for the large divergence case. Both consoles provide the surface a certain degree of thermal protection, especially in the upstream region. The distributions of heat flux ratio q/q0 are similar with those of g because the influence of g on q/q0 is much larger than that of h/h0 on q/q0. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction The desire for higher overall efficiency and higher specific thrust in air, land and sea gas turbines renders the need for increase in turbine entry temperatures. The turbine blades cannot withstand these temperatures and cooling technologies are required. Film cooling is one of the major cooling technologies for protection of turbine airfoils from the hot gas stream. Since the coolant that facilitates the film cooling is extracted from the compressor, poor management of the cooling air can be otherwise detrimental to the engine overall efficiency. Therefore, the design of film cooling systems should be optimized to produce the most effective film cooling with a minimum amount of coolant, and to minimize the aerodynamic loss associated with film cooling. Many studies show that employing shaped holes is an effective way to improve the film cooling performance (e.g. Bunker [1]). The most common configurations of shaped holes are expansion shaped holes which have expanded exits. Because expansion shaped holes can be manufactured easily in the blades, numerous studies on the film cooling performance of these expansion shaped holes have been/are being performed. The study of Goldstein et al. [2], which is recognized as the first to demonstrate and qualify the film cooling effects due to shaped holes, reported a significant increase in film cooling effectiveness in the near hole region as well * Corresponding author. Tel.: +86 029 88486075; fax: +86 029 88495911. E-mail address: [email protected] (C.-l. Liu). 0894-1777/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2009.02.010

as greater lateral coolant coverage for fan-shaped holes compared with standard cylindrical holes. The improvement in film cooling effectiveness brought by the expansion shaped holes has also been reported by Schmidt et al. [3], Gritsch et al. [4] and Yu et al. [5]. All these studies attributed the improvement primarily to the reduction in momentum flux ratio at the exit of the hole causing a decreased penetration of the jet into the mainstream. Film cooling flow field measurements made by Thole et al. [6] prove this idea. They measured velocity and turbulence flow field downstream of three hole configurations: cylindrical hole, fan-shaped hole and laidback fan-shaped hole. According to Thole et al. [6], expansion shaped holes produce less jet penetration, reduced velocity gradients, and lower turbulence production relative to cylindrical holes. Heat transfer coefficient measurements of expansion shaped holes were carried out by Makki and Jakubowski [7], Sen et al. [8], Gritsch et al. [9] and Yu et al. [5]. They found that expansion shaped holes generally produce lower spatially averaged heat transfer coefficient ratios than cylindrical holes. Furthermore, Saumweber et al. [10] investigated the effect of mainstream turbulence intensity Tu on the film cooling performance of shaped holes that had identical configurations as those of Gritsch et al. [4]. Investigations on film cooling performance of compound angle shaped holes can be found in Sen et al. [8], Schmidt et al. [3], Dittmar et al. [11] and Bell et al. [12]. Generally, compound angle shaped holes produce higher effectiveness and better protection over much wider ranges of blowing ratio than the axially oriented holes. More studies about expansion shaped holes can be found in the review paper of Bunker [1].

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Nomenclature a c D h I q ReD t T U X Y Z

square root of thermal diffusivity specific heat of wall, [J/kgK] film hole inlet diameter, [m] convective heat transfer coefficient, [W/m2K] momentum flux ratio (= qc U 2c =qg U 2g ) heat flux, [W/m2] Reynolds number based on film hole inlet diameter(=qgUgD/lg) time, [s] temperature, [K] velocity, [m/s] streamwise coordinate originating at the center of cooling hole exit, [m] normal coordinate, [ m ] spanwise coordinate originating at the center of cooling hole exit, [ m ]

