In previous research, the density ratio, blowing ratio, inclination angle, and diameter of the coolant tube hole are proven to have the most significant impact on the general temperature distribution near the external surface of the blade (24, 25). So in this paper, those four parameters were chosen to be researched. Given that successive combinations of several repeating units form the actual configuration of the blades, Figure 1 shows the three-view drawing of the minimal periodic reference model. The mainstream chamber and the coolant chamber are modeled as two cuboids of dimensions 39d ₀ × 6d ₀ × 5d ₀ and 8d ₀ × 6d ₀ × 11.5d ₀ , respectively. Here, d ₀ stands for the standard diameter of the circular cross-section of the coolant inlet tube, and d ₀ = 12.5 mm. 13d ₀ is measured from the mainstream inlet’s front side to the coolant tube’s central point. The coolant inlet tube is inclined at an angle θ concerning the x-z plane. The length of the coolant tube is 3.8d ₀ . The mainstream inflow has a density of ρ _m and velocity of V _m , and the coolant inflow has a density of ρ _c and velocity of V _c . Due to the deviation of the cross-section area, this paper guarantees that the velocity of the jet inflow right at the coolant exit would accelerate to V _c . The flow direction of the mainstream flow is defined as going right, and the coolant jet flows upwards. All values mentioned above remain stationary as referenced except for the diameter, inclination angle, density ratio, and blowing ratio. Table 1 summarizes the deviation interval of these four parameter inputs. These settings are the same as other scholars’ research (19, 21, 24, 25). When one parameter is adjusted, to keep mono-variate, other parameters remain unchanged. This is achieved by slightly adjusting V _c and T _c . For example, when dr is changed from 1.1 to 1.2, V _c and T _c are adjusted to ensure that br _, d, and θ are all unchanged.

Ansys Fluent software is proven to have excellent performance in solving cooling problems (28-30). The problem is solved by applying steady-state solvers, which function by performing several iterations until the result converges. The standard to determine convergence is by comparing the continuity residual with 10⁻⁴ (27). If using a machine with 56 CPU to obtain the 225 datasets, it takes approximately 675 (225 × 3 hours. The mainstream and coolant inlet are assumed as velocity boundary conditions, and the mainstream outlet pressure is set to be 1 atm. Adiabatic and no-slip wall boundaries are applied for the mainstream and coolant chamber walls. Both flows are treated as superheated ideal gas, which follows the ideal gas law and is incompressible for low speed. The buoyancy effect is not assumed following Wang et al. (22, 27) and Yang et al. (26, 29). The Prandtl number of the turbulent is set to 0.667.

The dr and br are defined as shown in Eqs 1, 2. dr is defined as the temperature ratio at the mainstream versus the coolant jet. ρ m a n d   ρ c mean the mainstream jet gas density and coolant jet gas density, respectively. The subscripts “m” and “c” denotes mainstream and coolant, which remain the same for temperature T and velocity V which are defined later. Furthermore, the definition of br can be interpreted as the product of dr and the velocity ratio at the coolant jet and mainstream. d r = ρ c ρ m = T m T c (1) b r = ρ c × V c ρ m × V m (2)

To avoid using a complicated high dimensional temperature matrix to represent the cooling efficiency, T ^* and T * ¯ are defined in Eqs 3, 4. T is the gauged temperature of the adiabatic and no-slip wall, with * denoting “dimensionless”, and short bar overhead “ − ” labeling “fixed-cord-averaged.” To represent the wall temperature, the output is interpolated as a 64 × 256 -dimensional matrix from 0 <z < 6d ₀ , 13d ₀ <x < 39d ₀ . T * x , z = T x , z − T c T m − T c (3) T * ¯ x , z = 1 64 ∑ 1 64 T * x , z d z (4)

Besides, the film cooling effectiveness at a single point, together with the fixed-cord-averaged and general film cooling effectiveness are derived according to dimensionless temperature T*, as shown below in Eqs 5–7. η x , z = T m − T T m − T c = 1 − T * x , z (5) η ¯ x = 1 64 ∑ z = 1 z = 64 η x , z (6) η a v = 1 64 × 1 256 ∑ z = 1 z = 64 ∑ x = 1 x = 256 η x , z (7)

