Mar 3, 2010 - avhandling har sitt fokus pådessa verktyg fö flygplanens ...... It would follow basically the same form as the conceptual design loop in figure 2, but without ...... Vortex filaments arranged on a straight line between points A and B, induces .... The free stream is represented by a uniform magnetic field, such as it.
Using Internet Interactions in Developing Vortex Lattice Software for Conceptual Design
Tomas Melin Department of Aeronautical and Vehicle Engineering Kungliga Tekniska Hogskolan ¨ SE-100 44 Stockholm, Sweden
Report: TRITA AVE 2003:12 ISSN 1651-7660
You can design and create, and build the most wonderful place in the world. But it takes people to make the dream a reality. Walt Disney.
Typsatt i LATEX med FLYG:s thesis-stil.
Akademisk avhandling som med tillstånd av Kungliga Tekniska Hogskolan i ¨ Stockholm framl¨agges till offentlig granskning for ¨ avl¨aggande av teknologie licentiatexamen tisdagen den 10:e juni 2003 kl 10:00 i rum S40, Teknikringen 8, Kungliga Tekniska Hogskolan, Stockholm. ¨ c
Tomas Melin 2002 Universitetsservice US AB, Stockholm 2003
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Acknowledgements The last two years have been a journey through the exiting world of graduate studies. When I set out on this expedition I had some initial guesses about the path ahead, but without the help and support of many people I would have got stuck in the shrubberies. Many thanks to my supervisor Arthur Rizzi, who has been my compass through the uncharted territories by guiding me around treacherous grounds. Many warm thanks to my: Fellow graduate students Yann & Stefan, on quests of their own but always ready to lend a hand. Thanks guys. To the other graduate students beyond the departmental bulkheads, for their help, assistance and friendly chats in corridors and coffee room. To my family: Mom n’ Pops, for their unconditional support and my sister for providing news from the real world. To my friends Daniel, Martin, Johan, Sara, Maria and the rest for their support, and proving that even graduate students can have a social live (at least on the side ). Språkkonsult Åsa Lindh, without whom my literal skills would not appear as such. To the voice from the Internet and smoke-filled dark rooms, Nella (aka Calyx). Who’s encouraging words helped me in times of shortage of Ommpfh . Åsa, for the only really important reason.
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Abstract Conceptual design is the process used by aircraft manufacturing companies to perform a rapid assessment of new ideas and concepts. In order to do so, they need an array of simplistic tools that can produce results quickly. The focus of this thesis is these tools of aircraft aerodynamic characteristics. In this thesis the role, and importance of predictive methods in conceptual design is shown. A discussion of the available design tools and design methodologies in conceptual design is covered. The fusion of disciplines in aeronautic design, aerodynamic, structural mechanic and power plant considerations are imposed on integrated design environments, a new class of design softwares, aimed at covering most factors in conceptual design. Together with the mathematical background of the ideal flow, the vortex lattice method has been explained. With the standard vortex lattice method explored, a description of the available method variations and the capabilities of the available softwares was explored. The Tornado vortex lattice method is discussed, together with a discussion of the importance of user feedback on software quality. The interaction between different design groups has become more integrated and more fast paced than before. Finally, some recommendations on different software licensing schemes are discussed, in particular the GNU general public licence.
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Sammanfattning Konceptuell design a¨ r en process som anv¨ands av flygplanstillverkare for ¨ att gora ¨ en snabb uppskattning av ideer och koncept. For ¨ att kunna gora ¨ detta behovs ¨ en upps¨attning enkla verktyg som ger resultat snabbt. Denna avhandling har sitt fokus pådessa verktyg fo¨ flygplanens aerodynamiska karakteristik. I denna avhandling påvisas vikten av prediktiva metoder inom konceptuell flygplans konstruktion. En diskussion fors ¨ om de tillg¨angliga konstruktionsverktygen och konstruktionsmetodologierna. Sammansm¨altningen av de olika disciplinerna flygteknisk konstruktion, aerodynamik, strukturmekanik och framdrivningsmaskineri diskuteras som forgrund till integrerade konstruktionsmilj¨oer, som a¨ r en ny klass av ¨ konstruktionsmjukvaror, avsedda att ta h¨ansyn till många faktorer inom konceptuell konstruktion. Tillsammans med en bakgrund av idealt flode, ¨ forklaras virvelgittermetoden. ¨ I och med utforskningen av standardformen av virvelgittermetoden, beskrivs de tillg¨angliga metodvariationerna och mojligheterna hos de tillg¨angliga ¨ mjukvarorna utvecklas. Interaktionen mellan olika designgrupper har blivit mer integrerad och fått ett snabbare tempo a¨ n tidigare. Slutgiltligen gors ¨ några rekommendationer angående olika stt att licensiera mjukvara, och speciellt GNUs generella publika licens.
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Symbols Symbol α A a β c c c CD CD0 CDG CL CLG CLmax
Quantity Angle of attack Area Lift slope Angle Chord Specific fuel consumption Constant Drag Coefficient Parasite Drag Coefficient Drag coefficient in ground effect Lift Coefficient Lift coefficient in ground effect Maximum lift coefficient
Dimension rad l2 rad−1 rad l −1 l Undetermined − − − − − −
CLα Cl
Liftslope Local Lift Coefficient
rad−1 −
Cmα Cp δ e η F Γ g k K µr m˙ f uel n n φ P
Pitching moment α derivative Pressure coefficient Small increment Ellipse factor Efficiency Force Circulation or vortex strength Gravitational constant Constant Constant Runway friction coefficient Fuel consumption Normal Subscript Velocity potential Power
rad−1 − Undetermined − −
q
Dynamic pressure
ml t22 l s l t2
Undetermined − − m/t l l2 t ml2 t3 m t2 l
Definition ∂CL ∂α
Length of airfoil m˙ f uel g P D qS D0 qS D qS L qS L qS Lmax qS ∂CL ∂α L qc ∂Cm ∂α p−p∞ q∞
Xreal Xideal
F⊥ Fk
2 ρV∞ 2
viii Symbol ρ R r r S s sr θ t U U∞ u V Vr W W0 W1 w X x y z
Tomas Melin Quantity Air density Range Coordinate, cylindrical Radius Reference area Length Runway length Coordinate, cylindrical Time Velocity Farfield velocity Velocity Volume Rotation speed Weight Empty weight Take off weight Induced velocity Parameter Coordinate, Cartesian Coordinate, Cartesian Coordinate, Cartesian and cylindrical
Dimension m l3
l l l l2 l l l t l t l t l t 3
l
l t
m m m l t
Undetermined l l l
Definition Distance travelled
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Contents Acknowledgements
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Abstract
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Sammanfattning
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Symbols 1.
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Introduction 1.1. Thesis overview . . . . . . . . . . . 1.2. Keyword introduction . . . . . . . . 1.3. The basic message of this thesis . . . 1.4. Historical overview of Aeronautics
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Aircraft Conceptual Design 2.1. Virtual prototyping . . . . . . . . . . . . . . . . . . . . . . . 2.2. The Design Process . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Conceptual design . . . . . . . . . . . . . . . . . . . 2.2.2. The design cycle . . . . . . . . . . . . . . . . . . . . 2.2.3. The pre-design loop . . . . . . . . . . . . . . . . . . 2.2.4. Parameter-Characteristic-Performance . . . . . . . 2.2.5. Design Optimization . . . . . . . . . . . . . . . . . 2.2.6. The output of conceptual design . . . . . . . . . . . 2.2.7. The other design phases . . . . . . . . . . . . . . . . 2.2.8. Overall design, from specification to rollout . . . . 2.3. The tasks of aerodynamics in conceptual aircraft design . 2.4. Aircraft Characteristics . . . . . . . . . . . . . . . . . . . . 2.4.1. Aerodynamic characteristics . . . . . . . . . . . . . 2.4.2. Structural characteristics . . . . . . . . . . . . . . . 2.4.3. Propulsive characteristics . . . . . . . . . . . . . . . 2.5. Aircraft Performance . . . . . . . . . . . . . . . . . . . . . . 2.5.1. Operational considerations . . . . . . . . . . . . . . 2.5.2. Handling qualities . . . . . . . . . . . . . . . . . . . 2.6. Design requirements . . . . . . . . . . . . . . . . . . . . . . 2.6.1. Customer requirements . . . . . . . . . . . . . . . . 2.6.2. Government requirements . . . . . . . . . . . . . . 2.6.3. Company requirements and technology constraints 2.7. Tools of the trade . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1. Engineering ”know-how”: . . . . . . . . . . . . . . 2.7.2. Tabulated data, handbook methods: . . . . . . . . . 2.7.3. Predictive methods, or linear CFD methods: . . . . 2.7.4. Integrated Design Environments . . . . . . . . . . . 2.7.5. Nonlinear Computational Fluid Dynamics (CFD) . 2.7.6. Experimental Aerodynamics . . . . . . . . . . . . . 2.8. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Formulation of Predictive Aerodynamic Methods and Models 3.1. Aerodynamic tool requirements . . . . . . . . . . . . . . . 3.2. Ideal flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Circulation . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Flow description . . . . . . . . . . . . . . . . . . . . 3.3.3. Irrotational flow . . . . . . . . . . . . . . . . . . . . 3.4. Potential Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1. Boundary conditions . . . . . . . . . . . . . . . . . 3.5. Elementary flows . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1. The uniform stream . . . . . . . . . . . . . . . . . . 3.5.2. The vortex singularity . . . . . . . . . . . . . . . . . 3.5.3. Electrical current analogy . . . . . . . . . . . . . . . 3.6. Virtues of predictive methods . . . . . . . . . . . . . . . . .
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The Classical Vortex Lattice Method and It’s Implementation 4.1. Historical Background . . . . . . . . . . . . . . . . . . . . . 4.2. The Vortex Lattice Method . . . . . . . . . . . . . . . . . . 4.2.1. Vortex Lattice Methods in Aerodynamic Design . . 4.2.2. Overall formulation . . . . . . . . . . . . . . . . . . 4.2.3. Panelling . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4. Vortex singularity distribution . . . . . . . . . . . . 4.2.5. System of equations . . . . . . . . . . . . . . . . . . 4.3. Standard implementation . . . . . . . . . . . . . . . . . . . 4.3.1. Required input for a simple test case . . . . . . . . 4.3.2. Computed properties . . . . . . . . . . . . . . . . . 4.4. Model Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Standard implementation . . . . . . . . . . . . . . . . . . . 4.5.1. Cambered wings . . . . . . . . . . . . . . . . . . . . 4.5.2. Trailing edge control surfaces . . . . . . . . . . . . 4.5.3. Cranked wings . . . . . . . . . . . . . . . . . . . . . 4.5.4. Multiple lifting surfaces . . . . . . . . . . . . . . . . 4.6. Intermediate level implementation . . . . . . . . . . . . . . 4.6.1. Twist . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2. Dihedral . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3. Flexible wake . . . . . . . . . . . . . . . . . . . . . . 4.6.4. Sideslip . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.5. Roll, Pitch and Yaw rates . . . . . . . . . . . . . . . 4.7. Advanced level implementation . . . . . . . . . . . . . . . 4.7.1. Ground effect . . . . . . . . . . . . . . . . . . . . . . 4.7.2. Formation flying effects . . . . . . . . . . . . . . . . 4.7.3. Time derivatives . . . . . . . . . . . . . . . . . . . . 4.7.4. Supersonic flow . . . . . . . . . . . . . . . . . . . . 4.7.5. Viscous effects estimation . . . . . . . . . . . . . . . 4.8. Available Classical VLM programs . . . . . . . . . . . . . . 4.8.1. VLMPc . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2. AVL . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3. Virgit . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.4. Wing-Body . . . . . . . . . . . . . . . . . . . . . . .
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Extended Vortex Lattice Implementation 5.1. Rom . . . . . . . . . . . . . . . . . . . . . . 5.1.1. Nonlinear VLM . . . . . . . . . . . 5.1.2. Pressure distribution . . . . . . . . 5.1.3. Wake treatment . . . . . . . . . . . . 5.2. Lan . . . . . . . . . . . . . . . . . . . . . . . 5.2.1. Singularity distribution . . . . . . . 5.2.2. High angle of attack aerodynamics 5.2.3. Boundary layer separation . . . . . 5.2.4. Wing thickness . . . . . . . . . . . . 5.2.5. Supersonic flow . . . . . . . . . . . 5.3. Continuation . . . . . . . . . . . . . . . . .
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The Vortex lattice implementation: Tornado. 6.1. Tornado features . . . . . . . . . . . . . . 6.1.1. Geometry . . . . . . . . . . . . . . 6.1.2. Flight conditions . . . . . . . . . . 6.1.3. Preprocessor . . . . . . . . . . . . 6.1.4. Postprocessor . . . . . . . . . . . . 6.1.5. The vortex sling . . . . . . . . . . 6.2. Validation examples . . . . . . . . . . . . 6.3. Code Development . . . . . . . . . . . . . 6.4. Choice of programming environment . . 6.4.1. Programming Language . . . . . 6.4.2. Data Visualization . . . . . . . . . 6.4.3. Program execution . . . . . . . . . 6.4.4. Portability . . . . . . . . . . . . . . 6.4.5. Numerical Libraries . . . . . . . . 6.4.6. Graphical User Interfaces . . . . . 6.4.7. Drawbacks . . . . . . . . . . . . .
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Internet Interactions in Software Development 7.1. Internet interactions . . . . . . . . . . . . . . . . . . . . 7.2. Scale-free networks . . . . . . . . . . . . . . . . . . . . . 7.2.1. User-Developer interaction . . . . . . . . . . . . 7.2.2. Software distribution . . . . . . . . . . . . . . . 7.2.3. A hub of applications . . . . . . . . . . . . . . . 7.3. Software development and Software Use . . . . . . . . 7.3.1. Third person developments done at KTH . . . . 7.4. Feedback from the Internet . . . . . . . . . . . . . . . . 7.4.1. Frequently asked questions . . . . . . . . . . . . 7.4.2. Requested features . . . . . . . . . . . . . . . . . 7.4.3. Software use . . . . . . . . . . . . . . . . . . . . 7.4.4. User developed extensions . . . . . . . . . . . . 7.5. Open source . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1. Software in education and engineering science 7.5.2. Assessing limitations in the implementation . . 7.5.3. Intellectual property rights . . . . . . . . . . . . 7.6. General software development strategy . . . . . . . . .
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Summary and Conclusions 8.1. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Future Work 9.1. Tornado Development . . . . . 9.1.1. User manual . . . . . . 9.1.2. Different vortex models 9.1.3. Trefftz plane analysis . 9.1.4. Body model . . . . . . . 9.1.5. Camber function . . . . 9.1.6. Supersonic capability . 9.1.7. Kirchhoff net . . . . . .
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Appendix A [GNU-GPL]
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1. Introduction This chapter contains a historical background of aeronautics and aerodynamics, and describes how it is related to the work presented in the other chapters. The thesis overview section below describes the different chapters, and how they are connected to each other.
1.1. Thesis overview This thesis will cover the phase of aircraft design known as the initial phase, or the conceptual aircraft design phase. It also covers the tools needed in the initial phases of an aircraft design project, the mathematical background to these tools, their implementation - the software packages available to the aeronautical engineer - and the concurrent tool development and finally the benefits of user interactions through the internet. Chapter 2 covers aircraft conceptual design. It contains a more detailed description of the product definition phase, together with examples of other nomenclature. The model used can be rough, since at this stage the aircraft will not be defined until the end of the phase. Hence the aircraft design tools used for assessment can be fairly coarse. One part of the aircraft conceptual design treats aerodynamics, and the aerodynamic tools used there are the focus of this thesis and these tools are described in later chapters. In chapter 3, the mathematical background of the aerodynamic tools gives the theoretical overview needed for the later chapters. The definitions of ideal flow are included together with the assumptions of potential flow and derivation of the core equations of the vortex lattice method. The main focus of chapters 4 to 5 is the existing tools used by aerodynamic designers in the product definition phase. These are potential flow methods such as the vortex lattice methods. The work done in developing the new implementation of the vortex lattice method, Tornado, is described in chapter 6 together with validation examples and descriptions of certain programming solutions. Chapter 7 covers post-release development work done with Tornado, based on the feedback from users. The importance of open-source in software development is discussed in relation to its importance for third party development. Chapter 8 contains the conclusions of the thesis which are carried on into chapter 9, the future works section, where new implementation plans are presented
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1.2. Keyword introduction The terminology of aeronautics in the popular press is often used in an ambiguous way. For the sake of clarity this section defines the standard nomenclature. According to Longman [Lon90], some of the keywords used in this thesis can be defined as follows: Aerodynamics: (1) n. The science that studies the forces that act on bodies moving through the air. (2) The qualities necessary for movement through the air. Aeronautics: n. The science of the operation and flight of aircraft. Design: (1) v. To make a drawing or pattern of something to be made or built; develop and draw the plans for... (3) n. The arrangement of the parts in any man-made product, such as a machine or a work of art, as this influences the product’s practical usefulness, artistic quality, etc... Model: n. A small representation or copy of something. When discussing computational aerodynamics, it is also useful to have a definition of some other concepts, such as: Mathematical model: n. A model of the reality (the aircraft, or of fluid motion [Authors note]). The axioms in the model are chosen so that they correspond to observations, made in the reality one wishes to describe. The degree of correlation gives a measurement of the applicability of the model. The theory, the model, may then be used for making an assertion about the reality and to make predictions about the future. (from [Tho96])
1.3. The basic message of this thesis Conceptual design is the process used by aircraft manufacturing companies to perform a rapid assessment of new ideas and concepts. In order to do so, they need an array of simplistic tools that can produce results quickly. The focus of this thesis is the tools of aerodynamic performance. One set of these tools is the predicting methods for aerodynamics known as potential flow methods, and in particular with regard to this thesis, the vortex lattice method (VLM). A new implementation of the VLM, called Tornado, has been created and the Internet has been used for feedback from users around the world. Tornado has been used for designing aircraft, for developing aeronautical tools and for tutoring in aerodynamic design classes. The source has been coded in Matlabr, an engineering programming language, and the the coding effectiveness of this approach is assessed. Open source software is discussed in the aerodynamic context, in particular the GNU general public license.
