Preface Prefatory Material Book text developed course applied aerodynamics Stanford University first year graduate course includes topics ranging review basic governing equations fluid flow practical issues airfoil wing design course approximately 26 lectures dealing basic topics weekly problem sets encourage students explore topics fully notes part course reading materials impossible applied aerodynamics computational methods course several programs aerodynamic analysis airfoils wings limited versions material presented complements separate course compressible flow theory important issues left course continuation material courses advanced aerodynamic topics aircraft design digital text intended supplement conventional textbook applied aerodynamics material composed based lecture notes last few years continually development comments suggestions helpful Digital Textbook? several reasons format course notes: updated changed -- important development process author students rewrite book places Analysis routines built notes directly (See example streamline calculations NACA airfoils wing analysis canards Java-version text ) format permits easy access information (through search command) organizes information way hardcopy inexpensive color pictures video possible providing couple custom pages tailor textbook particular course material potential flow theory appropriate class new outline contents page created avoids material I someday real "Hitchhiker's Guide Aerodynamics" complete cadre editors supplying section home institution moment I'm interested feedback edition Please send comments Desktop Aeronautics (info@desktopaero com) directly kroo@leland stanford edu Author Ilan Kroo Professor Aeronautics Astronautics Stanford University received degree Physics Stanford 1978 continued graduate studies Aeronautics leading Ph D degree 1983 worked Advanced Aerodynamic Concepts Branch NASA's Ames Research Center returned Stanford member Aero/Astro faculty Prof Kroo's research aerodynamics aircraft design focussed study innovative airplane concepts multidisciplinary optimization participated design high altitude aircraft human-powered airplanes America's Cup sailboats high-speed research aircraft principal designers SWIFT tailless sailplane design worked Advanced Research Projects Agency high altitude long endurance aircraft directs research group Stanford consisting ten Ph D students teaches aircraft design applied aerodynamics graduate level addition research teaching interests Prof Kroo president Desktop Aeronautics Inc advanced-rated hang glider pilot Acknowledgements I thank people contributed work students AA200A beta versions worked colleagues sent valuable suggestions Christine Beirne turned real product R T Jones Richard Shevell agreed review early version my family wonderful: thanks Copyright textbook copyright Desktop Aeronautics Inc Figures text either prepared originally book permission certain cases royalty payments arranged part document reproduced form express written permission from: Desktop Aeronautics P. O. Box 20384 Stanford CA 94309 (650) 424-8588 (Phone) info@desktopaero com Please contact Desktop Aeronautics information CD versions work Desktop Aeronautics Home Page World Wide Web
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Introduction Introduction Welcome Applied Aerodynamics Digital Textbook notes developed first-year graduate-level course applied aerodynamics Stanford University appropriate undergraduate aeronautics courses interested readers Please preface instructions notes electronic version text text meant introduce reader important concepts applied aerodynamics slightly different perspective books aerodynamics history section focuses earliest ideas aerodynamics links sites world-wide-web followed section fundamentals fluid dynamics readers familiar tried expanded coverage topics airfoil design wing lift distrubutions wing design configuration aerodynamics including brief discussion aerodynamics important stability control addition text pictures tried number javascript java-based tools examples illustrate material interactive fashion animations video interactive programs available edition version text updated site internet information current editions texts software visit Desktop Aeronautics site world-wide-web
History Historical Notes long ago people dream able fly dream subject great myths stories Icarus father Daedalus escape King Minos' prison Crete Legend difficulty structural materials aerodynamics (Picture woodcut 1493 ) few giant leaps little forward progress medieval times further work applied aerodynamics flight notable people climbed top convenient places intent commit aviation Natural selection survival fittest worked effectively preventing evolution human flight people started leaping several theories flight propounded (e g Newton) arguments impossibility flight research topic taken seriously late 1800's people observed flight Nature: birds bees (A Gallapagos hawk -- Photo Sharon Stanaway ) Papers suggested perhaps birds insects "vital force" enabled fly duplicated inanimate object Technical meetings held 1890's ability birds glide noticeable motion wings little negative altitude loss mystery time theory aspiration developed; birds way able convert energy small scale turbulence useful work theory fell out favor birds' ability attributed proficient seeking updrafts (Recently discussion birds fact able energy small scale air motion ) reproduced 1893 book First International Conference Aerial Navigation paper "The Mechanics Flight Aspiration " M Wellington flight path bird climbing flapping wings Today bird circling rising current warm air (a thermal) Designs people vaguest idea aircraft flew Leonardo Di Vinci designed ornithopters late 1400's modeled observations birds apart work designs pure fantasy first successes came gliders Sir George Cayley wrote book entitled "On Aerial Navigation" 1809 first successful glider 1804 full-size version five years age 36 years thereafter aeronautics taken seriously small group zealots William Henson patented Aerial Steam Carriage 1842 aircraft built publicized (with idea raising venture capital) design funding scheme ahead time ambitious designs actually built enormous aeroplane built 1894 Sir Hiram Maxim weighed 7 000 lbs spanned 100 ft Germany 1860's Otto Lilienthal took conscientious approach tests whirling arm ornithopter tests suspended barn finally flight tests glider design studied airfoil shape control surfaces propulsion systems detailed measurements bird flight book "Birdflight Basis Aviation" important influence pioneers Lilienthal's last flights killed 1896 gust-induced stall ground Lilienthal's first flights 1890's Wright brother's glider flights powered aircraft evolution quick Orville Wright soars glider 50 mile per hour winds 10 minutes Kitty Hawk Oct 24 1911 first applications aft horizontal tail Wright aircraft first 'Aerial Limousine' 1911 "The limousine doors mica windows seats four persons fitted pneumatic cushions pilot seats front number flights passengers entire success " Boeing 777 truly amazing quickly happened: tend think dawn flight something Greek mythology 100 years since people first flew airplanes course happen quickly 747 designed calculators big whirring contraptions sat desks square roots earlier transports flying today designed calculators women worked computing machines picture computational grid modern calculation flow 737 wing flaps slats deployed revolution computing changed way computational applied aerodynamics utilize variety methods Computation ground-based testing finally flight tests plot computer power required perform calculations 15 minutes 1985 algorithms modern supercomputers parallel machines time dropping dramatically long way routine applications direct Navier-Stokes simulations LES Cray C916 Supercomputer Projects NASA's Numerical Aerodynamic Simulation program continue develop simulation software takes advantage recent advances computer hardware software class talk methods compute aerodynamics flows simple methods personal computers design airfoil sections analyze wings talk elements wing design talking fundamental concepts demonstrated simple programs form basis modern computational methods methods work analytical studies wind tunnel tests CFD wing airplane design aircraft great concepts methods relevant wide range applications: Weather prediction boat design disk drive aerodynamics architectural applications land-based vehicles aerodynamics bumble bees disk heads weather solved problem impressive methods today able predict flows I Kroo 10/30/96
Early Attempts Early Attempts records people doing far back twelfth century: Saracen Constantinople first recorded would-be aviators Stephen Dalton "Miracle Flight" writes: "In 1503 Giovanti Dante Italian mathematician fixing wings arms jumped tower Perugia; seriously injured Four years John Damian Abbot Tungland physician Scottish court King James IV attempted fly wings battlements Stirling Castle " credited first fly
Birdflight as the Basis of Aviation Birdflight Basis Aviation Lilienthal's book full interesting comments introduction: "With advent spring air alive innumerable happy creatures; storks arrival old northern resorts fold imposing flying apparatus carried thousands miles lay back heads announce arrival joyously rattling beaks; swallows entry hurry streets pass windows sailing flight; lark appears dot ether manifests joy existence song; certain desire takes possession man longs soar upward glide free bird smiling fields leafy woods mirror-like lakes enjoy varying landscape fully bird "
References References Books Articles Anderson Fundamentals Aerodynamics 2nd Edition McGraw-Hill 1991 Shevell R S Fundamentals Flight Prentice-Hall 1983 Dalton S Miracle Flight McGraw-Hill 1980 Kuchemann J Aerodynamic Design Aircraft 1982 Taylor J Munson K eds History Aviation Crown Publishers 1978 Chanute O Progress Flying Machines American Engineer Railroad Journal N Y 1894 Lilienthal O Birdflight Basis Aviation first published German 1889 translation published Longmans Green & Co London 1911 Proceedings International Conference Aerial Navigation Chicago American Engineer Railroad Journal N Y 1893 Web Sites Boeing History Early Flight Invention Airplane Octave Chanute Pages AIAA 1903 Wright Flyer Project Wright Brothers Early Years Gallery First Flight Society
Fundamentals Fundamentals Fluid Flows chapter serves review fundamental concepts fluid dynamics including origin pressure shear forces basic equations fluid dynamics underlying approximations Applied aerodynamics concerned measurement prediction aerodynamic properties physical properties associated fluid as: pressure temperature density velocity gas properties forces bodies work applied aerodynamics concerned prediction forces several ways starting simple considerations moving detailed field equations
Origin Fluid Forces Origin Fluid Forces particularly interested forces moments applied bodies moving fluid divided two types: pressures shears Pressures created surface body due (nearly) elastic collisions molecules fluid surface body Shearing forces produced fluid viscosity quantity measure momentum transferred adjacent layers fluid types forces important applied aerodynamics pressures dominant type force shapes drawn scale drag reason streamlined airfoil larger force due shear stress pressure forces cylinder separated airflow large pressure forces give rise high drag fact amazing force generated differences pressure: 747 wing loading (Weight / Wing Area) 100 lbs/sq ft means takes section wing large book lift large dog (for example) possible normal atmospheric pressure 2116 lb / sq ft sea level fact 100 psf represents 5% change pressure upper side wing 68 000 ft create complete vacuum wing upper surface lift weight
Pressure Forces Origin Pressure Forces pressure arises molecule bounces off surface transfers momentum body particle mass m hits body "straight-on" bounces off transfers momentum amount 2mc c speed molecule pressure proportional number molecules striking unit area surface per unit time (Number density*c) times momentum transfer per particle (~mc) or: p = k1 r c 2 since temperature defined proportional mean kinetic energy molecules T = k2 c 2 expect: p = k r T perfect gas relation component molecular velocity normal surface really needed expression body moving add velocity molecular velocity measured "fluid-fixed" frame Typically consider direct interactions model molecules continuous fluid works flows interest rarefied flows those associated initial re-entry vehicles possible analyze aerodynamics kinetic theory keeping track molecular interactions calculation
Shear Forces Origin Shear Forces molecules adjacent layers different average velocities collide transfer momentum layers rate change momentum produces shear stress fluid surface body molecules transfer momentum surface collide resulting tangential shear force molecules hit surface body bounce around among surface molecules finally leave tangential velocity average surface itself average tangential velocity surface body zero respect body so-called no-slip condition layer slow moving air body surface boundary layer viscosity air causes distribution tangential velocity surface tangential momentum air molecules transferred surface shear stress produced shear stress viscosity velocity gradient expression: t = µ dU/dy quantitatively transfer tangential momentum fluid layers leads relation considering small section boundary layer Molecules starting top box moving bottom lose momentum amount: m h dU/dy Since shear stress t rate change momentum: t = n m h dU/dy n number molecules passing area per unit time n average molecular velocity c density r so: m n == r c shear stress is: t = r c h dU/dy that: µ == r c h height h molecules transfer momentum mean free path l detailed calculations showing that: µ = 0 49 r c l mean free path decreases proportion density average molecular speed varies T expect µ varies T depend pressure
Dimensionless Groups Dimensionless Groups forces body moving fluid depend body velocity V fluid density temperature viscosity size body l shape speed sound temperature* table: F = F ( V r µ l shape) M 1 0 1 0 1 0 0 L 1 1 -3 1 -1 1 0 T -2 -1 0 -1 -1 0 0 Buckingham pi theorem states number dimensionless parameters equal number parameters minus rank matrix case 7 - 3 = 4 exists functional relationship among four dimensionless groups express force body instance relationship four dimensionless parameters: dimensionless groups moment let's first functional relationship applied aerodynamics finding function f great say knowing example wide variety similar flows exist forces large slow-moving body predicted tests small higher-speed model long speed sound sufficiently high flow around small insect represented large model viscous fluid idea model testing simulate flow body matching dimensionless parameters easy -- possible Mach Reynolds number range several wind tunnels can't wind tunnels designed fully cover range parameters? alternatives exist wind tunnel tests? (See assignments ) Subsequent pages consider dimensionless groups detail First fluid properties specific heats lead additional dimensionless parameters Prandtl number important study compressible boundary layers heat conduction left out gravity important flow water around ships lead additional dimensionless parameter Froude number several ways combining parameters form dimensionless groups commonly aerodynamics I Kroo 11/20/95
Dimensionless Forces Dimensionless Forces dimensionless force moment coefficients defined by: CL lift coefficient Cm moment coefficient length^2 term first dimensionless parameter replaced area S area anything choose (the contact area nose wheel wing planform area fuselage cross-sectional area) particular application people generally agree area car drag coefficients frontal area aircraft wing area common area c (for chord) moment coefficient definition similarly agreed upon "agreement" area important seen advertisements cars (Automobile drag coefficients based frontal area numbers 0 4 mentioned car ads drag coefficient means nothing itself chose area floor area Fremont GM plant low drag coefficients )
Reynolds Number Reynolds Number quantity: r V l / m Reynolds number r fluid density V speed m fluid viscosity l characteristic length length areas definition dimensionless force coefficients agreed standard whoever chord Reynolds numbers based wing chord lengths Reynolds numbers based diameter sphere characteristic length devised Reynolds number important strange dimensionless numbers varies orders magnitude expresses importance viscosity: high Reynolds numbers achieved decreasing viscosity making length speed large Reynolds number sense represents ratio pressure shear forces: r V l / m = r V^2 / m (V/l) r V 2 pressure m V/l m dU/dy shear stress range Reynolds number Viscosity hence Reynolds number strongly affects performance wings airfoils making important parameter match wind tunnel tests possible match dimensionless parameters precisely plot Reynolds number maximum lift drag ratio two dimensional airfoil sections plight insects plot Reynolds number maximum section lift coefficient few typical airfoil sections necessarily best sections high lift Recent studies substantial changes CLmax seen high Reynolds numbers making difficult extrapolate data small wind tunnel models
Mach Number Mach Number Mach number ratio flow speed speed sound determines importance compressibility fluid directly compressibility fluid permits sound wave travel speed by: Assuming isentropic flow perfect gas: flow pattern pressures change dramatically Mach number applicable differential equation changes form chapters freestream Mach number denoted Mo increase local velocity parts airfoil local Mach number higher Mo fact compressibility effects important high-lift sections Mo low 0 3
Conservation Laws Conservation Laws derive equations motion fluid particles rely various intuitive conservation principles expressions entirely intuitive statement fact rate change mass momentum energy certain volume equal rate enters borders volume plus rate created inside first two extensively integral expressions combined divergence theorem fact hold arbitrary volumes differential form equations: momentum theorem itself useful example apply momentum theorem relate force body properties flow distance body technique useful wind tunnel tests basis several fundamental theorems lift induced drag wings take control volume bounded single surface S divide 3 parts: outer surface inner surface pieces surface connecting two write integral form momentum equation steady flow body forces below:
Simplifying Approximations Simplifying Approximations equations motion general fluid extremely complex problem formulated impractical solve outset certain simplifying approximations accurate assumptions Continuity Homogeneity fluid composed particles small plentiful statistically-averaged properties interest scale works gases fluids conditions work studying flow sand work fluid rarefied mean free path order dimensions interest problem mean free path varies altitude plot further medium treated single type fluid -- suspensions oil water Inviscid viscosity neglected modeled indirectly aerodynamic flows interest region high shear vorticity confined thin layer fluid Outside layer fluid behaves inviscid simpler equations inviscid fluid solved outside shear layers fluids completely inviscid Tests superfluid helium similar inviscid calculations Incompressible (constant density) fluid density change changes pressure fluid incompressible Water density changes little changes pressure generally treated incompressible fluid Air compressible pressure changes small comparison nominal value corresponding changes density small incompressible equations work describing flow degree fluid density changes pressure speed sound fluid assuming flow incompressible equivalent assuming speed sound infinite local Mach number 0 2 0 5 compressibility effects ignored reason further chapter compressibility qualitatively order appreciable change nominal 2116 lb/ft^2 air pressure sea level substantial speeds required Irrotational Circulation defined as: measure rotation area fluid integration contour shrunk down point ratio circulation area enclosed curve vorticity Fluid starts out rotational motion develop unless shear stress acting it* shear confined small region vorticity cases especially inviscid flow flow field treated irrotational: Del x V = 0 case vector field V written gradient scalar field f : V = grad f f potential simplifies equations subsequent sections velocity components then: u = d f / dx v = d f / dy Steady variables describing fluid properties point change time flow treated steady time derivatives equations motion zero condition depends chosen coordinate system system rest respect body uniform motion fluid equations system steady expressed system fixed respect undisturbed fluid flow unsteady convenient transform coordinate system flow steady course possible flow steady discussions course unsteady effects important study bird flight propellers aircraft gust response dynamics aeroelasticity study turbulence apply first assumptions adopt latter discussions *Some important exceptions idea viscosity irrotational flow remains irrotational: Vorticity created gravitational field density gradients exist rotating system (such earth) due Coriolus forces important sources vorticity meteorology
Equations of Fluid Flow Equations Fluid Flow conservation laws derive equations fluid flow supplemented constituitive relations perfect gas law: p = r R T isentropic relation pressure density: p 2 / p 1 = ( r 2 / r 1 ) g commonly-solved equations table corresponding assumptions Equation Inviscid Irrotational Small Perturbations Incompressible Notes Navier-Stokes - - - - Homogeneous Reynolds-Averaged Navier-Stokes - - - - Modeled Turbulence Euler X - - - Full Potential X X - - Transonic Small Disturbance X X X - Prandtl-Glauert X X X - Linearized Acoustic X X X - Linearized Laplace X X - X
Navier-Stokes Equations Navier-Stokes Equations Navier-Stokes equations describe flow continuous Newtonian fluid derived principal conservation momentum (For details Kuethe Chow Appendix B Moran Ch 6 Anderson Ch 15) X Y Z body forces per unit mass direction t stress tensor X Y Z associated gravitational forces neglected equations usable stress tensor expressed terms viscosity pressure pressure shear forces expanded NS equations are: l "bulk viscosity" relating normal stress rate change volume div(V) pressure function density rate change density then: l = - 2/3 m simplest case body force equations become: Solutions full Navier-Stokes equations onset turbulence interaction shear layers interesting aerodynamic phenomena (with exception interacting rarefied gas flows) Unfortunately equations difficult solve Reynolds number increased scale interesting dynamics gets smaller solutions full NS equations Reynolds numbers 1 1000 recent solutions flat plate boundary layer pushed calculations Reynolds number 1410 based boundary layer thickness calculations took hundreds hours Cray computer NASA Ames video direct simulation turbulence color-coded vorticity contours S Robinson NASA small Reynolds numbers geometries analyzed full NS equations simple currently sense consider solving equations realistic aircraft configurations reason case approximate equations work cases solved time averaged Navier-Stokes equations sufficient description problem resort "large eddy simulations" numerical solution time-dependent Navier-Stokes equations smaller scales turbulence modeled averaged way Larger scale turbulent motion way faster solving full equations slow large eddy simulation flow 2D circular cylinder simulation required approximately 300 CPU hours 10 megawords core memory Cray C-90 NASA / Parviz Moin
Navier-Stokes Derivation Navier-Stokes Derivation basic idea momentum equation x direction: F x represents general force fluid element might viscous forces gravity
Reynolds Averaged Navier-Stokes Equations Reynolds Averaged Navier-Stokes Equations popular simplifications Navier-Stokes Equations "Reynolds Averaging" simplification full Navier-Stokes equations taking time averages velocity terms equations writing: u = <u> + u ' v = <v> + v' etc (where <> represents time average) fluctuations having zero mean value: <u'> = 0 have: <u^2> = <u>^2 + <u'^2> <uv> = <u><v> + <u'v'> allows write time-averaged NS equations as: similarly y z components looks general Navier Stokes equations incompressible flow* hold steady laminar flow additional terms act additional stresses right hand side terms represent turbulence mean flow "Reynolds stresses" said caused "eddy viscosity" terms generally larger normal viscous terms business predicting stresses relating computed mean flow properties turbulence modeling accomplished empirically detailed time-dependent simulations Reynolds averaged NS solvers appropriate analysis viscous compressible flows applied general configurations careful assumptions turbulence model compatible characteristics flow interest *This clear manipulation equations work out Schlichting pgs 557-562 steady laminar flow extra terms - caused small scale unsteadiness associated turbulence