Equation 8 defines the Mean Absolute Error (MAE), which is the index chosen to quantify the performance of a method. In the equations, m denotes the total sample size, a _i is the training data gained by Reynolds-Averaged Navier-Stokes (RANS) model, and y _i is the data acquired via the Large Eddy Simulation (LES) carried out by Wang et al. (31). M A E = 1 m ∑ i = 1 m a i − y i (8)

Following other studies, the Fluent ^® 18.0 software is applied to all cases (21, 31). A validation experiment is conducted on the downstream central line’s temperature and jet velocity distributions. This validation experiment aims to find the most suitable turbulence model for the following ANN and uncertainty quantification (UQ) analysis. The reference conditions are d = d ₀ = 12.5 mm, θ = 35°, dr = 1.2, and br = 1.0, which follows Wang et al.’s settings (27). The three candidate numerical methods are the SA model, the RNG k-ε model, the SST k-ω model, the Realizable k-ε model, and the experimental data from Ito (32, 33) and Sinha et al. (6) are used for comparison. The plot is shown in Figure 2, where x denotes the x-coordinate of the coolant jet outlet. The ratio x/d ₀ is set as an x-axis parameter in two plots to realize parallel comparison under different d ₀ selections. The coordinate interval researched is concentrated from 13d ₀ < x < 39d ₀ , 0< z < 6d ₀ . The data located in the region where x < 13d ₀ is truncated because the constant mainstream temperature is assumed. In Figure 2A, the central-line film cooling effectiveness of the outer surface of the blade is computed via four numerical methods and two sets of experimental data. Figure 2B compares the four numerical models’ mean average values (MAE). The results indicate that the MAEs for the SA and Realizable k-ε models are larger than the SST k-ω and RNG k-ε models. Furthermore, the trend of the RNG k-ε model results matched better with the data obtained from the experiments. The MAEs for the SST k-ω and the RNG k-ε models are 0.0106 and 0.0117, respectively. In comparison, the MAE for the Realizable k-ε model and SA model are 0.0241 and 0.0228, respectively, which are almost twice as large as the RNG k-ε and the SST k-ω model. Therefore, the RNG k-ε model is utilized in this study.

In this study, the unstructured hybrid mesh is utilized. The y+ value for the near-wall cell is 1. Moreover, the grid stretch ratio is measured as 1.2 away from the solid wall. The grid cell number must be determined carefully since a massive number of grid cells raises the computational time meaninglessly, while too limited cell number conveys limited temperature distribution information and causes inaccuracy (34, 35). Therefore, the 2, 4.5, 6, and 7.5 million grid sizes are studied on the centerline of the flat plate model. The result is shown in Figure 3.

When the location is right downstream of the coolant hole, the 2-million case obviously differs from 4.5, 6, and 7.5-million cases, whereas the difference narrows as the distance increases. The 7.5 million grid cell case has the most significant fluctuation among all, which implies that the 7.5-million case is the most sensitive to react. This paper sets the grid cell number at a 6-million grid size for analyzing training and validation CFD data. Figure 4 is the mesh schematic from three views: Axonometric, Front, and Top. It can be observed that a more accurate mesh resolution is located near the coolant hole.

Given the enormous computational amount of the CFD method, this paper adopts ANN to reduce the computational burden. The input is a matrix containing coolant tube diameter, coolant tube inclination angle, density ratio, and blowing ratio. The output is a 64 × 256 -dimensional matrix, with each entry representing the grid’s temperature at the flat plate model’s external surface. Cooling effectiveness can thus be obtained from this output matrix by applying Eqs 3–7. The input data are normalized within (0, 1) before plugging into the input matrix. In Eq. 9, each parameter with the subscript “norm” stands for the corresponding parameter after normalization. The learning domain is set in the rectangular region, with x and z coordinates fulfilling 13d ₀ <x < 39d ₀ , 0<z < 6d ₀ . θ n o r m = θ − θ min θ max − θ min = θ − 15 55 − 15 = θ 40 b r n o r m = b r − b r min b r max − b r min = b r − 0.5 1.5 − 0.5 = b r − 0.5 d n o r m = d − d min d max − d min = d − 10.5 14.5 − 10.5 = d − 10.5 4 d r n o r m = d r − d r min d r max − d r min = d r − 1.1 1.3 − 1.1 = d r − 1.1 1.2 (9)