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1.4. Historical overview of Aeronautics Aircraft design is a young branch of engineering science with only a 100-year history. Aeronautics has come a long way since the first powered flight. The advances made in aerodynamics, flight mechanics, research methodology and not the least in the final product, the aircrafts, is only matched by the development in computer science. However, aeronautics as a scientific discipline is older than it’s engineering application. The history of aviation is as old as mans dream of flight. Classic legends tell stories about conquering the skies, stories created long before the oceans of the earth where vanquished. Holmquist [Hol40](pp 7-9) concludes that Leonardo da Vinci performed the first documented attempt to investigate aeronautics. With the book ”Codice sul volo degli uccelli” (The Codex on the Flight of Birds, [dV05] ) of 1505, now in the collection of Biblioteca Reale in Turin, da Vinci assesses the aerodynamic and flight mechanic properties of birds, and of the kite species in particular. In this work, da Vinci states that a soaring bird itself does not generate the power needed for sustained flight, but rather that the flapping of the wings are needed for control. These are two fundamental statements in aeronautics. The first one opening the scientific arena for glider aircraft, and the second one being the same paradigm that the Wright brothers exercised in their examination of controlled flight, four hundred years later. In his work, da Vinci came close to solving the engineering problem of glider aircraft design, with conceptual design sketches of various design implementations, both on a planform scale, and on a detailed design level (see figure 1) Holmquist [Hol40](pp 9-10) continues to describe the historical engineering attempts at aircraft design; he mentions Emanuel Svedenborg as the first to envision a fixed wing concept with a propeller device. Specifications stated a wing surface of 50 m2 and 2 to 3 humans providing power. The design was presented in the journal Dædalus Hyperboreus, 1716, one of Sweden’s first technical journals. Christopher Polhem, opposed the idea, not on the scientific base that human flight should be impossible, but rather from the point of view that the engineering problems would be impossible to overcome. In the eighteenth century, the English scientist Sir George Cayley identified a solution to the so called control problem by suggesting a fixed wing design with a separate tailplane. He also identified the need of a stronger power source than human power. Although steam engines where available for a multitude of applications at the time, Cayley rejected their use in aeronautic applications. In the introduction to Navier-Stokes solvers in European aircraft design the power problem of the 19th century flight is identified by Vos et. al as: Human power is too weak, steam power is too heavy. [JV02] (p604-605). With Cayley’s work there was a paradigm shift in aeronautical engineering. Before, scientists and engineers had been focused at imitating bird flight in the shape of ornithopters and winged garments. With the introduction of the Cayley design paradigm the engineering approach was to separate the functions
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Figure 1: Detail design sketch by da Vinci
needed for sustained flight. The different parts for aerodynamics and control, propulsion and structures can readily be identified on any conventional aircraft designs of today: The wing, control surfaces, engines and the fuselage. Cayley envisioned [Kc78] (p103) that the interference effects between these design modules would be small, and hence they could be assessed one at a time and then integrated into the final design. Cayley’s work was very advanced for the time, among his results was the importance of streamlining bodies to reduce drag. Cayley was also one of the first experimental aerodynamicists with his work with model aircraft. Operating outdoors, his last model had a wing surface of 30 m2 . At the beginning of the 20th century, aerodynamics had become an established branch of science. With the first powered flight of the Wright brothers’ Flyer in 1903 their solution to the engineering problems where proven to be correct. Ever since the first aircraft started to take shape at the Wright’s drawing board in Kitty Hawk, the analysis tools for aircraft performance have been developed alongside with the aircraft themselves. The more advanced tools where available, the more advanced the aircraft designs got. Successively, improvment of analysis tools have lead to more and more advanced aircraft designs. Interestingly, old tools where not always abandoned for new, more advanced tools. Instead the tools developed early in aircraft design history, have found
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their place in the early design stages of a new aircraft. In that sense, each new aircraft design project can be envisioned as a historical odyssey of aircraft analysis tools, starting with short, one-line formulas and ending in the large modern industrial software suites of CFD, FEM and CAD packages. Anderson [JA98] (p307) gives one famous example of early tools in the Breguet Range Formula 2.5.2, which gives a quick, practical estimate for the range of a propeller aircraft. R=
η CL W0 ln c Cd W1
(1.4.1)
Where the range R is dependent on: η : Propeller efficiency c : Specific fuel consumption CL : Cruise Lift coefficient CD : Cruise Drag coefficient W0 : Aircraft Take-off weight W1 : Aircraft loaded empty weight (Cargo, but no fuel) The Breguet Range Formula captures the three most important aspects of conventional conceptual aircraft design. The propulsion is covered by the first fraction. The aerodynamics by the second fraction and the structural considerations by the third fraction. In this thesis, the main focus will be on the aerodynamics and the aerodynamic tools available during conceptual design. The Cayley design paradigm has dominated aircraft design, and still does to a great extent. In the application of this paradigm in devising tools of aeronautical design, the focus came to rest on the traditional aircraft design: a cylinder with wings on it, a separate tailplane and a power plant, according to Cayley’s own early designs. Hence, the tools themselves where adapted to this type of aircraft. Recently, proposals for aircrafts not matching the conventional format have been put forward, box wings, flying elliptic wings etc. And it has become evident that some of the old tools cannot be used for analyzing these new designs. This is because of two factors: They are not conventional designs, therefore conventional design tools are not fully applicable. And the new designs are highly integrated, engines providing lift through trust vectoring or blown high lift systems, the wing itself is used to carry some or even all of the cargo, lifting bodies etc. The integrated design philosophy is detrimental to Cayley’s design paradigm of separate functions. With the research front moving toward unconventional designs we are seeing a breakdown of the Cayley design paradigm. It creates a need for developing conceptual design tools that model aircraft characteristics on a deeper scientific basis. This development is seen in the shape of integrated product development environments and predictive methods. Since the strive for better products, aircrafts, is a competition between individuals, companies and nations. The scientific tools become more advanced, the understanding of aeronautics grows, allowing for more and
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more elaborate aircraft designs. But in every case, new aircraft designs are born on a sheet of paper as a product of the designers imagination and knowledge, just as Leonardo da Vinci’s model of the kite.
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2. Aircraft Conceptual Design Or how to transform requirements into specifications. Form gives function in the aeronautics discipline, perhaps more than in any other engineering discipline. This chapter will deal with the initial phases of aircraft design, the product definition phase, where the shape of the aircraft is formed. Designing a new aircraft takes time and resources, and to be able to perform the design work in the most efficient manner, a multi-cycle design approach has evolved during the later half of the 20th century. The main idea is to make sure that the design team has access to as much high fidelity information - as early in the design process - as possible. By performing the design process in an iterative manner, designers can be sure that decisions of the same gravity are being assessed at the same time. It is important that the major decisions are correct, because changing them afterwards, when the design work has carried on to a more detailed level, is costly. Essentially, the design process should have a ”first-time-right” approach, any business relations after delivery should be considered as the start of a new project, either as a support mission or as a new design task. The design process may acquire this property by the design procedure of Virtual prototyping. Before the dawn of computers, aircraft designers where forced to design and manufacture prototypes, either in sub-scale for windtunnel testing, or full sized test-flight aircraft. Today, computer based simulations cover much of the research work earlier done in experiments. More and more of the information flow involved in aircraft design is being transformed into digital form, allowing a faster, cheaper and better, design process.
2.1. Virtual prototyping In describing the aircraft with aerodynamical data, together with structural estimations and estimations on the powerplants, the design engineer can construct a virtual prototype. A virtual prototype is a computer model that can be used for systems simulations, simulations of the aircraft on the ground (for handling near the terminals of an airport) or as a flight simulator. Vos et al. describes it in [JV02] as: due to the global competition, economical and ecological pressures the, the aircraft industry now needs to deliver products which are better, faster and cheaper to produce. Information technology allows the design teams to rely more and more on numerical simulations instead of tests as a way to reduce production costs. Additionally, the accurate data obtained through simulation allow for far-reaching optimization of the whole design as well as contributing sub-disciplines in a way which would not be possible without the information technology infrastructure. With the breakdown of Cayley’s design paradigm, a need for interdisciplinary optimization has been created.
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Holmberg [Hol00] implies that by the use of computers a digital mockup in an integrated product development environment enables the development engineers to see all of the design configurations that are collected in a database. In a well conceived environment, proper tradeoffs can be made in the early design stages, thus reducing the number of changes later. In such a context, design evolves into a multidisciplinary process supported by fast, accurate and cost effective means to generate data for each contributing discipline as well as for complex interdisciplinary relationships. The idea of a digital drawing board implies that design should be kept in a digital form for as long as possible. The reasons for this are mainly financial; keeping the design data in a digital format makes the whole design process more expedient than a paper-based approach. The digital format does not only allow a rapid transfer of data between engineers, but also between softwares - the number one tool of aircraft designers. Optimizations and trade studies can be made on-line etc. The main idea is to simulate the entire aircraft, in all concerns such as manufacturing, operational and servicing. It adds up to a Life-Cycle Assessment for which the cost (usually) should be minimized.
Figure 1: The digital drawing board, where the geometrical shape and it’s predicted performance interacts digitally. An example from Isikveren’s software Qcard [Isi02]
In an integrated product development environment as described, the workload of the aerodynamic designer increases as a vast multitude of design options must be simulated. The large quantity of data that needs to be handled in modelling, the interpretation and the synthesis of of aerodynamic shapes poses a new challenge. To allow aerodynamicists to concentrate on the essential tasks of studying and validating the design, the data processing
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operations must be made as transparent as possible. One approach to this problem are the integrated design environment tools described below, that in conjunction with predictive methods such as the vortex lattice method serves to give the designer access to high fidelity data as early in the design process as possible. The main design task is to produce as virtual product, a virtual aircraft which simulated characteristics and performance matches the customers requirements as close as possible. In performing this feat before any work has been undertaken at the manufacturing facilities, the quality of the final product is assured as no surprises are expected, neither in the manufacturing nor in operational phase of the aircrafts life cycle. Additionally, the digital drawing board gives a way of ensuring data security and data integrity. In this way, no one but the authorized design team has access to the aircraft data, and they can be sure that they are working with a current and correct version of the design. Much effort can be saved by keeping a coherent design database on file and thus not waste time in transferring data sets between paper and computers. The use of computers has become a central theme in conceptual design, without them the turn-around time for investigating new concepts would be prohibitively long.
2.2. The Design Process The design problem in aeronautics is defined by Kuchemann [Kc78] as: ¨ ”Design work is the ultimate goal of aerodynamics and all other activities should lead up to it. The design of an aircraft provides the most severe test of hypothesis, concepts and methods.” Kuchemann highlights the importance of having a suitable design strategy in ¨ order to create a good design. He argues that the base of all aerodynamic design work should be basic fluid mechanics and selecting the kind of flows that can be modelled with confidence. This would lead to the corresponding types of aircraft and the methods available for assessing them. Although the approach above would probably yield functioning aircraft if exercised, it lacks the motivation on why to start the design of an aircraft. In that sense Teichmann [Tei39] comes closer to the methodology used today. Teichmann too, acknowledges the necessity of an orderly form of procedure. However, he also recognizes that aircraft development is customer driven, and that customer requirements, more than fluid mechanics, determine the design of an aircraft. It is also an obvious fact that air traffic regulating authorities and their regulations ultimately govern the shape of new aircraft. The truth is that all of these background factors play an important role in the development of a new aircraft design.
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There are no standard definitions on when and how the design work can be said to have started, or what the different cycles in the project’s life time should be called. This thesis will use the nomenclature of Raymer [Ray99] with the following layout, calling the whole chain The product definition phase: • Conceptual design • Preliminary design • Detail Design This thesis discusses the part of aircraft design known as the conceptual design phase. During this phase, decisions are made on what requirements should drive the development work. These requirements can be a combination of: range, maximum number of passengers, pilot field of vision, terrain landing capabilities and cruise speed. Decisions are made on which technologies should be used, and what trade-offs could be accepted. Designers discuss the shape of the aircraft, the general layout, number of engines, maximum weight and the projected cost.
2.2.1. Conceptual design A characteristic of the conceptual design phase is that the aircraft exists in a multitude of different versions. Brandt [SB97] describes the aircraft geometry of this phase as ”fuzzy” to signify the level of uncertainty of the future aircraft’s shape. Of the many possible configurations, each version is evaluated against specific figures of merit, and compared to the customer’s specifications. The multitude of possible design approaches enables the design team to perform an evolutionary change of the design concepts. The design is in constant change during this phase and making sure that the entire design team is updated is an important task. The production of manufacturing drawings of the aircraft is a costly process, not to mention the metal cutting phase - the actual manufacturing of the airframe. The decision to book most of the costs in these phases are taken in the conceptual design phase. Hence, the goal is to make conceptual design decisions that renders the optimum performance, and a digital approach makes this easier. The essence of speed in the conceptual design phase is addressed by Chudoba [Chu01] (pp 39), who acknowledges that the accuracy of the results is indeed scarified in order to have an expedient investigation cycle. And rather than searching for absolute numbers, the aircraft engineer is more often interested in finding the trends and interactions between different disciplines. The time frame for the conceptual design phase ranges from a couple of weeks to several years, and in some cases, decades. Within this period, hundreds of design cycles are being performed and many different aircraft configurations are being assessed. A sketch of the design loop is shown in figure 2.
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Figure 2: Schematic of the iterations in the design procedure
2.2.2. The design cycle The design cycle is started by a market initiative. A customer’s, or the manufacturer’s own, market research department assesses the need for a new aircraft on the market. Today, this has more and more become a global market, with each aircraft manufacturer being able to deliver to a customer anywhere in the world. The result of the market study is the design requirements. The design requirements is a list of necessary specifications, usually several pages long, that the future aircraft must have in order to fill the market niche it is intended for. The design team then makes a quick assessment of the constraints on the new design imposed by available technology, company policies and legal requirements. When this is done the first centerline will be drawn, around which the designer shapes an initial design estimate. With that the conceptual design work starts, and the design is evaluated with regards to: • aerodynamics and flight mechanics • propulsion and range • structures and weight
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When the analysis has been done, the results are compared with the requirements. If there is a discrepancy the design team has to propose a change in either the requirements or in the design. Design changes are incorporated in the aircraft model and the analysis is performed once again. This loop continues until the analysis results match the requirements close enough, or when no new improvements are possible. When the iteration has ended, the design project can either be cancelled, shelved, or continued into the next design phase, the preliminary design. The main discussion of this thesis is the tools used in the aerodynamic analysis of a new conceptual aircraft design. In particular, the predictive methods called vortex lattice methods are considered. The other two important topics of the analysis, Propulsion and Structures (see figure 3) has to be assessed with other tools. These tools are often incorporated in handbook methods, or in the integrated design environments described below.
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Figure 3: The core of the conceptual design loop. The analysis is divided into the sub-disciplines: aerodynamics, propulsion and structures.
In the conceptual design phase, the whole design is composed of conceptual representations of all of the subsystems of the aircraft. The engine for example, may be modelled by nothing more than a volume, weight, the hard-points where the engine is intended to be attached to the fuselage and an assumed thrust and mass flow. As time progresses, the overall design will become more and more complex, as the level of detail of the subsystems increases. At the end of the conceptual phase, the design is fixed to a certain degree. What started with 3-view sketch has ended with a 3-view drawing of increased complexity. All discrete variables, such as number of engines, number of wings, and how many passengers seats the cabin should hold, are set to a certain number.
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Continuous variables, such as wing sweep, tail size etc are specified to within tens of percent. However, in some cases the external measurements of the aircraft may be set already in the specification. Examples of this are: the span requirement of the Boeing raptor (F22), where the total span was required not to exceed 13.5 meters, in order to make the aircraft fit in the same hardened shelters as it’s predecessors; or the external measurement limits of the Airbus 380 by 80 × 80 meters, in order to make it fit into ordinary airports’ parking spaces. From time to time, the design loop cannot be closed either by changing the design or by changing the specification. The requirements of the customer in terms of speed or other performance might not be compatible with the cost requirements, or the laws of physics. If this is the case, then the preliminary design has truly filled it’s purpose by hindering an impossible design to devour any more company resources. However, the data is not destroyed, it is instead added to the available technology database for reuse in later projects. An example of this is the Boeing Sonic Cruiser which was investigated in a conventional but area-ruled triple engine layout in the 1980s, and then shelved until the late 1990s when it morphed into a canard-delta, twin engine concept, and was shelved again. Compared with the definitions of the conceptual design suggested by Raymer, Isikveren suggests a further division [Isi02] of the conceptual design phase. By introducing an iteration loop named pre-design prior to the conceptual design phase, he addresses the the issue of how aircraft design specifications are created. As these requirements are a prerequisite to the conceptual design it is important to know how they are generated. The timescale of this pre-design loop ranges from one to a couple of weeks. It would follow basically the same form as the conceptual design loop in figure 2, but without being prompted by a market or customer initiative. Instead, the goal is to be able to react to every change in the aerospace market, doing a feasibility study and thus getting the initiative in starting a conceptual design phase, rather than waiting for customer requirements.
2.2.3. The pre-design loop The division of the conceptual design loop as proposed by Isikveren in [Isi03] is a two-tier approach captures the creation of the design requirements. The first tier being the pre-design loop and the second the conceptual design loop. The pre-design loop should take between one to seven days to complete, which stresses the importance of having tools that produce answers quickly, rather than with high accuracy. The tools should demonstrate robustness, consistency and applicability to the problem at hand. Due to the time constraint, numerical simulation of the flow field is not applicable. Instead the designer will have to rely on handbook methods. There is a cause for caution here, which the conceptual designer should be aware of. As the handbook methods are built on empirical observations, their results will favor conventional designs. This bias will not always account traditional designs as
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the best. Their lower accuracy of analysis of unconventional designs will yield stochastic results. The pre-design loop contains of four main sections: • Marketing Requirements and Objectives. In this section the work done includes assessing customer requirements, checking the company in-house requirements and policies that might influence the final design and inventorying the certification requirements stipulated by the civic government. • Design Concepts In this section, the design team investigates the available types of configurations on a macro scale such as, number of engines, fuselage volume etc. The possible types of appropriate engines are looked into, and the available technologies to be used in the overall project are explored. • Initial Baseline Design The design team then performs an initial design assessment, which is meant to answer questions such as: Should the new design be a ”Clean Sheet” design, or should older designs be reused as partial implementations to the new design? Is there a possibility to devise a concept family, built on a core design? Is there a need for provisions for fuselage stretching or different engine selections? A feasibility study is performed to get an estimate of the technical and financial risks of a continued design project. • Parametric Studies The Initial baseline is then subjected to a sensitivity analysis in order to capture the performance-critical design features, and what design features could be allowed to change in a trade-off study. The basic design parameter values are then used in the design trade-off study, where parameter values are pertubated in order to satisfy the requirements set in the marketing Requirements and Objectives section. 2.2.4. Parameter-Characteristic-Performance The actual number crunching in the conceptual design phase, consists of investigating the different features of the aircraft design. A number of parameters are chosen to be pertubated. Raymer selects the following [Ray99]: • • • • • • • •
Wingspan Wing sweep Fuselage diameter Tail size Tail placement Gross weight Engine power Fuel volume
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All of these parameters influence the characteristics of the aircraft. These characteristics are the figures that tell the designer a lot on how the individual parts of the aircraft behave. They could be aerodynamic characteristics such as lift slope, maximum lift coefficient or roll damping. They could also be propulsive characteristics such as static sea-level thrust or thrust specific fuel consumption (TSFC), or they could be structural characteristics such as the empty weight, fuel volume or moments of inertia. However, product requirements are not expressed in the form of these aircraft characteristics. Instead the requirements are expressed as performance, such as: • • • • • • • •
Stall speed Take off field length Cruise range Cruise ceiling Allowable change in center of gravity Approach angle Flight mechanic eigenmodes Pilot work environment
Hence, the aircraft characteristics need to be translated into performance. Usually this require input from all of the conceptual design disciplines: Aerodynamic, Propulsion and Structures. The Vortex lattice method discussed in this thesis is a tool for converting parameters into characteristics. To further the analysis and completing the conceptual design loop another type of tool such as the integrated design environment is needed. Examples of such integrated design environments, embodying the concept of the digital drawing board, are Raymer’s program RDS, Isikverens Qcard or Chudoba’s AeroMech. But it is not enough just converting characteristics to performance. In order to design a market competitive final design, the conceptual design has to be optimized. 2.2.5. Design Optimization The design optimization is an important part of the conceptual design phase. Mathematically it can be described as [Ray99] selecting a number of design parameters in such way that, the parameter X must reside within certain constraints: Xmin ≤ X ≤ Xmax The design parameters could be the ones listed in the parameter list above. Of these, the first five are evaluated with aerodynamic tools, the others with tools for structures and power plant, respectively. A baseline drawing is devised, with an initial guess, and the aircraft characteristics U of the design proposal is evaluated in such way that U = U(X). From this evaluation, specific measure-of-merit (MOM) is selected from the computed aircraft performance, which could be aircraft price, maximum speed or any other merit. This MOM, described as F(U(X)) is then optimized as: 2.2.1
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G(X, X(U)) > 0 H(X, X(U)) > 0 max(F(U(X)) I(X, X(U)) > 0
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(2.2.1)
The functions G, H and I are then the constraints to which the MOM (X) is subjected. Typical constraints can be takeoff field length, which is set by length of the the available runways around the world, or minimum range when taking off with maximum gross takeoff weight. 2.2.6. The output of conceptual design The final of conceptual design is to create a proposal design to serve as input for starting the next design phase, the preliminary design phase. The most concrete result coming out of the conceptual design phase, is the three view design drawings (see figure 4) out of [Ray99]. While in no way detailed enough to start manufacturing, the design drawings gives a fairly good estimate of the shape of the final product. But there is more to the conceptual design phase results: in the comparison of analysis results with specifications the flight mechanic derivatives play a major role. These numbers specify the handling qualities of the future aircraft. How heavy it will be on the stick, how fast it will recover from a gust disturbance and other properties relevant to the future pilot, and even more importantly relevant to the future control system designers and the structural designers.
