# The Rapid Data-Driven Prediction Method of Coupled Fluid–Thermal–Structure for Hypersonic Vehicles

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Rapid Data-Driven Prediction Method

#### 2.1. Proper Orthogonal Decomposition

#### 2.2. Radial Basis Function Interpolation

**.**

#### 2.3. Rapid Prediction Method Based on Data-Driven

**.**The pulsation value ${\mathit{Y}}^{\prime}$ is used to construct matrix $\mathit{M}$:

## 3. Numerical Simulation

## 4. Result of Prediction

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

${\mathit{P}}_{\mathit{\infty}}$$\left(\mathbf{P}\mathbf{a}\right)$ | ${\mathit{T}}_{\mathit{\infty}}\left(\mathbf{K}\right)$ | $\mathit{M}\mathit{a}$ | $\mathit{R}{\mathit{e}}_{\mathit{\infty}}$ |
---|---|---|---|

$648.1$ | $241.5$ | $6.47$ | $1.31\times {10}^{6}$ |

Elastic Modulus $\left(\mathbf{G}\mathbf{P}\mathbf{a}\right)$ | Poisson’s Ratio | Density $\left(\mathbf{K}\mathbf{g}/{\mathbf{m}}^{3}\right)$ | Coefficient of Linear Expansion $\times {10}^{-6}\left({\mathbf{K}}^{-1}\right)$ | Thermal Conductivity $\left(\mathbf{W}/\left(\mathbf{m}\mathit{\xb7}\mathbf{K}\right)\right)$ | Specific Heat Capacity $\left(\mathbf{J}/\left(\mathbf{k}\mathbf{g}\mathit{\xb7}\mathbf{K}\right)\right)$ |
---|---|---|---|---|---|