The ANN model is utilized to build the non-linear relationship between four flat plate configuration inputs, and the output temperature distribution matrix near the flat plate surface. The ANN model has seven layers in total, including one input layer, five hidden layers and one output layer. The first layer contains the four input parameters clarified above, and the information is propagated forward to the next layer through weighting, biasing, and activation (29). The activation of the first six layers is accomplished by applying the “Rectified Linear Unit” (ReLU) to intensify the non-linear regression before the output layer. A sigmoid function accomplishes the activation of the output layer.

Equation 10 shows the forward propagation process in the first six layers, where the weight matrix and bias matrix are denoted using W and b, respectively. The y _h denotes the output of the 2nd, 3rd, 4th, 5th, and 6th layers. Equation 11 shows the forward propagation in the output layer. The y _out stands for the output layer. In the output layer, final weighting and biasing are implemented. Then the result will be plugged into a sigmoid function to generate the predicted temperature distribution as the output value.

Batch normalization and dropout are implemented in the five hidden layers to enhance learning and avoid the neural network collapsing by big data. Mean square error (MSE) is deployed as a loss function to assess the difference between predicted temperature T and the CFD simulated value P, as shown in Eq. 12. The dropout is set to 0.1 in case of overfitting. The regression starts from 4 neurons and then multiplies until 256 neurons in the output layer. Figure 5 graphically illustrates the regression process. 45,176 parameters in total are yet to be defined. R e L U x = 0 , x < 0 x , x ≥ 0 y h = R e L U W x + b (10) σ x = 1 e − x + 1 y o u t = σ W x + b (11) M S E = T − P 2 (12)

In the training process, the learning rate is one of the most significant hyperparameters that needs to be determined when applying ANN (36, 37). In this study, the learning rate is set to 0.01, and it is reset to 10% of its former value after every 1,000 epochs to keep convergence. The MSE is used to be the error index to evaluate whether the convergence of the model is accomplished. Figure 6 shows that as the number of epochs increases, the MSE value converges. There is a noticeable fluctuation before the epoch reaches around 500. However, with the continuous growth of epochs, the value of MSE and the fluctuation keep falling.

According to the result, the 0.01 learning rate and reduction to 10% of its last value every 1,000 epochs is a good choice. Quoted error (QE) is proposed in Eq. 13 to measure the ANN model’s absolute error. QE also serves as an index to help find the optimal hyperparameter settings. In the expression of QE, T is the temperature derived from CFD simulation, and P is the predicted temperature result at the final layer of the ANN. η min * is a fixed number 0, and the upper bound of η max * i s   1 . The result is scaled to a hundred percent of the corresponding QE value. Q E = T − P η max * − η min * × 100 % (13)

The ANN model uses QE to find the optimized hyperparameter settings containing splitting ratio, dropout, and batch size. The splitting ratio is acquired by dividing the amount of validation dataset by the whole dataset. A higher splitting ratio means a relatively lower portion of data is utilized in the training process. Thus, a balance in training and validation must be determined to achieve better prediction efficiency and accuracy. An experiment aiming at finding the optimal splitting ratio is conducted, and the result is shown in Figure 7A. The QE value of the case with splitting ratio = 0.2 is the highest compared with the situations whose splitting ratios are 0.1 and 0.3, which indicates that there is no simple proportional relationship between the magnitude of the error and the magnitude of splitting ratio. Besides, it is found that the QE value of the validation dataset is always higher than the training dataset by the ANN algorithm, regardless of which splitting ratio is chosen. Results show that a splitting ratio of 0.3 is believed to be the best among the three splitting ratios researched.

Similar experiments are designed for dropout and batch size. Figure 7B is the result of finding the ideal dropout. There is an apparent disparity between the case where the dropout is 0.1 compared to cases with 0.2 and 0.3 dropouts. The QE for the training dataset with 0.1 dropout is 0.31%, and the QE for the validation dataset with 0.1 dropouts is 0.33%. The QEs for the other two cases vary from 0.52% to 0.55%, which are much larger than those with a dropout of 0.1. This proves that the case with a dropout of 0.1 has the least error between the predicted temperature and the simulated value.