Figure 4: Conceptual design 3-view drawings, from [SB97].
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2.2.7. The other design phases The Preliminarily design phase starts as soon as the conceptual design phase has ended by freezing the configuration. At this stage, the external surface of the aircraft is defined. Advanced testing and simulations start to create the necessary database, and all major items, wingbox, fuselage, empennage etc are designed. When this is done a statistical cost analysis can be made and presented to the customers. No real surprises are expected in this phase, but if they emerge, the problems must be rectified before proceeding to the next level, the detailed design phase. By continuing beyond the preliminary design phase, the designer-in-chief passes a threshold called ”You Bet Your Company!” The literal meaning of this is that the company costs for continued development and manufacturing will be higher than the total company net worth. This means that if the projected number of sold aircraft are not reached, the whole company will be in economic peril. The Detail design phase incorporates the making of the production drawings, with structural simulations down to the nuts and bolts level. This means designing tools, jigs and preparing the shop layout. Finalizing delivery dates with subcontractors and performing physical tests on the major items. In this phase, a full sale mockup might be built. With more detailed CAD drawings, the actual design weight can be assessed and the final performance predictions can be made.
2.2.8. Overall design, from specification to rollout There is much more to the entire design process than the product definition phase. Vos et. al. [JV02] makes another distinction than Raymer by excluding the production of the manufacturing drawings from the initial design work. Instead they suggest: 1. product definition a) conceptual design b) pre-design c) design 2. product development 3. product manufacturing 4. product support The concept design (1a) is more or less equivalent to both Isikverens pre-design phase and the conceptual design phase. The product development phase is equivalent to Raymer’s detail design phase, where the manufacturing drawings are produced. In addition, the two phases manufacturing and support, forms the base of the rest of the design work needed during the life-span of the product. There will be product upgrades such as retrofitting new engines, elongation of the cabin or the installation of new and improved sub systems. In the military sector, it is not uncommon for an airframe to muster on for 40 years or more and during that time it will constantly be upgraded.
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2.3. The tasks of aerodynamics in conceptual aircraft design The role of predictive aerodynamic methods in the conceptual aircraft design phase is to determine the forces and moments acting on a body in motion in the air. This is done by using the multitude of tools available to the aerodynamic designer, see figure 5. Vos et al. [JV02] states that the major task of aerodynamics is to define the overall aerodynamic shape of an aircraft in order to fulfill the aerodynamic performance demands of the sizing missions, which includes flyability and controllability of the vehicle.
Figure 5: The tools, tasks and results of the aerodynamic design process, from [JV02].
By employing some of the tools discussed in the tools of the trade section, the conceptual design engineer can estimate the aerodynamic characteristics of the future aircraft, and in doing so removing many of the uncertainties regarding the future behavior of the aircraft.
2.4. Aircraft Characteristics The aircraft characteristics are the performance numbers of the aircraft subsystems, describing the operational performance of an individual part of the aircraft. Following Cayley’s design paradigm, the characteristics can be divided up into three major topics: aerodynamic, structural and propulsion system characteristics. Later, these characteristics are joined together to form the figures of the whole aircraft, the aircraft performance. 2.4.1. Aerodynamic characteristics The predictive aerodynamic tools described later in this chapter are devised with the intention of describing the flow field around an aircraft. By entering simple geometrical data, the designer is able to simulate the flow field and make engineering decisions based on the results.
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When the flow has been estimated at a satisfactory level of fidelity, The resulting pressures on the surfaces of the aircraft can be computed. The aerodynamic data is then post-processed in order to extract the aerodynamic coefficients and their derivatives. If handbook methods had been used, the coefficients had been directly estimated without the external flow field being describing in between. It is these coefficients and derivatives that form the aircrafts aerodynamic characteristics. These numbers describe the aircraft behavior in flight in such way that different types of aircraft can be compared with each other. A non-exhaustive list of aerodynamic coefficients would be: • • • • • • •
Lift coefficient: CL Drag coefficient: CD Maximal lift coefficient: CLmax Pitching moment alpha derivative :Cmα Rolling moment due to sideslip: Clβ Roll damping: ClP Yaw moment due to sideslip: Cnβ
Each of these numbers, either by themselves or in combination with other coefficients or other aircraft data tell something specific about the aircraft. All in all, they add up to the aircraft flying qualities, or aircraft performance when they are used in the flight mechanic equations of a rigid aircraft determining it’s motions and responses. 2.4.2. Structural characteristics In the conceptual design phase, the structural is fairly limited in detail. Partly because some input regarding aerodynamic loads and weights of the power plant is needed before initial estimates on the structural design can be made. Nonetheless, structural considerations are integrated in the conceptual design. Important characteristics numbers are: • • • • •
Aircraft empty weight Fuel volume Payload weight and volume Moments of inertia Wing loading
2.4.3. Propulsive characteristics In most modern aircraft design projects, the power plant design is totally separated from the aircraft design, especially in the conceptual design stage. However, the integration of the aircraft engine in the overall design requires consideration. Important propulsive characteristics are: • • • •
Thrust specific fuel consumption Installed thrust Service ceiling Power
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2.5. Aircraft Performance The aircraft performance is simply the capabilities of the entire aircraft, rather than it’s individual subcomponents. Aircraft performance is usually divided into sections regarding the different operational phases of a standard mission as taxing, takeoff, climb, cruise, decent and landing. For military operations there are additional performance figures regarding, survivability, resilience, stealth etc. The performance numbers depend greatly on the characteristics numbers, and some examples of this will be given below. Performance is related to the flying qualities of the aircraft. Stinton [Sti96] defines flying qualities as the following collection of information sets. • Performance • Handling Qualities • Functioning of systems Performance numbers could be operational considerations as: Field distances, how long must the airfield be in order to allow the aircraft to take off? Climb rates and gradients, how fast will the aircraft reach cruise altitude? and at what angle to the horizon will this be? Decent rates, how steep will the approach for landing be? Handling numbers could be: Stability, will the aircraft maintain it’s angular orientation in space by itself, or will it divert? Controllability, is the standard pilot able to control the aircraft at all flight conditions within the flight envelope? Stalling, how does the aircraft behave when it stalls? Ability to trim, will the pilot be able to trim the aircraft in all flight conditions? System qualities incorporates the functioning of the aircraft subsystems such as control linkage, undercarriage and also radios, air conditioning etc.
2.5.1. Operational considerations Mair and Birdsall [MB92] describes various operational performance figures, and how to compute them. For example, the runway length from standstill to rotation can be described as follows: sr =
W 2g
Z 0
VR
(F0 − k1 F0
V2 )
d(V 2 ) − 0.5ρV 2 S(CDG − µr CLG ) − µR W
(2.5.1)
Were W is the aircraft weight, F0 − k1 F0 V 2 is a function relating to the available thrust, µr the rolling resistance and CDG CLG being the drag coefficient and lift coefficient in ground effect at rotation. In this equation (2.5.1) there are characteristics from the three areas described above. Weight from structures, thrust from propulsion and lift and drag from aerodynamics.
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In the same way, the cruise range of the aircraft can be modelled by the Breguet Range Formula 2.5.2, which gives a quick, practical estimate for the range of a propeller aircraft: R=
ηprop CL c Cd
ln
W0 W1
(2.5.2)
as specified in the introduction chapter. 2.5.2. Handling qualities The handling qualities are the performance figures that have an impact either on the pilot workload or on autopilot design. Among these are the static stability criterion, which according to Etkin [ER96](p.29) is defined as Cmα < 0 where Cmα is the pitching moment derivative with respect to the angle of attack. If the aircraft characteristics are evaluated with a predictive method Cmα would be one of the direct outputs. On the other hand, if handbook methods where used, it could be computed as 2.5.3: Cmα = −KCLα
(2.5.3)
Were K is the static margin. The aircraft itself is a tool for the pilot in performing certain operational procedures. The Cooper/Harper scale is used in flight tests as a measure of aircraft handling qualities performance. In order to assess these handling qualities before a test flight has been performed. The designer may use Isorating curves based on a large historical database of flight tests. One such curve from [ER96](p15) shows the pilot opinion contours of the longitudinal short-period oscillation (figure 6). However care should be taken when using such historical data in transport aircraft design. Many flight tests where done in developing fighter aircraft, and it might be so that high frequency pitch properties might be experienced differently in a very large transport such as the Airbus A380 than in a small fighter aircraft.
2.6. Design requirements Aircraft design is not initiated by itself, barring hobbyists. There has to be a need or a market niche in air transport that someone imagines being suitable for exploitation. This niche then forms the requirements. The need could come from a company, such as an airline company exploring new destinations; or a government institution, inquiring for a new military design project. The design work could also be initiated by the manufacturer on their own initiative. The design group might have identified a possible application for a breakthrough technology. The market research department might have identified a market and therefore initiates a preemptive design strategy. Or perhaps, some of the constraints on the company has changed, allowing more kinds of designs at that manufacturer.
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Figure 6: Longitudinal Short-period oscillation, pilot opinion contours, from [ER96].
Anyway, the aircraft manufacturing is customer driven, and the specifications have their base in the customers needs. Other factors are the government requirements, company requirements and technology constraints.
2.6.1. Customer requirements In the end, it is the customer, the aircraft passenger, that decides what is right or wrong in aircraft design. Although the main figure of merit seems to be the passenger ticket price, it is so only because all of the government requirements are fulfilled. An aircraft which is deemed unsafe by the passenger population, will not stay on the market for very long (but it will stay on the ground). However, the passengers are regarded as secondary or tertiary customers in this thesis. Since there is not much a designer can do in the conceptual design stage to address emotional arguments from a passenger who will travel in the aircraft, perhaps twenty years after the conceptual design phase. Instead, the requirements from the main customers, airlines and governments, are the primary requirements. Airline companies conduct market research before opening up new routes. In this analysis, the prospect of procuring a new aircraft model to suit market demands might lead to a new set of design requirements. An example specification set is available in Teichmann [Tei39] (p7-18). The specification was issued around 1935 by the Bureau of Air Commerce, Department of Commerce (and no longer in force). The main items of this Twin engine,
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six-place cabin monoplane specification where: 1. 2. 3. 4. 5. 6. 7. 8. 9.
General Specification Performance Requirements Special Service Requirements Fixed equipment Instruments General Design Requirements Design Requirements Mock-up Requirements Inspection and rests
In the general specification the two first requirements are: 1) ”The current issue of the Bureau of Air Commerce,’Airworthiness Requirements for Aircraft’, forms a part of this specification and ”Prior to delivery the airplanes shall have been granted an approved Type Certificate”. This indicates the impact of the governmental aircraft regulations. However, the author suspects that nowadays the phrase ”current issue” has been changed to ”The issue of 2003-02-30” or another appropriate date in order to steer clear of ambiguities. In the performance section, weights, speeds and service ceilings are specified as: With fixed equipment listed hereinafter and a useful load consisting of: 1 Pilot 170 lb. 5 passengers 850 lb baggage 200 lb cargo 200 lb fuel and oil sufficient for 500 miles at any altitude below 5000 ft. The minimum performance acceptable when using fuel of not more than 80 Octane will be: a. High speed in level flight 175 m.p.h. b. Landing speed with power off 65 m.p.h. c. Ceiling, one engine off 6000 ft. d. Distance from start to clear 50-ft. obstacle 1500 ft. e. Distance to stop after clearing a 50-ft. obstacle 1000 ft. And so it goes on for about ten pages. From requirements listed above it is also clear that the level of analysis, even at the conceptual stage must be thorough. The distance from start... requirement for example, must be assessed with a weight model, propulsor power output and an output efficiency model, and an aerodynamic model of lift and drag, a flight mechanic model of the minimum speed for control etc. To simulate this before the aircraft has been fully defined, there is a need of predictive methods. In general every word in the specification will have impact on the final design. Or as Wood puts it [Woo63] (p.VI) in his foreword about a day at work of the airplane designer. ”[The designer] Thinks it would be a good idea to underscore with red pencil the parts of the customers specification
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that will affect the design. After completing four pages [he] finds that he has underscored all but three words, so [he] throws down the specification in disgust.” Of course, modern designers do not underline with a red pencil, they use a designated marker for that. 2.6.2. Government requirements Most nations have some kind of government body responsible of exercising authority over air traffic. In the United States this is the Federal Aircraft Administration (FAA), and in Sweden the Luftfartsverket (LFV). These organizations control, by mandate, many aspects of air traffic. From running airports, issuing security tips for passengers, distributing accident data to issuing airworthiness certificates for specific aircraft designs. Without such an airworthiness certificate, the aircraft will not be allowed to fly in that country. Hence, it’s important that the procedure of acquiring such a certificate is an issue already in the design specification. Luckily, international cooperation through the International Civil Aviation Organization (ICAO), has ensured that the standards of different countries differ very little. The US regulations, the Federal Aircraft Regulations (FAR) is the de facto standard. The Swedish counterpart is the Best¨ammelser for ¨ Civil Luftfart (BCL). As always, special rules apply for military application. Below is an excerpt from the FAR part 25, Airworthiness Standards: Transport Category Airplanes [Adm03] (25.231) Sec. 25.231 Longitudinal stability and control. a) Landplanes may have no uncontrollable tendency to nose over in any reasonably expected operating condition or when rebound occurs during landing or takeoff. In addition– (1) Wheel brakes must operate smoothly and may not cause any undue tendency to nose over; and (2) If a tail-wheel landing gear is used, it must be possible, during the takeoff ground run on concrete, to maintain any [attitude up to thrust line level, at 75 percent of VSR1.] (b) For seaplanes and amphibians, the most adverse water conditions safe for takeoff, taxiing, and landing, must be established. These demands are by no means unreasonable, but the problem is to analyze a not yet defined geometry with sufficient accuracy to guarantee compliance. To be able to do so, the design group needs access to predictive methods which can be used to analyze a coarse representation of the concept. 2.6.3. Company requirements and technology constraints There are other limiting factors for what kind of design will come out of the conceptual design phase. Requirements from the aircraft design company itself is an important factor. One design group might be constrained by the physical size of the manufacturing plant, another might have a policy not to design jet aircraft. To this kind of limitations, the lack of know-how is also a factor.
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Technology constraints are simply the other bounds of capability of the available technology. In some areas, the design engineer can, and should, plan for an increase in specific technology performance by time. Jet engines are a good example of a design detail that is not designed by the aircraft manufacturer, but rather considered as an available technology. As most specifications for new aircraft designs are fairly modest in performance increase, when compared with previous models (barring those developed under government contract), the technology constraints may not be obvious in the first iterations of the conceptual design cycle.
2.7. Tools of the trade As stated earlier, the conceptual design phase is characterized by rapid changes in the design, many different concepts are being assessed at the same time. Since a 100% accuracy in the results is unnecessary, as the design itself might be specified to only 10% there is a need for tools that can produce results fast, with sufficient accuracy. In the conceptual design phase, there are a number of possible tools for performing an initial assessment of a particular design. In order to complete the conceptual design loop, the designer will have to use simplified tools for flight mechanics, operation maintenance and aerodynamics. Listed below are some of the aerodynamic tools available to the conceptual design engineer. 2.7.1. Engineering ”know-how”: One of the most valuable assets in an aircraft design team, is a person who has designed aircraft before. The value of engineering ”know-how” is not easily assessed, but it does have a major impact on project efficiency. Essential background data of legal, economic and standardization issues are managed within engineering know-how. It has happened more than once that a company fails with a project of a kind that they have been successful with in the past, only because the engineers involved in the earlier projects have left the company. In a sense, managing the human resources efficiently is the core of good management.
2.7.2. Tabulated data, handbook methods: Methods based on empirical or analytical formulae or are refereed to as handbook methods. Essentially, handbook methods are engineering know-how written down together with a statistical database of existing aircraft. These methods are used in the initial iterations of the conceptual design phase. As they are based on experiments and tests with existing aircraft, they are accurate when considering conventional designs. Interpolations for new designs is possible by the empirical formulae. Some of the handbooks contains isometric pictures of design details to give the conceptual designer an idea of the size and shape of a particular detail. An
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example of a handbook method is the prediction of CLMax for a power-off, one g stall. As done by Isikveren [Isi02](p76) where equation 2.7.1 is derived analytically, and then calibrated against measurement. CLMax = 0.224
dCL dα
(2.7.1)
These methods are often based on an actual book such as the USAF DATCOM (DATa COMpendium) [Ano99], Torenbeek’s Synthesis of subsonic airplane design [Tor76], Roscam’s Airplane design [Ros85] or Raymer’s AIRCRAFT DESIGN: A Conceptual Approach [Ray99]. In most cases the handbook method has an accompanying software which contains the design equations in a computer program. This approach rapidly accelerates the design process. The more advanced digitalized handbook methods are called integrated design environments, and they are described below. Unconventional designs In the conceptual phase conventional aircraft designs implies a cylinder with wings on it. As there is a great number of aircraft looking like this, the statistical database in a handbook method will provide reasonable answers for this kind of design. But in the case of unconventional designs such as canard configurations, unusual planforms etc, the extrapolating handbook methods do not describe the aircraft behavior accurately. This is due to the fact that most handbook methods uses background material from earlier design projects to calibrate the prediction formulae. This empirical approach does not hold when considering designs that are obvious extrapolations from the built-in database of conventional designs. Handbook methods, or digital handbook methods, which are based on conventional designs cannot be used for other designs than the standard, one wing and a tail design. 2.7.3. Predictive methods, or linear CFD methods: The predictive methods have an advantage over handbook methods, as they treat general configurations, hence also unconventional designs. Characteristic of these methods are that they are fast, and require little geometry input. Predictive methods go deeper in the mathematical model than do the handbook methods. Predictive methods typically assume an ideal flow, i.e. inviscid. Thus, the aircraft geometry can be modelled by solving a usually linear partial differential equation (PDE), often the LaPlaces equation. Because it is linear, it can be solved by superposition of elementary solutions. The boundary conditions are that no flow is allowed through the solid surfaces. The aircraft is represented by a distribution of singularities, and the flow is set to be parallel to the aircraft surfaces. As no assumptions are made regarding the similarity of the new design with old designs in the solution method, the predictive methods can successfully model unconventional designs.
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Predictive methods allow an instantaneous analysis of coarse models, where the airplane geometry is still fuzzy, and in addition to handbook methods extrapolations to previously unknown concepts yield good results. Among these methods are the vortex lattice methods and the panel methods. These models do not require the whole flow volume to be meshed, instead only the wetted surface is covered with a panel grid, or a vortex lattice. They do not require an exact, representative model of the final design. The degree of refinement of the models used in these methods range from very simple, as a single flat wing with some taper and sweep; to moderately advanced full aircraft representations with fuselage and viscous effects. This approach gives predictive methods that are linear, which run expediently on an ordinary desktop computer. Some predictive methods can also model the viscous effects of the boundary layer. The upper limit of accuracy is set either by the model, if it is a crude one, or by the method assumption.