Euler Equations Euler Equations momentum equation Euler's equation (There lots equations Euler equations!) people talk solving Euler equations days referring inviscid equations motion by: work* equation x direction becomes: vector notation: combined equations energy continuity equations solved finite differences whereby values velocity component density internal energy computed point flow quantities constituitive relations perfect gas law isentropic pressure relation find pressure Since Euler equations permit rotational flow enthalpy losses (through shock waves) useful solving transonic flow problems propeller rotor aerodynamics flows vortical structures field * Recall DF/Dt substantial particle derivative F defined by: DF/Dt = dF/dt + V · grad F derivation NS equations looks assumed constant density case
Full Potential Equation Full Potential Equation full potential equation derived assumption irrotational flow equations continuity momentum pressure density terms Euler equations combined perfect gas law isentropic relation pressure density Ashley Landahl derive vector form unsteady full potential equation: simplified case steady flow 2-D to: notation: flow irrotational Del x V = 0 definition curl gradient: Del x (grad f) = 0 f scalar field define nonphysical scalar potential f describes velocity field f velocities relation: V = grad f equations written terms unknown scalar 3 components velocity simplifies solution local speed sound x streamwise coordinate V vector velocity subscripts denote partial derivatives respect subscripted variables (e g U_x = du/dx)
FPE Derivation Derivation Potential Equation derivation full potential equation seen case 2-D steady flow case continuity equation is: steady inviscid flow Euler's relation p rho is: flow isentropic relation p rho is: Combining last two expressions: write: substituting continuity equation obtain: local speed sound written terms constants local velocities:
Transonic Small Disturbance Equation Transonic Small Disturbance Equation full potential equation simplified assuming perturbation velocities small relate local speed sound freestream value making isentropic relations small disturbance equation ( derivation ): freestream Mach number ignore last term equation classic transonic small disturbance equation: great written nonlinear equation variants (See Nixon ) frequently days since finite difference methods solve full potential equation directly
TSD Derivation TSD Derivation begin 2-D full potential equation: Ignoring terms second order perturbation velocities with: local speed sound local velocity isentropic relations algebra again dropping terms second order perturbations: Substituting: final term neglected
Prandtl-Glauert Equation Prandtl-Glauert Equation Prandtl-Glauert equation linearized form full potential equation Full potential: velocity perturbations smaller freestream velocity expression becomes: unsteady case: 3-D version constructed addition z derivatives corresponding y derivatives linearized form equation hold nose airfoil velocity perturbation order freestream unless freestream Mach number itself small expression holds subsonic supersonic flow (but transonic flow) forms basis aerodynamic analysis methods
Acoustic Equation Acoustic Equation acoustic equation full potential equation assuming freestream velocity perturbation velocities small changing coordinate system fixed body fixed respect undisturbed fluid Prandtl-Glauert equation transformed acoustic equation equation study sound propagation rotor aerodynamics; name
Laplace's Equation Laplace's Equation Laplace's equation Prandtl-Glauert equation limit freestream Mach number goes zero actually first derived Euler derivation simple requiring equation continuity assumptions irrotational constant density flow continuity equation then: Since flow irrotational: Substitution continuity equation yields: interesting Laplace's equation require assumption small perturbations Prandtl-Glauert equation fact stagnation point airfoil velocities small full potential equation reduces Laplace's equation Prandtl-Glauert equation time dependent terms full potential equation multiplied 1/a^2 form equation holds unsteady phenomena
Bernoulli Equations Bernoulli Equations Equations equations posed terms state variables pressures cases (e g potential flow equations) differential equations boundary conditions allow compute local velocities pressures Once velocities known momentum equation find local pressure equations known Bernoulli equations various forms depending assumptions flow conservation momentum principle source relation pressure velocity simply derive Bernoulli equation illustrate basic physics Bernoulli equations derive simple form: steady incompressible flow case streamline: flow steady Euler equations integrated general form result: Kelvin's equation Bernoulli equation irrotational flow Where: f body force per unit mass* F arbitrary function time flow irrotational introduce potential expression nicely integrable flow steady "body forces" necessarily irrotational write expression holds streamline: equations hold steady flows streamline irrotational flows hold throughout fluid derive useful form Bernoulli equation starting expression steady flow body forces flow assumed isentropic flow (no entropy change heat addition): substitution yields compressible Bernoulli equation: actually works adiabatic (no heat transfer) flows isentropic flows summary two simple forms Bernoulli equation Pressures incompressible compressible forms Bernoulli's equation 3 terms quantity p T total stagnation pressure pressure measured points flow V = 0 p expressions static pressure incompressible flow speed directly difference total static pressure measured directly pitot-static probe dynamic pressure defined as: static pressure coefficient defined as: p freestream static pressure incompressible flow expression Cp especially simple: local velocity expressed small perturbation freestream: incompressible Cp relation written: careful expression! good approximation correct expression difficult expression Cp compressible isentropic flow (sometimes isentropic pressure rule) derived compressible Bernoulli equation expression speed sound perfect gas terms local Mach number expression is: air gamma = 1 4 interesting follow expression tell flow supersonic looking value Cp critical value Cp denoted Cp* found setting M = 1 expression: minimum value Cp corresponding complete vacuum Setting local Mach number infinity yields: Cp negative Experiments airfoils get 70% vacuum Cp limit maximum lift supersonic wings Freestream Mach: Local Mach: Cp:
Simple Bernoulli Derivation Simple Bernoulli Derivation momentum equation flow written terms flow small control volume change momentum per unit time is: r S V (V+dV) - r S V 2 = r S V dV change momentum arises pressures acting faces control volume: pressure force (ends) = pS - (p+dp)(S+dS) = -p dS - S dp pressure force (sides) = (p + dp/2) dS = p dS (to first order) pressure (total) = -S dp Equating force due pressure force required produce momentum change yields: -S dp = r S V dV dp = - r V dV simple form Euler equation Integrating: p2 + r /2 V2 2 = p1 + r /2 V1 2 or: p + r /2 V 2 = pt
Solution Methods Solution Methods chapter brief overview methods investigate fluid flows includes discussion role experimental analytical computational methods outlines basic ideas computational approaches 1 Role Theory Experiment 2 Analytic Methods 3 CFD Overview 4 Panel Methods 5 Nonlinear CFD
Theory and Experiment Theory Experiment course focus computational methods aerodynamic analysis design Indeed CFD playing greater role computers powerful capable handling realistic cases Despite claims zealots computational methods eliminate need experimental cases change character tests Applied aerodynamics historically strong mix theory experiment partly experiments costly computations rarely sufficiently sophisticated continue case I believe recent simple wind tunnel model few control surfaces pressure measurements cost $150 000 build buy fair amount computer time took several months model delivered once "slot" tunnel wait 9 months "run" additional cases great motivation computational methods possible hand geometry complicated interested behavior leading edge vortex onset flow separation spanwise flow boundary layer features require solution complex Navier-Stokes equations NS code predict wing-alone characteristics takes weeks get converged solution great reservations results: grid fine enough capture important details off-body vortex flow? turbulence modeled properly? (We solving Reynolds Averaged NS equations ) pressure integration forces moments sufficiently accurate? error geometry input? People design business really don't believe answers codes time Indeed everyone should try get opportunity try out codes problem manual (there rarely manual) common problem computational methods users understand assumptions exact solutions simplified equations fluid dynamics wrong answer physical problem exactly lead 19th century mathematicians scientists conclude flight possible airfoils generate lift bodies moving fluid drag errors (usually obviously paradoxical) everyday Perhaps ideal method predicting aerodynamics vehicle flight test several reasons ideal method aerodynamic testing cost building changing full scale designs making repeated flights extremely high; instrumentation generally good ground-based instrumentation; atmosphere static take convective activity atmosphere introduce significant errors Experimental Methods wide range experimental techniques investigate problems applied aerodynamics include: Wind tunnel testing Scale models expensive full scale flight vehicles changing wind tunnel speed pressure Reynolds number matched difficult severe limitations especially considers modeling flows associated atmospheric entry hypersonic flight cost advantages working ground-based instrumentation technique staple applied aerodynamics investigation simple wind tunnel configuration consists open circuit contraction acting increase airspeed test section experiments conducted diffuser section limitations associated wind tunnels apparent basic calculations needs Mach number 15 test section pressure upstream contraction high room temperature upstream test section cold enough liquefy air heats incoming flow air disassociates closed circuit wind tunnel need accelerate air rest requires lower power Problems higher turbulence levels heating problems contamination moisture combustion materials powered tests addition problems matching flow parameters wind tunnel corrected effects wind tunnel walls support struts Wall effects blockage changes streamline curvature interaction boundary layers wall model Unitary Wind Tunnel Complex NASA's Ames Research Center Flight tests tests complete aircraft in-flight experiments conducted wind tunnel environment "Gloves" wings test new airfoil sections Reynolds number turbulence level wind tunnel inappropriate Scale models prop-fans tested flying aircraft scale models aircraft dropped larger aircraft fact X-airplanes tested way first flights Instrumentation Instrumentation wind tunnel flight testing cases similar Methods flow visualization - transition separation shock location streamline patterns - fluorescent oil shear sensitive liquid crystals particle suspensions sublimating chemicals optical methods interferometry holography view wing wind tunnel flourescent oil right wing mini-tufts left wing Quantitative measurement techniques - pressures velocities loads temperatures - pressure taps surfaces pitot probes yaw heads rakes hot wires flying hot wires laser Doppler velocimetry Simulators - translate wind tunnel flight computational data aircraft handling quality information simple batch integration vehicle equations motion real time simulations pilot-in-the-loop simulations visuals motion View NASA Ames Vertical Motion Simulator
Analytic Results Analytic analytical solutions solutions relevant differential equations exist great advantage equations linear added together form new solutions true linearized potential theory solutions theory complex variables (See potential theory chapter ) complex equations laminar Navier-Stokes equations solved analytically certain cases useful (e g analyzing slow motions concentric tubes fluid in-between them) careful solutions unique simple solutions solutions found nature
CFD Overview CFD* (Computational Fluid Dynamics) Pressure contours Mach 2 1 HSCT wing-body (computation Steve Ryan Intel iPSC/860 38 processors NASA Ames) CFD partnership experimental methods CFD detailed view flow field generating velocities pressures densities point field (whether interested not) - something expensive measure experimentally calculations approximate flow way either solving simplified equation introducing approximations numerical method itself wind tunnel solves correct equations right conditions (Reynolds number differences) right geometry (because model support interference wall effects) provides good measures integrated flow properties total forces moments acting body Flight test provides realistic solutions expensive well-suited flow field property measurements (e g velocity fields) compromised atmospheric disturbances ideal approach appropriate combination tools Computational methods predict wide range flows Click short QuickTime video clip showing comparison measured computed pressures shuttle (courtesy NASA) * Cynics speculated CFD really stands colorful fluid dynamics since helpful producing beautiful pictures contribute project actual data represented
Panel Methods Panel Methods Image courtesy Analytical Methods Inc simplest CFD methods linear solvers Panel methods solve linear differential equations Prandtl-Glauert equation represent flows complex geometries section describes panel methods detail including: Introduction Geometry AIC Matrix Boundary Conditions End Notes
Panel Methods -- Introduction Panel Methods -- Introduction Since equations solved panel methods linear multiply known solution scalar add together form general solutions work subsonic supersonic cases Panel methods based fundamental solutions Prandtl-Glauert equation Laplace's equation commonly source vortex doublet flows section potential theory basic idea add known solutions uniform flow point source produce streamline pattern matches flow interest add freestream source sink (negative source strength) produce flow oval (called Rankine Oval) superimpose sources sinks get nearly flow pattern desired: Panel methods based idea Sources (or doublets vortices) strength located flow combined solutions satisfy boundary conditions problem boundary conditions typically combined flow surface far body flow approaches freestream solution
Panel Methods -- Geometry Panel Methods -- Geometry first step panel method divide geometry panels straightforward types programs different panel geometries panels flat quadrilaterals fit together complex curved surfaces codes hyperbolic paraboloids warped plates subdivide panels triangles Calculation velocity influence coefficients difficult simple flat plate surface represented infinitely thin plate "actual surface" model combination Panel methods capable handling wide variety configurations Images courtesy Analytical Methods Inc
Panel Methods -- AIC Matrix Panel Methods -- AIC Matrix step dividing geometry panels compute flow pattern panel i associated source doublet vortex unit strength panel j component velocity normal panel denoted AIC(i j) element aerodynamic influence coefficient matrix flow panel i associated singularity unit strength panel j computed basic singularity solution result depends vector distance panels R specifically vector jth singularity control point panel i (often panel centroid) fundamental solution flow field distance singularity sections jth panel contains single point source vortex doublet several vortex lines distribution singularities form distribution differentiates various panel methods distribution singularity strength panel constant value vary linearly quadratically directions list panel codes choice singularity types distributions
Panel Methods -- Boundary Conditions Panel Methods -- Boundary Conditions AIC matrix computed specify boundary conditions total normal velocity panel i expression zero flow tangent surface body constitutes boundary conditions problem {n} vector surface unit normals {sigma} represents unknown singularity strengths element vectors associated panel geometry boundary conditions panel methods express requirement streamlines follow surface contour explicitly set V·n = 0 fact method currently vogue specify B C 's terms potential Dirichlet (as opposed von Neumann) type boundary condition works follows doublet panel method total potential interior section set 0 total potential 0 everywhere inside body (in practice set 0 inside panel control point) velocity 0 particular velocity normal panel inside panel 0 Since doublets produce jump normal velocity (see section) V·n = 0 external flow form B C 's better behaved (numerically) direct (Neumann) type B C
Panel Methods -- End Notes Panel Methods -- End Notes AIC matrix boundary conditions computed solve unknown singularity strengths finally complete flow field pressures practical cases invert AIC matrix solve linear system singularity strengths seems solution unique: linear system equations long number boundary conditions unknowns equal (AIC square matrix) matrix singular answer unique true simply decided panels decision unique chose panels body surface might rotational flows panels model vorticity shear layers boundary layers 3D lifting surfaces shed vortex wakes panels region location wake answer extent especially important position wake properly case interfering lifting surfaces necessarily restrict placement singularities surface panels thin wake cases vorticity exists flow possible model flow singularities flow idea vortex methods especially useful 2-D point vortices shed flow allowed move local flow velocity singularity others (and body) computed process integrated forward time (See Spalart 1988) Flow separation four element airfoil Vortex method 1300 vortices Memory requirements Panel methods advantage need solve quantities surfaces interested need keep track velocities throughout flow field finite difference methods mean memory requirements smaller problem need compute panel neighboring panels panels AIC matrix Npanels x Npanels wing 20 chordwise panels 40 spanwise 1600 panels (upper lower surfaces) 2 56 million influence coefficients requires 10 2MB store forcing codes disk storage slowing down process moral complex geometries panel methods faster finite difference methods require "serious" computing power
Nonlinear Methods Nonlinear Methods Nonlinear CFD methods predict complex flow fields those associated transonic separated flows recently predicting flows air hypersonic aircraft blood artificial hearts list includes links internet sites example applications: NAS Technical Summaries (NASA Ames) Flow turbine rotor number examples Fluent Inc Analytical Methods Inc suggestions general methods principle capable doing anything: simplest method model adequate job need sophisticated method solve fluid mechanics problems fashionable latest technology need submit Cray job find roots quadratic "outdated" 1960's technology needed Evaluate CFD critically accepting assumptions limitations method order magnitude analysis sense qualitatively? Neither computation nor experiment infallible "Nobody believes theoretical predictions engineer computed them; everybody believes experimental engineer conducted test " basis modern CFD techniques solution nonlinear equations fluid flow illustrated start differential equation as: partial differential equation solved two fundamentally different ways: Finite Difference: · Discretization differential form equations · Solutions unknowns computed node points Finite Volume: · Discretization integral form equations · Solution computed cell centroids requires flow field first divided grid difficult grid conform body dense regions large flow gradients two grid directions relatively orthogonal difference equations good approximations real PDE types grids common type grid structured grid grids sort simple structure approach especially useful adaptive grids grid systems complex multiply-connected domains might found multi-element airfoil computer power required solve various non-linear problems reasonable period time (~15 min) points optimistic vast amounts data generated codes displayed computer graphics illuminating simulated experimental techniques oil flow patterns smoke particle traces order visualize simulated oil patterns surface wing-body combination separation junction Click short Quicktime video clip streamline simulation wing-body model clip NASA Ames flow vertical take-off aircraft ground-effect 128 x 128 x 128 grid requires 2 1 M grid points (25 MB velocities 4 bytes per number) wing 20 x 40 panels might modeled 60x80x80 grid requiring 384K points
Finite Difference Methods Finite Difference Methods start differential equation as: matrix equations (tridiagonal system) solved time step matrix huge (perhaps million million) sparse
Finite Volume Methods Finite Volume Methods start differential equation as:
2-D Potential Flow 2-D Potential Flow chapter starts description solution methods detail Beginning simplest flows: two-dimensional inviscid irrotational chapter describes basic theoretical applied airfoil problems chapters modified effects compressibility viscosity Basic Theory Sources Vortices Interactive Calculations
Potential Theory Basic 2-D Potential Theory outline way "known" solutions panel methods generated useful solutions fundamental fluid flow problems known solutions out thin air applied approaches possible simplest case two-dimensional potential flow illustrates process shall 2-D incompressible potential flow mention extension linearized compressible flow case relevant equation Laplace's equation: several ways generating fundamental solutions linear homogeneous second order differential equation constant coefficients Two methods particularly useful: Separation variables complex variables Complex variables especially useful solving Laplace's equation following: theory complex variables region function complex variable z = x + iy analytic derivative respect z direction: dF / dx = dF / dy limit small d's leads famous Cauchy-Riemann conditions analytic function complex plane Consider complex function: Cauchy-Riemann conditions are: Differentiating first equation respect x second respect y adding gives: analytic function complex variable solution Laplace's equation part general solution complex potential consists velocity potential real part stream function imaginary part flow velocities written single complex number: dW/dz = u - iv (Try deriving ) consider simple analytic functions W great applied aerodynamics: Uniform flow: Line Source Vortex: Doublet:
Uniform Flow Uniform Flow U real flow x direction speed U flow direction adjusted changing real imaginary parts good example fact potential defined apart arbitrary constant flow uniform everywhere potential depends choice origin Differences potential physically meaningful depend choice origin
Line Source and Vortex Line Source Vortex expression describes "point" source vortex 2D (which thought vortex line line sources 3-D) K real expression describes source radially directed induced velocity vectors; imaginary values lead vortex flows induced velocities tangential direction Further discussion flows section
Doublet doublet formed superimposing source sink x-axis doublet strength S dx fundamental doublet singularity potential formed taking limit dx goes zero S goes infinity keeping product constant doublet commonly fundamental singularities panel methods
