# Aerodynamic Modeling and Simulation of Multi-Lifting Surfaces Based on the Unsteady Vortex Lattice Method

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## Abstract

**:**

## 1. Introduction

- Based on the idea of object-oriented programming, the mesh generation and part assembling for an arbitrary number of lifting surfaces can be easily performed as every surface can be defined as an object instance of the class to be quickly integrated into the model.
- The physical influence between multiple lifting surfaces is considered in the modeling process. With different built-in vortex core models, the wake–boundary interaction is also included to expand this tool to more engineering applications.
- Boundary conditions are directly updated based on the dynamic mesh, which can be used to model complex motions.
- Inspired by the concept of free flight, this tool generates not only the aerodynamic state-space data based on the frozen wake assumption, but also the six-degree-of-freedom (6DOF) rigid-body dynamics model. Nonlinear time domain simulations can be directly conducted using this integrated model, which gives a higher-fidelity flight simulation with the consideration of unsteady effects.
- The technique of parallel programming is utilized to improve the calculation performance of the problems involving large wakes. This makes it possible to efficiently conduct the modeling and simulation on a personal computer.

## 2. UVLM Formulation

#### 2.1. Basic Concepts of Potential Theory

#### 2.2. Vortex Element

#### 2.3. Wake Transport

#### 2.4. Vortex Core Model

#### 2.5. Aerodynamic Force

#### 2.6. State-Space and Time-Stepping Forms

## 3. Modeling Process and Algorithm Implementation

#### 3.1. Geometry Discretization

#### 3.2. Boundary Condition Integration with Kinematic States

#### 3.3. Dynamic Coupling

#### 3.4. Algorithm and Data Structure

#### 3.5. Parallel Optimization

## 4. Examples

#### 4.1. Benchmark

#### 4.2. Multiple Surfaces

#### 4.3. Wake–Surface Interactions

#### 4.4. Free Flight with Rigid Body Dynamic Coupling

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 7.**The illustration of the dynamic coupling using the state-space model (

**left**) and the time-stepping model (

**right**).

**Figure 10.**Diagrams of lifting coefficient ${C}_{L}$ for pitch oscillations at different reduced frequencies.

Processes | Without Parallel Optimization | With Parallel Optimization |
---|---|---|

Generate influence matrices for vortex strength calculation | 5 min 42 s | 1 min 22 s |

Generate influence matrices for force calculation | 5 min 37 s | 0 min 53 s |

Wake roll-up | 47 min 29 s | 1 min 20 s |

Total running time | 59 min 09 s | 3 min 40 s |

Variables | Values | Annotation |
---|---|---|

geo.RefPos | [0, 0, 0] | Reference point coordinate of the lifting surface |

geo.flag_half | 2 | Flag variable, 1 for single wing, 2 for symmetrical wing |

geo.SectionLength | [0.55, 2.09, 2.65, 0.07, 0.08, 0.06] | Length of each wing segment (m) |

geo.SweepAngle | [0, 0, 3.9, 0, 35.2, 64.5] | Sweep angle of each wing segment (deg) |

geo.DihedralAngle | [0, 1.7, 1.7, 0, 0, 0] | Dihedral angle of each wing segment (deg) |

geo.local_aoa | [0, 0, 0, 0, 0, 0, 0] | Local angle of attack of each wing section (deg) |

geo.airfoil | {“n2412”, “n2412”, “n2412”, “n2412”, … “n2412”, “n2412”, “n2412”} | Airfoil name of each wing section |

setting.uvw | [60, 0, 0] | Translational velocities in body coordinate system (m/s) |

setting.pqr | [0, 0, 0] | Angular velocities in body coordinate system (deg/s) |

setting.rot0 | [0, 0, 0] | Initial rotation of the whole lifting surface (deg) |

setting.tran0 | [0.42, 0, −0.71] | Initial translation of the whole lifting surface (m) |

setting.t_vec | [0: 0.005: 5] | Simulation time series determined by total time and time step (s) |

mesh.nx | 8 | Number of meshes in chord direction |

mesh.ny | [8, 20, 24, 2, 2, 2] | Number of meshes in span direction |

mesh.nw | 33 | Number of wakes in chord direction |

**Table 3.**Comparison of aerodynamic results of the stabilizer regarding the influence from other lifting surfaces.

Cases | Total Lift (N) | Maximum Pressure (Pa) | Minimum Pressure (Pa) |
---|---|---|---|

Considering influences from other surfaces | 3040.28 | 1395.19 | −1718.02 |

Without considering influences from other surfaces | 3401.06 | 1038.91 | −1863.84 |

**Table 4.**Geometric parameters of the propeller model [44].

Airfoil | Rotor Radius (m) | Root Cutoff Radius (m) | Blade Chord (m) | Number of Blades | Collective Pitch (deg) |
---|---|---|---|---|---|

NACA0012 | 1.143 | 0.1905 | 0.1905 | 2 | 8 |

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**MDPI and ACS Style**

Gao, W.; Liu, Y.; Li, Q.; Lu, B.
Aerodynamic Modeling and Simulation of Multi-Lifting Surfaces Based on the Unsteady Vortex Lattice Method. *Aerospace* **2023**, *10*, 203.
https://doi.org/10.3390/aerospace10020203

**AMA Style**

Gao W, Liu Y, Li Q, Lu B.
Aerodynamic Modeling and Simulation of Multi-Lifting Surfaces Based on the Unsteady Vortex Lattice Method. *Aerospace*. 2023; 10(2):203.
https://doi.org/10.3390/aerospace10020203

**Chicago/Turabian Style**

Gao, Wei, Yishu Liu, Qifu Li, and Bei Lu.
2023. "Aerodynamic Modeling and Simulation of Multi-Lifting Surfaces Based on the Unsteady Vortex Lattice Method" *Aerospace* 10, no. 2: 203.
https://doi.org/10.3390/aerospace10020203