19760004918_1976004918

download 19760004918_1976004918

of 209

Transcript of 19760004918_1976004918

  • 7/22/2019 19760004918_1976004918

    1/209

    C O N T R A C T O RR E P O R T

    I

    -LOAN COPY: RETURN TOAFWL TECHNICAL LrSRARY

    KIRTLAND AFB, N. M.

    OF S PANL O A D I N GF STRAIGHT-WING/PROPELLERTO STALL

    . A. McVeigh, L . Gmy and E . Kisielowski

    Bell Pa. 19422LangleyResearchCenter

    U T I C S N DP A C E D M I N I S T R A T I O N W A S H I N G T O N , D. C. 0 , OCTOBER ' 1975. I. .

    . . .

  • 7/22/2019 19760004918_1976004918

    2/209

    TECH LIBRARY KAFB. NM~ ~.'1. Report No. 2. Govcmnwnt Accession No.NASA CR-2602 . l l n l l l m l H l l l r l r r n l l l ~ ~ ~

    4. Title andSubtitle Predi ct i on of span l oadi ng of O b b L S L Lst rai ght - w ng/ propel l er combi nat i ons upto stal lOctober IY13

    6. Performing organization Coda- 7. A u t h o M M. A McVei gh 8. Performing Organization Report No.L. Gray

    10. Work Unit No.. Ki si el owski UTR- 004

    Uni ted Techhol ogy I nc. 11.Contract or Grant No.1777 Wal ton Road NAS1- 12238Bl ue Bei l Pa. 194229. Performing Organization Name and Address

    13. Type of Repor t and Period Coveredr12: SponsoringAgencyNameandAddressNational Aeronautics and Space AdministrationWashington, D.C. 20546 14. SponsoringAgency code

    I.__."15. NotesThe AG techni cal representat i ves were Mr. Robert T.Tayl or andMr Loui s P. Tost i . The cont r i but i ons of the NASAtechni cal personnel t o t hi s work are gratef ul l y acknow edged.

    16.Abstractdi st r i but i on on st rai ght - w ng/ propel l er combi nat i ons. The methodcombi nes a modi f i ed f orm of t he Prandt l w ng theory w th a rrepresentat i on of the propel l er sl i pstream di str i but i on. The sl i p-st ream anal ysi s permts cal cul at i ons of t he non- uni f orm axi al androtat i onal sl i pstr eam vel oci ty f i el d ofropeller/nacelle-combinations.Thi s non- uni f orm f i el d i s t hen used to cal cul ate the w ng l i f tdi st r i but i on by means of t he modi f i ed Prandt l w ng theory.f or both the w ng. and the propel l er bl ade ai r f oi l sect i ons and appl i cabl e up to stal l . The theory i s devel oped f or any numberfnon- over l appi ng propel l ers on a w ng w th par t i al or f ul l - span and i s appl i cabl e throughout the aspect rat i o range f rom 2. 0 anddi gi tal computer. The computer program i s usedo cal cul atesl i pst ream character i st i cs and w ng span l oad di st r i but i ons f or anumber of conf i gurat i ons f or whi ch exper i mental data are avai l ablFavorabl e compar i sons are demonst rated bet ween the theoreti calpredi ct i ons and the exi st i ng data.

    The method ut i l i zes non- l i near aerodynamc secti on data

    The anal ysi s i s programmed f or use on theDC 6600 ser i es

    17. Key Words Suggested by Author(s)ILoad di st r i but i onropel l eronWngngl e of at t ack

    18.DistributionStatement

    Ai rcraf t I Tw stStal l Taper rat i o Subject Category 011 I

    19. Sccurity Classif. (of this report ). . - . . 20. Security Classif. (of this page). . ~~~ 21. NO. of Pages 22. RiceUnclassified Unclassified $7.2508

    . - ... .. . . -For sale by the National Technical Information Service, Springfield, Virginia 22161

  • 7/22/2019 19760004918_1976004918

    3/209

    4

  • 7/22/2019 19760004918_1976004918

    4/209

    SUMMARY

    A method is presented for calculating the spanwise liftdistribution on straight-wing/propeller combinations. Themethod combines a modified form of the Prandtl ing .theorywith a realistic representation of the propeller slipstreamdistribution. The slipstream analysis pennits calculationsof the non-uniform axial and rotational lipstream velocityfield of propeller/nacelle combinations. This non-uniformfield is then sed to calculate theing lift distributionby means of he modified Prandtl wing heory.

    The method utilizes non-linear aerodynamic sectiondata for both the wing nd the propeller blade airfoilsections and is applicable up to stall. The theory isdeveloped for any number o f non-overlapping propellers, on awing with partial or full-span flaps, and is applicablethroughout the aspect ratio ange from 2 O and higher .The analysis is programmed for use on the CDC 6600 seriesdigital computer. The computer program is used to calculateslipstream characteristics and wing span load istributionsfor a number of configurations for which experimental dataare available. Favorable comparisons are demonstrated betweenthe theoretical predictions nd the existing data.

    iii

  • 7/22/2019 19760004918_1976004918

    5/209

  • 7/22/2019 19760004918_1976004918

    6/209

    CONTENTS PaqeS U M M A R Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLIST OF ILLUSTRATIONS. . viii

    S E C T I O N 1 INTRODUCTION........................... 1SECTION 2

    2.12.22.3

    3346

    SECTION 3 TiIEORETICAL ANALYSIS................... 83.1 PROPELLER SLIPSTREAMANALYSIS. . 83.1.1 Genera lropel le rolution. . . . . . . . . . . . . 83.1.2 I n i t i a l C a l c u l a t i o n fnflow Angle. . . . 13

    Solution....o....o...~................. 15D i s t r i b u t i o n s ......................... 18

    ,' 3.1.3 Con verg ence f the I t e r a t i v e P r o p e l l e r3 .1 .4 An al ys is or Slipstream V e l o c i t y

    3.2 WING-IN-SLIPSTREAMANALYSIS. . 21o r W i t h Full-SpanDeflectedlaps. . . . . .1Deflected Flaps . .~ . . . . . . . . . . 29Small Aspe'ct Ratios.................... 39

    3 .2 .1 An a ly s i s o r a Wing W i t h noFlaps3.2.2 A na ly s i s or a Wing With Part-Span3.2.3 Extensionof . t he Wing Analysis t o

    SECTION 4 D I G I T A L C O M P U m R PROGRAM............... 424.1 PROPEZLER SLIP-S'I'REAM COMPUTATIONS. . 42

    S l i p s t r e a mV e l o c i t yD i s t r i b u t i o n s . 424 .1 .1C o mp u t a t i o n a lP ro c e d u re s o rP ro p e l l e rV

  • 7/22/2019 19760004918_1976004918

    7/209

    . % . 4.x.24.1.3..

    1

    4.2,.. 4'.2.1

    4.2.24.2.34.2.4

    4.3

    4.4S E C T I O N 5

    5.1

    5.2 5.2.15.2.25.2.35.2.45.2.5

    S E C T I O N 6S E C T I O N 7

    Propel ler : B l a d e Sect ion C h a r a c t e r -ist ics. . . . . . . . . . . . . . . . . . . . . . . . . ......Table L o o k - U p Procedures f o r Pro-p e l l e r A i r f o i l C h a r a c t e r i s t i c s . . . . . .

    - .WING-IN-SLIPSTREAM COMPUTATIONS. .C o m p u t a t i o n a l Pracedures f o r Span-w i s e Loading on a Wing With no Flapso r w i t h Full-Span D e f l e c t e d Flaps.. .C o m p u t a t i o n a l Procedures f o r Span-w i s e Loading on a Wing With P a r t -Span D e f l e c t e d Flaps ................Wing Sec t ion Character is t ics . . . . . . . .T a b l eL o o k - U p Procedures for WingSec t ion Character is t ics . . . . . . . . . . . . .D E S C R I P T I O NO F THE: COMPUTER ROGRAMLOGIC.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .SAMPLE OUTPUT.......................V E R I F I C A T I O N O F THE DEVELOPED THEORYCORRE L AT I ONS EQR AN I S O L A T E DPROPELLER........................ ' . . .CORFSLATIONS FOR WING-IN-SLIPSTREAMC o r r e l a t i o n s f o r Low A s p e c t R a t i oWingsC o r r e l a t i o n ?or C e n & 3 r a l l y - M o u n t e dPr o p e l l e r s and Jets . . . . . . . . . . . . . . . . .C o r r e l a t i o n for Twin P r o p e l l e rC o n f i g u r a t i o n s ......................E f f e c t of Prope l l e r R o t a t i o n . . . . . . . .E f f e c t of Flap Def lec t ion . . .CONCLUSIONS ANDRECOMMENDATIONS.. .REFERENCES.. . . . . . . . . . . . . . . . . . . . . . . . .

    vi

    485254

    5 4

    576161

    616568

    6877

    \ 78808488969899

  • 7/22/2019 19760004918_1976004918

    8/209

    APP E ND IX A PR OPEL L ER T I P LOSS CORRECTION TABLES 104A P P E N D I X B P R O P E L L E R A I R F O I L TABLES............ 108A P P E N D I X C PROGRAM USER INSTRUCTI0,NS. . 121A P P E N D I X D I N T E R N A LI S T I N G O F THE COMPUTERPROGfZIIM............................. 146

    V i i

  • 7/22/2019 19760004918_1976004918

    9/209

    LIST OFLLUSTRATIONS

    F i g u r e 1

    234

    5

    6

    78

    9

    10

    11

    12

    No t a t i o n f o r a P r o p e l l e r O p e r a t i n gi n t h e Presence of a Wing .........Bl ad e Element Veloci ty Diagram ....Analy t i ca l Model f o r S l i p s t r e a mContraction.. . . . . . . . . . . . . . . . . . . . . .No t a t i o n f o r Wing-In-SlipstreamModel. . . . . . . . . . . . . . . . . . . . . . . . . . . . .Mathematical R e p r e s e n t a t i o n o fF l a p D i s c o n t i n u i t y ................Method forS u p e r p o s i t i o n ofSolut ions. . . . . . . . . . . . . . . . . . . . . . . . .Computer Program Flow Diagram. ....LogicDiagram orPrope l l e rSl ip-stream Subroutine. . . . . . . . . . . . . . . . .

    Paqe9

    12

    20

    24

    30

    3562

    64Co r r e l a t i o n Between P r ed i c t ed an dMeasuredElementalThrustandTorqueLoadings a t 75 Percentadius. . . . .9Cor re la t ion Be tween P r e d i c t e d andMeasuredElemental T h r u s t andTorqueLoadings a t 52 Percen tadius . . . . . 70Cor re la t ion Be tween P r e d i c t e d andMeasuredElementalTnrustandTorqueLoadings a t 25 Percen tadius . . . . . 7 1ComparisonBetween P r e d i c t e d andM s a s u r e dDi s t r i b u t i o n so fS l i p s t r eamAxia lVeloc i tyand Swirl Angle fo rth e P-2 Pro pe l l e ro fR e f e r e n c e 1 7 ,a t J = 0.12....................... 73

    v i i i

  • 7/22/2019 19760004918_1976004918

    10/209

    F i g u r e 13

    14

    15

    16

    17

    l a

    1 9

    20

    21

    , . PaqeComparisonBetween Pr ed ic t ed an dM e a s u r e d D i s t t i b u t i o n s . of Slipst ream . . ,Axia lVe . loc i tyand Swirl Angle f o rt h e P-1 P r o p e l l e r of Refe rence17 ,a t J = 0.26.. ...................... 74Comparison Between P r e d i c t e d a n dMeasured Dis t r ibu t ionso fS l i p s t r e a mSwirl Angle forT y p i c a l Tes tCondi t ionsfe fe rence 42........" 76V e r i f i c a t i o n of Low Aspect R a t i o~ a l y s i s......................... 79ComparisonBetween P r e d ic t e d Span-w i s e Loadingand.MeasurementsofRe fe ren c e 6 f o r a Rectangular WingWith End P l a t e s Subjec ted t o aUniform J e t ; Vs/Vo = 1.36... . . . . . .PredictedVersusMeasuredSpanwiseLo a d in g s o r t h e Rectangular Wingof Reference 29 With a C e n t r a l l y -Mounted Propel le r ; AR = 6.........P r e d i c t e dVersus MeasuredSpanwiseLo a d in g s o r t he Rectangular Wingof Reference29 With a C e n t r a l l y -Mounted Prope l le r ; R = 3.. .......P r e d i c t e d VersusMeasuredS,panwiseLoadings f o r theTwin-Prope l le rCo n f ig u ra t io no fReference 42;AR = 3-01 C T s = o.................Predic tedVersusMeasured S,panwiseLoad ings o r heTwin-Prope l le rCo n f ig u ra t io no fRe fe re n c e 42;AR = 3.0, C T ~ 0.36, p 75 = 25O..Predic tedVersusMeasuredSpanwiseLoadings f o r t he Twin-Propel lerCo n f ig u ra t io no fRe fe re n c e 42;R = 3.0, CT = 0.64, p75 = 25O.oS