For Batch size, three different batch sizes are studied, and Figure 7C plots the result. The QE values of three different batch sizes are all located at [0.31%,0.38%], which implies that the batch size does not serve as a significant parameter with a considerable influence on the output in the ANN model. In the case with 32 batch size, the MSEs for the validation and training datasets are 0.33% and 0.31%, respectively. Therefore, the optimal batch size of 32 is chosen. For learning rate, as shown in Figure 7D, it is easily found that the varied learning rate could provide the best performance over the fixed learning rate such as 0.01, 0.001, and 0.0001. In all, the optimal hyperparameters are: dropout = 0.1, batch size = 32, splitting ratio = 0.3, and a varied learning rate.

Moreover, the structure of the ANN has been investigated to obtain the best performance. With these optimal parameters, the number of layers is determined to be 7 and the modes are determined according to the symmetry aiming for best training results. The information of different layers and nodes selection is shown in Table 2 below. We chose 2, 3, 4, 7, and 9 layers because under these cases, the nodes arrangements are symmetric from 4 nodes to 256 nodes. In the experiment, we find that when the layer number comes to 7, the quoted error value (QE) reaches its minimum. However, we find that the quoted value of the 9-layer case increases, which is because too much layer increases the number of parameters and the iteration error. In the process of training, it is also reasonable in practice that some parameters are lost. After comparison, we find that the QE value of the 7-layer case is quite acceptable, so we consider 7 layers as the best layer number choice.

To further visualize the prediction accuracy using the designed ANN model, the fixed-cord-averaged film cooling effectiveness achieved by the ANN and CFD method simulation are compared for training and validation datasets. Six cases with diverse input parameter sets containing θ, d, dr, and br are randomly selected. The fixed-cord-averaged film cooling effectiveness on the upper surface of the turbine blade under six different cases are shown in Figure 8. All the six plots in Figure 8 are about the fixed-cord-averaged cooling effectiveness results. The blue dotted lines in the plots represent the CFD cooling results, while the red line indicates the ANN results.

Figure 8A is for training datasets, and Figure 8B is for validation datasets. The result shows that the temperatures predicted using the ANN model for the training and validation datasets almost overlap with the temperatures obtained from CDF simulation. For example, in the case where d = 10.5 mm, θ =15 ° , dr = 0.3, br = 0.5, the fixed-cord-averaged cooling effectiveness derived by CFD is 0.108 when x/d ₀ is positive zero, compared with the result derived by the ANN model of 0.130. While x/d ₀ increases up to 15, the two curves almost overlap, and the error is negligible. After x/d ₀ goes beyond 15, the error expands to its maximum value of 4.28%, where x/d ₀ equals 17.4. Then, as x/d ₀ keeps increasing, the error minimizes continuously. At the back end of the research region, the error is close to zero again. For fixed-cord-averaged film cooling effectiveness, the QE for training datasets is 0.29%, while for the validation dataset it is 0.32%. Conclusion can be drawn that even though fluctuation still occurs inevitably, the regression performance of the ANN model in fixed-cord-averaged film cooling effectiveness is delighted under all randomly selected cases.

Besides the fixed-cord-averaged cooling effectiveness, the general cooling effectiveness is also studied to enhance the above conclusion. Figure 9 compares the overall film cooling effectiveness from CFD and ANN methods downstream the coolant hole. The relationship between general film cooling effectiveness and computation numbers is plotted in Figure 9. 225 ( 3 × 5 × 5 × 3 ) cases of 0.3 splitting ratio are included, which means that Figure 9A includes the 156 training data, Figure 9B includes the 69 validation data. And they are ranked according to the magnitude of their general film cooling effectiveness to exhibit the results in a visual-friendly way. We labeled each data point using Computation No. from 1 to 156 and from 1 to 69 as shown on the horizontal axis. So, two lines are both discrete lines. Even though it is inevitable for the ANN and CFD results to have differences because the ANN’s training data are derived from CFD. The results show that the general film cooling effectiveness derived from the ANN model is accurate enough for both training and validation data, regardless of the computation number. For general film cooling effectiveness, the QE for training datasets is 0.35%, while for the validation dataset, it is 0.30%.

