Minimize the Aerodynamic Effect of a Strut on the Wing Optimization of Strut-Braced Wing Configuration for Concep- tual Design using Low-fidelity CFD Models and Optimization Algorithms Master’s thesis in Mobility Engineering ANDRES FELIPE SANTOS BLAIR MERT UTKU DEPARTMENT OF MECHANICS AND MARITIME SCIENCES CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2023 www.chalmers.se www.chalmers.se Master’s thesis 2023 Minimize the Aerodynamic Effect of a Strut on the Wing Optimization of Strut-Braced Wing Configuration for Conceptual Design using Low-fidelity CFD Models and Optimization Algorithms ANDRES FELIPE SANTOS BLAIR & MERT UTKU Department of Mechanics and Maritime Sciences Division of Fluid Dynamics Chalmers University of Technology Gothenburg, Sweden 2023 Minimize the Aerodynamic Effect of a Strut on the Wing Optimization of Strut-Braced Wing Configuration for Conceptual Design using Low- fidelity CFD Models and Optimization Algorithms ANDRES FELIPE SANTOS BLAIR & MERT UTKU © ANDRES FELIPE SANTOS BLAIR, 2023. © MERT UTKU, 2023. Supervisor: Alexandre Antunes, Heart Aerosapce AB Examiner: Carlos Xisto, Department of Mechanics and Maritime Sciences Master’s Thesis 2023 Department of Mechanics and Maritime Sciences Division of Fluid Dynamics Chalmers University of Technology SE-412 96 Gothenburg Telephone +46 31 772 1000 Cover: Picture of the ES-30. Typeset in LATEX, template by Kyriaki Antoniadou-Plytaria Printed by Chalmers Reproservice Gothenburg, Sweden 2023 iv Minimization of the Aerodynamic Effect of a Strut on the Wing ANDRES FELIPE SANTOS BLAIR & MERT UTKU Department of Mechanics and Maritime Sciences Chalmers University of Technology Abstract The field of aircraft design is continuously advancing through the utilization of scientific techniques and empirical methods. The incorporation of computational methods has facilitated the design process of new and complex aircraft, enabling more efficient conceptual design and optimization. These advancements have the potential to significantly reduce fuel consumption and emissions, making a pos- itive impact on the environment, a critical global concern. The development of battery-electric airplanes represents a significant step towards creating a more sus- tainable aviation sector. Among the various emerging concepts, the Strut-Braced Wing (SBW) has shown great promise in enhancing aerodynamic efficiency while reducing wing weight. However, the implementation of new concepts and technologies also presents new challenges and limitations that must be addressed, particularly the impact of aero- dynamics on the aircraft’s range, which can impose limitations on its maximum travel distance. The primary objective of this thesis is to minimize the aerodynamic effects of a strut and wing configuration by reducing total drag and increasing the Oswald efficiency of the Strut-Braced Wing during the conceptual design phase. To achieve this goal, Sequential Quadratic Programming (SQP) and Genetic Al- gorithm (GA) optimization algorithms are employed, utilizing low-fidelity Compu- tational Fluid Dynamics (CFD) methods. The airfoil data utilized in the study is obtained from the XFOIL tool, which provides important viscous aerodynamic characteristics. By implementing these methodologies, it is anticipated that the aerodynamic per- formance of the Strut-Braced Wing configuration can be optimized, leading to im- proved efficiency and weight reduction. The results obtained from this research will contribute to the advancement of aircraft design and promote the development of more environmentally friendly and efficient aircraft during the conceptual design phase. Keywords: SBW, CFD, aerodynamics, optimization, SLSQP, SGA, VLM, weight estimation v Acknowledgements First and foremost, we would like to extend our heartfelt thanks to the Heart Aerospace team for their unwavering support and for providing us with the op- portunity to collaborate on this master’s thesis. Their guidance and collaboration have been invaluable throughout the entire process. We would also like to express our deep appreciation to Alexandre Antunes, our su- pervisor, for his exceptional advice, guidance, and expertise during this thesis work. His extensive knowledge has played a pivotal role in our personal and professional growth. We are grateful to have had the privilege of working with him and for making this journey both enjoyable and enlightening. Furthermore, we extend our warmest gratitude to our examiner, Carlos Xisto, for his invaluable guidance and support during these past months. His insightful feedback has greatly contributed to the quality of our work. We also want to express our sincere thanks to Chalmers University of Technology and all the professors who have educated and nurtured us over the past two years. Their dedication and passion for teaching have made this journey both enjoyable and intellectually stimulating. Finally, we want to offer a special heartfelt thank you to our family and friends for their unconditional support and belief in us. Her şey için teşekkür ederim. ¡Gracias a todos los que me acompañaron en este viaje! In conclusion, we are immensely grateful to everyone who has contributed to our academic journey and made it a truly enriching experience. Thank you for your unwavering support, and for making this adventure fun and captivating! Andres Felipe Santos Blair & Mert Utku, Gothenburg, June 2023 vii List of Acronyms Below is the list of acronyms that have been used throughout this thesis listed in alphabetical order: 2D Two dimensional 3D Three dimensional AC Aerodynamic Center ARAC Aviation Rulemaking Advisory Committee AoA Angle of Attack AVL Anthena Vortice Lattice BMX Bending moment in x axis CO2 Carbon Dioxide CAD Computer-Aided Design CFD Computational Fluid Dynamics CoP Center of Pressure EASA European Union Aviation Safety Agency FN Surface Forces file FS Strip Forces file FT Total Forces file FVM Finite Volume Method GA Genetic Algorithm HWB Hybrid Wing Body I/O Input - Output IATA International Air Transport Association JSON JavaScript Object Notation LE Leading Edge LLT Lifting Line Theory MAC Mean Aerodynamic Chord MGC Mean Geometric Chord MTOW Maximum Take-Off Weight NACA National Advisory Committee for Aeronautics NASA National Aeronautics and Space Administration N-S Navier-Stokes OOP Object Oriented Programming RANS Reynolds-averaged Navier–Stokes equations SBW Strut Braced Wing ix SGA Simple Genetic Algorithm SLSQP Sequential Least Squares Programming TE Trailing Edge VLM Vortex Lattice Method WS Wing-Strut x xii Nomenclature Below is the nomenclature of parameters,variables and coefficients that have been used throughout this thesis. Lowercase greek letters α Angle of attack α∗ Angle of attack value for non-linear behaviour in Cl vs AoA graph αϕ Shift value for cambered airfoils in Angle of attack axis γ Strut dihedral angle ηz Ultimate load factor θ Angle between pressure vector and the shear stress vector angle vs chord line λ Taper ratio λW Wing taper ratio µ Freestream dynamic viscosity ν Kinematic viscosity ρ Density ρ∞ Freestream density τ Shear stress τl Shear stress in the airfoil’s lower surface (2D) τu Shear stress in the airfoil’s upper surface (2D) Uppercase greek letters Γ Circulation Λ Sweep angle xiii Λc/4 Sweep angle at quarter MAC Lowercase latin letters a Speed of sound a Axial force in airfoil (2D) al Axial force in the airfoil’s lower surface (2D) au Axial force in the airfoil’s upper surface (2D) b Span c Chord croot Chord in the wing root ctip Chord in the wing tip cMGC Mean geometric chord d Drag force in airfoil (2D) e Oswald efficiency factor l Lift force in airfoil (2D) m Moment in airfoil (2D) n Normal force in airfoil (2D) nl Normal force in the airfoil’s lower surface (2D) nu Normal force in the airfoil’s upper surface (2D) p Static pressure p∞ Freestream static pressure pl Static pressure in the airfoil’s lower surface (2D) pu Static pressure in the airfoil’s upper surface (2D) q Dynamic pressure q∞ Freestream dynamic pressure s Distance alongside airfoil’s surface from the LE sl Distance alongside airfoil’s lower surface from the LE su Distance alongside airfoil’s upper surface from the LE r Resultant force in airfoil (2D) t Thickness t Time t Parametric value for Bezier curve t/c thickness/chord ratio xiv y Spanwise axis y Position in spanwise axis ymotor1 Spanwise location of motor 1 ymotor2 Spanwise location of motor 2 Uppercase latin letters A Axial force in body (3D) AoA Angle of Attack Ainc Incidence angle AR Aspect ratio ARW Wing aspect ratio Cd Drag coefficient (2D) Cf Skin friction coefficient (2D) Cl Lift coefficient (2D) Cm Moment coefficient (2D) Cp Pressure coefficient Cr Resultant force coefficient (2D) CD Drag coefficient (3D) CDi Induced drag coefficient CDf Skin friction drag coefficient CDff Induced drag from Trefftz plane analysis CDtot Total drag coefficient CDvis Viscous drag coefficient CL Lift coefficient (3D) CLϕ Shift value for cambered airfoils in lift coefficient axis CM Moment coefficient (3D) CR Resultant force coefficient (3D) D Drag force in body (3D) Dn Drag force for the n-th body (3D) E Young‘s modulus F Factor for secondary structures I Geometrical inertia K Effective length factor xv L Lift force in body (3D) L Strut length M Mach number M Moment in body (3D) M∞ Freestream Mach number Mlocal Local Mach number N Normal force in body (3D) O Sample space P0 Left control point for bezier curve construction P1 Middle control point for bezier curve construction P2 Right control point for bezier curve construction P0x Left control point horizontal coordinate for bezier curve con- struction P1x Middle control point horizontal coordinate for bezier curve construction P2x Right control point horizontal coordinate for bezier curve con- struction P0y Left control point vertical coordinate for bezier curve con- struction P1y Middle control point vertical coordinate for bezier curve con- struction P2y Right control point vertical coordinate for bezier curve con- struction Pcr Critical load for buckling R Resultant force in body (3D) Re Reynolds number Si,n Cell in the i-th row and n-th column in the LHS cell matrix Sref Reference area SW Wing reference area T Target space V Flow velocity V Volume V∞ Freestream flow velocity W Weight W0 Maximum Take-Off Weight WF W Fuel stored in wing weight xvi Wmotor1 Weight of motor 1 Wmotor2 Weight of motor 2 WF W Fuel stored in wing weight WW Wing weight Xi Input variables Yn n-th strata in LHS method xvii xviii Contents List of Acronyms ix Nomenclature xii List of Figures xxiii List of Tables xxvii 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Strut-braced wings . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Theory 7 2.1 Fluid dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1.1 Navier-Stokes . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1.2 Potential Flow theory . . . . . . . . . . . . . . . . . 8 2.1.2 Subsonic flow and compressiblity . . . . . . . . . . . . . . . . 8 2.1.3 Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Numerical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Low-fidelity models . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1.1 Vortex sheet and horseshoe vortex . . . . . . . . . . 10 2.2.1.2 Lifting Line theory . . . . . . . . . . . . . . . . . . . 10 2.2.1.3 Vortex Lattice Method (VLM) . . . . . . . . . . . . 11 2.2.1.4 3D Panel Method . . . . . . . . . . . . . . . . . . . . 12 2.2.1.5 Viscous Drag Prediction with XFOIL . . . . . . . . . 12 2.2.1.6 Summary of low-fidelity models . . . . . . . . . . . . 12 2.2.2 High-fidelity models . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.2.1 Finite Volume Method (FVM) . . . . . . . . . . . . 12 2.3 Geometric Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.1 Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.2 Wing Planform . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3.2.1 Planform parameters . . . . . . . . . . . . . . . . . . 14 2.3.2.2 Planform types . . . . . . . . . . . . . . . . . . . . . 16 2.3.3 Strut and strut braced wing . . . . . . . . . . . . . . . . . . . 18 xix Contents 2.4 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4.1 Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4.2 Aerodynamic forces . