# Fast Aero-Structural Model of a Leading-Edge Inflatable Kite

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## Abstract

**:**

## 1. Introduction

## 2. Computational Approach

#### 2.1. Structural Model

#### 2.2. Aerodynamic Model

- Generate the wing geometry, along with the definition of the vortex filaments, control points, and the relevant vectors for each section.
- Start with an initial guess for the bound circulation ($\Gamma $) to initiate the iterative process to find a solution.
- Calculate the relative velocity (${\mathbf{U}}_{rel}$) at each control point (${P}_{j}$), i.e., the velocity seen by the airfoil, which is found by relating the inner 2D region (airfoil) to the outer 3D region (vorticity system) as follows:$$\left.\begin{array}{c}2\mathrm{D}:{\mathbf{U}}_{rel}+{\mathbf{U}}_{ind,2D}={\mathbf{U}}_{3/4c}\hfill \\ 3\mathrm{D}:{\mathbf{U}}_{\infty}+{\mathbf{U}}_{ind,3D}={\mathbf{U}}_{3/4c}\hfill \end{array}\right\}\Rightarrow {\mathbf{U}}_{rel}={\mathbf{U}}_{\infty}+{\mathbf{U}}_{ind,3D}-{\mathbf{U}}_{ind,2D},$$Induced velocities are calculated with the previous circulation distribution by using the Biot–Savart law, which relates the strength of a vortex filament to the magnitude and direction of the flow field that it induces. ${\mathbf{U}}_{ind,3D}$ is calculated as the sum of the velocities induced by all the sections, and ${\mathbf{U}}_{ind,2D}$ only takes into account each section’s own circulation.
- Calculate the effective angles of attack (${\alpha}_{eff}$) with the direction of the relative velocities with respect to the airfoil and interpolate the lift coefficients (${C}_{l}$) from the 2D airfoil polars.
- Recalculate the bound circulation at each section with the obtained lift coefficients using the Kutta–Joukowski law, which relates the lift force (L) with the bound circulation, formulated as$$\rho \left|{\mathbf{U}}_{\infty}\times \Gamma \right|=\frac{1}{2}\rho {\left|{\mathbf{U}}_{rel}\times {\widehat{\mathbf{z}}}_{airf}\right|}^{2}c{C}_{l}\left({\alpha}_{eff}\right),$$
- Compare the updated circulation to the last iteration and check if it falls below the convergence criteria. If it does not, go back to step 3. For the next iteration, the circulation is calculated using an under-relaxation factor to stabilize the solution.
- Recalculate the local angles of attack at the quarter-chord position using the converged circulation distribution.
- Convert the local forces into the freestream velocity direction and derive the global lift and drag coefficients by integrating the forces along the span.

#### 2.3. Aero-Structural Coupling

#### 2.4. Photogrammetry

## 3. Results

#### 3.1. Kite Deformation

#### 3.1.1. CAD Geometry versus Powered Wing Shape

#### 3.1.2. Powered versus Depowered Wing Shape

#### 3.1.3. Wing Shape for Powered Straight Flight versus Turning Maneuvers

#### 3.2. Kite Aerodynamics

#### 3.2.1. Computed Aerodynamic Properties

#### 3.2.2. Comparison with Experimental Results

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

2D | Two-dimensional |

3D | Three-dimensional |

AWE | Airborne wind energy |

CAD | Computer-aided design |

CFD | Computational fluid dynamics |

FE | Finite element |

FEM | Finite element method |

FSI | Fluid-structure interaction |

KCU | Kite control unit |

LE | Leading edge |

LEI | Leading-edge inflatable |

LLM | Lifting line method |

PSM | Particle system model |

TE | Trailing edge |

RANS | Reynolds-averaged Navier–Stokes |

VSM | Vortex step method |

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**Figure 1.**Front view (

**left**) and side view (

**right**) of the LEI V3 kite with geometric parameters, mass distribution, and definition of the reference chord ${c}_{\mathrm{ref}}$. The total wing surface area is denoted as S, and the projected value is denoted as A. The mass of the bridle lines is part of the wing mass. The red bridle lines denote the steering lines and the black ones the power lines. The explicit dimensions describe the unloaded design shape of the wing. Adapted from [5].