2.7.4. Integrated Design Environments The development of new tools have lead to integrated design environments, where the aerodynamic, the structural and the propulsion analysis is performed in parallel in a computer environment. These software packages are usually based on digitalized handbook methods such as Roskams AAA (Advanced Aircraft Analysis), or Raymer’s conceptual aircraft design software RDS, described in the AIAA paper [Ray92]. Due to their compactness, predictive methods are easily integrated into larger design system. These design systems are more than just a pre- and post-processor, since they can often be run without engaging the predictive methods. Two examples are the QCARD system [Isi02] by Askin Isikveren which uses the Tornado VLM core, or the AeroMech [Chu01] by Bernd Chudoba, that uses the Vorstab core, initially developed by C.E. Lan [Lan93] based on the quasi vortex lattice method [Lan] also by Lan.
2.7.5. Nonlinear Computational Fluid Dynamics (CFD) Another group of computer based methods are nonlinear CFD codes such as Euler codes and Navier-Stokes (NS) codes. Being more accurate than the predictive methods, they also require a much more detailed input and are much more platform-bound. For simple geometries, output data from these codes match experimental data closely. The upper limit of accuracy is set by the available computing resources and the long computation time. This constraint bars CFD from use in conceptual design to any great extent.
2.7.6. Experimental Aerodynamics Experimental methods in aerodynamics include wind tunnel testing and Test flights. This approach is useful, since any kind of geometry can be modelled.
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The experimental methods have their own restrictions, such as wind tunnel wall influence, or scaling effects. But ultimately the only upper limit for accuracy is set by the prohibitively high costs of flight testing. Experimental methods often serve to give validation data to computational methods. Experimental models are often too time consuming to be considered in a conceptual design project. Both Nonlinear CFD and experimental aerodynamics are used primarily in the detailed design phase and to some extent in the preliminary design phase.
2.8. Summary As shown above, there is a need for predictive methods in aircraft conceptual design. The following chapters will assess the special category of predictive methods, the ideal flow methods, and in particular vortex lattice methods.
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3. Formulation of Predictive Aerodynamic Methods and Models This chapter will investigate the predictive aerodynamic methods for use in conceptual aerodynamic design. In particular the potential flow methods, for which a short mathematical background will be displayed.
3.1. Aerodynamic tool requirements Most conceptual design systems, such as RDS, uses predictive methods that are semi-empirical. As discussed in the conceptual design chapter these semi-empirical models are good for conventional designs, but when they are applied to unconventional designs, the built in assumptions and limitations may be exceeded and the solution break down. However, resolving the aerodynamic characteristics of an unconventional geometry by a full Navier-Stokes (NS) simulation, is impossible in the conceptual design phase. In conceptual design, there is no CAD file yet, only a 3-view drawing, so there is no possibility to create the grid needed for the non-linear CFD methods (as NS simulations). Additionally, the workload of generating grids for each possible design permutation would require to much time. Due to the time constraint and to lack of accurate geometrical data, NS-simulations will not be able to provide the needed data. Hence, there is a need for a method that is: • Accurate enough • Fast, in respect of computation time • Compatible with the lack of geometry definition at the conceptual design stage The predictive method should be able to generate aerodata, or aerodynamic characteristics, for the following uses: Performance : Range, endurance, field length etc. Stability and control : Handling qualities, damping etc. Loads : Pressures, forces and moments. For this task, we have the following candidate methods: • Semi-empirical handbook methods • Vortex lattice methods • Panel methods As discussed earlier, the handbook methods are very fast and they produce fair results for performance of conventional designs. These results are stretched when considering handling qualities for any design, and are not applicable to unconventional designs. Aerodynamic loads as pressure distributions are not assessed.
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Panel methods do not require the entire flow volume to be meshed, and so they can be made fairly fast. They can be used to produce all of the numbers needed for performance computations, stability and control assessments as well as providing surface pressures on the entire aircraft. However they do require quite detailed geometrical input. A geometry which is not yet defined. Vortex lattice methods (VLM) can provide the same data as the panel methods, without being dependent on high-fidelity geometrical data. The VLM can be made very fast for the simplest geometries, and as it model the flow field it produces accurate data for unconventional designs. This makes the vortex lattice method ideal for conceptual aircraft design, and the following chapters of this thesis will cover the intricacies and possibilities of the vortex lattice method in general, and of the Tornado implementation in particular.
3.2. Ideal flow Vortex lattice methods are build on the theory of ideal flow. This section will cover the basic ideas and assumptions behind the ideal flow, and the equations that govern it. Ideal flow is a simplification of the real flow experienced in nature, however for many engineering applications this simplified representation has all of the properties that are important from the engineering point of view. And as we have learned from the conceptual design chapter, the aeronautical engineer should always select the tool that gives him an answer with sufficient accuracy within the shortest possible time span. The predictive methods such as the different vortex methods or the panel methods, are built on the assumption of ideal flow, also known as potential flow. This means that the methods neglect all viscous effects. Turbulence, dissipation and boundary layers are not resolved at all. However, lift induced drag can be assessed and, taking special care, some stall phenomena can be modelled.
3.3. Definitions Acheson’s [Ach90] way of defining an ideal flow, gives it the following properties: • It is incompressible, so that no ’dyed’ blob of fluid can change in volume while it moves. It may however, change in shape. • The density ρ (i.e. the mass per unit volume) is a constant, the same for all fluid elements and for all time t. • The force exerted across a geometrical surface element nδS is: pnδS
(3.3.1)
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where the scalar p(x, y, z, t) is the pressure, which is independent of the surface normal vector n(x,y,z) Assume a volume V created by a closed surface S being fixed in space, where the velocity field vector u(x,y,z,t) is not zero everywhere. The velocity component surrounding the volume along the outward normal will then be u · n, hence the fluid entering the volume through the small surface element δS in one unit time is: u · nδS. By integrating along the entire surface we get the net flow leaving the volume: u · ndS (3.3.2) S
But if we assume no mass production inside the volume, this must be identical to zero for an incompressible fluid. Using the divergence theorem we arrive at: $ ∇ · udV = 0 (3.3.3) V
As 3.3.3 must hold for every volume in the fluid we get: ∇·u=0
(3.3.4)
everywhere in the fluid. This mathematical formulation of the incompressibility constraint is the one used in the rest of this thesis.
3.3.1. Circulation Circulation is an important concept in the idea of ideal flow. It is coupled with the rotation of a velocity field as described below, in such way that if the rotation of the velocity filed is zero, then the circulation in the same field is zero. Figure 1 shows that the circulation is defined as the line integral of the → − → − projection of the vector field V on a small curve segment ds around any closed curve C as in 3.3.5. The integration is done across the complete closed path. By defining the circulation of a flow as done in Bertin and Smith [BS98] (pp74-76): I
− → − → V · ds
−Γ =
(3.3.5)
C
This line integral can be transformed into a surface integral in a three-dimensional space by Stokes theorem (3.3.6). I " − → − → → − ˆ V · ds = (∇ × V · ndA) C
(3.3.6)
A
ˆ where ndA is a vector normal to the surface enclosed by the curve C. Stokes → − theorem is valid for a simply connected region in which V is continuously differentiable, i.e. any points of infinite velocity must be excluded.
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Figure 1: Concept of circulation, from [BS98]. The line integral of the velocity field around the closed line C.
3.3.2. Flow description Complex flows can be regarded as being composed of two types of elementary flows. If the viscous effects in a flow are ignored, the possible distortions of a small control volume due to the shear forces can be neglected. Houghton and Carpenter [EH93] (figure 2) describes the only three ways by which the control volume can move: • Pure translation. The control volume can move along any path in the fluid, closed orbits included, and it keeps its axes parallel with the same directional vector at all times. • Pure rotation. The control volume rotates around an axis which is fixed in space. • General motion. A combination of translational and rotational motion. A flow which only exhibits the properties of the pure translation mode is called a potential flow, or an irrotational flow. The flow around an airfoil outside of the boundary layer and the wake is essentially irrotational. 3.3.3. Irrotational flow If we assume that the flow is irrotational, it has the mathematical implication → − that the curl of the velocity field ( V ) is zero (3.3.7) → − ∇× V =0
(3.3.7)
in all points on the surface A, bounded by the curve C. By refereing to Stokes theorem (3.3.6), it is apparent that the circulation of a closed loop (not encircling any singularities) is also zero: I − → − → − → V · ds = −Γ ≡ 0 (3.3.8) ∇× V =0⇔ C
This means that since the line integral of a closed path in the volume V is zero, the value of a line integral between two points in this volume is independent
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Figure 2: Motion of a small fluid volume in a) Irrotational flow and b) rotational flow. From [EH93]
of the integration path. However, the line integral between the points p to q 3.3.9: Z
q
− → − → V · ds
(3.3.9)
p
can only be zero if the integrand is an exact differential as in 3.3.10: − → − → V · ds = dφ
(3.3.10)
Or in three dimensions: udx + vdy + wdz =
∂φ ∂φ ∂φ dx + dy + dz ∂x ∂y ∂z
(3.3.11)
so that the Velocity potential φ can be defined as: → − V = ∇φ
(3.3.12)
Together with equation 3.3.7 it becomes evident that: ∇ × (∇φ) = (0, 0, 0)
(3.3.13)
Which always is true. Thus a velocity potential φ(x, y, z) exists, which means that the partial derivative of φ in any direction is the velocity component in that direction. An irrotational flow as described above is also termed a potential flow, due to the presence of the velocity potential φ. An interesting observation is that in an irrotational flow, that energy can only be extracted by two mechanisms: by moving in the flow field, or by generating vorticity.
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3.4. Potential Flow When considering a vector field where the rotation along any closed path is zero, the conclusion arises that the line integral between two points in the field is independent of the path, or constant. Such a flow field is called conservative according to Ramgard [Ram92]. The concept of conservative fields allows the definition of the potential. 3.3.13. This mathematical theory can be, and is, applied to many physical problems. Among these are the theory of electric potentials and the theory of velocity potentials in flow fields. In the case of a fluid flow the field is defined as follows for an irrotational (3.3.7), incompressible (3.3.4) flow by Andersson [JA98]: ∇·V =0
(3.4.1)
Further, as the flow is irrotational [3.4.2],
Hence [3.4.3], or [3.4.4]
V = ∇φ
(3.4.2)
∇ · (∇φ) = 0
(3.4.3)
∇2 (φ) = 0
(3.4.4)
Equation 3.4.4 is Laplace’s equation for which there is a number of solutions. Laplace’s equation is a second order linear equation, see Boyce and DePrima [WB92] (pp.113-130), and being so it is subject to the principle of superposition [WB92] (p123). This states that if y1 and y2 are two solutions of the differential equation L(Y), then the linear combination c1 y1 + c2 y2 is also a solution for any values of the constants c1 and c2 . Or, as Anderson [JA98](p180) puts it: ”A complicated flow pattern for an irrotational, incompressible flow can be synthesized by adding together a number of elementary flows, which are also irrotational and incompressible.”. Such elementary flows are the point source, the point sink, the doublet and the vortex line, each being a solution of Laplace’s equation. These may be superposed in many ways to create the formation of line sources, vortex sheets and so on. For this thesis, the line vortex is the most important one. As one may use an arbitrary number of singularities, or elementary flows; the concept of using numerical methods is close at hand. Today, a wide variety of methods exist and one of them is the vortex lattice method (VLM) which will be discussed in the later chapters.
3.4.1. Boundary conditions Laplace’s equation does not contain any time-dependence, hence the solutions will be time-independent. In fluid terms this means that the flow will be steady and will not change over time. There are different kinds of boundary
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conditions, but the one used throughout this thesis is the Neumann formulation. The Neumann boundary condition prescribes that the normal velocity at the boundary is zero. It is also known as the flow tangency condition, or no cross flow condition. It means that at the boundary (e.g the surface of a wing) the flow must be parallel to the surface. The mathematical formulation is: ∇φ · nˆ = 0
(3.4.5)
3.5. Elementary flows In the following section, the basic elementary flows that are solutions to Laplace’s equation will be presented. The uniform stream and the straight vortex segment are both solutions to Laplace’s equation and they the two elementary flows that by the principle of superposition can be combined in such way to create the geometrical make-up of an aircraft. 3.5.1. The uniform stream The uniform stream is used to describe the undisturbed flow. When it is aligned to the x axis it is defined in Cartesian coordinates as: φ = U∞ x
(3.5.1)
Or, if it is set at an angle of attack α relative to the x axis: φ = U∞ (x cos α + ysinα)
(3.5.2)
Proving that this is a solution to Laplace’s equation by ∇2 φ = ∇ · (∇φ) = ∇ · (U∞ (cos α, sin α, 0)) = (0, 0, 0) = 0
(3.5.3)
3.5.2. The vortex singularity Another solution to Laplace’s equation is the vortex. For the scope of this thesis, the vortex is the most important one. The vortex is the only flow besides the uniform stream above, that occurs in nature. Due to this natural occurrence, vortical flow has been thoroughly researched. This research has resulted in more requirements on the vortical flows, besides the no curl and no divergence criteria of the ideal flow. Vortex theorems The German mathematician Hermann von Helmholtz (1821-1894) was the first to make use of the vortex filament, or vortex line, concept. Anderson [JA91] (p323) lists Helmholtz’ vortex theorems as: • The strength of a vortex filament is constant along its length. • A vortex filament cannot end in a fluid; it must extend to the boundaries of the fluid (which can be at ±∞) or form a closed path. In addition, according to Acheson [Ach90] (pp. 162-163)
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• The fluid element that lie on a vortex line, continue to lie on a vortex element, i.e. the vortex lines ’move with the fluid’. The last remark is a direct result of the lack of viscosity, thus inexistent dissipative phenomena, and the conservation of energy. There are different ways of modelling vortex flows, the simplest one is the potential vortex, described below as the infinite vortex line. Finally, Kelvins circulation theorem states that: An inviscid irrotational flow will remain irrotational with the mathematical formulation: ∂Γ =0 (3.5.4) ∂t The physical interpretation is that along any closed circuit not encompassing any singularities, the rate of change of the circulation is zero. Which then induces that without viscous effects, the vortex lines will continue to exist forever. The Infinite Vortex Line The three dimensional potential vortex line, extending from infinity to infinity is derived from the two dimensional vortex. It has the following velocity potential in cylindrical coordinates [r, θ, z]: φ=
Γθ 2π
(3.5.5)
Thus, the velocity field is: → − Γ V = ∇φ = 0, ,0 (3.5.6) 2πr In equation 3.5.6 r is the radial distance from the vortex axis. A graphical representation of the tangential velocity is shown in figure 3 The velocity field equation 3.5.6 indicates that there is no flow in the radial direction, and no flow in the axial direction. The flow is centered around the vortex axis. Checking for compliance with Laplace’s equation gives: " # 1 ∂ Γ ∇ · (∇φ) = 0+ +0 =0 r ∂θ 2πr The rotation of the velocity field is: e → − 1 ∂r ∇ × V = ∂r r 0
reθ
∂ ∂θ Γ r 2πr
ez ∂ ∂z = [0, 0, 0] 0
As with the other elementary flows, the field is undefined in the flow-generating singularity itself. For the potential vortex, this means that all of the vorticity is concentrated in the vortex core of infinitesimal thickness.
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Figure 3: The radial dependence on tangential velocity of an infinite length vortex line of unit strength. For a real vortex in viscous flow, the maximum tangential velocity of would be limited by a core exhibiting solid rotation.
The finite length vortex According to the law of Biot and Savart, the velocity dV induced by a vortex filament of strength Γ and length dl is: Γ(dl × r) (3.5.7) 4πr3 Vortex filaments arranged on a straight line between points A and B, induces the velocity dV at a point p, (figure 5) according to the law of Biot and Savart (3.5.7): dV =
Γ sin(θ)dl (3.5.8) 4πr2 −→ Integrating along the finite length AB (R0 ) gives the induced velocity, or downwash, in the collocation point C by the entire vortex line R0 : Z α Γ Γn Vθ = (cos α − cos β) (3.5.9) sin(θ)dθ = 4πr β 4πr dV =
A special case is the infinitely long vortex line. In that case α = 0 and β = π for which the induced velocity becomes the infinite vortex line according to equation 3.5.6 If we take equation 3.5.9 and assume a straight vector of finite length, this would transform the expression into:
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Vθ =
Γ 4πr
Z
α
β
39
sin(θ)dθ =
ηΓn 4πr
(3.5.10)
Were: η = cos α − cos β The divergence of this velocity field is then: # " 1 ∂ ηΓ ∇ · (V) = +0 =0 0+ r ∂θ 4πr Checking the velocity field for rotation gives: er reθ ez → − 1 ∂ ∂ ∂ ∇ × V = ∂r ∂θ ∂z r ηΓ 0 r 4πr 0
= [0, 0, 0]
Hence, for the velocity field of a vortex filament of finite length, there is a velocity potential and this potential is a solution to Laplace’s equation. This opens up for the possibility of crafting a velocity field of arbitrary shape, using nothing else but segments of finite vortices. This is done by numerous formulations in different software implementations, some of which are discussed in the following chapters. However, in order to comply with the Helmholz vortex theorems, the vortex segments cannot be allowed to end inside the fluid. This can be taken care of by letting the segments be joined into a loop. By adding a number of nodes into which two or more vortex filaments can connect (see figure 4), a network of vortex filaments can be created. As no vorticity is produced nor destroyed in the node points, we can use Kirchoff’s junction rule 3.5.11 as formulated in Benson [H.B91](p 546) N X
Γn = 0
(3.5.11)
n=1
Were Γn is vortex segment n entering a node point. Thus, the algebraic sum of the vorticity entering and leaving a node is zero, as shown in figure 4. Useful vortex math When programming an aerodynamic software based on a method that uses vortices, it is good to have the vector relations expressed as explicitly as possible. The following section gives the equation for induced velocity of a vortex of finite length, expressed with vector multiplication, division and simple algebra. With the nomenclature of figure 5 and by using vector algebra and transforming we get: − − |→ r1 × → r2 | rp = (3.5.12) r0 cos β =
→ − − r0 · → r1 r0 r1
(3.5.13)
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Figure 4: The sum of all vortex strengths (Γ) entering a node (n) must be zero. Hence, P for node 2: 4n=1 Γn = 0
Figure 5: Nomenclature for calculating the induced velocity of a finite vortex segment according to Biot-Savart, from Houghton & Carpenter [EH93]
cos α =
→ − − r0 · → r2 r0 r2
Using these three and 3.5.9 we get the following expression: " !# − − → − − Γn → r1 × → r2 → r1 → r2 − V= r · − 0 − − 4π |→ r1 r2 r1 × → r2 |2
(3.5.14)
(3.5.15)
Which is very useful in the implementation of vortex lattice methods. 3.5.3. Electrical current analogy It is sometimes useful to have an analogy of thoughts. In the case of vortex lattice such an analogy is the electrical current analogy. Consider every vortex line to be an electrical conductor, then the magnetic field forming around this conductor would be the equivalent of the flow field of air forming around a vortex. The free stream is represented by a uniform magnetic field, such as it would be inside a solenoid. The cross product of current and magnetic field vectors gives, together with the length of the conductor, the force vector acting on the conductor. This implies that the vortex lattice method is not only
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suitable for numerical analysis, but that an analogy-engine could be built to solve the same problem.
