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## Numerical accuracy estimation

Eventually, the influence of the mesh has to be evaluated. First, it is necessary to check the influence of the sections number per half-wing on the computation time. The lifting line program has been run for a wing with between 10 and 500 sections, 50 times each. Figure 2.10 (a) shows the averaged computation time for 50 runs on a classic computer. The purpose here is not to give a precise estimation of the computation time as the program is not properly optimized but more to give an idea of the evolution of the computation time as a function of the number of sections on the wing. The number of iterations can be seen in Figure 2.10 (b).

As can be seen in Figure 2.10, the computation time and the number of iterations start to increase significantly for more than 100 sections per half-wing. Under 100 sections, the influence of the iterations number is harder to evaluate as the other parts of the program, especially files writing, take comparatively more time. Furthermore, as shown in Figure 2.11, for 30 sections per half-wing the estimation of the aerodynamic coefficients has three significant figure which is considered enough, taking into account the intrinsic accuracy of the lifting line model. Indeed, for a large number of sections, the numerical accuracy is certainly superior to the accuracy of the lifting line model itself. That is why, for the use of the lifting line program, a discretization between 30 and 50 sections per half-wing is recommended.

### 2D results on the NACA2412 profile

For validation purpose, it has been decided to work with the NACA2412 profile instead of a LEI kite profile. Indeed the NACA2412 can be more accurately and easily meshed. Furthermore in following simulations, the profile was kept constant along the span of the wing.

2D RANSE simulations were carried out on the NACA2412 section for 14 angles of incidence within the range of [−8°; 16°]. Numerical results were compared with experimental ones obtained in a wind tunnel (Abbott & Doenhoff (1959)) at the same Reynolds number, Re = 3.1 106. The parameters of the parametric polar curves used for the lifting line method were chosen to fit the RANSE results at best. All these results are presented Figure 3.6. The agreement between experimental and numerical results is excellent for the lift coefficient and satisfactory for drag and moment coefficients. One explanation for the differences encountered for drag and moment is that simulations were conducted in a fully turbulent regime, whereas experiments were performed on a smooth section in a low residual turbulence wind tunnel.

#### Estimation of the numerical accuracy of the RANSE simulations

An attempt was done to estimate a numerical accuracy of 2D and 3D RANSE results.

It was assumed that results from 2D simulations were sufficient for that purpose and that they could be extrapolated to the 3D cases. In addition, it was assumed sufficient to examine a single angle of incidence of 2 ° for the root kite section, to be representative. Three elementary variations of the general numerical set up were considered. Only results in terms of lift and drag coefficients are presented in Figures 3.7 and 3.8. Curves obtained are similar for the coefficients of moment. First (Figures 3.7 and 3.8 (a)), the size of the computational domain was varied, parameter N varying from 1 to 27 keeping constant the base size, the absolute targeted size at the section, the near wall mesh parameters, and the growth rate (see Figure 3.1). Second (Figures 3.7 and 3.8 (b)), the targeted size at each boundary being defined relatively to base size, the base size ranged from 0.6cr to 0.02cr, leading to a number of cells from 6 103 to 120 103, keeping constant the near wall mesh parameters and the size of the computational domain (N = 10). Third (Figures 3.7 and 3.8(c)), the number of layers of the near wall mesh ranged from 4 to 32, leading to mean y+ values from 156 to 0.5, keeping constant all the other mesh parameters, that means the size of the computational domain N = 10, and the base size 0.1cr. In this last case, two other turbulence model were also tested (k-! SST and Spalart-Allmaras).

**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 . .