    81

    0 2

    8 3

    85

    86

    8 7

    ix

  • 7/22/2019 19760004918_1976004918

    11/209

    23

    24

    25

    24

    27

    2 8

    29

    30

    PaqePredictedVersusMeasuredSpanwiseLoadings f o r the Twin-PropellerCo n f ig u ra t io n of Reference 44;R = 4.7, c = o................. 89

    Predic ted VersusMeasuredSpanwiseLoadings f o r the Twin-PropellerCo n f ig u ra t io n of Reference44; AR =3.26-, C = O..................... 90PredictedVersusMeasuredSpanwiseLoad ings o r t he Twin-PropellerConf igura t ion o fRefe rence 44 ;R = 2 .28 , C = O................ 91

    PredictedVersusMeasuredSpanwiseLoadings f o r theTwin-Propel lerConf igura t ion ofRefe rence 44 ;

    TS

    TS

    TS

    AR = 4-07, CT = 0.4...............S 92P r e d i c t e d VersusMeasuredSpanwiseLoadings f o r theTwin-Propel lerConfigura t ionofReference44:AR = 3.26, C T ~ 0.4..............3P r e d i c t e d VersusMeasuredSpanwiseL o a d i n g s o r t h e Twin-PropellerConf igura t ion o f Refe rence 44 ;AR = 2.28, CTs = 0.4.............. 94E f f e c t of P r o p e l l e r Rota t ion onSpan L oad ing or t h e C o n f i g u r a t i o nofReference42; AR = 3.0,C T ~ 0.64, 0 = 10 Degrees. ....... 95P r e d i c t e d SpanwiseLoadings f o r h eTwin-Propel ler Configura t ionofReference44 t o Show the E f f e c t ofFlap De f le c t io n ; AR = 4.7, Q = 10Degrees.. . ........................ 97Assembly of ComputerProgram InputData Cardeck....................23

    X

  • 7/22/2019 19760004918_1976004918

    12/209

    LIST OF TABLES

    Table I- I1

    I11

    IV

    V

    PaqeTypical. ropeller Blade Sections . 49

    . .

    . Summary .ofPropeller AirfoilSections Tabulated for use in theComputer Program............. ..... 51Sample' utput for Lift. istributionon a Wing-In-Slipstream........... 66Sample Output. or Propeller VelocityDistribution............'.......... 67Card Format for Wing Section AirfoilTables............................ 130

    xi

  • 7/22/2019 19760004918_1976004918

    13/209

    LIST O F . SYMBOLS

    AR wi ng aspectat i oa l i f t cur ve s l ope f or f i ni t e aspect r at i o perdegree. .

    BB,bC DcdCLC 1CQCT

    C

    CRDEF

    . .speed of sound msecsect i on l i f t cur ve sl ope per degr eenumber of bl ades per pr opel l ercoef f i c i ent s i n t r i gonomet r i c ser i eswi ng span mt ot al wi ng dr ag coef f i ci entsecti on dr ag coef f i ci entt ot al wi ng l i f t coef f i ci entsect i on l i f t coef f i ci entpr opel l er t or que coef f i c i ent , Q/, 2 D5pr opel l er t hr ust coef f i ci ent T/ pn2 D4pr opel l er t hr ust coef f i ci ent , T/ qs T R2wi ng l ocal chor d mwi ng r oot chor d mpr opel l er t i p di amet er medge vel oci t y f act orpr opel l er t i p loss f act ori ncl i nat i on of t he pr opel l er axi s t o t he f usel agecent er l i ne degr ees

    x i i

  • 7/22/2019 19760004918_1976004918

    14/209

    J1434 7nQqs ,RRerrSTu

    VVa

    Vn

    voVS

    propeller advance ratio, Vo/ nDwing section lift, per unit span, Nfreesteam Mach number, V d a slocal M a c h number for propeller blade element, V/asrotational speed, rev/secpropeller shaft torque, Nom.average slipstream dynamic pressure, N/m2propeller tip radius, D/2, mReynolds numberlocal radius in propeller disk plane, mlocal radius in slipstream, mpropeller thrust, Naxial component of velocity induced by a blade elementin the propeller disk plane, m/sec.local velocity, m/sec.component of freestream velocity along the propelleraxis, m/sec.component of local slipstream velocity normal to thezero-lift line, m/sec.freestream velocity, m/sec.component of local slipstream velocity parallel to thezero-lift line, m/sec.axial component of local velocity in the fully-developed

    I slipstream, m/sec.

    X i i P

  • 7/22/2019 19760004918_1976004918

    15/209

    VSt

    vW

    YY*yPQ

    at3

    P

    momentum-we&gh,ted. mean. . , a x i a l -. >veloci$y in , , the fu l ly- : .deve lopeds l ips . t ream, m/sec.t a n g e n t i a l c o m p o n e n t of locai . .veloci . ty in , t he f u l l y -d e v e l o p e ds l i p s t r e a m , m/sec.

    I

    j - . ._ . . . , . . :. . - . ... . _ . . :._ ~ I . . .. . . ~. I . .,. . . 4

    . . ,, . .. .% . . ~ , . ~ I . ; I .. upwash v e lo c i t y component ac t i n g i n t h e p r o p e l l e r d i s k

    p l a n e . .ue t o , h e . ,. . .p re se n c e , . o f . : a l i f t i n g . w in g ,/se,c,.j;

    sp a n wiseco-o rd ina teo f o c a lw in ge le m e n t , mspanwise o -o rd ina te flap end, ,m . .spanwise co -o rd ian te of p r o p e l l e r . a x i s , . m,,a n g l e o f a t t a c k r e l a t i v e . to. a i r f o i l s e c t i o n chord-l i n e ,d e g r e e sa n g l e o f a t t a c k r e 1 a t i v e t o u s e l a g ec e n t e r l i n e , d e g r e e s . I . . . .

    , ._. . .

    co r r ec t ed s e c t i o n a n g l e o f a t t a c k , , deg-reese f f e c t i v e a n g l e o f a t t a c k ofwing s e c t b n , d e g r e e s

    . .I . .. I : . l L ,. . , _ . .

    g e o m e t r i ca n g l eo f a t t a c k of w.ing se c t io n , de gr ee si n d u c e d a n g l e o f a t t a c k .ofw in g s e c t io n , d e g re e sa n g l e of a t t a c k o f a i r f o i l s e c t i o n a t z e r o lift , ;d e g re e ss e c t i o na n g l e 0.f a t t a c k for wo-dim ensional a i r - ~f o i l , d e g r e e s . . . . .

    . .. . . . . 2

    p r o p e l l e r a x i s a n g l e o f , a t t a c k , d e g r e e s.

    xiv

  • 7/22/2019 19760004918_1976004918

    16/209

    g e o m e t r i ca n g l e of a t t a c ko f . w i n g o o t ,d e g r e e s -.i n c l i n a t i o n o f t h e s l i p s t r eam a x i s to t h e f l i g h tpa th , d e g r e e sp i t c h a n g l e f o r p r o p e l l e r b l a d e e l e m e n t , d e g r e e sm u l t i p l i e r f o r n d u c e d a n g l e o f a t t a c k , d e g r e e sm a g n i t u d e o fd i s c o n t i n u i t y na b s o l u t ea n d n d u c e da n g l e s of a t t a c k , d e g r e e sg e o m e t r i c t w i s t a t a n ywin gse c t io n ,d e g re e sc i r c u l a t i o n a b o u t a n y w i n g s e c t i o n , m2/sec.wing t a p e r r a t i ob l a d e s p e e d r a t i o f o r p r o p e l l e r b l a d e e lemen tblade t i p speed r a t i ok i n e m a t i c v i s c o s i t y , m2/sec.i n f l o w a n g l e f o rp r o p e l l e rb l a d e e l e m e n t , d e g r e e si n f l o w a n g l e f o r p r o p e l l e r b la de e l e m e n t ,e x c lu d in gc o n t r i b u t i o n f r om n d uc e dv e lo c i ty n d i s k p l a n e ,d e g r e e sa m b ie n t d e n s i ty , kg/m3s o l i d i t y f o r p r o p e l l e r b l a d e elementwing panwisecoord ina te , COS"( 2 y / b )sp a n wisec o -o rd in a t e o rr o t a t i o n a l p e e d , a na n g u l a rv e l o c i t y n d u c e d

    *f l a p e n d , os" ( y l b )r a d i a n s / s e c .by a b ladee lemen tbeh ind

    the p r o p e l l e r d i s k p l a n e , r a d i a n s / s e c .f a c t o r t o a c c o u n t f o r low a s p e c t r a t i o e f f e c t s

    xv

  • 7/22/2019 19760004918_1976004918

    17/209

    P R E D I C T I O N OF SPAN L O A D I N G O FS T R A I G H T - W I N G / P R O P E L L E RC O M B I N A T I O N SUP To STALL

    M. A. McVeigh, L. Gray,and E. Kis i e lo wsk iU N I T E D T EC HN OL OG Y , I N C .

    S E C T I O N 1I N T R O D U C T I O N

    The p r o p e l l e r s l i p s t r e a m e x e r t s a n m p o r t a n t n f l u e n c eo n w i n g o a d d i s t r i b u t i o n , w h i c h n u r n a f f e c t s h e . a i r c r a f ts t a l l c h a r a c t e r i s t i c s .T h i se f f e c t i s in t roduced hrough a ni n c r e a se n o c a lv e l o c i t yo v e r h es l i p s t r e a m - i m m e r s e dp o r t i o n o f t h e wingand a changeofwing lo c a l a n g l e o f a t t a c kdue tos l i p s t r e a m o t a t i o n . While t h e n c r e a s e dv e l o c i t yt e n d s t o s t a b i l i z e the f lowover t h a t wing p o r t i o n , h es l i p s t r e a m r o t a t i o n may g i v e r i s e t o a n a s y m m e t r i c stallcond i t ion d u e t o i n c r e a s e d l o c a l a n g l e s o f a t t a c k of t he wings e c t i o n s b e h i n d t h e up-going p r o p e l l e r b l a d e s , a n d reduceda n g l e s of a t ta c k o f t h e wing sec t io ns beh ind the down-goingb l a d e s .

    A r e v i e w o f h e a v a i l a b l e e c h n i c a l l i t e r a t u r e i n d i c a t e st h a t there a r e no r e l i a b l e h e o r e t i c a l o r s e m i - e m p i r i c a lmethodswhichcanadequate lypredic t t h e e f f e c t s of a p r o p e l l e rs l i p s t r e a m on t h e s p a n w i s e l o a d d i s t r i b u t i o n o f a n e n t i r e w i n g .Many o f the e x i s t i n g m ethods a r e su i t a b l e o n ly fo r com putingt o t a l wing fo rces s i n c e t h e y a r e o f t e n base d on g ros ss i m p l i -fyingassumptions.Thus, orexample ,anassumption t h a t t h es l i p s t r e a m - im m e rse dpor t ions of the wingcan be t r e a t e d a si s o l a t e dp l a n f o r m sn e g l e c t s h es t r o n g n f l u e n c eo f t hes l i p s t r e a mo na d ja c e n t wing r e g io n s .O t h e r h e o r e t i c a lm e t h o d sa r e g e n e r a l l y c l a s sed a s r igorousmathemat ica lapproaches w h i c ha r e u s u a l l y v e r y c o m p l e x a n d a r e f r e q u e n t l y n o t n s u f f i ci e n tag reementwi thexper imenta l d a t a t o w a r r a n t t h e i r use a s adesign ool .Fur the rmore ,mos to f the a b o v e h e o r i e su s el i n e a r l i f t c u r v e s and a s a r e s u l t c an n ot be expected t o y i e l ds a t i s f a c t or y g r e e m e n t w i t h t e s t data n e a r wing s t a l l .

  • 7/22/2019 19760004918_1976004918

    18/209

    The l i m - i ta t i on imposedby the u s e of l i n e a r l i f t c u r v e sf d r the wing has b e e nsucc ess f u l ly removed i n t h e work repor tedi n Re fe re nc e 1. This r e f e r e n c ep r e s e n t s a computerized methodf o r p r e d i c t i n gs p a n w i s e o a d d i s t r i b u t i o n s of s t ra igh t -wing /fuse lagec o m b in a t io n s a t a n g l e s of a t t a c k u p t o stall. Thismethod, which i s based on t h e Pr a n d t l w ing o r " l i f t i n g l i n e 't h e o r y a s fo rm u la t e d by S i v e l l s nR e f e r e n c e 2, p r o v i d e sa r e l i a b l e a n a l y t i c a l tool fo r p r e d i c t i n g w i n g s t a l l i n gc h a r a c t e r i s t i c s of g e n e r a l a v i a t i o n y p e a i r c r a f t , b u t i s o n lya p p l i c a b l e t o pow er-off f l i g h t c o n d i t i o n s , s u c h a s might beencoun te reddur ing and ing .