Aerodynamic efficiency penalty is a big problem in applying expansion shaped holes. Measurements made by Day et al. [13] showed that the use of fan-shaped hole rows in the vanes essentially double the decrement in stage efficiency over that associated with the use of only cylindrical holes. To reduce aerodynamic efficiency penalty while producing high film cooling effectiveness, Sargison et al. [14,15] demonstrated a converging slot hole, or console, geometry in both flat plate and cascade testing. Console transits from circular to slot with convergence in the axial direction and divergence laterally. The hole area diminishes to cause flow accelerate. This accelerated flow is speculated to have lower jet turbulence and more stability. Flat plate tests showed the effectiveness of consoles to be similar to fan-shaped holes, while aerodynamic loss is less. Transonic cascade tests also showed less aerodynamic loss caused by console than that associated with fan-shaped hole and cylindrical hole. Sargison et al. [16] present flow visualization experiments for a row of consoles and cylindrical holes and fan-shaped holes and a slot at the same inclination angle of 35° to the surface. Flow visualization data demonstrated that the console film remained thin and attached to the surface for the momentum flux ratios of 1.1–40 and for a case with no crossflow because of Coanda effect. Detail flow structures of console film cooling flow field were investigated by Azzi and Jubran [17] and Liu et al. [18] through numerical simulations. They found that jets from consoles formed completely different flow structures from that of cylindrical holes and expansion shaped holes. Geometry parameters of shaped holes are important concerns of manufacturing and could significantly influence the thermal performance. Among the few published papers, only Kohli and Bogard [19] and Gritsch et al. [20] investigated the effect of hole geometry on the thermal performance of laidback fan-shaped hole in terms of film cooling effectiveness and discharge coefficient. To our knowledge, few published studies have been carried out on the effect of console’s geometry parameters. Moreover, studies of Sargison et al. [14–16] connect the exits of consoles to form a continuous slot exit. But discrete consoles are more favorable to maintain the strength of the blade. And when the discrete consoles are used, console’s geometry parameters, such as divergence angle, exit-entry area ratio, could have influence on the thermal performance of console. This paper presents flat plate thermal performance measurements of discrete converging slot holes with variation in a very important geometry parameter for console: divergence angle. Transient liquid crystal measurement technique has been used in the present paper to obtain detailed distributions of film cooling effectiveness and heat transfer coefficient.

Greek symbols a inclined angle, [°] b divergence angle, [°] g film cooling effectiveness q density, [kg/m3] l molecular viscosity, [Ns/m2] k thermal conductivity of wall, [W/mK] boundary layer displacement thickness d* Subscripts aw adiabatic wall c jet g mainstream i initial t = 0 s surface 0 without injection

2. Transient liquid crystal technique for film cooling measurements The transient heat transfer measurement is based on the assumption that the transient conduction in the test plate is onedimensional. When the test plate has a low thermal diffusivity (e.g. Perspex), one-dimensional assumption is often a good approximation, since the surface temperature response is limited to a thin layer near the surface and lateral conduction is small (Vedula et al. [24]). The one-dimensional transient conduction equation is:

@Tðy; tÞ @ 2 Tðy; tÞ ; ¼ a2 @t @y2

sffiffiffiffiffiffi k a¼ qc

ð1Þ

The boundary condition at the plate surface (y = 0) is obtained using an energy balance:

k

@Tðy ¼ 0; tÞ ¼ h½T s ðtÞ  T aw ðtÞ @y

ð2Þ

where the Taw(t) stands for the adiabatic wall temperature. This value can be replaced by the film cooling effectiveness, which is defined by:



 



g ¼ T aw ðtÞ  T g ðtÞ = T c ðtÞ  T g ðtÞ

ð3Þ

Here the variation of mainstream temperature and jet temperature with time are approximated by power series of the following forms,

T g ðtÞ ¼

M X m¼0

Bm

tm ; Cðm þ 1Þ

T c ðtÞ ¼

N X n¼0

An

tn

Cðn þ 1Þ

ð4Þ

usually chosen of 4th–5th order. In the former work including Vedula and Metzger [21], Drost et al. [22], Yu et al. [5] and Chambers et al. [23], the initial wall temperature was always considered as constant. In the present measurement, the initial wall temperature is measured and approximated by power series of 3rd order:

e þ By e 2 þ Dy e þ Cy e 3 T i ðyÞ ¼ f ðyÞ ¼ A

ð5Þ

In our experiments, the nonuniform initial temperature, which is caused by temperature difference between the test plate and the air in the test duct, is usually in a small region near the plate surface. The distribution in this region can be properly approximated by 3rd order power series. And it is sufficient to get accurate results according to the analysis in another paper [27].

