Basic Aeronautics for Modellers

Basic FO

ODELLERS SECOND EDITION

FOR MODELLERS

BY ALASDAIR SUTHERLAND BSc

© 2002 Traplet Publi cations Limited

All rights re serv ed . All trad em ark s and register ed nam es ac kno w ledge d . No part o f th is book ma y be co pie d, reprodu ced or tran smitt ed in any fo rm w ithout the wri tte n co nsen t o f the Publish e rs. The informati on in this book is tru e to the best o f o ur kn owl ed ge at the time o f co mpilatio n. Recommendati ons are made without any gua rantee, impli ed or o the rwise , o n the part o f the a utho r o r publish er, wh o also discl aim any liability incurred in co nnec tion with the use o f d ata o r specific informatio n co ntaine d within th is publicat ion .

First ed ition publish ed by Trapl et Publi cat ions Limited in 1995 Publi sh ed by Tra plet Publi cation s Limited 2002 Traplet House , Severn Drive , Up to n-up o n-Seve rn,

Wo rces te rsh ire . WR8 OJ L United Kingdom .

ISBN 1 9003 7 1 41 3

Front Couer. Stefan If/u rlll seen bere exercising some ofb is considerableflying skills ioitb b is 1:2 scale Pitts 51. Stefa n brought tbe Pitts backwards, balancing the thrust oftbe engine against tbe stlffbrecze, until tbe rudder tou ched b im! (Photo: Peter Dauison)

Tecbnical D ra uiings by Lee \\7isedale Ca rtoons by Simo n Bates

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Acknowledgements o n ve n tio n a lly th is is a p a ge o f syco p ha n tic ramblings wh er ein I thank everyo ne in my life from th e midw ife wh o d elivered m e to my dent ist's rece ptio nist. Well , thank yo u o ne and all. I o we my parents a small a po logy , as I rem ember bu ying a mod el ae ro plane a nd then promi sin g that it would be my last ; not o nce but thr ee o r four times. I made no suc h rash pr omi ses to my wife Ann e who unwittingly made th e mistake o f marr yin g a dormant Ae rom od elle r, who ever since then has been e rupting with increasing magnitude and frequen cy, sprinkling the hou se with successive layer s o f styrene bead s, wood shav ings, balsa dust, glass fibre stra nds and Solarfilm fragmen ts. Sorry Anne . As for my daughters Ron a and Shee na , if the y ever live in Ame rica th e ir a nalysts w ill ma ke mu ch of the socia l a nd paternal deprivat ion the y have e ndured by being the offspring o f a fervent aero mode ller. Passi ng q uic kly over m y educat io n a t Le n zie Acade my, Glasgow Universi ty a nd the Hambl e College of Air Training, the grea t mileston e in my modellin g life was when Jo hn Mich ie had the time a nd p at ien ce to teach me to fly proporti on al R/C aeroplanes. And it was Brian Davies who introduced me to aeroba tics and wo rd p rocessing, which is whe n this book ge rmina ted . I have learn ed a grea t deal from my frie nds in the Alde rsho t club and W'indsor Park, a nd co ntinue to learn from my present circle of friends in Scotland . It was du e to one of th ese , Bob McGill , th a t I became imme rsed in wa ter plan es. Finally , th ank yo u to Dr. Fra n k Cot on of th e Depa rtment of Ae ro s pace Eng inee ring at Glas g o w University wh o read throu gh th e manuscript to check th at I wo u ld not e m ba rr ass th e Dep artm ent to o e xte ns ive ly b y preach ing fund am enta l ae ro dy na mic fallacies.

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Foreword ne of the first technical qu estions my son ever aske d me was "How do plan es fly?" Well, we all know how plan es fly .. . don't w e? Th ink again! If you were asked that simpl e qu estion , co uld yo u give a co ncise comprehensible a ns wer? If yo u co u ld, how would yo u deal with the retor t, delivered by the son of on e of my colleagues . . . "How do plan es fly upside down?". On e of the most fascinating as pec ts of th e modern w orld is th e science of flight. Wh ether it be a bird , heli co pter , fighter aircr aft or e ve n th e marvellous bumble bee, people ha ve always been intrigued by the same basic qu estion - "How does it fly?". Unfortunately, th e a nswe r is n ot a lways s traig h tfo rward a n d is co mp lica te d b y the w id e varie ty of mechanisms a nd physical ph enomena which interact to produce flight. Man 's interest in model aircra ft is a lon g stand ing one. Over the yea rs, the mot ivation for this has largely be en recreation al altho ugh since scientific studies ha ve been co nduc ted, most notabl y those in Ge rmany between the Wor ld Wars. As a res ult, tod ay's aeromo de ller is a fairly well info rmed ind ivid ua l w ho , inste ad of ask ing th e bas ic nature of flight qu estion, is more inter est ed in how to improve the performance of a n ai rcraft o r how to avoid problems during fligh t. The answers to most of these question s can be found in Basic Aeronautics for Mod ellers. Thi s book skillfully guides th e reade r through th e bas ics o f a irc raft flight a n d p erform anc e before addressing issues specific to model aircraft. Alasdair Su the rl a n d draws on his p e rs onal e xperience as a stude nt, a pilot, and most imp ortantly a n aeromodeller, to pr esent fundamental informati on in a friendly and eas ily accessib le form . He does so b y building th e kn owled ge bas e of the read er in a steady progressive m ann er, h ighlightin g a numb er o f co m m o n miscon ception s along the way. In this wa y, he en sures that the rea de r is prepared for each new sectio n of th e book as it is reache d. Thankfully, the use of complicated equa tions or tedi ou s derivation s wh ich, if excessive, can ofte n det er th e laym an , is either avoided o r they a re provided in appendices . Th rou gh ou t th e book, use is mad e of observat ions from flow visua lisation ex peri me nts to illustrate asp ects of fluid be hav iour. Over the years, flow visua lisation has been o ne of the mo st p owerful too ls in the development of our current understanding of fluid dynam ics. Ind eed , smo ke flow visua lisatio n w ind tunnels are still used in ma ny un ive rsitie s for resea rc h a n d s tu d e n t dem on str ations. It is o bvio us th at the demonst rations given to Alasdair Sutherland in his stude nt days had a co nsiderable impact; after all seeing is believing! Whether you consider yoursel f to be a novice or a