    The c u r r e n t n v e s t i g a t i o n e x t e n d s t h e a n a l y s i s o fReference 1 to p e r m i t c a l c u l a t i o n s of s p a n o a d i n g a n ds t a l l i n gc h a r a c t e r i s t i c s under power-on co nd it io ns e . g . t ake -o f f ) f o rwings w i t h or wi tho u t f l a ps an d hav ing any . number o f non-o v e r l a p p i n gp r o p e l l e r s . The present method i s basedon employ-i n g n o n - l i n e a r a i r f o i l s e c t i o n c h a r a c te r i s t i c s f o r b o t h t h ep r o p e l l e ra n d the wing. The b a s i ca n a l y t i c a la p p r o a c h of th i smethod i s t o r e t a i n t h e i n h e r e n t s i m p l i c i t y o f the Prand t l wingtheory ,modi fy the t h e o r y a s r e q u i r e d oaccept non-uniform Is l i p s t r e a m v e l o c i t i e s , a n d e f f e c t i v e l y combine t h i s modif iedl i f t i n g l i n e t h e o r y w i t h a r e a l i s t i c p r o p e l l e r t h e o r y t o forma u n i f i e d a n a l y t i c a l t o o l .

    A d e t a i l e d d e s c r i p t i o n of t h i s a n a l y t i c a l m e t h o d ,t o g e t h e r w i t h t he spec ia l lydeve lopedd ig i ta lc o m p u te rp ro g ra mi s p r e s e n t e d i n t h e f o l l o w i n gp a g e s .

  • 7/22/2019 19760004918_1976004918

    19/209

    SECTION 2GENERAL REVIEW OF THE ANALYTICAL METHODS

    The prime o b j e c t i v e of t he cur ren t d e v e lo p m e n t i s t op r o v i d e a .p r a c t i c a l a n a l y t i c a l s o l u t i o n f o r d e t e r m i n i n g thel i f t d i s t r i b u t i o n a n d s t a l l i n g c h a r a c t e r i s t i c s ofwingsp a r t i a l l y o r t o t a l l y immersed in a p r o p e l l e r s l i p s t r e a m .I:n order t o dep ic t some of the h i g h l i g h t s o f t h e c u r r e n tw o r k r e l a t i v e t o o t h e r a p p r o a c h e s , t h i s s e c t i o n p r e s e n t s ab r i e f r e v i e w of the e x i s t i n g a n a l y t i c a l a n d e x p e r i m e n t a li n v e s t i g a t i o n s t h a t a t t e m p t s o l u t i o n s of the wing /p rope l le rproblem2.1 STATEMENT OF THE PROBLEM

    The ba s i c l i m i t a t i o n i n p r o v i d i n g r e l i a b l e s o l u t i o n s t ot he wing/propel le r p rob lem i s r e l a t e ' d t o a l a c k ofcompleteunders tand in g o f the f l o w f i e l d ge ne ra te d ''by the win g /p ro p e l l e ri n t e r a c t i o nu n d e rp r a c t i c a lo p e r a t i n gc o n d i t i o n s . The problemi s fur ther compounded by t he d i f f i c u l t y ok developing r ea l i s t i ca n a l y t i c a l r e p r e s e n t a t i o n s o f t h i s complex low f i e l d environ-ment so a s t o a c c o u n t f o r t h e m a j o r i n t e r a c t i o n e f f e c t s a c t i n g.on a wing/propel le rcombinat ion . A c o m p l e t eso lu t io n t o theproblem m u s t t h e r e f o r e a c c o u n t f o r a l l th ese e f f e c t s , which a sa minimum sh ou ld nc lu de t h e f o l l o w i n g

    ( a ) Local wing angle-of-attackchangesdue t o t h e meani n c l i n a t i o n a n d r o t a t i o n of t h e s l i p s t r e a m f l o w .( b ) Non-uniform s ,pnwised i s t r i b u t i o no fv e l o c i t yo v e r

    those p o r t i o n s of t h e wing w i t h i n t h e s l i p s t r e a m .( c ) Non-uniform ver t ica ld i s t r i b u t i o n of v e l o c i t y w i t h -i n t h e s l ips t ream- immersedregionsof t h e wing.( d ) Viscousmixingbetween t h e s l i p s t r e a ma n d freestreamf l o w a long t h e s l i p s t r ea m b o u n d a r y .I n viewof t he r e a l f l u i d f l o w e f f e c t s involved it i su n l i k e l y t h a t e v e r y a s p e c t of the problemcan be t r e a t e da d e q u a t e l y ,u s i n g t h e e s t ab l i sh ed a n a l y t i c a la p p r o a c h e s .H i s t o r i c a l l y , the approach has been t o i n t r o d u c e a se r i e s o fs i m p l i f y i n ga s s u m p t i o n s n o r d e r t o a r r i v e a t a s o l u t i o n .These approaches a re discussed be low.

    3

  • 7/22/2019 19760004918_1976004918

    20/209

  • 7/22/2019 19760004918_1976004918

    21/209

  • 7/22/2019 19760004918_1976004918

    22/209

    approaches i s s u i t a b l e f o r d i r e c t a p p l i c a t i o n t o the p r e s e n tp r o b l e m o f p r e d i c t i n g t h e e f f e c t s of p r o p e l l e r s l i p s t r e a mon t h e s t a l l c h a r a c t e r i s t i c s of s t r a i g h t wing a i r p l a n e s .E i t h e r t he e x i s t i n g t h e o r e t i c a l models a r e t o o s i m p l i f i e d a n dd i s r e g a r d e f f e c t s which a r e known t o be c r i t i c a l , (e .g .Refe rence 6 and 13), o r t h e a n a l y s e s a r e toocomplexanddon o t y i e l d p r a c t i c a l and r e l i a b l e s o l u t i o n s( e . g .R e f e r e n c e 9 ) .T h e r e f o r e , t h e r e e x i s t s a r e q u i r e m e n t t o developan mprovedmathematical model capable of p r o v i d i n g p r a c t i c a l and r e l i a b l ea n a l y t i c a l s o l u t i o n s t o t h e wing /p rope l le rp rob lem.

    "he a n a l y t i c a lm e t h o d sd e v e l o p e du n d e r the c u r r e n tp r o g r a mp o t e n t i a l l y e , p r e s e n ta na n s w e r t o t h i s problem.Althought h i s optimism i s based on a f e w i s o l a t e d c o r r e l a t i o n s w i t h thea v a i l a b l e t e s t d a t a , s u f f i c i e n t n d i c a t i o n o f the e f f e c t i v e n e s so f t h e d e v e lo p e dm e th o d o lo g yh a sa l r e a d yb e e no b ta in e d , a sc o nf irm e d by c o m p a r a t iv e r e s u l t sp r e s e n t e d l a t e r i n t h e t e x t .The b a s i s f o r t h i s improved mathematical model i s d e s c r i b e dbe low.-.3 BAS1 S FOR T H E PRESENT A N A L Y S I SA commm approach of p a s t i n v e s t i g a t i o n s i n v o l v e s a ni d e a l i z e d r e p r e s e n t a t i o n of t h e p r o p e l l e r s l i p s t r e a m nwhich the v e l o c i t y i s d i s c o n t i n u o u sa c r o s s t h e s l i p s t r e a mboundary. This m odel g e n e r a l l y e q u i r e sc o m p l e x o l u t i o n s o

    t h e b o u n d a r yc o n d i t i o n sa s s o c i a t e dw i t h t h e d i s c o n t i n u i t y ,The b a s i s o f the c u r r e n t a n a l y s i s l i e s i n h e o b s e r v a t i o nt h a t i n a r e a l s l i p s t r e a m t he v e l o c i t y d i s t r i b u t i o n r e m a i n sc o n t in u o u s h ro u g h o u t t h e s l i p s t r e a mb o u n d a ry .An ex am in at io nofexper imea ta l da ta ob ta ined on t h ev e l o c i t y d i s t r i b u t i o n s n t h e w a k e so fpropel le rs shows t h a t

    t he r e i s no sudd en jump i n v e l o c i t y a c r o s s t he s l i p s t r e a mboundary. There i s , however, a r a p i d i n c r e a s e nv e l o c i t y a sthe boundary i s c r o s s e d b u t t h e c o f i t i n u i t yo fv e l o c i t y i ss t i l l p re se rv e d .S in c e the l i f t d i s t r i b u t i o nm u s t be c o n t in -uousand the v e l o c i t y d i s t r i b u t i o n i s c o n t i n u o u s , . h e n thea s s o c i a t e d c i r c u l a t i o n d i s t r i b u t i o n m u s t a l s o be con t inuous .Th e re fo re , the s t r e n g t h of t h e s h e dvor t ic i ty may be o b t a i n e dby d i f f e r e n t i a t i n g t h e s p a n w i s ed i s t r i b u t i o n o fc i r c u l a t i o ni n t h e u su a lm a n n e r ,w i th o u t h ec o m p l i c a t io no fa c c o u n t in gf o r d i s c o n t i n u i t i e s n c i r c u l a t i o n .

    6

  • 7/22/2019 19760004918_1976004918

    23/209

  • 7/22/2019 19760004918_1976004918

    24/209

    SECTION 3THEORETICALANALYSIS

    This s e c t i o n p r e s e n t s a summary of t h e a n a l y t i c a l methodsdeveloped f o r , p r e d i c t i n g t h e p r o p e l l e r s l i p s t r eam e f f e c t s on.t h e s p a n w i s e o a d d i s t r i b u t i o n of win g so p e ra t in g a t a n g l e so fa t t a c ku p t o s t a l l . The analy t ica la , p p r o a c h p r e s e n t e dh e r e i n i s based upon f i r s t d e te rm in in g t h e v e l o c i t y d i s t r i b u t i o ni n t h e p r o p e l l e r wake a n d h e n c a l c u l a t i n g i t s e f f e c t on t h ew i n g l i f t d i s t r i b u t i o n . The a n a l y s i s p r o v i d e s f o r the u s e ofn o n - l in e a r l i f t c u r v e s f o r b o t h t h e p r o p e l l e r a n d t h e winq i no r d e r o r e a l i s t i c a l l y r e p r e s e n t h e p r o p e l l e r s l i p s t r e a md i s t r i b u t i o n a n d i t s e f f e c t o nwing oading a t a n g l e s of a t t a c ku,p t o s t a l l .

    Accord ing ly , he f i r s t p a r t o f t h i s s e c t i o n d e a l s w i t ht h ep r o p e l l e rS l i p s t r e a m c a l c u l a t i o n s ,a n d h e s e c o n d ,part pre -s e n t s t h e implementa t ionof t h e s l i p s t r e a m p a r a m e t e r s n themo dif ie d wing th eo ry .3 , 1 PROPELLER S L I P S m A M ANALYSIS

    The f i r s t p a r t o f t h e a n a l y s i s d e a l s w i t h t h e p r o p e l l e rs l i p s t r e a m e p r e s e n t a t i o n , n c l u d i n g t h e r e q u i r e d t e r a t i v eso lu t io na n dc o n v e rg e n c ep ro c e d u re s .3 1.1 G e n e r a l P r o p e l l e r. - S o l u t i o n

    Cons ide r a p r o p e l l e r o p e r a t i n g a t a n a n g l e of a t t a c k aPt o t h e remote f r e e s t r e a m o fv e l o c i t y Vo , a s shown inF i g u r e 1.The presenceof a l i f t i n g w ingbehind t h e p r o p e l l e r m o d i f i e st h e i n f l o w t o t h e p r o p e l l e r d i s k t h ro u g han nd uc ed upwashv e l o c i t y V . An a n p p ro x im a t io n t h i s upwash ve loc i ty i sassumed t o be uniform a c r o s s t h e p r o p e l l e r d i s k , a n d t o l i e w i t h h .t h e d i s kp l a n e . The methodused fo rc a l c u l a t i n g t h i s upwashv e l o c i t y i s , p r e s e n t e d nS e c t i o n 4.2.

    For t h e purpose of a n a l y z i n g t he wing l i f t d i s t r i b u t i o nit i s assumed t h a t t h e s l i p s t r e a m c a n be c o n s id e re d a s b e i n gf u l l yd e v e l o p e d . W i t h t h i s assumpt ion the a v e r a g e n c l i n a t i o no f t h e c o n t r a c t e ds l i p s t r e a mc a n be r e a d i l y c a l c u l a t e d u s i n g

    8

  • 7/22/2019 19760004918_1976004918

    25/209

  • 7/22/2019 19760004918_1976004918

    26/209

    s i m p l ea c t u a t o r d i s k t h e o ry e . 9 ,R e fe r e n c e 1 9 ) a n d , n then o t a t i o no fF i g u r e 1, i s obta ined f rom

    where u i s t h e a x i a l n d u c e dv e lo c it y n c re m e nt a t t h ep r o p e l l e r d i s k . Frommomentum theory, u i s r e l a t e d op r o p e l l e r h r u s t , T by

    where VI i s t he r e s u l t a n t v e l o c i t y a t t h e d i s k and i s givenby

    On co m bi ni ngequa t ions ( 2 ) and ( 3 ) a q u a r t i c n u i sobta inedand i s g e n e r a l l y o l v e d by i t e r a t i o n . However, s i n c ethe p r e s e n t a p p l i c a t i o n i s t o c o n v e n t i o n a l a i r c r a f t where (I pis s m a l l a n d V 5 Vo t h i s q u a r t i c may be r e d u c e d o aq u a d r a t i c whose s o l u t i o n i s

    With the abovevalueof u , e q u a t i o n (1) can be s o l v e dt o y i e l d the mean s l i p s t r e a m n c l i n a t i o n a s , r e l a t i v e othe f rees t ream.