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Eqs. (1)–(5) can be solved analytically using the Laplace transform technique. The solution is a function of wall surface temperature Ts with respect to time t. It is also the calculation formula of h and g in the transient measurement: " # pffiffi N 2n M X X X ðb tÞk k   E0 þ T s ðtÞ ¼ An gb2n Bm ð1  gÞb2m C 2þ1 n¼0 m¼0 k¼0 " # ( pffiffi k  pffiffi  2m 2e e X ðb tÞ e 0 þ a B ½1  E0  þ 2a C 2pffiffiffitffi  1 þ E0 k   E0 þ AE  b 2 p b b C 2þ1 k¼0 ) p ffiffi   e 6a3 D 2 t 1  E0 þ ð6Þ t  pffiffiffiffi þ b b p b2

different Tg(t) and Tc(t) which are measured by K type thermocouples. The initial wall temperature distribution Ti(y) is measured by the thermocouples shown in Fig. 2. The location of this cross-section which is perpendicular to the mainstream is 180 mm from the start of the test plate. The nonuniform initial wall temperature distribution is always in the region of [0, 10 mm] in the Y direction according to the measurement. And due to that the test region of interest is at least 10 mm from the edge of test plate in the X and Z direction, the temperature in the X-Z plane under the test region surface can be considered as uniform. 3.2. Film cooling hole configurations

pffiffiffiffiffiffiffiffi pffiffi 2 where b ¼ h= qck; E0 ¼ eb t  erfcðb t Þ. In the present work, a single layer of narrow-band liquid crystal was used as a surface temperature indicator. And 6–8 tests were carried out at identical momentum flux ratios, whereas the mainstream temperature and the jet temperature were varied. This multiple-test method is same with the method in Drost et al. [22].

3. Experimental apparatus and procedure

Configuration of the console and the console row in the present study is shown in Fig. 3. In the present study, two kinds of console are investigated with difference in the divergence angle b. Different divergence angles are caused by different hole lengths. The two tested divergence angles are: b = 11° and b = 21°. The other geometry parameters are the same for the two consoles. Hole entry diameter D is 10 mm. The hole exit length and the hole pitch and the inclined angle a is 3D, 3.5D and 35°, respectively.

3.1. Experimental apparatus

3.3. Operating conditions

Fig. 1 shows a schematic of the overall test setup. Mainstream passes through valves, settling chamber, contraction section before entering the mesh heater. Moreover, another contraction is placed after the mesh heater to ensure a uniform mainstream entering the test section which is a Perspex rectangular duct with 220 mm in width and 75 mm in height. There is knife bleed structure at the start of the test section to control the origin of the mainstream boundary layer. And there was a small step installed at 100 mm upstream of the hole exit to guarantee a turbulent boundary layer on the test plate. The test plate is a Perspex plate with size of 550 mm  40 mm  220 mm in X, Y and Z directions. A transient test is initiated by switching the solenoid valve and the butterfly valve simultaneously to introduce the mainstream and the secondary flow injection into the test section. Simultaneously, a CCD camera starts to record the vide image of the liquid crystal coated on the test surface. Different tests are distinguished by varying the power of heating mainstream and the secondary flow to produce

