List of Figures XI

Contents

Content III

List of Figures XI

List of Tables XIII

1 Static and Dynamic Aeroelasticity 1

1.1 Introduction to Aeroelasticity . . . . 1

1.2 Static Aeroelasticity . . . . 3

1.2.1 Divergence . . . . 3

1.2.2 Eect of Engine position on divergence . . . . 5

1.2.3 Aileron reversal . . . . 6

1.3 Dynamic Aeroelasticity: Flutter instability . . . . 8

1.3.1 Typical section model: equations of motion . . . . 9

1.3.2 Aeroelastic analysis of a Typical Section . . . 11

1.3.3 Fundamentals of Classical Flutter Analysis . . . 14

1.3.4 Flutter for a Typical section model: k and p-k method . . . 16

1.3.5 Flutter instability of real wings . . . 26

2 Swept and Curved Wing Models 27 2.1 Introduction . . . 27

2.2 CAD Models of the swept and curved wing . . . 27

2.3 Modied Geometry for FEM Model . . . 33

2.4 CAD models for CFD . . . 36

3 CFD Analyses: Lift-Curve and Rigid Polar 40 3.1 Introduction . . . 40

3.2 CFD Model: Blocking Technique . . . 40

3.3 CFD Analysis and Results . . . 49

3.4 Drag Reduction and Fuel Saving . . . 64

II

CONTENTS III

4 Structural Models and Modal Analysis 68

4.1 Introduction . . . 68

4.2 Structural Models of the Wings . . . 68

4.2.1 Engine and Nacelle . . . 69

4.2.2 Stringer and Spar Flanges . . . 70

4.2.3 Spar Web Thickness . . . 73

4.2.4 Skin Thickness Distribution . . . 73

4.2.5 Rib Thickness Distribution . . . 77

4.2.6 Fictitious Mass Moment of Inertia . . . 78

4.2.7 Material Properties . . . 79

4.2.8 Structural Model Mesh . . . 79

4.3 Modal Analysis . . . 82

4.4 Structural damping . . . 100

5 FSI Analysis: Swept Wing and Curved Wing 103 5.1 Introduction . . . 103

5.2 2-Way FSI Analysis in ANSYS Workbench . . . 103

5.3 ANSYS Mechanical Setup for FSI . . . 105

5.3.1 Transient Analysis: General Settings . . . 105

5.3.2 Transient Analysis: Boundary Conditions . . . 106

5.3.3 Transient Analysis: Solution Monitoring . . . 106

5.4 CFD Models for FSI . . . 107

5.5 ANSYS Fluent Setup for FSI . . . 119

5.6 ANSYS System Coupling settings . . . 121

5.7 Results of FSI Analyses . . . 124

5.7.1 FSI Results: Case 1 - h=0 m and h=10000 m . . . 124

5.7.2 FSI Results: Case 2 - h=10000 m . . . 151

6 Conclusions 218

Bibliography 221

List of Figures

1.1 The aeroelastic triangle of forces . . . . 1

1.2 Wing model mounted to elastic torsional support . . . . 3

1.3 Wing model with engine . . . . 5

1.4 Airfoil section of a apped two-dimensional wing . . . . 7

1.5 Airfoil section of a apped two-dimensional wing . . . . 9

1.6 Plot of modal frequencies vs V for a = −1/5, e = −1/10, µ = 20, r ² = 6/25 and σ = 2/5 . . . 13

1.7 Plot of modal damping vs V for a = −1/5, e = −1/10, µ = 20, r ² = 6/25 and σ = 2/5 . . . 13

1.8 Plot of the real and imaginary part of C(k) vs 1/k . . . 18

1.9 Plot of estimated value of Ω 1,2 /ω θ and g vs V = U/(bω) using k- method and Theodorsen aerodynamics. Parameters: a = −1/5, e = −1/10 , µ = 20, r ² = 6/25 and σ = 2/5. . . 21

1.10 Plot of extimated value of Ω 1,2 /ω _θ vs V = U/(bω) using p-k method and Theodorsen aerodynamics. Parameters: a = −1/5, e = −1/10, µ = 20 , r ² = 6/25 and σ = 2/5. . . 24