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well-season ed ae rornode ller, there is so meth ing in this b ook fo r yo u . Beginner s ca n le arn a bou t th e ba sic mech ani sms of lift generation and the manner in which for ces act on an aircraft. The more ex perience d , o n the othe r hand, can contemplate the detailed influ ence of model sca le and the role of the Re ynold s number. The book may even encou rage so me to raid the library for mor e informatio n or carry o ut so me res earch of their own. Most importantly though , this book was written by an e nthus iast for its readers to enjoy. I hope yo u do! Dr. Frank Cotton Department of Aerospace Engineering University of Glasgow. Alasd a ir Sutherl and w as b orn a n d e d ucated in th e Glasgow area , progressing from Lenzie Aca demy to Glasgow Univers ity wh ere he ea rned a B.Sc. w ith Honours in Aero nautical Engineering. Afte r training for a career as an airlin e pilot at Ham ble , near Southampton, he joine d BEA in 1973 to fly Trident aircraft arou nd Euro pe and Lockheed LlD11 aircraft wo rldwide. An aerorno delle r sinc e th e age of eleven, he flies most types of radio co ntrolled airc raft especi ally spo rts and aerobatic, and particul arly enjoys designin g models of va rio us typ e s. After man y years as a member o f Alder sh ot Mod el Club he mov ed back to Scotland as Captain o f British Airw ays turboprop aircra ft, first th e H.S. 748 a nd latt erly th e British Aerospace ATP. He is now a member of both the Clyde Valley Fliers and the Garn ock Valley !vIAe.

Tbe Author: Alasdair Sutherland

Contents Page Introduction

11 13

Chapter 1

The Aeroplane's Environment Tbe air. Mass toeigbt and grauity. Newton

Chapter 2

Requ ireme nt for Flight - Lift \fiatcbing tbe a irfloto. Pressure variation . Pressure exerts a force . Wind tunnel testi ng.

16

Chapter 3

The Stall's the Limit The lift cu rve. 17Je stall, tbe reason . Variation in sta lling cbaracteristtcs.

20

Chapter 4

The Drawback Drag 17Je boundary lay er. Wing drag; drag polar , effect of tbickness and ca m ber, la m tn a rfloui sections. Fuselage drag , strea mlin ing . A bit for golfers.

23

Cha pter 5

Have you a Moment? 17Je m om ent on tbe wing . Centre ofpressure. Aero dyna mic centre. A erofoil section su m m a ry, tbe effect oftbickness and ca m ber. Section classification and use.

26

Chapter 6

The Vortex Syste m The uortex around tb e wing. Seeing tb e cortices. Even m ore drag, tbe reason . Complications. Simp lifica tio ns. 17Je importa nce ofAspect Ratio. Lessons forpra ctical modellers . Ground effect,

30

Chapter 7

Planform and Twist Elliptical lift distribution. Local angle ofattack. Different planform shapes. Tipstalling . Wasbout, aerodynamic ioasbout. Sweep ba ck. Mean cho rd . Horses for courses.

35

Chap ter 8

CG and Stability 17Je CG. Stability in gen eral. Motio n ofan aeroplane. Stability ofaerop lan es in Pitcb , CG Position . Complica tion s. We can work it out? Simpler equations. Va riations on tbe formu la .

.41

Chapter 9

Directional and Late ral Stability Directional sta bility , the fin. Lateral sta bility, sideslip. Fin sideforce, wing p osition , d ihedral, sweep back . Aspects ofdesign . Directional and lat eral interaction, spiral divergen ce, dutch ro ll.

.49

Chapter 10

Control Rudder. Elevators. A ilerons, a ileron drag, aileron alternatives. Control su rface balances. Control effective ness, rotational inertia, sta bility, a erodynamic damping. Otberflying con trols, throttle, air brakes, flaps, sla ts. Control combinations, ta ilerons, flaper ons, eleuons, V-ta il.

.56

Cha pter 11

Turning Flight Mecbanics of turning . Turning aeroplan es, loa d f a ctor in a turn, refinem ent, stdeslipp tng and skidd ing, drag in a turn, stalling speed. Higb aspect ratio. Turning using rudder. Specia l effects. Wben is a rudder an eleva tor?

63

Chapt er 12

A Delicate Balan ce Equilibrium. Tail lift to trim. Elevator ang le to trim . Ta il Setting angle . The effec t of thru st o n trim.

67

s Laws. Inertia . Vectors. Moments.

Cha p te r 13

Glid er Performance Lift/Drag rati o. Speed ra nge. Ae rodyna m ic da ta . Optimising performance, strea mlin ing , toeigbt. Iiffect ofto ind on perfo rmance, down trim , ballast.

72

Cha p te r 14

Power ed Performance Propeller thrust, slipstrea m effects. Level flight, top speed, sta lling speed, effect on toeigbt. Take oJ(. Clim b. Descent and landing .

76

Cha p ter 15

Th e Ae ro dyn am ics of Aeroba tics 77Je sta ll. Sp in . Snap . Loop . In oerted. Roll. Yatu. Aero bat ic trim set up.

80

Chap te r 16

Special Cases Low asp ect ratio, handling, CG position . Ca na rd . sta bility, CG Position , Tail-less aeroplane, sta bility, trim, co n trol. Multitoing, performance, CG p osition .

85

Cha p te r 17

Reyn olds Number Definition, importance, n ontogra nt. In tbe bou ndary lay er, situation normal, laminar separation , separation bu bble, tbe underside. Re-eff ect on a erodynamic da ta . 77Je p roblem area . Hysteresis loop . 77Je effec t 0 11 m odel desig n a nd performance, wing tips, class rules, optim u m weigbt. Tu rbu lator strips. surface fi nish , Using publisbed data .

90

Chapte r 18

Aeroelasticity Effect on stability , ta il bend, wing twist. A ileron reversal. Wi ng divergen ce. Aileron flutter , tbe ca use, tbe cure. 117ingflutter. Tail Flutter .

96

Cha p te r 19

Tu ck Under Description . 77Je villa in unmasked. Wing twist. ta il bending.flexible con trols. 77Je elevator trim g rap h . Critica l speed . Tuck under speed . Getting away witb it. Tailplane insta bility. Rem edies/or tu ck under. Conclusions .

102

Cha p te r 20

Th e Air o n th e Move Navigatio n . Slop e lift . Tbennal lift , Windsbear a nd Win d Grad ien t. Gusts. Mytbs a nd miscon ceptions. Momentum . Kin etic energy . A nalogies. 77Je meaning ofl ife?

109

Cha p te r 21

Mod el Aircraft Structures Defining some words, composite structures, tobat a ir does to wings, bending m om ents, stru tted wings, torsional stiff ness, fuselages, tailplanes.