    To o b t a i n the d e t a i l e d v e l o c i t y d i s t r i b u t i o n w i t h i nt he i n c l i n e d s l i p s t r e a m it i s assum ed t h a t , t o a goodapprox ima t ion , t h i s can be o b t a i n e d d i r e c t l y from t h e s o l u t i o nf o r a n s o l a t e d p r o p e l l e r o p e r a t i n g n a x i a l f l o w a t speed

    The c a l c u l a t i o n o f n o n -u n i fo rms l i p s t r e a mv e l o c i t y

    LO

  • 7/22/2019 19760004918_1976004918

    27/209

  • 7/22/2019 19760004918_1976004918

    28/209

    F i g u r e 2. Blade Eleme nt Velocity Diagram

    12

  • 7/22/2019 19760004918_1976004918

    29/209

    To a c c o u n t f o r t h e s i g n i f i c a n t a n d w e l l known loss ofl i f t toward the b l a d e i p e q u a t i o n s ( 6 ) and ( 7 ) a r e r e w r i t t e ni n the f o r m

    i

    and

    where F i s t h e t i p - l o s sc o r r e c t i o n a c t o rg i v e n b y Lock i n .Refe rence 2 1 and u*, 1/2(w*r) a r e m o d i f i e d v a l u e s o f t h e inducedve loc i ty componen ts .E q u a t i o n s ( 1 0 ) and 11)yield improved valueso fthe i n f l o w n g l e C I a n d e c t i o n charac te r i s t ic s C$ andC d , a s a f f e c t e d b y the t i p - l o s s o r r e c t i o n a c to r F .These v a l u e s a re then used t o o b t a i n ' a b e t t e r approx ima t ionfo r s l i p s t r ea m- i n d u c e d e l o c i t y o mp o n e n t s u and w r / 2 ,u s i n ge q u a t i o n s ( 6 ) and ( 7 ) .

    3.1.2 I n i t i a lC a l c u l a t i o no f n f l o w AnqleThe i t e r a t i v e s o l u t i o n f o r t h e system of e q u a t i o n s ( 8 )t h ro u g h (11)

    be o b t a i n e d .t o t a l t h r u s tis g e n e r a l l ys a t i s f a c t o r y

    r e q u i r e s t ha t a n i n i t i a l a p p r o x i m a t e v a l u e of 4E q u a t i o n ( 4 ) can be used i f the p r o p e l l e ri s known. However, s i n c e the p r o p e l l e r t h r u s tn o t known in a d v a n c e , a method t h a t y i e l d s as t a r t i n gv a l u e f o r $ i s developed a s fo l lows:

    From Figure 2 the i n f l o wa n g l e C I can be expressed a s+ = U*where the r e s u l t a n t n d u c e dv e l o c i t y n c r e m e n t i s assumedto e n orma l t o the l o c a l b l a d e v e l o c i t y .

    On making the as sumpt ion t h a t c d

  • 7/22/2019 19760004918_1976004918

    30/209

    By s o l v i n g q u a t i o n (13) f o r u * / a r a n d u b s t i t u t i n gt h i s v a l u e n q u a t i o n (12) a n n i t i a lv a l u e o r + i sobtained. However, the r i g h t hand s i d e of e q u a t i o n 13)must f i r s t be r e d u c e d o a t r ac t ab l e form. This i s accom-p l i s h e d byapp ly ing the f o l l o w i n g r e l a t i o n s h i p s :

    ( a ) A l i n e a r i z e de x p r e s s i o n o r the b l a d e s e c t i o nl i f t curv e , g ive n by

    cL~ = oo ( Q -Q.) (14)where a 0 i s a r e p r e s e n t a t i v e l i f t s l o p e

    Q i s given by equa t i on ( 9 ) , andQ~ i s the a n g l eo f a t t ack a t z e r o l i f t .

    ( b ) An e x p r e s s i o n o r V, o b t a i n e d ro mFi g u re 2 a s ,

    v = J v c + ( s l r ) 2( c ) P r a n d t l ' se x p r e s s i o n o r the t i p l o s s f a c t o r Fobta inedf ro mRefe rence 2 1 a s P

    Fp = o s ' [exp{-+ ( , - + ) J q y } ]where B i s the number of b lades

    Combining equations ( 1 2 ) th rough ( 1 5 ) and ub-s t i t u t i n g f o r the t i p loss f a c t o r , Fp, g i v e n by e q u a t i o n (16)l eads t o the f o l l o w i n ge x p r e s s i o n :

    from which the s o l a t i o nf o r */a,s o b t a i n e d as14

  • 7/22/2019 19760004918_1976004918

    31/209

    * VQ (18)nr =12 [ . ( ~ + x ) 2 + . . ( p - + o - . o ) - ( z . x ) ]where

    I

    3.1.3 Convergence of t h e I t e r a t i v eP r o p e l l e rS o l u t i o nT h e i t e r a t i v e s o l u t i o n o e q u a t i o n s ( 8 ) th rough 11)i s n a t u r a l l y d i v e r g e n t w i t h i n t he normal- rangeof t h eb l a d e e c t i o n l i f t curves .There fo re , onve rgence of thes o l u t i o n m u s t b e f o r c e d by a p p l y i n g a c o r r e c t i o n t o eachnew computed value of #I . A c o m e c t i o np r o c e d u r e whichy i e l d sr a p i dc o n v e r g e n c e i s d e r i v e d by the methodpresen tedbelowL e t t he e x a c t o l u t i o n o r n f l o w n g l e , $I , beexpressed a s

    where i s the v a l u es e d a s i n p u t o the n t h i t e r a t i o n n dS I i s a sm al l unknown in crem ent.I n the g e n e r a lt e r a t i o nr o c e d u r e , i s f i r s t usedi n q u a t i o n ( 9 ) t oo b t a i n a v a l u e of Q from which C$ andCd may be determinedknowing the b l a d e a i r f o i l s e c t i o n char-a c t e r i s t i c s . Next, e q u a t i o n s (1 0 ) and 11)are u s e d o o l v ef o r u * and u* and these v a l u e s a r e t h e n u b s t i t u t e d ne q u a t i o n ( 8 ) t o o b t a i n a new value of i n f l o wa n g l e ,d e n o t e dby I 1 . It i s t h is new value of i n f l o wa n g l e w h i c h must bec o r r e c t e db e f o r ep r o c e e d i n g o t he n + l t h i t e r a t i o n .m e r e f o r e , l e t the exact s o l u t i o n f o r i n f l o wa n g l e , $Ia l s o b e e x p r e s s e d a s

    15

  • 7/22/2019 19760004918_1976004918

    32/209

    where 82 i s a second smal l unknown increment.Combining equat ions (19) a n d 2 0 ) , oe l i m i n a t e + ,y i e l d s

    S u b s t i t u t i n ge q u a t i o n ( 2 1 ) i n t oe q u a t i o n (19), here f o l l o w s :

    Equa t ion ( 2 2 ) forms t h e bas i s o f a method f o r c a l c u l a t i n ga n m p r o v e d v a l u eo f n f l o w a n g l e f o r n p u t o the n e x ti t e r a t i o n c y c l e by u si n g t he g u e s s e da n dc a l c u l a t e dv a l u e sf r o mt he p r e v i o u s y c l e . The r a t i o , 82/81, r e ma i n s ob ed e t e rmi n e df r o ma n a p p r o x i m a t e e r r o ra n a l y s i s n the fo l l o w i n g manner:

    From equat ion ( 2 0 ) t he v a l u e of t a n 4 i s expressedt o f i r s t o r d e r n S2 by

    From equat ions ( 9 ) a n d ,419) the exac t s o l u t i o n f o rblade l i f t c o e f f i c i e n t , - Cd , i s e x p r e s s e d n terms o f thev a l u e C& c a l c u l a t e d n the n t h i t e r a t i o n y c l e from

    where a, i s a mean value of l i f t - c u r v es l o p e .Equa t ion (10) w r i t t e n i n terms of v a l u e s f o r t he exac t

    16

  • 7/22/2019 19760004918_1976004918

    33/209

  • 7/22/2019 19760004918_1976004918

    34/209

    where p i s the l o c a l f o r w a r dp e e da t i o ( V a / n r ) .. .R e w r i t i n ge q u a t i o n ( 2 9 ) i n terms of t h e v a l u e s k,,kyI Ic a l c u l a t e dn t h e n t h i t e r a t i o n y c l e i e l d s

    I

    tan ' I = p I t k + k yCombin ingequat ions ( 2 9 ) a n d 3 0 ) e a d s o t h e f o l l o w i n gr e l a t i o n s h i p

    F i n a l l yu s i n ge q u a t i o n s 2 3 ) , ( 2 6 ) ( 2 8 )a n d (31) asolution f o r t h e r a t i o 8,/8, i s o b t a i n e d a s f o l l o w s :

    E qua t i on ( 3 2 ) t h u sp r o v i d e s t h e e s s e n t i a l e l a t i o n s h i pby which e q u a t i o n ( 2 2 ) i s a p p l i e d oo b t a i na n m p r o v e dv a l u eof i n f l o wa n g l e o r n p u t o t he n e x t t e r a t i o nc y c l e . np r a c t i c e , h e t e r a t i o n p r o c e d u r e i s t e rmina ted when thed i f f e r e n c e . (r$'-#) for e a c h s u c c e s s i v e i t e r a t i o n c y c l e h a sc o n v e r g e d o w i t h i n a p r e s c r i b e d m a r g i n o f e r r o r .3.1.4 Analysis f o rS l i p s t r e a mV e l o c i t yD i s t r i b u t i o n s

    ~ ~ ~~ ~-

    Upon reaching a c o n v e r g e ds o l u t i o nf o r t he i n f l o wa n g l e# , t h e f i n a l a l u e s o f (#I , CIQ and cd a r e then sub-s t i t u t e d ne q u a t i o n ( 6 ) and ( 7 ) t o 'solve f o r h e r u e n -duced ve loc i tyc o m p o n e n t s n t h e p r o p e l l e r d i s k plane,^and 1/2 w r .The l o c a l a x i a lve l oc i t y c om pone n t VSa i n t h e f u l l y

    d e v e l o p e ds l i p s t r e a m i s o b t a i n e d r o mF i g u r e 1 a svo COS ap + 2uV s a =

    18

    ( 3 3 )

  • 7/22/2019 19760004918_1976004918

    35/209

    The l o c a l ro t a t i o n a l v e l o c i ty component, V s t , i n thefu l ly d e v e l o p e d s l ips t ream i s o b ta in e df ro mc o n se rv a t io no fangular momentum and i s g iven by

    v s t = w r (+-) ( 3 4 )where r s i s the l o c a l r a d i u s n t h e s l i p s t r e a m o r thes t r e a m tu b e l e m e n t which has a l oca l r a d i u s r i n t hep r o p e l l e r d i s k p l a n e .

    The l o c a l r a d i u s rs f o r each f l o we le m e n t n thes l i p s t r e a m i s d e r i v e d from a s i m p l i f i e d a p p l i c a t i o n o f t h ec o n t i n u i t ye x p r e s s i o n os u c c e s s i v es t r e a m t u b ee l e m e n t s .For the n t h b l a d e e l e m e n t t a t i o n a t r a d i u s r n the corres-p o n d i n ga d i u s r s n i n t h e s l i p s t r e a m i s given by

    where, i n t h e no ta t ionfF ig u re 3 , r l and r ~ l re thev a l u e s o f h u b a n d n a c e l l e r a d i u s , r e s , p c t i v e l y ., The v a l u e of rsn g iven by eq ua t i on (35) i s basedu p o n r e p re se n t in g t h e s l i p s t r e a m bya s e r i e s of c o n c e n t r i ca n n u la rs t r e a m tu b e s w i t h un i fo rmveloci tybe tween each e lemen ts t a t i o n . This s o l u t i o n , while approx imate , i s found t o be

    more thana d e q u a t efo r a l l r e a s o n a b l ev a r i a t i o n sb e t w e e n u r land Un .

    1 9

  • 7/22/2019 19760004918_1976004918

    36/209

    r-tkE

    R l

    kE

    \

    ~-

    Propeller and Nacelle Axis

    Figure 3 . Analytical Model For Slipstream Contraction.

