J. Cent. South Univ. (2017) 24: 1183-1189

DOI: 10.1007/s11771-017-3521-x

Data mining optimization of laidback fan-shaped hole to improve film cooling performance

WANG Chun-hua(王春华)¹, ZHANG Jing-zhou(张靖周)^{1, 2}, ZHOU Jun-hui(周君辉)¹

1. College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics,

Nanjing 210016, China;

2. Collaborative Innovation Center of Advanced Aero-Engine, Beijing 100191, China

Central South University Press and Springer-Verlag Berlin Heidelberg 2017

Abstract: To improve the cooling performance, shape optimization of a laidback fan-shaped film cooling hole was performed. Three geometric parameters, including hole length, lateral expansion angle and forward expansion angle, were selected as the design parameters. Numerical model of the film cooling system was established, validated, and used to generate 32 groups of training samples. Least square support vector machine (LS-SVM) was applied for surrogate model, and the optimal design parameters were determined by a kind of chaotic optimization algorithm. As hole length, lateral expansion angle and forward expansion angle are 90 mm, 20° and 5°, the area-averaged film cooling effectiveness can reach its maximum value in the design space. LS-SVM coupled with chaotic optimization algorithm is a promising scheme for the optimization of shaped film cooling holes.

Key words: gas turbine; laidback fan-shaped film cooling holes; optimization; support vector machine (SVM); chaotic optimization algorithm

1 Introduction

Film cooling scheme is a typical external cooling method for gas turbines. In film cooling, cooler air is ejected through the blade surface into the external boundary layer, and forms a relatively cool insulating film on the blade surface, which protects the blade surface coming in contact with high gas temperature effectively [1]. To improve the film cooling performance, recent attention has been given to contouring the hole geometry. Film cooling holes with a diffuser-shaped expansion at the exit portion of the hole are believed to yield higher cooling effectiveness than traditional round holes. The flow deceleration in the diffuser section of the hole allows the coolant jet attach more closely to the wall surface. Moreover, the diffuser promotes lateral spreading of the coolant, and thus provides a more uniform film on the surface [2-5].

Many experimental and numerical investigations of shaped film cooling holes have been carried out to analyze cooling efficiency and aerodynamic performance. SAUMWEBER et al [6, 7] reported that the effects of large free-stream turbulence intensity and periodic unsteady wakes on the film cooling performance of fan-shaped holes are always detrimental. The experiments of GRITSCH et al [3] showed that the increase of forward expansion angle of shaped holes can improve the film cooling performance. Moreover, GRITSCH et al [3] indicated that coolant passage cross-flow Mach number and orientation have active effects on film cooling performance. COLBAN et al [8] developed a semi-empirical correlation to predict film cooling effectiveness of laidback fan-shaped holes. They indicated that the film cooling effectiveness is strongly affected by pitch-to-diameter ratio, blowing ratio, hole size, area ratio and width of hole at trailing edge of hole breakout. Film cooling of shaped holes is a complex system affected by a large number of geometric parameters [9]. Designers have to deal with a high- dimensional design space, where a global optimum solution needs to be found for a given set of requirements. So, it is necessary to develop an optimization tool for the design of shaped hole [10, 11]. Based on this background, authors propose a data method for shape optimization of a laidback fan-shaped hole.

In the present optimization method, least square support vector machine (LS-SVM) is used for surrogate model. LS-SVM, one kind of SVM, is a mathematical model with associated learning algorithms that used for regression and classification [12-14]. Compared with traditional surrogate model, LS-SVM takes less computation time, and shows better prediction performance as the size of training sample is small. Furthermore, authors apply a kind of chaotic algorithms for global optimization. Because of the non-repetition of chaos, chaotic optimization algorithm can carry out global searches at higher speeds than the random ergodic searches [15].

In this work, the optimization problem of laidback fan-shaped hole is introduced firstly; and then the CFD model is established; finally, the optimization by LS-SVM and chaos optimization algorithm is performed, and the optimization results are analyzed in detail.