1.11 Plot of estimated value of Γ 1,2 /ω _θ vs V = U/(bω) using p-k method and Theodorsen aerodynamics. Parameters: a = −1/5, e = −1/10, µ = 20 , r ² = 6/25 and σ = 2/5. . . 24

1.12 Plot of extimated value of Ω 1,2 /ω _θ and Γ 1,2 /ω _θ vs V = U/(bω). Comparison between p-k method (Theodorsen aerodynamics) and p-method with steady aerodynamic theory. Parameters: a = −1/5, e = −1/10 , µ = 20, r ² = 6/25 and σ = 2/5. . . 25

2.1 Planform shape of the swept wing and the curved wing . . . 28

2.2 Supercritical airfoil SC(2)-0410 . . . 29

2.3 Wing reference geometry in CATIA V5R20 . . . 30

2.4 Swept wing CAD model . . . 31

2.5 Curved wing CAD model . . . 32

2.6 Stringers and Spars - Top View . . . 33

2.7 Swept Wing - CAD model for FEM (CATIA V5R20) . . . 34

IV

LIST OF FIGURES V

2.8 Curved Wing - CAD model for FEM (CATIA V5R20) . . . 35

2.9 Base geometry in the Center line plane . . . 36

2.10 Base Geometries in the four reference planes - Curved wing . . . 37

2.11 Swept Wing - CAD model for CFD (CATIA V5R20) . . . 38

2.12 Curved Wing - CAD model for CFD (CATIA V5R20) . . . 39

3.1 Base Geometries imported in ANSYS Gerometry - Curved wing . . 41

3.2 Curved wing surface and some control volumes in ICEM CFD . . . 41

3.3 Curved wing: control volume block meshing ICEM CFD . . . 42

3.4 Blocking strategy for the curved wing in ICEM CFD . . . 43

3.5 Blocking near the Curved Wing . . . 44

3.6 External surfaces of the uid domain (ICEM CFD) . . . 45

3.7 Named surfaces of the curved wing (ICEM CFD) . . . 45

3.8 Swept Wing CFD Model . . . 46

3.9 Curved Wing CFD Model . . . 47

3.10 Wing Surface Meshes . . . 48

3.11 Convergence for C L : Curved Wing and Swept Wing . . . 50

3.12 Convergence for C D : Curved Wing and Swept Wing . . . 51

3.13 Curved vs Swept Wing: C L vs α - Mesh: 6.57M nodes . . . 52

3.14 Curved vs Swept Wing: C D vs α - Mesh: 6.57M nodes . . . 52

3.15 Curved vs Swept Wing: Drag Polar - Mesh: 6.57M nodes . . . 53

3.16 Curved vs Swept Wing: Eciency vs α - Mesh: 6.57M nodes . . . 53

3.17 Curved vs Swept Wing: Eciency vs C L - Mesh: 6.57M nodes . . . 54

3.18 Pressure coecient on upper skins. M = 0.85; C L = 0.4 . . . 55

3.19 Pressure coecient on lower skins. M = 0.85; C L = 0.4 . . . 56

3.20 Pressure coecient on cross section planes. M = 0.85; C L = 0.4 . . 57

3.21 Pressure coecient on root prole and y = 12 m. M = 0.85; C L = 0.4 58 3.22 Pressure coecient on proles at y = 18 m, and y = 24 m. M = 0.85 ; C L = 0.4 . . . 59

3.23 Pressure coecient on proles at y = 29.995 m. M = 0.85; C L = 0.4 60 3.24 Swept Wing & Curved Wing: Supersonic region. M = 0.85; C L = 0.4 61 3.25 Mach number on cross section planes. M = 0.85; C L = 0.4 . . . 62

3.26 Swept Wing & Curved Wing: Root plane Mach number and Tur- bulent Kinetic Energy on planes x = −29 m; −39 m; −49 m . M = 0.85 ; C L = 0.4 . . . 63