As the designing and manufacturing processes are deterministic, the variety of structures is usually not considered. For products with simple designs, the function could be achieved. However, the performance of sophisticated appliances such as gas turbines could vary significantly due to the uncertain deviation of their parameters. The best way to analyze and reduce unexpected uncertainty is to conduct an uncertainty quantification analysis of all related parameters (38, 39). In this paper, the parameters studied are d, θ, dr, and br.

Monte Carlo (MC) simulation is an extensively utilized technique to quantify the engineering field’s uncertainty among various kinds of uncertainty quantification methods (40). MC simulation conducts statistical analysis to sample datasets obtained by repeated random sampling (41). MC simulation shows excellent advantage in mathematics and engineering due to its concise methodology, broad application domain, and various software choices (38). The mean and standard deviation of the mean squared pure error (MSPE), namely M S P E μ   a n d   M S P E σ are studied instead to impair its harmful effect. The random samples are generated in the following way: Firstly, a sample vector x → is formed from the normalized training dataset, which is x → = (d, θ, dr, br), and each entity is normalized with a value ranging from 0 to 1 according to Eq. 9. Secondly, enough sample vectors x → are generated in the same way and are arranged to form a distribution X for the following MC simulation. The sample size is defined as the number of sample vectors in each distribution. Therefore, parallel experiments are conducted, and the MSPE is proposed to quantify how well the MC simulation behaves. The mean and the standard of MSPE are defined in Eq. 14. The experiment results are shown in Figure 10. It can be observed that even though the M S P E σ   a n d   M S P E μ almost remain at a steady level, when the sample size is below 10,000, the fluctuation of both M S P E σ   a n d   M S P E μ are very large, which represents that a sample size smaller than 10,000 is not suitable for MC simulation. As the sample size increases, however, the undulations of both the mean and the standard deviation narrow and remain stable continuously. Therefore, 10,000 is chosen as the optimal sample size for this research. y a v g = ∑ i = 1 n y i n M S P E σ = 1 n − 1 × ∑ i = 1 n y i − y a v g 2 M S P E μ = 1 n − 1 ∑ i = 1 n y i − y a v g 2 (14)

After the parameter distribution is settled, the Sobol method is utilized to investigate how the sensitivity of br, d, θ influences the film cooling effectiveness under three different values of dr. The Sobol method shows excellent performance in the sensitivity analysis (42, 43). The sample vector x is firstly transferred into the uncertainty input vector {X ₁ , X ₂ , X ₃ }, where X ₁ , X ₂ , and X ₃ denote coolant hole diameter, inclination angle, and density ratio, respectively. A functional mapping Y = f(X) is constructed to represent the relation between X and Y, where Y is the general film cooling effectiveness, η a v , defined in Eq. 7.

Sobol indices are defined in Eq. 15, where S _i and S _Ti are the first and total-effect variance-based Sobol indices, respectively. The Sobol index is used to quantify the sensitivity of a parameter. The larger the Sobol index is, the more significant impact its corresponding parameter has. X _i is the input parameter among the uncertainty input set {X ₁ , X ₂ , X ₃ }, X _∼i denotes the set of all the input variables but X _i . By applying uncertainty deviation to each of the four-parameter sets, Y is affected to vary accordingly. Hence, the variances of Y under diverse scenarios are defined in Eq. 16. S i = 1 C o v Y , Y × C o v X i E X ∼ 1 Y | X i , E X ∼ 1 Y | X i S T i = C o v Y , Y − C o v X i E X ∼ i Y | X i , E X ∼ i Y | X i C o v Y , Y (15) V i = C o v X i E X ∼ i Y | X i , E X ∼ i Y | X i V i j = C o v E X ∼ i j Y | X i , X j , E X ∼ i j Y | X i , X j − V i − V j C o v Y , Y = ∑ i = 1 n V i + ∑ i < j n V i j + V 1,2 , … , n (16)