2 Design variables and objective function

The geometry of the computation domain in current research is shown in Fig. 1. The computation domain consists of mainstream duct, coolant plenum, and a laidback fan-shaped hole. The hole diameter (D) is 10 mm, and the incline angle is 30°. The mainstream duct length, width and height are 42D, 6D and 8D, respectively. The coolant plenum length, width and height are 10D, 6D, and 5D respectively. The mainstream and coolant are both compressible air. The mainstream velocity is 130 m/s, and the mainstream total temperature is 540 K and the coolant temperature is 310 K. The blowing ratio is set to 1.0. Three geometrical parameters, including hole length (L), hole forward expansion angle (β_for) and lateral expansion angle (β_lat), are selected as design variables. The definitions of these three parameters are shown in Fig. 2. The lower and upper limits of these three parameters are listed in Table 1.

Area-averaged adiabatic film cooling effectiveness (0avg, is defined as follows:

                      (1)

           (2)

           (3)

where T_ad is the adiabatic wall temperature; T_c is the coolant temperature; T_∞,rec is the recovery temperature of mainstream, and can be determined by energyconservation equation:

             (4)

where T_∞ is the mainstream static temperature; v_∞ is the mainstream velocity; c_p is the specific heat capacity at constant pressure; γ is adiabatic constant.

Fig. 1 Geometry of computational domain:

Fig. 2 Definition of design variables

Table 1 Design space of design variables

3 Methods for solving optimization problem

3.1 Numerical analysis

To investigate the influences of the design variables on the film cooling effectiveness, numerical experiments by Ansys-Fluent 14.0 are performed. The working fluid is ideal gas. At the gas inlet, constant mass flow rate and temperature are specified. At the mainstream outlet, the gauge pressure is set to 0 Pa. At the wall, adiabatic and non-slip condition is specified. The turbulence is solved by standard k-ω equations. The turbulence intensity of inlet gas is estimated by the following empirical correlations [16]:

I=0.16Re^-1/8                                          (5)

where Re is Reynolds number of inlet gas. Mesh grids are generated using GAMBIT with structured tetrahedral topology grids except for cooling hole (shown in Fig. 3). Near the wall, the first grid points are placed at y+>30 so that empirical wall-function can be implemented. Grid independent test is used to determine the optimal grid number in current research.

Fig. 3 Grid used in present work:

3.2 Surrogate model based on LS-SVM

SVM model uses the spirit of the structural risk minimization theory, and builds the input prototypes in a space with greater dimensions by employing a nonlinear mapping method. The major shortcoming of traditional SVM technique is its privileged computational load as the constrained optimization programming is needed in this model. To lower the computation time of SVM, a better technique called LS-SVM was developed by SUYKENS and VANDEWALLE [17]. The solution in LS-SVM is obtained by solving a linear series of equations, instead of inequality constraints. In the present work, LS-SVM is used for surrogate model.

Knowing the approximation issue, the provided dataset of {(I_i, O_i), i=1, 2, …, N}through a nonlinear function is expressed by [17-19]

                    (6)

where I_i is the input vector; O_i is the output value; N is the total number of training points; k(·) denotes the kernel function; a and b can be determined by solving the following equations:

                    (7)

where c denotes the penalty factor; O=[O₁, O₂, …, O_Ns]^T; A_l=[1, 1, …, 1]; a=[a₁, a₂, …, a_N]^T; E is the unit matrix; and Ω(i,j)=k(I_i, I_j). In the present work, radial basis function is used for kernel function:

                    (8)

where δ denotes the kernel parameter.

3.3 Chaos optimization algorithm

The optimization problem can be modeled as follows:

max F(L, β_lat, β_for)

s.t.                   (9)

where F(·) is the objective function. Logistic model is selected as chaotic model to generate chaotic time series by iteration. The expression of Logistic model is shown as follows:

                 (10)

The basic steps of adaptive mutative scale chaos optimization algorithm are listed as follows [20, 21]:

Step 1: Assign two large positive integers to N₁ and N₂ respectively. Set the global searching times n=1, and the detailed searching times m=1.