3.27 Curved vs Swept Wing: C D vs C L - h = 10000 m - M = 0.85 . . . . 65

3.28 Swept Wing & Curved Wing Conguration: X = 3500 nm (6483 km).

W (X) , C L (X) , C D (X) and Drag Polar - Steady Level Flight: h =

10000 m−M = 0.85 - W 0 = 210000 kg - S ref = 379 m ² - c = 0.545 h ⁻¹ 67

4.1 Line bodies for Nacelle and Engine . . . 69

LIST OF FIGURES VI

4.2 Wing-box sections and line bodies nomenclature . . . 70

4.3 Cross-section dimensions . . . 71

4.4 Spar anges and stringers, upper side - Curved Wing . . . 72

4.5 Spar thickness . . . 73

4.6 Thickness distribution outside the wing-box . . . 74

4.7 Thickness distribution inside the wing-box . . . 75

4.8 Thickness distribution inside the wing-box, Section 5 . . . 75

4.9 Second layer thickness distribution for Fuel . . . 76

4.10 Fuel Mass distribution. Total mass: 20000kg . . . 76

4.11 Rib numbering . . . 77

4.12 Moment of Inertia distribution . . . 78

4.13 Swept Wing Structural Model . . . 80

4.14 Curved Wing Structural Model . . . 81

4.15 Swept Wing - Case 0: modes 1, 2 and 3 . . . 85

4.16 Swept Wing - Case 0: modes 4, 5 and 6 . . . 86

4.17 Swept Wing - Case 0: modes 7, 8 and 9 . . . 87

4.18 Curved Wing - Case 0: modes 1, 2 and 3 . . . 88

4.19 Curved Wing - Case 0: modes 4, 5 and 6 . . . 89

4.20 Curved Wing - Case 0: modes 7, 8 and 9 . . . 90

4.21 Swept Wing - Case 1: modes 1, 2 and 3 . . . 91

4.22 Swept Wing - Case 1: modes 4, 5 and 6 . . . 92

4.23 Curved Wing - Case 1: modes 1, 2 and 3 . . . 93

4.24 Curved Wing - Case 1: modes 4, 5 and 6 . . . 94

4.25 Swept Wing - Case 2: modes 1, 2 and 3 . . . 95

4.26 Swept Wing - Case 2: modes 4, 5 and 6 . . . 96

4.27 Curved Wing - Case 2: modes 1, 2 and 3 . . . 97

4.28 Curved Wing - Case 2: modes 4, 5 and 6 . . . 98

4.29 Damping factor ζ vs ω n (α = 0.431, β = 1.82 · 10 ⁻³ ) . . . 102

5.1 2-Way FSI Flow Chart . . . 104

5.2 FEM Boundary Conditions . . . 105

5.3 FEM Boundary Conditions . . . 106

5.4 Control Points for FSI Solution Monitoring . . . 107

5.5 CFD Mesh Sensitivity - Swept Wing C L (N ) , M = 0.85, h = 10000 m108 5.6 CFD Mesh Sensitivity - Curved Wing C L (N ) , M = 0.85, h = 10000 m108 5.7 CFD Mesh Sensitivity - Swept Wing C D (N ) , M = 0.85, h = 10000 m109 5.8 CFD Mesh Sensitivity - Curved Wing C D (N ) , M = 0.85, h = 10000 m109 5.9 Mesh Sensitivity - Lift and Drag Curves - M = 0.85, h = 10000 m . 110 5.10 Mesh Sensitivity - Drag Polar - M = 0.85, h = 10000 m . . . 111

5.11 Curved Wing vs Swept Wing - Drag Polar - 400k and 6.5M meshes.

M = 0.85 - h = 10000 m . . . 113

LIST OF FIGURES VII 5.12 Swept Wing & Curved Wing - Best-t of Lift and Drag Curves.