114

Cha p te r 22

Centre of Grav ity Pos ition Rigbt and wrong CGs, Fligb t testing, p opula r m isu nderstand ings, tobat m a tters, m ean cbo rds, tbe flying toing , biplanes, tb e neutral point , adjustm ents, p u tting it togeth er, sta bility m arg in .

.123

Appendices

131 A Bemoulli 's equation B Boundary Laye r C vortices D Dib edral and sweep E Usefu l Nomogra ms

143

Glos sary Symbols, Abb rev iations a nd Co mmo n Aero dy na m ic Terms Ind ex

145

No tes

147

Introduction hen the cold raw wind howls down from the North bringing grey fragme nted clouds which sc ud low o ver th e d amp d a rk fo rbidd in g landscap e like a demon arm y. When sheets of icy rain deluge incessantly from a leaden sky and the puddles join fo rces to threaten us with an oth er great flood . When the gre at oak trees bow down to the un seen forces of th e w ind like frightened peasants befor e th eir Gods . When ever the outside environment beco mes hostile to man and his aeroplane, I curl up in a cha ir by the fire with so me books and magazines, to absorb all the fact, fiction and folklore of o ur fascinating hobby. It is o n nights like the se as I lie in be d listening to the w ind howlin g or the rain lash ing o r the deathl y silence of th e s nowfa ll that I h e ar voices , vo ices from my pas t. They are the vo ices of aerodyna mics lecturers and au thors a n d the y remi nd m e h o w littl e acc u ra te knowledge of aerodyna mics is ava ilable to the ave rage modeller, and they tell me w hose fault it is. Mine! My fault for not writing this book soone r! I have three main aims in wri ting this boo k. The first is to disp el the half-truths and old wives tales passed on , usu ally in go od faith , into the folklore of the hobby. I o nce ha d a very pu zzling conversa tio n w ith a modell e r a bout th e u se of "flap s", until he cla rifi ed matt e rs by explai ni ng that he me ant th e "bac k flaps" (e leva tors) . So the se cond aim is to ge t us all speaking th e sa me langu age a s fa r as p o ssible so that our in e vit able di s cussions and a rg u me nts can b e more meaningful. The third aim of my book is an introduction to aerodynam ics so that you ca n understand how to make use of th e data available elsewhere wh en designing your own mod els . Und erstanding some simp le theory will not turn you o vern ight into the design e r of the most elegant and super-efficient models (that still requires experience, in s piration a n d talent) , but yo u ca n le arn what is p o ss ibl e und er the laws of Ph ys ic s , a n d w h a t is impossible - unlike the alche mists of o ld w ho was ted their lives trying to turn lead into go ld . Now let me plea for pati ence es pecially fro m th e more knowledgeable readers. I have started off with a simple , rosy , idealised view of the wo rld and I introduce the rea l co mplications little by little.

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Basic Aeronautics for Modellers

11

Chapter I

The Aeroplane's Environment The Air Please try this simple experiment. Take a can of beer, open it, and drink the contents. Now w ha t are you left with? Most peopl e say "an e mpty can " but th at is wrong . If you answered "a can full of air" give yo urself a pat o n the back . We ae ro mode llers mu st be co nscious of the air. We are depending o n it to su pp ly the lift for our aerop lanes. Next time yo u see a Jumbo jet lumberin g off th e runw a y reme m ber t ha t th e air is provi di ng th e upward force of up to 400 to ns. So how heavy is, say a room full of air, 4 met res by 3 an d 2.36 metres high? Wo uld yo u believe 35 kg or 77 Ib? At abou t 1.22 o u nces p e r c ubic foot ai r is not very d ens e , b u t yo u w ou ldn 't ca ll a roo m empty if it co nta ine d 77 Ib of balsa wood! Now, how stro ng is the air? In a school ex periment the halves of a four inch (lOO mm) diamet er hollow stee l sphere were pla ced togeth er and as mu ch as possible of the air ins ide wa s remo ved. The air hel d th e halv es togethe r. It too k a lot of effort from the four strongest lad s in the class to pull the tw o halves apa rt. Pres sure is defined as a force per un it area . The force which the air pressu re exerts on a surface wi th a vacu um o n th e ot he r side is 14.7 pounds per square inch or nearly a ton pe r sq u are foo t! Th e pull n e ed ed to se pa rate the h emispheres in sc hool was almost 180 Ib (8 00 N). Natura lly the air exerts its fo rce on a surface w he ther the re is a vac uum on the other side or not. Hold up a sq uare foot of paper and there is a ton of force on ea ch side , but so wh at? Th e two forces cancel o ut. Pressure is not direction al, or rather it is omnidirection al; it acts in all direction s at on ce . And it acts perpendicular to the surface at every point. So w hichever way up yo u hold the paper th e re is exactly th e sa me o ne ton for ce o n each side . You can see the air pressure vary ing slightly from day to day o n yo ur barometer. Both density and p ressure reduce w ith altitude but we aeromo de lle rs can igno re these small differences. The reduction in air pressure is about a tenth of one p er ce nt for every 30 feet climb ed . Incid entally it is by measuring that reduction in pressure that an aeropl an e's altim eter works. Low sp eed airflow is called "incompressible " because, although the pressure wiII vary, density does not. We all kno w air ca n be co mpressed, and its den sity change d, but only in a co ntainer. Aeroplanes in free air do no t co mpress it unl ess they travel at ne ar so nic speeds .

Mass, Weight Gravity An o bject's mass is the amo unt of mate rial which it contain s. Becau se we live o n the earth's surface we tend

Basic AeronauticsforModellers

to use the word weight instead and to us there is no differ ence. Whe re a n object's mass (as opposed to its weig h t) s hows itse lf is in its resistan ce to b e ing accelerate d . Take an iron ca nno nba ll into space and it wiII be "weightless" but try kicking the ca nno nball and yo u will bre a k yo u r fo ot. It s resistan ce to being accelerated , its mass, has not changed . Th e weight of th e ball is just th e for ce o f th e earth 's gra vi tatio n al a ttractio n o n its ma ss. To ge t th e w eight o f a body, multiply its mass times "g", the "gravitatio nal co nsta nt" which on th e earth' s surface is 32.2 It/ sec/ sec or 9.8 1 m/ sec/ se c. The we ight of a "kilogram" of mass is a force of 9.81 Newtons an d the we ight of a "slug " (yes really) of mass is a force of 32.2 pounds. (But you don 't ne ed to remember all th at) .