Step 2: Assign two random real number to t⁽¹⁾=(t₁⁽¹⁾, t₂⁽¹⁾, t₃⁽¹⁾), and generate 3-dimensional time series t^(j+1), j=1, 2, …, N₁ by iteration of Eq. (11). Transform the chaotic time series t_i^(j+1) from the range (0,1) to (t_i,min, t_i,max) by the following equation:

                 (11)

where i=1, 2, 3.

Step 3: Set (L, β_for, β_lat)=and calculate Fⁿ by Eq. (9). If n=1, then F^*=Fⁿ and go to Step 4; If n>1, then go to Step 4 directly.

Step 4: If F^*>Fⁿ, then set F^*=Fⁿ and t_i^*=t_i^’(n).

Step 5: n=n+1. If n1, return to Step 3, or else narrow the range of chaotic variables from (t_i,min, t_i,max) to by the following equations:

                   (12)

where φ is the shrinkage factor, and φ∈(0,0.5). If then If then

Step 6: Generate the new chaotic variable by the following equation:

        (13)

where β_i is adaptive adjustment coefficient and β_i∈(0,1). The value of β_i is determined by the following equation:

                            (14)

where l=2.

Step 7: Set t_i^(m)=t_i^*, and (L, β_for, β_lat)=(t₁^(m), t₂^(m), t₃^(m)). Calculate Fⁿ by Eq. (8). If m=1, then F^*=Fⁿ and go to Step 8. If m>1, then go to Step 8 directly.

Step 8: If F^*>Fⁿ, then F^*=Fⁿ and t_i^*=t_i ^(m).

Step 9: m=m+1. If m2, then return to Step 6. If m=N₂, then end the calculation process.

4 Results and analysis

4.1 Validation of CFD model

To validate CFD model, the simulation results are compared with the experimental data from SAUMWEBER et al [7] in Fig. 4. For cylinder-shaped hole, adding the blowing ratio (M>1.0) leads to pronounced lift-off effects of coolant jet, resulting in the deterioration of cooling performance. However, the flow deceleration in the diffuser section of the shaped hole can mitigate the lift-off effects of coolant jet effectively. So, the film cooling effectiveness of shaped holes always increases as blowing ratio increases from 0.5 to 2.5. Overall, the CFD calculated results show good agreement with the experimental results.

Fig. 4 Comparison of CFD and experimental results [7]:

In the present work, 32 groups of the CFD calculated results are used for the training samples for LS-SVM, and 10 groups of the CFD calculated results are used for the testing samples. These samples are listed in Table 2.

4.2 Prediction performance of LS-SVM

The prediction performance of LS-SVM depends on penalty factor and kernel parameter. In current research, the trial-and-error method is used to determine the value of these two parameters. Figure 5 shows the influences of penalty factor and kernel parameter on the prediction error. The prediction error is defined as

                        (15)

where C_d,cal is the output results of LS-SVM, and C_d,tst is the reference value in the testing samples. The optimal value of the kernel parameter is 0.4, and the optimal value of the penalty factor is 11.

4.3 Analysis of LS-SVM output results

Figure 6(a) shows the influence of hole length on film cooling effectiveness. In current model, the rise of hole length leads to the increase of the area of hole exit, which reduces the coolant jet velocity and allows the jet attach closer to the wall [3]. So, adding the hole length improves the film cooling performance. However, as the area of hole exit exceeds a critical value, the coolant becomes difficult to cover the whole wall surface because of the low momentum. So, for large expansion angle (β_lat=β_for=20°), the film cooling effectiveness decreases with increasing hole length from 70 mm to 90 mm.

Table 2 Training and testing samples

Fig. 5 Influence of SVM parameters on prediction performance

Figure 6(b) shows the influence of lateral expansion angle on film cooling effectiveness. As the lateral expansion angle increases, film cooling effectiveness also increases. The rise of lateral expansion angle mitigates the lift-off effect of coolant jet, which prevents the separation of coolant jet from the wall surface. Moreover, adding lateral expansion angle promotes the lateral spreading of the coolant, and thus provides a more uniform film on the surface [5].

