1 Introduction

With the advancement of gas turbine power technology, the gas temperature approaching the turbine inlet is continuously increased. In the future, the gas temperature approaching the turbine inlet will be further promoted to meet the increasingly demand of high-efficiency gas turbine power, leading to more and more serious challenges in the thermal protection of the hot section components from overheat. Therefore, developing innovative high-efficient cooling technology or the bringing out the latent potentiality of the existing cooling configuration is a remarkable issue for the researchers.

It is well known that the film cooling is an indispensable means to the high efficient cooling configuration, which has been widely used in the advanced gas turbine engines [1]. As the film cooling air is generally bled from the compressor of the engine, vast amount utilization of the coolant for the cooling purpose must be otherwise detrimental to the engine overall efficiency. For this cause, considerable efforts have been devoted to the film cooling improvement over past decades. One major development for the film cooling technology is the shaped-hole concept by the incorporation of exit shaping to the film holes [2].

Previous works on the shaped holes had revealed that the mechanism of film cooling enhancement related to the conventional holes lie mainly on the anti-kidney vortices effect. Over the past decades, a lot of shaped film cooling holes were proposed [3,4,5,6,7,8,9,10], such as fan-shaped hole, branch or sister-shaped hole, crescent-shaped hole, trench-shaped hole, tabbed-shaped hole and converging-slot hole (CONSOLE), etc. Among these shaped holes, fan-shaped hole should be one of the most common shaped holes, which was presented early in the middle of last century. Goldstein et al. [11] carried out an experiment on the film cooling effectiveness to demonstrate the film cooling effect of a fan-shaped hole with an axial incline angle of 35 degree and lateral diffusion angle of 10 degree. A significant increase in the film-cooling effectiveness immediately downstream of the shaped holes as well as lateral coolant coverage was illustrated. Thole et al. [12] presented a detailed flow field measurement on fan shaped holes. It was illustrated that the fan-shaped hole leads to less shear mixing of the coolant jet with the primary flow, less penetration of the coolant jet into the primary flow, and wider lateral-coverage of the coolant jet related to the conventional hole. In order to evaluate the effects of hole geometry on the performance and characteristics of fan-shaped film cooling, many experimental and numerical studies had been conducted, e.g., Gritsch et al. [13,14,15], Mahmood et al. [16], Leedom and Acharya [17], Bonanni et al. [18, 19], Saumweber and Schulz [20]. Due to the complicity of multi-parameter influence on the shaped-hole film cooling system, it was difficult to evaluate the effect of single geometric factor (such as forward diffusion angle, lateral diffusion angle, length-to-diameter, etc.) on the film cooling performance, leading to that the existing correlations for predicting the film cooling performance are limited to narrow ranges of parameters [21].

In the view of bringing out the latent potentiality of the existing cooling configuration, multi-parameter optimization of the shaped-hole geometry by using optimization strategies is certainly an important issue. Lee and Kim [22,23,24] performed a series of single-objective optimizing investigations to evaluate the effects of geometric variables of a laidback fan-shaped hole on the film-cooling effectiveness using a Reynolds-averaged Navier-Stokes analysis, and to optimize the shape of the hole using various surrogate models with the objective to maximize spatially averaged film cooling effectiveness. Naghashnejad et al. [25] made numerical simulations for the effects of interest parameters on the film cooling performance through a single row of inclined cylindrical holes. A predictive model was implemented by using GMDH-type (group method of data handing) neural networks to successfully present a correlation of area-weighted average adiabatic film cooling effectiveness. Wang et al. [26] and Huang et al. [27] made an optimization of a fan-shaped hole and round-to-slot hole respectively to improve film cooling performance by using radial basis functions neural network (RBF-NN) and genetic algorithm (GA). By using single-objective optimization, the optimal film cooling hole geometry for achieving the maximum spatially averaged film cooling effectiveness was able to be determined corresponding to its application conditions.