Sources and Vortices Sources Vortices solutions 2-D potential equation proposed singular fact source solution seems ultimate way violating continuity vortex essence rotational (not irrotational assumed) flow solutions indeed singular point satisfy differential equation point singularity perfectly adequate solutions seen evaluating integral forms continuity irrotationality conditions flow field source satisfies continuity: flow field vortex satisfies irrotationality: solutions singular point singular point strange happen: velocity gets large real life large velocities region give rise compressibility effects; viscous effects smear discrete vortex distribution vorticity viscous core actual velocity distribution core free vortex behaves solid body velocity distribution V(R) = kR (This result assuming Gaussian distribution distributed vorticity core region size viscous core depends Reynolds number taken G / n ) 1/r behavior vortex induced velocity mathematical result essential flow exist equilibrium velocity vary 1/r pressure gradients balance centrifugal force acting fluid derivation combine singularities different locations produce desired flow pattern Since solution Laplace's equation uniquely determined regions singularities solution boundaries specified combinations singularities model flows interest Method Images : Ground Effects Wall Interference Source Doublets Circular Cylinder Ellipses Blasius Theorem Groups Vortices Far Field Flow Stokes Theorem Free Vortex Motion
Method of Images Method Images flow field created singularities presence solid boundaries simulated superimposing "image vortices" works symmetry problem right ensures flow plane symmetry boundary problem left Since problems boundary conditions satisfy linear differential equation flow technique useful simulating effects ground aerodynamics cars airplanes low altitude complex situations three images required simulate boundary conditions associated corner technique predict effects wind tunnel walls flow field models tested Imagine system image vortices required simulate wall effects 2D airfoil test Yes 2 images required 3-D situation general solved images
Cylinders Cylinders flow circular cylinder computed uniform stream doublet (See 4 1 6) interesting conclusions generalizations follow expressions velocity potential circular cylinder surface cylinder tangential velocity is: V = 2U sin theta maximum velocity twice freestream value general forms hold ellipsoids: Vmax = V (1 + t/c) V surface = n x (n x Vmax)) holds exactly incompressible potential flow ellipse t/c larger 1 course case real flow probably different potential flow solution force general 2-D cylinder computed calculating velocities Bernoulli's law compute pressures integrating surface pressures total forces moments derived directly complex potential result Blasius theorem derived result follows theory residues complex potential incompressible Bernoulli equation (Or might momentum equation compute net force far field integrals ) Gamma total circulation S net source strength case net source strength net force exerted collection sources vortices flow freestream velocity U perpendicular freestream proportional U total circulation
Circulation and Vorticity: Stokes Theorem Circulation Vorticity Stokes Theorem Stokes' theorem integral identity written: vector function F taken velocity field V relation 2-D restated as: result implies circulation around contour contains group vortices equal sum enclosed vortex strengths (See 2 3 4 ) allows application Blasius theorem find force acting group vortices result Kutta-Joukowski law: treat flow field far group vortices created single vortex strength equal sum individual vortices far field solutions especially simple useful check complex Far field solutions boundary conditions complex field solution reducing required extent computational grids should find superposition singularities satisfies boundary conditions differential equation mean found solution problem example add vortex doublet model circular cylinder find flow went around cylinder non-unique solutions problemsome appeal additional considerations find one(s) actually appear nature auxiliary condition Kutta condition provided viscous effects determine value circulation
Free Vortices Free Vortices Singularities free move flow behave response F = ma (what m?) move local flow velocity vortices sources convected downstream flow interacting singularities produce complex motions due mutual induced velocities pair counter-rotating vortices moves downward mutual induced velocities Co-rotating vortices orbit influence mutual induced velocities
Streamlines Past Sources and Vortices Streamlines Past Sources Vortices Drag singularities right main computation area Set freestream speed (the flow left right) click Compute marks page simulate small tufts direction local flow Experiment multiple singularities simulate pair wing trailing vortices source/sink doublet spinning baseball
Airfoils Part I Airfoils Part I: Introduction chapter introduction airfoils airfoil theory followed application potential flow methods analysis airfoils purpose section relation airfoil geometry airfoil performance methods compute distribution pressures airfoil surface relation pressures airfoil performance Outline Chapter chapter divided several sections first consist introduction airfoils: history basic ideas latter sections simple analyses relate airfoil geometry basic aerodynamic characteristics History Development Airfoil Geometry Pressure Distributions Relation Cp Performance Relating Geometry Cp Methods Airfoil Analysis
Airfoil History History Airfoil Development earliest serious work development airfoil sections began late 1800's known flat plates produce lift set angle incidence suspected shapes curvature closely resembled bird wings produce lift efficiently H F Phillips patented series airfoil shapes 1884 testing earliest wind tunnels "artificial currents air (were) produced induction steam jet wooden trunk conduit " Octave Chanute writes 1893 " seems desirable further scientific experiments concavo-convex surfaces varying shapes impossible difference success failure proposed flying machine depend upon sustaining plane surface properly curved get maximum 'lift' " nearly time Otto Lilienthal similar ideas carefully measuring shapes bird wings tested airfoils (reproduced 1894 book "Bird Flight Basis Aviation") 7m diameter "whirling machine" Lilienthal believed key successful flight wing curvature camber experimented different nose radii thickness distributions Airfoils Wright Brothers closely resembled Lilienthal's sections: thin highly cambered possibly early tests airfoil sections extremely low Reynolds number sections behave better thicker erroneous belief efficient airfoils thin highly cambered reason first airplanes biplanes sections gradually diminished decade wide range airfoils developed based primarily trial error successful sections Clark Y Gottingen 398 basis family sections tested NACA early 1920's 1939 Eastman Jacobs NACA Langley designed tested first laminar flow airfoil sections shapes extremely low drag section achieved lift drag ratio 300 modern laminar flow section sailplanes illustrates concept practical applications thought practical years Jacobs demonstrated wind tunnel utility concept wholly accepted "Laminar Flow True-Believers Club" meets year homebuilt aircraft fly-in reasons modern airfoils different designers settled best airfoil flow conditions design goals change application right airfoils designed low Reynolds numbers low Reynolds numbers (&10 000 based chord length) efficient airfoil sections peculiar suggested sketch dragonfly wing thin highly cambered pigeon wing similar Lilienthal's designs Eppler 193 good section model airplanes Lissaman 7769 designed human-powered aircraft Unusual airfoil design constraints arise leading unconventional shapes airfoil designed ultralight sailplane requiring high maximum lift coefficients small pitching moments high speed possible solution: variable geometry airfoil flexible lower surface airfoil Solar Challenger aircraft flew across English Channel solar power designed totally flat upper surface solar cells mounted wide range operating conditions constraints generally existing "catalog" section best days airfoils designed especially intended application remaining parts chapter describe basic ideas
Airfoil Geometry Airfoil Geometry Airfoil geometry characterized coordinates upper lower surface summarized few parameters as: maximum thickness maximum camber position max thickness position max camber nose radius generate reasonable airfoil section parameters Eastman Jacobs early 1930's create family airfoils known NACA Sections NACA 4 digit 5 digit airfoils created superimposing simple meanline shape thickness distribution fitting couple popular airfoils time: +- y = (t/0 2) * ( 2969*x^0 5 - 126*x - 3537*x^2 + 2843*x^3 - 1015*x^4) camberline 4-digit sections defined parabola leading edge position maximum camber parabola back trailing edge NACA 4-Digit Series: 4 4 1 2 max camber position max thickness % chord max camber % chord 1/10 c 4-digit sections came 5-digit sections famous NACA 23012 sections thickness distribution camberline curvature nose cubic faired straight line 5-digit sections NACA 5-Digit Series: 2 3 0 1 2 approx max position max thickness camber max camber % chord % chord 2/100 c 6-series NACA airfoils departed simply-defined family sections generated prescribed pressure distribution meant achieve laminar flow NACA 6-Digit Series: 6 3 2 - 2 1 2 Six- location half width ideal Cl max thickness Series min Cp low drag tenths % chord 1/10 chord bucket 1/10 Cl six-series sections airfoil design specialized particular application Airfoils good transonic performance good maximum lift capability thick sections low drag sections designed wing design begins definition several airfoil sections entire geometry modified based 3-dimensional characteristics I Kroo 11/28/95
Airfoil Pressures Airfoil Pressure Distributions aerodynamic performance airfoil sections studied distribution pressure airfoil distribution expressed terms pressure coefficient: Cp difference local static pressure freestream static pressure nondimensionalized freestream dynamic pressure (See discussions Cp Bernoulli equation ) airfoil pressure distribution like? generally plot Cp vs x/c x/c varies 0 leading edge 1 0 trailing edge Cp plotted "upside-down" negative values (suction) higher plot (This upper surface conventional lifting airfoil corresponds upper curve ) Cp starts 1 0 stagnation point leading edge rises rapidly (pressure decreases) upper lower surfaces finally recovers small positive value Cp trailing edge Various parts pressure distribution sections Upper Surface upper surface pressure lower (plotted higher scale) lower surface Cp case doesn't Lower Surface lower surface carries positive pressure design conditions actually pulling wing downward case suction (negative Cp -> downward force lower surface) present midchord Pressure Recovery region pressure distribution pressure recovery region pressure increases minimum value value trailing edge area known region adverse pressure gradient sections adverse pressure gradient associated boundary layer transition possibly separation gradient severe Trailing Edge Pressure pressure trailing edge airfoil thickness shape trailing edge thick airfoils pressure slightly positive (the velocity freestream velocity) infinitely thin sections Cp = 0 trailing edge Large positive values Cp trailing edge imply severe adverse pressure gradients CL Cp section lift coefficient Cp by: Cl = int (Cpl - Cpu) dx/c (It area curves ) Cpu = upper surface Cp recall Cl = section lift / (q c) Stagnation Point stagnation point occurs leading edge place V = 0 incompressible flow Cp = 1 0 point compressible flow larger get intuitive picture pressure distribution looking examples sections chapter
Pressures and Performance Airfoil Pressures Performance shape pressure distribution directly airfoil performance features considerations airfoil boundary layer characteristics take inviscid case draw conclusions compute maximum local Mach numbers relate those lift thickness; compute pitching moment decide acceptable inviscid pressures form qualitative conclusions section input detailed boundary layer calculation first investigate close relation airfoil geometry pressures
Pressures and Geometry Relating Airfoil Geometry Pressures discussing detail methods predict airfoil pressure distributions let's consider intuitively relationship airfoil geometry airfoil pressure distributions first changes surface curvature (Click detail ) airfoil pressures vary angle attack "nose peak" extreme angle increases clear small java program change angle attack Cp Cl Cm Click upper half plot increase angle attack alpha lower portion decrease Let's consider detail relationship airfoil geometry airfoil pressure distributions few examples effects changes camber leading edge radius trailing edge angle local distortions airfoil surface reflexed airfoil section reduced camber aft section producing lift region nose-down pitching moment case aft section actually pushing downward Cm zero lift positive natural laminar flow section thickness distribution leads favorable pressure gradient portion airfoil case sharp nose leads favorable gradients 50% section symmetrical section 4° angle attack pressure peak nose thicker section prominent peak thicker section 0° line plot zero lift upper lower surface pressure coincide conventional cambered section aft-loaded section opposite reflexed airfoil carries lift aft part airfoil Supercritical airfoil sections best way develop feel airfoil geometry pressures interactively modify section watch pressures change Program ANalysis Design Airfoils (PANDA) available Desktop Aeronautics simple version program built text allows vary airfoil shape effects pressures (Go Interactive Airfoil Analysis page clicking ) full version PANDA permits arbitrary airfoil shapes permits finer adjustment shape includes compressibility computes boundary layer properties
Interactive Airfoil Analysis Interactive Airfoil Analysis Introduction program built page allows experiment airfoil shape angle attack pressure distribution Instructions Click top part plot increase angle attack; clicking lower portion reduces alpha Drag handles upper lower surfaces modify shape section watch effects Cp Suggested Exercises Change airfoil thickness upper lower surface pressures thickness affects Cp trailing edge Create pressure peak nose upper lower side changing angle attack Change camber nose remove pressure peak Try create positive pitching moment section thin highly cambered section symmetrical section Technical Details program uses combination thin airfoil theory conformal mapping quickly compute pressures airfoil method 1950's compute airfoil pressure distributions Java invented section shape simple well: upper lower surfaces consist quadratic sqrt(x) quadratic (1-x) patched together control points provides 4 degrees freedom lead curves airfoils
Airfoil Analysis Airfoil Analysis Several methods available predicting pressure distribution around airfoils sections 1 Conformal Mapping uses fact solutions Laplace's equation constructed analytic functions complex variable 2 Thin Airfoil Theory textbook aerodynamics derived briefly analysis days high-speed computers leads important insights airfoils 3 Surface Panel Methods accurate analysis method airfoils widely limited assumptions inherent linear differential equations solve
Conformal Mapping Conformal Mapping analytic function complex variable satisfies equation incompressible irrotational flow: Why? relate flow field setting: z' z analytic function z z' = f(z) (Recall z = x + iy ) idea airfoil analysis conformal mapping relate flow field around shape already known (by whatever means) flow field around airfoil circle first shape problem find analytic function relates point circle corresponding point airfoil Joukowski found simple function: z' = z + 1/z transforms circle shape looks airfoil taking origin circle various points different airfoil-like shapes produced 2 degrees freedom (the coordinates circle origin) number airfoil shapes represented simple Joukowski transformation limited thickness greatest 25% chord - far forward constraint led number generalizations Joukowski mapping produce practical shapes methods based basic idea illustrated simple mapping begin circle center circle Xc Yc point z mapped point z' relation: z' = z + 1/z W(z) = F(z) + i G(z) complex potential function velocity by: w = u - iv = dW/dz set W(z') = W(z) velocity airfoil velocity cylinder by: w(z') = dW/dz' = (dW/dz) (dz/dz') = w(z) / ( 1-z -2 ) relates velocity airfoil directly circle relation blows z 2 = 1 velocity airfoil infinity anywhere reason mapping work points mapping analytic mean means sure points flow: inside airfoil choose circle encloses point -1 0 choose circulation velocity 0 point 1 0 point 1 0 maps airfoil trailing edge stagnation point cylinder 1 0 certain amount circulation origin circle determines lift airfoil angle attack (Also point -1 0 enclosed ) troubles conformal mapping methods parameters xc yc airfoil shape analyze particular airfoil iteratively find values produce desired section technique doing developed Theodorsen technique superposition fundamental solutions governing differential equation method subsequent sections thin airfoil theory
Thin Airfoil Theory Thin Airfoil Theory Thin airfoil theory predict forces moments pressures thin cambered surfaces; basic apply airfoils thickness section aspects theory Classical Theory --a conventional derivation Basic Inverse Design Method Thickness Synthesis general airfoil camber thickness
Thin Airfoil Theory Derivation Thin Airfoil Theory Derivation start analysis thin cambered plate build solution arbitrary airfoil distribution vorticity airfoil solution Laplace's equation satisfy boundary conditions combination velocity induced vortices cancels component freestream normal plate: (where small angle approximations introduced) basic approximation thin airfoil theory velocity induced point x due vorticity x' approximated velocity induced x position x axis due vortex x axis: velocity induced vorticity computed basic vortex singularity formula known Biot-Savart Law 2-D element vorticity x' reads: total induced velocity point x by: Combining expression flow-tangency boundary condition basic integral equation solved unknown vorticity distribution: approach solving equation change variables: write g Fourier series: Substituting (4) (2) trigonometric relations: yields: Finally multiply sides cos m q integrate 0 p : substitute coefficients expression g (4) find vorticity pressures lift moment function surface slope distribution dz/dx particular expressions local pressure difference integrated lift moment leading edge are: (Note expression relating Cp g applies thin airfoils ) method computing circulation distribution permits compute pressure difference across airfoil pressures upper lower surfaces computed noting surface perturbation velocities x direction caused singularities zero those due local vorticity: contribution du(x) local vorticity (at x) perturbation velocity du = ± g / 2 + sign upper surface - sign lower surface
Thin Airfoil Theory Thin Airfoil Theory basic equations derived thin airfoil theory below: Several important derived expressions sections Ideal angle attack constants depend airfoil shape - Ao depends angle attack Ao term strange term Fourier series g since leads singularity leading edge ( g -> infinity) angle attack ideal angle attack Ao = 0 vorticity goes 0 leading edge angle ideal angle attack course real flows vorticity infinite (why?) concept ideal angle attack important identifying flow conditions leading edge pressure peaks avoided Lift curve slope rate change lift coefficient angle attack dC L /d inferred expressions result CL changes 2 p per radian change angle attack ( 1096/deg) far measured slope airfoils effects thickness viscosity ignored cancel out extent result airfoils lift curve slope within 10% 2 p value thin airfoil theory Aerodynamic center pitching moment L = 0 expression pitching moment coefficient measured leading edge measure moment center position xo/c expression becomes: choose point xo = 0 25c lift dependence drops out moment coefficient measured point independent angle attack* point dCm/dCL = 0 aerodynamic center according thin airfoil theory quarter chord point airfoil Experiments close *The Cm point denoted Cma c equal Cm zero lift Cmo written independent moment location particularly useful quantity: Cmo = Cma c general example + Rules Thumb parabolic camber meanline example thin airfoil theory case serve useful first approximations thin cambered airfoil Assume: z(x) = 4 h x (1-x) Thin airfoil theory estimate flap deflection airfoil lift moment provides estimate hinge moments vs deflection angle angle attack good problem work are: where: effects viscosity tend overestimate extent lift moment due flap deflections
Thin Airfoil Inverse Design Thin Airfoil Inverse Design basic thin airfoil theory formulation design airfoils desired pressure distribution process actually easier direct analysis integral equation circulation distribution airfoil shape: evaluated directly circulation distribution evaluate integral several values x find surface slope dz/dx airfoil Actually have: integrated obtain: C alpha chosen z(0) = z(1) = 0
Thickness Effects Thickness Effects ideas predicting characteristics thin airfoils camber applied study airfoils thickness placed vortices airfoil chord line model camber line place sources x axis model thick symmetric airfoil source strengths computed simply vortex strengths satisfy continuity region ABCD: sigma dx source strength contained box first order then: induced velocity chord line then: condition tangent flow automatically satisfied solution source strength (2) total x component velocity then: nose airfoil approximation small angles longer valid answers previous expression accurate 1950's Riegels introduced correction thin airfoil theory greatly improves accuracy fact Riegels correction produces work ellipses exceeding 100% thick previous theory works 10 15% velocity surface airfoil Riegels modification is:
Moderately Thin Airfoils Moderately Thin Airfoils general cambered airfoil represented sources vortices source strength symmetrical airfoil thickness distribution vorticity distribution cambered plate desired angle attack solution lead singularity leading edge ideal angle attack better estimate velocity distribution adding velocity distribution around thick symmetrical airfoil angle attack perturbations associated camberline ideal angle attack simple since thin airfoil theory velocity distribution thick symmetrical airfoil angle attack find distribution experiment theory combine thin airfoil general solution
Surface Panel Methods Surface Panel Methods Airfoils assumption thin airfoil theory singularities boundary conditions evaluated x-axis panel method compute forces vorticity thin airfoil approach place vortices (·) mean line boundary conditions tangential flow satisfied control points (x) Vortices placed 1/4 chord point panel control points 3/4 chord point chosen solely lift moment out right few panels details method left homework exercise place vorticity actual surface (not mean line) thick section common approach distribution linearly-varying vorticity panels solves vorticity strength panel edges requires boundary conditions Kutta condition satisfied detail method Kuethe Chow Moran Recall PDE BC's sufficient produce unique solution gotten explicit extra condition tricks thin airfoil theory assumed form circulation distribution vorticity trailing edge discrete vortex panel model previous page selection control points one-half panel vortices effectively true surface panel method auxiliary condition explicitly forcing vorticity upper surface equal magnitude opposite sign lower surface flows satisfy Kutta condition condition enforced reality viscosity: flow continue around sharp trailing edge (at high speeds) boundary layer separate upper surface Fortran subroutine computes pressure distribution arbitrary airfoils linear surface vorticity distribution
Compressibility in 2D Compressibility 2D section basic equations compressible flow compressibility airfoil performance analysis design transonic supersonic airfoils Basic Compressible Flow Theory Effects Airfoils Linear Theory Transonics Supersonic Flow