Figure 6(c) shows the influence of forward expansion angle on film cooling effectiveness. For low hole length and lateral expansion angle, adding the forward expansion angle results in the rise of film cooling effectiveness. However, for high hole length and lateral expansion angle, adding the forward expansion angle leads to the deterioration of film cooling performance.

4.4 Chaos optimization process

The mathematical model of optimization problem of film cooling effectiveness can be expressed as follows:

max η_avg(L, β_lat, β_for)

s.t.                   (16)

Set the maximum global search step N₁=4000, and the maximum detailed search step N₂=2000. Two different optimization processes are carried out. In the first optimization process shown in Fig. 7(a), the initial value of (L, β_lat, β_for) is (70 m, 15°, 15°). By the optimization, η_avg increases from 0.367 to 0.433, and the optimal value of the shape parameters is (89.9 mm, 19.7°, 5.0°). In the second optimization process shown in Fig. 7(b), the initial value of (L, β_lat, β_for) is (60 mm, 10°, 10°). By the optimization, η_avg increases from 0.291 to 0.433, and the optimal value of the shape parameters is (90.0 mm, 19.9°, 5.1°). It illustrates that the optimization results are independent on the initial values. Figure 8 shows the distributions of local film cooling effectiveness on the flat plate before and after optimizations. By the optimization, the lift-off effect of coolant jet is mitigated, and the lateral spreading of coolant is promoted effectively. The distribution of film cooling effectiveness after optimization is more uniform than that before optimization. Overall, the film cooling performance after optimization is improved obviously.

Fig. 6 Influence of SVM input parameters on film cooling performance:

Fig. 7 Variation of objective function with iteration step:

Fig. 8 Distribution of local film cooling effectiveness on flat plate before and after optimization:

5 Conclusions

1) Adding the area of hole exit can improve the film cooling performance. However, as the hole exit area exceeds a critical value, the film cooling effectiveness starts to decrease with increasing the hole exit area.

2) LS-SVM can model the film cooling system with high accuracy. Moreover, as the training sample size is not large, the prediction performance of LS-SVM is better than that of ANN.

3) As L=90 mm, β_lat=20° and β_for=5°, the film cooling effectiveness of laidback fan-shaped hole can reach its maximum value in the design space.

Overall, LS-SVM coupled with chaotic optimization algorithm is a promising method for shape optimization of laidback fan-shaped film cooling holes.

References

[1] RALLABANDI A P, GRIZZLE J, HAN J C. Effect of upstream step on flat plate film-cooling effectiveness using PSP [J]. ASME Journal of Turbomachinery, 2011, 133: 241-247.

[2] BUNKER R S. A review of shaped hole turbine film-cooling technology [J]. ASME Journal of Heat Transfer, 2005, 127: 441-453.

[3] GRITSCH M, SCHULZ A, WITTING S. Adiabatic wall effectiveness measurements of film-cooling holes with expanded exits [J]. ASME Journal of Turbomachinery, 1998, 120: 549-556.

[4] LIU Cun-liang, ZHU Hui-ren, BAI Jiang-tao, XU Du-chun. Film cooling performance of converging slot-hole rows on a gas turbine blade [J]. International Journal of Heat and Mass Transfer, 2010, 53: 5232-5241.

[5] YANG Cheng-feng, ZHANG Jing-zhou. Experimental investigation on film cooling characteristics from a row of rigid-shaped tabs [J]. Experimental Thermal and Fluid Science, 2012, 37: 113-120.

[6] SAUMWEBER C, SCHULZ A. Free-stream effects of the cooling performance of cylindrical and fan-shaped cooling holes [C]// Proceedings of ASME Turbo Expo 2008: Power for Land, Sea, and Air. Berlin, Germany: ASME, 2008: GT2008-51030.

[7] SAUMWEBER C, SCHULZ A, WITTING S. Free-stream turbulence effects on film cooling with shaped holes [C]// Proceedings of ASME Turbo Expo 2002: Power for Land, Sea, and Air. Amsterdam, The Netherlands: ASME, 2002: GT2002-30170.