  • 7/22/2019 19760004918_1976004918

    37/209

    -.2 . ING-IN-SLIPSTREAM- ANALYSISI n t h e s e c o n d p a r t of t h e a n a l y s i s a m o d i f i e d form ofl i f t i n g l i n e t h e o r y i s p r e s e n t e d wh i c h u s e s the nonuniforms l i p s t r e a m v e l o c i t y d i s t r i b u t i o n ,a s d e t e r m i n e d above, t oc a l c u l a t e t h e l i f t d i s t r i b u t i o n on wings w i t h p r o p e l l e r s .

    The approadh p r e s e n t e d r e l i e s on t h e u s e of a s implephys ica l mode l to o b t a i n a s o l u t i o n f o r wing- in -s l ips t reaml o a d i n g s f o r a widerange of r e a l a i r c r a f t p r o p e l l e r - w i n gcombinations.

    F i r s t t he method i s ,p resented for anunf lappedwingimmersed ino n e o r more non-overlapping sl ipstreams. Follow-in g t h i s , t h e m o d i f i c a t i o n sr e q u i r e d t o i n c l u d e f l a p s a r ed e s c r i b e d .F i n a l l y ,a ne x t e n s i o n of the a n a l y s i s t o i n c l u d el o w a sp e c t - r a t i op ro p e l l e r -w in gc o m b in a t io n s i s d i s c u s s e d .3.2.1 A n a l y s i s o r a Winq w i t h no Flapso r w i t h Ful l -SpanDef lec tedFlaps

    Cons ide r t h e b a s i c case of a wing i n a uniform streamof v e l o c i t y , v o . If t h e l o c a l i r c u l a t i o n i s ro , t h e n t h es p a n o a d d i s t r i b u t i o n a t a n ys p a n w i s es t a t i o n i s given by

    90 = P vo roTh e su p e rp o s i t i o n of a p r o p e l l e r s l i p s t r e a m f l o w g i v e s r i s et oa n n c r e a s e d o c a lv e l o c i t y vd a n da n n c r e a s e dc i r c u l a t i o nr; , which canbe expressed, r e s p e c t i v e l y , a s

    v i = v o -t n v (37)

    where n V and Ar a r e t h e i n c r e m e n ta lc h a n g e s n l oca lv e l o c i t ya n dc i r c u l a t i o n ,r e s p e c t i v e l y ,d u e t o p r o p e l l e r s l i p -s t r e w . Now t h e c o r r e s p o n d i n g p a n w i s e o a dd i s t r i b u t i o n f o rt h e b a s i c wing mmersed i n the p r o p e l l e r s l i p s t r e a m can bew r i t t e n a s

    1 = v, rsl 39)21

  • 7/22/2019 19760004918_1976004918

    38/209

  • 7/22/2019 19760004918_1976004918

    39/209

    Using t h e n o m e n c la tu r eo fF ig u re 4 , t h i s v e l o c i t y componentcan be expressed a s

    A l s o , t he corr espo ndin g component of he o t a l f l o wn o r m a lto the w i n g s e c t i o n z e r o - l i f t i n e i s given by

    IThe q u a n t i t i t e s "ha and V s t i n t h e a b o v e q u a t i o n s e p r e s e n tthe a x i a l and s w i r k v e l o c i t yc o m p o n e n t sof he combined f re es t rea na n d 's l i p s t r e a mf l o ba n d a r e g iven by e q u a t i o n s( 3 3 ) a n d ( 3 4 )r e s p e c t i v e l y . AlsD, t h ea n g l e s a s a n d Q e a r e known quan t i t i e swhich r e p r e s e n t i n c l i n a t i o n s o f t h e s l i p s t r e a m a n d t h e z e r o - l i f tl i n e r e l a t i v e t o t h e r e m o t ef r e e s t r e a mv e l o c i t y ,r e s p e c t i v e l y .

    The e x t r a , i n g i r c u l a t i o n r2 , caused by t he a c t i o no f t he p ro p e l l e rs , l i p s t r e a m , i s de te rmined by e q u a t i n g h ere su l t i n gc h a n g e i p wing u,pwash t o the downwash change a s s o c i a t e dw i t h . This upwash change,non-dimensionalorm, i sd e f i n e d a s

    S u b s t i t u t i n ge q u a t i o n 4 3 ) n t oe q u a t i o n 4 4 )y i e l d s t h eex t ra dpwashdue to t h e s l i p s t r e a m a s

    v = - s in ( a s + Qe) +- O S s t Qe) -sin Q e (4 5 )VOI n o r d e r t o s a t i s f y t h e wingboundarycondit ionof no

    f l o w t h ro u g h t h e s u r f a c e , this extra upwash orc ro s s f lo wm u s tbebalanced by the combined inf luence o f the extra boundv o r t i c i t y , l-2 , and t h e a s s o c i a t e d streamwise ( i . e . chordwiseand t r a i l i n g ) v o r t i c i t y , - d r2d Y dY.

    23

  • 7/22/2019 19760004918_1976004918

    40/209

  • 7/22/2019 19760004918_1976004918

    41/209

  • 7/22/2019 19760004918_1976004918

    42/209

    Combining equations ( 4 8 ) w i t h ( 4 9 ) andperforming t h er e q u i r e dm a t h e m a t i c a lo p e r a t i o n s , h eF o u r i e rc o e f f i c i e n t sB n can be e x p re s se d a s

    By l i m i t i n g h e e r i e s o r r2 t o r - l terms, e q u a t i o n ( 4 9 )be comes

    r- Ir Z m = 112 b V o x Bn s in n-r r

    rn = l

    andequat ion ( 5 0 ) i s reduced t o h e summation

    m = lFrom equation ( 4 0 ) t h e i f t d 2 a s s o c i a t e d w i t h t h e l i p s t r e a mmay be ex .p ressed a s

    where the l i f tc o e f f i c i e n t i s based on V, . T h e r e f o r e

    6

  • 7/22/2019 19760004918_1976004918

    43/209

    C o m p a r i n ge q u a t i o n s 5 1a n d 5 4 ) h e r e e s u l t s

    If t h e r e l a t i o n s h i p o r t h e c o e f f i c i e n t s , 8, i s sub-s t i t u t e d n t oe q u a t i o n ( 5 5 ) , h e l i f t d i s t r i b u t i o n a s s o c i a t e dw i t h t h e s l ips t ream i s o b t a i n e d i n t h e form

    ( 5 6 )Havingdetermined the l i f t a s s o c i a t e d w i t h the s l ips t ream upwashthe o v e r a l l wing l i f t i s c a l c u l a t e d a s f o l l o w s .L e t ail and ai2 be t h e n d u c e d a n g le s o f a t t a c h e d a s so c -

    i a t e dw i t h t h e l i f td i s t r i b u t i o n s d l and 4 r e s p e c t i v e l y ,a s g iv e nbye q u a t io n ( 4 0 ) . Then the t o t a l n d u c e da n g l eo fa t t a c k a t a n y o i n t k i s given by

    Also, r e - e q r e s s i n ge q u a t i o n ( 4 0 ) i n terms o f h e l i f t c o e f f i c -i e n t sa s e dn V y i e l d s

    u s in g the m u l t i p l i e r s P m k fromReference 1 i ne q u a t i o n 5 8 )t he re f o l l o w s

    Now, by d e f i n i t i o n

    27

  • 7/22/2019 19760004918_1976004918

    44/209

  • 7/22/2019 19760004918_1976004918

    45/209

    3.2.2 A n a l y s i s~ . f o r a Winq w i t h Part-Span Deflected F l a p sThe d e f l e c t i o n of a p a r t - s p a n f l a p c a u s e s a d i s c o n t i -n u i t y 8 i n t he d i s t r i b u t i o n of a b s o l u t e n g l e of a tt a ck a tthe end of the f l a p , andproduces a c o r r e s p o n d i n gd i s c o n t i n u i t yi n the s l i p s t r e a m - in d u c e dc ro s s f lo w . The e f f e c t of these d i s -c o n t i n u i t i e s on the span load d i s t r i b u t i o n i s t reated below.Th e a n a ly s i s i s d e v e l o p e d for a winghaving a d e f l ec t edp a r t - s p a nl a px t e n d i n gr o m y = - b / 2 t o y = y * .The m o s t g e n e ra l case i s t h a t of a f l a p whoseend l i e s w i t h i n

    t h e s l ips t ream, a s i l l u s t r a t e d n F i g u r e 5.Following the p r e c e d i n g t r e a t m e n t o f a wing w i t h n o f l a p sor w i t h f u l l - s p a n d e f l e c t e d f l a p s , the t o t a l wing l i f t distri-b u t i o ng i v e nb ye q u a t i o n ( 4 0 ) can be d i v i d e d n t o w o p o r t i o n sandcan be expressed i n non-dimensional orm a s

    where C & . c / b i s t he l i f t d i s t r i b u t i o na s s o c i a t e d w i t h s l i p - .strea m-in duce d upwash and CJ, .c /b i s t h e remainder of t h ed i s t r i b u t i o n .I n the p r e s e n t case , however, the s l i p s t r e a m - in d u c e d

    upwash VOv , g i v e n by e q u a t i o n (48), i s d i s c o n t i n u o u s a t t h eendof t h e f l a p a s shown i nF i g u r e 5. The n e td i s c o n t i n u i t yi nc r o s s f l o w a t t h e edge of t h e f l a p , y = y * , can be o b t a i n e dfrom equat ion ( 4 5 ) by cons ide r ing the upwashon b o t h s i d e s of t h ef l a p end.Thus ,us ing quat ion (45 ) f o r t h e f l a p p e d s i d e oft h e wing, a t y = y * - o , t h i s e x t r a upwash can be expressed a sf o l l o w s

    vo.v = vS% sin (a,*+ $ +8) + v t t . cos(Y 0 ) (a*, .:,ss, ( 4)- V s in ( ~ g 8)29

  • 7/22/2019 19760004918_1976004918

    46/209

    Propeller SlipstreamDiameter\ / iameter ? . . ,

    -I-TI

    Distxibution (a) = (b) + ( c )Fig.ure 5. Mathematical Representation of Flap Discontinuity

    0

  • 7/22/2019 19760004918_1976004918

    47/209

  • 7/22/2019 19760004918_1976004918

    48/209

  • 7/22/2019 19760004918_1976004918

    49/209

  • 7/22/2019 19760004918_1976004918

    50/209

  • 7/22/2019 19760004918_1976004918

    51/209

  • 7/22/2019 19760004918_1976004918

    52/209

    I I I I Ill11111II I I 'I I I I I I I I I II I I I II I I

    d i s c o n t i n u i t y 8 , and Cd, '/b , i s the remainder .Applying t he m u l t i p l i e r s P m k t oe q u a t i o n ( 7 7 ) y i e l d s

    the f o l l o w i n g

    /A l s o , a p p l y i n g t h e m u l t i p l e r s Prnk t o e q u a t i o n (63)y i e l d s t h e t o t a l l i f t d i s t r i b u t i o n C d c /b a s

    S u b s t i t u t i n ge q u a t i o n ( 7 8 ) i n t oe q u a t i o n ( 7 9 ) , y i e l d s\

    rn = II n o rd er t o o b t a i n t h e i t e r a t i v e s o l u t i o n t o e q u a t i o n ( 8 0 ) ,it i s now necessary t o r e l a t e t h e l i f t d i s t r i b u t i o n s g i v e n ne q u a t i o n s (63) and ( 7 7 ) t o t h e i r c o r r e sp o n d in g n d u c e dang le ofa t t a c kd i s t r i b u t i o n s . If Q i i s the i n d u c e d n g l e o f a t t a c kd i s t r i b u t i o n o r r r e s p o n d i n g o h e t o t a l l i f t d i s t r i b u t i o n ,

    C& d b and Qi i and Q i 2 a r e the i n d u c e dn g l e of a t t a c kd i s t r i b u t io n s o r r e s p o n d i n g t o t h e l i f t components C d l c / b andC & c / b thenro mqua t ion 6 3 ) t h e r e f o l l o w s

    A l s o , a p p l y i n g s i mi l a r c o n s i d e r a t i o n s t o e q u a t i o n ( 7 7 ) theze

    36

  • 7/22/2019 19760004918_1976004918

    53/209

    r e s u l t s

    S u b s t i t u t i n ge q u a t i o n (82) i n t oe q u a t i o n (81) y i e l d s ther e l a t i o n s h i p f o r the i n d u c e d o t a l a n g l e of a t t a c k a s follows:

    where ail' = 8 o v e r h e l a p p a no o u t s i d e the f l a p spanI t can be n o t e d n e q u a t i o n ( 8 3 ) t h a t t h e i n d u c e d , a n g l e

    of a t t a c k d i s t r i b u t i o n ai l ' must be c o n t in u o u s , i n c e i t scor respond ing l i f t d i s t r i b u t i o n Cdl' c/b a s g i v e n n q u a t i o n( 7 7 ) i s con t inuous .Th e re fo re , t h i s induced ng le of a t t a c kd i s t r i b u t i o n i s o b t a i n e d d i r e c t l y u s i n g h e m u l t i p l i e r s , h u s