Related to the single-objective optimization, multi-objective optimization is able to present more reasonable insight into the comprehensive effects, both in heat transfer performance and in aerodynamic performance. For this cause, multi-objective based optimization was now attractive to many researches in the enhanced film cooling applications. Lee et al. [28, 29] performed multi-objective optimization investigations for a row of fan-shaped holes film cooling on flat surface by using an evolutionary algorithm and surrogate modelling. In their works, two objective functions were selected as the spatially averaged film cooling effectiveness over an area in 40 hole diameters in the streamwise length and aerodynamic loss. Proietti et al. [30] carried out a fan-shape optimization on flat plate with the aim at achieving the highest adiabatic cooling effectiveness while minimising the coolant mass flow rate by using mesh morphing method. Ayoubi et al. [31] conducted an optimization of film cooling on the suction surface of a high pressure turbine blade to optimize the film coolant flow parameters by selecting the coolant-to-mainstream temperature ratio and total pressure ratio as two design variables. Recently, Jiang et al. [32] performed an aero-thermal optimization on multi-rows film cooling of a high pressure turbine vane to determine the optimal cooling structure that yields better cooling performance with an acceptable aerodynamic penalty.

It is well known that surface curvature and pressure gradient in the main channel have great influence on the cooling performance, which leads to different roles of shaped-hole on film cooling improvement on the curved surface and in the pressure-gradient primary flow related to that on a flat plate with nearly zero pressure-gradient [33,34,35]. Therefore, the optimization of the shaped-hole geometry on the turbine vane should be conducted under the representative thermal-aerodynamic conditions. In the current work, an effort is paid to optimize the laidback fan-shaped film cooling holes on the suction side of a turbine guide vane near throat location, in order to obtain an optimum compromise between a high film cooling effectiveness and a low pressure drop of coolant flow passing through the film hole. The optimization procedure is realized by using radial basis function neural network (RBFNN) model surrogate and a variant of non-dominated sorting genetic algorithm (NSGA-II).

2 CFD modelling and assessment

2.1 Brief description of physical model

A specified high pressure turbine guide vane used in a real aero-engine is selected for investigation. In the current study, a simplification is made in the physical model of turbine guide vane by simply stacking a two-dimensional mid-span profile extracted from a three dimensional geometry of the original turbine guide vane. Its mid-span profile is displayed in Fig. 1a, which is taken from the previously researched model presented by Yao et al. [36]. The guide vane has a chord length (C) of 74.4 mm and an axial chord length (Cx) of 42.3 mm. The cascade pitch (P) is 53.6 mm. The inflow angle (δ) of cascade or primary flow is set as 90°, and the stagger angle between the guide vane chord and the cascade inlet plane (θ) is set as 35.7°. A single row of laidback fan shaped film cooling holes is distributed on the guide vane suction side, corresponding to the axial location of 69% axial chord length. Nearby this axial location, a significant pressure variation on the suction surface generally occurs [36, 37]. This region is a concerned zone in the film-cooled turbine vane. The film-hole diameter (d) is 0.6 mm and the hole-to-hole pitch (Phole) is set as 4.5d.

Fig. 1
figure 1

Schematic of guide vane and laidback fan-shaped film cooling hole

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The laidback fan-shaped film-cooling hole is taken into consideration in the current investigation, as illustrated in Fig. 1b. Beside the film-hole diameter and the hole-to-hole pitch, the other main geometric parameters for this fan-shaped hole include the height (t), the inclination angle (α), the lateral expansion angle (β) and forward expansion angle (γ), the orientation position of lateral expansion from the hole inlet (l1), length of lateral expansion section (l2) and the orientation position of forward expansion from the hole outlet (l3). In really, as the total length of the film-hole is determined by the film-hole height and inclination angle, one of the li (i = 1, 2, 3) is an un-dependent geometric parameter. Among the above geometric parameters, only three parameters (such as α, β and γ) are selected as the design variables and the others are fixed in the current study, as summarized in Table 1. The film-hole height, length of lateral expansion section and the orientation position of forward expansion section are specified as 2.5d, 2d and d respectively. The lower and upper bounds for the film-hole inclination angle, lateral expansion angle and forward expansion angle are designed as α∈(25°, 55°), β∈(10°, 20°) and γ∈(3°, 15°), respectively.