Newton's Laws If a body is in "equilibrium" it is e ither at res t or moving at co nstant speed in a straight line (tha t is, not acc elerating). Man y years ago Sir Isaac Newto n put into wo rds three funda mental Law s of Motion . • 1. The first says that a bo dy wiII be in eq uilibri um if a ll the fo rces o n it cancel ou t, Le . if there is n o resultant force . • 2. The second says th at the force nee ded to cause an acceleration equals the mass times the acce leration . • 3. Th e th ird is the old favo uri te ab o u t each fo rce ha ving an eq ua l and opposite rea ction.

Inertia When yo u kicked the canno nba ll in sp ace , it app lied an eq ua l and opposite fo rce to yo ur foot. Tha t kind of for ce is ca lled an "inertia force", and is the for ce w ith which a body res ists being accelera ted . Similarly, w hen yo u catc h a ba ll yo u appl y a force to s low it do wn , overcoming its "ine rtia" which makes it wa nt to carry on the way it was go ing .

Vectors A riddl e! The re was a car sitting on a level roa d with th e brakes off and three men pu shing it but it wasn't mo ving! Why not? One w as pu shing the front, o ne the back , and o ne was pushing the side. An important little de tail! Any quantity w hos e direct ion must be specified as well as its a mou n t, for exa m p le for ces, is ca lle d a "Vector". O the r examp les of vec tors are distance moved, acceleratio n and velocity. I prefer the word velocity to speed because it is a rem ind e r that it is a ve ctor.

13

_....

..

. .: . .......

Vec to rs ca n b e added to g eth er b y a d d ing th eir am ounts o n ly if th ey are in th e sam e d irec tion . If two Fig u re 1.1

B

10

c

force s are in o p pos ite dir ections, like tw o men pu s hin g a t e ithe r e n d o f a c a r, th e y w ill ca nce l each o ther o ut. If ve ct ors a re at a n a ngl e to ea c h o the r th e y ca n b e added by drawing a "vecto r di agram" using a ru ler a n d protract or. A vecto r dia gr am is a scale drawin g in whi ch th e len g th o f th e lin e s re p rese n ts th e a mo u n t, and th e direction represe nts the d irection o f th e vectors . Figure 1.1 co u ld re prese n t a trea sure map . "Starting at A wa lk ten metres n o rth to B, th en go ten metres ea st to c." The e q uivale nt, or re s u lt ant , o f the tw o vec to rs AB and BC a d de d to get h er is the vecto r AC which is 14.14 metres to the northeast. Figure 1.1 co u ld just as ea sily hav e represented t h e addit ion of two for ces or veloci ties . Ve ctors ca n al s o be s p lit up , or "reso lve d" , int o two o r more "co mpon ents " whi c h wil l h a v e th e s a m e e ffec t (F ig u re 1. 2) . The tr e a sure is in a ca v e , "C ". The ins c rip tio n o n th e Azt e c Temp le , "A" says ; Go five kilometres on a bearing 037 0 East of No rth (b ut beware of th e Dragon at "0 "). Preferring an ea sy life to he ctic adve ntur e, o ur hero "Tri gon o metry" ]ones instead goes 4 km du e No rth, sto ps for a few beers at "B", and th en goes 3 km du e East w here he finds th e cave , treasure et c . e tc . Very precise and sc ie n tific but no u se for a mo vie script. From th e vector d iagram in Figure 1.2, vec to r AC can b e s p lit in to its tw o co m p o ne n ts , AB th e No rthe rly co mpo ne nt and BC the Easte rly co mp onent. The bigger a ng le A is, th e smaller AB become s as a proportion of AC and th e bigger BC become s as a p roportion of AC. The ratio of BC to AC is called th e sine of th e angl e , the ra tio of ve ctor AB to AC is ca lle d the co s in e of the a ngle , and the rati o of BC to AB is called th e tange nt of th e angle A. These ratios are usually sho rte ne d to sin, cos a nd tan a n d ca n be lo o ke d up in table s for an y angle. Us ing his ma themat ica l tabl e s "Tr ig " jones co u ld work out t he c omponents for a n y a n gl e with ou t reso rting to sca le drawi ng . Th e sine of 37 d egrees is 0.6 a nd cos 37 0 = 0.8. Of co urse th e same go es fo r other vecto rs like force s or ve locities e tc.

Moment A

14

Th e "mo ment " of a forc e abo ut a point is the size of the force times the di sta nce of the for ce from the p oint.

eastc Aero nauticsfor Mode/!el :~

Figure 1.3

Figure 1.2 B

3

100

C

~

Easterly Co mponent

Nortbernly Component

50

5

10

~

J:

Figure 1.4

5 4

10

~- - - - --- -- -- --- - - --- --- ------- - ---- - --~ P ivo t

Mo ment = 5 .'\" 10 = 50

A

5

groundsp eed vec tor. Wind ha s no o ther e ffec t (b ut se e the cha pte r o n wind near the e nd anyway). To save any argument I shall I ass ume still air conditions in all th e cha pte rs until th en.

Figure 1.3 represents a seesaw th e pl ank o f w h ich is exactly balan ced . Th er e is a ch ild w eighing 100 lb 5 feet from th e pivot and a ch ild w eighing 50 Ib 10 feet from th e pi vot. Th e ch ild o n the right has a moment o f 500 ft. Ib clock wise abo ut th e pivot , and th e ch ild o n th e left has a mom ent of 500 ft. lb anticloc kw ise ab out th e piv ot. Th e tw o moments are equa l but in opposite d irections a nd so th e y c a nce l o u t whi ch le a v e s th e seesa w balan ced . It is in eq u ilibr ium as th er e is zero resu ltant mom en t. In Figure 1.4 two eq ua l but opposite fo rces act o n a b od y. Th e two fo rce vec tors ca nc el out, th ey h ave no resultant but th ey will o bvious ly tend to turn th e body. Th e turning effect , o r moment, o f the pair o f for ces is th e sa me about any p o int yo u care to choose. The tot al moment is Force tim es th e d istan ce b etween th em . Th is kind o f syste m is called a co up le and its moment is th e sa me 5 x 10 = 50 about a ny pi vot point. In Cha p ter 5 I'll remi nd yo u th at you ca n hav e a for ce sys te m w ith no res u lta nt excep t a mom ent wh ich is the sa me about an y point. You will ofte n see so me qu anti ty lik e a irspeed (V) w ith a number s u persc rip t. Fo r exam p le V3 me an s V "c u bed " o r V "to th e p owe r 3" or s peed x s peed x speed. Similarly th e "cu be root" of V (w ritte n 3jV) is th e numb er which , when mu ltiplied together th re e tim es , . gives V.