    The a b o v ere l a t i o n sh ip c a n be r e - e x p r e s s ed n terms o f t h e t o t a linducedng le of a t t a c k i s t r i b u t i o n ai by us ingq u a t i o n(83), t h u s

    Now, equation 8 5 ) can be s u b s t i t u t e d n t oe q u a t i o n ( 8 0 ) t oe l i m i n a t e the C$/ c/b d i s t r i b u t i o na n d oy i e l d t h e t o t a ll i f t d i s t r i b u t i o n CJ c /b i n h e d e s i r e d m u l t i p l i e r f o r m as follows

    37

  • 7/22/2019 19760004918_1976004918

    54/209

    F i n a l l y ,r e a r r a n g i n g the e q u a t i o n (86), h e t o t a l i n d u c e do f a t t a c k a t a n yp o i n t k on the wing an be r e l a t e d t o _.l i f t d i s t r i b u t i o n a n d t h e k n o w d i s t r i b u -nducednglef a t t a c k a i l t 1 and Qi2 and t he i rl i f t d i s t r i b u t i o n s Cd,llc/b and Cd2 c/b , r e s p e c t -. The r e s u l tn ge l a t i o n , s h i p i s \

    > I

    .. .. ._ .:. . ..

    from t h e a n a l y s i s of Reference 1

    Cd2c/bhas b e e n a l r e a d y d e t e r m i n e d i n e q u a t i o n ( 7 6 ) . 1Equa t ion ( 8 7 ) i s a n a l o g o u s oe q u a t i o n 6 1 )d e v e l o p e d n3.2.1 f o r n o f l a pd e f l e c t i o n . This e q u a t i o n i s a l s oy a n t e r a t i o n p r o c e d u r e , s i mi l a r t o t h a t used fo r s o 1 v i . q

    Thus, upon ob ta in ing the requ i redc o n v e rg e n c e ofs o l u t i o n ,e q u a t i o n ( 8 7 ) y i e l d s t h e t o t a l liftC$ c/b f o r a wing w l t h a d e f l e c t e d f l a pw i t h i n t h e . .

    38

  • 7/22/2019 19760004918_1976004918

    55/209

    3.2,3 Ex te n s io no f t h e Winq A na ly s i s o Small Aspec t R a t i o sTo p r o v i d e a d d e d f l e x i b i l i t y t o t he methodologyd e v e l o p e dh e r e i n , the wing a n a l y s i s r e a t e d n S e c t i o n s 3.2.1and 3.2.2 i s ex tended to nc lu de w i n g s o f s m a l l a s p e c t r a t i o .This a n a l y s i s i s p a r t i c u l a r l y u s e f u l f o r t h e c u r r e n t a p p l i c a -t i o n , s i n c e much o f t h e a v a i l a b l e t e s t d a t a on spanw ise oad-

    i n g s for w i n g s i n s l i p s t r e a m f a l l s w i t h i n t h e low a s p e c t r a t i or a n g e . The c o r r e l a t i o n so f t h i s e x t e n d e da n a l y s i s w i t h thec o r r e sp o n d in g t e s t da ta w h e r ea p p r o p r i a t e i s shown in S ec t i o n 5.0.The m o d i f i c a ti o n o f t h e p r e s e n t a n a l y s i s t o s m a l l a s p e c tr a t i o wings i s based on th e wing th eory of Kuchemann (R ef er en ce 22),a s o u t l i n e db e lo w .I ne q u a t i o n s (6 1 ) and ( 8 7 ) a s e t o fm u l t i p l i e r s wa su s e d o o b t a i n the inducedang leof a t t a c k d i s t r i b u t i o n s f o ra wing w i t h no f l a p s an d w i t h p a r t - s p a n d e f l e c t e d f l a p s ,r e s p e c t i v e l y . These m u l t ip l i e r swe reo b ta in e d ro m h e u n d -amenta lequa t ionof t he h i g h - a s p e c t - r a t i o , i f t i n g - l i n e h e o r yw h i c he x p r e s s e s h e n d u c e da n g l eo fa t t a c k n terms o f t he spanl o a d i n g ,

    b/2 d C J c/b)d Y lY, - Y

    d y IQ i --b/2

    Kuchemann, Reference 2 2 , h a s shown t h a t t h i s e q u a t i o n may beg e n e r a l i z e d t o w i n g s o f a n y a s p e c t r a t i o by w r i t i n g

    where w ' i s a f a c t o r w h ic hvar ie sbe tween 1 f o rh i g ha s p e c tr a t i o ( A R + a ) , and 2 f o r low a s p e c t a t i o ( A R + O ) .39

  • 7/22/2019 19760004918_1976004918

    56/209

  • 7/22/2019 19760004918_1976004918

    57/209

  • 7/22/2019 19760004918_1976004918

    58/209

    SECTION 4D I G I T A L COMPUTER PROGRm

    The t h e o r e t i c a l a n a l y s i s p r e s e n t e d n S e c t i o n 3 w a sprogrammed foru se o n the CDC 6600 s e r i e s d i g i t a l com puter.This was accompli shed by ex t ens ive ly mo di fy ing t h e computer.programofReference 1 t o i n c l u d e the p r o p e l l e r s l i p s t r e a mand t h e w i n g n - s l i p s t r e a m a n a l y s i s .This s e c t i o n p r e s e n t s a d e s c r i p t i o n o f h e com binedco m p u t e rp r o g r am o g i c , h ese l ec t i o nan dassem b l yo f h ep e r t i n e n t a i r f o i l s e c t i o n c h a r a c t e r i s t i c s , and a sampleco m p u t e ro u t p u t .Wh er ev e r p p r o p r i a t e , t h e d i s c u s s i o n i s

    d i r e c t e d t o w a r d s h o s e f e a t u r e so f h e m o d i f i e d p r o g r a m t h a ta r e d i r e c t l y r e l e v a n t o h e r e a t m e n t o f h e p r o p e l l e r s l i p -stream and i t s e f f e c t on t h e w in g p a nw is e o a di ng .Addi t ion-a l i nf or m at io np e r t a i n i n g oc o m p u t a t i o n so f t he wing loadingf o r a b as i cwi n g / f u se l ag eco m b i n a t i o ncan be obta inedf romReference 1.4 1 PROPELLER SLIPSTREAM COMPUTATION-S

    This s u b s e c t i o n p r e s e n t s h e m e t h o d o l o g y a n d the a s s o c i a t e da i r f o i l s e c t i o n d a t a u s e d n c o m p u t a t i o n so f the p r o p e l l e rs l i p s t r e a m v e l o c i t y d i s t r i b u t i o n s , w h i c h a r e l a t e r implementedi n t h e o v e r a l l s o l u t i o n f o r t h e w ingspanwise oadingof ag en e r a lwing /p rope l l e r omb ina t ion . The bas ic om put a t io na ls t e p sf o r m p l e m e n t i n g h es l i p s t r e a mv e l o c i t yd i s t r i b u t i o n si n t o the w i n ga n a l y s i s a r e summarized ins u b s e c t i o n 4 . 2

    4 . 1 . 1C o m p u t a t i o n a lP r o c e d u r e s o rP r o p e l l e r Sl ips t r e a mV e l o c i t y D i s t r i b u t i o n s( a ) C a l c u l a t e t h e p r o p e l l e ra n g l eo fa t t a c ka n d i p

    speed r a t i o f r o m

    42

  • 7/22/2019 19760004918_1976004918

    59/209

    J COS U pPLT = 7r

    J COS a pP = 7 r r

    (c) o b t a i n n p p r o x i m a t e o l u t i on o r t h e t i p lossf a c t o ru s i n g

    where, b y d e f i n i t i o n

    k = FP

  • 7/22/2019 19760004918_1976004918

    60/209

    and a o , Q&) a r e the l i f t - c u r v es lo pe and angleo fa t t ack a tz e r o l i f t , - r e s p e c t i v e l y , f o r a l inear izedapproximat ion t othe t a b u l a t e d a i r f o i l s e c t i o n c h a r a c t e r i s t ic s .( e ) :Compute a n i n i t i a l n f l o wa n g l e a t eachbladeelement s t a t i o n from

    U *(p' = A + x( f ) O b t a i n a n i n i t i a lv a l u e o r h eq u a n t i t yd e f i n e das

    ( 9 ) As t h e f i r s t s t e p i n t h eb a s i c t e r a t i o n o u t i n e ,ca lcu la te a b e t t e r approx imation for t h e t i p loss factor from

    where F /F p i s ob ta ined by in te rp o l a t in g he r e s u l t s fromthe t i p loss c o r r e c t i o n t a b l e s f o r s p e c i f i e d v a l u e s of B , Tand sin . A l i s t i n g o f h e i p loss co r r ec t i o n ab l e s t o r edand u t i l i z e d by the computer program i s presen ted i n Appendix A.(h) Calcu la te heb ladesection angleof at ta ck and theb l ad e s ec t i o n Mach number rom

    44

  • 7/22/2019 19760004918_1976004918

    61/209

    Then o b t a i n the s e c t i o n c h a r a c t e r i s t i c s C& and c d b y i n t e r -p o l a t i o n a n d / o r e x t r a p o l a t i o n o f t he d a t a p r e s e n t e d i n thepropel ler a i r f o i l t a b l e s , f o r t h e s p e c i f i e d a i r f o i l s e c t i o ngeomet ryndaluesf Q b and M, .i) Compute t h e

    k, = L [F

    f o l l o w i n g q u a n t i t i e s d e f i n e d a s

    and then c a l c u l a t e a new value of 4 from

    (j) If t h e abso lu temagn i tude of (t#)"-~#)')>O.l d e g r e e s ,t h e n t h e s o l u t i o n f o r # r e q u i r e s e i t e r a t i o n . n t h i s case t h ev a l u e of + t o be s u b s t i t u t e d o r # I i n t e p s ( 9 ) th rough i )is obta ined f rom

    45

  • 7/22/2019 19760004918_1976004918

    62/209

    . . . .

    where .kc i s given by

    ( k ) If the abso lu tem a g n i tu d e of ( I1- I ) 5 0.1d e g re e s h e n t h e f i n a l s l i p s t r e a mv e l o c i t yc o m p o n e n t s f o r t h es t r e a m tu b ee le m e n tpass ing h rough the . spec if ied b la de e lemen ts t a t i o na r ed e t e r m i n e da s o l l o w s . F i r s t , c a l c u l a t e t h e t r u ei n d u c e d a x i a l v e l o c i t y r a t i o i n t h e p r o p e l l e r d i s k p l a n eu s i n gU 4 F k y k z

    S l r 2 i- I -I-k x ) 2 - 4a n d h e no b t a i n t h e a x i a l v e l o c i t y r a t i o i n t h e f u l l y con-t r a c t e d s l i p s t r e a m f ro m

    1) Obtain t he l o c a l a d i u s n the f u l l yc o n t r a c t e ds l i p s t r e a m which cor responds t o t h e s p e c i f i e d b l a d e e l e m e n ts t a t i o n f rom

    -where r s p , r p and V S a p / v a a re t he v a l u e s o r r e s p o n d i n g t othe immedia te lypreceding nboard b l a d e s l e m e n t t a t i o n . Thev e l o c i t y r a t i o a t t h e o u t e rs l i p s t r e a mb o u n d a r y i s t aken a su n i t y , a s i s t h a t a t fhe hub /nace l leb o u n d a ryun less a bladee l e m e n ts t a t i o n i s s p e c i f i e d a t t h e hub.