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4.2.1 Aerodynamic coefficients in 2D surfaces . . . . . . . 23 2.4.2.2 Aerodynamic coefficients in 3D bodies . . . . . . . . 24 2.4.3 Aerodynamic lift . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4.4 Aerodynamic drag breakdown . . . . . . . . . . . . . . . . . . 26 2.4.4.1 Drag due to lift . . . . . . . . . . . . . . . . . . . . . 27 2.4.4.2 Parasite drag . . . . . . . . . . . . . . . . . . . . . . 27 2.4.5 Efficiency in lifting bodies . . . . . . . . . . . . . . . . . . . . 30 2.4.5.1 Lift distribution and Wing types . . . . . . . . . . . 31 2.5 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5.1 Modelling structure . . . . . . . . . . . . . . . . . . . . . . . . 32 2.5.2 Optimization classes . . . . . . . . . . . . . . . . . . . . . . . 33 2.5.3 Globalized Sequential Quadratic Programming Algorithm . . . 34 2.5.4 Simple Genetic Algorithm . . . . . . . . . . . . . . . . . . . . 34 2.6 Latin Hypercube Sampling (LHS) . . . . . . . . . . . . . . . . . . . . 35 2.7 Bezier curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3 Methods 39 3.1 Conceptual Design and Wing Design . . . . . . . . . . . . . . . . . . 39 3.2 Airfoil selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3 Low-fidelity CFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3.1 OpenVSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3.1.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3.1.2 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.3.1.3 Setup - VSPAERO . . . . . . . . . . . . . . . . . . . 41 3.3.1.4 Output - VSPAERO . . . . . . . . . . . . . . . . . . 41 3.3.2 XFOIL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3.3 AVL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3.3.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . 42 3.3.3.2 Mesh generation . . . . . . . . . . . . . . . . . . . . 44 3.3.3.3 Output . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 High-fidelity CFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4.1 StarCCM+ (Euler equation) - 2D . . . . . . . . . . . . . . . . 48 3.4.2 StarCCM+ (Euler equation) - 3D . . . . . . . . . . . . . . . . 49 3.4.2.1 Geometry generation and Boundary conditions . . . 49 3.4.2.2 Mesh generation . . . . . . . . . . . . . . . . . . . . 50 3.4.2.3 Setup and Post process . . . . . . . . . . . . . . . . 51 3.5 Optimization and analysis framework . . . . . . . . . . . . . . . . . . 51 3.5.1 Wing-strut class . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.5.2 Optimizer class . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.5.2.1 SLSQP . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.5.2.2 SGA . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.5.3 Objective function . . . . . . . . . . . . . . . . . . . . . . . . 58 3.5.4 Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 xx Contents 3.5.5 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.5.5.1 Area constraint . . . . . . . . . . . . . . . . . . . . . 59 3.5.5.2 Twist constraint between sections . . . . . . . . . . . 61 3.6 Weight estimation methodology . . . . . . . . . . . . . . . . . . . . . 62 3.6.1 Semi-empirical formula . . . . . . . . . . . . . . . . . . . . . . 62 3.6.2 Bending moment and simplified strut methodology . . . . . . 62 3.6.2.1 Load factor . . . . . . . . . . . . . . . . . . . . . . . 64 3.6.3 General Assumptions of weight calculation . . . . . . . . . . . 65 3.6.4 Bending Moment . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.6.5 Force on the strut . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.6.6 Total Weight of the SBW . . . . . . . . . . . . . . . . . . . . 68 4 Results and Discussion 69 4.1 XFOIL Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.2 Tools and methods comparison . . . . . . . . . . . . . . . . . . . . . 72 4.3 Wing planform comparison . . . . . . . . . . . . . . . . . . . . . . . . 74 4.4 Relation between Reference Area and Drag for SBW . . . . . . . . . 75 4.5 Oswald efficiency investigation . . . . . . . . . . . . . . . . . . . . . . 77 4.6 Strut dihedral influence in aerodynamic coefficients investigation . . . 78 4.7 Strut dihedral and local geometrical incidence influence in aerody- namic coefficients investigation . . . . . . . . . . . . . . . . . . . . . . 82 4.7.1 Design of Experiments (DOE) - incidence . . . . . . . . . . . 83 4.8 Design of Experiments (DOE) - Bezier curve . . . . . . . . . . . . . . 85 4.9 Optimization Investigation for SLSQP . . . . . . . . . . . . . . . . . 88 4.9.1 Optimizer settings investigation . . . . . . . . . . . . . . . . . 88 4.9.2 Unit and Initial point effect . . . . . . . . . . . . . . . . . . . 89 4.9.2.1 Strut dihedral angle . . . . . . . . . . . . . . . . . . 89 4.9.2.2 Bezier curve . . . . . . . . . . . . . . . . . . . . . . . 91 4.9.3 Fairing load . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.9.4 Objective functions effect investigation . . . . . . . . . . . . . 97 4.9.5 Summary and selection of design . . . . . . . . . . . . . . . . 100 4.9.5.1 Design 1 . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.9.5.2 Design 2 . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.9.5.3 Comparison of obtained designs . . . . . . . . . . . . 107 4.10 Optimization Investigation for GA . . . . . . . . . . . . . . . . . . . 109 4.11 Comparison between GA and SLSQP . . . . . . . . . . . . . . . . . . 110 4.12 Weight estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.12.1 Bending Moment Results . . . . . . . . . . . . . . . . . . . . . 112 4.12.2 Wing weight results . . . . . . . . . . . . . . . . . . . . . . . . 114 4.12.3 Strut weight results . . . . . . . . . . . . . . . . . . . . . . . . 114 4.12.4 Total weight results . . . . . . . . . . . . . . . . . . . . . . . . 115 4.12.5 Methodology comparison . . . . . . . . . . . . . . . . . . . . . 115 5 Conclusion 117 5.1 Research Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.2 Future Development . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 xxi Contents Bibliography 121 A Appendix 1 I A.1 Mesh dependency study . . . . . . . . . . . . . . . . . . . . . . . . . I A.2 Tool comparison dCP graphs . . . . . . . . . . . . . . . . . . . . . . . III A.3 Strut dihedral and local AoA influence . . . . . . . . . . . . . . . . . IV A.4 Further discussion about Oswald efficiency . . . . . . . . . . . . . . . IV xxii List of Figures 1.1 Air Travel between 1985-2050 [3] . . . . . . . . . . . . . . . . . . . . 2 1.2 CO2 emissions between 2005-2050 [7] . . . . . . . . . . . . . . . . . . 2 1.3 CO2 emission goals for 2050 [8] . . . . . . . . . . . . . . . . . . . . . 2 1.4 Expected contribution of new technologies in CO2 emissions [3] . . . 3 1.5 Joined Wings (Box Wing) configuration [3] . . . . . . . . . . . . . . . 3 1.6 Hybrid Wing Body configuration designed by DLR [3] . . . . . . . . . 3 1.7 Strut-braced Wing designed by NASA/Boeing [3] . . . . . . . . . . . 3 1.8 Hurel-Dubois HD31 [12] . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.9 ATR-42 FAA drop test [18] . . . . . . . . . . . . . . . . . . . . . . . 5 1.10 Shorts 3-30 FAA drop test [19] . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Comparison of different governing equations . . . . . . . . . . . . . . 7 2.2 Replacement of finite wing with horseshoe vortex . . . . . . . . . . . 10 2.3 Superposition of a finite number of horseshoe vortices along the lifting line. Source: [30] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Airfoil geometric parameters. . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Wide-body aircraft scheme with basic wing parameters labeled. . . . 14 2.6 Taper ratio effects in lift distribution. Source: adapted from [39] . . . 15 2.7 Wing sweep in planform. . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.8 Wing with positive dihedral angle, front view. . . . . . . . . . . . . . 16 2.9 Constant chord planform . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.10 Trapezoidal-straight planform . . . . . . . . . . . . . . . . . . . . . . 17 2.11 Elliptic planform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.12 Strut braced wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.13 Streamlines scheme in a NACA 6409 airfoil with an angle of attack of 0 degrees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.14 Streamlines scheme in a NACA 6409 airfoil with an angle of attack of 15 degrees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.15 Pressure and shear stress in an airfoil. Source: adapted from [30] . . . 20 2.16 Airfoil free body diagram. Source: adapted from [30] . . . . . . . . . 21 2.17 Scheme for pressure and shear stress integration over an airfoil. Source: adapted from [30] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.18 Airfoil free body diagram for 2D analysis . . . . . . . . . . . . . . . . 23 2.19 Body free body diagram for a 3D analysis . . . . . . . . . . . . . . . 24 2.20 Airfoil lift coefficient for uncambered airfoil and cambered airfoil for Reynolds number approximately 2.2 x 106. . . . . . . . . . . . . . . . 25 xxiii List of Figures 2.21 Drag classification scheme . . . . . . . . . . . . . . . . . . . . . . . . 26 2.22 Pressure distribution in an airfoil. Source: XFOIL [25] . . . . . . . . 29 2.23 Aerodynamic efficiency for two different airfoils . . . . . . . . . . . . 31 2.24 Drag polar for two different airfoils . . . . . . . . . . . . . . . . . . . 31 2.25 Local and global minima and maxima plotted in a function. Where; light blue dots represent local minima, blue dots represent global minimum, orange dots represent local maxima, and red dots represent global maximum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.26 Simple Genetic Algorithm flowchart . . . . . . . . . . . . . . . . . . . 35 2.27 LHS with low number of samples number for three input variables . . 36 2.28 LHS with low strata number for three input variables. . . . . . . . . . 37 2.29 LHS with high strata number for three input variables . . . . . . . . 37 2.30 Example Bezier curve and control points. . . . . . . . . . . . . . . . . 38 3.1 NACA0012 airfoil [54]. Source: XFOIL [25] . . . . . . . . . . . . . . . 40 3.2 Wing planform used in OpenVSP simulations. Source: OpenVSP [35] 40 3.3 Mesh dependency study for OpenVSP . . . . . . . . . . . . . . . . . . 41 3.4 File format of AVL geometry file . . . . . . . . . . . . . . . . . . . . 42 3.5 Front view of the SBW geometry . . . . . . . . . . . . . . . . . . . . 43 3.6 Isometric view of the SBW geometry . . . . . . . . . . . . . . . . . . 43 3.7 Mesh dependency study for AVL for trapezoidal wing based on CDi . 44 3.8 Mesh dependency study for AVL for elliptical wing based on CDi . . . 45 3.9 Mesh dependency study for AVL for rectangular wing based on CDi . 45 3.10 Mesh of "Rectangle wing with strut" case . . . . . . . . . . . . . . . . 46 3.11 Mesh of "Rectangle wing with strut-re-meshed" case . . . . . . . . . . 46 3.12 Mesh dependency for strut . . . . . . . . . . . . . . . . . . . . . . . . 47 3.13 FT-Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.14 FN-Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.15 FS-Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.16 2D mesh dependency study . . . . . . . . . . . . . . . . . . . . . . . 49 3.17 Fluid domain with basic setup and properties. Source: StarCCM+ [55] 49 3.18 Additional relevant setup and properties in the fluid domain. Source: StarCCM+ [55] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.19 Mesh dependency study for StarCCM+ for trapezoidal wing . . . . . 50 3.20 Mesh on sections 12.5 m away from wing root and 1.2 m away from the leading edge of the wing . . . . . . . . . . . . . . . . . . . . . . . 50 3.21 I/O scheme for airfoil class . . . . . . . . . . . . . . . . . . . . . . . . 52 3.22 I/O scheme for comparator class . . . . . . . . . . . . . . . . . . . . . 52 3.23 I/O scheme for DOE class . . . . . . . . . . . . . . . . . . . . . . . . 53 3.24 I/O scheme for SGA class . . . . . . . . . . . . . . . . . . . . . . . . 53 3.25 I/O scheme for wing-strut class . . . . . . . . . . . . . . . . . . . . . 55 3.26 I/O scheme for optimizer class . . . . . . . . . . . . . . . . . . . . . . 