Basic Compressible Flow Theory Basic Compressible Flow Theory Compressibility alters previous theories important density variation significant - situation several parameters First important gasses liquids require large pressure changes cause mild density variations great importance aircraft propulsion systems rockets vehicle air large pressure variations Pressure variations produce density changes speed sound: Since sea level air pressure 2116 psf changes order 150 psf (required airplane fly) produce small change overall pressure hence density 35 000 ft pressure 500 psf disturbances created airplane greater compressibility effects important connection Mach number clear following: sea level air pressure 2116 psf density = 002377 sl/ft^3 1/2 rho V^2 = 1 p M = 378 35 000 ft air pressure 499 psf density 000738 sl/ft^3 1/2 rho V^2 = 1 p M = 378 (Same number!) importance density changes directly Mach number M < 3 effects small M > 7 effects large (But careful rule thumb uncommon Cp -7 airfoil M=0 3 implies local Mach number 1 0 ) density constant assumptions deriving incompressible Bernoulli equations longer valid effects density changes incorporated follows: consider adiabatic (no heat addition) reversible (no friction dissipation) flow: isentropic flow flows outside boundary layer isentropic basic thermodynamics have: together equation state perfect gas: p = rho R T isentropic relations: Combining Euler equation: dp = - rhoV dV leads compressible Bernoulli eqn dynamic pressure written: compressibility seen directly way flow responds changes distances streamlines: continuity equation: rho S V = constant together Euler's equation expression speed sound derive: M gets closer 1 0 small changes area produce large changes velocity reflected Prandtl-Glauert compressibility correction factor M=0 1% reduction area produces 1% increase speed M = 0 9 1% reduction area produces 5% increase speed M>1 sign velocity increment changes converging flows speed subsonically slow down supersonic
Compressibility Effects on Airfoils Compressibility Effects Airfoils Compressibility produces changes flow patterns resulting airfoil forces moments low Mach numbers local flow velocities airfoil changed compressibility overall shape changed Mach number increased local zone supersonic flow develop changes pressure distribution significantly airfoil lift upper surface supersonic first transition subsonic supersonic flow occurs smoothly transition back subsonic flow occurs shock local Mach number appreciably greater 1 shock wave strong pressure distribution radically altered happens quickly (small changes M) characteristic lambda shape caused subsonic flow boundary layer exaggerated Mach number further increased shocks develop lower surface sudden increase pressure shock cause boundary layer separate correspondingly large increases drag shock moves back Mach number increased shock wave gradually moves back trailing edge airfoil surrounded supersonic flow condition drag high airfoil operating far design point Further increases Mach number (supersonic incident flow) change flow character totally freestream flow supersonic airfoil type behaves differently supersonic flow able suddenly change direction happens onto airfoil leading edge bow wave formed extending far flow field small region airfoil subsonic bow wave bent back Mach number increased further speed small area nose remains subsonic finds experiments increase wing lift curve slope Mach number increased increase Cl Mach number reverses itself strong shocks form wing effects moving aft center pressure causes nose-down pitching moment rapid increase drag shocks form drag increases Mach number CL data based flight tests medium size airliner
Linear Compressibility Linear Compressibility effects compressibility predicted simply Mach number range flow satisfies linear differential equations so-called compressibility corrections simplest Prandtl-Glauert correction derived governing differential equation change variables: incompressible potential equation new variable x': boundary conditions need revised terms x' resulting velocity be: revealing approach comes introducing variables: terms variables write Prandtl-Glauert equation as: describes incompressible potential flow phi'(x y') original (linearized) boundary conditions are: imagine fictious airfoil coordinates: y' (x) incompressible flow field potential phi' boundary conditions are: Noting follows x location velocity component y direction airfoil surface equal compressible fictious incompressible flows fictious airfoil shape actual section solve incompressible potential phi' compute compressible potential desired location velocity perturbation component x direction u computed by: u' x velocity perturbation component airfoil incompressible flow assumes velocity perturbations small compared freestream velocity pressure coefficient written approximately* as: Hence compressible Cp Cp airfoil incompressible flow by: Prandtl-Glauert compressibility correction variety alternative compressibility corrections proposed higher order approximations Cp formula attempt nonlinear effects full potential equation complex corrections lead nonlinear relation compressible Cp incompressible useful nonlinear relations Karman-Tsien compressibility correction slightly complex Prandtl-Glauert rule: expression derived incorporating increased u perturbation velocity second order expression Cp plots incompressible Cp distributions compared those predicted two correction rules Cp's predicted full potential equation experiments simple rules minimum local Cp approaches critical value Cp* air gamma = 1 4: Plots Mach Number predicted measured airfoil pressures * linearized expression Cp terms perturbation velocity u: Cp = - 2u/U derived incompressible flow follows isentropic expression Cp derivation obvious (see Anderson) assumption small perturbations consistent assumptions led PG equation first place
Compressibility Effects on Airfoil Cp Compressibility Airfoil Cp Mach number pressure distribution NACA 0012 airfoil Comparison various compressibility correction methods Mach > 0 3 0 5 0 7 0 8
Transonic Airfoil Pressures Transonic Airfoil Pressures Transonic airfoils designed achieve desired thickness lift coefficient developing strong shock waves upper surface larger t/c larger Cl lower freestream Mach number order avoid shocks accompanying drag increase Actually need entirely avoid supersonic flow avoid high transonic drags keeping maximum local Mach number something 1 2 trying keep maximum speeds forward airfoil "crest" low drag airfoil supercritical flow designed way keep upper surface Cp's getting low carry positive pressures lower surface trailing edge technique aft-loading together minor changes airfoil nose lead achievable low drag Mach numbers 0 06 conventional sections "supercritical" airfoils discussion transonic airfoil design chapter 8 detail Karman-Tsien correction works transonic speeds PANDA incompressible KT correction FLO-6 full potential program Mach > 0 70 0 72 0 74 0 76 0 78 0 80
Supersonic Airfoils Supersonic Airfoils Fundamentals flow bodies supersonic speeds different subsonic speeds earlier differential equations inviscid flow hyperbolic elliptic Mach number exceeds 1 0 disturbances propagate speed slower freestream speed waves form angle waves generated disturbances small Mach angle: Sin beta = dt / V dt = 1/M Mach wave boundary areas affected presence small disturbances those unaffected shock wave produced larger disturbances flow compressed temperature changes speed sound changes shock angle greater Mach angle flow forced turn corner shock wave created relationship initial Mach number Mach number shock turning angle delta shock angle theta computed shock jump conditions: relationships derived continuity momentum energy relationship turning angle shock angle textbooks compressible flow important point two solutions corresponding weak shock waves corresponding strong shocks weak shock solution actually occurs external aerodynamics problems solution maximum value d q incident Mach number means maximum value turning angle possible Attempting turn flow angle greater detached bow shock higher drag leading edges supersonic airfoils sharp sharp leading edges lead great simplifications analysis supersonic airfoils entire flow approximated small perturbations Further details left courses compressible flow theory should familiar oblique shocks shock jump conditions Prandtl-Meyer expansions etc Additional information sonic booms linear interference effects edition notes Linear Supersonic Theory Solutions Prandtl-Glauert equation supersonic flow particularly simple Since differential equation linear superimpose known solutions vortices sources doublets example 3-D supersonic source by: change character equation write down simple general solution equation (in 2-D now): verify solution: satisfy differential equation F G arbitrary functions! solution different character subsonic solution fact disturbances propagate ahead wave front known law forbidden signals two functions F G solution represent characteristic lines moving upper lower surfaces body respectively Consider part describes upper surface flow field: expressions perturbation velocity components then: boundary condition approximated by: Then: total velocity airfoil surface approximately by: pressure computed compressible Bernoulli equation Since assumed small perturbations freestream linearized form Bernoulli equation appropriate: Airfoil pressure surface airfoil thus: theta positive flow turned freestream direction negative turned back Since solution applies characteristic line pressure flow field changes due dispersion dissipation Unlike incompressible case require supersonic airfoils produce small perturbations satisfy equations means small surface slopes everywhere: blunt le's means express surface slope as: Cp equations linear theta net Cp sum due thickness camber angle attack net lift moment sum those associated 3 components flat plate angle attack component contributes integral result supersonic airfoil then: (The zero lift angle attack 0) camber line contributes moment result moment coefficient leading edge is: three components contribute drag: cross terms angle attack * thickness slope useful canceling cross terms occurs total drag sum drags component (the drag due flat plate angle attack drag symmetrical thickness form zero lift drag camber line zero lift drag flat plate is: implies net force plate acts normal plate seems obvious: integrating surface pressures act normal surface way? fact explain subsonic flow Cd = 0 flat plate angle attack? reason get zero drag subsonic flow thick airfoils indeed surface area facing freestream direction airfoil thinner forward facing area gets smaller Cp gets negative (a pressure peak develops) limit t/c goes 0 singularity develops combined effects cancel Cd = 0 supersonic sections singularity form; fact pressure peak finite limit Cd <> 0 forward facing pressure component leading edge suction amount leading edge suction achieved depends angle attack Mach number sharpness leading edge appreciate airfoil shape angle attack performance supersonic speeds try changing airfoil examine lift moment L/D Click upper half plot increase angle attack lower half decrease pitching moment measured 50% chord point
Viscosity and Boundary Layers Viscosity Boundary Layers chapter effects viscosity two dimensions sections describe basic phenomena simple theory estimate boundary layer properties Boundary layers appear surface bodies viscous flow fluid seems "stick" surface (*see note) Right surface flow zero relative speed fluid transfers momentum adjacent layers action viscosity thin layer fluid lower velocity outer flow develops requirement flow surface relative motion "no slip condition " velocity boundary layer slowly increases reaches outer flow velocity Ue boundary layer thickness delta defined distance required flow nearly reach Ue might take arbitrary number (say 99%) define mean "nearly" certain definitions frequently (see theory section ) boundary layer concept attributed primarily Ludwig Prandtl (1874-1953) professor University Gottingen 1904 paper subject formed basis future work skin friction heat transfer separation subsequently fundamental contributions finite wing theory compressibility effects (His name appears 30 times notes ) Theodore von Karman Max Munk among famous students R T Jones student Max Munk I subsequently learned great R T Jones- readers notes great-great grandstudents Prandtl character boundary layer changes develops surface airfoil Generally starting out laminar flow boundary layer thickens undergoes transition turbulent flow continues develop surface body possibly separating surface certain conditions laminar flow fluid moves smooth layers lamina relatively little mixing consequently velocity gradients small shear stresses low thickness laminar boundary layer increases distance start boundary layer decreases Reynolds number fluid sheared across surface body instabilities develop eventually flow transitions turbulent motion Turbulent boundary layer flow characterized unsteady mixing due eddies scales result higher shear stress wall "fuller" velocity profile greater boundary layer thickness wall shear stress higher velocity gradient wall greater effective mixing associated turbulent flow lower velocity fluid transported outward result distance edge layer larger Several fundamental effects produced viscosity: Drag : Skin friction drag caused shear stresses surface contribute majority drag airplanes pressure distribution changed presence boundary layer significant separation present changes CL Cm Flow separation : Viscosity responsible flow separation causes major changes flow patterns pressures compute characteristics basic boundary layer theory detailed computational methods laminar turbulent boundary layers *Actually zero slip condition surface arises roughness surface molecular scale Fluid molecules hitting surface impart net momentum surface mean velocity molecules hitting surface surface velocity surface extremely smooth electrostatic forces exist surface air molecules introducing shear stress surface interaction reduced reduction skin friction result found way (Yet )
Viscous Drag Viscous Drag Skin Friction shearing stresses surface body produce skin friction drag define skin friction coefficient Cf by: shear stress viscosity by: Cf drag coefficient CD (skin friction) = Cf*Swetted/Sref Swetted area "wetted" air Sref area define drag coefficient expression applies flat plate body thickness local velocities surface higher freestream velocity skin friction increased write: CD = k * Cf * Swetted / Sref k "form factor" depends shape body skin friction coefficient varies Reynolds number Mach number character boundary layer momentum transferred air body surface appears velocity deficit viscous wake body plot Reynolds number location transition laminar flow turbulent flow affects skin friction coefficient basic boundary layer theory combined experimental fits obtained: laminar boundary layers flat plates: fully-turbulent flat plate boundary layers: Pressure Drag addition direct skin friction presence boundary layer creates pressure form drag bodies appear flat plate pressure acts perpendicular drag direction case adverse pressure gradient skin friction drag reduced pressure drag increases increase pressure drag compensates reduction skin friction combined drag estimated handy expression derived Squire Young (see Thwaites Incompressible Aerodynamics) gives amazingly good estimates total profile drag: theta nondimensionalized chord length velocity outside boundary layer trailing edge Ue normalized freestream U Hte shape factor boundary layer trailing edge (See section Boundary Layer Theory ) Ue 1 0 drag twice momentum thickness upper lower surfaces H = 2 Ue = 0 9 (Cp = 2) CDu = 1 38 theta
Boundary Layers and Pressures Boundary Layers Pressures addition direct skin friction drag presence boundary layer changes effective shape body leading changes pressure distribution overall lift drag effective shape approximate boundary layer inviscid analysis methods combined boundary layer equations Outside boundary layer flow behaves inviscid (and irrotational) fluid simpler analysis methods outside boundary layer compute streamlines outside boundary layer boundary conditions (This actually easier said coupled boundary layer - inviscid solutions poorly conditioned numerically ) leads changes lift drag moment compared inviscid solution change pressure distribution leads non-zero pressure drag addition skin friction drag previously sum skin friction pressure drag termed "profile drag " angle attack changes boundary layer shape changes thicker boundary layers developing toward aft part airfoil higher angles attack (because severe adverse pressure gradients) effective shape airfoil changes angle attack mean line effective shapes clear viscous effects cause effective decambering airfoil shape leads changes lift curve slope (up 10% reduction Cl Reynolds numbers millions) aerodynamic center forward predicted inviscid theory increasing importance Reynolds number reduced
Separation Separation flow surface reverses direction flows upstream place generally upstream streamlines meet leave surface separation caused presence adverse pressure gradient occurs assumptions u component velocity larger v component certain derivatives x direction ignored longer valid coupling inviscid analysis boundary layer calculation work resort experiment Navier-Stokes solutions* changes flow pattern associated forces moments large Drag increases substantially airfoil lift drops generally Reynolds number dependent presence adverse pressure gradient (increasing pressure) tends cause deceleration fluid coasts uphill fluid starts (pressure) hill little speed starts rolling backward picture explains flow separate readily higher Reynolds numbers case velocity profile "fuller" high external velocities extending down closer surface Turbulent boundary layers greater velocity surface better able handle adverse pressure gradients Since velocity surface laminar boundary layer lower velocity turbulent counterpart laminar boundary layer likely separate occurs laminar boundary layer leaves surface undergoes transition turbulent flow surface process takes place certain distance inversely Reynolds number happens quickly enough flow reattach turbulent boundary layer continue surface phenomenon significant effects airfoil pressure disctributions low Reynolds numbers compute separation occur solve Navier-Stokes equations apply several separation criteria solutions boundary layer equations Laminar Separation Criteria Turbulent Separation Criteria
Laminar Separation Criteria Laminar Separation Criteria Since Thwaites method essentially shape profile tell flow boundary layer reverses happens when: exact value important since lambda changes quickly area separation criterion require numerical integration boundary layer equations due Stratford criterion asserts laminar separation occurs when: Cp_bar x_bar canonical pressure coefficient effective boundary layer length Stratford's laminar separation criteria appropriately reflects deleterious adverse gradient's severity length laminar boundary layer prone transition adverse gradient difficult predict flow transition separate first flow separates transitions reattaches laminar separation bubble length bubble function pressure gradient Reynolds number growing longer Reynolds number reduced case laminar separation avoided airfoil design several ways including forcing transition surface roughness elements (grit) building special transition region pressure distribution Once flow turbulent apply entirely different set separation criteria sections
Canonical Pressure Distribution Canonical Pressure Distribution Separation criteria stated terms so-called canonical pressure coefficient Cp_bar defined as: canonical pressure coefficient varies 0 start pressure recovery 1 stagnation
Effective Boundary Layer Length Effective Boundary Layer Length x_bar effective length boundary layer laminar flat plate x_bar equal x boundary layers pressure gradients effective length modified fact favorable pressure gradients good boundary layer health boundary layer traveling favorable gradient similar traveled shorter distance favorable gradient laminar boundary layers: turbulent boundary layers Stratford derived equations assuming flat turbulent rooftop distribution start pressure recovery case xbar = x distance leading edge pressure gradient exists flow laminar Stratford's criterion long effective turbulent boundary layer length effective distance x effective distance start recovery xm plus actual distance traveled recovery section begins turbulent flow pressure gradient difference expression laminar boundary layer previous card case great interest effective boundary layer length laminar flow followed transition turbulent boundary layer basic idea case: problem computing presence transition straightforward Ue = constant special case card flat rooftop section pressure gradient estimate equivalent turbulent boundary layer length follows: momentum thickness changed transition process compute thickness laminar layer transition length turbulent boundary layer lead momentum thickness laminar flat plate momentum thickness is: turbulent flat plate (from Schlichting -- Kuethe Chow 408): effective turbulent length then:
Turbulent Separation Criteria Turbulent Separation Criteria Turbulent separation criteria useful since pressure recovery turbulent boundary layers criteria Minimum Cp simplest criterion estimate flow separate leading edge airfoil rule thumb states minimum value Cp tolerated Numbers -10 -13 crude rule applies cases leading edge separation Loftin's criterion sophisticated method attributed Loftin states maximum value Cp_bar canonical pressure coefficient start recovery +0 88 conservative estimate relied wide range airfoils Shape Factor Perhaps reliable criterion based computed boundary layer quantities separation likely value shape factor H exceeds 2 2 2 4 Stratford's Criterion 1959 Stratford devised simple criterion separation turbulent boundary layers Similar laminar separation criterion rule states separation occur when: constant S 0 35 d^2p/dx^2 < 0 (concave recovery) 0 39 d^2p/dx^2 > 0 (convex recovery) Reynold's number Stratford formula based local effective length boundary layer x_bar maximum velocity Um formula based great empirical data valid Cp_bar < 4/7 useful design airfoil sections Stratford's method conservative predicting separation methods based explicit computation shape factor Comparison corresponding laminar flow formula turbulent boundary layers resistant separation expression x different turbulent boundary layers detail definition effective boundary layer length presented end section Stratford's criterion compute shape pressure distribution everywhere edge separation useful distribution reasons importantly permits rapid possible recovery minimum pressure something approaching design sections maximum extent laminar flow sections maximum lift maximum thickness subsequent section particular Cp distribution derived start taking Stratford's criterion differential equation describing Cp variation integrate expression resulting Cp straightforward appears since formula valid Cp_bar < 4/7 Stratford effectively assumed constant value boundary layer shape factor (e g H = 2 0) section derived: values b chosen slope value Cp_bar match Cp_bar = 4/7 Upper surface pressures Stratford recovery Cp = 0 20 trailing edge Laminar Rooftop Re = 5 x 10^6 Upper surface pressures Stratford recovery Cp = 0 20 trailing edge Turbulent Rooftop Re = 5 x 10^6 Liebeck airfoils Startford pressure recoveries designed maximum lift
Boundary Layer Theory Boundary Layer Theory basic boundary layer theories divided sections below: Basic Theory Definitions Laminar Boundary Layers Transition Turbulent Boundary Layers Summary Interactive Calculation
Boundary Layer Theory and Definitions Boundary Layer Theory Definitions Boundary Layer Thickness boundary layer thickness delta defined distance required flow nearly reach Ue might take arbitrary number (say 99%) define mean "nearly" certain definitions frequently displacement thickness momentum thickness alternative measures boundary layer thickness calculation various boundary layer properties displacement thickness defined considering total mass flow boundary layer mass flow boundary layer completely rest thickness delta*: laminar boundary layers delta* third distance edge boundary layer delta momentum thickness theta defined similarly momentum flux mass flux: laminar boundary layers delta tends order magnitude greater theta ratio delta* theta termed shape factor H: Boundary Layer Equations Newton's law applied fluid element 2-D is: x direction: leads directly to: boundary layer equation combined continuity equation dp/dy = 0 3 equations unknowns: p u v basic (assumptions?) development boundary layer theory static pressure constant boundary layer (dp/dy = 0) important result exactly true* start 2-D boundary layer equation steady flow reduces to: approach solution equation pressure vary y specified external velocity distribution v computed continuity equation leaving PDE u integrated holds laminar flow since turbulent boundary layers inherently unsteady Subsequent sections solution equations detail *If considers balance normal forces fluid element dp/dn = rho V^2/R R radius curvature holds viscous fluids since expect viscosity produce shear stresses normal stresses Cp (p-p0)/( 5 rho Uo^2) dCp/dn = 2 (V/U0)^2 / R change Cp boundary layer is: Cp ~ delta/R * (V/U0)^2 long radius curvature larger boundary layer thickness local velocities large true conditions met though!
