# Performance Enhancement by Wing Sweep for High-Speed Dynamic Soaring

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Flight Mechanics Modellings of High-Speed Dynamic Soaring

- (1)
- Windward climb where the wind shear layer is traversed upwards.
- (2)
- Upper curve in the region of high wind speed, with a flight direction change from windward to leeward.
- (3)
- Leeward descent where the wind shear layer is traversed downwards.
- (4)
- Lower curve in the no wind region, with a flight direction change from leeward to windward.

#### 2.1. 3-DOF Dynamics Model

#### 2.2. Energy Model

- (1)
- wind speed, ${V}_{w}$
- (2)
- maximum lift-to-drag ratio, ${({C}_{L}/{C}_{D})}_{max}$

#### 2.3. Straight Wing Reference Configuration (Aerodynamics, Size and Mass Properties)

## 3. Maximum Speed Achievable with Straight-Wing Configuration

#### 3.1. Maximum-Speed Performance of Straight Wing Reference Configuration

#### 3.2. Maximum-Speed Performance and Related Key Factors

## 4. Increase of Maximum-Speed Performance by Wing Sweep

#### 4.1. Aerodynamic Characteristics of Swept Wing Glider Configurations

- (1)
- Energy based model

- (2)
- Critical Mach number, $M{a}_{cr}$

- (3)
- Maximum lift-to-drag ratio, ${({C}_{L}/{C}_{D})}_{max}$

#### 4.2. Swept Wing Configurations

#### 4.3. Potential of Wing Sweep for Enhancing the Maximum-Speed Performance

- −
- Sweep angles up to around 15° yield only a very small improvement.
- −
- The area between the ${V}_{inert,max}$ curve for $\Lambda =15\xb0$ and the ${V}_{inert,max}$ curve for $\Lambda =30\xb0$ shows that there are efficient possibilities for enhancing the maximum-speed performance by wing sweep. The area between the ${V}_{inert,max}$ curve for $\Lambda =30\xb0$ and the ${V}_{inert,max}$ curve for $\Lambda =40\xb0$ can serve as an indication for the performance potential of larger sweep angles.

## 5. Further Effects of Wing Sweep Important for High-Speed Dynamic Soaring

- −
- cycle time
- −
- load factor
- −
- trajectory radius

#### 5.1. Optimal Cycle Time

#### 5.2. Load Factor

#### 5.3. Trajectory Radius of Maximum-Speed Cycle

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

a_{ij} | coefficients |

b | wing span |

C_{D} | drag coefficient |

C_{L} | lift coefficient |

D | drag |

E | energy |

g | acceleration due to gravity |

h | altitude |

J | performance criterion |

L | lift |

Ma | Mach number |

m | mass |

n | load factor |

R_{cyc} | loop radius |

S | wing reference area |

s | length |

t | time |

u, v, w_{i} | speed components |

u | control vector |

V_{a} | airspeed |

V_{inert} | inertial speed |

V_{w} | wind speed |

x | longitudinal coordinate |

x | state vector |

W | work |

y | lateral coordinate |

z | vertical coordinate |

A | aspect ratio |

χ | azimuth angle |

γ | flight path angle |

$\Lambda $ | sweep angle |

μ | bank angle |

ρ | air density |

## Appendix A. Formulation of Optimal Control Problem

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**Figure 2.**Speed vectors and inertial coordinate system. (The $x$ axis is chosen parallel to the wind speed vector ${V}_{w}$ and pointing in the opposite direction).

**Figure 3.**Oblique view on high-speed dynamic soaring loop. (The inclination of the trajectory relative to the horizontal is shown exaggerated for representation purposes).

**Figure 4.**Drag coefficient of straight wing configuration dependent on lift coefficient, ${C}_{L}$ and Mach number, $Ma$, [7].

**Figure 5.**Maximum lift-to-drag ratio, ${({C}_{L}/{C}_{D})}_{max}$ and associated lift coefficient, ${C}_{L}^{*}$, dependent on Mach number, $Ma$, (relating to drag polar in Figure 4.

**Figure 6.**Dynamic soaring loop optimized for maximum speed, ${V}_{inert,max}=271.8\text{}\mathrm{m}/\mathrm{s}$.

**Figure 7.**Side and top views of dynamic soaring loop optimized for maximum speed, ${V}_{inert,max}=271.8\text{}\mathrm{m}/\mathrm{s}$.

**Figure 8.**Time histories of speeds, ${V}_{inert}$ and ${V}_{a}$, and Mach number, $Ma$, of dynamic soaring loop optimized for maximum speed, ${V}_{inert,max}=271.8\text{}\mathrm{m}/\mathrm{s}$.

**Figure 10.**Lift-to-drag ratio in maximum-speed dynamic soaring.$\text{}{({C}_{L}/{C}_{D})}_{av}$: actual lift-to-drag ratio, 3-DOF dynamics model; ${({C}_{L}/{C}_{D})}_{max,av}$: maximum lift-to-drag ratio, 3-DOF dynamics model.

**Figure 11.**Maximum speed, ${V}_{inert,max}$, dependent on wind speed, ${V}_{w}$ and on maximum lift-to-drag ratio, ${({C}_{L}/{C}_{D})}_{max}$. Results obtained using the energy model, Equation (20). Results obtained using the 3-DOF dynamics model.

**Figure 15.**Effect of wing sweep on maximum speed, ${V}_{inert,max}$, for swept wing configuration 1.

**Figure 16.**Effect of wing sweep on maximum speed, ${V}_{inert,max}$, for swept wing configuration 2.

**Figure 17.**Effect of wing sweep on the cycle time for swept wing configuration 1 in maximum-speed dynamic soaring.

**Figure 18.**Effect of wing sweep on the load factor for swept wing configuration 1 in maximum-speed dynamic soaring.

**Figure 19.**Effect of wing sweep on the trajectory radius for swept wing configuration 1 in maximum-speed dynamic soaring.

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**MDPI and ACS Style**

Sachs, G.; Grüter, B.; Hong, H.
Performance Enhancement by Wing Sweep for High-Speed Dynamic Soaring. *Aerospace* **2021**, *8*, 229.
https://doi.org/10.3390/aerospace8080229

**AMA Style**

Sachs G, Grüter B, Hong H.
Performance Enhancement by Wing Sweep for High-Speed Dynamic Soaring. *Aerospace*. 2021; 8(8):229.
https://doi.org/10.3390/aerospace8080229

**Chicago/Turabian Style**

Sachs, Gottfried, Benedikt Grüter, and Haichao Hong.
2021. "Performance Enhancement by Wing Sweep for High-Speed Dynamic Soaring" *Aerospace* 8, no. 8: 229.
https://doi.org/10.3390/aerospace8080229