206 | 0.3 | 8030 | 17.5 | 16.27 | 502.48 |

## References

- Zel’dovich, Y.B.; Raizer, Y.P. Elements of Gasdynamics and the Classical Theory of Shock Waves. In Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena; Hayes, W.D., Probstein, R.F., Eds.; Dover Publications: Mineola, NY, USA, 1967; pp. 1–104. [Google Scholar]
- Hirschel, E.H. The Flight Environment. In Basics of Aerothermodynamics; Springer: Berlin, Germany, 2005; pp. 15–26. [Google Scholar]
- Anderson, J.D. Hypersonic Shock and Expansion-Wave Relations. In Hypersonic and High-Temperature Gas Dynamic; AIAA Press: Palo Alto, CA, USA, 2019; pp. 39–53. [Google Scholar]
- Shang, J.S.; Surzhikov, S.T. Radiative Energy Transfer. In Plasma Dynamics for Aerospace Engineering; Cambridge University Press: New York, NY, USA, 2018; pp. 304–336. [Google Scholar]
- Gui, Y.; Yuan, X. Numerical simulation on the coupling phenomena of aerodynamic heating with thermal response in the region of the leading edge. J. Eng. Thermophys.
**2002**, 23, 733–735. [Google Scholar] - Chen, F.; Liu, H.; Zhang, S. Time-adaptive loosely coupled analysis on fluid–thermal–structural behaviors of hypersonic wing structures under sustained aeroheating. Aerosp. Sci. Technol.
**2018**, 78, 620–636. [Google Scholar] [CrossRef] - Geng, X.; Liu, L.; Gui, Y.; Tang, W.; Wang, A. Studying the Temperature Field of Hypersonic Vehicle Structure with Aero-Thermo-Elasticity Deformation. Int. Sch. Sci. Res. Innov.
**2015**, 9, 1434–1437. [Google Scholar] - Zhao, X.; Sun, Z.; Tang, L.; Zheng, G. Coupled flow-thermal-structural analysis of hypersonic aerodynamically heated cylindrical Leading Edge. Eng. Appl. Comput. Fluid Mech.
**2011**, 5, 170–179. [Google Scholar] [CrossRef] [Green Version] - Korkegi, R.H. Survey of Viscous Interactions Associated with High Mach Number Flight. AIAA J.
**1971**, 9, 771–784. [Google Scholar] [CrossRef] - Wieting, A.R. Experimental Study of Shock Wave Interference Heating on a Cylindrical Leading Edge. Ph.D. Thesis, Old Dominion University, Norfolk, VA, USA, 1987. [Google Scholar]
- Thornton, E.A.; Dechaumphai, P. Coupled Flow, Thermal and Structural Analysis of Aerodynamically heated Panels. J. Aircr.
**1988**, 25, 1052–1059. [Google Scholar] [CrossRef] - Dechaumphai, P.; Thornton, E.A.; Wieting, A.R. Flow-Thermal-Structural Study of Aerodynamic Heated Leading Edges. J. Aircr.
**1989**, 26, 201–209. [Google Scholar] - Li, J.; Wang, J.; Yang, L.; Shu, C. A hybrid lattice Boltzmann flux solver for integrated hypersonic fluid-thermal-structural analysis. Chin. J. Aeronaut.
**2020**, 33, 2295–2312. [Google Scholar] [CrossRef] - Liu, H.; Wang, Z. Fluid–thermal–structural coupling investigations of opposing jet in hypersonic flows. Int. Commun. Heat Mass Transf.
**2021**, 120, 105017. [Google Scholar] - Sirovich, L. Turbulence and the dynamics of coherent structures. Part 1: Coherent structures. Quart. Appl. Math.
**1987**, 45, 561–571. [Google Scholar] [CrossRef] [Green Version] - Duan, Y.; Cai, J.; Li, Y. Gappy proper orthogonal decomposition-based two-step optimization for airfoil design. AIAA J.
**2012**, 50, 968–971. [Google Scholar] [CrossRef] - Tan, B.; Damodaran, M.; Willcox, K. Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition. AIAA J.
**2004**, 42, 1505–1516. [Google Scholar] - Marc, O.; Markus, B.; Kilian, O.; Sonja, S. Statistical characterization of horizontal slug flow using snapshot proper orthogonal decomposition. Int. J. Multiph. Flow
**2021**, 134, 103453. [Google Scholar] - Huang, D.; Jan, N.; Christian, W.; Peter, W. A machine learning based plasticity model using proper orthogonal decomposition. Comput. Methods Appl. Mech. Eng.
**2020**, 365, 113008. [Google Scholar] [CrossRef] [Green Version] - Legresley, P.; Alonso, J. Airfoil design optimization using reduced order models based on proper orthogonal decomposition. Proc. AIAA Space
**2000**, 40, 1954–1960. [Google Scholar] - Esmaeilbeigi, M.; Garmanjani, G. Gaussian radial basis function interpolant for the different data sites and basis centers. Calcolo
**2017**, 54, 155–166. [Google Scholar] [CrossRef] - Wang, X.D.; Ding, Y.H.; Shao, H.H. The improved radial basis function neural network and its application. Artif. Life Robot.
**1998**, 2, 8–11. [Google Scholar] [CrossRef] - Tyshchenko, O. A reservoir radial-basis function neural network in prediction tasks. Autom. Control Comput. Sci.
**2016**, 50, 65–71. [Google Scholar] [CrossRef] - Cheng, Q.S.; Wu, L.W.; Wang, S.Z. Fusion prediction based on the attribute clustering network and the radial basis function. Chin. Sci. Bull.
**2001**, 46, 789–792. [Google Scholar] [CrossRef] - Liu, Z.; Ning, F.; Zhai, Q.; Ding, H.; Wei, J. Study on the flow characteristics in the supersonic morphing cavities using direct numerical simulation and proper orthogonal decomposition. Wave Motion
**2021**, 104, 102751. [Google Scholar] [CrossRef] - Di, G.; Dakun, S.; Ruize, X.; Daniel, B.; Sun, X.F.; Ni, S.L.; Du, J.; Zhao, D. Experimental investigation on axial compressor stall phenomena using aeroacoustics measurements via empirical mode and proper orthogonal decomposition methods. Aerosp. Sci. Technol.
**2021**, 112, 106655. [Google Scholar] - Lukas, K.; Cyrille, V.; Christian, B. Using a Proper Orthogonal Decomposition representation of the aerodynamic forces for stochastic buffeting prediction. J. Fluids Struct.
**2020**, 99, 103178. [Google Scholar] - Chen, X.; Liu, L.; Yue, Z. Reduced order aerothermodynamic modeling research for hypersonic vehicles based on proper orthogonal decomposition and surrogate method. Acta Aeronaut. Astronaut. Sin.
**2015**, 36, 462–472. [Google Scholar] - Kou, J.; Zhang, W. Dynamic mode decomposition and its applications in fluid dynamics. Acta Aerodyn. Sin.
**2018**, 36, 163–179. [Google Scholar] - Wang, Y.; Han, R.; Liu, Z. Progress of deep learning modeling technology for fluid mechanics. Acta Aeronaut. Astronaut. Sin.
**2021**, 42, 524779. [Google Scholar] - Cui, E. MEMS and Intelligent Fluid Mechanics. Acta Aerodyn. Sin.
**2000**, 18, 52–59. [Google Scholar] - Karami, S.; Soria, J. Analysis of Coherent Structures in an Under-Expanded Supersonic Impinging Jet Using Spectral Proper Orthogonal Decomposition (SPOD). Aerospace
**2018**, 5, 73. [Google Scholar] [CrossRef] [Green Version] - Hu, Z. Research on Intelligent Prediction and Machine Learning Based on Big Data. Digit. Technol. Appl.
**2021**, 39, 84–86. [Google Scholar] - Ren, F.; Gao, C.; Tang, H. Machine learning for flow control: Applications and development trends. Acta Aerodyn. Sin.
**2021**, 42, 524686. [Google Scholar] - Lin, Y.; Guan, Z. The Use of Machine Learning for the Prediction of the Uniformity of the Degree of Cure of a Composite in an Autoclave. Aerospace
**2021**, 8, 130. [Google Scholar] [CrossRef] - Hashemi, S.; Botez, R.; Grigorie, T. New Reliability Studies of Data-Driven Aircraft Trajectory Prediction. Aerospace
**2020**, 7, 145. [Google Scholar] [CrossRef] - Uzun, M.; Demirezen, M.; Inalhan, G. Physics Guided Deep Learning for Data-Driven Aircraft Fuel Consumption Modeling. Aerospace
**2021**, 8, 44. [Google Scholar] [CrossRef] - Lerro, A.; Brandl, A.; Battipede, M.; Gili, P. A Data-Driven Approach to Identify Flight Test Data Suitable to Design Angle of Attack Synthetic Sensor for Flight Control Systems. Aerospace
**2020**, 7, 63. [Google Scholar] [CrossRef]