Laminar Boundary Layer Theory Laminar Boundary Layer Theory Flat Plate Flow previous section boundary layer equations reduce boundary layer assumed steady: especially simple case incompressible flow pressure gradient laminar * flat plate boundary layer flow satisfies: equations solved introducing variable eta: introducing function: continuity then: boundary layer equation becomes: f''' + ff'' = 0 ordinary differential equation accompanied boundary conditions state velocity right surface 0 far velocity approaches specified Ue B C 's written: f = f' = 0 h = 0 f' = 2 h = infinity problem first formulated way Blasius 1908 equation seems simple nonlinear closed form solution known problem solved assuming series approximation f said relations found: Cf twice momentum thickness flat plate expect? Thwaites Method Thwaites method computing boundary layer characteristics laminar flow pressure gradients based steady boundary layer equations specified external pressure gradient gives approximate solution Starting Navier-Stokes Equations boundary layer assumptions flow steady incompressible ignore higher order terms (See K&C pg 461 330 314) x-direction: rewritten terms boundary layer variables theta H Ue Cfl: theta momentum thickness: H shape factor: Cfl local skin friction coefficient: basic idea laminar boundary layer methods method particular form variation u y: i e u/ue = f(y) Pohlhausen's method based quartic polynomial u(y) Thwaites' method direct uses exact solutions certain laminar boundary layers approximate relationship H Cf Re theta Substituting theta H expression Thwaites obtained: integrated theta(x)
Transition Transition boundary layer changes character smooth steady motion (laminar flow) turbulent motion unsteadiness eddies small scale Several responsible transition two states include: surface roughness adverse pressure gradients wing sweep NASA TP 2256 contains flight data allowable surface roughness waviness laminar flow NACA TN 4363 Simplified Method Determination Critical Height Distributed Roughness Particles Boundary Layer Transition contains discussion rule thumb Hoerner roughness element should least 25% boundary layer displacement thickness cause transition Wing sweep encourages transition cross-flow instability discussion appears NASA TP methods predicting general complex People run large computer codes take time master questioned reason transition airfoils presence adverse pressure gradient (Increasing Cp x ) severe enough long enough transition likely occur rule thumb boundary layers high Reynolds numbers withstand little adverse gradient once gradient adverse transition occur quantitative basis several transition criteria proposed wide empirically-based work restricted Reynolds number range Two popular Michel's Granville's criteria Michel's criteria states transition occur Reynolds number based momentum thickness exceeds certain value depends local Reynolds number: relationship holds Rex values 10^5 40x10^6 Unfortunately practice left side increases rate right hand side making determination transition location inaccurate adverse gradient exists left side grows rapidly transition reasonably predicted (Note gradients large transition expected values Rex 3 million ballpark estimate course ) refined approach transition prediction analysis stability boundary layer equations disturbances basic ideas approach Osborne Reynolds (of Reynolds number fame) Lord Rayleigh (of Rayleigh number fame) hypothesis critical Reynolds number small disturbances laminar boundary layer damped viscosity amplified laminar character flow disappears starts unsteady Navier-Stokes equations assumes unsteady terms small written Fourier series unknown coefficients Upon substitution equations unsteady part NS equations 4th order linear ODE known Orr-Sommerfeld equation mean velocity profile Reynolds number specified leads eigenvalue problem assumed value disturbance wavelength 1957 M O Smith several experiments correlate location transition value real part eigenvalue found transition generally occurred disturbances grown 8000 times (e^9) value point neutral stability sort linear stability theory (e^n method) routinely estimate transition location n 9 11
Turbulent Boundary Layer Theory Turbulent Boundary Layer Theory equations turbulent flow full Navier-Stokes equations simplify equations empirical models complex aspects turbulence making assumption thin layer method compute characteristics "steady" turbulent boundary layer pressure gradient case solve two coupled first order ODE's first von Karman integral equation: theta Ue dimensionless chord length freestream velocity H shape factor delta*/theta second expression describing entrainment flow boundary layer: H1 mass flow shape factor: F entrainment parameter variables expressions skin friction coefficient H theta Ludwieg-Tillman skin friction law: mass flow shape factor H1 H fits: Finally entrainment parameter F approximated by: integrate system equation numerically variation theta H position airfoil expressions evaluated numerically useful flat plates (Re = 0 5 10 million): formulas involving Re^ 2 based assumption 1/7th power law shape boundary layer profile (u/Ue = (y/d)^1/7 agree experiments Re = 20 million skin friction underestimated based logarithmic distribution agree limited test data out Re = 500 million cases flow starts laminar undergoes transition continues turbulent boundary layer details transition process matter current research estimate gross features flow matching assuming virtual start turbulent layer upstream transition momentum thicknesses laminar turbulent layers match transition "point"
Summary of Boundary Layer Results Summary Boundary Layer useful expressions flat-plate boundary layers: Laminar Turbulent Interactive Boundary Layer Calculations java applet computes various properties flat plate boundary layer Specify transition location dragging transition point forward (left) aft computations equations momentum thickness laminar turbulent layers matched transition point Ilan Kroo 11/96
Airfoils Part II Airfoils Part II: Design chapter introduction problem airfoil design typical design goals constraints brief discussion design methodology Design Methods Objectives Typical Design Problems High Lift Systems
Airfoil Design Methods Airfoil Design Methods process airfoil design proceeds knowledge boundary layer properties relation geometry pressure distribution goal airfoil design varies airfoils designed produce low drag (and required generate lift ) sections need produce low drag producing amount lift cases drag doesn't really matter - maximum lift important section required achieve performance constraint thickness pitching moment off-design performance unusual constraints further section historical examples approach airfoil design airfoil already designed someone knew she doing "design authority" works goals particular design problem happen coincide goals original airfoil design rarely case existing airfoils good enough cases airfoils chosen catalogs Abbott von Doenhoff's Theory Wing Sections Althaus' Wortmann's Stuttgarter Profilkatalog Althaus' Low Reynolds Number Airfoil catalog Selig's "Airfoils Low Speeds" advantage approach test data available surprises unexpected early stall likely hand available tools sufficiently refined reasonably sure predicted performance achieved "designer airfoils" specifically tailored needs project common section notes process custom airfoil design Methods airfoil design classified two categories: direct inverse design Direct Methods Airfoil Design direct airfoil design methods specification section geometry calculation pressures performance evaluates shape modifies shape improve performance two main subproblems type method identification measure performance approach changing shape performance improved simplest form direct airfoil design starting assumed airfoil shape (such NACA airfoil) determining characteristic section problemsome fixing problem process fixing obvious problems airfoil repeated major problem section design airfoils require specific definition scalar objective function require expertise identify potential problems considerable expertise fix Let's simple (but real life!) example company business building rigid wing hang gliders low speed requirements decide version Bob Liebeck's high lift airfoils pressure distribution lift coefficient 1 4 small amount trailing edge separation predicted Actually airfoil works achieving Clmax 1 9 Reynolds number million glider actually built flown fact won 1989 U S National Championships terrible high speed performance lower lift coefficients wing seemed fall out sky plot pressure distribution Cl 0 6 pressure peak lower surface causes separation severely limits maximum speed hard fix reducing lower surface "bump" leading edge increasing lower surface thickness aft bump pressure peak low Cl removed lower surface flow attached remains attached down Cl 0 2 check hurt Clmax new section original design condition (still Clmax) modification lower surface upper surface pressure peak Clmax turns out changed little section better match application demonstrates effective small modifications existing sections new version glider section designed scratch lower drag objective airfoil design stated positively "fix worst things" might try reduce drag high speeds trying keep maximum CL greater certain value slowly increasing amount laminar flow low Cl's checking maximum lift objective defined numerically actually minimize Cd constraint Clmax maximize L/D Cl^1 5/Cd Clmax / Cd@Cldesign selection merit airfoil sections important generally considering rest airplane example wish build airplane maximum L/D build section maximum L/D section Cl best Cl/Cd different airplane CL best CL/CD Inverse Design type objective function target pressure distribution possible specify desired Cp distribution least squares difference actual target Cp's objective basic idea variety methods inverse design example thin airfoil theory solve shape camberline produces specified pressure difference airfoil potential flow second part design problem starts somehow defined objective airfoil design stage design changing airfoil shape improve performance several ways: 1 hand knowledge effects geometry changes Cp Cp changes performance 2 numerical optimization shape functions represent airfoil geometry letting computer decide sequence modifications needed improve design
Airfoil Design Problems Typical Airfoil Design Problems Regardless design goals constraints faced common problems airfoil design difficult section common issues arise design problems: Design maximum thickness Design maximum lift Laminar boundary layer airfoil design High lift thickness transonic design Low Reynolds number airfoil design Low positive pitching moment designs Multiple design points
Thick Airfoil Design Thick Airfoil Design difficulty thick airfoils minimum pressure decreased due thickness severe adverse pressure gradient need start recovery sooner maximum thickness point specified section maximum thickness recover point steepest possible gradient sort problem addressed Liebeck connection maximum lift thickest possible section boundary layer verge separation throughout recovery thickest section Re = 10 million 57% thick course separate suddenly angle attack I Kroo 11/28/95
High Lift Airfoil Design High Lift Airfoil Design produce high lift coefficients require negative pressures upper surface airfoil limit suction associated compressibility effects imposed requirement boundary layer capable negotiating resulting adverse pressure recovery maximize lift starting specified recovery height location best keep boundary layer verge separation* distributions Re 5 million difference laminar turbulent thickest section Re = 10 million 57% thick course separate suddenly angle attack maximum airfoil lift best recovery location chosen airfoil thin lower surface produces maximum lift (Since upper surface Cp specified increasing thickness reduces lower surface pressures ) upper surface Cp negative -3 0 perturbation velocity greater freestream means thin section lower surface flow upstream cause separation maximum lift achieved upper surface velocity 2U thickness keep lower surface stagnation pressure detailed discussion topic found section high lift systems *This conclusion Liebeck derived Stratford's criterion laminar boundary layer method Thwaites turbulent boundary layer criteria conclusion obvious indeed suggested (Kroo Morris) case
Laminar Airfoil Design Laminar Airfoil Design section boundary layers laminar flow useful reducing skin friction drag increasing maximum lift reducing heat transfer achieved work low Reynolds numbers maintaining smooth surface airfoil favorable pressure gradient section pressures tailored achieve long runs laminar flow upper and/or lower surfaces Again Stratford-like pressure recovery helpful achieving maximum run favorable gradient either upper lower surfaces
Transonic Airfoil Design Transonic Airfoil Design transonic airfoil design problem arises wish limit shock drag losses transonic speed effectively limits minimum pressure coefficient tolerated Since lift thickness reduce (increase magnitude) minimum Cp transonic design problem create airfoil section high lift and/or thickness causing strong shock waves generally tolerate supersonic flow drag increase sections operate efficiently "supercritical airfoils" rule thumb maximum local Mach numbers should exceed 1 2 1 3 well-designed supercritical airfoil produces considerable increase available Cl compared entirely subcritical designs Supercritical sections refer special type airfoil designed operate efficiently substantial regions supersonic flow sections take advantage design ideas maximize lift thickness Mach number: Carry lift practical aft potion section flow subsonic aft lower surface obvious candidate increased loading (more positive Cp) several considerations limit extent approach sure sufficient lift carried forward portion upper surface Mach number increases pressure peak nose diminished additional blunting nose possible extra lift lost region lower surface nose loaded reducing lower surface thickness leading edge provides lift positive pitching moment Shocks upper surface leading edge produce wave drag shocks aft airfoil crest feasible best design sections forward shocks sections known "peaky" airfoils transport aircraft idea carefully tailoring section locally supersonic flow shockwaves (shock-free sections) pursued years sections designed tested practical cases range design CL Mach number sections weak shocks favored cautious supercritical airfoil design Several sections looked promising initially led problems actually incorporated aircraft design Typical difficulties aft loading produce large negative pitching moments trim drag structural weight penalties adverse pressure gradient aft lower surface produce separation extreme cases thin trailing edge difficult manufacture Supercritical especially shock-free designs sensitive Mach CL perform poorly off-design conditions appearance "drag creep" common situation substantial section drag increase Mach number occurs speeds design value section pressures typical modern supercritical section weak shock design condition rooftop Cp design minimum Cp considerably greater Cp* I Kroo 11/28/95
Low Reynolds Number Airfoil Design Low Reynolds Number Airfoil Design Low Reynolds numbers problem airfoil design difficult boundary layer capable handling adverse pressure gradient separation low Reynolds number designs severe pressure gradients maximum lift capability restricted Low Reynolds number airfoil designs cursed problem laminar flow difficult assure boundary layer turbulent steepest pressure recovery regions Laminar separation bubbles common unless properly stabilized lead excessive drag low maximum lift low Reynolds numbers boundary layer laminar conditions boundary layer handle gradual pressure recovery Based expressions laminar separation finds all-laminar section generate CL 0 4 achieve thickness 7 5% (Try PANDA )
Low Cm Airfoil Design Low Moment Airfoil Design airfoil pitching moment constrained possible carry lift far back airfoil desired situations arise design sections tailless aircraft helicopter rotor blades sails kites giant pterosaurs airfoil Liebeck section designed perform low Reynolds numbers positive Cmo performance bad clearly inferior Clmax compared sections Cm constraint (Clmax = 1 35 vs 1 60 conventional sections Re = 500 000 ) thickest section Re = 10 million 57% thick course separate suddenly angle attack I Kroo 11/28/95
Multiple Design Point Airfoils Multiple Design Point Airfoils difficulties designing good airfoil requirement acceptable off-design performance low drag section hard design separate angles attack slightly design point Airfoils high lift capability perform poorly lower angles attack approach design airfoil sections multiple design points well-defined way clear upper surface critical points design upper surface condition lower surface designed section behave properly second point Similarly constraints Cmo effected airfoil trailing edge geometry compromise possible variable geometry employed (at expense) case high lift systems
High Lift Systems High Lift Systems wing designed efficient high-speed flight different designed solely take-off landing Take-off landing distances strongly influenced aircraft stalling speed lower stall speeds requiring lower acceleration deceleration correspondingly shorter field lengths possible reduce stall speed increasing wing area desirable cruise hundreds square feet extra wing area (and associated weight drag) area needed few minutes Since stalling speed wing parameters by: possible reduce stalling speed reducing weight increasing air density increasing wing CLmax latter parameter interesting design wing airfoil compromises cruise efficiency good CLmax efficient movable leading and/or trailing edges good high speed performance achieving high CLmax take-off landing primary goal high lift system high CLmax; desirable maintain low drag take-off high drag approach necessary system low weight high reliability generally achieved incorporating form trailing edge flap perhaps leading edge device slat triple-slotted flap system 737 double-slotted flap slat system (a 4-element airfoil) increase CLmax associated increase chord length (Fowler motion) provided motion flap track rotation axis located wing Modern high lift systems complex elements multi-bar linkages double-slotted flap system DC-8 time Douglas resisted temptation tracks resorted elaborate 4-bar linkages idea reliable practice seems schemes reliable Current practice simplify flap system (or single) slotted systems preferred Flaps change airfoil pressure distribution increasing camber airfoil allowing lift carried rear portion section maximum lift coefficient controlled height forward suction peak flap permits lift peak height Slotted flaps achieve higher lift coefficients plain split flaps boundary layer forms flap starts flap leading edge "healthier" traversed entire forward part airfoil reaching flap forward segment achieves higher Clmax flap pressure trailing edge reduced due interference reduces adverse pressure gradient region favorable effects slotted flap Clmax known early development high lift systems 2-slotted flap better single-slotted flap triple-slotted flap achieved higher Cl's suggests might try slots Handley Page 1920's Tests showed Clmax 4 0 6-slotted airfoil Leading edge devices nose flaps Kruger flaps slats reduce pressure peak nose changing nose camber Slots slats permit new boundary layer start main wing portion eliminating detrimental initial adverse gradient Today computational fluid dynamics design complex systems; prediction CLmax direct computation difficult unreliable Wind tunnel tests difficult interpret due sensitivity CLmax Reynolds number freestream turbulence levels Navier Stokes computations flow 4-element airfoil section (NASA)
3-D Potential Flow 3-D Potential Flow chapter introduces useful techniques dealing three dimensional flow fields including finite span wings slender bodies General Theory Finite Wings Aerodynamics Slender Bodies