56 3.27 Detailed flowchart for SLSQP optimization scheme . . . . . . . . . . 57 3.28 Simplified flowchart for SGA optimization scheme . . . . . . . . . . . 58 3.29 Forces on the wing and distributions . . . . . . . . . . . . . . . . . . 63 3.30 Constant inboard loading assumption . . . . . . . . . . . . . . . . . . 63 xxiv List of Figures 3.31 Strut beam shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.32 Generic Load factor graph . . . . . . . . . . . . . . . . . . . . . . . . 65 3.33 Assumed Spanwise BMX distribution . . . . . . . . . . . . . . . . . . 65 3.34 Regions and point forces for Bending Moment calculation . . . . . . . 66 3.35 Force acting on strut . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.1 Result section flowchart . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2 Cl vs. AoA comparison between XFOIL and wind tunnel data for NACA0012 @ Re = 2 · 106, M = 0.15 . . . . . . . . . . . . . . . . . . 70 4.3 Cd vs. AoA comparison between XFOIL and wind tunnel data for NACA0012 @ Re = 2 · 106, M = 0.15 . . . . . . . . . . . . . . . . . . 70 4.4 Cd vs. Cl comparison between XFOIL and wind tunnel data for NACA0012 @ Re = 2 · 106, M = 0.15 . . . . . . . . . . . . . . . . . . 70 4.5 NACA0012 Pressure distribution over airfoil from evaluated sources . 71 4.6 Cl vs. AoA comparison generated with XFOIL NACA0012 @ Re = 2.24 · 106, M = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.7 Cd vs. AoA comparison generated with XFOIL NACA0012 @ Re = 2.24 · 106, M = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.8 Cd vs. Cl comparison generated with XFOIL NACA0012 @ Re = 2.24 · 106, M = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.9 CL vs. AoA graph for different tools . . . . . . . . . . . . . . . . . . 73 4.10 CDi vs AoA graph for different tools . . . . . . . . . . . . . . . . . . . 73 4.11 StarCCM+ velocity scalar for 16 AoA. Source: StarCCM+ [55] . . . 73 4.12 Streamlines over the airfoil for 16 AoA. Source: StarCCM+ [55] . . . 73 4.13 Lift distribution comparison for different Wing Planforms . . . . . . . 74 4.14 SBW design for different strut dihedral angles . . . . . . . . . . . . . 75 4.15 Oswald efficiency values for wing and wing+strut surfaces from FN file 77 4.16 Oswald efficiency values for strut surface from FN file with different scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.17 Oswald efficiency of different strut dihedral angles . . . . . . . . . . . 78 4.18 Dihedral change in the strut . . . . . . . . . . . . . . . . . . . . . . . 79 4.19 CD and strut length vs strut dihedral angle for the case CL = 0. . . . 79 4.20 CL and strut length vs strut dihedral angle for the case CL = 0.8 . . . 79 4.21 CDvis and strut length vs strut dihedral angle for the case CL = 0.8 . 80 4.22 Wing and strut CL and CD in the NACA0012 profile Cd vs. Cl graph for four different values of strut dihedral angle . . . . . . . . . . . . . 80 4.23 CDi , CL vs strut dihedral angle for the case CL = 0.8 . . . . . . . . . 81 4.24 DOE for Total Drag. Values are expressed as drag counts and degrees 83 4.25 DOE for Induced Drag. Values are expressed as drag counts and degrees 84 4.26 DOE for Viscous Drag. Values are expressed as drag counts and degrees 84 4.27 Bezier curve for different control points . . . . . . . . . . . . . . . . . 85 4.28 DOE for control points of Bezier curve for the strut. Values are expressed as drag counts . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.29 DOE for control points of Bezier curve for the wing. Values are expressed as drag counts, degrees for the incidence and unit-less for the coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 xxv List of Figures 4.30 Spanwise local geometrical distribution of the wing for initial points of both designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.31 Spanwise local geometrical distribution of the strut for initial points of both designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.32 Spanwise lift distribution for loaded and unloaded cases . . . . . . . . 94 4.33 Local geometrical incidence distribution through span for wing when the fairing is counted in total drag . . . . . . . . . . . . . . . . . . . . 94 4.34 Local geometrical incidence distribution through span for strut when the fairing is counted in total drag . . . . . . . . . . . . . . . . . . . . 95 4.35 Local geometrical incidence distribution through span for wing when the fairing is not counted in total drag . . . . . . . . . . . . . . . . . 95 4.36 Local geometrical incidence distribution through span for strut when the fairing is not counted in total drag . . . . . . . . . . . . . . . . . 96 4.37 Spanwise lift distribution for both objective properties . . . . . . . . 98 4.38 Local geometrical incidence distribution through span for wing for Oswald objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.39 Local geometrical incidence distribution through span for strut for Oswald objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.40 Local geometrical incidence distribution through span for wing for total drag objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.41 Local geometrical incidence distribution through span for strut for total drag objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.42 Local geometrical incidence distribution through span for wing . . . . 103 4.43 Local geometrical incidence distribution through span for strut . . . . 103 4.44 Local geometrical incidence distribution through span for the wing . . 106 4.45 Local geometrical incidence distribution through span for the strut . . 106 4.46 Spanwise lift distribution for both SLSQP designs . . . . . . . . . . . 108 4.47 DOE for optimized wing. Values in drag counts for drag coefficients, degrees for incidence and unit-less for coordinates . . . . . . . . . . . 108 4.48 SLSQP vs. GA Lift distubution . . . . . . . . . . . . . . . . . . . . . 111 A.1 dCp for different tools . . . . . . . . . . . . . . . . . . . . . . . . . . III A.2 dCp for different tools . . . . . . . . . . . . . . . . . . . . . . . . . . III A.3 dCp for different tools . . . . . . . . . . . . . . . . . . . . . . . . . . III A.4 dCp for different tools . . . . . . . . . . . . . . . . . . . . . . . . . . III A.5 Lift distribution when the dihedral angle of the strut is 10 and 50 degrees for the case CL = 0.8 . . . . . . . . . . . . . . . . . . . . . . . IV A.6 Induced drag factor [39] . . . . . . . . . . . . . . . . . . . . . . . . . V xxvi List of Tables 2.1 Flow regimes for compressible flow [33]. . . . . . . . . . . . . . . . . . 9 3.1 Mesh dependency table for trapezoidal wing . . . . . . . . . . . . . . 41 3.2 Mesh generation strategy of wing strut combination . . . . . . . . . . 46 3.3 Values used to limit the search space for both optimization algorithms 59 3.4 Additional values used to limit the search space for both optimization algorithms for ’bezier’ mode . . . . . . . . . . . . . . . . . . . . . . . 59 3.5 Assumptions for motor . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.1 Comparison table for different Wing planforms . . . . . . . . . . . . . 75 4.2 Change in total drag when the reference area is defined as the area of the wing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.3 Change in total drag when the reference area is defined as the sum of the areas of the strut and wing . . . . . . . . . . . . . . . . . . . . 76 4.4 Variation in total drag comparison between reference area based on the wing vs when the reference area is based on wing and strut . . . . 76 4.5 Oswald efficiency values for surface and surface group for both sources 77 4.6 Induced and viscous drag breakdown for each surface and joint surfaces 82 4.7 Local geometrical incidence of wing, local geometrical incidence of strut and dihedral of strut . . . . . . . . . . . . . . . . . . . . . . . . 82 4.8 Initial conditions and restrictions . . . . . . . . . . . . . . . . . . . . 88 4.9 Total drag and normalized Bezier curve control point for different Optimization settings . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.10 Optimizer setting for Unit effect on strut dihedral angle . . . . . . . . 89 4.11 Restriction table for Unit effect on strut dihedral angle investigation . 90 4.12 Objective function unit effect table . . . . . . . . . . . . . . . . . . . 90 4.13 Restrictions and initial points . . . . . . . . . . . . . . . . . . . . . . 91 4.14 Results of the optimization . . . . . . . . . . . . . . . . . . . . . . . . 92 4.15 Optimizer settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.16 Fairing load investigation initial conditions and restrictions . . . . . . 93 4.17 Aerodynamic coefficients and geometry of the cases with ’LOAD’ and ’NOLOAD’ conditions applied to fairing surface after optimization process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.18 Objective function investigation initial conditions and restrictions . . 97 4.19 Objective function investigation results . . . . . . . . . . . . . . . . . 98 4.20 Restrictions and initial points . . . . . . . . . . . . . . . . . . . . . . 101 4.21 Optimizer settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 xxvii List of Tables 4.22 Geometric parameters for Design 1 . . . . . . . . . . . . . . . . . . . 102 4.23 Aerodynamic coefficients for Design 1 . . . . . . . . . . . . . . . . . . 102 4.24 Geometric parameters for Design 2 . . . . . . . . . . . . . . . . . . . 104 4.25 Aerodynamic coefficients for Design 2 . . . . . . . . . . . . . . . . . . 105 4.26 Aerodynamic coefficients for Design 1 and Design 2 . . . . . . . . . . 107 4.27 GA analysis fixed variables and values . . . . . . . . . . . . . . . . . 109 4.28 GA optimizer settings . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.29 Aerodynamic coefficient results for GA and SLSQP . . . . . . . . . . 110 4.30 Parameters used for Semi-empirical formulas . . . . . . . . . . . . . . 112 4.31 Semi-empirical formula results for cantilever and strut-braced wing . 112 4.32 Variables used for weight and bending moment calculation . . . . . . 113 4.33 Bending moment contributions for Region 1 . . . . . . . . . . . . . . 113 4.34 Bending moment contributions for Region 2 . . . . . . . . . . . . . . 113 4.35 BMX for each considered region . . . . . . . . . . . . . . . . . . . . . 114 4.36 Geometric and material variables of the strut . . . . . . . . . . . . . . 114 4.37 Total weight of cantilever and strut braced wing configuration . . . . 115 4.38 Weight methodology comparison Table . . . . . . . . . . . . . . . . . 115 A.1 Mesh dependency table for Trapezoidal Wing . . . . . . . . . . . . . I A.2 Mesh dependency table for Elliptical Wing . . . . . . . . . . . . . . . II A.3 Mesh dependency table for Rectangular Wing . . . . . . . . . . . . . II A.4 Mesh dependency table for Strut Wing . . . . . . . . . . . . . . . . . II A.5 Oswald efficiency estimation for different Wings and methodologies . V xxviii 1 Introduction This master’s thesis was conducted in collaboration between Chalmers University of Technology and Heart Aerospace, a Swedish aviation startup with the goal of elec- trifying regional air travel. The collaboration lasted for five months. This chapter provides a brief background and description of the problem and aims of the study. Additionally, the section defines the specific airplane type that is the focus of this work. Heart Aerospace ES-30 is a reserve-hybrid airplane designed for commercial short- haul flights [1]. The aircraft is specifically designed to accommodate short takeoff and landing operations in regions with complex topography and short runways and is capable of performing steep approaches. Furthermore, ES-30 is a high-wing, T- tailed airplane with strut-braced wings. Such configuration is commonly used for this type of aircraft [2]. The airplane will be certified under Aviation Safety Agency (EASA) CS-25 certification standards. The focus of this thesis, written for Heart Aerospace, is to create and analyze a generic optimization process for wing-strut configuration design. 