    -

    46

  • 7/22/2019 19760004918_1976004918

    63/209

    ( m ) Compute t h e t a n g e n t i a lv e l o c i t y a t i o n t h ef u l l y c o n t r a c t e d s l i p s t r e a m as

    ( n ) H a v i n go b t a i n e d o l u t i o n s o r the f l o w c o r r e s -ponding t o all p r o p e l l e rl a d el e m e n tt a t i o n s m = l ( a tt h e hub) th rough m = M ( a t t h e b l a d e i p ) ,c a l c u l a t e t h ev a l u eo f t h e i n t e g r a t e d p r o p e l l e r h r u s t c o e f f i c i e n t f r o m

    1

    where 3 F k y k z-T = (7r Td 7 ( I + k )0 ) Obtain the momentum value of p r o p e l l e r h r u s tc o e f f i c i e n t from

    ( p ) Compute t h e i n t e g r a t e dp r o p e l l e r o r q u ec o e f f i c i e n t u s i n g

    47

  • 7/22/2019 19760004918_1976004918

    64/209

    ~ ~

    where

    9) C a l c u l a t e the va lueo f t he momentum-weighteda v e ra g e a x i a l v e l o c i t y r a t i o i n t h e f u l l y c o n t r a c t e d s l i p -stream from

    4 .1 .2 P ro p e l l e r B l a d e S e c t i o n - C h a r a c t e r i s t i c sThe a n a l y t i c a l m e t h o d s d e v e l o p e d h e r e i n r e q u i r e t h a ts u i t a b l e a e r o d y n a m i c c h a r a c t e r i s t i c s be employed for t he

    b l a d e s e c t i o n s o f p r o p e l l e r s u s e d on g e n e r a la v i a t i o n - t y p ea i r c r a f t . The i n f o r m a t i o n o n t y p i c a l b l a d e s e c t i o n s wasob ta ined rom t h e a v a i l a b l e e c h n i c a l i t e r a t u r e a n d i ssummarized i n Table I .A s can be noted from t h i s t a b l e , e a r l y blade . s ec t ionsu s e d n y p i c a l p r o p e l l e r s a r e of the U S N P S and Cl a rk Ya i r f o i l s e r i e s . These s e c t i o n sh a v ev e r y similar p r o f i l e s

    and members of each s e r i e s a r e u n i q u e l y d e n t i f i e d by thev a l u eo f h i c k n e s s / c h o r dr a t i oa l o n e .La te r b l a d e s e c t i o n s a r e of t h e NACA 1 6 - s e r i e sf a m i l y ,which have a w i d e r a ,p ,p l i c a t io n n m odern p ro p e l l e r d e s ig nbecause of t h e i r super ior low-drag c h a r a c t e r i s t i c s ( s eeReference25) . These c o n s i d e r a t i o n sa l s oa p p l y o t he useo f NACA 64and65 a i r f o i l s e r i e s . All of the l a t t e r air-f o i l s a r e s p e c i f i e d n terms ofbo th a d e s i g n l i f t c o e f f i c i e n tand a t h i c k n e s s / c h o rd r a t i o .Basedon a rev iewofpubl ished exper imenta l measurementso f p r o p e l l e r a i r f o i l s e c t i o n c h a r a c t e r i s t i c s , it i s e v i d e n tt h a t the most r e l i a b l e d a t a f o r t h e c u r r e n t a p p l i c a t i o ncan be obta ined f rom t e s t s c o n d u c te d n t h r e e wind unne l

    48

  • 7/22/2019 19760004918_1976004918

    65/209

  • 7/22/2019 19760004918_1976004918

    66/209

    f a c i l i t i e s only . These are the Langley Low TurbulencePressureTunne l R efe rence 261, f o r s e c t i o nd a t a a t l o ws p e e d o n d i t i o n s ( M e .15) 8 and b o t h the Langley ndAmes HighSpeed Wind Tu nne ls Re fer enc es 27 and 2 8 , respect-i v e l y ) f o r s e c t i o n da t a a t h i g h s p e e d s (0.3< M < 0 . 8 5 ) .E x p e r i m e n t a ld a t aava i lab le f rom t e s t s i n t h e s e f a c i l i t i e swere t h e r e f o r eu s e d a s the b a s i s f o r p r e p a r a t i o n of t herequ i red s e c t i o n c h a r a c t e r i s t i c s f o r a l l s e l e c t e d a i r f o i l sw i t h the e x c e p t i o n of t h e USNPS andClark Y s e r i e s . Thes e c t i o n d a t a f o r the l a t t e r two a i r f o i l s w a s genera ted f romt h e m e a su re m e n t so b ta in e d n the Lang leyVariable-Densi tyTunnel .

    Ap p l i c a t io no f t he p r e s e n ta n a l y t i c a lm e t h o d sr e q u i r e sinformat ion on t h e two-dimensionalbehavior of b o th l i f t andd r a g o r the s p e c i f i e d b l a d e a i r f o i l s . However, a n import-a n ts i m p l i f i c a t i o n np r e , p a r i n g t h e s e a i r f o i l c h a r a c t e r i s t i c si s r e a l i z e d h r o u g h t h e useof a c o n s t a n t v a l u e f o r d ra gc o e f f i c i e n to n t he b a s i s of the fo l lowingapproximat ion .

    From t h e p r o p e l l e r a n a l y s i s it can be n o t e d t h a t t h ec o n t r i b u t i o n so f t he b l a d e s e c t i o nd r a g o e f f i c i e n t c d t ot h e a x i a l a n d s w i r l v e lo c i ty com ponents i n the s l i p s t r e a ma r e g iven , approx im ate ly , b y ( Cd/Cd) an d and (Cd/Cd) o t $r e s p e c t i v e l y , where c i s t h e i n f lo wngle .or lowpeedf l i g h t c o n d i t i o n s a p p r o p r i a t e o g e n e r a l a v i a t i o n y p e a i r -c r a f t , t h e c o n t r i b u t i o n so f b l a d e s e c t i o nd r a g o t h e l o c a la x i a l v e lo c i ty com ponent i n t h e s l i p s t r e a m a r e f o u n d o ben e g l i g i b l e ,w h e r e a s the c o n t r i b u t i o n s o h e o c a l s w i r lv e l o c i t y a r e t y p i c a l l yn o t m o re h an a f e w percen t . Thus iti s c o ns id e re d a j u s t i f i a b l e s i m p l i f i c a t i o n n t h e computerprogram t o s u b s t i t u t e a r e p r e s e n t a t i v ec o n s t a n t v a l u e f o r Cdi n p l a c e o f t h e a c t u a l v a r i a t i o n s a s a f u n c t i o n o f a n g l eo fa t t a c k and Mach number.

    I t i s t h u s e v i d e n t h a t r e a l i s t i c a p p l i c a t i o n of t h ep r o p e l l e r - s l i p s t r e a ma n a ly s i s demands t h a t s e l e c t e d da ta onb l a d e s e c t i o n l i f t c h a r a c t e r i s t i c s be a c c u r a t e l yd e f i n e d a sa f u n c t i o no f o c a la n g le -o f - a t t a c k an d Mach number f o r h o s et y p i c a l a i r f o i l s e c t i o n s i d e n t i f i e d a b o v e .Table I1 summarizes t h e a i r f o i l s e c t i o n s f o r whicha e r o d y n a m i c c h a r a c t e r i s ti c s h a ve b e e n o b t a i n e d a n d d e n t i f i e st h e s o u r c e e f e r e n c e s . I n g e n e r a l it i s a p p a r e n t t h a t in su f -f i c i e n t d a t a e x i s t t o e n a b l e a thoroughcoverageof a l l thep o s s i b l ev a r i a t i o n s n s e c t i o ng e o m e t r y , a n g l e - o f - a t t a c ka n d50

  • 7/22/2019 19760004918_1976004918

    67/209

  • 7/22/2019 19760004918_1976004918

    68/209

  • 7/22/2019 19760004918_1976004918

    69/209

    d a t a a b l e s , it i s e s s e n t i a l t h a t t h e y be assembled i n as p e c i a lo r d e r . The assembly of a l l t a b l e s o r each g iv e na i r f o i l f a m i l y must be i n a s c e n d i n g o r d e r o f Mach number,t h i c k n e s s / c h o rd a t i o and d e s i g n l i f tc o e f f i c i e n t . H o w e v e r ,t h e s e t s of t a b l e s f o r a ny a i r f o i l f a m i l y may be assembledi n a nyorder .As a n i n i t i a l s t e p i n t he t a b l e look-upprocedure,

    the computerprogram f i r s t s e a r c h e s t h r o u g h t h e t a b l e s tol oca t e a n d n d e x h o s e p a r t i c u l a r t a b l e s r e q u i r e d f o r i n t e r -p o l a t i o n as e a c h p r o p e l l e rb l a d e e l e m e n t s t a t i o n i s s p e c i f i e d .The a c t u a l o o k - u p p r o c e d u r eu t i l i z e s i n e a r n t e r p o l a t i o nthroughoutand i s perform ed f i r s t f o r t h e r e q u i r e dv a l u e ofa , se c o n d ly o r h eva lueof Mach number, t h i r d l y o r t hes e c t i o n h i c k n e s s / c h o r d r a t io and f i n a l l y f o r the d e s ig n l i f tc o e f f i c i e n t o f the a i r f o i l f a m i l y s p e c i f i e d .To p e r m i t s a t i s f a c t o r yo p e r a t i o no f h e c o m p u t e rprogram f o r c o n d i t i o n s o u t s i d e t he r a n g e o f h e d a t a a b l e s

    a s e r i e s o fs i m p l ee x t r a p o la t io n p r o c e d u r e sh a v e been developede m p i r i c a l l y from t h ea v a i l a b l ee x p e r i m e n t a ld a t a . These pro-c e d u r e s a r e o u t l i n e d b e l o w .Fo r a n g le s o f a t t a c k o u t s i d e the t a b u l a t e d r a n g e neach t a b l e it i s assumed tha t t h e va lueo f CJ remainsc o n s t a n t ,a n df o r a Mach number o u t s i d e h eg i v e nr a n g e thee x t r a p o la t io n p r o c e d u r ed e te r mi n es a c o r r e c t i o n t o t h e r e q u i r e dva luef Q , d e f i n e ds ac , t h u s

    where t h e s u b s c r i p t T d e n o t e sv a l u e s o r t h e t a b l e t o bee x t r a p o l a t e d .This method i s based o n a n a p p l i c a t i o n o f h e s t a n d a r dP r a n d t l - G l a u e r tr u l ef o r t he change i n i f t - c u r v e s l o p e w i t hMach number and assumes t h a t t h e e x t r a p o l a t e d f a m i l y o f l i f tcurvescan be r e p r e s e n t e d by a s implea d ju s tm e n to f the a n g l e

    of a t t a c k s c a l e a b o u t ~1~ p o i n t .For s e c t i o n h i c k n e s s / c h o r d r a t i o so u t s i d e the g i v e nr a n g e o f a b l e s a t e a c hva lueofd e s ig n l i f t c o e f f i c i e n t iti s assumed t h a t t h e a i r f o i l c h a r a c t e r i s t i c s w i l l be i n v a r i a n t .While t h i s a s s u m p t i o n d o e s n o t s a t i s f a c t o r i l y r e p r e s e n t t h e

    53

  • 7/22/2019 19760004918_1976004918

    70/209

    g e n e r a l e d u c t i o n n i f t - c u r v es l o p e f o r t h i c k sec t ions . , , .-..-.the e x i s t i n g d a , t a . d o e s . n o t - p r o v i d e . : a ,,ase . . . . If o r a be t t e r approx imat ion .( ' t i c ,> 0.2 1. . . .- ,

    .. , . . . . , . 2. .For a s e c t i o n e s i g n l i f t c o e f f i c i e n t " c&i , o u t s i d e. .. . . - .the t a b u l a t e d r a n g e ; a n e x t r a p o l a t i o n p r o c ; e d u r e i s . 'used t oo b t a i n a cor rec ted v a l u e - o f C& d e f i n e d as :-C& , h u s- I.. kc& - , 8. . , , ,. ' , .

    . . ccec = CLeT + (c9 - CLeiTj :.. : (126. . _ . ,

    where t h e s u b s c r i p t T d e n o t e sv a l u e s ' o r the t a b l e to bee x t r a p o l a t e dn d kCdi i s a nm p i F i c a 1o n s t a n t whichg e n e r a l l y v a r i e s f a r each a i r f o i l ' f a k i i y a n d h i c k n e s s / c h o r dr a t i o . This c o n s t a n t has beendetermined f o r each a i r f o i lf a m i l y u s e d h e r e i n , a n d c o n s t i t u t e s a n i n h e r e n t p a r t of thecomputerprogram t a b l e look-upsubrou t ine . . ..~ ..... .

    4.2 WI NG. IN-SLIPSTREAM -COMPUTATIONS . . . ,This s u b s e c t i o n p r e s e n t s t h e m ethod o f implementat ionof the p r o p e l l e r s l i p s t r e a m d i s t r i b u t i o n s o b t a i n e d a,bove i n t o

    the s p a n w i s e o a d c a l c u l a t i o n s of a pro,peller/wingcombination., . . ,The e s s e n t i a l c o m p u t a t i o n a l s t e p s a r e d e s c r i b e d below.'4 .2 .1Computat ionalProcedures orSpanwiseLoadinqon aWinq w i t h no F l a p s , o r w i t h Ful l -Span Def lec ted Flaps~ . - ..

    ( a ) Obta in t h e wing b a s i c geometr ic parametersnamely , , . .s e c t i o n h o r d r a t i o c / C R , t w i s t d i s t r i b u t i o n . ' E 8t h i c k n e s s - c h o r da t i o t / c , and camber d i s t r i b u t i o n . T h e nc a l c u l a t e the wing ec t ionReynolds number R e based on thel o c a lh o r d c and the l o c a le s u l t a n t e l o c i t y V , t h y s . .

    . . c

    , , . . .

    . .Pi , , . : , . :' ... ~.. . .' - .. . .

    where V i s t he combined f re es t re am nd sl ipstr 'eam v e l o c i t yg i v e n n q u a t i o n ( 3 ) and Y i s t he k i n e m a t i c , i s c o s i t y .A l s o , o b t a i n t h e s e c t i o ne r o - l i f tn g l e a d o . . . . .

    , .. .I. .

    %.

    ( b ) Compute t h e wing-induced upwash funct ion, f ,. .