General Theory General Theory start discussion three dimensional aerodynamics considering simple case irrotational inviscid flow compressibility neglected consider solutions Laplace's equation 3D: 2-D case since equation linear might construct solutions superimposing known solutions 2-D sources vortices extensively construct flow airfoils 3-D sources vortices model flow wings bodies section starting fundamental 3-D singularities Fundamental Singularities 3D Potential Flow derive fundamental solutions Laplace's equation 3-D 2-D (although complex variables useful) 3-D Source discovered potential: phi = -k/r satisfied Laplace's equation 3-D Since V = grad phi velocity associated solution directed radially magnitude: V = k/r^2 constant k volume flow rate S by: k = S/4pi so: V = S / 4 pi r^2 velocity distribution associated 3-D source dies off r^2 r 2-D case Point Doublet basic solution success supersonic aerodynamics programs point doublet moving point source sink together keeping product strength S separation L constant mu = SL velocity associated point doublet is: Vortex Filament useful fundamental solutions 3-D Laplace equation vortex filament vortex filament visualized thin tube flow vorticity w limit diameter tube small circulation Gamma held fixed region vorticity vortex filament Helmholtz Vortex Theorems Helmholtz summarized properties vortex filaments vortices 1858 vortex theorems three theorems govern behavior inviscid three-dimensional vortices: 1 Vortex strength constant 2 Vortices forever (end boundaries form closed path) 3 Vortices move flow Vortex strength constant: vortex line fluid constant circulation proved imagining closed 3-D loop around vortex line shown: integral around closed loop b c d cuts vorticity Stokes theorem integral zero slit small integral approaches sum integral b c integral d local circulations around vortex line circulations constant line Since vortex strength constant vortex line strength suddenly zero vortex end fluid end boundary extend infinity course real viscous fluid vorticity diffused action viscosity width vortex line large hardly recognized vortex line tornado interesting example end twister boundary; end vortex diffuses large area vorticity section sources vortices singularities vortices flow move local flow velocity interactions vortices trailing wake cause curve around form nonplanar wake Biot-Savart Law Biot-Savart law relates velocity induced vortex filament strength orientation expression frequently electromagnetic theory derived basic equations 3D potential result is: simple case infinite vortex 2-D result: case horseshoe vortex two trailing legs contribute: simple subroutine compute velocity components due vortex filament length Gx Gy Gz start vortex rx ry rz point interest
Biot-Savart Derivation Biot-Savart Derivation determine velocity associated vortex line consider expression vorticity (see definition theorems ): flow incompressible write vector potential free choose satisfies Poison equation well-known solution: expression integrated vortex line velocity induced filament Biot-Savart law:
Finite Wing Theory Finite Wing Theory section several aspects wing theory development theoretical models finite wing simple computational methods: Wing Models Lifting Line Theory Induced Drag Trefftz Plane Computational Models Simple Sweep Theory
Wing Models Wing Models apply 3-D potential theory several ways first consider theory finite wings might start out saying section finite wing behaves 2-D analysis true find lift curve slope 2 per radian drag 0 distribution lift vary distribution chord Unfortunately work way several reasons this: explanation high pressure lower surface wing low pressure upper surface causes air leak around tips causing reduction pressure difference tip regions fact lift zero tips model 3-D wing differently 2-D take naive view 2-D model work 3-D might picture right section distribution vorticity chord 2-D lift proportional chord drop off tips sort model conform physical picture happens wing tips indeed satisfy equations 3-D fluid flow reason work case streamlines confined plane move 3-D flow pattern different back governing equations start simply linear Laplace equation superimposing known solutions simple model 3-D wing might start superimposing vortices wing itself: strip theory model work reason model (which seems superposition known solutions) adequate violates governing equations certain regions model satisfy Helmholtz laws since vorticity ends flow tips additional requirements imposed model requirements model Helmholtz vortex theorems previously simple 3-D model modified satisfy first Helmholtz theorems fact seen picture vortex model far reality downwash field existence trailing vortices strange mathematical result necessary conservation mass 3-D flow Air pushed downward wing downward velocity persist far wing Instead move outward outward-moving air squeezed upward outboard wing flow pattern develops trailing vortex visualized NASA engineers flying agricultural airplane sheet smoke main vortex wake produce downwash field wing downwash field several significant effects: changes effective angle attack airfoil section changes lift curve slope implications Induced drag: Lift acts normal flow 2D accounts 40% fuel commercial airplane 80% drag critical climb segments produces interference effects important analysis stability control magnitude downwash estimated Biot-Savart law previously applied simple model two discrete trailing vortices equation predicts infinite downwash wing tips result clearly wrong fact induced downwash large failure simple model led Prandtl develop slightly sophisticated 1918 representing wing horseshoe-shaped vortex wing represented several them: way circulation wing vary root tip strength trailing vortex filaments circulation wing by: vortex shed wing whenever circulation changes limit number horseshoe vortices goes infinity trailing wake sheet vorticity trailing vortex strength per unit length y direction (vorticity) derivative total circulation wing station model derive basic relations finite wings vorticity strength trailing vortex sheet by: gamma = dGamma/dy since wing circulation changes quickly tips trailing vorticity strongest region tip vortices complete vortex sheet NASA photo F-111 4-g turn vortices visible picture low pressure region lowers temperature condensed water vapor
Lifting Line Theory Lifting Line Theory Basic Theory try 2-D flow section correct influence trailing vortex wake downwash idea lifting line theory 2-D result that: together relation: obtain: angle attack reduced effects downwash effective angle attack true angle* minus downwash angle: induced downwash Wind Biot-Savart Law: Combining expression gamma: expression downwash angle: provides integral equation circulation distribution wing thin airfoil theory integral equation solved assuming Fourier series representation distribution Substitution expression circulation integral equation leads to: integrating have: solution equation values y easy case thin airfoil theory get closed form expressions An's generally numerically several interesting simple appear analysis ever actually computing An's distribution local angle attack section Elliptic Wing example represent lift distribution single term Fourier series then: represents elliptic distribution lift downwash angle case: integral constant |y| < b/2 domain: Since downwash distribution constant Cl distribution just: angle attack constant wing (no twist) Cl constant since: case section Cl equal wing CL and: or: Recall holds unswept elliptical wings General Lift Distributions lift distribution compute An's Fourier expansion once Fourier coefficients compute downwash distribution induced drag: Substitution evaluation definite integral** leads to: formula gives downwash plane wing arbitrary load distributions simple elliptical case closed form solutions downwash sidewash start wake sheet exist simple relation velocity induced elliptic wing tailing vortex sheet is: variable Z complex coordinate y + iz wo downwash wing root: y = z = 0 formula permits computation induced velocities wing downstream surfaces horizontal tails downwash constant plane wing wing move outboard wing out plane wake downwash varies considerably large upwash beyond wing tips downwash field produces several important effects changes lift surfaces surfaces wakes important analysis airplane stability effectiveness horizontal tails seen downwash plot interference canard wake wing extreme: wing lift reduced canard part wing outboard canard increased lift downwash produces induced drag section *Note geometric angle attack flat plates general angle attack zero lift **The integral tables integrals Glauert integral example Kuethe Chow page 146
Induced Drag Trefftz Plane Induced Drag Trefftz Plane Fundamentals 2-D paradox surfaces inviscid flow produce drag longer applies 3-D downwash created trailing wake changes direction force generated section: three dimensions force per unit length acting vortex filament local velocity V includes component freestream component induced downwash latter component produces component force direction freestream: induced drag induced drag lift by: lifting line theory lift downwash terms Fourier coefficients lift distribution: have: induced drag written as: written coefficient form as: definition e e simply depends shape lift distribution span efficiency factor Oswald's efficiency factor induced drag force depends principally lift per unit span L/b determine quickly expression induced drag drag minimum lift span Fourier coefficients A1 term (which produces lift) zero corresponds elliptic loading case mention previously case downwash constant e = 1 Far Field Analysis theTrefftz Plane analysis works analyzing drag wings distribution lift invented Prandtl Betz around 1920 hand-waving requires approximate idealizations flow example: talk downwash induced wake wing wing mean? single bound vortex line compute velocities real wing single bound vortex velocity induced wake varies chord Fortunately answers model general model itself appears induced drag formulas derived fundamental momentum considerations box large contributions certain sides vanish* limit box sides infinity expressions lift drag: u v w perturbation velocities induced wing wake drag depends velocities induced "Trefftz Plane" -- plane far wing drag expressed integral infinite plane perturbation velocities squared Gauss' theorem ( derivation ) expressed line integral wake itself: simplifies calculation drag normalwash Vn downwash wake flat downwash far wing wing itself expression lifting line theory downwash due wake wing half downwash infinity indeed case unswept wings modeled lifting line Far-field velocities compute lift subtle interesting Munk's Stagger Theorem result drag lifting system depends distribution circulation shed wake leads useful classical aerodynamics Perhaps useful Munk's stagger theorem states that: total induced drag system lifting surfaces changed elements moved streamwise direction theorem applies distribution circulation surfaces held constant adjusting surface incidences longitudinal position varied implies drag elliptically-loaded swept wing unswept wing useful study canard airplanes canard downwash wing complicated Moving canard far wing change drag computation easier stagger theorem prove several useful mutual induced drag theorem states that: interference drag caused downwash wing equal produced second wing first surfaces unstaggered (at streamwise location) especially useful analyzing multiple lifting surfaces
Trefftz Plane Drag Derivation Trefftz Plane Drag Derivation contribution drag front back sides vanish? pressure terms clearly 0 since faces parallel x direction momentum terms easy argue contribution goes zero walls recede faster area goes infinity why: far field induced velocities lifting system represented velocities induced single transverse vortex filament two trailing vortices two trailing vortices cancel out far field leaving piece bound vorticity piece induces velocity varies 1/r^2 area increasing r^2 flux (V·n) top bottom opposite first order term exists surfaces vorticity induces V·n sides start expression drag terms perturbation velocities: result comes directly application conservation momentum incompressible Bernoulli equation Actually assumes wake extens infinitely far downstream trails back wing freestream direction far downstream wake assumned straight general expression be: streamwise perturbation velocities small great simplifications possible: Now: outside wake: So: Gauss' theorem states that: contour integral taken below: obtain: jump potential location y wake integral V·ds point wake point Since normal velocity continuous across wake integral equal circulation enclosed loop circulation wing point part wake left trailing edge Similarly derivative phi normal wake induced normalwash Vn So: last expression recognized result lifting line theory derived general way
Trefftz Plane Lift Derivation Trefftz Plane Lift Derivation calculation drag based velocities induced Trefftz plane lift calculated similar way? answer easy start expression force based momentum equation Let's (naively now) contribution integrals goes zero side box back side dimensions box increased leaves contribution Trefftz plane due wake Trefftz plane induced velocities normal plane lift becomes: evaluation integral seems straightforward Consider integral wake modeled simply pair vortices induced velocity w by: inner part integral becomes: integral lift is: looks exactly right; let's consider integral order integration reversed: inner part integral becomes: integrand antisymmetric plot integral lift is: L = 0 paradox get two values sameintegral Actually paradox; function Lebesgue integrable order evaluate integral unambiguously function satisfy two conditions: 1 continuous countable number points 2 integral absolute value finite avoid problem integral lift first evaluated finite limits Taking z -A y -B B find: L = (2 rho U Gamma / pi) ((B+s)atan(A/(B+s) - (B-s) atan(A/(B-s) + A/2 ln( (A^2+(B+s)^2) / (A^2 + (B-s)^2)) ) limit B get large depends ratio B A>>B value goes 2 rho U Gamma s B>>A value 0 = B value rho U Gamma s integral remains ambiguous evaluated infinite domain dilemma resolved different model flow field vertical velocity associated vortex system integrated Trefftz plane ambiguities arise surprising result contributions finite length trailing vortices goes zero contribution bound vortex found independent length trailing vortices is: starting vortex contribution similarly independent trailing vortex length equal bound vortex contribution lift due momentum flux trailing vortices finally need pressure term upper lower sides control volume might expected integrals once again unambiguous depend relative sizes box sides everything infinite careful analysis leads basic results: 1 wake length small compared box width height lift associated momentum term starting bound vortices 2 wake length large box height large compared width lift associated momentum term trailing vortices 3 wake long width large compared height lift associated pressure terms top bottom
Computational Models Computational Models Panel Methods computational models analysis methods based linear three-dimensional potential flow theory overview panel methods earlier chapter section take simplest panel method detail Weissinger Method Weissinger theory extended lifting line theory differs lifting line theory several respects really simple panel method (a vortex lattice method chordwise panel) corrected strip theory method lifting line theory model works wings sweep converges correct solution high low aspect ratio limits version model Wing Design program actually variant Weissinger's method: uses discrete skewed horseshoe vortices horseshoe vortex consists bound vortex leg two trailing vortices arrangement automatically satisfies Helmholtz requirement vortex line ends flow (The trailing vortices extend infinity wing ) basic concept compute strengths "bound" vortices required keep flow tangent wing surface set control points vortex unit strength station j produces downwash velocity AIC_ij station i linear system equations representing boundary conditions written: {alpha} represents angle incidence sections span (assumed flat plates) section camber local angle attack taken angle zero lift line section linear system equations solved written terms angle attack wing root twist amplitude wings linear distribution twist (washout): where: {alpha_r} vector containing root angle attack element {y} spanwise coordinate varying 0 root b/2 {theta} total twist (washout) wing root tip wing circulation distribution written sum two distributions: Since section lift (lift per unit length span) circulation by: lift distribution expressed as: l1 l2 independent incidence angles depend planform shape wing Since lift coefficient wing CL linearly angle attack write lift distribution form: first term known additional lift distribution second term basic lift distribution scale linearly wing lift coefficient twist angle respectively Additional information basic additional lift distributions available section wing design interactive computation based idea available internet investigate wing shape lift distributions design wings sections source code available
Wing Analysis Program Wing Analysis Program Java application computes lift C l distribution wing sweep twist increase angle attack click upper part plot; reduce alpha click lower area Details: analysis discrete vortex Weissinger computation Pitching moment based mean geometric chord measured root quarter chord point twist assumed linear taken positive washout (tip incidence root incidence)
Simple Sweep Theory Simple Sweep Theory Lifting line theory works unswept wings Weissinger theory provides means computing distribution lift swept wings chordwise distribution pressures Vortex lattice models panel methods nonlinear CFD pressure distrtibutions swept-back wings insight simpler models mostly reason partly historical reasons simple sweep theory interesting invented Buseman around 1935 independently R T Jones Consider infinite wing ignore effects viscosity wing painted white nothing distinguish section slide wing sideways tell moving sideways air tell either pressure distribution remain unchanged created infinite obliquely-swept wing moving respect air speed: fact design wing fly high speed pressure distribution associated lower speed main idea sweeping wing reduce effects compressibility component flow parallel wing effected presence wing; normal component decoupled tangential component true according linear flow theory case nonlinear compressible flow shock waves interesting exercise full potential equations decompose normal term tangential term asserts nothing changes tangential direction idea simple sweep theory consider sections normal wing edges operating flow lower Mach number dynamic pressure effective normal Mach number then: reduced normal dynamic pressure section lift coefficient based component freestream velocity increased total lift fixed: angle attack lift reduced reduction lift angle attack swept wings important implications: airplane incidence angle higher causing several problems aircraft landing approach (e g Concorde's drooped nose long nose gear F-8 variable incidence wing) reduced lift curve slope due sweep improve ride quality gusty air basic idea permits subsonic sections supersonic freestream Mach numbers transonic airfoils Mach numbers higher otherwise able operate works Mach numbers 1 0 recent airplane design I worked design Mach number 1 4 airfoils designed operate Mach number 0 7 (normal Mach number) wing swept 60° airplane lift coefficient meant exceed 0 25 airfoils design Cl 1 0 reduced normal dynamic pressure simple sweep theory reason designing supersonic wing sweep subsonic airfoil sections seen thin airfoil theory predicted drag subsonic sections supersonic airfoil sections produce drag due thickness camber lift Since varies cosine sweep angle expect either forward aft swept wings realize similar benefits basically true section wing design forward-swept wings important differences Similarly wings oblique sweep designed tested Further discussions oblique wings section supersonic wings
Forward-Swept Wings Forward-Swept Wings Since sweep produces effects vary cos(sweep) might expect either forward aft sweep yield first approximation true; considerations important comparing designs forward aft sweep Historically led designers adopt aft-swept wings aircraft universally true Hansa Jet forward-swept wing business jet designed 1960's forward swept wing permitted larger cabin wing spar interrupting floor sailplanes slight forward sweep better visibility Recently renewed interest forward swept wing concept aerodynamic reasons demonstrator / research aircraft X-29 built Grumman NASA DARPA Air Force Several aerodynamic advantages forward swept wing suggested interesting illustrated claim lower surface swept forward wing contributes larger share total lift lower surface aft-swept wing exaggerated predicted observed due part perturbation velocities induced 3-D thickness distribution part velocities induced streamwise vorticity advantages disadvantages forward sweep: Advantages · Better off-design span loading (but taper: Cl advantage weight penalty) · Aeroelastically enhanced maneuverability · Smaller basic lift distribution · Reduced LE sweep structural sweep · Increased TE sweep structural sweep - lower Cdc · Unobstructed cabin · Easy gear placement · Good turboprop placement · Laminar flow advantages? Disadvantages · Aeroelastic divergence penalty avoid · Lower |Cl / ß| (effective dihedral) · Lower Cn / ß (yaw stability) · Bad winglets · Stall location (more difficult) · Large Cmo flaps · Reduced pitch stability due additional lift fuse interference · Smaller tail length???
