This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
Nowadays, gas turbines intake jet air at high temperatures to improve the power output as much as possible. However, the excessive temperature typically puts the blade in the face of unpredictable damage. Film cooling is one of the prevailing methods applied in engineering scenarios, with the advantages of a simple structure and high cooling efficiency. This study aims to assess the uncertain effect that the three major film cooling parameters exert on the global and fixed-cord-averaged film cooling effectiveness under low, medium, and high blowing ratios
In gas turbine applications, since the intake gas temperature positively correlates with the power output, the temperature of the intake gas is expected to be as high as possible in pursuit of a better power output of a gas turbine. However, such temperature typically exceeds the melting temperature of turbine blades, which would cause the blade to melt and even lead to potential dangers in a gas turbine (
Nonetheless, the process of predicting the cooling effects from a given flat plate hole parameter set is complicated by the complexity and unpredictability of the vortex structure and gas mixing motion. Previous studies have shown that two categories of input parameters matter to the resulting film cooling efficiency. They include the property of the coolant jet and the coolant injection method. The thermal property of the coolant includes coolant temperature (
Uncertainty analysis is of great importance in gas turbines (
However, the computation time and computation load increase exponentially in cases of higher dimensions, and the traditional PCE methods are commonly used to solve the “single output” problem, i.e., it is more widely used to obtain the overall cooling temperature of the research region only, instead of the fix-cord-averaged temperature analysis. Even though theoretically, the PCE methods can also be utilized to produce laterally averaged results, the computational cost is relatively higher. Many beneficial attempts are conducted to solve the difficulties in high-dimensional cases, such as the surrogate-based optimization method and artificial neural network et al., to conclude the complex non-linear correlation between the input coolant parameter configurations and the resulting cooling effectiveness using semi-empirical correlations. Mellor et al. (
In recent years, deep learning has emerged and is making a favorable contribution in pushing the process of various application fields forward (
This paper constructed and validated a deep-learning-based ANN model to obtain the dataset to identify a non-linear mapping to link the four cooling parameters to cooling effectiveness. The application of the ANN model greatly enhanced the reliability of the correlation between parameters and the performance of the flat film cooling. The four cooling parameters include coolant hole diameter
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 (
Schematic of the flat plate film-cooling configuration:
Values of the film cooling parameters used in CFD.
Cooling parameters | Values |
---|---|
Blowing Ratio, |
[0.5,1.0,1.5] |
Film Cooling Diameter, |
[10.5, 11.5, 12.5, 13.5, 14.5] |
Coolant Inclination Angle, |
[15, 25, 35, 45, 55] |
Density Ratio, |
[1.1, 1.2, 1.3] |
Mainstream Temperature, |
313 |
Mainstream Velocity, |
20 |
Ansys Fluent software is proven to have excellent performance in solving cooling problems (
The
To avoid using a complicated high dimensional temperature matrix to represent the cooling efficiency,
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
Equation
Following other studies, the Fluent ® 18.0 software is applied to all cases (
Comparison between model validation:
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 (
Centerline dimensionless temperature with 2, 4.5, 6, and 7.5 million grid cells.
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.
Mesh schematic with enlarged part.
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
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 (
Equation
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
Structure of the 7-layer ANN model.
In the training process, the learning rate is one of the most significant hyperparameters that needs to be determined when applying ANN (
Loss of the ANN model.
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.
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
QE values under different hyperparameters setups:
Similar experiments are designed for dropout and batch size.
For Batch size, three different batch sizes are studied, and
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
Layers and nodes information and corresponding QE values.
Number of layers | Nodes details | QE for training group (%) | QE for validation group (%) |
---|---|---|---|
2 | 4-256 | 0.91 | 0.96 |
3 | 4-128-256 | 0.35 | 0.40 |
4 | 4-16-64-256 | 0.32 | 0.36 |
7 | 4-8-16-32-64-128-256 | 0.31 | 0.33 |
9 | 4-8-16-16-32-64-64-128-256 | 0.34 | 0.45 |
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
Fixed-cord-averaged cooling effectiveness results comparison by the ANN and CFD model;
Besides the fixed-cord-averaged cooling effectiveness, the general cooling effectiveness is also studied to enhance the above conclusion.