400k mesh - M = 0.85, h = 10000 m . . . 114

5.13 CFD Meshes for FSI . . . 116

5.14 CFD Mesh for FSI - Swept Wing . . . 117

5.15 CFD Mesh for FSI - Curved Wing . . . 118

5.16 FSI Project Schematic . . . 121

5.17 Curved Wing - Case 1 - M = 0.45 - h = 0 m - stable solution . . . . 124

5.18 Results of FSI analyses - Case 1 h=0 m . . . 125

5.19 Reference Proles and Fz vs Vertical Displacement - Case 1 h=0 m 126 5.20 Nodal Force and Lift Coecient - Case 1, h=0 m - Unstable Solutions128 5.21 F z (t) - Case 1, h=0 m - Unstable Solutions . . . 129

5.22 C L (t) - Case 1, h=0 m - Unstable Solutions . . . 130

5.23 C D (t) - Case 1, h=0 m - Unstable Solutions . . . 131

5.24 Fluid Power - Case 1, h=0 m - Unstable Solutions . . . 132

5.25 Mechanical Energies - Case 1, h=0 m - Unstable Solutions . . . 133

5.26 Swept Wing Tip Section - Last Cycle Frames Case 1, h=0 m, M=0.45 - Unstable Solution: Flutter . . . 134

5.27 Curved Wing Tip Section - Last Cycle Frames Case 1, h=0 m, M=0.5 - Unstable Solution: Flutter . . . 135

5.28 Swept Wing - Case 1 - h=0 m - M=0.45 - Unstable Solution. Lead- ing Edge/Trailing Edge Mean Vertical Displacement, W mean , and Vertical Displacement Dierence, ∆W , at the Tip Prole. . . 136

5.29 Curved Wing - Case 1 - h=0 m - M=0.50 - Unstable Solution. Leading Edge/Trailing Edge Mean Vertical Displacement, W mean , and Vertical Displacement Dierence, ∆W , at the Tip Prole. . . . 137

5.30 Results of FSI analyses - Case 1 h=10000 m . . . 139

5.31 Nodal Force and Lift Coecient - Case 1, h=10000 m Unstable Solutions . . . 140

5.32 F z (t) - Case 1, h=10000 m - Unstable Solutions . . . 141

5.33 C L (t) - Case 1, h=10000 m - Unstable Solutions . . . 142

5.34 C D (t) - Case 1, h=10000 m - Unstable Solutions . . . 143

5.35 Fluid Power - Case 1, h=10000 m - Unstable Solutions . . . 144

5.36 Mechanical Energies - Case 1, h=10000 m - Unstable Solutions . . . 145

5.37 Swept Wing Tip Section - Last Cycle Frames Case 1, h=10000 m, M=0.80 - Unstable Solution: Flutter . . . 146

5.38 Curved Wing Tip Section - Last Cycle Frames Case 1, h=10000 m, M=0.9 - Unstable Solution: Flutter . . . 147

5.39 Swept Wing - Case 1 - h=10000 m - M=0.80 - Unstable Solution.

Leading Edge/Trailing Edge Mean Vertical Displacement, W mean ,

and Vertical Displacement Dierence, ∆W , at the Tip Prole. . . . 148

LIST OF FIGURES VIII 5.40 Curved Wing - Case 1 - h=10000 m - M=0.90 - Unstable Solution.

Leading Edge/Trailing Edge Mean Vertical Displacement, W mean ,

and Vertical Displacement Dierence, ∆W , at the Tip Prole. . . . 149

5.41 FSI analyses for Case 1: Damping ratio vs Mach (the damping data are represented with the opposite sign) . . . 150

5.42 Supersonic Zone around the Swept Wing and the Curved Wing Front View . . . 152

5.43 Supersonic Zone around the Swept Wing and the Curved Wing Back View . . . 153

5.44 Results of FSI analyses of the swept wing (Case 2, cruise altitude) . 154 5.45 Results of FSI analyses of the swept wing Case 2 - cruise altitude - M=0.9 - α = 0.977 ^◦ . . . 154

5.46 Results of FSI analyses - Case 2 - h=10000 m . . . 155

5.47 Nodal Force and Lift Coecient - Case 2, h=10000 m Unstable Solutions . . . 157

5.48 Curved Wing - Case 2 - h=10000 m - M=0.948 - Unstable solution Lift Coecient Cycles - zoomed Views . . . 158

5.49 F z (t) - Case 2, h=10000 m - Unstable Solutions . . . 159

5.50 C L (t) - Case 2, h=10000 m - Unstable Solutions . . . 160

5.51 C D (t) - Case 2, h=10000 m - Unstable Solutions . . . 161

5.52 Fluid Power - Case 2, h=10000 m - Unstable Solutions . . . 162

5.53 Mechanical Energies - Case 2, h=10000 m - Unstable Solutions . . . 163

5.54 Swept Wing Tip Section - Last Cycle Frames Case 2, h=10000 m, M=0.90 - α = 0.977 ^◦ - Unstable Solution: Flutter-Buet . . . 164