Aerodynamics of Slender Bodies Aerodynamics Slender Bodies interested generating lift reducing drag reduce drag volume best shape slender body -- nearly body revolution aerodynamics shapes different airfoils wings follows basic principles Bodies including fuselages important produce drag lift moment produce important interference effects wings substantially change stability airplane flow general fuselages bodies predicted way flow airfoils wings Superposition sources doublets form panel methods simpler forms (ala thin airfoil theory) Navier Stokes equations flow separation suspected first part section examine calculations methods themselves second part briefly introduces concept slender body theory Flow Bodies Slender Body Theory
Flow over Bodies Flow Bodies closed-form solutions potential flow bodies revolution available useful noted 2-D maximum velocity ellipse by: umax/U = 1 + t/c 3-D surface velocity ellipsoid revolution by: x distance midpoint units length 2 0 r radius (or units ratio diameter length ) maximum velocity by: (from Schlichting) illustrates pressure distribution bodies revolution D/L = 0 1 perturbation velocities smaller 2-D velocities estimated superimposing point sources case ellipsoid: maximum velocity sensitive actual shape paraboloid having 50% larger perturbations distribution sources axis slightly underpredicts velocity perturbations pressure distribution typical fuselage shape D/L = 0 09 computed source distribution x-axis pressure falls center cylindrical portion fuselage fuselages inviscid flow produce nose-up pitching moment angle attack increased destabilizing important consideration sizing horizontal tail inviscid picture suggests lift produced viscous flow actually separates back fuselage making moment smaller lift larger predicted inviscid theory lift produces induced drag fuselage behaves low aspect ratio wing angle attack fuselage lift drag moment based experimental data estimated moment based inviscid theory
Slender Body Theory Slender Body Theory simple theory estimate aerodynamic characteristics bodies vary slowly X-direction: wings low aspect ratio high sweep slender fuselages cases rate change quantities x direction small governing equation becomes: (1-M^2) du/dx term dropped everything varies slowly X direction remaining equation equation 2 dimensional incompressible flow cross-plane since Mach dependence dropped out valid M<1 M>1 2-D cross flow computed conformal mapping method 2-D panel method cross flow plane flow unsteady span swept wing diameter fuselage changes cuts plane solution Laplace's equation provides correct velocity distribution pressures computed unsteady Bernoulli equation particularly simple useful case highly-swept low aspect ratio wing slender body concept solve lift low aspect ratio wing low aspect ratio wings tend produce constant downwash nearly elliptic loading angle attack gets large rate change momentum air cross-flow plane equal force per unit length wing: consider certain area cross plane velocity w ( ) force becomes: 2-D unsteady theory (not here) effective area circle around plate diameter equal local span Y total force wing by: force acts normal wing plane small angles attack: low aspect ratio limit lift curve slope differs lifting line theory predicted factor 2 expression lift curve slope derived second order corrections lifting line theory Jones: p ratio wing semi-perimeter wing span rectangular wing p = (b+c) / b = 1 + 1/AR CL = 2AR / (AR+3) expression close agreement experiment wide range aspect ratios similar analysis slender bodies revolution leads result lift produced body cut-off base by: again independent Mach number Prandtl-Glauert correction applies reduction forces due effective stretching x-direction cancels increase velocities (1/1-M^2) applied 2-D
Compressibility in 3-D Compressibility 3-D role compressibility aerodynamics wings bodies introduced chapter including brief discussion linear compressibility effects supersonic wave drag Subsonic Effects Supersonic Aerodynamics
Subsonic Compressibility 3D Subsonic Compressibility Effects compressibility 3-D flows dramatic 2-D flows effects important techniques predicting linear compressibility effects work 3-D example transform 3-D Prandtl-Glauert equation 3-D Laplace equation incompressible flow changing variables 2-D Whereas 2-D effects stretching x coordinate analyzed 3-D stretching x coordinate lower aspect ratio wing larger sweep -- effects produce nonlinear changes wing forces moments Nonetheless changing lift curve slope Prandtl-Glauert factor badly: better approximation applying Prandtl- Glauert correction 2-D lift curve slope applying downwash correction lifting line theory: or: Applying 2-D Prandtl-Glauert correction sections based normal component freestream Mach number better basic idea formula lift curve slope widely preliminary design calculations DATCOM formula includes effects finite aspect ratio sweep Mach number correction factor viscous effects 2-D lift curve slope section illustrated course solving correct 3D Prandtl-Glauert equation preferred approach 3D linear compressibility transonic computations better result close cases right expecting linear theory
3-D Supersonic Aerodynamics 3-D Supersonic Aerodynamics Sweep produce subsonic characteristics wing supersonic flow point sweep longer effective delaying effects compressibility difficulties associated sweep outweigh advantages required sweep angle gets large Mach number normal leading edge greater 1 airfoil sections behave according linear supersonic theory associated wave drag 2D supersonic wings analyzed subsonic counterparts Consider point (A) wing flow propagate upstream disturbances travel speed sound freestream traveling faster fact law forbidden signals implies disturbances originating (A) darker shaded area Similarly points outside forward-going Mach cone (lightly shaded area) flow point means points tips supersonic wing small part wing rest wing behaves wing tips (except effects sweep taper) rest wing treated set 2-D sections detailed analysis tip regions behave 2-D sections lift curve slope reduced 50% avoid loss lift tip sections supersonic wings truncated part wing affected tips: Supersonic Drag 2-D supersonic wing lift volume-dependent wave drag addition skin friction induced drag terms: approximate expression derived R T Jones Sears Haack minimum drag supersonic body fixed lift span length volume expression holds low aspect ratio surfaces unlike subsonic case supersonic drag depends strongly airplane length l equation gives minimum wave drag length actual wave drag computed calculating surface pressures (by solving Prandtl-Glauert equation panel method example) compute wave drag body revolution relatively paraboloid revolution drag coefficient based frontal area is: body minimum drag fixed length maximum diameter result is: fineness ratio (L/D = length / diameter) 10 drag coefficient 0 1 -- large number considering typical total fuselage drag coefficients based frontal area around 0 2 Sears-Haack Bodies Computing Supersonic Volume Wave Drag body Sears-Haack shape volume dependent wave drag computed linear supersonic potential theory result known supersonic area rule says drag slender body revolution computed distribution cross-sectional area according expression: A'' second derivative cross-sectional area respect longitudinal coordinate x configurations complicated bodies revolution drag computed panel method CFD solution simple means estimating volume-dependent wave drag general bodies creating equivalent body revolution - Mach 1 0 body distribution area length actual body higher Mach numbers distribution area evaluated oblique slices geometry body revolution distribution area oblique cuts actual geometry created drag computed linear theory angle plane respect freestream Mach angle Sin beta = 1/M M=1 plane normal flow direction M = 1 6 angle 38 7° (It inclined 51 3° respect M = 1 case ) actual geometry rotated longitudinal axis 0 2 drag associated equivalent body revolution averaged comparison actual estimated drags method Wave Drag Due Lift expression wave drag due lift: holds wings low aspect ratio general expression derived R T Jones "Minimum Drag Airfoils Supersonic Speeds" J Aero Sciences Dec 1952 combined vortex wave drag written: expression approaches correct limits M-> 1 AR -> assumption lift distribution elliptical directions assumption realized exactly practice Jones gives expression wave drag due lift yawed ellipse showing optimum sweep angle M = sqrt(2) 10:1 yawed ellipse 55° 1/2 wave drag ellipse 0° 90° yaw basic ideas formulate simple method estimating wing supersonic drag lead interesting configuration concepts R T Jones' oblique wing
Sears-Haack Bodies Sears-Haack Bodies Sears Haack derived shapes bodies revolution produce minimum wave drag due volume several solutions: 1 maximum diameter length: 2 volume length:
Supersonic Drag Estimation Supersonic Drag Estimation inviscid wing drag estimated Euler full potential analyses useful employ simple formulas bounds achievable minimum drag pages describe method Lift-dependent wave drag Based R T Jones' expression ellipses approximate lift-dependent wave drag general shape considering ellipse area S length l: choice preserves average wing pressure difference agrees experimental data well-designed supersonic wings Supersonic Drag Due Lift Computed Present Method (*) Boeing Optimization Volume-dependent wave drag area rule estimate volume-dependent wave drag wing equivalent ellipse idea useful J H B Smith derived expression volume-dependent wave drag ellipse minimum drag volume: t maximum thickness b semi-major axis semi-minor axis limit high aspect ratio (a -> infinity) result approaches 2-D result minimum drag thickness: CD = 4 (t/c)^2 / beta ellipse area length volume drag is: works Volume-dependent wave drag slender wings area distribution Data Kuchemann program <in edition> compute various components supersonic drag wing area length span t/c specified Mach number CL program uses formulas drag equivalent ellipse
Oblique Wings Oblique Wings suggested R T Jones obliquely-swept wings ideal shape supersonic aircraft wings first proposed concept 1940's flew flying wing models first ICAS meeting Madrid 1958 great work since oblique wing aircraft including design work Boeing General Dynamics Lockheed wind tunnel testing analysis NASA flight testing models piloted aircraft picture AD-1 low speed oblique wing demonstrator principal advantages reduced supersonic wave drag concept merits compared aircraft symmetric variable sweep (such B-1 F-111) oblique wing little change aerodynamic center position keeps stability reasonable levels avoids large trim changes complex fuel-pumping schemes addition several structural advantages suggested especially variable-sweep aircraft all-wing version oblique wing first proposed G H Lee 1962 idea revived advent active control systems recent artist concept oblique flying wing
Viscosity in 3-D Viscosity 3-D brief chapter viscous effects peculiar three-dimensional boundary layers including separation high low aspect ratio surfaces 3-D Boundary Layer Effects Wings High Angles Attack
3-D Boundary Layers on Wings 3-D Boundary Layers Wings presence lateral induced velocities spanwise pressure gradients 3-D wings means simply 2-D boundary layer streamwise strips actually frequently today since 3-D boundary layer codes slow cumbersome cases 2-D boundary layer equations solved direction inviscid streamlines better streamwise strips misses important spanwise pressure gradient effects boundary layer especially significant swept wings cases strong pressure gradients cause boundary layer flow outward piling tired slow air tips contributing premature tip stall streamwise growth boundary layer tends cause early stall tips boundary flows direction pressure gradients oil flow photographs means travel traveled streamwise Consequently boundary layer gets tired capable surviving adverse gradients separating Techniques improving problem vortilons fences vortex generators photograph outboard flow boundary layer shock-induced separation outer portion wing Mach = 0 85 spanwise flow wing tends create streamwise vorticity boundary layer cross-flow instability damaging laminar boundary layers quickly causes transition turbulent flow Wings sweep angles excess 30 40° require sort boundary layer control (e g suction) maintain laminar flow 3-D approximate boundary layer equations appear: align x-coordinate flow direction flow direction change substantially boundary layer three dimensional boundary layer calculations require work 2-D equations
Wings at High Angles of Attack Wings High Angles Attack high angles attack several phenomena distinct cruise flow appear part wing begins stall (separation occurs lift section reduced) approximate way predict occur well-designed high aspect ratio wings Cl distribution wing determine wing CL section (the critical section) reaches 2-D maximum Cl example outboard sections Clmax = 1 5 wing begins stall tip CL = 1 24 effects wing sweep taken account critical section theory outboard flow boundary layer acts reduce maximum Cl available outboard sections sweep large separation tends occur leading edge wing unlike low sweep situation separated region large reduce lift Instead flow rolls vortex lies wing surface reducing lift wing leading edge vortices increase wing lift nonlinear manner vortex viewed reducing upper surface pressures inducing higher velocities upper surface net result large seen plot predicted quantitatively computing motion separated vortices nonlinear panel code Euler Navier-Stokes solver computations unsteady non-linear panel method Wakes shed leading trailing edges allowed roll-up local flow field good thin wings vortices unstable "burst" - phenomenon predicted methods simple methods computation-intensive simple method estimating so-called "vortex lift" Polhamus 1971 Polhamus suction analogy states extra normal force produced highly swept wing high angles attack equal loss leading edge suction associated separated flow according idea leading edge suction force present attached flow (upper figure) transformed lifting force flow separates forms leading edge vortex (lower figure) suction force includes component force drag direction component difference no-suction drag: CDi = Cn sin full-suction drag: CL^2 / pi AR angle attack alpha total suction force coefficient Cs then: Cs = (Cn sin - CL^2/pi AR) / cos L L leading edge sweep angle acts additional normal force then: Cn' = Cn + (Cn tan a- CL^2/pi AR) / cosL = Cn + (Cn sin - CL^2/pi AR) / cosL Cn' = Cn + (Cn sin - CL^2/pi AR) / cosL attached flow: CL = CLa sin Cn = CL cos Cn' = CL cos + (CL cos sin - CL^2/pi AR) / cosL = CLa sin cos + (CLa sin cos sin - (CLa sin a)^2/pi AR) / cosL = CLa sin cos + CLa/ cosL sin^2 cos - CLa^2/(pi AR cosL) sin^2 CL' = CLa [sina cos^2 + sin^2 cos^2 /cosL - CLa/(pi AR cosL) cosa sin^2 a] = CLa sin cos (cos + sin cos a/ cosL - CLa sin /(pi AR cosL)) take low aspect ratio result: CLa = pi AR/2 then: CL '= pi AR/2 sin cos (cos + sin cos a/ cosL - sin /(2 cosL) ) plot computation compared experiment 80° delta wing (AR = 0 705) Attached flow computations Polhamus suction analogy experiment lift 80° delta wing flow pattern similar highly swept delta wing found tips low aspect ratio wings fuselages vortex formation significantly increases lift cases Especially case fuselage vortices airplane stability affected interaction downstream surfaces important hard predict Vortices generated fuselage leading edge stakes F-18 visible photo QuickTime video clip NASA's High Alpha Research Vehicle investigate phenomena ways control
Wing Design Wing Design essentially two approaches wing design direct approach finds planform twist minimize combination structural weight drag CLmax constraints approach selecting desirable lift distribution computing twist taper thickness distributions required achieve distribution latter approach generally analytic solutions insight important aspects design problem difficult incorporate certain constraints off-design considerations approach direct method combined numerical optimization latter stages wing design starting point established simple (even analytic) chapter considerations wing design including selection basic sizing parameters detailed design chapter begins general discussion goals trade-offs associated wing design initial sizing problem illustrating complexities associated selection several basic parameters parameter affects drag structural weight stalling characteristics fuel volume off-design performance important characteristics Wing lift distributions play key role wing design lift distribution directly wing geometry determines wing performance characteristics induced drag structural weight stalling characteristics determination reasonable lift Cl distribution combined way relating wing twist distribution provides good starting point wing design Subsequent analysis baseline design quickly might changed original design avoid problems high induced drag large variations Cl off-design conditions description detailed methods modern wing design examples followed brief discussion nonplanar wings winglets Wing Design Parameters Lift Distributions Wing Design Detail Nonplanar Wings Winglets
Wing Design Parameters Wing Design Parameters Span Selecting wing span basic decisions design wing span constrained contest rules hangar size ground facilities might decide largest span consistent structural dynamic constraints (flutter) reduce induced drag directly span increased wing structural weight increases point weight increase offsets induced drag savings point rarely reached several reasons optimum flat stretch span great reach actual optimum Concerns wing bending affects stability flutter mount span increased cost wing itself increases structural weight increases spend 10% wing order save 001% fuel consumption volume wing fuel stored reduced difficult locate main landing gear root wing Reynolds number wing sections reduced increasing parasite drag reducing maximum lift capability hand span greater benefit might predict based analysis cruise drag aircraft constrained second segment climb requirement extra span help great induced drag 70-80% total drag selection optimum wing span requires analysis cruise drag structural weight Once reasonable choice basis considerations sensitivities changes span assessed Area wing area span chosen based wide variety considerations including: Cruise drag Stalling speed / field length requirements Wing structural weight Fuel volume considerations lead wing smallest area allowed constraints true; wing area increased reasonable CL selected cruise conditions Selecting cruise conditions integral part wing design process should dictated priori wing design parameters strongly affected selection appropriate selection knowing parameters wing designer complete freedom choose either Cruise altitude affects fuselage structural design engine performance aircraft aerodynamics best CL wing best aircraft whole example seen considering fixed CL fixed Mach design fly higher wing area increased wing drag nearly constant fuselage drag decreases though; minimize drag flying high large wings feasible considerations engine performance Sweep Wing sweep chosen exclusively desirable transonic wave drag (Sometimes reasons c g problem move winglets back greater directional stability ) permits higher cruise Mach number greater thickness CL Mach number drag divergence increases additional loading tip causes spanwise boundary layer flow exacerbating problem tip stall either reducing CLmax increasing required taper ratio good stall increases structural weight - increased tip loading increased structural span stabilizes wing aeroelastically destabilizing airplane sweep difficult accommodate main gear wing section 9 2 5 detail simple sweep theory effects sweep sweep varies cosine sweep angle making forward aft-swept wings similar important differences further section forward swept wings Thickness distribution thickness wing root tip selected follows: t/c large possible reduce wing weight (thereby permitting larger span example) Greater t/c tends increase CLmax point depending high lift system gains 12% small Greater t/c increases fuel volume wing stiffness Increasing t/c increases drag slightly increasing velocities adversity pressure gradients main trouble thick airfoils high speeds transonic drag rise limits speed CL airplane fly efficiently Taper wing taper ratio (or general planform shape) determined considerations: planform shape should give rise additional lift distribution far elliptical required twist low cruise drag large off-design penalties chord distribution should cruise lift distribution distribution lift coefficient compatible section performance Avoid high Cl's lead buffet drag rise separation chord distribution should produce additional load distribution compatible high lift system desired stalling characteristics Lower taper ratios lead lower wing weight Lower taper ratios result increased fuel volume tip chord should small Reynolds number effects cause reduced Cl capability Larger root chords accommodate landing gear again diverse set considerations important major design goal keep taper ratio small possible (to keep wing weight down) excessive Cl variation unacceptable stalling characteristics Since lift distribution nearly elliptical chord distribution should nearly elliptical uniform Cl's Reduced lift t/c outboard permit lower taper ratios Evaluating stalling characteristics easy low speed configuration something high lift system: flap type span deflections flaps- retracted stalling characteristics important (DC-10) Twist wing twist distribution perhaps least controversial design parameter selected twist chosen cruise drag excessive Extra washout helps stalling characteristics improves induced drag higher CL's wings additional load distributions highly weighted tips Twist changes structural weight modifying moment distribution wing Twist swept-back wings produces positive pitching moment small trimmed drag selection wing twist accomplished examining trades cruise drag drag second segment climb wing structural weight selected washout higher improve stall
Wing Lift Distributions Wing Lift Distributions design airfoil sections easier relate wing geometry performance intermediary lift distribution Wing design proceeds selecting desirable wing lift distribution finding geometry achieves distribution section describe lift lift coefficient distributions relate wing geometry performance Wing Lift Cl Distributions Relating Wing Geometry Lift Distribution Lift Distributions Performance
Lift and Cl Distributions Lift Cl Distributions distribution lift wing affects wing performance ways lift per unit length l(y) plotted wing root tip case distribution roughly elliptical general lift goes zero wing tip area curve total lift section lift coefficient section lift by: lift distribution planform shape find Cl distribution lift lift coefficient distributions directly chord distribution examples: lift Cl distributions divided so-called basic additional lift distributions division allows examine lift distributions couple angles attack infer lift distribution angles especially useful process wing design discussion lifting line theory Weissinger theory saw distribution lift written: where: alpha_r angle attack root theta twist angle {l_1} wing lift distribution twist alpha_r = 1 {l_2} lift distribution zero angle attack unit twist lift distribution written conventional way: distributions {la} {lb} wing lift distributions twist CL = 1 unit twist zero lift respectively first term CL {la} additional lift lift distribution added increasing total wing lift theta {lb} basic lift distribution lift distribution zero lift useful? Consider example data two angles attack learn great wing expression above: or: additional lift distribution CL {la} interpreted graphically additional lift coefficient distribution CL = 1 0 plotted rises upward toward tip -- indicative wing low taper ratio wing sweep-back basic lift distribution negative tip implying wing washout