Table 1 Fixed geometric parameters and design variables
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2.2 Computational domain and grids

According to periodicity of the cascade flow, one cascade pitch is chosen as the computational domain, as seen in Fig. 2a. In the lateral or spanwise direction (z-direction), three film cooling holes are involved and the symmetric planes are applied at both spanwise ends of the computational domain. The front section upstream the turbine vane has an axial length of 35d and the rear section downstream the turbine vane has an axial length of 45d respectively. The coolant flow is fed into the coolant plenum inside the turbine vane from two inlets located at both spanwise ends. This coolant plenum has a constant sectional area (about 5d in width and 12d in length) in the spanwise or lateral direction, taking on a similar racetrack shape. The coordinate system is originated at the trailing edge of film-hole outlet at the middle-span sectional plane, as seen in Fig. 2a. The s-direction and z-direction denote the streamwise direction along the blade curved surface and spanwise direction, respectively. The normal direction related to the blade surface is denoted as y-direction.

Fig. 2
figure 2

Computational domain and grid sensitivity test

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In the current study, the aerodynamic parameters, either for the primary flow or for the cooling air, are chosen according to the engine representative conditions. The boundary conditions of computational domain are specified as the following.

The cascade channel: the primary flow inlet is defined as a velocity-inlet boundary with Re∞ = 400,000. Here, the Reynolds number of primary flow (Re∞) is defined on the base of the blade chord length and the primary flow inlet velocity, as seen in Eq. (1). The total temperature at the cascade inlet (\( {T}_{\infty, in}^{\ast } \)) is 2100 K. A turbulence intensity level of 5% and a turbulence length scale of 3% of the blade chord length are used. The cascade outlet is set as a pressure-outlet boundary with the static pressure (poutlet) of 1.3 MPa. For the periodic planes and symmetric planes enclosing the cascade channel, the correspondingly periodic conditions and symmetric conditions are applied respectively.

$$ {\operatorname{Re}}_{\infty }=\frac{{\left({\rho}_{\infty }{u}_{\infty}\right)}_{cascade- inlet}C}{\mu_{\infty }} $$
(1)

where ρ∞ and u∞ are density and velocity of the primary flow at the cascade inlet respectively, C is the vane chord length, and μ∞ is dynamic viscosity of the primary flow.

Cooling air plenum: the cooling air is fed into the coolant plenum from two inlets located at both spanwise ends by adopting a velocity-inlet boundary. The inlet velocity is determined according to the blowing ratio (M) as defined in Eq. (2). The total temperature of cooling air (\( {T}_{\infty, in}^{\bullet } \)) is set as 900 K. A turbulence intensity level of 5% and a turbulence length scale of 3% of the inlet hydraulic length are used. The internal surface of the cooling air plenum is treated by thermally-adiabatic non-slip boundary condition.

$$ M=\frac{{\left({\rho}_{\infty }{u}_{\infty}\right)}_{hole- inlet}}{{\left({\rho}_{\infty }{u}_{\infty}\right)}_{cascade- inlet}} $$
(2)

where ρc and uc are density and mean ejecting velocity of the coolant flow at the film-hole inlet respectively.

It is noted that the cross-sectional area of fan-shaped film cooling hole is varied from its inlet to exit due to the diffusing exit geometry. Therefore, under that same coolant usage, the mean injection velocity at the film-hole exit is changed with the varying of fan-shaped hole. To ensure the same comparison benchmark for the shaped holes on the account of coolant usage, in the current definition of blowing ratio, the coolant parameters are defined at the film-hole inlet, and the primary flow parameters are defined at the cascade entrance. Considering that the film cooling holes at the suction side are located in the accelerated primary flow surrounding, the blowing ratio of film cooling on the suction surface is generally high. Therefore, a typical blowing ratio of M = 1.5 is selected for the multi-objective optimization of laidback fan-shaped film cooling hole in the current study.