5.55 Curved Wing Tip Section - Last Cycle Frames Case 2, h=10000 m, M=0.948 - α = 1.15 ^◦ - Unstable Solution: Flutter . . . 165

5.56 Swept Wing - Case 2 - h=10000 m - M=0.80 - Unstable Solution. Leading Edge/Trailing Edge Mean Vertical Displacement, W mean , and Vertical Displacement Dierence, ∆W , at the Tip Prole. . . . 166

5.57 Curved Wing - Case 2 - h=10000 m - M=0.948 - Unstable Solution. Leading Edge/Trailing Edge Mean Vertical Displacement, W mean , and Vertical Displacement Dierence, ∆W , at the Tip Prole. . . . 167

5.58 FSI analyses for Case 1 and Case 2 (the damping data are repre- sented with the opposite sign) . . . 169

5.59 Swept Wing - Case 2 - h=10000 m - M=0.9 - α = 0.977 ^◦ Supersonic region, prole at y % = 93% and Tip LE Vertical Displacement . . . 170

5.60 Swept Wing - Case 2 - h=10000 m - M=0.9 - α = 0.977 ^◦ Pressure coecient C p (t) on prole at y % = 93% . . . 171

5.61 Swept Wing - Case 2 - h=10000 m - M=0.9 - α = 0.977 ^◦ Supersonic

Region and ∆p(t) on cross section at y % = 93% . . . 172

LIST OF FIGURES IX 5.62 Curved Wing - Case 2 - h=10000 m - M=0.952 - α = 1.15 ^◦ Super-

sonic region, prole at y % = 94% and Tip LE Vertical Displacement 173 5.63 Curved Wing - Case 2 - h=10000 m - M=0.952 - α = 1.15 ^◦ Pressure

coecient C p (t) on prole at y % = 94% . . . 174 5.64 Curved Wing - Case 2 - h=10000 m - M=0.952 - α = 1.15 ^◦ Super-

sonic Region and ∆p(t) on cross section at y % = 94% . . . 175 5.65 Swept Wing - FSI - Case 2 - PSD of Tip LE Vertical Displacement 178 5.66 Curved Wing - FSI - Case 2 - PSD of Tip LE Vertical Displacement 179 5.67 Swept Wing - Case 2 - h=10000 m: Time History of C L (t) . . . 180 5.68 Swept Wing - FSI - Case 2 - h=10000 m: PSD of C L (t) . . . 181 5.69 Curved Wing - Case 2 - h=10000 m: Time History of C L (t) . . . . 182 5.70 Curved Wing - FSI - Case 2 - h=10000 m: PSD of C L (t) . . . 183 5.71 Swept Wing - Case 2 - h=10000 m: Time History of C D (t) . . . 184 5.72 Swept Wing - FSI - Case 2 - h=10000 m: PSD of C D (t) . . . 185 5.73 Curved Wing - Case 2 - h=10000 m: Time History of C D (t) . . . . 186 5.74 Curved Wing - FSI - Case 2 - h=10000 m: PSD of C D (t) . . . 187 5.75 Swept Wing & Curved Wing - FSI - Case 2 - h=10000 m. RMS of

Tip LE Vertical Displacement . . . 188 5.76 Swept Wing & Curved Wing - FSI - Case 2 - h=10000 m. RMS of

C _L and C D . . . 189 5.77 Swept Wing - Vertical Displacement of Deformed Shape from FSI:

Case 2 - M=0.9 - α = 0.977 ^◦ - ¯t=14.30s . . . 190 5.78 Swept Wing - Monitor surface for F z . . . 191 5.79 Swept Wing - Time History of F z Monitor - α = 0 ^◦ , 0.5 ^◦ , 0.75 ^◦ . CFD

analysis on deformed shape from FSI: Case 2 - M=0.9 - α = 0.977 ^◦ - ¯t=14.30s . . . 192 5.80 Swept Wing - Time History of F z Monitor - α = 0.977 ^◦ , 1.025 ^◦ ,