1.1 Background The aviation industry is constantly evolving, but it faces a significant challenge in reducing its carbon footprint to address the global climate crisis. Commercial air travel has been steadily increasing over the years, with projections indicating continued growth in the future, despite the impact of the COVID-19 pandemic as seen in Figure 1.1. Unfortunately, this growth is expected to lead to higher CO2 emissions as shown in Figure 1.2, with short-haul flights accounting for a significant portion of these emissions [3]. Studies show that 2% of all man-made CO2 emissions are generated by the aviation industry [4]. To address this challenge, the International Air Transport Association (IATA) and National Aeronautics and Space Administration (NASA) have set am- bitious goals to reduce global net aviation carbon emissions by 50% by 2050 relative to 2005 as indicated in Figure 1.3. Some countries, such as Sweden and Norway, have even more ambitious targets [3, 5, 6]. 1 1. Introduction Figure 1.1: Air Travel between 1985-2050 [3] Figure 1.2: CO2 emissions between 2005-2050 [7] Figure 1.3: CO2 emission goals for 2050 [8] 2 1. Introduction In this context, new technologies and concepts are crucial to achieving these goals. The IATA predicts that they will contribute 13% to the route to net-zero emissions as can be observed in Figure 1.4. One of the promising solutions to achieve low emissions and high performance for regional flights in the aviation industry is electric aircraft. Figure 1.4: Expected contribution of new technologies in CO2 emissions [3] In recent years, various designs have emerged with the goal of reducing fuel consump- tion, emissions, and noise, in line with the target of achieving net-zero emissions by 2050 or NASA’s HR2454 Goals [9]. IATA has summarized different concepts under ’The Revolutionary Aircraft Technologies’, which includes innovative designs such as Joined Wings (Box Wing), Hybrid Wing Body, and Strut-braced wings [3] as seen in Figure 1.5, Figure 1.6 and Figure 1.7. This thesis will concentrate on the Strut-braced wings design concept, which has the potential to bring a step-change in sustainability for the aviation industry, especially in the transonic flow regime. Figure 1.5: Joined Wings (Box Wing) con- figuration [3] Figure 1.6: Hybrid Wing Body configuration designed by DLR [3] Figure 1.7: Strut- braced Wing designed by NASA/Boeing [3] 1.1.1 Strut-braced wings Improving the aerodynamic efficiency of the aircraft leads to reductions in fuel con- sumption, emissions, and noise [10]. One approach to improve aerodynamic effi- 3 1. Introduction ciency is through the design of a high aspect ratio wing. Previous studies have shown that increasing the aspect ratio can reduce induced drag by up to 75% [20]. However, high aspect ratio wings imply higher bending moments as the wings are longer. Engineers are faced with a trade-off between increasing stiffness, which would result in a heavier wing, or adopting a wing-strut configuration to decrease the bending moment. The maximum bending moment using a wing-strut configura- tion can lead to a reduction of up to 50% experienced by the wing when compared to an equivalent cantilever wing.[11]. The design of strut-braced wings has a long history in aviation, dating back to the Hurel-Dubois HD-31 aircraft in 1956 which utilized strut-braced wings to achieve higher aerodynamic efficiency, as can be seen in Figure 1.8. At first, designers fa- vored wing-bracing over wing-box design. However, the use of wing-braced concept resulted in an increase in profile drag from the struts, leaving space for the develop- ment of the wing-box concept with thicker profiles to define the wing shape. Figure 1.8: Hurel-Dubois HD31 [12] In recent years, strut-braced wings have once again gained attention as a way to im- prove efficiency by increasing the aspect ratio of the wing while maintaining a lower thickness which allows for decreasing wave drag and induced drag. This reduction in wave drag during transonic flight, combined with a lower sweep angle and natural laminar flow, leads to improved overall efficiency [13]. Moreover, the lighter weight of strut-braced wings which can reach a decrease of 70% in comparison with cantilever wings [20], and the advancements in computational power and numerical methods, have enabled the optimization of these wings for both transonic and subsonic flight. As a result, there has been a trend towards the use of strut-braced wings in subsonic flights to reduce induced drag with higher aspect ratios and improve fuel efficiency towards net-zero emission goals [14, 15]. Moreover, strut-braced wings have certifi- cation benefits in CS 25.25 Weight Limits and CS25.305 Strength and deformation [16]. The Aviation Rulemaking Advisory Committee (ARAC) has issued a report on a new airframe crash-worthiness rule that mandates adequate fuselage resistance to loads during emergency landings or survivable crash events [17]. A Wing-Strut configuration can help satisfy this certification requirement, as the struts distribute the load onto the fuselage and prevent the structural deformation caused by the wing, as illustrated in Figures 1.9 and 1.10. 4 1. Introduction Figure 1.9: ATR-42 FAA drop test [18] Figure 1.10: Shorts 3-30 FAA drop test [19] Prior research has shown that various methods can be used for the aerodynamic analysis of wing-strut configurations, including semi-empirical formulas [20, 21], low- to-medium fidelity models [13], high-fidelity models [15, 22, 23], and multi-fidelity methods [24]. While high-fidelity methods are capable of capturing flow separations and details, they are computationally expensive and their accuracy is relatively more sensitive to mesh quality than low-fidelity methods, as discussed in Section 4.3. On the other hand, low-fidelity methods are generally less accurate in capturing details and separations but still provide accurate results depending on the simulation objective. Additionally, during the conceptual or preliminary design phase of an aircraft, design variables are not fully defined, and investing significant engineering time in high-fidelity methods may not be practical since the design is not mature enough. Therefore, in the optimization process of this thesis, low-fidelity methods have been utilized, as they offer a time-efficient and reliable approach to achieving the objective described in Section 1.2. 1.2 Objective The objective of this Master’s thesis project is to explore various integration strate- gies for implementing a Wing-Strut configuration while reducing the computational cost of designing strut-braced wings. An optimization process has been established using low-fidelity methods for airfoils and planforms, a panel method with a transi- tion model to include viscous effects for the former, and Vortex-Lattice method for the latter. Such a combination sets the aerodynamic analysis tool for the optimiza- tion process. The optimization process employs both gradient-based algorithms and stochastic algorithms, to achieve the best results. The optimization process will be conducted using XFOIL [25], AVL [26], and the SciPy [27] library module for op- timization algorithms in the Python programming language [28]. The optimization process will be implemented in Python, and the results will be evaluated against the design requirements. To assess the effectiveness of the low-fidelity tool, the op- timization process was compared to high-fidelity tools available in the market using a generic wing as a benchmark. The optimization process considered various con- straints and factors, with the objective of achieving the most optimal and realistic 5 1. Introduction wing, while maximizing the benefits of the strut-braced wing concept within the given scope. The ultimate goal is to achieve increased aerodynamic efficiency, de- creased weight, and reduced fuel consumption, thereby aligning with sustainability goals. 1.3 Scope This thesis focuses on the investigation and optimization process of wing-strut con- figurations for aircraft operating in subsonic flow regime conditions during the con- ceptual design phase. The study is limited to a duration of five months, which affected the complexity of the defined variables, constraints, and optimization algo- rithms. Additionally, due to cost in terms of time constraints, high-fidelity RANS simulations were not included. However, the investigation and optimization con- ducted in this study are adequate for the conceptual design phase. This thesis provides valuable insights into the design and optimization process of wing-strut configurations, which can inform future research in this area. 6 2 Theory 2.1 Fluid dynamics 2.1.1 Governing equations The governing equations are fundamental equations used to describe the motion of fluids. The accuracy of the numerical solutions obtained from solving these equations depends on the underlying assumptions and theoretical frameworks employed. Figure 2.1: Comparison of different governing equations 2.1.1.1 Navier-Stokes Velocity and pressure field are found with Navier Stokes equations where the flow is incompressible. Continuity equation is given in [29, Equation 2.1] which ensures the mass conservation. ∂vi ∂xi = 0 (2.1) The momentum equation as index notation is given in [29, Equation 2.2] for constant viscosity (incompressible). ∂vi ∂t + ∂vivj ∂xj = −1 ρ ∂p ∂xi + ν ∂2vi ∂xj∂xj (2.2) 7 2. Theory Where xi and vi are the positions and velocity in the direction i, ρ is density, ν is the kinematic viscosity, and p is the static pressure. Euler equations are the Navier-Stokes equations with inviscid assumption. 2.1.1.2 Potential Flow theory Potential flows use the inviscid Euler equations, along with irrotational assumptions, which state that the flow velocity is equal to the gradient of the velocity potential as shown in equation 2.3. V⃗ = ∇ϕ (2.3) The main idea behind potential flow theory is to calculate the forces acting on a given geometry by modelling the circulation using the velocity potential ϕ [30, 31]. The circulation Γ is defined as the closed loop integral of velocity over the geometry. The continuity equation, also known as Laplace’s equation, is given as: ∇2Φ = 0 (2.4) The velocity potential can be solved numerically by implementing the boundary conditions. These boundary conditions ensure that the flow does not penetrate the geometry and that the velocity normal to the surface is zero [31]. The circulation can be converted into a force by using the Kutta-Joukowski theorem, as shown in equation 2.5. L = ρ∞V∞ × Γ (2.5) Where L is the Lift force, v∞ is the free stream velocity and Γ is the circulation. Potential flow theory is valid in situations where vorticity is not important, such as in thin boundary layers or where there are no wakes. Additionally, the theory does not consider viscous effects due to the inviscid assumption, which assumes no viscous drag over the geometry [30]. Furthermore, the Kutta condition is enforced to ensure that the solution has a "physical sense." The Kutta condition states that an airfoil creates lift by deflecting the flow, which adds a velocity field to the free-stream velocity. The deflection of the flow is such that the total flow should leave the trailing edge smoothly. For more details and an extensive explanation of the methodology, refer to [30] and [32]. The sections below will focus on the applicability of these methods. 2.1.2 Subsonic flow and compressiblity The fluid flow can be separated into five regimes due to free-stream Mach number and local Mach number as seen in Table 2.1. 8 2. Theory Flow regime Description Incompressible M∞ < 0.3 Subsonic M∞ < 1 and M < 1 Transonic case 1: M∞ < 1 and M > 1 locally case 2: M∞ > 1 and M < 1 locally Supersonic M∞ > 1 and M > 1 Hypersonic M∞ > 5 and M > 5 Table 2.1: Flow regimes for compressible flow [33]. Mach number is defined as in equation 2.6 M = V a (2.6) Where V is local velocity, a is the speed of sound as a function of flow density, temperature, and molar gas constant. 2.1.3 Reynolds number Free stream Reynolds number is defined as : Re = ρ∞V∞c µ∞ (2.7) Where c is the reference length, µ∞ is the freestream dynamic viscosity, ρ∞ is the freestream density, V∞ is the freestream velocity. Typically, the reference length for the airfoil is the chord for lifting bodies. Reynolds number symbolizes the ratio between the inertial forces and viscous forces. 