Commercial software GAMBIT is used to generate the computational meshes. Overall multi-block grids are generated for the whole computational domain, covering the cascade flow channel, coolant plenum and film holes. Viscous clustering is employed at the solid surface to guarantee that the neighboring-cell y+ meets the requirement of enhanced wall treatment in the near-wall region. The grid is stretched away from the viscous wall with a stretching ratio less than 1.2. Figure 2b presents the computational grids at the film-hole centerline plane. In order to eliminate the influence of the number of grids on the calculation results, systematic mesh sensitivity is performed in advance. Four sets of meshes are selected with approximate 1.63 million, 2.38 million, 3.24 million and 3.92 million grids respectively. As seen in Fig. 2c, the increase of grid number has nearly no influence on the computed results when the grid number is beyond 3.24 million. Consequently, approximately 3.24 million grids are finally involved in the whole computational domain. Figure 3 shows the y+ values downstream of film cooling hole on the suction surface. It is seen that y+ values are scattered mainly around 0.8 and the maximum y+ is about 1.05 at the location tightly close to the film cooling exit.

Fig. 3
figure 3

y+ values downstream of film cooling hole on the suction surface

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2.3 Computational scheme and validation

The numerical simulation of the film cooling flow field and the film cooling effectiveness is carried out by using the commercial software (ANSYS Fluent 14.0) based on the Reynolds time averaged (RANS) method [38]. According to the precious works [39,40,41], the realizable k-ε turbulence model with the incorporation of enhanced wall treatment is used to for the turbulence closure. The second order upwind scheme is used for the spatial discretization of convective terms in the conservation equations and the SIMPLEC algorithm for pressure-velocity coupling. In the computations, both the primary flow and the secondary flow are treated as ideal air. In ANSYS Fluent software, the physical parameters of are set as follows: the ideal-gas based density; Kinetic-theory based specific heat and thermal conductivity; Sutherland law based viscosity convergence is achieved when both of the following criteria have been met: (a) reduction in all residuals of five orders of magnitude, and (b) the temperature change not exceeding ±0.1 K in local temperature prediction on the protected surface for an additional 30 iterations.

For the validation purpose, an example is performed in advance, which is taken from the experimental tests of the converging slot-hole film cooling on a guide vane suction surface presented by Yao al. [35]. The validating computational model is built up according to the corresponding experimental model. Figure 4 shows the comparison between the computed laterally-averaged adiabatic film cooling effectiveness distributions and the corresponding experimental data. It is confirmed that the computed results are generally in good agreements with the corresponding experiment results, with a maximum difference within 10%.

Fig. 4
figure 4

Comparison of current simulations and experimental data

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3 Optimization methodology

3.1 Objective function

For the multi-objective optimization of the film cooling holes, undoubtedly, the spatially-averaged film cooling effectiveness is necessarily selected as one of the objective functions to be maximized, which is averaged in a specified zone. According to previous works [26,27,28,29], the specified zone for the average purpose is generally within 10d~20d downstream of the film-hole outlet. Considering that multiple rows of film cooling holes are involved in the real applications of film-cooled gas turbines, the row-to-row pitch is generally less than 15d in the focused film-cooled region. Beside, as the film cooling enhancement in the near filed of film cooling holes is the main concern of shaped-hole, in the current study, the spatially-averaged adiabatic film cooling effectiveness is defined in the specific zone between s/d = 0 and s/d = 12, as seen in Eq. (3).