**Figure 2.**Two photos of the LEI V3 kite in flight: the red quadrilateral indicates the structural model of one of the wing segments, and the red circles indicate Y-splits of the bridle close to the tubes to distribute the transmitted load [25].

**Figure 3.**Particle-system representation of the LEI V3 kite [26].

**Figure 5.**Aerodynamic model of the LEI V3 kite, illustrated for a coarse spanwise discretization of the wing [26].

**Figure 7.**Structural and aerodynamic meshes with largely different spanwise resolutions [26]. While not illustrated in this schematic, the quadrilateral defined by the structural mesh nodes is not necessarily planar but skewed in the general case.

**Figure 8.**In black, the wing shape in a fully powered state (${u}_{p}=1$), and in red, the shape of the CAD model, displayed in an orthographic view (

**top left**), a top view (

**top right**), a side view (

**bottom left**), and a front view (

**bottom right**) [26].

**Figure 9.**In black, the wing shape is in a fully powered state (${u}_{p}=1$), and in red, the wing shape is in a fully depowered state (${u}_{p}=0$), for $\Delta {l}_{d}=8\%$, displayed in an orthographic view (

**top left**), a top view (

**top right**), a side view (

**bottom left**), and a front view (

**bottom right**) [26].

**Figure 10.**Evolution of the LE widths (

**a**) and TE widths (

**b**) as a function of the power setting ${u}_{p}$.

**Figure 11.**Frontview of the depowered kite (${u}_{p}=0$) indicating the tensioning state of the bridle-line system [26].

**Figure 12.**In black, the wing shape for no steering input (${u}_{s}=0$), and in red, the wing shape for the maximum steering input (${u}_{s}=0.4$), displayed in an orthographic view (

**top left**), a top view (

**top right**), a side view (

**bottom left**) and a front view (

**bottom right**). Adapted from [26].

**Figure 13.**Front view of the kite for maximum steering input (${u}_{s}=0.4$), indicating the tensioning state of the bridle-line system [26].

**Figure 14.**Lift coefficient (

**a**), drag coefficient (

**b**), and lift-to-drag ratio (

**c**) as functions of the angle of attack for different kite geometries.

**Table 1.**Steering and relative flow inputs representing the different flight maneuvers of a pumping cycle.

Angle of Attack $\mathit{\alpha}$ (°) | Angle of Sideslip $\mathit{\beta}$ (°) | Power Setting ${\mathit{u}}_{\mathit{p}}$ | Steering Setting ${\mathit{u}}_{\mathit{s}}$ | |
---|---|---|---|---|

Straight-powered | 4–14 | 2–8 | 0.7–1 | 0 |

Depowered | 0–8 | 0–4 | 0–0.6 | 0 |

Turn-powered | 9–15 | 2–10 | 1 | 0.1–0.4 |

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## Share and Cite

**MDPI and ACS Style**

Cayon, O.; Gaunaa, M.; Schmehl, R. Fast Aero-Structural Model of a Leading-Edge Inflatable Kite. *Energies* **2023**, *16*, 3061.
https://doi.org/10.3390/en16073061

**AMA Style**

Cayon O, Gaunaa M, Schmehl R. Fast Aero-Structural Model of a Leading-Edge Inflatable Kite. *Energies*. 2023; 16(7):3061.
https://doi.org/10.3390/en16073061

**Chicago/Turabian Style**

Cayon, Oriol, Mac Gaunaa, and Roland Schmehl. 2023. "Fast Aero-Structural Model of a Leading-Edge Inflatable Kite" *Energies* 16, no. 7: 3061.
https://doi.org/10.3390/en16073061