2.2 Numerical Models Various methods exist to predict the aerodynamic performance of a wing, each with different levels of complexity and accuracy. It is crucial to choose a methodology that matches the required accuracy for a given phase in the development process. For example, during the conceptual phase, the selected methodology should keep up with the design changes and ensure accuracy. Most of the commercial finite volume codes are based on Navier-Stokes (N-S) equa- tions, providing the ability to analyze fully described fluid flow. However, using N-S based finite volume tools comes at a higher computational cost compared to low-fidelity methods. 9 2. Theory 2.2.1 Low-fidelity models Low-fidelity methods mentioned in this Section are based on Potential Flow theory. 2.2.1.1 Vortex sheet and horseshoe vortex The concept of a vortex sheet plays a big role in the analysis of flow around airfoils and finite wings. A vortex sheet contains an infinite number of straight vortex fila- ments with infinitesimally small strength. A vortex filament is a line with the same strength (Γ) along it, and the filament’s tangent vector is aligned with the direction of the local vorticity vector. According to Hermann von Helmholtz’s theorem, the circulation (Γ) of a vortex filament is constant through time, and the vortex fila- ment should end up at a solid surface or in a closed form. Therefore, Helmholtz’s theory suggests that there will be the presence of tip vortices and induced drag in a three-dimensional fluid flow, but in a two-dimensional fluid flow, there will be no tip vortices and no induced drag [30, 32, 34]. A vortex filament is fixed in location and experiences a force due to the Kutta- Joukowski theorem (bounded vortex). For this reason, Prandtl modified a wing with a bounded vortex, where the length of the vortex is equal to the span. However, the vortex filament should be continued with two free vortices that move fluid elements through the flow. The shape that contains a bounded vortex and two free vortices is named a horseshoe vortex, as seen in Figure 2.2. Figure 2.2: Replacement of finite wing with horseshoe vortex 2.2.1.2 Lifting Line theory Lifting Line Theory (LLT) is a three-dimensional application of potential flow theory, incorporating multiple horseshoe vortices to predict the spanwise lift distribution across the wing as seen in Figure 2.3. The circulation is modeled as a velocity potential in the horseshoe vortices using the Biot-Savart law [30]. The strength of the trailing vortex is directly proportional to the variation in circulation along the lift line [30]. The trailing edge vortices create downwash which reduces circulation in the spanwise direction, resulting in a non-uniform spanwise lift distribution as circulation must be conserved along the wing [31]. 10 2. Theory Figure 2.3: Superposition of a finite number of horseshoe vortices along the lifting line. Source: [30] 2.2.1.3 Vortex Lattice Method (VLM) In VLM, the geometry is discretized with quadratic panels by dividing it into finite spanwise and chordwise elements. Each panel has a ring or horseshoe vortex that enables analysis of the camber, sweep, and dihedral of the wing, which is not possible with LLT as the wing is modeled as a straight lift line [31]. Additionally, VLM allows for the calculation of moment coefficients by determining the pressure distribution over the wing [30]. The panels in VLM are placed on the mean chamber line, which means that if the static pressure distributions on the suction and pressure side are of interest, VLM cannot capture this effect. The wake is created using only chordwise vortices, as the spanwise vortices are not effective downstream through the trailing edge [30]. The strength of the trailing vortices is defined by the trailing edge vortices and remains constant in the longitudinal direction [30]. The total lift force is calculated by adding up the lift force in each panel. The main limitations of VLM are that trailing vortices should not intersect with other vortices, and panels should not intersect with each other. AVL [26] and OpenVSP [35] are some of the commercial tools that use the VLM method [32]. Trefftz plane analysis is an alternative method for calculating induced drag, which provides more accurate results than the wake integral method [26]. Trefftz plane analysis calculates induced drag by tracing the wake from far downstream while integrating kinetic energy. More information and details about Trefftz plane analysis can be found in [26, 36, 37]. AVL uses Trefftz plane analysis to calculate induced drag as an alternative method. Moreover, AVL uses slender body theory, which enables the modelling of the fuse- lage, unlike other commercial VLM solvers. Further details about slender body theory are not introduced in this chapter, as the point of interest of this thesis is the Wing and Wing-strut combination. More details about slender body theory can be found in [37, 38]. 11 2. Theory 2.2.1.4 3D Panel Method The 3D panel method is an advanced form of VLM that covers the entire geometry (including top and bottom surfaces) with panels, enabling it to capture thickness changes through the chord in airfoils. In thicker airfoils, thickness across the chord and camber have a significant effect on the static pressure distribution, which can be captured in panel methods but not in VLM [31]. Additionally, the panel method allows the modelling of the fuselage, which also generates lift. However, as with VLM, the panels in panel methods must not coincide [32]. 2.2.1.5 Viscous Drag Prediction with XFOIL XFOIL is a popular well-known tool for analyzing airfoil characteristics, particularly for low Reynolds number flows. It utilizes the panel method coupled with a bound- ary layer subroutine with transition prediction [39]. The software incorporates 2D Boundary Layer Integral equations as a transition model, which can capture tur- bulence and laminar separation bubbles [40]. This enables XFOIL to calculate the pressure distribution over the airfoil [40]. In addition, XFOIL can estimate limited trailing edge flow separation and predict the maximum lift coefficient [25]. The soft- ware also separates skin friction drag and pressure drag contributions for a given airfoil [25]. 2.2.1.6 Summary of low-fidelity models The VLM method has a huge advantage in conceptual design due to its low compu- tational cost and relative ease of adapting to geometry changes compared to CAD and high-fidelity CFD. These two factors make VLM suitable for optimization in the conceptual design phase. However, the VLM method cannot model viscous effects, which is its main limitation. As a result, VLM tools cannot capture bound- ary layer effects and wing stall. In AVL, the VLM method can be extended with skin friction correlations and form factors for viscous drag. However, this extension still cannot fully capture the effect of boundary layer and viscous drag. To address this limitation, the extension can be made with interpolation of viscous drag from two-dimensional flow codes such as XFOIL. 2.2.2 High-fidelity models In this section, high-fidelity models refer to those based on the numerical solution of Navier-Stokes equations in discretized form using computational mesh. These mod- els are capable of capturing complex flow phenomena such as turbulence, boundary layer effects, and flow separation, but they are computationally expensive and re- quire significant computational resources. 2.2.2.1 Finite Volume Method (FVM) The governing equations shown in Section 2.1.1 are discretized with Finite Volume Method (FVM). Moreover, differencing scheme is utilized to calculate the convective and diffusive effects that result from neighboring cells for the cell that is considered 12 2. Theory in the computation. Furthermore, a pressure-velocity coupling is needed as the pressure term is only found in momentum equation but velocity is found in both of the equations. 2.3 Geometric Parameters As stated in Chapter 1, the wing-strut design is the core of this thesis, as all the parametric studies were carried out for a particular setup of both surfaces. In this section, the relevant concepts and parameters will be covered and briefly explained to the reader. 2.3.1 Airfoil The most relevant geometric parameters for this surface will be briefly introduced to the reader, as explained by [39, 41, 43]. Figure 2.4 depicts an NACA6409 airfoil, exhibiting the specified parameters. Figure 2.4: Airfoil geometric parameters. • Leading Edge: is the foremost point of an airfoil. It serves as the reference point for the coordinate system used to define the airfoil geometry. • Trailing Edge: is the after-most point of an airfoil. • Chord (chordline): it is commonly defined as the shortest distance between the leading edge and the trailing edge. The chord is known as the length of the chord line Can be seen as in Figure 2.4, and as ’c’ in Figure 2.5. • Thickness (maximum thickness): defined as the distance between the lower and upper surface perpendicular to the chordline of the airfoil. Specif- ically, the thickness of the airfoil is commonly defined as its maximum thick- ness, and it is accompanied by the corresponding location along the chordline denoted as tmax. • Camber (mean line): determined by calculating the average of the lower and upper surfaces coordinates along the thickness axis at each point along the 13 2. Theory chordline. The camber is defined as the maximum computed value of the mean line, and similar to the thickness, its location along the chordline is commonly indicated and defined as cammax. 2.3.2 Wing Planform In conceptual aircraft design, four main parameters should be estimated in the earlier stages to size up the aircraft; takeoff weight (MTOW), Thrust, reference area, and aerodynamic efficiency [41]. The wing is the main driver of aerodynamic efficiency, as it is usually the only or the main lifting surface of the aircraft. The aerodynamic properties of the wing are defined by its geometry. This geometry is typically defined as joining two different shapes; the airfoil and the planform [39]. The wing has a large number of parameters, as throughout history the design com- plexity has been considerably increased in order to achieve higher aerodynamic ef- ficiencies for specific purposes. Therefore, the need to establish several parameters was raised looking forward to addressing, identifying, defining, and estimating the effects in the aircraft design stages. The information given in this subsection was summarized in concordance with the objective, scope and theoretical background of the project, which are defined in subsections 1.2, 1.3, and 2.1.2 respectively. 