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## The lifting line method

The 3D non-linear lifting line is based on an extension of the Prandtl’s lifting line theory. This extension is intended to address cases of wings with variable dihedral and sweep angles. Leloup introduced this technique in Leloup et al. (2013) with a linear implementation while the present method is taking into account the non-linearity of the lift coefficient, generalizing the work of Anderson (Anderson (2011)) and Graf (Graf et al. (2014)) for straight wings. Assuming a rigid kite, the wing is supposed # » # » to fly in a given wind VRW , with a given velocity VK at the quarter chord point K in #» its symmetry plane, and with a given turn rate Ω. In this general case, the kite global # » # » # » # » apparent wind is Va = VRW − VK and its direction is denoted by xa. The finite wing and its wake are represented by a set of horseshoe vortices of different strengths = ( i)i=1…2n. The aim of the algorithm presented thereafter is to calculate the circulation i of each horseshoe vortex. Once these strengths are obtained, the local effective flow for each wing section allows local aerodynamic forces and torques calculation along the span of the wing. The numerical iterative solution is taken from Anderson (Anderson (2011)), but the calculation of the local effective angles of incidence is adapted to the cases of wings which are non-straight and non-planar. The horseshoe vortices used for discretization, and the calculation of their influences, are for their part derived from Katz and Plotkin (Katz & Plotkin (2001)).

The wing is divided in a finite number of plane sections normal to #»i, as defined 2n t in section 2.2.1, each one is represented by a horseshoe vortex. The horseshoe vortex number consists of six vortex segments. The bound vortex i #»i i is located at the i t l local quarter chord length, perpendicular to the plane of the considered section. Each of the two trailing vortices are separated into two parts: the first which extends parallel to the chord over one chord length, ± i #»i i, and the second which extends parallel xs c to the global relative free stream direction # » and over several chords length leading xa to length, ± i # » . The starting vortex − i #»i i is finally used to close the Lwake xaLwake t l horseshoe vortex. An example of the discretized model is presented in Figure 2.5 and the geometry of the section is described in Figure 2.6. It can be noticed that even with a swept wing, the bound vortex along the lifting line is orthogonal to the two adjacent trailing vortices. This leads to a piecewise constant discretization of the lifting line as it can be seen in Figure 2.5, but a correct match is needed between the local lift calculated from the Kutta formula and from the polar of the section, as theoretically required by such models.

### 3D Non Linear Lifting Line model

This part is devoted to the presentation of the 3D non-linear lifting line algorithm. Section 2.2.1 is committed to the presentation of the lifting line geometry while Section 2.2.2 presents the way chosen to solve the circulation distribution. Section 2.2.3 details the computation of the aerodynamic coefficients of the wing.

#### Parametric definition of the lifting line

The kite geometry is defined in the Cartesian reference frame (K, xb , yb , zb) (see Figure #» #» #» , zb), 2.3). A generatrix line is given as a parametric planar curve M(s) in the plane (K, yb #» #» M denoting the current point and s the curvilinear abscissa. This curve is elliptical and symmetrical with respect to the axis #» . At the current point , the section of the (K, zb) M(s) kite is defined in the normal plane with respect to the tangent vector to the generatrix line #» # » #» #» #» #» #» t(s) = dM /ds. This plane is denoted by (M(s), xb , n(s)), where n(s) = xb × t(s) is the normal vector to the generatrix line in the #» #» plane. The local chord (K, yb, zb) direction of the section #» is obtained from the rotation of the vector #» of an angle xs(s) xb αv(s) around #» t(s), where αv(s) is a given twist law. Since M(s) is supposed to be the quarter chord point of the current section, the leading edge A(s) and the trailing edge F(s) are then located along the chord axis (M(s), xs(s)) according to a given chord law» c(s), following AM(s) = 0.25c(s)xs (s) and M F(s) = 0.75c(s)xs(s). This finally leads to the local chord reference frame of the kite section (A(s), xs (s), zs (s)) with zs (s) = xs (s) × t(s), where the points of the extrados and intrados can be placed; for example from a given non dimensional section definition, scaled with the chord law c(s). In the case of a kite wing with sweep angle, the local section points are finally translated by the vector #», with a given sweep law. Appendix 7.4.3 details the different twist, sweep and chord laws implemented in the house made LP3DNL program as well as the various geometrical parameters used for the construction of the geometry.

Lastly each section must be coupled to its aerodynamic properties. For the model, these aerodynamic specs consist of three 2D polar curves, lift, additional drag and moment, with respect to the angle of incidence. These 2D polar curves are described by a combination of polynomials of various degrees, which have been chosen in order to make easier the copy of polar curves obtained via experimental measures or CFD simulations. Figure 2.4 shows an example of the division of the lift polar curve and the different parameters needed for the lifting line algorithm. The details on the definition of polar curves are given in Appendix 7.4.3.

**Table of contents :**

**1 Introduction **

**2 Fluid model presentation **

2.1 Introduction to the Lifting Line problem

2.2 3D Non Linear Lifting Line model

2.2.1 Parametric definition of the lifting line

2.2.2 The lifting line method

2.2.3 Post-processing

2.3 Numerical tests

2.3.1 Initial conditions and computation parameters

2.3.2 Numerical accuracy estimation

2.3.3 Verification again an analytical case

**3 Fluid model comparison against RANSE simulations **

3.1 Numerical settings

3.2 2D results on the NACA2412 profile

3.3 Estimation of the numerical accuracy of the RANSE simulations

3.4 3D numerical results

3.4.1 Un-twisted and un-swept semi-circular wing

3.4.2 Un-twisted and linearly swept semi-circular wing

3.4.3 Linearly twisted and non-linearly swept wing

3.5 Conclusion

**4 Fluid model application **

4.1 Introduction

4.2 Kite in circular flight

4.3 Results

4.4 Discussion

**5 Structure models presentation **

5.1 Introduction

5.1.1 Inflatable beam modeling

5.1.2 Modeling of the canopy

5.2 Complete Finite Element Model

5.3 Kite as a Beam

5.3.1 The elementary cell concept

5.3.2 The equivalent beam properties

5.3.3 Kite construction

5.3.4 Numerical tests for the elementary cell

**6 Structure models comparison **

6.1 Description of the comparison cases

6.1.1 Kite under pressure

6.1.2 The four elementary comparison cases

6.2 Comparison of the two models

6.2.1 Comparison for a LEI kite geometry

6.2.2 Comparison with a cylindrical geometry

6.3 Discussion

**7 Fluid-structure interaction **

7.1 Introduction

7.2 Fluid-structure coupling

7.2.1 Bridles and tether implementation

7.2.2 Data exchange

7.3 Comparison between IDK and KaB models

7.3.1 Global results

7.3.2 Local results

7.4 Discussion

7.4.1 Mesh influence on the KaB model

7.4.2 Influence of the work pressure on the KaB model

7.4.3 Possibilities of model calibration