In addition to S _i and S _Ti , the temperature distribution near the blade also needs attention. If the blade surface is well covered by the cooling jet, its life span and reliability could be extended and increased heavily. To set up computation, the Probability Distribution Function (PDF) is introduced to represent the distribution of cooling effectiveness of every grid point on the flat surface of the turbine blade. The PDF are defined in Eq. 17. Where y denotes the general film cooling effectiveness predicted by the ANN model, μ and σ is the mean value and standard deviation of the y. P D F y = 1 σ 2 π e − y − μ 2 2 σ 2 μ = 1 n ∑ i = 1 n y i σ = 1 n ∑ i = 1 n y i − μ 2 (17)

By applying the uncertainty Sobol method to represent the three geometric parameters at blowing ratios of [0.5, 1.0, 1.5], the results of experiments are plotted in Figures 11A–C.

Figure 11A shows that at the low blowing ratio, the coolant hole diameter d has the largest value for both the S _i and S T i . It means that among the three geometric parameters, the deviation of coolant hole diameter has the largest uncertainty influence on the global film cooling effectiveness at a low blowing ratio. Followed by the hole diameter is the coolant jet inclination angle. Regardless of S i or S T i , the uncertainty Sobol indices are both around 35%. The deviation of the density ratio is the weakest of all three parameters, the Sobol indices of which are less than 2%. In all, when the blowing ratio is set to 0.5, the trends of S _i and S _Ti for d and dr show high similarity. d is the dominant index which composites nearly 64%, θ yields around 35%, and dr has the most minimal effect on the uncertainty result.

For the medium-blowing ratio of 1.0, the order of the impact of the three geometric parameters changes utterly. Figure 11B shows that the difference between the S i and the S T i is tiny. Still, the θ affects the result far more than the other two parameters. The S i and S T i of θ both exceed 80%, which are approximately eight and fifteen times more than those of d and dr. It can be concluded that when the br is 1.0, the uncertain deviation of θ has the most significant impact on the general film cooling effectiveness.

Figure 11C indicates that for a large blowing ratio of 1.5, the corresponding result can be interpreted as a magnified one of a blowing ratio of 1.0. The coolant hole inclination angle is still the primary parameter, but its Sobol index is furthermore prominent than the medium-blowing ratio case. This time, the first-order and overall Sobol indices for θ outstrip 95%. Compared with the Sobol indices under the medium blowing ratio br, the hole diameter d and density ratio dr, which are all less than 3%, show negligible influence on the general cooling effectiveness. Moreover, it is found that when the blowing ratio increases, the S _i and S _Ti for all parameters increase accordingly.

In the gas turbine application, in a low blowing ratio case, both the coolant hole diameter and inclination angle significantly impact cooling effectiveness. However, the uncertain deviation of coolant hole diameter has a more significant effect. As for a high blowing ratio, usually more than 1.5, the coolant hole inclination angle needs special attention. In medium and high blowing ratio cases, the coolant inclination angle dominates the results, and its dominant effect increases as the blowing ratio increases, in all three cases. The trends of the first-order and the total-effect index are very similar. Thus, the same conclusion can be drawn.

Figure 12 shows the flow field of the flat plane. Since there are up to 225 cases in total, a reference case for the flow field example is defined with the following parameters: d = 14.0 mm, θ = 40 ∘ , dr = 1.2, br = 0.5. Figure 12A is the top view. It can be found that the flow field is highly symmetric on the top surface, and the green region is the widest temperature region, which occupies more than half the length of the region downstream of the confluence on the centerline. Moreover, the lateral diffusion is not evident since the width of the coolant stream keeps nearly equal to the hole diameter. Figure 12B is the front view of the flow field. It is detectable that right downstream the coolant hole, a reverse vortex pair is generated that drives the coolant jet away from the blade. A portion of the coolant on the top blends into the mainstream. The flow field shows that as the two jets move forward, the convergence of the two streams becomes more pronounced. Thus it is true that the coolant jet performs well only at a limited distance downstream of the confluence point. The reference case has a blowing ratio of less than 1, which means that the mainstream moves faster than the coolant jet, so Figure 12C indicates in the cross-section view that the general effect of gas motion is moving from the coolant jet to the hot mainstream. This leads to the coolant jet’s diffusion and thus significantly impairs the cooling effectiveness. While in the cross-section view, we can also find that the coolant jet, which stays near the surface of the turbine blade, is rotating in a closed loop, that ensures that coolant is not lifted away from the blade, and achieves better cooling effectiveness at the surface layer of the blades.