Win d I co uld have used th e w ind as another exam p le on vec tors. To find the e ffec t of th e wind , just ad d th e wi nd vector to th e ae ro p la ne 's ve loci ty ve ct or to ge t th e

Basic Ae rona utics f or Modellers

15

Chapter 2

Requirementfor Flight - Lift hat makes a n aer oplane s pecia l is its wing. The qu estion is, ho w does it produce lift? I wish I co uld tak e yo u to a wind tunnel with a p prop riate mode ls and mea surement eq u ip me n t. I co uld then dem on strate ho w lift is produced just as it wa s shown to me . Instead I sha ll have to att empt to describ e it in words and diagrams.

W

Wa tching th e Airflow It is interesting to watch the flow in a smo ke tunnel , wh ich is a specia l low speed wind tunnel in w hich many s ma ll s tre a ms o f s mo ke a re fe d in to t he ai rstream up wind o f th e wi ng. T h e t h in s t re a ms o f s m o ke travellin g wit h the air as it flows over the wi ng help to visualise the airflow . Figure 2.2 is a dia gram sho wing a typ ical flow pa ttern aro und a win g. Th e lines sho w the position of the smo ke streams . Th is is a co mmo n way o f s ho w ing a n airflow a nd th e lin e s drawn a re ca lle d "streamlines" . Strea mlines are lines drawn in the direction of th e airflow suc h th at no wh er e does th e air flo w across a line. As the airflow approaches the Lead ing Edge (L.E.) of th e w ing it s p lits in two, part going a bove a nd p a rt below. The strea mline which d ivides the air w hich w ill go over the w ing fro m the air whic h w ill flow unde r it meets the w ing at poi nt A. Air molecules flo win g exactly alo ng th is line will me et th e wi ng hea d on a n d be b rou ght to a d e ad s top a t A. Po int A is ca lled th e "stagnatio n poi nt" becau se the air's ve locity is red uced to ze ro. . Wa tching th e s mo ke st rea ms over th e top surface very closely, it ca n be seen that the air speeds up as it

Definitions Figure 2.1 sho ws th e cross-sectio n o f a wing. Th e straig ht line from the ce ntre of the leadi ng edge (L.E.) the trailing edge (T.E.) is the chord line . The len gth of the chord line is the cho rd of the w ing (the w ing tip to wi ng tip distan ce is the spa n) . Th e maximum distan ce b et w e en th e to p a n d b ottom su rfaces is th e win g th ickness , usu all y ex p ressed as a percen tage o f th e cho rd. The line drawn midway be tween top an d botto m surfaces is ca lle d th e mean line or ca mber lin e . Th e maximu m distan ce between the mean line and the cho rd line is the ca mbe r of the sec tio n and it too is give n as a pe rce ntage of th e chord. Th e leading edge is always smoothly ro unded and the trailin g edge is always sha rp. A typica l test wing fo r a w ind tunnel has a uniform chord and aerofoil sectio n from o ne e nd to the o the r and fits e xactly in th e width of th e tunn el F ig ure 2.2 wh ich do es a wa y with th e co mp licat ion of tip effec ts w hich w e don 't need at this stage. I s ha ll give you fair w arning w he n I come to a win g w ith tips. For the mom ent the flow is ass ume d to be the same a t an y p o sit ion a lo ng th e s pa n ( two dimensional flow).

Figure 2.1 Cam ber L ine

• Ca ll/be"

.• L E.

Chord Lbw

T.E.••

•

•

~--------------- ----- -------- --------- - - ---- ------- ----- - -~ CIJOI'd

16

Basic Ae rona uticsfar Modellers

pa sses over th e thick Figure 2 ,3 pa rt of the w ing a nd resumes its p re vi o u s speed by th e Trai ling Edge (T.E .) . Under th e wing the smoke bu nches u p as it slows down , an d then it accelerates to its or iginal speed at the T.E. If the smoke strea ms are pu lsed, Le. re leased in s ho rt burs ts , it ca n be seen that the start of the smoke pulse above the w ing re ach es the trailing edge before the smo ke Figure 2.4 be low th e w ing a s illustrated in Figure 2.3. Obv io us ly the air over the top surface has had to speed u p to cover a longer pat h in the same ti m e . No tice a lso t h a t w here th e f low h a s speede d up the streamlin e s a re close r a nd w he re t he flo w is slo we r th e streamlines are furthe r apart. As th e a ng le o f a ttack is in cre ased th e stag nat io n p o int A mov es down around the cu rve of the leading edge increasing the dis tance the air travels over the to p , a nd re duci ng the dis tance alo ng the unde rside . On a w ing w ith a symme trica l sectio n a t a n ang le to th e airflow, the stag na tio n poi nt is be low the ce nt re of the le ad ing e dge (as in Figure 2.4) so jus t as wi th th e cambered sect ion the air flowi ng over the to p surface has fu rther to go in the same time , and must therefore speed up .

~ ---------------

Pressure Variation You can 't get a change in velocity wi thou t a pp lying a force (Newton's First Law). The on ly force ava ilable to t h e free air is its press u re so th e p re s su re mu st be changing as speed cha nges across the chord of the wing (See App en dix A, Bern oull i's equation) . If we wish to measur e accurately the pressure cha nges we have dedu ced mu st b e occurri ng o ve r o ur Figure 2.5 aerofo il, we ca n drill a row of tiny holes in the top and bo ttom surfaces and connec t eac h one to a p re ss u re measur ing d e v ice . Eac h pres su re meas ured ac ts at right ang les to th e surface at th e po int w he re it was measured . The pressur e is, as ex pected , less on th e upper surface th a n on th e und e r s u rf ace a n d th e re is a h igh pre s su re p e ak a t t he stag na tio n p oint w here

Basie Aero nalilies fo r Modellers

---

the ai r me ets the wing head o n. See Figure 2.5 in whic h the len gth of each arrow represents the pr essure at tha t po int.

Pressure Exerts a Force Pressure is de fine d as force per uni t area . Imagine in Figure 2.5 that th ese pr essure arrows , o ne inch apart, each represent the fo rce o n the o ne squa re inch around eac h hole . If all those force vecto rs are added togeth er, the resu ltan t will be the total force on a o ne inc h wide strip of w ing . Its size and di rec tion de pend upon th e aerofoil section , the ang le to the air flow , the speed of the a irflow , ete. See Figure 2.6 in wh ich the res ultant force is sho w n as force F. Th e p o int where th is force crosses the chord line of the section is ca lled the Centre of Pressur e (or C P.) . It is the poi nt th rough w hich the total pressure effec t on the w ing ca n be repl aced by a sing le force .