3.6. Virtues of predictive methods Predictive methods can be used for other applications than conceptual aircraft design, this section deals with some of these other applications. Regarding predictive methods as coarse tools they have a natural place early in the design process, when the design outline is still rough and undetailed. The biggest virtue is the speed of the computations. The predictive methods give an instantaneous analysis of the aerodynamic in the conceptual design phase. However, there are other applications, besides conceptual design, to the predictive methods as well. These are three major applications for predictive methods software: • Industrial design applications, as in aircraft conceptual design. • Educational applications, tutoring in the relation between shape and aerodynamic response. • Scientific applications, in interdisciplinary research e.g. fluid-structure interaction. In industrial applications, time is often an important factor; and since most industrial projects tend to be large, one can accept a certain amount of approximations in the computations in order to have results ready for dissemination quickly. The predictive methods are very compatible with the ”fuzzy” geometry at the conceptual level, e.g. the airfoil is not defined yet, hence a panel method cannot be used. These approximations then require an iterative design process where more computational accuracy can be added in the later stages. Three important aspects of software for conceptual design in industry are: • Fast answers. • Approximations accepted. • Simplicity of software, in order to have engineers focusing on design problems, rather than software problems. At universities teaching aerodynamics, the need for computational aerodynamics is apparent. Mason [Mas02] addresses the issue of software in education. He concludes that teaching the use of computers is necessary, but takes time from the basic topic, aerodynamics. Hence the softwares in aerospace education have a somewhat different set of requirements than the industrial applications. For pedagogic reasons, it is often very useful to be able to show partial results during a computation. An open source approach is suitable, regardless if the material to be taught is the theory, the method or the programming application. By supplying the source, rather than a black-box software, the student can control the learning curve himself by following the computation at different levels of the software. Three important aspects of software for aeronautical education are: • Detailed partial results during a computation.
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• Provisions to examine and change the code. • Linear learning curve, with not-to-high steps along the way. When predictive methods are used for scientific applications, the accuracy of the results are important. Just as in the educational aspect, there is a need to be able to examine the code in detail and also to be able to modify the code to suit the research project at hand.
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4. The Classical Vortex Lattice Method and It’s Implementation This chapter will deal with the classical vortex lattice method (VLM), the assumptions behind them and how they are programmed into computer software. The restrictions on their use, and what the expected computational results may look like will also be assessed.
4.1. Historical Background A historical background to the vortex lattice method is given by Mason in the on-line textbook Applies Computational Aerodynamics Text/Notes [Mas]. Due to the relatively compact code of the VLM, the method and was among the first computational schemes to assist aerodynamicists in estimating aircraft aerodynamics. Mason describes the main features of the classical vortex lattice method as: • The singularities (combinations of vortex lines) are placed on a surface. • The non-penetration condition is satisfied at a number of control points. • A system of linear algebraic equations is solved to determine singularity strengths. • It is oriented toward lifting effects, and classical formulations ignore thickness. • The Boundary conditions (BC:s) are applied on a mean surface, not the actual surface (not an exact solution of Laplaces equation over a body, but embodies some additional approximations, i.e., together with the first item, we find δCp , not CPupper and CPlower ) • The singularities are not distributed over the entire surface • It is oriented toward combinations of thin lifting surfaces The term Vortex lattice was dubbed by Falkner in 1946. The method itself had been developed since the 1930s, however due to the numerical property of the method, it required a, for the time, large amount of hand computations to produce any results. Bertin and Smith [BS98] (pp301-305) shows an example of a hand computation of the lift produced by a swept wing modelled by four span-wise vortex segments. Although the computation disregards the induced drag, and all of the steps in the computation are not written out in full, this relatively simple case covers four pages. The vortex lattice method has been thoroughly investigated since computers became available as scientific tools in the 1960s. In the United States, NASA developed the NASA-Langley Vortex Lattice FORTRAN computer program as described in [JL82] (NASA TM 83303 ). This code can handle four lifting surfaces in the same simulation. Due to it’s availability, it has become a standard code in the US. In 1975, NASA held a workshop dedicated to vortex lattice methods. The proceedings [Ano76] cover the vortex lattice method in great detail. The historical evolution of the vortex lattice method was covered by John
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DeYoung in the opening talk of the workshop. DeYoung marks the beginning of the Computational Fluid Dynamics (CFD), and vortex lattice era, as being a paper by L.F. Richardson in 1910. Richardson was the first to formulate a numerical approach of solving partial differential equations (PDE:s). As PDE:s are used in modelling various types flow, this paper had impact on the numerical modelling used in aerodynamics. In 1918 Prandtl formulated the lifting-line theory, where the chord loading on a wing is concentrated on a single load vortex, with a trailing vortex sheet. The vortex system formed a ”Horseshoe” arrangement. Hence, the lifting line model is a similar model to the of vortex lattice methods. V. Falkner extended the lifting line theory by covering the entire wing with a grid of horseshoe vortices (see figure 5) and predicting the surface loadings. In one report he uses the title ”The solution of Lifting Plane Problems by Vortex Lattice Theory”, which is the first use of the term Vortex Lattice. In Sweden, the work on vortex lattice methods was pioneered by Sven Hedman who developed the method further and used it to study the loadings on a thin elastic wings [Hed66]. Hedman also developed the VLM software Virgit. The Swedish term for vortex lattice methods is VirvelGitterMetod. The Virgit code has been used in the aerodynamics education in the aeronautics department at KTH. The Vortex lattice method is built on the assumption of potential flow as described in chapter three. The inviscid velocity field of the air around the aircraft can be modelled by a distribution of discreet vortex segments. The VLM is useful when describing fluid motion around different kind of surfaces, in different kinds of fluids. In the case of aerodynamics the fluid would be air, and in the case of hydrodynamics the fluid would be water. The vortex lattice method is not sensitive to the type of fluid, as it neglects viscosity. However, the density of the fluid enters the equations at some point, which will be discussed below. VLM:s can compute the flow around a wing with rudimentary geometrical definition. For a rectangular wing it is enough to know the span and chord. On the other side of the spectrum, they can describe the flow around a fairly complex aircraft geometry, with multiple lifting surfaces with taper, kinks, twist, camber, trailing edge control surfaces and many other geometric features. By simulating the flow field, one can extract the pressure distribution or as in the the case of the VLM, the force distribution, around the simulated body. This knowledge is then used to compute the aerodynamic coefficients and their derivatives that are important for assessing the aircrafts handling qualities in the conceptual design phase. With an initial estimate of the pressure distribution on the wing, the structural designers can start designing the load bearing parts of the wings, fin and tailplane and other lifting surfaces. Additionally, while the VLM cannot compute the viscous drag, the induced drag stemming from the production of lift can be estimated. Hence as the drag must be balanced with the thrust in the cruise configuration, the
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propulsion group can also get important data from the VLM simulation. In this sense, the aerodynamics group has to provide initialization data to the other design groups. In the section below, the basics of the vortex lattice method will be explained. How the vortex equations of the preceding chapter can be put into useful service will also be discussed.
4.2. The Vortex Lattice Method In its simplest form, the computer implementation of a VLM does not require more than 100 lines of software code in an advanced programming language. This simplicity makes it a good exercise for engineering students to create their own Vortex lattice implementation. From the most rudimentary programmes more features can be added on with an almost linear increase of software complexity and programming workload. Below, the history of the VLM connects to the first types of implementations and how thy have been expanded to allow more simulation features, both in aircraft geometry, and in post processing capabilities. 4.2.1. Vortex Lattice Methods in Aerodynamic Design The method is widely used in industry and in government institutions for aerodynamic computations. However, it’s use in integrated design environment is limited. There are two examples of vortex lattice method integrated in an aircraft development system, and those are Isikveren’s Qcard [Isi02] and Chudoba’s AeroMech [Chu01]. The vortex lattice method provides good insight into the aerodynamics of wings, including interactions between lifting surfaces. Typical analysis uses (in a design environment) include (from: Mason [Mas]) • Predicting the configuration neutral point during initial configuration layout, and studying the effects of wing placement and canard and/or tail size and location. • Finding the induced drag, CDi , from the spanload in conjunction with farfield methods. • With care, estimating control and device deflection effectiveness. • Investigating the aerodynamics of interacting surfaces. • Finding the lift curve slope, CLα , approach angle of attack, etc. Typical design applications include: • Initial estimates of twist to obtain a desired spanload, or root bending moment. • Starting point for finding a camber distribution in purely subsonic cases. 4.2.2. Overall formulation Vortex lattice methods can be formulated in a number of ways. But they all share the common trait that the induced flow field is generated by
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superposition of straight vortex elements. By assuming a distribution of N vortices of unknown strength over a surface and applying an impermeability, or tangential flow, boundary condition on the surface; the vortex strengths can be solved for. With the vector length of the vortex segments, the resulting flow field and the vortex strengths the forces acting on each vortex segment can be computed. The vortex lattice method is described in detail by Bertin and Smith [BS98], and by Moran [Mor84]. In the vortex lattice method, lifting surfaces are modelled by two dimensional surfaces. The surfaces does not have to be flat, but they must be infinitely thin. These surfaces are then divided into panels, which could be distributed evenly or by some analytical scheme. Each panel is equipped with a horseshoe vortex which is central to the standard implementation of the vortex lattice method (See the vortex singularity distribution section 4.2.4). For extended implementation other vortex distributions may be used. En example is the Tornado software developed by the author, which uses vortex slings (see the Tornado chapter). These vortices are composed of the potential flow vortex lines described in chapter three. Furthermore, each panel is equipped with a Collocation point where the flow tangency, a.k.a. the impermeability boundary condition, should be satisfied. . The induced velocity from each horseshoe vortex (strength unknown) at each collocation point is computed as a function of the distance from the vortex to each collocation point. These induced velocities forms the influence matrix. The system of equations to be solved is then the vortex strengths, the influence matrix and the right hand side being the onset air, the freestream’s projection on the surface normal in the collocation point. If we allow ourselves to neglect the induced drag, the lift acting on each panel equals the panel vortex strength times the freestream velocity according to Kutta-Joukovsky’s theorem 4.2.1. If we want to resolve the induced drag as well, the induced velocity at each vortex needs to be computed (i.e. computing a new influence matrix). The induced velocity at the vortex, Inwash is added to the freestream and the resulting total force is then the cross product of the local velocity and the span-wise vortex vector.
L = ρVΓl
(4.2.1)
When the local forces have been resolved, it is just a matter of post-processing to acquire the pressure distribution, aerodynamic coefficients etc. By performing computations at different flight conditions or with different geometries, the aircraft performance sensitivity to different design parameters can be assessed.
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4.2.3. Panelling Once the planform is defined by the user input, the method (henceforth referred to as the program) must have a provision to divide the quadrilateral planform shape into panels. This usually means for-loops and space stepping in the implementation layer. There are some different ways to layout panels, the first one is the linear panel distribution described in figure 1.
Figure 1: Linear panel distribution
Another way of distributing panels, is by the span-wise cosine spacing proposed by Moran [Mor84] (p. 132). This approach will yield a faster lift convergence with respect to the number of panels as it resolves the rapidly changing pressure coefficient at the leading edge with higher resolution. An example distribution is shown in figure 2 The cosine spacing scheme can also be applied in the chordwise direction, as done by G.R. Hough in [Ano76] (pp325). Or chordwise as well as spanwise, as done by C.Lan [Lan] to yield even better lift convergence. Additionally, by inserting a strip of a quarter-panels width at the wing tip which should not be meshed (see figure 3) G.R. Hough showed in [Ano76](pp325) that the error in lift slope dropped from 4% to 0.5% for a wing with ten span-wise panels. The insert strip makes the Kutta condition, -no edge circulation, to be valid at the wing tip as well as the trailing edge. Figure 3 shows a wing with an inserted unpanelled strip.
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Figure 2: Cosine span-wise panel distribution
4.2.4. Vortex singularity distribution Once the panels have been defined on the lifting surface, the singularities needed to induce the necessary flow field is placed at appropriate locations. The standard vortex lattice method uses the horseshoe vortex. Horseshoe vortex Since solutions to Laplace equation are possible to superposition, it is also possible to create complex flow fields using line vortices, as long as all vortex paths are closed or goes off to infinity. By using equation the law of Biot-Savart, Moran [Mor84] models the induced velocity of a horseshoe vortex by combining two semi-infinite vortex elements with one finite element. All three vortex segments have the same strength to satisfy the Kirchhoff junction rule (eqn. 3.5.11). In the standard model the induced velocity, or downwash from the entire horseshoe system is combined into one analytical expression. To have one expression for the downwash of one panel saves computation time. The alternative would be having three simpler expressions for the downwash of the line vortices, and then combining their downwash numerically. Figure 4 shows the position of the horseshoe vortex on a panel. The span-wise vortex segment should cross the panel at it’s quarter chord point, this was initially a rule-of-thumb used in Prandtl’s lifting line theory selected on it’s property to reproduce both the lift and the pitching moment of a flat plate. Bertin and Smith [BS98] evolves the c1/4 rule which is motivated from thin
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Figure 3: Insert of an un-panelled strip at the wing tip
airfoil theory results, where the c1/4 point is the position on a wing where the pitching moment is independent of the angle of attack. In a sense then, the vortex position is governed by the speed of convergence for the pitching moment. According to Mason [Mas] (p.6-23) The quarter chord rule was first proposed by the Italian Pistolesi. The position of both the collocation point and the vortex position is assessed more mathematically by Lan [Lan]. The position of the collocation point, where the boundary condition should be satisfied, in relation to the cross-span vortex segment is examined by Bertin and Smith [BS98] (p192). When considering the downwash from a vortex line on a panel (see figure 6), the induced velocity in the collocation point is defined as: V=
Γ 2πr
(4.2.2)
with Γ being the vortex strength and r being the distance from the vortex to the collocation point. If the flow is to be parallel to the surface of the panel, the induced velocity in the collocation point must equal: α ≈ sin α =
U Γ = U∞ 2πrU∞
(4.2.3)
Then, by using another result from the thin airfoil theory, the lift equation for a thin wing of infinite span, the sectional lift per unit span l is:
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Figure 4: Panel layout, with horseshoe vortex and collocation point
2 l = ρ∞ U∞ cπα = ρ∞ U∞ Γ
(4.2.4)
Substituting the angle of attack α of (4.2.3) into equation (4.2.4) yields: 2 πρU∞ c
Γ = ρ∞ U∞ Γ 2πrU∞
and solving for r gives: c (4.2.5) 2 The result in equation 4.2.5 then depends on the assumption of a small angle of attack as in equation 4.2.3. Hence, the vortex lattice is also limited to small angles of attack. Additionally, equation 4.2.4 stems from the thin wing theory, thus limiting the thickness of the lifting surfaces to about 12% of the wing chord. r=
By setting up the geometry of each panel, with the position of each vortex and collocation point, we can set up a system of equations to solve for the unknown vortex strengths. 4.2.5. System of equations The core of the vortex lattice method is the formation of the influence matrix (D), which contains the induced velocity field at every collocation point by each vortex of unit strength. The influence matrix together with the unknown vortex strength vector, and the boundary condition vector - the free stream velocity vector projected on the panel normal in the collocation point forms the algebraic system of equations to be solved: D · Γ = V∞ · nˆ
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Figure 5: Panel layout, with vortices and collocation points on all panels
Or: w11 . . . wn1
... .. .
w1n wnn
Γ1 .. .
Γn
=
b1 .. . bn
(4.2.6)
The vortex strengths are solved for, and a new influence coefficient matrix (I) is created. The new matrix does not contain the induced velocity in the collocation points, instead it contains the downwash as the midpoint of the cross-panel vortex. This new influence matrix is then multiplied with the known vortex strengths to yield the local induced velocity at each panel: I · Γ = V induced i11 . . . in1
... .. .
i1n inn
Γ1 .. .
d1 . = . . Γn dn
(4.2.7)
This local induced velocity is then added to the free stream velocity vector to form the local, or effective, flow field at the mid point of each panels cross-vortex. V e f f = V induced + V ∞
(4.2.8)
Then, by employing the Kutta-Joukovski theorem, the force acting on each panel can be resolved by:
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Figure 6: Planar airfoil indicating location of control point where the flow in parallel to the surface.
F = ρV e f f × (Γl)
(4.2.9)
Were Γ is the vortex strength, l the direction and length of the cross-span vortex segment and ρ is the density of the air.
4.3. Standard implementation In the section below, the standard vortex lattice formulation will be discussed together with the kind of restrictions that guides the expected output. The standard implementation is the simplest possible implementation of the VLM and it is the one described in most textbooks such as [Mor84], [BS98], [EH93] and [JA91]. The standard implementation will only cover about 100 lines of code. The core of the extended vortex lattice methods are however, not much larger. As in many software applications most of the programming effort goes into user interface and pre- and postprocessing. 4.3.1. Required input for a simple test case This section deals with the required input for the standard implementation of the vortex lattice method. The parts of the aircraft which is represented in the tree-view drawing of conceptual design: It’s wings, tailplane, fin, fuselage and control surfaces, are the parts of the design that are subject to aerodynamic forces. The thin, lifting surfaces are the parts of the aircraft experiencing the largest aerodynamical load are the ones suitable for modelling in the VLM, hence the aircraft wing geometry has to be entered into the program. The geometry of a real aircraft can be quite complex to describe, albeit for the conceptual design phase the geometry description is simplified. The text
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below describes the simplest possible geometry worth examining. In order to be useful for design purposes, the method needs input in the form of aircraft geometry parameters such as: • • • •
Root chord Taper ratio Span Sweep
With respect to this thesis, a standard vortex lattice method will be regarded as one which can model the following geometrical features of an aircraft: Aircraft geometry • • • • • •
One lifting surface Span Root chord Simple taper Simple sweep Flat wing, no camber
In addition, the aircraft has to be set at a certain flight condition. Depending on whether absolute units (i.e. forces and moments in N an Nm rather than non-dimensional coefficients) are needed, some extra input is required. The standard model needs the following data: Flight conditions • Angle of attack • Velocity* • Air density* (pressure height) *Needed for absolute dimensional results. Obviously, an aircraft has many more features and may fly at many other flight conditions than the ones listed above. However, in order to model these, the standard model needs to be modified slightly. These modifications will be discussed later in this chapter .
4.3.2. Computed properties When the forces acting on each panel have been resolved, the post processing operations that need to be done are pretty straight-forward. The area of each panel is easily computed, and with that the local pressure coefficient can be assessed. Note that since the model wing in the vortex lattice method does not model any thickness, the resulting pressure coefficient is the difference pressure coefficient ∆CP between the upper and lower sides of the airfoil. For this standard implementation, the output is somewhat limited from a design point of view. The number of design parameters are insufficient to describe a generic aircraft well enough, more so because the flight condition is
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limited to different angles of attack, the only derivative possible to deduce is the α-derivatives. None the less, the standard implementation can answer some of the big questions regarding a one wing planform. These are described below: Lift coefficient Besides the pressure distributions, the lift coefficient for a specific angle of attack is one of the most important numbers for the aerodynamic designer. By summing up all the forces that are perpendicular to the free stream, F⊥ , i.e. the local lift, in the aircraft symmetry plane into one number, the lift coefficient is defined as follows: CL =
F⊥V∞ qS
(4.3.1)
Were q is the dynamic pressure in the flow, and S the wing surface area. Drag coefficient The drag coefficient is an equally important number in aerodynamic design. However, as the drag is composed of many different factors that depend on, among other things viscous effects (which are neglected in potential flow) and thickness effects (which are neglected in the standard vortex lattice method), the drag computation will not be complete. The drag component that can be assessed with a vortex lattice method, is the induced drag CDi . The induced drag contributes to about 40% of the total drag in the cruise flight condition, and about 80% of the total drag in landing and take-off configurations [Kro03]. The drag coefficient is the un-dimensionalized force component that is parallel to the free-stream (Fk ): CD =
FkV∞ qS
(4.3.2)
Drag Polar Both the lift and drag are functions of the angle of attack. This makes it possible to describe the drag as a function of the lift coefficient. An example of a drag polar for an Airbus A320 wing is shown in figure 7. Usually this relation is described as:
CD = CD0 + kC2L
(4.3.3)
The factor k can be expanded to give: CD = CD0 +
C2L πeAR
(4.3.4)
Were AR is the aspect ratio, and e is the planform efficiency factor or ellipse factor (for an entire aircraft, Oswald efficiency factor). The ellipse factor equals one for a wing with an optimum L/D, and less for less optimum designs. Hence, the ellipse factor is a good measure of merit of aerodynamic efficiency.
