# Aerodynamic Uncertainty Quantification of a Low-Pressure Turbine Cascade by an Adaptive Gaussian Process

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Simulation

## 3. Adaptive Gaussian Process

#### 3.1. Gaussian Process

#### 3.2. Adaptive Sampling

**Step 1**: The initial training sets are prepared for GP training, and a batch of test sets are prepared for prediction.

**Step 2**: Train the GP model by hyperparameter optimization, which is then used for function predictions of the test sets.

**Step 3**: Calculate the acquisition function ${f}_{acq}$ at each test point and determine the maximum ${f}_{acq,max}$, which is then compared with the threshold.

**Step 4**: If ${f}_{acq,max}\le \u03f5$, model training can be completed, where $\u03f5$ is the threshold. If ${f}_{acq,max}>\u03f5$, the current GP model does not meet the accuracy requirement. The first selected test point is added to the training sets. Calculate the ${f}_{acq}$ for the second time and determine the maximum ${f}_{acq,max}$, which is then compared with the threshold.

**Step 5**: If ${f}_{acq,max}\le \u03f5$, model training can be completed. If ${f}_{acq,max}>\u03f5$, the second selected test point is added to the training sets and goes to

**Step 2**.

**Step 2**, while only the covariance matrix needs to be updated. In this way, two new training samples can be selected per each iteration.

#### 3.3. Function Test

#### 3.4. ADGP for Aerodynamic Parameters

## 4. Results and Discussion

#### 4.1. Uncertainty Quantification and Statistical Analysis

#### 4.2. Sobol Sensitivity Analysis

## 5. Conclusions

- (1)
- By comparing the methods of adaptive NIPC and GP with static sampling, the prediction accuracy of ADGP introduced in the study is proved to be higher through a function experiment. The machine-learning-based model training can find the optimal hyperparameters. The ADGP is then further verified and validated by accurately predicting the performance parameters of an LPT cascade.
- (2)
- For this LPT cascade, the total pressure-loss coefficient and Zweifel number are sensitive to the uncertain variations of inlet flow parameters, while the outlet flow angle is almost insensitive to the uncertainties. Statistical analysis of the flow field demonstrates that flow transition on the suction side of the LPT cascade and the viscous shear stress are rather sensitive to uncertainties, which can be regarded as the main sources of the increased mean flow losses and performance dispersion.
- (3)
- By the Sobol method, the effects of each uncertainty on performance changes are quantified by sensitivities. The contributions of each inlet flow parameter variation to the changes in the total pressure-loss coefficient are almost the same. However, most of the changes in outlet flow angle and Zweifel number are attributed to the variation of inlet total pressure. For this LPT cascade, the contributions of the pairwise uncertainties to performance changes are quite different. The impact on performance changes may be strengthened or weakened, considering the effects of pairwise uncertainties.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ANN | artificial neural networks |

ADGP | adaptive Gaussian process |

CFD | computational fluid dynamics |

GP | Gaussian process |

LPT | low-pressure turbine |

MM | method of moment |

MAPE | mean absolute percentage error |

MCS | Monte Carlo simulation |

ML | machine learning |

NIPC | non-intrusive polynomial chaos |

PC | polynomial chaos |

probability density function | |

RANS | Reynolds-averaged Navier–Stokes |

UQ | uncertainty quantification |

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**Figure 3.**Performance parameter changes versus inlet flow parameter variations: (

**a**) $\Delta \zeta $; (

**b**) $\Delta \beta $; (

**c**) $\Delta Zw$.

**Figure 4.**Contours of intermittency factor: (

**a**) Ref; (

**b**) $\Delta \alpha $; (

**c**) $\Delta {P}_{t,in}$; (

**d**) $\Delta $Tu.

**Figure 7.**Sample distributions of the 2-d function experiments: (

**a**) function; (

**b**) static GP; (

**c**) adaptive NIPC; (

**d**) ADGP.

**Figure 11.**Contours of the statistics of relative variation of intermittency factor: (

**a**) mean; (

**b**) standard deviation.

**Figure 12.**Distributions on the suction side of the statistics of the relative variation of viscous shear stress.

**Figure 14.**Pressure distributions on the blade considering the effects of: (

**a**) $\alpha $; (

**b**) ${P}_{t,in}$; (

**c**) $Tu$.

Parameter | Value |
---|---|

inlet total pressure ${P}_{t,in}$ | 294,679.4 Pa |

inlet total temperature ${T}_{t,in}$ | 1238.9 K |

inlet flow angle $\alpha $ | 33.50° |

inlet turbulence intensity $Tu$ | 0.025 |

outlet static pressure ${P}_{out}$ | 201,017 Pa |

axial chord ${c}_{x}$ | 0.0165 m |

total pressure-loss coefficient $\zeta $ | 0.0317 |

outlet flow angle $\beta $ | −61.20° |

Zweifel number $Zw$ | 0.8682 |

${\mathit{n}}_{0}$ | $\mathit{\u03f5}$ | ${\mathit{n}}_{\mathit{t}}$ | |
---|---|---|---|

2-d | 6 | 0.02 | 36 |

4-d | 12 | 0.05 | 66 |

6-d | 16 | 0.15 | 66 |

2-d | 4-d | 6-d | |
---|---|---|---|

GP | 10.594 | 8.873 | 4.933 |

ADGP | 1.764 | 4.248 | 3.760 |

ANIPC | 216.7 | 683.2 | 194.2 |

Parameter | MAPE(%) |
---|---|

$\zeta $ | 0.2420 |

$\beta $ | 0.0062 |

$Zw$ | 0.0089 |

Parameter | ${\mathit{\mu}}_{\mathbf{\Delta}\mathit{f}}$(%) | ${\mathit{\sigma}}_{\mathbf{\Delta}\mathit{f}}$(%) | ${\mathit{\epsilon}}_{\mathit{\mu}}$(%) | ${\mathit{\epsilon}}_{\mathit{\sigma}}$(%) |
---|---|---|---|---|

$\zeta $ | 4.943 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 1.759 | −3.797 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 2.403 |

$\beta $ | −4.656 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 0.162 | 7.602 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | −2.231 |

$Zw$ | 1.112 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 3.080 | 3.747 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 0.178 |

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**MDPI and ACS Style**

Fu, W.; Chen, Z.; Luo, J.
Aerodynamic Uncertainty Quantification of a Low-Pressure Turbine Cascade by an Adaptive Gaussian Process. *Aerospace* **2023**, *10*, 1022.
https://doi.org/10.3390/aerospace10121022

**AMA Style**

Fu W, Chen Z, Luo J.
Aerodynamic Uncertainty Quantification of a Low-Pressure Turbine Cascade by an Adaptive Gaussian Process. *Aerospace*. 2023; 10(12):1022.
https://doi.org/10.3390/aerospace10121022

**Chicago/Turabian Style**

Fu, Wenhao, Zeshuai Chen, and Jiaqi Luo.
2023. "Aerodynamic Uncertainty Quantification of a Low-Pressure Turbine Cascade by an Adaptive Gaussian Process" *Aerospace* 10, no. 12: 1022.
https://doi.org/10.3390/aerospace10121022