1.1 ^◦ . CFD analysis on deformed shape from FSI: Case 2 - M=0.9 - α = 0.977 ^◦ - ¯t=14.30s . . . 193 5.81 Swept Wing - PSD of F z Monitor CFD analysis on deformed shape

of FSI - Case 2 - M=0.9 - α = 0.977 ^◦ - ¯t=14.30s . . . 194 5.82 Swept Wing - Time History of C L - α = 0 ^◦ , 0.5 ^◦ , 0.75 ^◦ . CFD

analysis on deformed shape from FSI: Case 2 - M=0.9 - α = 0.977 ^◦ - ¯t=14.30s . . . 195 5.83 Swept Wing - Time History of C L - α = 0.977 ^◦ , 1.025 ^◦ , 1.1 ^◦ . CFD

analysis on deformed shape from FSI: Case 2 - M=0.9 - α = 0.977 ^◦ - ¯t=14.30s . . . 196 5.84 Swept Wing - PSD of C L CFD analysis on deformed shape of FSI -

Case 2 - M=0.9 - α = 0.977 ^◦ - ¯t=14.30s . . . 197

LIST OF FIGURES X 5.85 Swept Wing - Time History of C D - α = 0 ^◦ , 0.5 ^◦ , 0.75 ^◦ . CFD

analysis on deformed shape from FSI: Case 2 - M=0.9 - α = 0.977 ^◦ - ¯t=14.30s . . . 198 5.86 Swept Wing - Time History of C D - α = 0.977 ^◦ , 1.025 ^◦ , 1.1 ^◦ . CFD

analysis on deformed shape from FSI: Case 2 - M=0.9 - α = 0.977 ^◦ - ¯t=14.30s . . . 199 5.87 Swept Wing - PSD of C D CFD analysis on deformed shape of FSI

- Case 2 - M=0.9 - α = 0.977 ^◦ - ¯t=14.30s . . . 200 5.88 Swept Wing - RMS of F z , C L and C D vs α - CFD Analyses on

deformed shape from FSI - Case 2 - M=0.9 - α = 0.977 ^◦ - ¯t=14.30s 201 5.89 Swept Wing - mean C L vs α - mean C D vs α. CFD on Fixed

Deformed Shape from FSI: Case 2 - h = 10000 m - M = 0.9 - α = 0.977 ^◦ - t = 14.30 s . . . 202 5.90 Swept Wing - Drag Polar - mean C L vs mean C D . CFD on Fixed

Deformed Shape from FSI: Case 2 - h = 10000 m - M = 0.9 - α = 0.977 ^◦ - t = 14.30 s . . . 203 5.91 Curved Wing - Vertical Displacement of Deformed Shape from FSI:

Case 2 - M=0.948 - α = 1.148 ^◦ - ¯t=9.04s . . . 204 5.92 Curved Wing - Vertical Displacement of Deformed Shape from FSI:

Case 2 - M=0.952 - α = 1.148 ^◦ - ¯t=6.58s . . . 205 5.93 Curved Wing - Monitor Surfaces for F z1 and F z2 . . . 206 5.94 Curved Wing - Time History and PSD of F z1 , C L and C D . CFD

analysis on deformed shape of FSI - Case 2 - M=0.948 - α = 1.148 ^◦ - ¯t=9.04s . . . 207 5.95 Curved Wing - Time History and PSD of F z2 , C L and C D . CFD

analysis on deformed shape of FSI - Case 2 - M=0.952 - α = 1.148 ^◦ - ¯t=6.58s . . . 208 5.96 Swept Wing PSD and Phase of Cl, CD and FZ (Rigid CFD) . . . . 210 5.97 Curved Wing PSD and Phase of Cl, CD and FZ (Rigid CFD) . . . 211 5.98 Swept Wing: Pressure RMS - Rigid CFD Analysis. Case 2 - M=0.9

- α = 0.977 ^◦ - Deformed shape at Time = 14.30 s . . . 212 5.99 Curved Wing: Pressure RMS - Rigid CFD Analysis. Case 2 -