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Figure 7: Drag polar of an Airbus A320, in three different flap settings.
Glide slope, L/D Another important measure of merit is the glide slope, of the quota L/D. The glide slope is simply the slope of the drag polar. The maximum value of the glide slope corresponds to the angle of attack for which the wing will generate the most lift, for the least drag. Hence the designers try to incorporate this angle of attack as the cruise angle of attack, since the least drag corresponds to the least fuel consumption. Span load The local lift coefficient (Cl ) is not equal along the span. It may vary from tip to root in different manners, and as the airflow will separate at the span wise section with the highest Cl . It is important for the designer to keep the stalled section clear of any control surfaces, or clear of the wing tips. The separated flow, or stall, will create a drop in the Cl , and since this phenomenon is usually asymmetric a large rolling moment may result. The span load is connected to the ellipse factor, a wing with an ellipse factor of one has an elliptic lift distribution along the span. Although this is the most beneficial loading from a drag point of view, it is not ideal from a structural point of view. An elliptic lift distribution has it’s centroid of lift far out from the centerline, which creates a large bending moment at the wing root. The structures necessary to bear these loads will be heavy, thus requiring a larger
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wing surface than a wing with a parabolic lift distribution. This larger wing surface will penalize the design by increasing parasitic drag, which will not be seen in a vortex lattice simulation. An example of lift distribution is shown in figure 8.
Figure 8: Spanload of an Airbus A320, computed with the VLM implementation Tornado. Both in cruise configuration and with inner flaps deflected 22 degrees.
Lift slope As potential flow is linear, the lift slope for a wing of moderate to high aspect ratio modelled with a potential flow method will be a linear function. This is valid below the start of the stalled region (se figure 9), as stall is not modelled by potential flow. As the lift at zero degrees angle of attack is zero for a flat wing, we can assess the CLα derivative, the change in lift coefficient due to a change in angle of attack, by computing the lift coefficient at a specific angle of attack (in the linear region). The lift slope is then defined as:
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CLα =
∂CL Cl − 0 ∆CL = ≈ α−0 ∆α ∂α
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(4.3.5)
Figure 9: Linear region of the lift curve for different wings.
The lift slope is an important variable, together with the span loading (the span wise lift distribution) and data regarding the maximum angle of attack of the intended wing profile, the maximum lift coefficient CLmax of the planform can be assessed. The CLmax together with an initial guess of the aircraft weight can give the conceptual designer a first guess of the future landing speed, takeoff field length etc. Summary The aerodynamic properties:
• • • • •
Lift coefficient Drag coefficient Drag polar Lift slope Span load
are all important to the aerodynamic designer. By using a predictive simulation tool, he can determine the sensitivity of these parameters with respect to the design variables. Questions like ”What will happen to the drag polar if I change the taper ratio by 10%?” are crucial in the tradeoff involved in the conceptual design. By changing the geometrical features by a small amount, the designer can create Carpet plots as described by Raymer [Ray99] to perform the tradeoffs that will yield the best product. The example with the elliptic/parabolic lift distribution is a good example of a design tradeoff. All of the aerodynamic properties above can be simulated with the standard vortex lattice method. While not enough to design an entire aircraft, it has important educational properties in describing basic relations of a simple wing. In the following section, additions to the standard vortex lattice method will be discussed both in terms of their implementations and in what way they increase the simulation possibilities of the method.
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4.4. Model Hierarchy The standard vortex lattice model represents the simplest implementation of the vortex lattice method. There is however many additions that can be made that gives the program increased capabilities. Many of them are connected to geometric feature not yet discussed, and some with the flight condition. The list below starts with the standard formulation and then lists geometry or flow features that require increases in code complexity. • Standard implementation Cambered wing Trailing edge control surfaces Cranked wings (span-wise sections with different properties, e.g. sweep) Multiple lifting surfaces • Intermediate level implementation Twist Dihedral flexible wake Sideslip Roll, Pitch and Yaw rates (derived from results) • Advanced level implementation Compressibility effects (by the Prandtl-Glauert correction) Trefftz-Plane induced drag assessment Ground effect Formation flying effects Supersonic flow Time derivatives, such as depending on the rate of change of the angle of attack. Viscous effects estimation
4.5. Standard implementation To the standard formulation that is expressed in the aerodynamics textbooks, almost every available software, commercial or public domain has additional features. The entry level complexity section describes features that can be implemented with a two-dimensional approach. Since the wing is modelled as a flat plate throughout, there is no need to describe the z-position of the vortices and collocation points. Therefore, time and memory can be saved during computations by simply ignoring the third coordinate. The features of the entry level complexity vortex lattice softwares are described below. 4.5.1. Cambered wings To extend the geometry, the simulated wing could also be cambered. In a an implementation that uses horseshoe vortices, the wing is still regarded as a flat plate. As the wing is thin, the boundary conditions can be shifted to the chord line (see figure 10). That is, the normal of the cambered surface is calculated and the non-flow-through boundary condition is employed at the chord line. This approximation is common and used in a variety of methods.
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Figure 10: Boundary conditions shifted from the camber line to the chord line. The slope of the camber line is assumed to act on the chord line.
4.5.2. Trailing edge control surfaces Trailing edged control surfaces can be simulated by rotating the normals of the panels at the trailing edge (see figure 11). This method does not work for large control surface deflections, as the vortices themselves are not deflected. However, important coefficient derivatives due to flap deflections close to zero deflection can be assessed. A control surface deflection if often defined as δ with a subscript to denote which type of control it as, δ f lap δelevator etc. Hence, the change in pitching moment due to flap deflection can be defined as Cmδ f lap
Figure 11: A320 wing with trailing edge control surfaces deflected. Ailerons 20 degrees and flaps 30 degrees.
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4.5.3. Cranked wings Most modern aircraft have some kind of cranked planform. The wing is cranked when either the sweep or the taper ratio is changed along the span. Figure 12 shows a wing which is cranked in both sweep and taper.
Figure 12: Cranked planform
4.5.4. Multiple lifting surfaces Often, the designer is interested in interactions between lifting surfaces, such as main wing and tailplane, or canard and main wing. As the second set of wings will have their own geometrical features, the number of parameters, that the designer has to modify in order to assess the aircraft handling qualities, increases.
Figure 13: Multiple lifting surfaces of a Cessna 172. Picture composed of: Main wing, horizontal stabilizer and vertical fin.
One issue with two serial sets of lifting surfaces is the wake handling. As the
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wake vortices from the forward wing will pass over the panels on the second, special care must be taken to ensure that the wake vortices of the two wings line up. Otherwise, a vortex passing close to a collocation point will induce an infinite velocity in that point and by that causing the system of equations to have a very poor condition and yielding poor results. This issue will be further discussed in the Tornado implementation section.
4.6. Intermediate level implementation Most commercial VLM software packages contain the features in this section. The implementation of the intermediate complexity level is a discrete step more advanced than the entry level implementations. This is because all of the feature simulations capabilities in this section require that the code has provisions for a tree dimensional geometry (i.e. that all vectors are saved as three element variables). 4.6.1. Twist When adding twist to the layout, it implies that the geometric angle of attack varies with span, the design is no longer a flat plate but a mildly skewed surface. The twist will cause the two outgoing vortex legs from a panel to no longer be parallel, see figure 14. This too, could be approached by just twisting the collocation point normals. However, by using a three dimensional vortex, where the trailing vortices no longer stays in the xy-plane, the twist can be simulated with higher fidelity. Such a distribution is called a vortex sling.
Figure 14: Twisted wing partition.
4.6.2. Dihedral The dihedral is the angle by which the wing is inclined to the ground, or x-y plane. See figure 15. Winglets and vertical fins can be modelled by a 90 degrees dihedral. 4.6.3. Flexible wake The wake coming off the trailing edge does not necessarily continue in the direction of the chord line direction. More intricate implementations allow
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Figure 15: Wing with dihedral.
some controls over the wake model and it’s direction, either by manual control or by some automatic function. 4.6.4. Sideslip The angle of attack α is the angle the oncoming air has to the wing in the xz plane, the sideslip β is the angle the oncoming air has to the wing in the xy plane. In a sense, the oncoming air is defined in spherical coordinates with strength, angle of attack and angle of sideslip. 4.6.5. Roll, Pitch and Yaw rates The standard vortex lattice method can simulate any steady flight condition. Rotations around the aircraft principal axis are also steady flight conditions. Hence roll, pitch and yaw rates can be simulated by applying a rotational airflow in the setup up the boundary conditions. In simulating these rotational airflows, a flexible wake is needed to evaluate the secondary effects such as wake-fin interaction in yaw.
4.7. Advanced level implementation Advanced vortex lattice methods may also include the special features listed below. Some of the features will only be discussed briefly here, to be assessed more thoroughly in the extended vortex lattice chapter. 4.7.1. Ground effect When a wing operates close to the ground, interference effects will change the aircraft flying qualities. As the ground hinders the downwash from the wing, the induced drag will decrease. Mathematically, the vortex lines are mirrored by the flat ground as in figure 16. The solid ground can be modelled by adding the mirror vortex with opposite sign. This will yield the same flow field, as if the entire ground below the vortex was modelled in all directions.
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The flow field around an aircraft close to the ground can be modelled in the same way. The ground effect can be modelled by adding a mirror image of the aircraft vortex system, as in figure 17
Figure 16: An potential flow vortex and it’s mirror image, replacing the flat ground.
Figure 17: An entire aircraft, and it’s mirror image, replacing the flat ground.
4.7.2. Formation flying effects Recent accidents initialed by the interaction of an aircraft and the vortex system of another aircraft highlights the need for simulating formation flying aerodynamics. As in the case with the ground effect, the vortex system of another aircraft can easily be included in the simulations. 4.7.3. Time derivatives Time varying properties such as CLα˙ , the change in lift coefficient due to rate of change of the angle of attack, cannot be modelled with the standard wake
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model with infinite flow-wise straight vortex lines. Instead, the wake must contain span wise vortices, either as sheets or as lumped lines going span wise across the wake. They are needed as an increment in lift requires that span wise vorticity is dumped into the wake, the span wise vortex lines in the wake closes the vortex circuit in order to satisfy Kelvins circulation theorem. If the increase in lift is done as a unit step, the resulting vortex system will equal the starting vortex as described by Anderson [JA91] (pp 264). 4.7.4. Supersonic flow Vortex lattice methods can also be used in supersonic flow. Bertin and Smith [BS98] (pp483) evolves an implementation where each panel only receives downwash from the panels that are inside the sonic cone of influence. 4.7.5. Viscous effects estimation While not a part of the vortex lattice method itself, various viscous effects can be modelled by subroutines in the aerodynamic design program that contains a vortex lattice method. This is done to give an estimation of e.g. parasite drag (CD0 ) or maximum lift.
4.8. Available Classical VLM programs There is a number of vortex lattice implementations available for the design engineer. The following section will cover some of the program that are regarded as classical vortex lattice methods. 4.8.1. VLMPc The VLMpc program is the IBM/Microsoft compatible version of Lamar’s vortex lattice method. The code is written in Fortran, and the source is described in [RM71]. The VLMpc software is also described in Mason’s web-cased course notes [Mas] (ppD40) as: The VLMpc program can handle two complex planforms, i.e. two cranked wings, and estimating their subsonic aerodynamic characteristics. The configuration can be modelled with up to two planforms, all of which must extend to the plane of symmetry (xz-plane). The fuselage is represented by its planar projection; experience to date indicates that this produces acceptable global forces and moments for most wing-body-tail configurations. Winglets can be modelled, but the dihedral angle must be less than 90.0 degrees and greater than -90.0 degrees. Both upper (positive dihedral) and lower (negative dihedral) winglets can be accounted for in this code. The program uses the chord plane as its solution surface which may be inclined due to dihedral. Moreover, the only out of ”X-Y plane” displacement specifically allowed for is the dihedral. Local camber and twist is assumed to be small and can be represented by its slope projection to the local solution surface. The wind and body axes are assumed to be coincidental in the code,
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which indicates that the horseshoe vortex distribution distribution has been used, and that the induced drag must be estimated by Trefftz-plane analysis. 4.8.2. AVL The Athena Vortex Lattice program, developed by Harold Youngren and Mark Drela at MIT. This software handles multiple lifting surfaces and utilisers the Prandtl-Glauert compressibility correction for high subsonic Mach numbers. Induced drag is computed by the Trefftz-plane analysis. The singularity distribution used is the standard horseshoe vortex, which means that the wake is parallel to the geometrical x-direction. Thus the angle of attack is limited to small values. AVL handles any type of wing camber for the lifting surfaces. By interpolating a spline as a camber surface from user-defined points on the camber line, the program computes the camber normal at the collocation points. The user interface allow the user to investigate trimmed CL at a specific angle of attack, or to create the stability derivative matrix for any flight condition. The derivatives are expressed entirely in the AVL stability axes (both the forces as well as rates). AVL and the NASA standard stability axes differ in having the x and z directions reversed. 4.8.3. Virgit Virgit is a vortex lattice method developed by Sven Hedman. It’s name comes from the Swedish word for vortex lattice VIRvel GITter. This implementation too, is made in Fortran. 4.8.4. Wing-Body This code is popularly known as the NASA Ames Wing-Body program. It is the outgrowth of a series of contracts with the Boeing Company for the development of a method of computing optimum camber and twist distributions on wing-body combinations at supersonic speeds. There are many local implementations of this program, such as the FFA-WingBody. Of the standard vortex lattice methods, Wing-body is special since it can model a circular fuselage.
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5. Extended Vortex Lattice Implementation This chapter will cover methods and codes that that are extensions to the vortex lattice method. The previous chapter started with the primitive end of ideal flow methods, the lifting line and the standard vortex lattice method. The wing model in these methods are thin wings, ignoring thickness effects. This topic will be carried on in this chapter closing the ideal flow methods discussion with the most advanced method of this family. The next step up in modelling fidelity would be panel methods. However, panel methods are not suitable for conceptual design due to the amount of data they require in the geometry definition. For example: the wing profile is usually not determined until the preliminary design stage. Hence the amount of available information in the conceptual design phase is too low to allow successful modelling of an aircraft with panel methods.
5.1. Rom In his book High angle of attack aerodynamics : subsonic, transonic and supersonic flows [Rom92], Rom introduces the Nonlinear vortex lattice method (NLVLM). The method utilizes the horseshoe vortex system to simulate lifting effects and also a source distribution to simulate thickness effects. The horseshoe vortices are modified into vortex slings, where vortices shed at the trailing edge and at the wingtip are divided into segments that follow the streamlines behind the wing. This approach allows the simulated wake to roll up in the same way as the real wake behind an aircraft wing. Some of the features of the NLVLM is described below. 5.1.1. Nonlinear VLM Even though the laplace equation is linear, the problem of evaluating the the potential φ becomes nonlinear because the spatial position and shape of the free vortex wake is dependent on the induced velocity from the wake itself. The wake is allowed to roll-up in the same way as observed in real flow. This phenomenon has impact on the aerodynamic results, among others that the lift slope becomes a function of the total lift. 5.1.2. Pressure distribution In the NLVLM the forces on each panel are not evaluated with the Kutta-Joukovski theorem, instead the local tangential velocity at the collocation point is computed, and from that the surface pressure is assessed by the Bernoulli equation. 5.1.3. Wake treatment In the standard vortex lattice method, the wake was modelled as a plane sheet of vortex lines coming off the trailing edge. The span-wise vorticity at the trailing edge must be zero (defined as the Kutta condition), as the flow must leave the airfoil smoothly. By the formulation of the standard method, the
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Kutta condition is automatically taken care of. In the NLVLM on the other hand, the Kutta condition is extended to be valid for all sharp edges, such as the wing tips or the leading edge. This in turn means that vorticity has to be shed from these edges, forming a non-planar wake. The wake is modelled as a series of segmented vortex lines, see figure 1 coming off all sharp edges on the planform. Each of the segmented vortex elements are allowed to move in the total flow field. As the flow field itself is dependent on the shape and the strength of vortex lines, the NLVLM requires an iterative approach to compute the flow field and the singularity distribution. The solution iteration has two loops, one inner and one outer. In the outer loop, the distribution is initialized with a flat wake and the influence matric is computed and the vortex strengths are solved for. Then, the wake is updated in the inner loop, by moving the segmented vortex line’s node points in the total velocity field until the vortex path is converged. Then, a new influence matrix is computed in the outer loop, and the vortex strengths are solved for. When the bound vortex strength is converged the computation continues as with the standard implementation. In this way, the wake roll up as observed in real flow, is captured in the simulation.
Figure 1: Flexible wake.
5.2. Lan One of the most important improvements of the vortex lattice method has been done by C.Lan in developing the Quasi-Vortex-Lattice Method (QVLM) [Lan], [Lan93] and [CL89]. The method addresses some singularity issues neglected in the standard VLM, and the chordwise position of the collocation point and lifting vortex is solved with an analytical approach. The leading edge suction of delta wings is modelled with much more accuracy in the QVLM than in the standard VLM, which does not resolve separates leading edge vortices. The method was implemented in a computer program, which was available through the Internet, both as a commercial software and as a public domain software. However, support for this program has ceased.
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The program, known as VORSTAB, was coded in Fortran 77, and had many interesting features, some of which will be discussed below. The code handles (from an advertisement listing, [Hsi03]): with lifting surfaces such as wings, vertical tails, horizontal tails, leading-edge flaps, strakes, ailerons, trailing-edge flaps, winglets, and fuselage, body of revolution, shape and user definable wing cambers, twisting, tapered wings and airfoil thickness. It covers both subsonic and supersonic (compressible flow) flows, and includes asymmetrical forebody vortex separation, vortex breakdown effect and vortex wake effect, etc. 5.2.1. Singularity distribution The singularity distribution proposed by Lan in [Lan] differs from the earlier Vortex lattice methods in that the vortex distribution is continuous in the chordwise direction, and discrete in the span wise direction. Thus, the wing is covered with a number of distributed vortex strips with the associated trailing vortices. in the span wise direction Besides the singularity distribution on the wing, the method allows for a body with circular cross section. 5.2.2. High angle of attack aerodynamics By the additions proposed in [CL89], the method became able to predict non-linear high angle of attack aerodynamics. The flow coming from the separation along the leading and side edges of a low aspect ratio wing significantly influences the flow field around the aircraft. These vortices are simulated by free vortex filaments emanating from the sharp edges (see figure 2). An iterative scheme is employed to give trajectory of the free vortex.
Figure 2: Vortex filaments shed at the leading edge, from [Lan]
An empirical model is used to predict the angle of attack for which vortex breakdown occurs, and the consequent decreasing effect on lift is assessed from analysis of experimental data.
5.2.3. Boundary layer separation Another limiting factor in high-angle of attach aerodynamics besides vortex burst, is the separation of the boundary layer. The VORSTAB code has provisions for simulating boundary layer separation by assuming that a separated flow will decrease the local angle of attack on the separated span
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wise station. The viscous two dimensional sectional lift coefficient is used for assessing the local angle of attack. 5.2.4. Wing thickness The VORSTAB code also models the wing thickness. The effect of thickness on the free vortex trajectories is computed by a source distribution inside the wing. The source strength themselves are computed with a Source lattice method. 5.2.5. Supersonic flow Late versions of the software also had provisions for supersonic flight conditions, but as for most ideal flow methods the results in the transonic region where poor.