The mainstream velocity was maintained at 17 m/s, the corresponding Reynolds number based on film hole inlet diameter, ReD, was 10000. The turbulence intensity of mainstream was of the order of 2%. Four momentum flux ratios were tested: 0.5, 1, 2, 4. The calculation of the momentum flux ratio I is based on the hole inlet cross -sectional area. Because the temperature difference between mainstream and jet is not very large, the density ratio qc/qg is nominally equal to one. 3.4. Experimental uncertainties and validation test Error analysis has been performed with the method proposed by Kline and McClintock [25]. In the present experiment, the measurement uncertainties include temperature uncertainties: DTg, DTc, DTi, DTs; and time uncertainty pffiffiffiffiffiffiffiffi Dt in Tg(t), Tc(t), Ts(t); and material property uncertainty D qck. The estimated uncertainty intervals for the present experiment are: DTg = DTc = DTi = DTs = ± 0.2oC;

Temperature acquisition

Computer Blower

Air tank

valve

CCD camera

Mesh heater

-

TC Test section Butterfly valve

Test plate Settling Contraction + Secondary chamber section contraction Solenoid valve

Blower Air tank Flow meter

valve

-

+ Air heater

Fig. 1. Sketch of experiment system.

TC

controller

Solenoid valve

Tungsten halogen lamp

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100mm

100mm

20mm

X Z 40mm

11 1 2 3 4 5 6 7 8 9 10

Y

Fig. 2. Sketch of thermal couples’ distribution in the test plate.

3.5D

0.5

3D

β

C-HOLE: I=1 (Present study) C-HOLE: I=1 (Ref. [26]) C-HOLE: I=1 (Ref. [5]) C-HOLE: I=4 (Present study) C-HOLE: I=4 (Ref. [26])

X 0.4

Z 0

ave-η

0.3

D

Y

0.2

X

α

0.1

Fig. 3. Configuration of the consoles in the present study.

0

5

10

15

20

25

30

35

X/D 2 C-HOLE: I=1 C-HOLE: I=1 C-HOLE: I=4 C-HOLE: I=4

1.8

(Present study) (Ref. [26]) (Present study) (Ref. [26])

1.6

ave-h/h0

pffiffiffiffiffiffiffiffi Dt = ± 0.1s; D qck ¼ 20. The temperature and time uncertainties in Tg(t) and Tc(t) have been assessed to be the uncertainties of the fitting coefficients in Eq. (4). According to the uncertainty intervals given above, the relative uncertainties in heat transfer coefficient is about 6%, and in local film cooling effectiveness is about 15% at g = 0.1 and 3% at g = 0.7. Moreover, to validate the present measurement technique, film cooling effectiveness and heat transfer coefficient of cylindrical hole film cooling were measured and compared with the data of Goldstein et al. [26]. The operating conditions are the same with the description in Section 3.3. The cylindrical hole diameter is 10 mm in current study. And the pitch-diameter ratio and the inclined angle are 3 and 35°, respectively which are same with the data in Goldstein et al. [26]. Fig. 4 shows the currently measured results and the published results in the form of laterally averaged values in one hole pitch. Good agreement was obtained, especially for the film cooling effectiveness results. However, the h/h0 results are a little larger for the current measurement. According to the analysis in Bunker [1], this is due to the smaller d*/D in the current experiment, because the hole diameter in the our study is larger than that used in Goldstein et al. [26] which is 6.35 mm. The validation of the measurement technique gives confidence in the subsequent console results.

1.4

1.2

1 5

10

15

20

25

30

35

X/D 4. Results and discussion Fig. 4. Cylindrical hole results compared with published data.

4.1. Film cooling effectiveness Figs. 5 and 6 present the measured film cooling effectiveness distributions of the two consoles with different divergence angles.

The effectiveness distributions are displayed in one hole pitch [1.75D, 1.75D] as well as the distributions of the other two quantities showing in the following sections.

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Fig. 5. Distributions of local film cooling effectiveness of console with divergence angle 11°.