$$ {\eta}_{ad, av}=\frac{1}{12d}{\int}_{0d}^{12d}{\eta}_{ad, av s} ds $$
(3)

here ηad is the local adiabatic film cooling effectiveness. ηad,avs is the laterally-averaged adiabatic film cooling effectiveness along the streamwise direction, which is line-averaged within one hole-to-hole pitch (Phole = 4.5d).

$$ {\eta}_{ad}=\frac{T_{\infty }-{T}_{aw}}{T_{\infty }-{T}_c} $$
(4)
$$ {\eta}_{ad, avs}=\frac{1}{4.5d}{\int}_{-2.25d}^{2.25d}{\eta}_{ad} dz $$
(5)

Another objective function should be associated with the aerodynamic performance. In the previous works [28, 29, 32], the aerodynamic loss or penalty is adopted. In the current study, the discharge coefficient of film-hole is selected as the objective function to be maximized. To our knowledge, the effect of film-hole on the discharge coefficient is more direct related to the aerodynamic penalty inside a cascade channel, because the latter is tightly dependent on many factors, such as the losses due to the flow turning, multiple-rows coolant jet s injection and mixing, wall fraction, etc. Besides, the greater discharge coefficient of film-hole indicates the smaller pressure drop across the film hole, which is helpful for reducing the bleed pressure of the cooling air for a given coolant mass flow rate. According to the study of Gritsch et al. [14], the discharge coefficient is defined as

$$ {C}_d=\frac{m_{inlet}}{p_c^{\ast }{\left(\frac{p_c}{p_c^{\ast }}\right)}^{\frac{k+1}{2k}}\sqrt{\frac{2k}{\left(k-1\right)R{T}_c^{\ast }}\left({\left(\frac{p_c^{\ast }}{p_c}\right)}^{\frac{k-1}{k}}-1\right)\frac{\pi }{4}{d}^2}} $$
(6)

where minlet is actual mass flow rate of coolant flow. pc* and Tc* are the total pressure and the total temperature at the film hole inlet, respectively. pc is static pressure at the film-hole outlet. R is the gas constant and k is the ratio of specific heat.

Consequently, two independent objective functions that are simultaneously optimized are constructed as

$$ {\displaystyle \begin{array}{l}\Big\{\begin{array}{l}\min {F}_1\left(\alpha, \beta, \gamma \right)=\frac{1}{\eta_{ad, av}}\\ {}\min {F}_2\left(\alpha, \beta, \gamma \right)=\frac{1}{C_d}\end{array}\\ {}s.t.\Big\{\begin{array}{c}{\alpha}_{\mathrm{min}}\le \alpha \le {\alpha}_{\mathrm{max}}\\ {}{\beta}_{\mathrm{min}}\le \beta \le {\beta}_{\mathrm{max}}\\ {}{\gamma}_{\mathrm{min}}\le \gamma \le {\gamma}_{\mathrm{max}}\end{array}\end{array}} $$
(7)

where F1(α, β, γ) and F2(α, β, γ) denote the fitness functions of surrogate model.

3.2 Optimization scheme

Figure 5 describes the flow chart of a complete optimization cycle used in the present work. Once the optimization problem is set up, three main steps are then performed for the optimization process.

Fig. 5
figure 5

Flow chart of optimization process

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The first step is the construction and validation of a surrogate model. In the present study, radial basis function neural network (RBFNN) is applied for constructing the surrogate model, which is established by a customized command ‘newrbe’ in Matlab software [42]. During the RBFNN construction, the Latin Hypercube Sampling (LHS) algorithm is used to select the appropriate design points over the design space. Totally 25 design points are selected as training samples and the other 8 design points are selected as testing samples randomly within the scope of design variables, as illustrated in Tables 2 and 3 respectively. By CFD simulations, the corresponding spatially-averaged adiabatic film cooling effectiveness and discharge coefficient are obtained, as also illustrated in the above tables. Figure 6 presents the comparison between the RBFNN output and the CFD calculation results for the testing samples. It is confirmed that the RBFNN outputs are in good agreements with the corresponding CFD calculation results, with the maximum errors less than 3% both for the spatially-averaged adiabatic film cooling effectiveness and for the discharge coefficient.