17

Figu re 2.6

Ail flow ~

Figure 2. 7

Ailflow ~

L

,, ,,

, ,, , ,, , ,

c.r.

Figure 2.8

lV

It is in c on veni ent to hav e a fo rc e ac ting in an arbitrary direction like that and so it is split up into two co mpone nts at right an gles to each othe r. Th e d ire cti ons chose n a re th e obvio us on es fo r a w ind tunnel. Th e co mpone nt in th e dire cti on o f th e airflow is called Drag, and the co mpo ne nt at right ang les to the a irflow is called Lift (See Figure 2.7) . No te that I d id not sa y ve rtical and hor izontal! It is tru e if the w ind tunne l is built horizontal , but lift w ill not b e ve rtical wh en we co me to an aeroplane climbing or desce ndi ng o r ban kin g. Figures 2.8 and 2.9 show what I mean . No te that it is a mat hematical co nven ience to sho w forces like F, or L and D at the ce ntre of pressure . They are merel y representing the tru e situa tion of Figure 2.5. So me p re ssure m ea suring d e v ic e s m ea sure the diffe rence in p ressur e between the desired point and the static pr essure of the a ir in the room . Or if you like the pressure differen ce between the insid e and outside of a hollow win g. Figure 2.10 is simila r to Figure 2.5 but this tim e sho w ing th e pr essure difference betwe en inside and outside . Th e reduction in pressure whe re the air is speeded up ca uses an upward force over the top surface and w he re the air is slowed down ther e is an upward force on the lo wer surface . Thi s is a co mmo n meth od of sho w ing the lift d istribution whi ch yo u may have co me across before (so me times o nly the line joining the tops

18

of the arrows is shown) . The resultant of all thes e force s (o r pressur es ) is exac tly the sa me as in Figure 2.6. Ju st to get all this in perspective , co nsider how mu ch pressure cha nge is needed to su pport the w eight of a m od e l with a typ ical win g lo a d in g of 20 o z./ft -. Atmospheric pressure is ab out 14.7 pounds per sq uare inc h . An ave rage press ur e rise on th e und ers id e of 0.02%, and an averag e pr essure reduction of 0.04% on the top surface will suffice. We are n ot asking mu ch a re w e ? To ca ll t hi s a "va c u u m" w ou ld b e mi sleadin g. I ex ag gera te d enormously the arrows o n my diagram s 2.5 and 2.10 to mak e them mean ingful.

Wind Tu nnel Testing Of c o urse w e don 't real ly g o th rou gh a ll thi s r ig m a ro le o f m e a surin g pre ssure s a n d in vol v ed ca lc u la tio n to w ork out th e lift and dra g in a wind tunn e l. Beside s th e co mp lica tio n in volved , th e s kin friction drag has been igno red . The w ing co uld simply be mounted o n a bal an ce to mea sur e the forces directly. The force mu st be mea sured through the attac hme nt point (e .g. the L.E. or qu arte r cho rd point) together with the mom ent abo ut this poin t. This mom ent is ca lled the

Basic Ae rona uticsfor Modellers

Pitch in g Mom ent. As m om ent e q ua ls fo rce tim e s di st a nce , if th e lift a nd moment a re kn o wn th en th e position whe re the lift ac ts (the Centre of Press ure) can be calculated . Th e w ind tunnel sho uld be eq uipp ed w ith a ba la nce ca pa b le o f me asu rin g h o ri zontal force s , vertica l forces , an d p itch ing moments a ll at th e sa me time . Thi s eq uipment can be used to test a wing, adjus ting o ne variable at a time and kee ping everythi ng else the same to find out the effect of each variable. For instan ce test ing the sa me wing in the sa me position at d ifferent airs peeds s hows th at Lift, Dr ag a nd Moment are a ll prop ort ional to the sp eed sq uared. In o ther words at twic e the speed yo u ge t four times the force , and at three times the speed, nine times the force etc. By s imilar means it is found th at Lift a nd Drag are also proport ion al to the air den sity p and the wing area. The mom ent is proportional to the sp eed squared, the air den sity and the wing are a times the chord . To turn these relationships into use ful eq ua tions for es tima ting the lift from a wing , a co ns ta nt has to be intro duced and its valu e mu st be found ex pe rime ntally. 50 for exa mple • L

=

A d ifferent co ns tant is ne ed ed in each case but to sa ve running o ut of suitable lett ers, the letter C is used in all three equations w ith a d ifferent subscri pt. The people w ho mad e up the eq ua tions put in a !1 as we ll becau se the te rm !1 p V2 had turn ed up in Berno ulli's equ ation (see Appendix A agai n). We e nd up with these three familiar eq ua tions • L = !1 P V2 5 CL • D = !1 pV2 5 CD • M = !1 P V2 5 C C~ I

Wh ere CL is th e lift coe ffic ie n t a nd CD is th e dra g coefficie nt an d CM is th e pitching mom ent coefficient. Th ey all vary with ang le of attack as you w ill see.

P V2 5 x co nst.

Figure 2.9

Figure 2.10

! 1I t Basic Aeronautics forModellers

t t t t

+

~ 19

Chapter 3

The Stall's the Limit n w ind tunn els the win g is stationa ry and the air is drawn over it, so that is how it is usu ally describ ed in th e ory. It is just as valid to th ink of the a ir as stat ionary and the wing moving. Its directio n of motion is exact ly opposite to th e arrow marked "a irflo w ". The dir ection of the a irflow must be measured far enough ahead o f the wing so that it is not affected by the wing's approach.

I

Definitions Figure 3.1 s ho ws a wing se ctio n in an airflow. Th e angl e between the chord line and and the airflow is called the angle of attack . It is usually represent ed by the greek lett er a (alpha). Occa sionally a different datum line is used instead of the cho rd line. It may be a straight line o n the und erside of a flat bottom ed or und ercarnbered Wing, or the wing 's zero lift line. As the nam e suggests, if the airflow is parallel to the ze ro lift line, the lift is zero (usefu l in mathem atical formulae). Th e inciden ce of the wi ng is the a ng le betwe en its cho rd line (or oth er datum line) and the fuselage datum line . It bears no relation to the airflow and angle of attack at all. It is just a riggin g angle. It may be measured o n the aeroplane with an incidence meter or on the plan with a protractor. Those are the usu al definition s and I shall stick to t he m , but it is not uncommon to see th e w ord incide nce used mea ning angl e of atta ck .