General results comparison by the ANN and CFD model;
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 (
Monte Carlo (MC) simulation is an extensively utilized technique to quantify the engineering field’s uncertainty among various kinds of uncertainty quantification methods (
MSPE values for MC for diverse sample size:
After the parameter distribution is settled, the Sobol method is utilized to investigate how the sensitivity of
Sobol indices are defined in Eq.
In addition to
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
Sodol Indices for
For the medium-blowing ratio of 1.0, the order of the impact of the three geometric parameters changes utterly.
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.
Details of flow fields under different views:
During the experiment and analysis, it is found that under the small blowing ratio case of
Comparison of the top-view flow behavior at
Comparison of the side-view flow behavior at
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(
PDF distribution of general film cooling effectiveness at three blowing ratios:
As shown in
For the hole inclination angle, as shown in
The uncertainty of the combined effect decreases as
This study aims to improve the gas turbine performance by strengthening the film cooling effectiveness, especially by focusing on the uncertainty of the three significant parameters, including single hole diameter, density ratio, and inclination angle, on the film cooling effectiveness under low, medium, and high blowing ratios. The uncertainty analysis was conducted using a deep-learning-based ANN model and uncertainty quantification method. Firstly, all related indices and research regions are defined at the beginning. Due to its best performance, the six-million grid size and the RNG k-ε model are chosen for the turbulence model. Secondly, a high-performance ANN model is delicately constructed for training and to seek the non-linear correlation between the parameter input and the cooling effectiveness output. CFD provides training and validation datasets. Finally, the sensitivity of three parameters is quantified, and uncertainty quantification is conducted to quantify the single and combined effect of the uncertainty of these three parameters on the general cooling effectiveness. The following conclusions are drawn. 1. After careful hyperparameter selection and training, the ANN model built in this study shows excellent performance in predicting the general and fixed-cord-averaged film cooling effectiveness according to input parameters compared with the data simulated by the CFD method. The QE value for fixed-cord-averaged film cooling effectiveness in training and validation datasets are 0.29% and 0.32%. The QE value for general film cooling effectiveness in training and validation datasets are 0.35% and 0.30%. 2. The Sobol method based on MC simulation shows that at a small blowing ratio, the coolant tube’s diameter and inclination angle are two main factors to the cooling effectiveness, and the former has a more dominant effect. At medium and large blowing ratios, the inclination angle is the only leading factor to the film cooling effectiveness. Furthermore, the maximum effect of the inclination angle increases as the blowing ratio grows. 3. Uncertainty quantification reveals that the uncertainty of hole diameter, inclination angle, and density ratio all decrease as the blowing ratio rises. Moreover, the combined effect shows a higher impact on the general cooling effectiveness than any single effect. Within three parameters, the variation of the uncertainty interval of the hole diameter at three blowing ratios is the most obvious. Furthermore, the inclination angle
The data that support the findings of this study are available from the corresponding author upon reasonable request.
YW: Conceptualization, data curation, formal analysis, investigation, methodology, resources, software, validation, and writing–original draft. XQ: Data curation, formal analysis, investigation, resources, visualization, and writing—original draft. SQ: Investigation, resources, visualization, and writing—original draft. YS: Data curation, formal analysis, methodology, visualization, and writing—original draft. WW: Resources, software, and validation. JC: Conceptualization, formal analysis, funding acquisition, investigation, project administration, software, supervision, visualization, writing—review and editing.
This study was supported in part by State Key Laboratory for Aerodynamics, the Zhejiang University/University of Illinois at Urbana-Champaign Institute and National Natural Science Foundation of China (Grant No. 52106060 and 92152202). It was led by Supervisor JC.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
film cooling diameter
standard Value of
coolant tube inclination angle
coolant-to-mainstream blowing ratio
coolant-to-mainstream density ratio
density of coolant jet
density of mainstream jet
velocity of coolant jet
velocity of mainstream jet
gauged temperature
temperature of coolant jet
temperature of mainstream jet
dimensionless temperature
fixed-cord-averaged
film cooling effectiveness
fixed-cord-averaged
general film cooling effectiveness
mean square error
quoted error
first-order sensitive index
total-effect sensitive index
mean squared pure error
standard deviation of MSPE
mean of the MSPE
mean
standard deviation
probability distribution function