Table 2 Training samples
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Table 3 Testing samples
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Fig. 6
figure 6

Comparison of RBFNN output and CFD results

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The second step is the searching of the optimal global point. In the current study, a variant of non-dominated sorting genetic algorithm (NSGA-II) [43] coupled with the RBFNN-based surrogate model is adopted for obtaining the global Pareto optimal solutions by using the function ‘gamultiobj’ in Matlab software. The population size and the Pareto front population fraction are set as 200 and 0.35 respectively. The optimization procedure achieves the convergence criterion after 103 iterative times.

The final step is the CFD assessment of optimum candidates, such as the optimal points corresponding to the maximum spatially-averaged adiabatic film cooling effectiveness, the maximum discharge coefficient, and the compromise of both aspects.

4 Results and analysis

4.1 Optimization results

Figure 7 presents the Pareto front of optimal solutions, including all the optimal design points obtained from the two-objective optimization. The horizontal and vertical coordinates represent the objective function of discharge coefficient, given by Cd, and the objective function of spatially-averaged adiabatic film cooling effectiveness, given by ηad,av, respectively. This Pareto front represents the competing nature of the two independent objectives, provides a variety of options for the designer, such as for seeking the maximum spatially-averaged adiabatic film cooling effectiveness (as an example of optimal point A), the maximum discharge coefficient (as an example of optimal point B), or the compromise of both aspects (as an example of optimal point C). Some clusters corresponding to the above options are summarized in Table 4. It is seen that a large lateral expansion angle of the laidback fan-shaped film-cooling hole is necessary in the shape optimization for all of the optimal options, which is documented to be nearly the same as the upper bound of this design variable in the current scope. However, with regard to the other design variables, their selections are very distinct following the optimal option. In the viewing of seeking maximum film cooling effectiveness, a large inclination angle and a moderate forward expansion angle are favorable. In the viewing of seeking maximum discharge coefficient, a small inclination angle and a moderate forward expansion angle are favorable. While for seeking the compromise of both aspects, a small inclination angle and also a small forward expansion angle seem to be more favorable.

Fig. 7
figure 7

Pareto front of optimal solutions searched by NSGA-II with RBFNN-based surrogate model

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Table 4 Some clusters corresponding to different optimal options
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According to the previous works, a typically reference structure of the laidback fan-shaped film-cooling hole is selected for comparison where the (α, β, γ)ref is set as (30°, 14°, 15°)ref. Table 5 presents a comparison of CFD results between the reference case and the optimization cases. Related to the reference case, the Opt-A achieves approximately 18% increase in the spatially-averaged adiabatic film cooling effectiveness but 7% decrease in the discharge coefficient. The Opt-B produces approximately 8.8% increase in the discharge coefficient but 2% decrease in the spatially-averaged adiabatic film cooling effectiveness. For the Opt-C, both the spatially-averaged adiabatic film cooling effectiveness and the discharge coefficient are all improved, with about 10.5 and 1.6% relative increases respectively. Defining a pressure-drop ratio \( \psi =\frac{\left({p}_c^{\ast }-{p}_c\right)}{{\left({p}_c^{\ast }-{p}_c\right)}_{\mathrm{ref}}} \), it is determined that the pressure-drop ratios for Opt-A, Opt-B and Opt-C are 1.15, 0.845 and 0.963 in turns.

Table 5 Comparison between reference case and optimization cases
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4.2 Detailed film cooling behaviors

In this section, the detailed film cooling behaviors for three optimal design points (such as Opt-A, Opt-B and Opt-C) are analyzed.