Although the consoles in this study are discrete and there is space about 0.5D between adjacent consoles, the jets from both consoles cover the entire spanwise region in the streamwise from 3.5D to 36D at all the four momentum flux ratios. Maybe in the upstream of 3.5D the jets can’t cover the entire spanwise region, discrete consoles still show the ability to form continuous film in most of the downstream region. This ability should be owed to the divergence angle to some degree because the lateral divergence of console can make jets have lateral velocity component and cover wider than the exit width. Also due to this reason, consoles with different divergence angles can produce different film coverage as well as different effectiveness distributions. The comparisons between Figs. 5 and 6 do show that the effectiveness distribution features under different momentum ratios are similar for the same console, but the distributions of different divergence cases are different, especially in the upstream region, say X/D < 8. For consoles with large divergence angle, the effectiveness peak is located in the midspan between adjacent holes. The location of peak value of consoles with small divergence angle is still at the edge of the jets, but not in the midspan between holes. It slightly moves towards the hole centerline. According to the analysis above, large divergence angle can make jets have relatively large lateral velocity component, so adjacent jets will intersect and accumulate in the midspan to produce very high effectiveness. When the divergence angle is small, jets cannot penetrate the mainstream in the lateral with the small lateral velocity component. So the effectiveness in the midspan is

Fig. 6. Distributions of local film cooling effectiveness of console with divergence angle 21°.

0

Z/D Fig. 7. Secondary vortices at cross-flow downstream of console.

lower relatively. But due to the couple vortices formed by interaction between the mainstream and the jet, film is thickened near the jet edge, and thinned near the hole centerline, and the effectiveness at the jet edge is higher than that at the hole centerline. Fig. 7 shows the typical form of secondary vortices downstream of console which is obtained by Azzi and Jubran [17] and verified by Liu et al. [18]. The couple vortices also exist in the flow field

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of consoles with large divergence angle. Because of different divergence angle, the effect of couple vortices is different. They reduce the effectiveness in the region between jet edge and jet centerline and form a ‘‘W” shaped high effectiveness region in the upstream which is notable under large momentum ratios for large divergence angle case. Different from the distributions in the upstream, the effectiveness distributions of the two consoles are similar in the downstream. The effectiveness decreases gradually from midspan of two holes to hole centerline in the spanwise direction. The reason for the above is that the effect of divergence angle on the film fades and couple vortices flow structures begin to dominate the film. The couple vortices make the film and the effectiveness in the midspan thicker and higher than those in the center region. Fig. 8 gives the laterally averaged cooling effectiveness in one hole pitch. From Figs. 5, 6 and 8 we can see that the cooling effectiveness of both consoles decreases along streamwise direction under all the four momentum ratios. This means the jets ejected from both consoles provide very good coverage on the surface under all the momentum ratios and no obvious lift-off and reattachment occur that always exist in the large momentum ratios’ film cooling of cylindrical hole. However, slight lift-off occurs when the momentum ratio is four in authors’ opinion, because comparisons of effectiveness values shown in Figs. 5, 6 and 8 show that the effectiveness increases as momentum ratio increasing when I 6 2, but decreases when I P 2 in the upstream region. According to authors’ analysis, it is the slight lift-off of jets of I = 4 that lets small amount of mainstream penetrate underneath the jets causing the effectiveness lower than that of I = 2 in the upstream. Moreover, the fact that the highest average effectiveness of both consoles is obtained at I = 2 indicates that the best momentum ratio for console’s cooling effectiveness distribution is around two. The differences in the laterally averaged effectiveness between the two consoles are very small or even within the range of measurement uncertainty when the momentum ratio is 1 or 2 or 4. Lower average effectiveness in the region X/D < 20 for consoles with large divergence angle at I = 0.5 is caused by the jet’s intense lateral spreading which can significantly reduce the film’s capacity to resist the mixing of mainstream when the coolant flux is small. However, the differences are still not very large. So we can come to the conclusion that the influence of the divergence angle on the laterally averaged cooling effectiveness of console is very small, at least within the range of the two divergence angles in the present study.