During the experiment and analysis, it is found that under the small blowing ratio case of br = 0.5, the coolant hole diameter d has more impact sensitivity on the general film cooling effectiveness, compared with the coolant inclination angle. However, under a larger blowing ratio of 1.0, the coolant inclination angle has more effect than hole diameter. To further explain the flow physics that account for this result, the flow field of the cases under low and high blowing ratios, including d = 10.5 mm and 11.5 mm, coolant inclination angles of 20 ° and 30 ° , and in both x-z and y-z direction are compared and analyzed. From a top view, by comparing Figures 13A,B and Figures 13A,C, we find that the low-temperature region at the center of the coolant hole expands to a larger degree under the influence of diameter change. This discovery implies that under a small blowing ratio of br = 0.5, the change in cooling effectiveness due to diameter change is larger than the change due to inclination angle. As for the case where the blowing ratio is as large as br = 1.0, Figures 13D,E and Figures 13D,F shows that the influence of the inclination angle is larger than the influence of the diameter, because the coolant flow in (F) shrinks dramatically, while the ones in (D) and (E) are similar to each other. Figure 14 shows the cross-section view of the same 6 cases in Figure 13. When the blowing ratio is low, the difference of width of the CVP between Figures 14A,B is larger than the difference between Figure 14A,C; whereas, under a large blowing ratio, the coolant CVP is lifted to a larger degree in the case of increasing the angle instead of increasing thecoolant hole diameter. As shown in Figure 14F, the height of the CVP center climbs higher under the influence of the inclination angle compared with Figure 14E. Furthermore, the CVP is rotating upward, which results in the additional lifting of the coolant flow at the location of x/d = 19. This impairs the cooling effectiveness as well.

In addition to utilizing Sobol indices on behalf of the sensitivity, the control variate method is deployed to further research the individual effect of the three independent geometric parameters on the general effectiveness of cooling under three different blowing ratios in terms of the probability distribution(44). The reference values for d, θ, and dr are 12.5 mm, 35°, and 1.2 respectively. While varying one of the three, the others are set as the value in the reference case to achieve uniformity. Figure 15 shows the probability distribution of general cooling effectiveness at different blowing ratios. The four subplots study the three single-parameter effects and the combined effect.

As shown in Figure 15A, the PDF distribution of the coolant hole diameter is studied. When the blowing ratio is 0.5, the 95% confidence interval due to hole diameter is [0.070, 0.113]. However, those for blowing ratios of 1.0 and 1.5 are [0.030, 0.049] and [0.014, 0.017], respectively. The corresponding interval length for the three blowing ratios are 0.043, 0.019, and 0.003, respectively. The result shows that as br is increasing, the 95% confidence interval due to hole diameter narrows rapidly, almost halving each time, and the mean of general cooling effectiveness decreases continuously, indicating that the uncertainty caused by hole diameter decreases.

For the hole inclination angle, as shown in Figure 15B, when the br increases, the uncertainty effect maintains a high level due to the hole inclination angle. The confidence intervals remain wide for all low, medium, and high blowing ratios. This is different from the case for hole diameter. With the increase of br, the range of variation of general film cooling effectiveness of inclination angle decreases from 0.030, 0.029 to 0.023 after simple calculation. It can be concluded that with the increasing of the blowing ratio, the uncertainty caused by the inclination angle θ reduces, but with a lower decreasing speed than the hole diameter.

Figure 15C shows that the magnitude of the length of the confidence interval of the density ratio is the smallest among all parameters. However, the PDF value of the density ratio is the highest, the minimum value of which still exceeds 160 in the case of the br = 1.0.

The uncertainty of the combined effect decreases as br rises, and the combined effect is more prominent than all three single effects. In Figure 15D, for example, the 95% confidence intervals are [0.062, 0.112], [0.023, 0.058], [0.004, 0.027], and the interval lengths are 0.060, 0.035, and 0.023, respectively. Compared with the single effect caused by the inclination angle θ, the interval length of the combination becomes wider.