# Prediction of Transonic Flow over Cascades via Graph Embedding Methods on Large-Scale Point Clouds

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## Abstract

**:**

## 1. Introduction

- A novel framework has been devised to predict flow fields over the cascade, combining GCN with point clouds to enhance prediction accuracy;
- This innovative framework facilitates swift and precise predictions across an extensive grid containing 295,035 flow-field points, ensuring large-scale flow-field analysis efficiency;
- A detailed investigation has been conducted to unravel the underlying mechanisms of GCN in the context of flow-field prediction, shedding light on its intricate understanding and application.

## 2. Numerical Methods and Dataset Generation

#### 2.1. Cascade Geometry Generation

#### 2.2. CFD Simulation and Dataset Generation

^{6}. The grid over the blade surface is controlled as y

^{+}≈ 1/2, with the size on the order of 10

^{−6}m. Over the surface of the cascade, 1603 grid points are set, and the far-field length is nearly four times the length of the cascade.

_{st}decreases to within 0.4%, meeting the grid independence requirements. To accurately predict the cascade flow field based on GCN, a grid number of 295,035 is ultimately selected for the subsequent optimization database construction, as the results are basically unchanged with the increase of the grid numbers.

## 3. Deep-Learning GCN-Based Framework and Model Training

#### 3.1. The Structure of the Framework

_{k}and B

_{k}stands for the calculating matrix, and AGG stands for the generalized aggregation function. In this study, aggregation and update functions can be expressed as:

#### 3.2. Training

## 4. Results

#### 4.1. Fields Prediction Performance

#### 4.2. Prediction of the Trained Model on Cascade with Different Nodes Selection Approach

#### 4.3. Explanation of Graph Embedding Approach Based on the Framework

## 5. Discussion and Limitations

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Identifications of the details in the flow field over the cascade based on different models.

**Figure 7.**Flow-field prediction based on the models utilizing MAE and Huber loss function as loss functions, respectively. (

**a**,

**f**) are the reference pressure field and turbulent viscosity field based on the CFD solution. (

**b**,

**c**,

**g**,

**h**) are the predicted flow fields and absolute error using MAE as the loss function. (

**d**,

**e**,

**i**,

**j**) are the predicted flow fields and absolute errors based on the Huber loss function.

**Figure 10.**Flow prediction based on a set model over different geometry. (

**a**,

**c**,

**e**,

**g**), respectively, display the CFD (

**left**) and predicted pressure fields (

**middle**), along with the absolute errors (

**right**) for different geometries. (

**b**,

**d**,

**f**,

**h**), respectively, display the CFD (

**left**) and predicted turbulent viscosity fields (

**middle**), along with the absolute errors (

**right**) for different geometries.

**Figure 11.**Comparison of predicted fields and CFD fields value. (

**a**,

**b**) sequentially display the results of the pressure field and turbulent viscosity field.

**Figure 12.**Comparison of CFD fields, GCN-based predicted fields, and CNN-based predicted fields. (

**a**–

**c**) sequentially display the results of CFD, GCN-based model and CNN-based model.

**Figure 13.**The contribution based on prediction over the cascade leading edge and wake regions with different intervals of global nodes removed. (

**a**,

**b**) represents the results of the pressure and turbulent viscosity field, respectively.

**Figure 16.**The contribution based on prediction over the cascade leading edge with continuous steps of global nodes removed, where (

**a**–

**d**) represent the results with steps 20, 50, 100, and 200.

**Figure 17.**The contribution shown in the same intervals based on prediction over the cascade leading edge with continuous steps of global nodes removed, where (

**a**–

**d**) represent the results with steps 20, 50, 100, and 200.

**Figure 18.**The contribution based on prediction over the wake region with continuous steps of global nodes removed, where (

**a**–

**d**) represents the results with steps 20, 50, 100, and 200, while (

**e**–

**h**) stands for the plotting intervals of 10 with different steps.

Number of the Nodes | η | P_{st} |
---|---|---|

32,573 | 0.0176191 | 85214.731 |

101,570 | 0.0163585 | 80511.061 |

174,568 | 0.0163082 | 80328.973 |

295,035 | 0.0162869 | 80060.078 |

408,914 | 0.0162905 | 80058.009 |

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**MDPI and ACS Style**

Lan, X.; Wang, L.; Wang, C.; Sun, G.; Feng, J.; Zhang, M.
Prediction of Transonic Flow over Cascades via Graph Embedding Methods on Large-Scale Point Clouds. *Aerospace* **2023**, *10*, 1029.
https://doi.org/10.3390/aerospace10121029

**AMA Style**

Lan X, Wang L, Wang C, Sun G, Feng J, Zhang M.
Prediction of Transonic Flow over Cascades via Graph Embedding Methods on Large-Scale Point Clouds. *Aerospace*. 2023; 10(12):1029.
https://doi.org/10.3390/aerospace10121029

**Chicago/Turabian Style**

Lan, Xinyue, Liyue Wang, Cong Wang, Gang Sun, Jinzhang Feng, and Miao Zhang.
2023. "Prediction of Transonic Flow over Cascades via Graph Embedding Methods on Large-Scale Point Clouds" *Aerospace* 10, no. 12: 1029.
https://doi.org/10.3390/aerospace10121029