Fig. 9. Distributions of local heat transfer coefficients of console with divergence angle 11°.

4.2. Heat transfer coefficient Figs. 9 and 10 show the distributions of the two consoles’ heat transfer coefficient. The heat transfer coefficient in the present 1

1 β=21 , I=0.5 o β=11 , I=0.5 β=21 o, I=4 o β=11 , I=4

β=21 o, I=1 o β=11 , I=1 β=21 o, I=2 β=11 o, I=2

o

0.8

0.8

0.6

ave-η

ave-η

0.6

0.4

0.4

0.2

0.2

0

5

10

15

20

25

30

35

0

5

10

15

20

X/D

X/D Fig. 8. Streamwise distribution of laterally averaged g.

25

30

35

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Fig. 10. Distributions of local heat transfer coefficients of console with divergence angle 21°.

paper is displayed in the normalized form: h/h0. The value of h0 was obtained using the same experimental setup and under the same mainstream, but without injection. The test under this condition is a conventional two-temperature convection problem and it requires only one test to reveal the heat transfer coefficient. The

calculation formula is also Eq. (6), but the g in it is zero in this case. When measuring h0, film holes were closed and the transient test was initiated by only introducing the mainstream into the test section. After obtaining the distribution of h0, dividing h with h0 which is at the same position to obtain the distribution of h/h0. Fig. 11 also gives the laterally averaged heat transfer coefficient in one hole pitch. Comparisons of the two consoles’ results shown in Fig. 11 indicate that the smaller the momentum ratio is, the smaller the differences of the averaged heat transfer coefficient are between the two consoles. Under I = 0.5 condition, the laterally averaged heat transfer coefficients are almost the same along the entire streamwise region for the two consoles. With the momentum ratio increasing, the averaged heat transfer coefficient of both consoles increases in the entire streamwise region, but the heat transfer coefficient value of console with small divergence angle is higher and gets larger and larger than that of large divergence angle case. The discussion in Section 4.1 shows that the jets from both consoles cover on the wall very well. So the wall surface is scoured directly by the jets. The larger the momentum ratio is, the larger the jet velocity is and the higher the heat transfer coefficient is. Large divergence angle means relatively large flow area as well as relatively small jet velocity. So the heat transfer coefficient of large divergence case is lower relatively. However, the velocity difference caused by the different divergence angle will decrease when the momentum ratio becomes smaller according to mathematical analysis. So the average heat transfer coefficient gets closer and closer as the momentum ratio decreasing. Actually not only the average heat transfer coefficient, but also the heat transfer coefficient distributions are more similar under smaller momentum ratios according to comparisons between Figs. 9 and 10. Under I = 0.5 condition, the heat transfer of the two consoles is enhanced in the upstream X/D < 6 due to the intense disturbance caused by the interaction between the mainstream and the jets. Moreover because the mainstream velocity is larger than that of jet, the normalized heat transfer coefficient in the region between holes is larger than that right downstream of hole. As flowing downstream, the boundary layer becomes thicker and the disturbance fades. The normalized heat transfer coefficient decreases gradually, even below one because of the smaller jet velocity. However, the couple vortices shown in Fig. 7, which can thicken the boundary layer in the region between holes and thin the boundary layer right downstream of the hole, begin to display effect from X/D = 10. They enhance the heat transfer right downstream of the hole,

3

3 β=21 , I=1 β=11 o, I=1 β=21 o, I=2 β=11 o, I=2 o

β=21 , I=0.5 β=11 o, I=0.5 β=21 o, I=4 o β=11 , I=4 o

2.5

ave-h/h0

ave-h/h0

2.5

2

2

1.5

1.5

1

1 5

10

15

20

25

30

35

5

10

15

X/D

20

X/D Fig. 11. Streamwise distribution of laterally averaged h/h0.