Figure 8 presents the Mach number contours in the cascade channel and static pressure coefficient distribution (\( {c}_p=\left({p}_s-{p}_{t\infty}\right)/\left(0.5{\rho}_{\infty }{U}_{\infty}^2\right) \)) on turbine vane surface without the presence of film cooling, here Ps is the static pressure on the turbine surface) at the turbine vane middle plane. Due to the flow turning of primary flow in the cascade channel, the primary flow experiences significant acceleration over nearly entire suction surface. While at the pressure side, the primary flow acceleration mainly occurs near the turbine vane trailing, as seen in Fig. 8a. At the leading edge of turbine vane, the static pressure is the same as the total pressure of primary flow at the cascade entrance due to the flow stagnation, as illustrated in Fig. 8b. From the stagnation point on, the pressure coefficient distributions on the turbine vane pressure and suction surfaces take on distinct variations. At the suction side, the static pressure decreases rapidly from the leading edge to the middle chord, owing to the rapid acceleration of the primary flow in the vicinity of turbine vane suction surface. The primary flow reaches the maximum speed around x/Cx = 0.67 and then decelerates toward the trailing edge. While at the pressure side, the static pressure changes a little from the leading edge to x/Cx = 0.67 and then falls off rapidly toward the trailing edge. The above flow features in a turbine cascade are in good agreement with the previous finding of Arts et al. [37].

Fig. 8
figure 8

Mach number contours and static pressure distribution on turbine vane surface

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Figure 9 presents the local streamlines and static pressure distributions at the vertical plane crossing the hole-centerline on the suction side. It is seen clearly that the static pressure inside the coolant plenum for the Opt-A is the highest but lowest for the Opt-B, indicating that a greater bleed pressure of the cooling air is needed for the Opt-A shaped-hole to overcome the bigger pressure drop across the film hole for providing such a coolant injection. It is also seen that a jet lift-off effect is occurred inside the film cooling hole due to the separation and flow turning near the film-hole inlet, which is a common flow feature for the coolant plenum-inflow scheme [44, 45]. By comparison, the jet lift-off effect inside the laidback fan-shaped film-cooling hole corresponding to the Opt-A case seems to behave more obviously. As the concerned film cooling holes have nearly the same lateral expansion angle (as illustrated in Table 4), the large inclination angle for the Opt-A makes a strong transition of lateral expansion, resulting in a great flow loss across the film cooling hole.

Fig. 9
figure 9

Local static pressure distributions and streamlines at the cross section traversing hole centerline

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Figure 10 presents the streamlines and dimensionless temperature contours in the normal section at s/d = 2 and s/d = 8. Here the dimensionless temperature is defined as \( \varTheta =\frac{T_{\mathrm{aw}}-{T}_{\mathrm{c}}}{T_{\infty }-{T}_{\mathrm{c}}} \). Figure 11 presents the local adiabatic film cooling effectiveness distributions downstream of film-hole outlet. From these figures, it is confirmed that all the fan-shaped holes behaves the anti-kidney vortices effect on the mutual interaction between the primary flow and the coolant jet, reflected by the less penetration and wider lateral-coverage of the coolant jet. Of particular interest is that the Opt-A seems to produce more stable film layer over the protected surface, as seen in Figs. 10a and 11a. As the fan-shaped hole for the Opt-A has a large inclination angle and a moderate forward expansion angle, the outlet area is much less than that for the Opt-B. Consequently, the mean exiting velocity of the coolant jet is significantly higher for the Opt-A than the Opt-B. At the suction side, because the coolant jet penetration is effectively suppressed by both the anti-kidney vortices effect of shaped hole and the accelerated primary flow effect, the coolant jet flow momentum is suggested to be transferred toward the streamwise direction. Therefore, the film layer development along the streamwise direction is stronger than the other optimal cases. On the other hand, the accelerated primary flow also aggravates the shear mixing between the coolant jet and the primary flow. For the Opt-B, as the coolant jet exiting momentum is relative low, its film layer is easy to be destroyed by the primary flow, as seen in Figs. 10b and 11b.