2.3.2.1 Planform parameters • Span: it is defined as the measured distance perpendicular to the flight di- rection of the aircraft between both tips. Can be seen as ’b’ in the figure 2.5. Figure 2.5: Wide-body aircraft scheme with basic wing parameters labeled. • Reference area: this parameter is often referred in the literature as one of the initial drivers for the conceptual design stage [41], as it is the base parameter used to define important aerodynamic properties such as Lift, Drag and moment coefficients [39]. The reference area is defined as the sum of the area of the simplified surfaces that compose the wing, including a virtual section which is made by extending the geometries of the left and the right 14 2. Theory sections that are joined with the fuselage as shown in Figure 2.5. Alternatively, it can be defined in its integral form as [39, Equation 2.8]. Sref = ˆ b/2 −b/2 c(y)dy (2.8) • Aspect ratio: it is used to infer conceptual design parameters such as in- duced drag, maneuverability, structural weight, flutter speed, etc [39, 41]. It is defined as the ratio of the squared span to the reference area as in [39, Equation 2.9]. AR = b2 Sref (2.9) • Taper ratio: denoted as the ratio between the tip chord and the root chord, plays a significant role in aerodynamic design. It is mathematically expressed as shown in [41, Equation 2.10]. Figure 2.10 in Section 2.3.2.2 depicts a ta- pered wing planform. The taper ratio is commonly employed to define the lift distribution characteristics during the conceptual design phase of an aircraft. This is partially shown in Figure 2.6. However, it is crucial to analyze and consider its behavior in conjunction with the sweep angle, as emphasized in references such as [39, 41]. Figure 2.6: Taper ratio effects in lift distribution. Source: adapted from [39] λ = ctip croot (2.10) • Wing sweep: it is defined as the angle formed by the line perpendicular to the airflow and the leading edge. However, in the industry, the latter is replaced by the quarter chord line. Both definitions of this property can be seen in figure 2.7. A sweep angle different than 0 affects negatively weight, 15 2. Theory lift and the control surfaces properties [41]. Nonetheless, it is required for high-speed flights starting in the subsonic regime to mitigate adverse effects from shock generation, and is useful to correct CG displacement of an aircraft [39]. Figure 2.7: Wing sweep in planform. • Dihedral angle: it is defined as the angle formed between the surface of the wing and the horizontal plane. It is usually considered positive if the tip is located higher than the root and is named ’dihedral’. Alternatively, if the tip is located in a lower position than the root the angle is considered negative and named ’anhedral’ [41, 39]. It is a crucial property influencing the stability behavior of aircraft, and is intricately tied to the vertical position of the wing, as its value is determined based on this parameter [39] Figure 2.8: Wing with positive dihedral angle, front view. • Incidence angle: used to establish the lowest drag configuration for the aircraft, typically for cruise conditions [39]. It is defined as the difference between root wing chord pitch angle, and the longitudinal axis of the fuselage. [41]. • Twist: defined as the difference of the local incidence angles of the wing’s surface. Typically defined between the tip and the root sections. Frequently used to prevent the stall of the wing area close to the tip, and its consequences to the maneuverability capabilities of the aircraft [41]. It could be used to slightly modify the lift distribution if required [39]. If the angle of attack of the root is higher than the tip is called ’washout’. Alternatively, if the angle of attack of the tip is higher than the one of the root is called ’washin’ [39, 41]. 2.3.2.2 Planform types In this subsection, a typical set of wing planforms will be introduced to the reader, alongside the equations for the most relevant properties. 16 2. Theory • Constant chord: for this planform type the most relevant geometric prop- erties are defined in the set of equations in [39, Equation 2.11], and the geo- metrical layout can be seen in figure 2.9. Figure 2.9: Constant chord planform S =b · croot AR = b croot cMGC =croot (2.11) • Trapezoidal-straight: for this planform type the most relevant geometric properties are defined in the set of equations in [39, Equation 2.12], and the geometrical layout can be seen in figure 2.10. Figure 2.10: Trapezoidal-straight planform S = b 2 · (croot + ctip) AR = 2b (croot + ctip) cMGC =2 3croot 1 + λ + λ2 1 + λ (2.12) • Elliptical: for this planform type the most relevant geometric properties are defined in the set of equations in [39, Equation 2.13], and the geometrical layout can be seen in figure 2.11. 17 2. Theory Figure 2.11: Elliptic planform S =π 4 · b · croot AR = 4b π · croot cMGC =0.9055croot (2.13) By adjusting variables such as aspect ratio, taper ratio, and twist, the spanwise lift distribution can be approximated as an elliptical wing, which can help to reduce induced drag. The Oswald efficiency factor can be used as a metric for optimizing the geometry of the wing to minimize induced drag. 2.3.3 Strut and strut braced wing The idea behind the strut braced wing (SBW) is to have additional airfoil-shaped trusses for supporting the Wing. This configuration comes with coupled aerody- namic effects and weight effects. Figure 2.12: Strut braced wing One of the advantages of the strut-braced wing is the relief of bending moment on the wing. When the bending moment is reduced, the weight of the wing can be decreased since lighter reinforcements can be used. Additionally, the strut-braced wing enables a higher aspect ratio and span which increases aerodynamic efficiency (refer to Section 2.4.5) and lowers induced drag. These benefits lead to lighter aircraft and lower fuel consumption or longer range for electric aircraft [2]. 18 2. Theory On the other hand, the strut-braced wing has disadvantages. Although the strut reduces the weight of the wing, the strut’s own weight is significant. If the decrease in weight of the wing is lower than the strut’s own, the aircraft will be heavier. If the dihedral angle of the strut decreases, the bending moment relief in the wing will increase, but as the length of the strut increases, the weight of the strut will also increase. Moreover, the strut causes a drag penalty due to its existence, which can be optimized with lift contributions of the wing and strut as well as their position. If the dihedral angle decreases, the skin friction drag contribution from the strut increases as the length of the strut increases. For these reasons, the position of the strut has significant importance in the design. Furthermore, the strut creates a negative pressure on the lower surface of the wing as the airflow accelerates between strut and wing which creates a Laval nozzle effect. Previous studies show this effect could create a significant negative effect on Lift [42]. In summary, strut braced wing designs have a coupled effect on aircraft design, which can be both advantageous and disadvantageous. The additional strut structure adds complexity and weight to the aircraft, potentially creating an overall disadvantage. However, on the other hand, the strut-braced wing design has the potential to significantly improve aerodynamic efficiency, allowing for weight reduction through optimized design. Therefore, a thorough investigation and optimization of strut- braced wing designs could lead to a substantial improvement in aircraft performance and design methodology. 2.4 Aerodynamics This section provides an overview of the fundamental aerodynamic principles related to wing design. Subsection 2.4.1 covers basic airfoil concepts, including the physics based on inviscid and incompressible flow assumptions. Subsection 2.4.2 introduces the basic concepts for aerodynamic forces. Subsections 2.4.3 and 2.4.4 define the aerodynamic Lift and Drag forces for 3D bodies. The contents of this section are limited to the scope of the project as detailed in Subsection 1.3. 2.4.1 Airfoil The airfoil plays a critical role in aeronautics, as it is the core concept used to define the cross-section of the lifting, and control surfaces [41]. The primary function of an airfoil in a lifting surface is to generate lift with the lowest associated drag [43]. The pressure distribution is created due to the airfoil’s shape. This can be explained by the concept of a streamline, which refers to the path that a massless particle follows in a region of interest [44]. The airfoil’s shape creates a physical constraint that alters the streamlines of the incoming airflow compared to the freestream direction [34]. As a result, a difference in velocity occurs in the airflow that follows these streamlines on both surfaces of the airfoil, which generates a static pressure difference [31]. The direction of the incoming airflow, or freestream direction, has a significant 19 2. Theory impact on the airfoil’s pressure distribution. When the angle between the freestream direction and the chord line of the airfoil changes, the streamlines are considerably modified, as illustrated in Figures 2.13 and 2.14. This angle is known as the angle of attack and is a key parameter for aerodynamic coefficients [37, 39, 41, 43]. In addition to the angle of attack, the freestream velocity also plays a crucial role in determining the airfoil’s pressure distribution, as it drives the magnitude of the pressure on both surfaces of the airfoil according to Bernoulli’s principle applied to streamlines [41]. The Bernoulli equation for low-speed flows is shown in [37, Equation 2.14], and the Bernoulli equation for every point on a streamline for low- speed flows is given in [37, Equation 2.15]. Figure 2.13: Streamlines scheme in a NACA 6409 airfoil with an angle of attack of 0 degrees. Figure 2.14: Streamlines scheme in a NACA 6409 airfoil with an angle of attack of 15 degrees. p1 + 1 2ρv2 1 + ρgz1 = p2 + 1 2ρv2 2 + ρgz2 (2.14) p∞ + 1 2ρv2 ∞ = constant (2.15) 2.4.2 Aerodynamic forces The flow around the body gives rise to two distinct types of forces: the pressure force and the skin friction force. The pressure force is a consequence of the variations in fluid pressure around the body, while the skin friction force is attributed to the adherence of the flow to the body surface due to the no-slip condition, or shear stress. This can be seen in Figure 2.15. Figure 2.15: Pressure and shear stress in an airfoil. Source: adapted from [30] These forces can be mathematically expressed as presented in [37, Equation 2.16]. The resulting forces from this interaction can be defined as the integration of both sources over the body surface [30]. These equations take into account the geometry 20 2. Theory of the body, the properties of the air, and the velocity of the air mass relative to the body. Rtotal = Rpressure + Rfriction (2.16) Typically, the resulting aerodynamic force (R) exerted on a body is concentrated at a specific location known as the center of pressure (CoP). This force can be divided into two distinct sets, each comprising two components. The first set separates the resulting force into a normal force (N), which acts perpendicular to the chord, and an axial force (A), which acts parallel to the chord. The second set represents the resulting force as the lift force (L), which acts perpendicular to the freestream direction, and the drag force (D), which acts parallel to the freestream direction. To establish a connection between the two sets, the angle of attack is defined as the angle formed between the freestream direction and the chord of the airfoil. Consequently, the angle of attack also corresponds to the angle between N and L, as well as A and D. This relation can be seen in [30, set of equations 2.17]. L = N cos α − A sin α D = N sin α + A cos α (2.17) However, in the aerospace industry, it is more customary to consider the resulting force to be located at the aerodynamic center (AC). In this representation, a pitch- ing moment (M) arises from the displacement of the resulting force. This concept has been defined in literature [39, 43]. Figure 2.16 depicts a free-body diagram il- lustrating the acting forces on the airfoil, with the forces positioned at the center of the AC. Figure 2.16: Airfoil free body diagram. Source: adapted from [30] To compute the resultant force acting on an airfoil, it is necessary to integrate the pressure and shear stress distributions along its surface. Figure 2.17 presents a schematic representation of the variables and parameters involved in determining the aerodynamic forces. The pressure distribution (p) and shear stress distribution (τ) are considered functions of the distance ’s’ measured from the leading edge (LE) to a specific point of interest (A) along the airfoil surface. Furthermore, an angle θ is defined to denote the orientation between the pressure vector and the shear stress vector with respect to lines perpendicular and parallel to the chord, respectively. 