**Figure 9.**Relative error contour of the structural temperature and stress: (

**a**) temperature; (

**b**) stress.

$\mathbf{Mach}\mathbf{Number}\left(\mathbf{M}\mathbf{a}\right)$ | $\mathbf{Altitude}\left(H/km\right)$ | $\mathbf{Angle}\mathbf{of}\mathbf{Attack}\left(\alpha /\xb0\right)$ |
---|---|---|

$3\le Ma\le 8$ | $20\le \mathrm{H}\le 40$ | $-8\le \mathsf{\alpha}\le 8$ |

Method | Maximum Temperature of Structure/K | $\mathbf{Maximum}\mathbf{Heat}\mathbf{Flux}/\left(\mathbf{k}\mathbf{W}/{\mathbf{m}}^{2}\right)$ | Maximum Pressure of Flow Field/Pa |
---|---|---|---|

Numerical Simulation in this paper | 436 | 663 | 35,386 |

Experiment [10] | 465 | 670 | 37,815 |

Test Condition | $\mathbf{Altitude}\left(\mathbf{H}/\mathbf{k}\mathbf{m}\right)$ | $\mathbf{Mach}\mathbf{Number}\left(\mathbf{M}\mathbf{a}\right)$ | $\mathbf{Angle}\mathbf{of}\mathbf{Attack}\left(\mathbf{\alpha}/\xb0\right)$ |
---|---|---|---|

1 | 38 | 4.2 | −3.2 |

2 | 26 | 5 | −6.4 |

3 | 30 | 3.4 | 3.2 |

4 | 34 | 6.6 | 0 |

5 | 22 | 5.8 | 6.4 |

Method | Number of Snapshots | CPU Time for One Snapshot/h | CPU Time for Predicting one Test Condition/s |
---|---|---|---|

Prediction method | 5 | $\approx 12$ | 0.14 |

Numerical simulation | 60 | $\approx 12$ | $-$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, J.; Wang, M.; Li, S.
The Rapid Data-Driven Prediction Method of Coupled Fluid–Thermal–Structure for Hypersonic Vehicles. *Aerospace* **2021**, *8*, 265.
https://doi.org/10.3390/aerospace8090265

**AMA Style**

Liu J, Wang M, Li S.
The Rapid Data-Driven Prediction Method of Coupled Fluid–Thermal–Structure for Hypersonic Vehicles. *Aerospace*. 2021; 8(9):265.
https://doi.org/10.3390/aerospace8090265

**Chicago/Turabian Style**

Liu, Jing, Meng Wang, and Shu Li.
2021. "The Rapid Data-Driven Prediction Method of Coupled Fluid–Thermal–Structure for Hypersonic Vehicles" *Aerospace* 8, no. 9: 265.
https://doi.org/10.3390/aerospace8090265