[8] COLBAN W F, THOLE K A, BOGARD D. A film-cooling correlation for shaped holes on a flat-plate surface [J]. ASME Journal of Turbomachinery, 2011, 133: 011002.

[9] SAUMWEBER C, SCHULZ A. Effect of geometry variations on the cooling performance of fan-shaped holes [J]. ASME Journal of Turbomachinery, 2012, 134: 061008.

[10] LEE K D, KIM S M, KIM K Y. Multi-objective optimization of a row of film cooling holes using an evolutionary algorithm and surrogate model [J]. Numerical Heat Transfer, Part A, 2013, 63: 623-641.

[11] LEE K D, KIM K Y. Shape optimization of a fan-shaped hole to enhance film-cooling effectiveness [J]. International Journal of Heat and Mass Transfer, 2010, 50: 2996-3005.

[12] CORTES C, VAPNIK V. Support-vector networks [J]. Machine Learning, 1995, 20: 273-297.

[13] LIU Jin-peng, NIU Dong-xiao, ZHANG Hong-yun, WANG Guan-qing. Forecasting of wind velocity: An improved SVM algorithm combined with simulated annealing [J]. Journal of Central South University, 2013, 20(2): 451-456.

[14] ZHANG Wei-ping, NIU Pei-feng, LI Guo-qiang, LI Peng-fei. Forecasting of turbine heat rate with online least square support vector machine based on gravitational search algorithm [J]. Knowledge-Based Systems, 2013, 39: 34-44.

[15] SHAYEGHI H, SHAYANFAR A H, JALILZADEHh S, SAFARI A. Multi-machine power system stabilizers design using chaotic optimization algorithm [J]. Energy Conversion and Management, 2010, 51: 1572-1580.

[16] E Jia-qiang, ZHAO Xiao-huan, DENG Yuan-wang, ZHU Hao. Pressure distribution and flow characteristics of closed oscillating heat pipe during the starting process at different vacuum degrees [J]. Applied Thermal Engineering, 2016, 93: 166-173.

[17] SUYKENS J A K, VANDEWALLE J. Least squares support vector machine classifiers [J]. Neural Processing Letters, 1999, 9: 293-300.

[18] WANG C H, ZHANG J Z, ZHOU J H, ALTING S A. Prediction of film-cooling effectiveness based on support vector machine [J]. Applied Thermal Engineering, 2015, 83: 82-93.

[19] THISSEN U, BRAKEL R V, WEIJER A P D, MELSSEN W J, BUYDENS L M C. Using support vector machines for time series prediction [J]. Chemometrics and Intelligent Laboratory Systems, 2003, 69: 35-49.

[20] WANG He-jun, E Jia-qiang, DENG Fei-qi. A novel adaptive mutative scale optimization algorithm based on chaos genetic method and its optimization efficiency evaluation [J]. Journal of Central South University, 2012, 19(9): 2554-2560.

[21] YANG Hai-Dong, E Jia-qiang. An adaptive chaos immune optimization algorithm with mutative scale and its application [J]. Control Theory & Application, 2009, 26(10): 1069-1074.

[22] QIN Yan-min, LI Xue-ying, REN Jing, JIANG Hong-de. Prediction of the adiabatic film cooling effectiveness influenced by multi-parameters based on BP neural network [J]. Journal of Engineering Thermophysics, 2011, 32(7): 1127-1132. (in Chinese)

(Edited by DENG Lü-xiang)

Cite this article as: WANG Chun-hua, ZHANG Jing-zhou, ZHOU Jun-hui. Data mining optimization of laidback fan-shaped hole to improve film cooling performance [J]. Journal of Central South University, 2017, 24(5): 1183-1189. DOI: 10.1007/s11771-017-3521-x.

Foundation item: Project(U1508212) supported by the National Natural Science Foundation of China; Project(2015M570448) supported by the Postdoctoral Science Foundation of China

Received date: 2015-10-12; Accepted date: 2016-03-24

Corresponding author: WANG Chun-hua, PhD; Tel: +86-25-83794700; E-mail:chunhuawang@nuaa.edu.cn