21 2. Theory This angle is employed to project the magnitudes of both pressure and shear stress in the directions of the normal and axial forces, thereby contributing to the overall resultant force computation [30]. Figure 2.17: Scheme for pressure and shear stress integration over an airfoil. Source: adapted from [30] The process of calculating the aerodynamic forces on an airfoil is commonly divided into two sets: the upper surface and the lower surface. For each set, the values of the normal force and axial force are determined for every point along the respective surface, starting from the leading edge and extending to the trailing edge. The mathematical expressions, presented in equations [30, set of equations 2.18 and 2.19], provide the per-unit-span formulation of these forces for each surface. dnu = −pudsu cos θ − τudsu sin θ dau = −pudsu sin θ + τudsu cos θ (2.18) dnl = pldsl cos θ − τldsl sin θ dal = pldsl sin θ + τldsl cos θ (2.19) Let dnu represent the elemental normal force acting on the upper surface, while pu and τu denote the elemental pressure and shear stress at point A on the upper surface. Additionally, dsu represents the elemental distance measured from the leading edge along the upper surface to point A. The same notation applies to the corresponding quantities for the lower surface. The total normal and axial forces per unit span can be determined by integrating the expressions given in sets of equations 2.18 and 2.19 from the leading edge (LE) to the trailing edge (TE). The integration yields the desired values of the normal and axial forces, as demonstrated in [30, set of equations 2.20]. Consequently, the lift and drag values per span unit are calculated using the set of equations 2.17. n = − ˆ TE LE (pu cos θ + τu sin θ) dsu + ˆ TE LE (pl cos θ − τl sin θ) dsl a = ˆ TE LE (−pu sin θ + τu cos θ) dsu + ˆ TE LE (pl sin θ + τl cos θ) dsl (2.20) Alternatively, the aerodynamic forces can be computed using the models presented in 2.2. 22 2. Theory 2.4.2.1 Aerodynamic coefficients in 2D surfaces Dimensionless aerodynamic coefficients are fundamental parameters that hold greater significance than the aerodynamic forces alone [30]. These coefficients provide a standardized representation of aerodynamic forces and can be expressed in relation to various factors, including the target force, airfoil chord, air density, freestream velocity, and angle of attack [39, 43]. The set of equations in [39, Equation 2.21] defines each resulting force and the moment in terms of the previously discussed parameters and properties. Figure 2.18 shows a free-body diagram of the acting forces of interest on the airfoil according to the most common representation. Figure 2.18: Airfoil free body diagram for 2D analysis r = 1 2ρ∞V 2 ∞cCr or Cr = 2r ρ∞V 2 ∞c l = 1 2ρ∞V 2 ∞cCl or Cl = 2l ρ∞V 2 ∞c d = 1 2ρ∞V 2 ∞cCd or Cd = 2d ρ∞V 2 ∞c m = 1 2ρ∞V 2 ∞c2Cm or Cm = 2m ρ∞V 2 ∞c2 (2.21) Where ρ∞ is freestream density, V∞ is freestream velocity or airspeed, S is reference area, c is chord, Cr is the dimensionless coefficient that relates the angle of attack to the resulting force, Cl is the dimensionless coefficient that relates the angle of attack to the lift force, Cd is the dimensionless coefficient that relates the angle of attack to the drag force, Cm is the dimensionless coefficient that relates the angle of attack to the pitching moment. To enhance simplicity and clarity, the set of equations could be reformulated by introducing the concept of dynamic pressure. Dynamic pressure, a fundamental property in aerodynamics, quantifies the impact of air motion on the airfoil and is mathematically expressed as follows: q∞ = 1 2ρ∞V 2 ∞ If it is within the interest of the reader to go deeper into the theoretical background 23 2. Theory of airfoils, it is recommended to look out for chapter 4 in [41], chapter 8 in [39] and chapter 5 in [43]. 2.4.2.2 Aerodynamic coefficients in 3D bodies In a manner analogous to the approach discussed in the previous section, the resul- tant forces acting on a three-dimensional (3D) body are defined, as illustrated in Figure 2.19. Furthermore, the resulting force can be decomposed into three com- ponents: the lift force, the section drag force, and the pitching moment. These components can be used to define the dimensionless aerodynamic coefficients, as shown in the set of equations in [39, Equation 2.22]. It is relevant to highlight that for 3D calculations the letter used to define the component is written in upper case, while for 2D calculations the letter used to address the component is written in lower case. Figure 2.19: Body free body diagram for a 3D analysis R = q∞SrefCR or CR = R q∞Sref L = q∞SrefCL or CL = L q∞Sref D = q∞SrefCD or CD = D q∞Sref M = q∞SrefcMGCCM or CM = M q∞SrefcMGC (2.22) Where q∞ is the dynamic pressure, Sref is the reference area, cMGC is the mean geometric chord, CL is the dimensionless coefficient for the lift force, CD is the dimensionless coefficient for the drag force, CM is the dimensionless coefficient for the pitching moment. The modelling of the dimensionless coefficients for each property can be as extensive and comprehensive as required. Basic models are commonly defined as functions of geometry, Reynolds number, Mach number, angle of attack, and angle of yaw. On the other hand, advanced models typically include the different components and bodies that contribute to the property of interest [39]. 24 2. Theory 2.4.3 Aerodynamic lift The lift force is defined as the component of the resulting force perpendicular to the freestream [30], which for a 3D body can be extended to a plane perpendicular to the freestream. As mentioned in section 2.3.1, the lift force is highly dependent on the angle of attack and the geometry of the airfoil, and on the planform shape whose effects are briefly discussed in subsection 2.3.2. Figure 2.20: Airfoil lift coefficient for uncambered airfoil and cambered airfoil for Reynolds number approximately 2.2 x 106. The relationship between the lift force and angle of attack is typically proportional, although it is subject to aerodynamic limitations that impose a limit on the amount of lift that can be generated [39]. The lift versus angle of attack curve can be divided into three sections. In the first section, which usually ranges from -7 to 9 degrees, the curve has a linear shape that can be extended based on the airfoil geometry [39]. In the second region, a non-linear behaviour appears between lift force and angle of attack, in this region the maximum lift is achieved and the value of angle of attack on such condition is achieved receives the name of stall angle of attack. The start of this region is typically referred to as α∗. This behaviour has a strong dependence on the Reynolds number [39, 41]. Beyond the stall angle of attack, the lift force rapidly decreases. This behaviour is evident in the lift curve for the ’NACA0012’ series, as shown in Figure 2.20. Furthermore, it is crucial to emphasize the considerable influence of airfoil camber on the generation of lift force. A cambered airfoil exhibits a translation effect in the lift force versus angle of attack (AoA) relationship, leading to enhanced lift force generation compared to a symmetrical airfoil at the same AoA. It is often expressed as CLϕ or αϕ according to the axis. This effect is attributed to the curved shape of the cambered airfoil, which results in favourable pressure distributions and improved aerodynamic performance [41]. Furthermore, it is important to highlight the significant role of the airfoil camber in lift force generation. A cambered airfoil generates a translation effect in the lift 25 2. Theory force vs. angle of attack curve, which can result in higher lift force generation at the same AoA when compared with a symmetrical airfoil [41]. 2.4.4 Aerodynamic drag breakdown In basic terms, drag force is defined as the component of the resulting force parallel to the freestream [30]. The accurate calculation of drag force is of paramount im- portance in aircraft design, as it affects the performance of the aircraft which leads to the need for changes and thereby, affects other areas and departments within an aircraft company. Additionally, drag modelling is one of the most complex tasks in aircraft design since it is difficult to predict accurately [39]. As a result, it is one of the main drivers or indicators of the feasibility of an aircraft project. As discussed in section 2.4, the sources for the resulting force are pressure and skin-friction. Consequently, the overall drag force can be further decomposed into various components. In this document, the drag breakdown scheme depicted in Figure 2.21 is adopted. It is important to note that this scheme does not consider the drag effects resulting from shock generation in supersonic flow regimes, nor does it account for additional drag effects that may arise when multiple distinct bodies are interconnected. Figure 2.21: Drag classification scheme Therefore, the total drag can be mathematically defined as in equation 2.23. Dtotal = Ddue to lift + Dparasite (2.23) Where: Ddue to lift = Dinduced + Ddue to lift,wave Dparasite = Dpressure + Dskin friction + Dinterference + Dparasite,wave However, according to the scope of this work, the total drag forces could be defined as in equation 2.24, as the aircraft will operate at a range M <= 0.32. Dtotal = Ddue to lift + Dpressure + Dskin friction + Dinterference (2.24) 26 2. Theory 2.4.4.1 Drag due to lift As it can be inferred from the name of the drag component, this class of drag is meant to gather all drag components associated with the generation of lift of any type of body. The most known and relevant drag component of this class is the "induced drag", which refers to the drag caused by the consequence of lift and the pressure difference on the tip of the Wing. The induced drag is proportional to the square of the lift force, according to the elemental derivation for the induced drag coefficient from lifting line theory which is briefly discussed in section 2.2.1.2. The induced drag mathematical expression that relates to the aforementioned concepts is given in [39, Equation 2.25]. CDi = C2 L π · AR · e (2.25) Where e is Oswald efficiency factor, CDi is the induced drag coefficient, CL is the lift coefficient, and AR is the aspect ratio. Additionally, induced drag is inversely proportional as well to the wing span, with the latter being the main driver to decrease the former, as shown in [45]. Nonetheless, the reference area of the wing is an important constraint for several stages of aircraft design, and varying the value of the wing span will cause a considerable change in the former. However, this effect could be prevented if the chord in the spanwise direction is modified to account for this constraint. Therefore, an additional property is often used to account for the induced drag reduction, aspect ratio [39, 41]. The wing tip shape plays a relevant role in the induced drag as the vortex generation starts in that zone, implementing different types of defined geometries in the wing tip is a common and effective strategy to achieve a relevant decrease in the induced drag [45]. The dihedral angle also has a proportional correlation with the induced drag, as the lift generated by the lifting body will decrease as the dihedral angle increase, but the drag will remain the same [45]. Drag reduction is intertwined with the body parameters and properties, for the wing-body such elements are discussed in section 2.3.2. 2.4.4.2 Parasite drag This drag class covers all the drag components that are not included in the ’drag due to lift’ class. In this section, the concepts of skin-friction, pressure, interference and wave drag will be briefly discussed. 