5.3. Continuation In a sense the VORSTAB code, with a body model, and wing thickness closes the gap between lattice and panel methods. As stated in the opening section of this chapter, the next step in modelling fidelity would be moving on to panel methods. But then we are moving away from the quick turnaround computation time in the order of minutes in the case of vortex lattice methods, to the order of hours or days for panel methods.
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6. The Vortex lattice implementation: Tornado. The Tornado code is a vortex lattice method programmed to be used in conceptual aircraft design and in aerodynamics education. The work on the code begun in 1999 at the Department of Aeronautics (http://www.flyg.kth.se/) at the Royal Institute of Technology, in Stockholm, Sweden. The first version was released in 2001 together with the users manual and code description. This was a part of the Masters Thesis of the author, likewise the code developer. The aircraft geometry in Tornado is fully three dimensionally oriented lifting surfaces, with a flexible, free-stream following wake. Tornado allows a user to define most types of contemporary aircraft designs with multiple wings both cranked and twisted with multiple control surfaces. Each wing may have taper of both camber and chord, which may wary spanwise. The Tornado solver solves for forces and moments, from which the aerodynamic coefficients are computed. Aerodynamic derivatives can be calculated with respect to: angle of attack, angle of sideslip, roll-, pitch-, and yaw- rotations, and control surface deflection. If necessary all of these conditions may be applied simultaneously. Any user may edit the program and design add-on tools as the program is coded in MATLAB (tm), and the source code is provided under the GNU-General Public License. The core method stems from Moran [Mor84], but is modified according to Melin [Mel01b] in order to accommodate a three dimensional solution and trailing edge control surfaces. The most notable change is the extension of the theory of the horseshoe vortex into the vortex-sling concept, which is described later. The vortex sling is essentially an seven segment vortex line, which, for each panel, starts in the infinity behind the aircraft, reaches the trailing edge, moves upstream to the hinge line of the trailing edge control surface, then forward to the quarter chord line of the panel in question, going across the panel and then back downstream in an analogous way. The issue of the wake passing through the geometry at certain flight conditions is resolved by a collocation point proximity detection routine which automatically removes the influence from a vortex thread passing too close to a collocation point. The code is generally distributed in the hope that it will be useful, but without any warranty; without even the implied warranty of merchantability or fitness for a particular purpose. However, validation comparisons have been made by the author in which the code output is compared with experimental data in [Mel01b]. The test case is a Cessna 172, for which the Cessna Aircraft Company have released flight test data [ea57] and for which computation results from AVL and PMarc, a vortex lattice method and a panel method respectively, were available.
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6.1. Tornado features This section deals with different Tornado features. The current version of Tornado negotiates all of the features listed under the entry level and the intermediate level implementations in chapter four. 6.1.1. Geometry Tornado supports multi-wing designs with swept, tapered, cambered, twisted and cranked wings with or without dihedral. Any number of wings may be utilized as well as any number of control surfaces. Canards, flaps, ailerons, elevators and rudders may be employed. Winglets, fences and engine mounts may also be incorporated in the design.
6.1.2. Flight conditions Tornado allows variations in angle of attack, angle of sideslip and rotational rates in roll, pitch and yaw. 6.1.3. Preprocessor Tornado has a pre-processing capability, which is more user friendly than competition codes. An an example, the Vorstab code require the user to specify wing corner coordinates directly. This works fairly well for an untapered wing without sweep. As soon as the geometry becomes slightly more complex, the hand calculations the user must perform becomes prohibitive. The Tornado preprocessor allow the user to enter wing data in the standard aeronautical engineering format, namely by: • • • • • • • • • • • • • • •
Apex coordinates Span Taper Sweep Camber Dihedral Twist Symmetry Root chord Flaps Flap symmetry (If flap is present) Flap chord in parts of root chord (If flap is present) Number of panels in chord (X) direction Number of panels in span (Y) direction Number of panels on the flap (If flap is present)
6.1.4. Postprocessor Due to the accessibility of the Matlab plotting function, the Tornado post processor is able to compute and display: Angular sweeps, different rotation rates, derivative extraction spanload etc.
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6.1.5. The vortex sling The wake coming off the trailing edge of every lifting surface is flexible and changes shape according to the flight condition. For example: a rolling aircraft will have a cork-screw shaped wake, which will influence the aerodynamic coefficients. The classical horseshoe arrangement of other vortex lattice programs has been replaced with a vortex sling arrangement. It basically works in the same way as the horse shoe procedure with the exception that the legs of the shoe are flexible and consist of seven (instead of three) vortices of equal strength. This is shown in figure 1. Five vortex segments would suffice on order to cover the flexible wake, but two extra segments are added to allow for a deflecting trailing edge control surface.
Figure 1: The vortex sling, as implemented in Tornado. Seven vortex segments on each panel instead of three.
While the flexible wake is implemented in the extended versions of the vortex lattice method such as AVL, the allowance for the trailing edge control surface is unique.
6.2. Validation examples This section shows some computational examples of the Tornado code in comparison with established data. Prandtl’s Lifting line and Jones small aspect ratio theory. To give a test of Tornado capabilities, a calculation has been made on how the lift-curve slope depends on the aspect ratio for a flat rectangular wing. Figure 3 shows the computation done by Tornado as well as the values obtained using the Prandtl’s lifting line approximation, which is shown in figure 2, and the Jones’ small aspect ratio method. Prandtl’s lifting line approximation is
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Figure 2: Prandtl’s lifting line theory. The entire wing is modelled by a series of vortices running through the wing quarter-chord line. Vorticity leaves the wing in a continuous trailing edge vortex sheet.
more restrictive as it only is valid. for high aspect rations and does not allow taper or sweep.
Figure 3: Tornado, Prandtl’s lifting line and Jones’ small aspect ratio theory comparison.
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Swept wing example In [BS98] (p 291) there is a computational example of the VLM . The case is a swept wing with a total of 8 panels where the lift-curve slope should be determined. Figure 4 shows the plan form of the wing, as well as the panel layout for the wing. The same geometry was fed into Tornado and the results were: The lift-curve slope computed by Bertin and Smith equals 3.443 per radian, while the lift-curve slope calculated by Tornado equals 3.450, which represents a difference of 0.2%. This is considered a good correlation, since the two methods differ slightly.
Figure 4: Tornado, Prandtl’s lifting line and Jones’ small aspect ratio theory comparison.
6.3. Code Development The development of the Tornados software is ongoing, and through the user feedback described in Internet interaction chapter, new possible development areas emerge an a daily basis. The Tornado software is distributed according to the GNU General Public Licence (See appendix) in order to allow users to make their own code development and code extensions.
6.4. Choice of programming environment When developing a software suite, it is important to choose the of platform programming environment wisely. Essentially, the programming environments under consideration for the Tornado project where: FORTRAN, C or Matlab. Each of them having their own pros and cons. For the discussion relating to the choice of platform, the FORTRAN and MATLAB options will
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be discussed. The choice of platform for the Tornado software is the Matlab environment due to it’s beneficial characteristics described below. Matlab is a programming environment developed by The MathWorks. Matlab has a range of features which makes it ideal for a project such as the Tornado development project. Some of these features are described below. 6.4.1. Programming Language MATLAB integrates mathematical computing, visualization, and a powerful language to provide a flexible environment for technical computing. The open architecture makes it easy to use MATLAB and its companion products to explore data, create algorithms, and create custom tools that provide insights into complex simulations. The Matlab programming language is a high level language initially intended for numerical simulations. It’s name is derived from MATrix LABoratory. Since the VLM is a matrix problem, the Matlab capabilities are very useful. Matlab supports online execution and variable manipulation, as well as scripting or the creation of procedures. This makes Matlab ideal for code prototyping and development. Fortran on the other hand, does not have the capacity to let the user modify variables, in other ways than the executed program allows, during runtime. Code readability is also an important factor, as source code sometimes becomes the only available software documentation. This is an increasingly important factor if the software is to be distributed and developed by other persons than the original programmer. It is common knowledge that Matlab code is more readable than FORTRAN code. 6.4.2. Data Visualization Another trait of Matlab, is the availability of data visualization routines. Line plotting, bar graphs, contour plots, etc, all of them are predefined in Matlab making data visualization easy and fast. By no means is visualization of data generated in a Fortran program impossible. It just takes more time, and usually involves a third party software. For a visualization essential task such as numerical simulation of flow fields and aerodynamic characteristics, Matlab is clearly the preferred choice over Fortran. The Tornado code makes many function calls to Matlab’s built-in plotting functions. Matlab also has a provision for developing graphical user interfaces. Granted, this is a feature of Fortran as well, but certainly belonging to a professional programming level. 6.4.3. Program execution The program execution in Matlab can be done in several ways. Either by command-line manipulation of variables, or by running scripts or functions.
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There is the built in capability in Matlab to call routines written in C, C++ or Fortran. The Tornado software can be run in either mode, and in such allowing the user to inspect certain variables, or to extract them for other user-developed subfunctions. Many available Vortex lattice methods have been programmed in Fortran, and quite often the codes are a decade or more older. The VLM:s that are in use today are then ’Dusty decks’ and the task of re-implementing the method serves to refresh and modernize the method. Tornado is currently the only VLM available that is implemented in Matlab (barring simple method demonstration codes.) 6.4.4. Portability Although in theory Fortran should be as portable as Matlab, it has been the authors experience that Fortran codes often have some lingering mis-feature which prevents compilation in an cross platform environment. Matlab on the other hand, has proven to be well behaved-portable, with the notable exception of user designed graphical user interfaces, where different font-size standards can cause problems. 6.4.5. Numerical Libraries Programming in Matlab is an intuitive process, due to the extensive numerical libraries. Excerpt from MathWorks website: With more than 600 mathematical, statistical, and engineering functions, MATLAB gives you immediate access to high-performance numeric computing. The numerical routines are fast, accurate, and reliable. These algorithms, developed by experts in mathematics, are the foundation of the MATLAB language. The core math engines incorporate the well-respected LAPACK and BLAS linear algebra subroutine libraries . 6.4.6. Graphical User Interfaces A big advantage of Matlab is the possibility to create graphical user interfaces. In release 12 of Matlab, the design tools for graphical user interfaces are powerful and allow for fast design of complex user interfaces. 6.4.7. Drawbacks There are of course some drawbacks with Matlab. Codes in Matlab are inherently slower than their compiled Fortran counterparts, and due to the forgiving nature of the language it is easy to produce spaghetti code if one does not have a structured development plan.
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7. Internet Interactions in Software Development This chapter deals with the use of Internet in software design projects in general, and how user feedback has influenced the Tornado software development in particular. Some examples of user feedback will be shown on the background of open source and the GNU General Public License. The Tornado software core functions was developed for the first release between 1999 and 2001. Since then the code has been distributed via the Internet and today the program has many users worldwide. Some of them have either helped in the development of Tornado or have developed their own modules. As the Tornado code is distributed under an open source agreement, the code is accessible to all users. The sections below will discuss the code development work done after the first release.
7.1. Internet interactions Due to the Internet, the development interaction between aerodynamic tool developers and aerodynamic tool users has become more integrated than before. Traditionally, software development for aerodynamic applications followed a linear path. The tool developer constructed the tool according to specifications and tested it with a few test cases. Usually, one or two publications in the scientific press would follow. With that, the code development ceased and the software was put aside and left on a shelf. Users often had good use of the software, when applied to the same design problems as the original programmer envisioned. But, when users needed to develop new capabilities of the software, there was little room for feedback. In some cases, the original programmer was unavailable for development, or in the case of compiled codes, the source was unavailable. In both cases, software development progressed slowly, if at all. With the Internet, users actively search the net for software applicable to their design problems. If the program developer shares the source, and or has an open dialogue with the users, code changes can be made in a rapid fashion. Thus the software can be adapted to have the capability to assess a much larger set of design problems than originally envisioned. This coupled feedback between users and developers also helps in expediently finding cures for bugs and mis-features in the software. An example of this rapid design cycle, is the case of a bug found by Luca di Rauso, at the University of Naples. The bug manifested itself as displaying a wrongful number as the total lift coefficient in the span load plot. The cause was that the entire aircraft lift coefficient was used as the total lift coefficient, when the interesting number for comparison is the wing-only total lift coefficient. The case was quickly resolved and the reporting user received a patch the same day, and the code correction was included in the new release
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the following day.
7.2. Scale-free networks The Internet is a scale free network, according to Barabasi and Bonabeau [BB03]. A scale-free network is characterized, as any network, by its nodes and links and in particular the presence of hubs. Hubs are particular nodes that have many more links than the average node. In contrast, random networks contains a set of nodes, each having links close to the average number of links in the network. As an example, a typical scale free network is an airline system, with some nodes (airports) acting as hubs with a large number of links (routes). The highway road system is a traditional random network, with every city having approximately the same number of highways entering and exiting the city as the average city. This kind of network analysis can applied to the Tornado project at different level of abstraction in describing how: • Interaction between code developers and software users work. • To select a strategy for software distribution. • Tornado functions can serve as hubs in new applications. 7.2.1. User-Developer interaction As described in the section above, the advent of the Internet has opened a new communications channel for researchers around the world. This has the implication that each potential Tornado user has fewer step to take before actually executing the software than he would have if the software would not have been available thorough the Internet. The old way of designing computational software meant that the development often was based on a need from a user located geographically close to the development site. In this sense, the code development described a random network, with each software having few potential users. As described in [BB03](p.57) the random network is prone to create isolated non-communicating islands of nodes (potential users) if random nodes fail. The same phenomenon can be seen in the case of older vortex lattice methods, where users are aware of a specific code, but is discouraged from using it by problems in acquiring the code. In a scale free network environment as the Internet however, the proliferation of a software is not dependent on the neighboring nodes. In particular for the Tornado code, this means that even if there would not be an immediate interest of the code at other research establishments in Sweden, users from other parts of the world would still have an easy access to it.
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7.2.2. Software distribution Network analysis could also be used for assessing a strategy for distributing the software in the most efficient way. By identifying web sites that act as hubs in spreading information about aerodynamic design software. The web masters of these sites can then be approached with information about Tornado, with the intention of creating a link between the hub and the Tornado project web page. Increasing the number of links to hubs will give potential users an even easier access to the code. An example of such a aerodynamic software hub web site is W. Mason’s Aircraft Design Software web page at: www.aoe.vt.edu
7.2.3. A hub of applications The development of the Tornado code, is intended to make the central vortex lattice method a hub in a network of nodes consisting of third party developed softwares. This has been successful to some degree, as is shown in the next section. In conclusion, the key of creating a software which is well used, is to create a software that can serve as a hub for other applications. And the key of becoming a hub is to be accessible, something which would be much harder without the Internet.
7.3. Software development and Software Use The development work done can be categorized in two classes: The work done at KTH, and the work done outside of KTH including code feedback from the Internet. 7.3.1. Third person developments done at KTH Askins Isikverens PhD thesis [Isi02] was founded on an integrated aircraft development software, QCARD. One of the core functions in this program was the Tornado software, which is used to predict the aerodynamic coefficients needed for evaluating stability and control issues with a new conceptual aircraft design. In his Masters thesis, Shahram Naimi [NM00] coupled the Tornado software with the Michell stability and control software for a rapid assessment of aircraft handling qualities. Another type of application, besides aeronautical, of the vortex lattice methods, was given by Bruno v. Sicard who used Tornado for nautical applications. In his masters thesis [vS02], he estimates the loads on a T-shaped aqua foil intended for use as a pitch and heave damper for ships in rough seas. Among the ordinary users at the department, Charlotte Mattheu and Ulf Ekengren separately used Tornado for acquiring initial aerodynamic figures for the two aircraft designs they where examining with the panel method Cmarc. Additionally the software is used in the aerodynamics class at the
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department of aeronautical and vehicle engineering.
7.4. Feedback from the Internet The feedback from the Internet has also been plentiful. Often it is ordinary users asking ordinary questions, the most common of these are collected below in the frequently asked questions section. Sometimes, the feedback comes in the form of new requested features and these are collected in the corresponding section. 7.4.1. Frequently asked questions Camber line slope The current Tornado release (Release 123b) features only one type of camber line, the NACA four-digits series (definition in Appendix B). The NACA four digits wing profile is suitable from a programming point of view in the sense that both the camber line and thickness are defined by continuous functions. As the tangential flow boundary condition is defined for the camber line in the current Tornado implementation, a well formulated camber line is beneficiary. More modern airfoils are defined by their upper and bottom side curvatures, which often are defined by curve sections, or by their coordinates only. This makes the camber line slope to be implicitly defined. Requests for information regarding how to implement other wing camber lines into the the code are the most common ones. Often the users wish to be able to include modern profiles intended for high performance gliders, information which gives some indication on uses of the Tornado code. A feature for the inclusions of other profiles has been considered and a possible implementation is listed in the future works section. Manual References Some questions have been regarding the users manual [Mel01a] and some problems occurring when the users try to follow the manual in some example runs. The manual was written for Tornado version 100a (in former version numbering, version 2.1), and it does not fully cover all the features oh the latest version. One change is the modification of the main menu in Tornado, where the lattice generation has become an explicit function rather than an implicit one. This proves the necessity of an updated users manual, but considering that the project is unfunded the software development comes as a first priority. The planned way of reducing the cost of user manual production and distribution is to move away from a fixed, printable document to a web-based solution. 7.4.2. Requested features Besides the ”How-To” questions, suggestions for code improvement, requests for new features and bug reporting are among the incoming correspondence.
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Some examples are: Body model A number of users have requested a provision to create a body model in Tornado. While this is geometrically possible to do in the current version, it is not yet included as a feature. As the validation example in the validation example section below shows, it is possible to model circular slender bodies at low angles of attack. A future goal is to allow arbitrary bodies to be defined in the geometry editor. More about this in the future works section. Propeller model Another request has been to allow the code to model propellers and rotors. This is a classical application of vortex lattice methods. Modelling a propeller or rotor require the wake to be able to form a helical downstream pattern with wake roll-up. While this is only an implemental problem, the development of such a flexible wake system is not planned at this time. Propeller slipstream The effects of the propeller slipstream has also been requested. The code changes needed to model the propeller slipstream are few. The slipstream should be modelled together with the other flow conditions to form the right hand side of the central system of equations in the method. Supersonic capability The current version of Tornado is intended for subsonic, incompressible flow conditions. Users have requested both a compressibility correction subroutine as well as a supersonic implementation. Although this feature was included already in the original development plan for Tornado, the user requests has upgraded the priority of the supersonic add-on. Example aircraft Users have also requested more example aircrafts, besides the current Cessna 172 and Boeing 747. This will be done eventually, with the latest addition being the SAAB B18. One of the big problems in adding new example aircraft is to acquire all of the necessary geometrical data. Some of the geometry may be acquired through planform sketches available in the open literature, but details such as twist distribution are hard to determine. Even though the geometrical representation may be very accurate in some cases, the lack of reference point position makes comparison with the manufacturers released aerodynamic data hard to perform.
7.4.3. Software use Users have also reported different kind of uses they have had of the code. The examples reported in this section are the ones that where reported via email.
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Education Tornado is also used overseas in several educational application, either in class lab tutoring or in more independent student projects. One example comes from a joint project of Loughborough University and Virginia Tech, where students under Prof. William Mason designed a general aviation aircraft using a box-wing design [Mas02]. A render of the aircraft, the Iskelos, together with the Tornado model are available in figure 1 below.
Figure 1: Iskelos, Mason’s students box wing design
Engineering science Several PhD students have reported using Tornado, either as a preliminary design tool or as validation method of their own tools. At the University of Naples a PhD project aiming at the development of a lifting-body UAV with canards uses Tornado with an in-house developed camber function for the body. During this development, important feedback regarding tip vortex influence on cambered trailing surfaces resulted in new design routines and an improvement of the code.