25

30

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C.-l. Liu et al. / Experimental Thermal and Fluid Science 33 (2009) 808–817

Fig. 12. Distributions of local heat flux ratios of console with divergence angle 11°.

but reduce the heat transfer in the region between holes. And the traces of the couple vortices are more clear for large divergence angle case because larger divergence can produce intenser couple vortices. Fig. 10 indicates that in the downstream region the effect of the couple vortices on the heat transfer coefficient distributions of large divergence angle case is significant for all the four momentum ratios. In the upstream region, jets from large divergence angle consoles intersect in the midspan region between holes. Intense interaction between jet and jet, and jet and mainstream makes normalized heat transfer coefficient there higher than that right downstream of the hole. However, the couple vortices reduce the heat transfer of the region between holes gradually and make it lower than that of the region near the hole centerline. But for small divergence angle case, the effect of the couple vortices on the heat transfer coefficient distributions is not very significant, especially under large momentum ratios. According the analysis in Section 4.1, in the upstream region jets of small divergence angle case can not penetrate the mainstream to intersect in the lateral direction. The midspan region between holes is influenced more by mainstream. So except I = 0.5 case, the highest heat transfer coefficient doesn’t locates at the midspan region, but locates at the edge of jet due to the intense interaction between jet and mainstream in this region. This distribution character is different from that of large divergence angle case, but somewhat similar with the cooling effectiveness distribution of small divergence angle case. With flowing downstream, only under I = 1

815

Fig. 13. Distributions of local heat flux ratios of console with divergence angle 21°.

condition the couple vortices reduce the heat transfer of the region near the jet edge to be lower than that of the region near the hole centerline. Under the other two larger momentum ratios, the couple vortices is nearly ineffective, especially under I = 4 condition. The reason for the insignificant effect of couple vortices is that the velocity of jet of small divergence angle case is so high that the heat transfer is mainly dominated by the jets’ scouring and the influence of secondary vortices on the heat transfer is too small. 4.3. Heat flux ratio According to Sen et al. [8], the ratio of heat flux on a film-protected surface to the corresponding baseline value without film cooling can be expressed by:

q=q0 ¼ ðh=h0 Þð1  g=uÞ;

ð7Þ

where / is a dimensionless temperature ratio given by / = (Tw  Tg)/ (Tc  Tg) which represents the overall cooling effectiveness. Typical values of / in actual engines range from 0.5 to 0.7. The present study uses / = 0.6, as the mean value of the actual range. Figs. 12 and 13 display the distributions of the two consoles’ heat flux ratios. Fig. 14 also gives the laterally averaged heat flux ratio in one hole pitch. According to Fig. 14, the two consoles, in average, provide the surface with a certain degree of thermal protection, as the values of laterally averaged heat flux ratio are less than one in the entire

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1.5

1.5 β=21 , I=1 o β=11 , I=1 o

β=21 , I=0.5 β=11 o, I=0.5 o

1

0.5

ave-q/q0

ave-q/q0

0.5

1

0

0

-0.5

-0.5

-1

-1

-1.5

10

15

20

25

30

35

-1.5

5

10

15

X/D

20

25

β=21 , I=2 o β=11 , I=2

β=21 , I=4 β=11 o, I=4

o

1

o

1

0.5

ave-q/q0

0.5

ave-q/q0

35

1.5

1.5

0

0

-0.5

-0.5

-1

-1

-1.5

30

X/D

5

10

15

20

25

30

35

-1.5

5

10

15

20

25

30

35

X/D

X/D Fig. 14. Streamwise distribution of laterally averaged q/q0.