1. Skin-friction drag: This drag component arises from the effect of viscos- ity in the flow. It manifests as a tangential force exerted on the surface of a body when a fluid passes over it, as the molecular interactions within the fluid impede the relative motion between its molecules, resulting in shear stresses [45]. The magnitude of skin-friction drag is closely associated with the surface roughness of the body. Specifically, it is directly proportional to the rough- ness regardless of the flow regime. However, it is worth noting that surface 27 2. Theory roughness can have advantageous effects in certain scenarios, as it can delay flow separation by promoting the transition of the boundary layer from lam- inar to turbulent, thereby reducing drag [46]. The skin-friction drag shows different behaviours according to the fluid flow regime, as well as the value of the Reynolds number. However, it is inversely proportional to the Reynolds number. The skin friction force and skin friction drag coefficient are given in [39, Equations 2.26 and 2.27]. Dskin friction = 1 2ρ∞V 2 ∞CfSref (2.26) CDf = 2Dskin friction ρ∞V 2 ∞Sref = Cf ( Swet Sref ) (2.27) Where CDf is the skin friction drag coefficient, Cf is the skin friction coefficient, Df is the skin friction drag force, ρ∞ is the air density, V∞ is the far-field airspeed, Swet is the wetted area. 2. Pressure drag (form drag): the origin of this drag type is due to the pres- sure distribution normal to the body surface [45]. The shape of the body surface dictates the magnitude and the rate of change of the pressure gradient alongside the axis parallel to the fluid flow according to Bernoulli’s principle. To illustrate the relationship between pressure and geometry, an airfoil can be considered with a flow at an angle of attack (AoA) of 0 degrees. In this scenario, the pressure gradient from the stagnation point to the maximum thickness region would exhibit a negative value. Conversely, beyond the max- imum thickness point, the pressure gradient becomes positive, leading to a deceleration of the flow. This positive pressure gradient is commonly referred to as an adverse pressure gradient. A visual representation of this pressure distribution can be observed in Figure 2.22. In such a scenario, flow separation occurs as the momentum of the decelerated fluid in the boundary layer is too low in order to move against the adverse pressure gradient. The separation of the boundary layer gives rise to a shear layer at the detachment point, leading to the formation of a turbulent wake behind the maximum thickness region (or in close proximity to it). In this wake, the pressure is expected to be equivalent to the local pressure within the boundary layer at the point of separation. The difference between the pressure acting in the front and rear zone of the body would be highly related to the pressure drag. In the case of flows characterized by high Reynolds numbers, the transition of the boundary layer from a lami- nar to a turbulent regime is expected to occur prior to reaching the maximum thickness point in the geometric configuration. However, the exact location of this transition point is influenced by factors such as the pressure distribution along the body geometry and the angle of attack. Hence, the momentum and energy of the boundary layer will be higher due to the turbulent mixing which would delay the separation when compared to the laminar flow regime [46]. 28 2. Theory Figure 2.22: Pressure distribution in an airfoil. Source: XFOIL [25] For lifting bodies as a wing, the angle of attack (detailed in section 2.3.2) would vary according to the mission profile of the aircraft. Therefore, the contribution of the pressure drag to the total drag will be increased when compared to a 0 angle of attack scenario [34], as the thickness of the boundary layer is proportional to the AoA. 3. Interference drag: arises when two or more bodies are in close proximity, connected, or even intersecting. The interaction between these bodies leads to a combined drag that exceeds the sum of the individual drag values for each body. This phenomenon can be mathematically represented by [45, Equation 2.28]. In cases where one body intersects another, the boundary layers of both bodies merge. The resulting interference drag depends on the thickness of the boundary layer of the larger body and the thickness-to-chord ratio (t/c) of the smaller body. Notably for this case, experimental data cited by [45] indicates that the interference drag can even become negative when the t/c ratio is below 8%. Interference drag is strongly influenced by the generation of lift, especially when one of the bodies involved acts as a lifting body. This correlation stems from the interaction between the pressure gradient and its effects on the upper and lower surfaces at the junction. Importantly, the interference drag exhibits a direct proportionality to the square of the lift coefficient [45]. When considering a body functioning as a strut, the effects of its orientation can be observed in both the spanwise axis (tilting the strut in the wing direction) and the longitudinal axis (tilting the strut in the direction of flow). Specifically, increasing the angle in the longitudinal axis has the effect of reducing interference drag. Conversely, increasing the angle in the spanwise axis leads to an increase in interference drag [45]. However, there is an exception for this case, two bodies that are near each other and one ahead in the flow direction. The drag versus the distance between them will have three different regions. In the first region, the distance between the bodies is inversely proportional to the drag, thus the combined drag will be 29 2. Theory lower than the sum of the individual values. In the second region, the distance between the bodies is now proportional to the drag, hence the drag value will increase. In the third region, the distance between the bodies will be too large and then the total drag would be equal to the sum of the individual drag values. The layout of the regions will be dependent on the Reynolds number and the geometry of the bodies [45]. Dinterference = Djoint bodies − #bodies∑ n=1 Dn (2.28) 4. Wave drag: this drag type originates from the generation of shock waves in the aircraft bodies due to local flow velocities greater than the local speed of sound (Mlocal>1). Shocks generate an abrupt rise in pressure and other flow properties, this receives the name of compressibility effects by some authors. Thus, wave drag is a form of pressure drag for a particular flow regime. For transonic and supersonic wing design, a parameter named ’critical mach num- ber’ is used to refer to the aircraft airspeed on which the compressibility effects start to appear for a given airfoil [39]. 2.4.5 Efficiency in lifting bodies In the field of aircraft design, the notion of "efficiency" plays a crucial role in quanti- fying the relationship between lift and drag forces or coefficients, depending on the context. This concept serves as a means to compare and evaluate various aerody- namic geometries. Three key parameters that contribute to this evaluation will be discussed in the following sections: aerodynamic efficiency, drag polar, and Oswald factor. • Aerodynamic efficiency: an indirect comparison between lift and drag. It is defined as the ratio of the former divided by the latter. Typically plotted versus the angle of attack. The aerodynamic coefficient (L/D) is important as it is an efficiency metric adopted to compare different aerodynamic geometries, such as airfoils, wings, lifting body, among others. In figure 2.23 can be observed the aerodynamic efficiency for two different airfoils. • Drag polar: it can be considered as a direct comparison between lift and drag in a graphical form. It is typically plotted as Cl (2D) or CL (3D) in the vertical axis, and Cd (2D) or CD (3D) in the horizontal axis. In figure 2.24 can be observed the drag polar for two different airfoils. • Oswald efficiency: is a factor that measures aircraft efficiency in producing lift. It relates to how the geometry behaves in terms of induced drag. There- fore, if the Oswald factor has a higher value, the induced drag is lower. This can be observed in [39, Equation 2.29]. e = C2 L π · AR · CDi (2.29) 30 2. Theory Figure 2.23: Aerodynamic efficiency for two different airfoils Figure 2.24: Drag polar for two dif- ferent airfoils 2.4.5.1 Lift distribution and Wing types The distribution of lift over the wing span plays a crucial role in the level of induced drag. The elliptical lift distribution is known to yield the minimum amount of induced drag, as it maintains a constant downwash across the span of the wing. This results in a lower level of induced drag compared to other wing planforms. The Oswald efficiency factor is a measure of the efficiency of a wing’s planforms in terms of induced drag, and the elliptical wing has the highest Oswald efficiency factor without ant wing-tip devices. Higher Oswald efficiency factors could be obtained with wing tip devices. However, due to its high manufacturing cost, the tapered wing is commonly used as an approximation to the spanwise lift distribution of the elliptical wing [30]. 2.5 Optimization In this section, an outline for optimization will be discussed. It would be followed by a brief review of the optimization types in terms of objective functions and constraints. A special focus on the gradient-based, and stochastic optimization techniques is given as well. Optimization, also known as mathematical programming, encompasses the theoret- ical background and methodologies used to reach the optimal solution according to a required goal and given constraints [47, 48, 49]. It involves a combination of an- alytical and numerical methods, including algorithms used to calculate or compute the solutions. Mathematical structure and properties are used to define and model the problem in order to solve it [50]. Optimization is classified based on various perspectives. The most common ones are; regarding the continuity of the domain (continuous or discrete), the order of the objective and constraint functions (linear and non-linear), and the differentiable na- ture of the objective and constraint functions (differentiable and non-differentiable), among others [48]. However, for the scope of this master’s thesis, the classification 31 2. Theory will be based on how much randomness can be observed. Therefore, two types can be identified; deterministic and stochastic [49]. 2.5.1 Modelling structure The optimization problems are typically defined using the following structure ac- cording to [49]: 1. Decision variables: input variables for the optimization model. Usually defined as x. 2. Objective function: the objective function is a fundamental component of the optimization scheme, defining the relationship between the decision variables and the optimization goal, which is typically to minimize or maximize a certain value. It is denoted as f(x), where x represents the decision variables. In certain cases, when a function cannot be directly formulated, an equation involving the decision variables is employed as a surrogate representation of the objective function. 3. Constraints: function or set of functions that must be satisfied in order for a solution to be considered feasible. Constraints are typically expressed as equalities or inequalities in terms of the decision variables of the problem. Constraints can be categorized as soft or hard, with the former allowing for solutions that do not fully satisfy the constraint while penalizing the objective function, and the latter enforcing the constraint strictly to label a solution as feasible. A common way to express an optimization problem is given: minimize (min) f(x) with respect to (w.r.t.) x = [x1, x2, ..., xn] subject to (s.t.) gi(x) ≤ 0, i = 1, . . . , m. hj(x) = 0, j = 1, . . . , m. Where f(x) is the objective function, x is the decision variables vector, gi(x) and hj(x) are functions used to define inequality and equality constraints respectively. In the field of optimization theory, the notions of locality and globality play a crucial role in determining the best solutions for a given problem. These concepts revolve around how objective functions and decision variables behave within a specific do- main or range. By examining the local and global properties of these functions and variables, insights can be gained into the nature of the problem and effective strate- gies can be devised to find the most favorable solutions. A brief introduction of the most relevant concepts is summarized as follows: 1. Minimum: point of the feasible set that has the lowest value of a given function. 2. Maximum: point of the feasible set that has the highest value of a given function. 32 2. Theory 3. Local minimum/maximum: point of the feasible set that has the lowest- /highest value for a subset of possible solutions of the given function. 4. Global minimum/maximum: point of the feasible set that has the lowest- /highest value for the whole set of possible solutions of the given function. Figure 2.25: Local and global minima and maxima plotted in a function. Where; light blue dots rep