7.4.4. User developed extensions Some users have developed their own add-on extensions for Tornado, as these too are available trough the GNU-GPL licence, the code has started to evolve in more directions than one. As yet, no third party patch as been bundled with the tornado package as they will all require some modification to be conformant with the Tornado format. User interface At the Royal Thai Airforce Academy (RTAFA), Pakadaet Pakawan devised a graphical user interface that was developed for a one-wing design assessment. A picture of the user interface is shown in figure 2. More Airfoils Besides being one of the most requested new features, different functions allowing arbitrary camber lines have been developed by different users. Most notably is the body camber function by Luca di Rauso at the University of Naples described above.
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Figure 2: Graphical User interface developed at the Royal Thai Air force Academy.
7.5. Open source Open source means that the source code for a specific program or method implementation is available to the public in the for humans readable ASCII format. There are a number of formats on how to do this, and one of the most common is the GNU-General Public Licence [Appendix A]. The licence agreement grants copyright to everyone who wishes to use the code for non-profit causes, but it also imposes restrictions on new codes based on the original, namely that they too are available with an open source agreement. The agreement does not in anyway transfer the immaterial ownership of the code. Software code needs to be developed, always! There is no, and will never be an Ultimate Implementation That Makes All Code Development Redundant. By employing an open source strategy the code developer opens the possibility for other creative minds to make modifications to the original code. By doing so the original programmer makes sure that the code, if it is useful, will spread and evolve into forms unimaginable to him.
7.5.1. Software in education and engineering science By supplying the source code, the instrinct knowledge in the implementation becomes publicly available, and if not being the essence of science, the spreading of knowledge is the essence of teaching. By this it is safe to say that all software in use in education must have open source, lest the teaching becomes only a manual training in tool use an not really a knowledge transfer.
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A program used for scientific research should have open source. By publishing the source code, the researcher makes sure that the computations and experiments are possible to repeat. It is no different from standard laboratory research, where both the method and the laboratory set up are described when new findings are reported. Sometimes only the method is published, leaving the implementation to no more than just a mention by name. This is often true in the case of the use of commercial software, when the source code by necessity is hidden from the end user.
7.5.2. Assessing limitations in the implementation Methods always impose some kind of assumptions on the input data; the standard vortex lattice method for example, assumes that the angles of attack in the flow problem are small. Method assumptions are well known and well documented. Implementation assumptions on the other hand, are not as well documented, and in the worst cases, unknown. Again, an example from vortex lattice codes is how the code handles the singularity problem when a vortex passes straight through a collocation point. The problem could be taken care of in a number of different ways, and it influences the computation either by imposing restrictions on the input data or by creating validity limits on the output data. With open source it becomes possible to determine output validity (perhaps in many cases that it is not). Commercial software has a tendency to behave as black-boxes in the research process. Commercial software often produces very accurate results when examining the manufacturers test examples. But when the internal implementation is unknown, the researcher just cannot be sure that the code output is valid for a certain kind of input. Since research is about exploring the unknown, the computational problem will by definition be untested. For industrial applications, most of the computations will be in, so to say, charted territory, where the implementation induced errors can be evaluated and corrected for. Employing an open source strategy means that the commercial value of the code itself drops to almost nothing. However, this does not mean that open source software is a purely pro-bono publico affair. For the business-inclined researcher, potential business plans incorporate user support, software education and old fashion consulting. A good example of this is the Linux operating system, which regardless of it’s open source character has proven to be a financial asset.
7.5.3. Intellectual property rights The distribution of open source codes needs to be governed by a license agreement in the same way as with propriety software. The use of the internet as a distribution channel sometimes invokes a more relaxed attitude towards copyright, both from the copyright owner and the licensees. Even though a software developer releases all copyrights to the public, this should be stated
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explicit. Otherwise the usefulness of the source becomes lower, because if another software developer wishes to use the code he’ll be confronted with three problems: • Without a copyright release writ, the copyright status of the software is unknown and hence the developer risks juridical issues with the copyright holder in the future. • The copyright of the newly designed software also becomes ambiguous, since the developer cannot claim copyright of the assimilated work. • Responsibility tracking in case of juridical action from a user. Most open source license agreements contains a liability limitation, often where the original programmer takes no responsibility for the software whatsoever.
7.6. General software development strategy Computer based methods require hardware and software, and by that a certain amount of computer science becomes involved. When developing software, the programmer must take into account in what way the software is intended to be used in the future. This is however hard to do, and by narrowing the options for future users, the overall usefulness of the code drops. Closed source is the ordinary way that software is distributed. It means that the software is delivered as a pre-compiled executable binary. This binary is then closed for inspection. The wrapping of a programs procedures into an executable, means adding some sort of user interface. This interface to a large extent determines what the future uses of the software will be, and it makes it harder for the users to apply the code to tasks which the original programmer did not intend. However, pre-compiled software is in widespread use in the aerospace community today. One of the largest workload among computational engineers today is managing output data between different applications and managing different formats to fit into different programs. By distributing the software by an open source agreement, making the source code available to the public, the code will be more useful to a larger audience. The users are not forced into a particular modus operandi, but the drawback is that it assumes that the user is able to perform basic programming. Open source makes large scale modularity possible, where programmers can design partial codes for each other. The main idea is to build building blocks, to be recycled both in the development project itself and in later projects. Using modularity in software development ensures that the final product is adaptable, robust, and resource saving.
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8. Summary and Conclusions This chapter covers the conclusions of this thesis.
8.1. Summary The role, and importance of predictive methods in conceptual design has been shown. Together with the mathematical background of the ideal flow, the vortex lattice method has been explained. From that a description of the available method variations and the capabilities of the available softwares was explored. The Tornado vortex lattice method was discussed, together with a discussion of the importance of user feedback on software quality. Finally, some recommendations on different software licensing schemes where discussed.
8.2. Conclusions The findings of this thesis is that the Tornado software has become wide spread with the use of Internet and the GNU General Public License. Although other vortex lattice codes has been available for general use, they have not been available for general modifications. This has given the Tornado software a competitive edge. The user interface of Tornado allows unexperienced users to explore the software themselves, and as the source is available they may also examine and develop the implementation of the method. The different singularity distributions of the vortex lattice method can all be described in the same form by using the Kirchhoff junction rule. This opens the possibility of the Kirchhoff net distribution as described in the future works section. A distribution which will allow much more complex geometries than the standard horseshoe vortex distribution. There is a need in aircraft conceptual design for the vortex lattice method, which is fast and does not require a highly detailed aircraft geometry to model the aircraft behavior.
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9. Future Work This chapter describes the planned, and the possible future work of the Tornado software. Besides continuing in proving user support for the software, some development work is also planned.
9.1. Tornado Development Among the planned future developments of the Tornado software, some of the additions mentioned in the internet feedback chapter will be programmed. 9.1.1. User manual As with any software development project, documentation is of the outmost importance if the code is going to be successfully used by other people than the original programmer. The development plan for the Tornado users manual is to create a web based users manual. Both to ease the documentation maintenance, and hopefully to provide a more accessible user manual. 9.1.2. Different vortex models The potential vortex used in the current implementation could be replaced by a Rankine vortex, or a Burgers vortex for more high-fidelity simulations. This needs to be investigated more thouroughly. 9.1.3. Trefftz plane analysis The induced drag computed by the Kutta-Joukovski theorem is sensitive to lattice refinement. By Including a Trefftz plane analysis, the induced drag computations could be made less sensitive. 9.1.4. Body model There is already provisions for modelling a circular body in the current Tornado release. However, this model has not been validated thoroughly enough. Validating the circular body and investigating the possibilities of a arbitrarily shaped body is in the development plan. 9.1.5. Camber function A camber function allowing input from standard airfoil libraries is desirable. It could be implemented by applying two splines, one to the airfoil upper surface and one to the lower surface. The camber line would then be close to the arithmetic mean value of the upper and lower surface. 9.1.6. Supersonic capability Including a supersonic capability to the vortex lattice method would greatly increase the areas of applicability of the code. Initial tests on a two dimensional case has come out favorable.
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9.1.7. Kirchhoff net The most interesting new feature is the Kirchhoff net formulation of the singularity distribution. The Kirchhoff net formulation is a new vortex lattice formulation under development by the author. It differs from the earlier vortex lattice formulations, the horseshoe vortex and the vortex ring in the way that the distributed singularities are plain straight vortex segments. Interestingly enough, this formulation is a superset of the horse-shoe vortex, vortex sling and vortex ring formulations, which are all special cases in the Kirchhoff net formulation. The Kirchhoff net formulation allows for a more complex geometry. By using only straight vortex elements instead of horseshoe vortices a more complex geometry model is allowed, hence a more accurate flow field. Instead of integrating line vortices into the standard horseshoe arrangement, line vortices run along the edges of a panel. The vortex layout follows the same pattern as the standard vortex lattice with the cross-panel vortex at the quarter chord-point, and the collocation point at the three quarter chord point. To form the basic cell, six vortices are needed (See figure 1). Two wake vortices (v5 and v6 ), one trailing edge vortex (v4 ), two panel side edge vortices (v1 and v3 ) and one cross panel vortex (v2 ). The vortices meet in the node points n1 to n4 . In total, this approach uses more vortices than the standard implementation, hence more equations are needed in solving for vortex strengths. Six vortices give six unknowns in the system of equations, and the corresponding six equations comes from: • The tangential flow boundary condition at the collocation point (1 equation). • Applying Kirchhoffs junction rule 9.1.1, stating that the sum of all vortices going in and out from a node is zero (4 Equations). • Applying the Kutta condition for the trailing edge element, setting it’s strength to zero (One equation). N X
Γn = 0
(9.1.1)
n=1
Basic cell A one cell Kirchhoff grid is stylized in figure 1. showing the edge vortices v1 to v4 and the wake vortices v5 and v6. The vortex nodes are denoted n1, n2, n3 and n4 and. The collocation point c1 is marked by a star. Examining the system of equations we find: • Number of nodes: 4 • Number of collocation points: 1 • Number of trailing edge elements: 1 • Total number of equations: 6 • Number of vortices: 6
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Figure 1: Kirchhoff grid with 1 cell. Setting the trailing edge vortex strength to zero gives the standard horseshoe layout.
Juxtaposed cells A complete lattice can be constructed by adding neighboring cells to the basic cell. The number of equations N scale with the number of panels (n) as equation 9.1.2 N = n + n2 + (n + 1)2
(9.1.2)
4 Cells • Number of nodes: 9 • Number of collocation points: 4 • Number of trailing edge elements: 2 • Total number of equations: 15 • Number of vortices: 15 9 Cells • Number of nodes: 9 • Number of collocation points: 16 • Number of trailing edge elements: 3 • Total number of equations: 28 • Number of vortices: 28 By extending the basic cell to a 3 by 3 lattice, it has been shown that the lattice can be expanded with edge cells (the 4 cell case), and edge cells that create inner cells (the 9 cell case). By induction the lattice can be extended to n by n cells. Cell splitting One cell can also be split into four cells (figure 4). The equation/unknown coherence is kept since: By removing one cell, we remove four vortices and
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Figure 2: Kirchhoff grid with f cells
one collocation point. Then by superposing the four new cells in the same place as the removed one we gain twelve vortices, five nodes and four collocation points. In total the operation yields eight equations and eight unknowns. Vortices: -4 + 12 = 8 Equations: -1+5+4 =8 Application The Kirchhoff grid formulation may allow the vortex lattice method to be extended to be in full compliance with panel methods in regard of geometry modelling capability, while still being able to model wings as flat two dimensional surfaces when needed.
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Figure 3: Kirchhoff Grid with 9 cells
Figure 4: Cell splitting
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Appendix A [GNU-GPL] GNU GENERAL PUBLIC LICENSE Version 2, June 1991 Copyright (C) 1989, 1991 Free Software Foundation, Inc. 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. Preamble The licenses for most software are designed to take away your freedom to share and change it. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change free software–to make sure the software is free for all its users. This General Public License applies to most of the Free Software Foundation’s software and to any other program whose authors commit to using it. (Some other Free Software Foundation software is covered by the GNU Library General Public License instead.) You can apply it to your programs, too. When we speak of free software, we are referring to freedom, not price. Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for this service if you wish), that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs; and that you know you can do these things. To protect your rights, we need to make restrictions that forbid anyone to deny you these rights or to ask you to surrender the rights. These restrictions translate to certain responsibilities for you if you distribute copies of the software, or if you modify it. For example, if you distribute copies of such a program, whether gratis or for a fee, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must show them these terms so they know their rights. We protect your rights with two steps: (1) copyright the software, and (2) offer you this license which gives you legal permission to copy, distribute and/or modify the software. Also, for each author’s protection and ours, we want to make certain that everyone understands that there is no warranty for this free software. If the software is modified by someone else and passed on, we want its recipients to know that what they have is not the original, so that any problems introduced by others will not reflect on the original authors’ reputations. Finally, any free program is threatened constantly by software patents. We wish to avoid the danger that redistributors of a free program will
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individually obtain patent licenses, in effect making the program proprietary. To prevent this, we have made it clear that any patent must be licensed for everyone’s free use or not licensed at all. The precise terms and conditions for copying, distribution and modification follow. GNU GENERAL PUBLIC LICENSE TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION AND MODIFICATION 0. This License applies to any program or other work which contains a notice placed by the copyright holder saying it may be distributed under the terms of this General Public License. The ”Program”, below, refers to any such program or work, and a ”work based on the Program” means either the Program or any derivative work under copyright law: that is to say, a work containing the Program or a portion of it, either verbatim or with modifications and/or translated into another language. (Hereinafter, translation is included without limitation in the term ”modification”.) Each licensee is addressed as ”you”. Activities other than copying, distribution and modification are not covered by this License; they are outside its scope. The act of running the Program is not restricted, and the output from the Program is covered only if its contents constitute a work based on the Program (independent of having been made by running the Program). Whether that is true depends on what the Program does. 1. You may copy and distribute verbatim copies of the Program’s source code as you receive it, in any medium, provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice and disclaimer of warranty; keep intact all the notices that refer to this License and to the absence of any warranty; and give any other recipients of the Program a copy of this License along with the Program. You may charge a fee for the physical act of transferring a copy, and you may at your option offer warranty protection in exchange for a fee. 2. You may modify your copy or copies of the Program or any portion of it, thus forming a work based on the Program, and copy and distribute such modifications or work under the terms of Section 1 above, provided that you also meet all of these conditions: a) You must cause the modified files to carry prominent notices stating that you changed the files and the date of any change. b) You must cause any work that you distribute or publish, that in whole or in part contains or is derived from the Program or any part thereof, to be licensed as a whole at no charge to all third parties under the terms of this License.
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c) If the modified program normally reads commands interactively when run, you must cause it, when started running for such interactive use in the most ordinary way, to print or display an announcement including an appropriate copyright notice and a notice that there is no warranty (or else, saying that you provide a warranty) and that users may redistribute the program under these conditions, and telling the user how to view a copy of this License. (Exception: if the Program itself is interactive but does not normally print such an announcement, your work based on the Program is not required to print an announcement.) These requirements apply to the modified work as a whole. If identifiable sections of that work are not derived from the Program, and can be reasonably considered independent and separate works in themselves, then this License, and its terms, do not apply to those sections when you distribute them as separate works. But when you distribute the same sections as part of a whole which is a work based on the Program, the distribution of the whole must be on the terms of this License, whose permissions for other licensees extend to the entire whole, and thus to each and every part regardless of who wrote it. Thus, it is not the intent of this section to claim rights or contest your rights to work written entirely by you; rather, the intent is to exercise the right to control the distribution of derivative or collective works based on the Program. In addition, mere aggregation of another work not based on the Program with the Program (or with a work based on the Program) on a volume of a storage or distribution medium does not bring the other work under the scope of this License. 3. You may copy and distribute the Program (or a work based on it, under Section 2) in object code or executable form under the terms of Sections 1 and 2 above provided that you also do one of the following: a) Accompany it with the complete corresponding machine-readable source code, which must be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or, b) Accompany it with a written offer, valid for at least three years, to give any third party, for a charge no more than your cost of physically performing source distribution, a complete machine-readable copy of the corresponding source code, to be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange; or, c) Accompany it with the information you received as to the offer to distribute corresponding source code. (This alternative is allowed only for noncommercial distribution and only if you received the program in object code or executable form with such an offer, in accord with Subsection b above.) The source code for a work means the preferred form of the work for making modifications to it. For an executable work, complete source code means all the source code for all modules it contains, plus any associated interface
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definition files, plus the scripts used to control compilation and installation of the executable. However, as a special exception, the source code distributed need not include anything that is normally distributed (in either source or binary form) with the major components (compiler, kernel, and so on) of the operating system on which the executable runs, unless that component itself accompanies the executable. If distribution of executable or object code is made by offering access to copy from a designated place, then offering equivalent access to copy the source code from the same place counts as distribution of the source code, even though third parties are not compelled to copy the source along with the object code. 4. You may not copy, modify, sublicense, or distribute the Program except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense or distribute the Program is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance. 5. You are not required to accept this License, since you have not signed it. However, nothing else grants you permission to modify or distribute the Program or its derivative works. These actions are prohibited by law if you do not accept this License. Therefore, by modifying or distributing the Program (or any work based on the Program), you indicate your acceptance of this License to do so, and all its terms and conditions for copying, distributing or modifying the Program or works based on it. 6. Each time you redistribute the Program (or any work based on the Program), the recipient automatically receives a license from the original licensor to copy, distribute or modify the Program subject to these terms and conditions. You may not impose any further restrictions on the recipients’ exercise of the rights granted herein. You are not responsible for enforcing compliance by third parties to this License. 7. If, as a consequence of a court judgment or allegation of patent infringement or for any other reason (not limited to patent issues), conditions are imposed on you (whether by court order, agreement or otherwise) that contradict the conditions of this License, they do not excuse you from the conditions of this License. If you cannot distribute so as to satisfy simultaneously your obligations under this License and any other pertinent obligations, then as a consequence you may not distribute the Program at all. For example, if a patent license would not permit royalty-free redistribution of the Program by all those who receive copies directly or indirectly through you, then the only way you could satisfy both it and this License would be to refrain entirely from distribution of the Program. If any portion of this section is held invalid or unenforceable under any particular circumstance, the balance of the section is intended to apply and the section as a whole is intended to apply in other circumstances.
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It is not the purpose of this section to induce you to infringe any patents or other property right claims or to contest validity of any such claims; this section has the sole purpose of protecting the integrity of the free software distribution system, which is implemented by public license practices. Many people have made generous contributions to the wide range of software distributed through that system in reliance on consistent application of that system; it is up to the author/donor to decide if he or she is willing to distribute software through any other system and a licensee cannot impose that choice. This section is intended to make thoroughly clear what is believed to be a consequence of the rest of this License. 8. If the distribution and/or use of the Program is restricted in certain countries either by patents or by copyrighted interfaces, the original copyright holder who places the Program under this License may add an explicit geographical distribution limitation excluding those countries, so that distribution is permitted only in or among countries not thus excluded. In such case, this License incorporates the limitation as if written in the body of this License. 9. The Free Software Foundation may publish revised and/or new versions of the General Public License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. Each version is given a distinguishing version number. If the Program specifies a version number of this License which applies to it and ”any later version”, you have the option of following the terms and conditions either of that version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of this License, you may choose any version ever published by the Free Software Foundation. 10. If you wish to incorporate parts of the Program into other free programs whose distribution conditions are different, write to the author to ask for permission. For software which is copyrighted by the Free Software Foundation, write to the Free Software Foundation; we sometimes make exceptions for this. Our decision will be guided by the two goals of preserving the free status of all derivatives of our free software and of promoting the sharing and reuse of software generally.
NO WARRANTY 11. BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW. EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM ”AS IS” WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
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PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU. SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION. 12. IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS), EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES. END OF TERMS AND CONDITIONS