Fig. 10
figure 10

Streamlines and dimensionless temperature contours in the normal section

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Fig. 11
figure 11

Local adiabatic film cooling effectiveness distributions

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Figure 12 presents the local adiabatic film cooling effectiveness distributions along the spanwise direction. Form this figure, it is seen directly that the fan-shaped hole for the Opt-A produces more stable and uniform film coverage over the protected surface than the other fan-shaped holes.

Fig. 12
figure 12

Local adiabatic film cooling effectiveness along spanwise direction

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The laterally-averaged adiabatic film cooling effectiveness (ηad,avz) along the streamwise direction is shown in Fig. 13. In this figure, the reference case is also displayed for comparison. Compared to the reference case, the laterally-averaged adiabatic film cooling effectiveness for the Opt-A and Opt-C is improved among the specified s/d range, especially obvious at the near field immediately downstream of the film-hole outlet. While for the Opt-B, the laterally-averaged adiabatic film cooling effectiveness is improved a little within s/d = 4 but reduced in the region of 4 ≤ s/d ≤ 12, related to the reference case. Consequently, the spatially-averaged adiabatic film cooling effectiveness in the region between s/d = 0 and s/d = 12 is highest for the Opt-A. The nest is the Opt-C, as illustrated in Table 5.

Fig. 13
figure 13

Laterally-averaged adiabatic film cooling effectiveness distribution

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5 Conclusions

This paper summarizes a multi-objective optimization research for improving the film cooling performance of the laidback fan-shaped holes on the suction surface of a turbine guide vane. Among the main geometric parameters, the inclination angle (α), lateral expansion angle (β) and forward expansion angle (γ) are selected as the design variables, with respective lower and upper bounds of (25°, 55°), (10°, 20°) and (3°, 15°) in turns. Two independent objective functions that are simultaneously optimized are selected as the spatially-averaged adiabatic film cooling effectiveness (ranging from s/d = 0 to s/d = 12) and the discharge coefficient. By using a variant of non-dominated sorting genetic algorithm (NSGA-II) coupled with the RBFNN-based surrogate model, the Pareto front of optimal solutions is obtained, providing a variety of options for the designer, such as for seeking the maximum spatially-averaged adiabatic film cooling effectiveness, the maximum discharge coefficient, or the compromise of both aspects.

In general, a large lateral expansion angle of the laidback fan-shaped film-cooling hole is necessary in the shape optimization for all of the optimal options. With regard to the other design variables, a large inclination angle and a moderate forward expansion angle are favorable in the viewing of seeking maximum film cooling effectiveness. In the viewing of seeking maximum discharge coefficient, a small inclination angle and a moderate forward expansion angle are favorable. While for seeking the compromise of both aspects, a small inclination angle and also a small forward expansion angle seem to be more favorable.

Corresponding to three typically optimal design points, such as Opt-A with the most maximum film cooling effectiveness, Opt-B with the most maximum discharge coefficient and Opt-C with the compromise of both aspects, the optimal geometers of (α, β, γ) are (50.3°, 19.5°, 9.8°), (25°, 18.7°, 11.8°) and (27.3°, 19°, 5.1°) respectively. By comparison, the jet lift-off effect inside the laidback fan-shaped film-cooling hole corresponding to the Opt-A case seems to behave more obviously. For the Opt-B, as the coolant jet exiting momentum is relative low, its film layer is more easily destroyed by the shear mixing with the accelerated primary flow. Related to a reference case of the laidback fan-shaped film-cooling hole where the (α, β, γ)ref is set as (30°, 14°, 15°)ref, the Opt-A achieves approximately 18% increase in the spatially-averaged adiabatic film cooling effectiveness but 7% decrease in the discharge coefficient. The Opt-B produces approximately 8.8% increase in the discharge coefficient but 2% decrease in the spatially-averaged adiabatic film cooling effectiveness. For the Opt-C, both the spatially-averaged adiabatic film cooling effectiveness and the discharge coefficient are all improved, with about 10.5 and 1.6% relative increases respectively.