Wing Geometry and Lift Distribution Wing Geometry Lift Distribution wing geometry affects wing lift Cl distributions mostly intuitive ways Increasing taper ratio (making tip chords larger) produces lift tips might expect: section Cl lift divided local chord taper different Cl distribution Changing wing twist changes lift Cl distributions Increasing tip incidence respect root wash-in Wings incidence tip root (wash-out) reduce structural weight improve stalling characteristics Since changing wing twist chord distribution lift Cl similar Wing sweep produces intuitive change lift distribution wing downwash velocity induced wing wake depends sweep lift distribution affected result increase lift tip swept-back wing decrease root (as compared unswept wing large causes problems swept-back wings greater tip lift increases structural loads lead stalling problems increasing wing aspect ratio increase lift angle attack saw discussion lifting line theory changes shape wing lift distribution magnifying effects parameters Low aspect ratio wings nearly elliptic distributions lift wide range taper ratios sweep angles takes great twist change distribution high aspect ratio wings sensitive easy depart elliptic loading picking twist taper ratio right effects similar combining right twist taper sweep achieve desirable distributions lift lift coefficient example: swept back wing tends extra lift wing tips wash-out tends lower tip lift swept back wing washout lift distribution unswept wing twist Lowering taper ratio cancel influence sweep lift distribution Cl distribution different Today relate wing geometry lift Cl distributions quickly means rapid computational methods intuitive understanding impact wing parameters distributions remains important skill
Lift Distributions and Performance Lift Distributions Performance Wing design several goals wing performance lift distribution distribution Cl(y) relatively flat airfoil sections area "working hard" others low Cl case airfoils Cl higher average likely develop shocks sooner start stalling prematurely induced drag depends solely lift distribution achieve nearly elliptical distribution section lift hand structural weight affected lift distribution ideal shape depends relative importance induced drag wing weight taper sweep twist "play with" goals achieved design point difficulty appears wing perform range conditions interesting tradeoffs required design wing drag structural weight several ways problems solved include: Minimum induced drag span -- Prandtl Minimum induced drag root bending moment -- Jones Lamar others Minimum induced drag fixed wing weight constant thickness -- Prandtl Jones Minimum induced drag wing weight specified thickness-to-chord ratio -- Ward McGeer Kroo Minimum total drag wing span planform -- Kuhlman problems sort left solve approaches solution problems closed-form analytic analytic together iteration finally numerical optimization best wing design depend construction materials arrangement high-lift devices flight conditions (CL Re M) relative importance drag weight say difficult design wing designing entire airplane job minimizing cruise drag wing high aspect ratio add constraint wing's structural weight based trade-off cost fuel savings problem better posed select wing small taper ratio High t/c high sweep suggested studies weight drag high lift characteristics design force taper ratio sweep values fundamental consideration early stages wing design Unfortunately estimation CLmax difficult parts preliminary design process example sensitivity high lift constraint optimal wing designs Wing sweep area span twist chord t/c distributions optimized minimum drag structural weight constraint (Results work Sean Wakayama
Wing Design in Detail Wing Design Detail determination reasonable lift Cl distribution combined way relating wing twist distribution provides good starting point wing design Subsequent analysis baseline design quickly might changed original design avoid problems high induced drag large variations Cl off-design conditions Once basic wing design parameters selected detailed design undertaken following: Computation selection desired span load distribution inverse computation required twist Selection desired section Cp distribution several stations span inverse design camber and/or thickness distribution All-at-once multivariable optimization wing desired performance examples approaches illustrated illustrates inverse wing design DISC (direct iterative surface curvature) method starting pressures (top) followed target (middle) design (bottom); light yellow = low pressure green = high pressure inverse technique successfully Navier-Stokes computations design wings transonic viscous flows example wing design based "fixing" span load distribution 737 re-engined high bypass ratio turbofans drag penalty avoided changing effective wing twist distribution details pressure distribution modify camber shape wing thickness best performance sounds straightforward difficult accomplish especially takes hours days examine proposed change simple methods fast turnaround times wing design process computers faster feasible full 3-D optimization early efforts applying optimization nonlinear CFD wing design Cosentino Holst J Aircraft 1986 problem few spline points several stations wing allowed move optimizer tried maximize L/D inviscid code design variables limited objective function simplistic current research realistic objectives design degrees freedom better analysis codes --but long way having "wings designed computer "
Nonplanar Wings and Winglets Nonplanar Wings Winglets begins wing design problem specifying target Cp distribution and/or span loading modifying wing geometry (either manually direct inverse nonlinear optimization) case planar wings elliptic loading useful benchmark creation target loadings (For high aspect ratio wings 2D airfoil useful chordwise loading ) complex methods creating target Cp's beyond scope discussion little guidance wing nonplanar section problem optimal loading nonplanar lifting surfaces generalized multiple surfaces wing planar previous simple longer valid Elliptic loading lead minimum drag span efficiency greater 1 0 describe method computing minimum induced drag planar nonplanar wings First consider distribution downwash minimum drag method restricted variations follows consider arbitrary variation circulation distribution represented dGamma1 dGamma2 change lift: implies: drag minimized initial distribution: downwash constant planar wing minimum drag general case multiple surfaces nonplanar wings approach case condition constant lift is: theta local dihedral angle lifting surface minimum drag: Vn induced velocity Trefftz plane direction normal wake sheet (the normalwash) case Vn = k cos theta normalwash proportional local dihedral angle sidewash optimally-loaded winglets 0 example solve distribution circulation produces distribution normalwash Alternatively direct optimization approach circulation distribution represented row vector {Gamma} wake modeled collection line vortices strength {Gammaw} write wake vorticity terms surface circulation based discrete vortex model drag by: D = rho/2 {Vn} · {Gamma} Vn normal wash Trefftz plane computed Biot Savart law {Vn} circulation strengths by: {Vn} = [VIC] {Gamma} [VIC] function geometry D = rho/2 [VIC] {Gamma} · {Gamma} lift function circulations: L = rho U {Gamma} · {cos theta} theta local dihedral angle Finally optimal values {Gamma} setting (D+lambda(L-Lref)) /Gammai = 0 lambda Lagrange multiplier problem homework summarized below: · wing/winglet combination optimized minimum drag fixed span achieves drag planar wing span increased 45% winglet height · wing lift distribution increased lift outboard compared winglet case increased tip loading extra bending moment winglet leads increased structural weight bending moment constraint replaces span constraint wings winglets seen minimum drag stretched-span planar wings Induced drag wings winglets planar wings integrated bending moment (related structural weight) solutions left span ratio = 1 0 line meaningful approach taken general nonplanar wake shapes summarizes showing maximum span efficiency nonplanar wings various shapes height span ration 0 2 Several points should preceding 1 result sidewash winglet (in Trefftz plane) zero minimum induced drag means self-induced drag winglet cancels winglet thrust associated wing sidewash Optimally-loaded winglets reduce induced drag lowering average downwash wing providing thrust component 2 inviscid flow nonplanar wings slight difference optimal loading viscous case due lift-dependent viscous drag Moreover planar wings ideal chord distribution achieved section maximum Cl/Cd inviscid optimal lift distribution nonplanar wings longer case optimal chord load distribution minimum drag complex 3 considerations primary importance include: Stability control Structures pragmatic issues details design nonplanar wings found recent paper " Highly Nonplanar Lifting Systems " accessible
Configuration Aerodynamics Configuration Aerodynamics chapter few considerations required dealing single wing body complex configuration particular multiple lifting surfaces introduce stability control issues key elements configuration aerodynamic performance sections represent introductions issues; complete discussions aircraft design text Multiple Lifting Surfaces Longitudinal Stability Trim Horizontal Tails Canard Aircraft Tailless Aircraft
Multiple Lifting Surfaces Multiple Lifting Surfaces Introduction Multiple Lifting Surfaces combination several lifting surfaces single surface Applications include: Balancing moments: Trim tail surfaces Structures: Biplanes increasing effective structural depth Stability: Tail surfaces increase stability airplane (or boat arrow) Drag: Additional surfaces reduce overall drag design Winglets case point Interference aerodynamics multiple lifting surfaces sum isolated parts several types interference important case velocities induced bound vorticity wing seen 2-D image effects previously case pictured rear wing increases velocity forward wing increasing lift case velocities induced wake surface lift forward wing smaller span downwash inboard reduces lift upwash outboard increases lift region common example multiple surface interference interaction wing tail surface downwash wing tail reduces tail lift value wing (i t incidence zero lift line tail respect angle attack line) value wing's presence: rate tail lift changes airplane angle attack reduced downwash to: important aircraft stability reducing effectiveness tail 40% wing AR = 8 Induced Drag Prandtl Biplane Equation derived integrating wing downwash compute induced drag two elliptically-loaded wings including interference effects interference factor sigma depends span ratio vertical gap surfaces varies b2/b1 gap 0 0 gap infinite comparing induced drag two systems necessary fix total lift: L= L1+L2 biplane equation ratio drag 2 wings total lift maximum span single wing total lift span is: define span efficiency e equation induced drag: b1 span surface largest span Then: biplane equation provides simple method computing induced drag two planar wings assumption wings elliptically-loaded leads minimum drag single planar wing multiple lifting surfaces nonplanar wings wings winglets compute minimum induced drag general arrangement nonplanar lifting surfaces compute optimal circulation distribution compute minimum drag method doing revised version section induced drag find similar expression drag two surfaces: constant sigma* sigma depends span ratio vertical gap sigma* goes 0 vertical gap large approaches (b2/b1)^2 limit small vertical gap minimum drag zero gap combined loading elliptical span efficiency 1 0
Longitudinal Stability and Trim Longitudinal Stability Trim drag system dependent distribution loads bewteen surfaces order determine properly size tail surface consider aircraft's stability trim Stability tendency system return equilibrium condition disturbed point Two types stability instability important static instability: dynamic instability: airplane stable system acceptable time constants assure careful analysis dynamic response controllability required simplest case: static longitudinal stability trim tell something aerodynamic design surfaces - load carry airfoil properties drag associated surfaces displace wing airplane equilibrium flight condition higher angle attack higher lift coefficient: return lower lift coefficient requires pitching moment rotation point* Cm negative increase CL: that: x distance system's center additional lift c g x 0 system neutrally stable x/c represents margin static stability static margin Typical values stable airplanes range 5% 40% airplane stable desired moving c g forward (by putting lead nose) moving wing back needs tail stability right position c g configuration stable tend nose down whenever lift produced addition stability require airplane trimmed (in moment equilibrium) desired CL implies that: single wing generating sufficient Cm zero lift trim reasonable static margin CL easy (Most airfoils negative values Cmo ) tailless aircraft generate sufficiently positive Cmo trim conventional solution add additional lifting surface aft-tail canard sections considerations design configurations
Horizontal Tails Horizontal Tails Trim achieved setting incidence tail surface (which adjusts CL) Cm = 0 Stability simultaneously assured appropriate location c g : stability constraint trim requirement determine c g located adjust tail lift trim lifts interfering surface compute combined drag system theory interfering surfaces previous section Horizontal tails generally trim control range conditions Typical conditions tail control power critical determine required tail size include: take-off rotation (with ice) approach trim nose-down acceleration stall Despite drawing tail surfaces normally loaded downward cruise commercial aircraft tail download 5% aircraft weight stability requirements relaxed application active controls size tail surface and/or magnitude tail download reduced
Canard Aircraft Canard Aircraft stability trim equations satisfied negative values tail length designs "canard" designs (Canard French word duck early canard designs bore strange resemblance namesakes Oddly enough canard English means gross exaggeration hyperbole ) notes basic analyses required design canard aircraft analyses differ several ways those methods initial design aft-tail configurations several differences apparent address detail general question relative merits canard aft-tail designs (See Refs 1 2 ) large variety canard designs illustrates unlike case aft-tail designs consensus optimal canard size emerged possible explanation lies fact aircraft performance sensitive canard size tail size proper canard size depends strongly critical design condition (e g climb unimportant high speed cruise critical vs long endurance design high speeds unnecessary ) consider simple methods estimating key performance parameters design particular project several canard sizes tried achieve best compromise Stability Trim Drag Summary: Canard Advantages Disadvantages References Interactive Calculations Canard Performance
Canard Stability and Trim Canard Stability Trim performance canard design depends strongly amount lift canard carry set stability trim requirements first necessary determine position c g relative loads carried wing canard ratio lift carried canard carried wing: depends several parameters listed Stability: moving c g back sm * cref airplane neutrally stable then: Trim: Summary: value static margin sm should large enough acceptable handling qualities aft center gravity position require analysis aircraft dynamics various flight conditions Since destabilizing fuselage explicitly value sm expressions should increased appropriately Lift Curve Slopes difficulty equations ratio canard wing lift curve slopes calculated wings interfere approximate relations: surfaces produce upwash downwash effective lift curve slope changed Unless canard wing close together major canard wing canard produces downwash inner part wing upwash outboard canard tip vortices net reduction wing lift estimated roughly formula based Hayes Reverse Flow Theorem (see Ref 3): kappa correction applied canard close wing lie plane wing kappa should computed 2-surface lifting line lifting surface program
Canard Drag Canard Drag Standard methods parasite drag prediction work canards: Compute drag fuselage etc desired Reynolds number Compute profile drag wing based airfoil data appropriate Cl canard profile drag computed airfoil data CL=CLc tricky part drag prediction induced drag includes generally termed "trim drag" (A part trim drag appear profile drag ) lift surfaces induced drag computed expressions derived earlier multiple lifting surfaces e overall span efficiency by: surfaces individually elliptically-loaded twisting wings right way induced drag reduced value (see Ref 5) preceding equations (which elliptical distribution lift surfaces) good practical estimate especially true structural design considered: distribution wing lift minimum drag fixed span carries large fraction total lift outboard wing sections leading larger bending moments greater weights drag minimized fixed weight optimal loading nearly elliptic particular set cases Unrelated canard system drag maximum CL systems based total area sensitive assumed parameters representative general conclusions
Canard Advantages and Disadvantages Summary Canard Advantages Disadvantages Advantages: Possibility good stalling characteristics elevator stops desirable layout packaging standpoint: Main wing carry-through cabin pusher engine installation simplified Synergistic winglets directional stability certain cases simplified control linkage possible wing flaps desired (for simplicity ultralights competition rules standard class sailplanes example) CLmax canard exceed aft-tail airplane unstable aircraft canard designs CLmax and/or drag advantage Control authority larger unstable canard aircraft high CL unstable aft-tail designs Disadvantages: Fuel center gravity lies aircraft c g conventional designs means large c g range produced fuel held elsewhere (e g strakes wing root ) CLmax problems flaps margin entire wing: Flaps produce larger pitching moment c g canard aircraft need large canard aerodynamic incidence change high maximum canard lift coefficient since value S larger canard designs Cmo greater impact L aft-swept designs Induced drag / CLmax incompatibility: Canard designs achieve equal better CLmax values conventional designs similar values span efficiency configurations high CLmax values terrible values e those respectable e 's low maximum lift coefficients Directional stability: distance aircraft c g aft part airplane smaller canard aircraft poses problem locating vertical stabilizer result large vertical surfaces (Note winglets advantage case ) Wing twist distribution strange CL dependent: wing additional load distribution distorted canard wake Power effects canard - deep stall: Accidents associated tractor canard configurations propeller slipstream prevented canard stall wing stall result possible deep-stall problem Finally perhaps importantly canard sizing critical aft tail sizing choosing canard big small aircraft performance severely affected easy bad canard design
Canard References Canard References 1 Kroo I McGeer T Optimization Canard Configurations 13th ICAS Congress ICAS-82-6 8 1 1982 2 McGeer T Kroo I Fundamental Comparison Canard Conventional Configurations J Aircraft Nov 1983 3 Jones R T Cohen D High Speed Wing Theory Princeton Univ Press 4 Abbott I VonDoenhoff Theory Wing Sections 1949 5 Kroo I Minimum Induced Drag Canard Configurations J Aircraft Sept 1982 6 Kroo I General Approach Multiple Lifting Surface Design Analysis AIAA paper 84-2507 Nov 1984
Wing/Tail Analysis Program Wing/Tail Analysis Program Java application computes trimmed performance wing canard tail Enter parameters text boxes drag tail forward aft canard watch required tail load induced drag change flow left right Details: calculations section pitching moment surface individually elliptically-loaded vertical gap surfaces static margin based chord defined total (wing+tail) area divided wing span
Tailless Aircraft Tailless Aircraft Northrop YB-49 Prototype W = 192 000 lbs Flies 3500 miles 390 mph pointed out tail needed stability locating c g far enough forward possible level stability tail canard pitching moment needed trim: stable airplanes x/c positive trim positive CL's Cm0 positive Typical airfoil sections negative pitching moments difficulty designing tailless aircraft obtaining sufficiently positive Cm0 Since static margin sm defined as: Trim requires that: several ways illustrated wide range tailless aircraft designs simple means obtaining positive value Cm0 choose airfoil section negative camber reflex good reason airfoils negative pitching moments: concentrating lift forward airfoil's 1/4 chord point long adverse pressure gradients correspondingly low CLmax short stretches laminar flow recent progress area low positive moment airfoil design fundamental limits area example tailless aircraft trims positive Cmo airfoil section: AeroVironment Pathfinder solar-powered aircraft flight 50 000 ft degradation airfoil performance associated positive Cm0 approaches trim tailless aircraft pursued best sweep twist Basic lift distribution consists lifting root downloaded tip -> positive Cm0 Horten IV uses lift distribution actually negative tips helps lateral stability poor span efficiency course peculiar lift distribution span efficiency low fact solve distribution lift minimizes induced drag providing amount pitching moment amount moment location lift centroid eta_c expression result elliptical load distribution lowest induced drag lift centroid 42 4% semispan Twisting wing move centroid say 33% semi-span (as recommended Horten's) span efficiency 72% fact measured span efficiency Horten IV 63% airplane trimmed c g located lift centroid question arises: find wing aerodynamic center 42 4% semi-span? stable trimmed wing minimum induced drag answer question yes aerodynamic center wing falls outboard elliptical lift centroid (at 42 4%) wing stable trimmed (Actually slight correction due section pitching moment moves required c out ) analysis easy wing c lie sufficiently far outboard 42% point reasonable stability levels wing aspect ratio sweep taper ratio sufficiently large practice means aspect ratios greater 8 sweep greater 25° (a lower high AR) taper ratios range 0 75 1 0 first swept slightly tapered wing seems inefficient Textbooks suggest taper ratios 0 25 0 3 conducive low induced drag reason design works out basic lift distribution carries lift far inboard additional loading far outboard result case two wrongs right difficulties tailless aircraft associated controls Northrop flying wings problems area order nose aircraft positive pitching moment introduced reducing airfoil camber increasing washout swept-back wing methods elevons deflected Elevons several problems however: initially decrease wing lift increase trimmed angle attack located wing tip region spanwise boundary layer flow problem require mixing surfaces deflected symmetrically elevators antisymmetrically ailerons reduces maximum control effectiveness design incorporating ideas designed Stanford foot-launched sailplane SWIFT swept wing inboard flap trim flaps extend half span deflected 50° approach landing clean laminar flow design lightweight composite construction outperforms foot-launched aircraft wide margin flown altitudes 15 000 feet distances 140 miles 100 aircraft produced Click short QuickTime video clip Swift recent motivations tailless aircraft associated reducing radar cross-section B-2 aircraft illustrates active controls unconventional design acceptable compromise aerodynamic efficiency low observability
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