streamwise test region. Moreover, negative values of q/q0, which mean the direction of heat transfer changes from film to wall to wall to film, appears in the upstream region shown in Figs. 12– 14. The reason for the negative value is that the used value of / is too small for the upstream region where the actual value of / should be close to one. But this also indicates that the consoles can provide very effective thermal protection in the upstream region. Comparisons between the distributions of heat flux ratio and the distributions of cooling effectiveness and heat transfer coefficient show that the shape of heat flux ratio distribution is similar with the shape of film cooling effectiveness distribution under all the four momentum ratios for both consoles. This can be explain it in a mathematical way. The absolute partial derivatives of q/q0 with respect to h/h0 and g can be expressed as:

  @ðq=q0 Þ   @ðh=h Þ ¼ j1  g=uj; 0   @ðq=q0 Þ    @ g  ¼ ðh=h0 Þ=u:

reasons: the first is the average film cooling effectiveness of the two consoles are very close according to Fig. 8, and the other is the same with the analysis in the above paragraph that the influence of g on q/q0 is large. However, larger normalized heat transfer coefficient of the small divergence console still produces higher heat flux ratio in the X/D > 12 region for I = 2 and I = 4 cases and in the X/D < 20 region for I = 1 case. For I = 0.5 case, lower heat flux ratio of small divergence console in the X/D < 20 region is caused by the larger film cooling effectiveness shown in Fig. 8. For the same console, the differences of average heat flux ratio between different momentum ratios are notable in the upstream region, because the differences of normalized heat transfer coefficient between different momentum ratios are larger in the upstream region according to Fig. 11. Higher normalized heat transfer coefficient and higher film cooling effectiveness produces lower heat flux ratio in the upstream region for I = 2 and I = 4 cases.

ð8Þ 5. Conclusion

ð9Þ

In the present paper, / = 0.6, so |1  g//| is always less than one, but (h/h0)// is always much larger than one. This means the influence of g on q/q0 is much larger than that of h/h0 on q/q0. So the distributions of q/q0 are more similar with those of g. The differences between the two consoles’ average heat flux ratios are not very notable according to Fig. 14. There are two

A transient liquid crystal measurement technique which can process the nonuniform initial wall temperature has been successfully carried out for film cooling measurements on a flat plate. Thermal performances, including film cooling effectiveness and normalized heat transfer coefficient and heat flux ratio, of two kinds of converging slot-hole with different divergence angles have been measured. The following conclusions can be drawn:

C.-l. Liu et al. / Experimental Thermal and Fluid Science 33 (2009) 808–817

1. Lateral divergence can enable lightly discrete consoles produce continuous film in most of the downstream region. The film cooling effectiveness distributions of different momentum ratios are similar for the same console. But consoles with different divergence angles produce different effectiveness distributions in the upstream region. However, the effectiveness distributions of the two consoles are similar in the downstream because similar flow structures begin to dominate the film. Moreover, the differences in the laterally averaged effectiveness of the two consoles are very small. Slight lift-off of jets of I = 4 makes the effectiveness lower than that of I = 2 in the upstream. And the fact that the highest average effectiveness of both consoles is obtained at I = 2 indicates that the best momentum ratio for the film cooling effectiveness distribution of console with 35° incline angle is around two. 2. With the momentum ratio increasing, the normalized heat transfer coefficient of both consoles increases, but the value of small divergence case is higher and gets larger and larger than that of large divergence case. The heat transfer coefficient distributions are more similar under smaller momentum ratios for the two consoles. The effect of the couple vortices on the heat transfer coefficient distributions of large divergence angle case is significant for all the four momentum ratios. But for small divergence angle case, the effect of the couple vortices is not very significant, especially under large momentum ratios. 3. Both consoles provide the surface with a certain degree of thermal protection, especially in the upstream region, as the values of heat flux ratio q/q0 are less than one in the entire test region and negative in the upstream. The distributions of q/q0 are similar with those of g because the influence of g on q/q0 is much larger than that of h/h0 on q/q0. The differences between the two consoles’ average heat flux ratios are not very large. For the same console, the differences of average heat flux ratio between different momentum ratios are notable only in the upstream region.

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