Notice the shape of the graph! It is straight from A to C and then curves up to a maximum at D th en down to E a nd be yond . At point B the an gle of attack is ze ro as the wing has be en arranged as in Figure 3.3 suc h that the chord line is paralle l to the airflow. Although the an gle of attack is ze ro , the wing is still producing lift. At point A the wing has been tilted further le ad ing edge down as in Figure 3.4 and is now producing no lift. Th e ze ro lift angle of attack is written as a o (the s u bsc rip t 0 den oting n o lift ) . T he normal wa y of measuring a ng le o f atta ck is to mea sure UP from the direction of moti on to the cho rd line . Because the chord

E

The Lift Curve Testing a wing at man y different ang les of attack and worki ng out th e Cl. e ac h time ( fro m the for mu la in Chapter 2) enables a graph of lift coefficient again st angl e of attack to be drawn for that particular section . For most normal sectio ns the graph loo ks like Fig ur e 3.2. Th is g ra p h is true for this sectio n regardless of the size or s pee d a nd c a n b e u s ed to es tim a te th e lift in an y co nd ition.

0< =0

Figure 3.1

C

--- - -Cb';;"';-;' - - - - - - Airflo w

~"~ _ _ ~

Zero lift l .

Direction of Motion

""

A ng le ofAttack (measured from z ero lift U1Ie)

20

, , A ngle ofA tta ck (measur ed f rom c h o r d U1Ie)

Basic Aerol/l/ /Ifics/or Modellers

li n e is DOW N in t h is case the ang le of attack is a negative ang le (for e x a m p le th e a ng le o f attack for ze ro lift on an Eppl e r 19 5 s ectio n is g ive n as - 3 d e gre es) . The zero lift line (ZLL) d rawn on the w ing is by definitio n parallel to the airflow . At p o int D th e lift c oe ffic ie n t is C Lm a x w hich is the maximum lift coe fficient wh ich the section ca n prod uce and oc curs at as the stalling angle of attac k.

The Stall

Figu re 3.3

Figu re 3.4

----:----c-_ _ _.~~

ZLL

-0: "\ //

I

,'

1/ /

. . --, 3

,/

1

I,'

1//

1:/ '

,:

1,/ '

~,r

:~

Even More Drag Th e ASPECT RATIO of a 3-D win g is defin ed as the spa n d ivided by the ave rage chord. It is found tha t when a rea l wing with tip s is tested in a w ind tunne l its dra g is more than if it fitted perfectly fro m wa ll to wa ll, and the lift is less. Th e loss in performance depends on its as pect ratio as illustrated in Figure 6.5 . The high e r the Aspect Ratio of the wing, the ne are r is its pe rformance to that of the ideal tw o di mensional wing (infinite aspect ratio) .

The Reason This sho rtfall in p erformance is caused by the trailing vortices whic h create a reg ion of descending air be hind the w ing, a fter all the energy to crea te the m mu st be p aid for so me ho w . Th at th es e vo r tices a ls o c a use d ownwash in the airflo w as it ap proaches the wing ca n be proven by the ory , or dem on strated at hom e by filling a tall glass w ith wa ter and placin g a few g rains of rice at the bottom . \Vith a spoo n , stir the wa ter in the glass near the top and yo u w ill soo n see th e rice g ra ins begin to

sw irl. Bec ause of the fluid 's viscosity a SWirling motion is ind uce d rig ht to the bo ttom of the glass. If the spoo n is the wingtip vo rtex stirring the air be hind the wing , the rice is be ing sw irle d ro und a head of th e wing, in the sa me d irection , bu t to a lesse r ex te nt. Fig ure 6.6 s hows th e airflow aro und a re a l thre e d imen sion al wi ng in more deta il. A long wa y ahead of the wi ng the airflow is undisturbed by its presence. As the air approac hes , it is angled down s ligh tly by the d ownwash ahead of the wi ng ind uced by the tra iling vo rtices and th e n jus t in fro nt of th e wi ng the air is swept up and over by the boun d vort ex as in 2-D flow . We started by defining the ang le of attack as the angle be twee n th e w ing and th e und isturbed a irflo w aw ay ahe ad of the w ing . No w we ca n see that th e "real" angle of attac k of the air meeting the wing has been red uced by the downwash. And the lift re lates well to the lift p redic ted from 2-D tests at this reduced an g le of attack. So t h e lo s s o f li ft is e xp la in e d b y t h e d ownwa s h red u ci n g th e a n g le of a tt a ck . But w h a t abo u t t he increase in d rag? Look ing back at Figure 6.6 agai n, the ae roplane mu st

'~OIlIlY, TIIAT ~1I0ULD BE 'NOW TIlYA BUNTCAIlEFULLY, WATCIIING rse~TIlEAMEIl~"

..

"..~

.. • .•

32

~. .

. C'"

Basic Aeronautics/or Modellers

Figure 6.6

Measw'ed

Cl.

Real

Real Lift

Cl.

---

---

Doumuiasb All

-------=======:----Undisturbed Ab'

Doumuiasb III Front Of lVi1lg

think it is constantly flying uphill, or rather flyin g le vel throu gh the sinking air of its o wn do w nwash . The lift for ce has been tilted back a little , by the amou nt of the dow n was h a ngle . Th at means th e lift ha s a s ma ll component in a d irec tio n opposite to the directio n of motio n . If it opposes motion it is Drag , isn't it? This compone nt of the total d rag is called "ind uce d drag" because it is caused by the tilting back of the lift ca use d by th e dow nwas h ind uced b y th e tra iling vo rtices. You may also see indu ced d rag referred to as "active drag". No amoun t of strea mlining or fiddling w ith th e section will red uce it. It is inevitable as it co mes from the lift.