M=0.948 - α = 1.15 ^◦ - Deformed shape at Time = 9.04 s . . . 213 5.100Swept & Curved Wing: Rigid CFD Analysis - Isosurface Pressure

RMS = 0.8 Pa - Isosurface Mach = 1 . . . 214 5.101Swept & Curved Wing: Rigid CFD Analysis - Isosurface Pressure

RMS = 0.15 Pa - Isosurface Mach = 1 . . . 215 5.102Swept Wing PSD Analysis: FSI vs Rigid CFD FSI: Case 2 - M=0.9

- α = 0.977 ^◦ . Rigid CFD: Deformed Shape at T ime = 14.30 s . . . 216

LIST OF FIGURES XI 5.103Curved Wing PSD Analysis: FSI vs Rigid CFD FSI: Case 2 -

M=0.948 - α = 1.15 ^◦ . Rigid CFD: Deformed Shape at T ime = 9.04 s217

List of Tables

2.1 Geometrical data of the two half wing models . . . 29

3.1 Flow Conditions and Flight Data . . . 49

3.2 ANSYS Fluent - Boundary Conditions . . . 50

3.3 Swept & Curved Wing: CFD results (ANSYS Fluent) . . . 51

3.4 Swept & Curved Wing: Drag Reduction . . . 54

3.5 Drag coecient for C L = 0.4 , M = 0.85, h = 10000 m. (Fluent) . . 60

3.6 Swept & Curved Wing conguration - Steady Level Flight - Fuel Saving . . . 66

4.1 Flanges cross-section distribution . . . 71

4.2 Stringer cross-section distribution . . . 72

4.3 Spar web thickness distribution . . . 73

4.4 Skin Thickness distribution outside the wing-box . . . 74

4.5 Skin Thickness distribution inside the wing-box . . . 74

4.6 Skin second sublayer thickness for Fuel . . . 76

4.7 Rib thickness distribution . . . 77

4.8 Fictitious Mass Moment of Inertia . . . 79

4.9 Aluminium Alloy Properties . . . 79

4.10 Structural Model Data . . . 84

4.11 Results of modal analysis - Case 0 (No ctitious moment of inertia) 84 4.12 Results of modal analysis - Case 1 and Case 2 . . . 99

4.13 Structural damping Mass and Stiness coecients . . . 101

4.14 Modal Analysis Results for Case 1 and Case 2 . . . 102

5.1 Swept & Curved Wing: CFD results (ANSYS Fluent) h = 10000 m - M = 0.85 - 6.5M Mesh. . . 112

5.2 Swept & Curved Wing: CFD results (ANSYS Fluent) h = 10000 m - M = 0.85 - 400k Mesh. . . 113

5.3 Swept & Curved Wing: Drag Reduction (C D

−C _D

)/C _D

h = 10000 m - M = 0.85 - C L = 0.4 - 6.5M Mesh vs 400k Mesh . . . 115

XII

LIST OF TABLES XIII

5.4 Flow Conditions and Flight Data . . . 119

5.5 ANSYS Fluent - Boundary Conditions . . . 120

5.6 ANSYS Fluent - Dynamic Mesh Zones parameters . . . 120

5.7 FSI Analysis - General Settings . . . 122

5.8 FSI Analysis - Initial Mach number . . . 123

5.9 FSI Analysis - Unstable solution frequencies . . . 150

5.10 Swept Wing & Curved Wing: Types of Dynamic Aeroelastic Insta- bility . . . 177

5.11 FSI Analyses - RMS of Tip LE V. Displ., C L and C D . . . 177

5.12 Swept Wing - RMS of F Z , C L and C D - CFD Analyses on deformed shape from FSI - Case 2 - M=0.9 - α = 0.977 ^◦ - ¯t=14.30s . . . 191 5.13 Curved Wing - RMS of F Z1 , F Z2 , C L and C D . CFD Analyses on:

Deformed shape from FSI - Case 2 - M=0.948 - α = 1.148 ^◦ - ¯t=9.04s

Deformed shape from FSI - Case 2 - M=0.952 - α = 1.148 ^◦ - ¯t=6.58s 206