Complications Figure 6.6 shows the airflow at o ne particul ar place on th e wing . Near the tip s w here th e vortices a re centred the downwash is grea ter than o n the ce ntreline, as in Figure 6.7. Th us o n th is recta ng u lar wi ng each point from the roo t to tip has a d ifferent "real" o r "local"

Doumuiasb Behind Wing

ang le of attack and th erefore a d iffere nt "loc a l lift coeffic ient ", u su all y d e n ot ed Cl (w it h a s ma ll I su bscript). The local lift coefficient red uces towards the wing tips. Th e situa tion even on a straight w ing is not as simple as the p icture I have pai nted up to now. On a ta p e red t w isted swep t wi ng the si t ua tio n is as co mplicate d as yo u ca n imagine , if not m o re ! Th e variatio n of the "local" ang le of attack de pends on the variatio n of the downwash w hich depends on . . . well just about everything , includ ing the variatio n of the local ang le of attack . So becau se the downwas h angle, lift coefficient, drag coefficien t, lo cal ang le of attack , ce n tre of pressure positio n, etc, all vary wi th po sition along the spa n , that makes it very difficult to use the section characteristics me asured in two dime nsional flow . It is just too muc h for the hu man bra in to cope wi th and is be st le ft to co mputers with time on the ir han ds. \'\fe co uld give u p the who le messy bu siness here and no w , o r we co uld just step back and look at it from a d istance.

Figure 6.7

~

Direction OfMotion Wi1lg's Apparent Angle OfAttack

Root Doumuias

Tip

Cl.

Direction OfMotiotl WhIg's Apparent Angle OfAttack

Basic Aero na ut ics fo r Mode llers

Tip Doumuiasb

33

F igu re 6.8

L

~ -- -- ~Airfloto

Simpiifications

Lessons f o r P r a ctica l Modellers

Thi s looks like a cla ssi c case for the bla ck box system . I shall draw an imaginary black bo x around the wing and care not a wh it for wha t is happening insid e . Air e nte rs th e fron t of the bo x a nd co mes o ut o f the back an gled d own slightly by the downwash, Th e angle of attac k is mea sured between the direction of motion and a referen ce line d rawn o n the outside of the box. Th e refer en ce line may be e ither the cho rd line at the ro ot sectio n o r th e ze ro lift lin e of th e who le w ing . Th er e is a Lift for ce perpendicu lar to the d ire ction o f motion and a total Drag for ce , including induced d rag, op posi te to the d ire ction of motion. Th e re w ill be an overall Centre o f Pressur e but I wouldn't care to guess at its p ositi on so I prefer to p ut th e lift a t th e w in g 's ae rody na mic ce ntre (25% mean chord) and apply a ze ro lift pit ch ing mom ent. See Figure 6.8. \Xrhe n I refe r to lift coefficient , or drag coefficient, I mean an ave rage for the wh o le win g wor ke d o ut fro m tests and th e Cha pter 2 formula e . The lift co efficient Cl. is the overall average for th e wh ole w ing and has a ca p ital L su bsc ript. Its grap h will still be a familiar sha pe but need not be the sa me as the section's curve . The wing would hav e to be tested to get acc ura te graphs but a re asonable guess could be mad e by making "allowances" for as pect ratio, tap e r and twi st. Th e slo pe of the straight bit will dep end o n the as pect ratio (as in Figure 6.5). And the position of aD and the stall will dep end o n the wing's planforrn a nd twi st as mu ch as its sec tio n.

Reducin g induced d rag is important fo r aeroplanes which cru ise a t a la rge lift coefficie nt (Le low speed) therefore glid ers mu st have as high a n Asp ect Ratio as is pra cticab le . Also a lig hte r ae ropl ane has less induced dr ag than an identical he avy o ne a t th e sa me spe ed be cau se of the lower lift co efficient. Any lift, even downward or s ide ways lift, will cause vor tices whi ch cause down wash which will tilt the lift back wh ich co ntributes to induced drag . If a tailp lan e is carrying a download , not o nly d oes the downward lift on th e tail produce indu ced dr ag but th e Wing must produce ex tra lift to co unterac t the download and that means ex tra indu ced dra g o n the wing as well. Most wind tunn el sectio n test s ar e d on e o n tw odim en sional models. If these resu lts a re used to es tima te th e p e rform an ce of a re al ae ro p la ne th e y will g ive opt im isti c a nswe rs becau se the y d o not include the induced drag. It is possib le to es tima te the charac teristics of a w ing from se ctio n da ta usin g var iou s fidd le factors but that is o utside the sco pe of this bo ok.

Th e Importa nce ofAspect R a tio A sho rt span broad win g mu st have a stro nger vort ex to g ive the sa me lift as a lo ng narrow w ing, a nd w ill co nseq ue ntly produce mor e downwash . Th er efor e the induced drag is greate r for a win g of low Asp ect Ratio than for o ne of high Asp ect Ratio (AR) becau se of the g re a te r d o wnwa s h , An d as th e a n g le of a ttac k is inc reased , bo th Lift a nd d ownwash a ng le in cr e as e (because the vo rtex stre ng th increa ses ). Thus induced dr ag increases rap id ly. In math ematical shorthand th e induced dra g coefficie nt (C n) is given by;

34

Grou nd Effect Lo oking at Figure 6.6 you probably thought that the wind tunnel wall s would co nstrain the downwash effect, and yo u we re right. Th e ind uced drag of a finite Wing will be underestimat ed in a wind tunnel becau se th e do wnwash is reduced . Th e sa me thing will happen if an aeroplane is flown very low over the gro und . Th e proxim ity of the gro und will reduce th e downw ash and therefore th e ind uced dra g will be reduced . Th e effec t is to prolon g the flare . For this rea son , flight testin g mod els with in a few win g cho rds of the grou nd will give misleading result s.

Basic Aeronauticsfor Modellers

Chapter 7

Planform and Twist s yo u sa w in the Figure 7. 1 previo us chapter, the Aspect Ratio Elliptical Lift Distrilnttton Loading - Lift/Unit Span o f th e wi ng lar g el y _-.,-.....-rTTT"'T"TTT"TT"T...........--,-.">----d et e rm in es ho w mu ch in d u c e d dra g it wi ll c a use . Bu t th e littl e e q ua tio n for in d u ce d drag coefficient w hich I ju st s lip pe d in th e re co ntaine d a co nstant, K, as we ll. Thi s K (ca lle d the Indu ce d Drag Factor) depen ds upo n how the load is sha red alo ng the wing . T he lift ma y Figure 7.2 be eve n ly sp read , o r mostly near the roo t, o r what eve r. Fig u re 7. 1 s hows w ha t theor y says is the Rectangular Area Dtstribution id e a l lift d istribu ti on w hich gives K its min imu m value o f 1. Th is is a d iagram of the Lift Per Unit Spa n . Each arrow rep re s ent s th e li ft o n a o ne inch wide strip of win g , and th e lin e joi ning the to p s of a ll the arro ws is an e llipse . Loading