    .s4

  • 7/22/2019 19760004918_1976004918

    71/209

    from the f o l l o w i n g . e q y a t i o n which i s based on a s implehorse -shoemodel of t h e wake (Refe rence 1 9 )

    J (7?4-$2+ xp - p2 -

    are the non-dimens ionalspanwiseandchordwise oca t ions oft h e r i g h t - h a n d p r o p e l l e r hub.

    ( c ) C a l c u l a t e the g e o m e t r i ca n g l eo f a t t ack a t eachw ing s t a t i o n from

    where ag i s the f u s e l a g en g l ef a t t a c kaR i s the w i n g / f u s e l a g e r o o t s e t t i n gE i s the l o c a l g e o m e t r i c t w i s tT [E$-,] i s the c o r r e c t i o na c t o ro ru s e l a g e

    upwash g i v e n n R e f e r e n c e 1) and A E n i s the s e t t i n g of thee q u i v a l e n t c h o r d l i n e , of the n a c e l l e a b o v e t he wingchordl i n e a t the n a c e l l es t a t i o n . The q u a n t i t y A E n i s o n l y obe included when a c o m p u t a t i o n s t a t i o n c o i n c i d e s w i t h ' t hen a c e l l e l o c a t i o n .(a) C a l c u l a t e t h e f o l l o w i n g n i t i a la p p r o x i m a t i o n othe o v e r a l l w i n g l i f t c o e f f i c i e n t

    - ICLAPPROX- (I + %j ("B "R -0.4 adoTIP-0.6 doROOT)(130)5-5

  • 7/22/2019 19760004918_1976004918

    72/209

  • 7/22/2019 19760004918_1976004918

    73/209

    ( j ) C a l c u l a t e an i n i t i a la p p r o x i m a t i o n o t he spanwise Il o a d i n gd i s t r i b u t i o nu s i n g

    AR + 1.8137)where X i s the wing t a p e r a t i o .

    ( k ) Compute the v a l u e so f n d u c e da n g l eo f a t t a c kf o r t h i s l o a dd i s t r i b u t i o nu s i n ge q u a t i o n ( 6 1 ) a nddeterminet h e r e s u l t a n t s e c t i o n a n g l e s o f a t t a c k f r o m e q u a t i o n ( 6 2 ) .1) From the s e c t i o n da ta o b t a i n t h e v a l u e s of l i f tc o e f f i c i e n t c o r r e s p o n d i n g t o the r e s u l t a n t a n g l e s of a t t a c kf r o ms t e p ( k ) a n d c a l c u l a t e the new valueso f me span oad-

    i n g , c(e c/b .m ) Compare the approx ima teva luesof p an oad ingw i t h the c a l c u l a t e dv a l u e s . If these a re n o t ns u f f i c i e n t l y

    c l o s eagreement, compute a new s e t ofappro.ximate values ofo fR e fe re n c e 1. Repeat the i t e r a t i o n p r o c e s s u n t i l t he r e q u i r e dconvergence i s a c h i e v e d .

    CCe c / b u s i n g the p r o c e d u r e sp r e s e n t e d ns u b s e c t i o n 3.2.2

    ( n ) n t e g r a t e t he new s pa n o add i s t r i b u t io n o o b t a i nt he o v e r a l l wing l i f t c o e f f i c i e n t CL a n dc a l c u l a t e a newv a l u e of wing-induced upwash a t t he p r o p e l l e r d i s c u s i n ge q u a t i o n s ( 1 2 8 ) and (131) .0 ) R e p e a t t e p s (f), ( 9 ) , ( h ) , i ) , k), (11, (m)( n ) u n t i l the a p p ro x i ma t ea n dc a l c u l a t e dva lues o f span

    l o a d i n g a r e i n s a t i s f a c t o r ya g r e e m e n t .( p ) Havingdetermined t h e l i f t d i s t r i b u t i o no b t a i n

    t h e s e c t i o n p ro f i l e d r a g a n d p i t c h i n g moment v a l u e s from t hes e c t i o n d a t a a n d c a l c u l a t e the o v e r a l lw i n g l i f t , drag ,andp i t c h i n g moment c o e f f i c i e n t s .

    4.2.2 Computa t ionalProcedures orSpanwiseLoadinqon a W i n q w i t h Part-Span Deflected F l a p s( a ) Calcu l a t e a n n i t i a la p p r o x i m a t i o n o the f l a p p e dwing l i f t d i s t r i b u t i o nf r o m the f o l l o w i n ge q u a t i o n s

    57

  • 7/22/2019 19760004918_1976004918

    74/209

    where C ~ Rs the v a l u eo f the l i f t c o e f f i c i e n t a t t h e r o o to b t a i n e d from the f l a p p e d s e c t i o n d a t a a t t h e a n g l e o f a t t ack

    ( b ) Determine t h e u n c o r r e c t e dv a l u e so f l i f t c o e f f i -c i e n t c>, a t each f l ap end a s f o l l o w s

    where FF i s the c o r r e c t i o n f a c t o r w h i c h a c c o u n t s f o r thechange in the two-d imens iona lsec t ion d a t a a t t h e f l a p end.The c a l c u l a t i o n p r o c e d u r e f o r o b t a i n i n g these c o r r e c t i o nf a c t o r s i s d e s c r i b e d n d e t a i l i n s u b s e c t i o n 4.1.3 ofRefe rence 1, and w i l l n o t be d u p l i c a t e d here .( c ) For t he v a l u e s f c 2 , o b t a i n e d n s t ep ( b )a b o v eo b t a i n t h e c o r r e s p o n d i n ga n g l e so f a t t a c k Q o from t h e

    data f o r f lapped s e c t i o n s ,C a l c u l a t e the cor respond ing o r -rected a n g l e s of a t t a c k Qc8 a t each endof t he f l a p from

    (d )U s i n g t he same p ro c e d u re a s i ns t e p ( c ) above ,ca lcu la te the v a l u e so fa n g l e of a t t a c k a c S=O on theunflapped s i d e so f the wing. Then ob ta in the f i r s t approx-i m a t i o n f o r the v a l u e so f the d i s c o n t i n u i t i e s n a n g l e ofa t t ack 8 , h u s

    58

  • 7/22/2019 19760004918_1976004918

    75/209

    ( e ) I n t e g r a t e the l i f t d i s t r i b u t i o ng i v e n by e q u a t i o n(138) t o o b t a i na n a p p r o x i m a t e v a l u e o f the o v e r a l l f l a p p e dl i f t c o e f f i c i e n t , C L , a n d s i n g q u a t i o n ( 3 ) , d e t e rmi n ethe wing-induced upwash a t the p r o p e l l e r d i sc . T h e n c a l c u l a t ethe v a l u e f slipstream i n c l i n a t i o n , a s , u s i n gq u a t i o n(5)

    I ( E ) ~ Using the v a l u e s of 8 and as f r o mt e p s ( d ) ,and *Fej, r e s p e c t i v e l y , c a l c u l a t e the d i s t r i b u t i o n of slip-stream c ro s s f l o wf r o m the f o l l o w i n g e q u a t i o n :

    +(%) COS (as + .e) - in Q ea n du s ee q u a t i o n s 6 4 )a n d ( 6 5 ) to de te rmine t he d i s c o n t i n -u i t i e snr o s s f l o w Av*= v I I . . . " .-OTE: - I n t h e m o s t g e n e r a l case of a wing hav ing t w o :p r o p e l l e r s , ' (one mounted on' each w i n g p a n e l ) , r o t a t i n g i n & e . .same d i r e c t i o n , the s l i p s t r e a m - i n d u c e dc r o s s f l o wd i s t r i b u t i o n ..w i l l be d i f f e r e n t a t ' he same s p a n w i s e t a t i o n Y on eachs i d e of the f u s e l a g ec e n t e r l i n e . This d i f f e r e n c e i s causedby pward s l ip s t re am s w i r l v e l o c i t i e s o n o n ewing.pane1 nd . .downward on t he other , o c c u r r i n g a t the ' same s p a n w i s e s t a t i o n son each s i de of the f u s e l a g e , i . e . v ( y ) # v (-y) . I n thecase of t w o prope l l e r s r o t a t i n g n o p p o s i t e d i r e c t i o n s , eachs l i p s t r e a m - i n d u c e dc r o s s f l o w i s symmet r ica labou t the f u s e l a g ec e n t e r l i n ea n de q u a t i o n 1 4 3 )n e e do n l y be a p p l i e do n c e ,s i n c e v'(y)= v(-y) .

    " . .

    ~.

    ( 9 ) U s i n g * i h ea p p r o p r i a t ev a l u e so f the d i s c o n t i n -u i t i e s 8 and nv = v i r from s teps ( d ) and ( f ) , r e s p e c t i , v e l y , ..compute the l i f t d i s t r i b u t i o n 'Cd2.c/b u s i n gq u a t i o n ( 7 6 ) .-OTE: For the m o s t g e n e r a l case, as d i s c u s s e d i n s tep . . -( f ) above, t h i s l i f t d i s t r i b u t io n m u s t be c a l c u l a t e d ' s e p a r a t e l yfor each wing pane l .

    (h ) Determine the l i f t d i s t r i b u t i o n CdZ,lI-c/b 8c o r r e s p o n d i n g t o th e l e f t a n d r i g h t s p a n w i s e d i s c o n t i n u i t i e sf ro me q u a t i o n (88) . _ _59

  • 7/22/2019 19760004918_1976004918

    76/209

    i) C a l c u l a t e the o v e r a l l n d u c e d n g l e - o f - a t t ac kd i s t r i b u t i o n Q i f ro mq u a t i o n (87), u s i n g the approximates p a n o a dd is t r i bu t io n computed i n s t e p ( a ) above.(j) Compute the e f f e c t i v e e s u l t a n ts e c t i o na n g l e ofd i s t r i b u t i o n f r o m the f o l l o w i n ge q u a t i o n

    Qe

    where Q g i s the g e o me t r i c n g l eo f a t t a c k , C d m a x i s thev a l u e of CLemax obta ined rom the c o r r e c t e ds e c t i o n da ta and( C d ? m a x ) o i s the u n c o r r e c t e d a l u e f CJma .

    (k )Us ing the v a l u e s of Q e from s t e p ( j) above ,o b t a i n the c o r r e s p o n d i n gv a l u e s o f l i f t c o e f f i c i e n t c& fromthe u n c o r r e c t e d w o -d i me n s io n a l e c t i o n l i f t data . Thend e t e rmi n e t he c o r r e c t v a l u e s of l i f t c o e f f i c i e n t C& b y s c a l i n g ,as f o l l o w s :

    1) C a l c u l a t e the d i s t r i b u t i o n C&c/b from 145) ndcompare th is ca lcula ted d i s t r i b u t i o n w i t h the approximated i s t r i b u t i o n . If agreementbetween t h e d i s t r i b u t i o n s i s n o ts u f f i c i e n t l y c l o s e , c a l c u l a t e a new and b e t t e r approx ima t ionu s i n g the p r o c e d u r e sp r e s e n t e d ns u b s e c t i o n 3,2.2 ofRefe rence 1..( m ) Repeat s teps ( b ) t h ro u g h 1) a b o v e , u n t i la g r e e -ment i s reachedbetween the approximate a n d c a l c u l a t e d v a l u e sof the s p a n l o a d d i s t r i b u t i o n .(n)Havingdetermined t h e l i f t d i s t r i b u t i o n ns t e p(m), c a l c u l a t e t h e c o r r e s p o n d i n gv a l u eo f t he o v e r a l l n t e -g r a t e d w i ng l i f t c o e f f i c i e n t CL .

    60

  • 7/22/2019 19760004918_1976004918

    77/209

    4.2.3 Winq Section Character is t icsThe wing a i r f o i l s e c t i o n c h a r a c t e r i s t i c s f o r t y p i c a lg e n e r a l a v i a t i o n a i r c r a f t a r e p r e s e n t e d i n S e c t i o n 4.2 o fRefe rence 1, and w i l l n o t be d u p l i c a t e d i n t h i s r e p o r t .These ch a rac t e r i s t i c s a r e u s e d d i r e c t l y n h e c u r r e n tcompute rp rogramandcons t i tu te a p a r t of t h e o v e r a l l t o o lf o r p r e d i c t i o n o f s t a l l i n g charac te r i s t ic s o f g e n e ra l w i n g /p r o p e l l e r c o m b i n a t i o n s .

    4.2.4 Table Look-Up Pr d ce d u re s o r Winq Sec ti onC h a r a c t e r i s t i c s 'The ta b le ook- up s u b r o u t i n ef o rw i n g s e c t i o n charact-e r i s t i c s u s e d n the c u r r e n tp r o g r a m i s i d e n t i c a l t o t h a td e s c r i b e d n S e c t i o n 4.2 ofReference 1.

    4.3 DESCRIPTION OF THE COMPUTER PROGRAM L O G I CThe c o m p u t a t i o n a lp r o c e d u r e sd e s c r i b e d nS e c t i o n 3.0have been programmed f o r u s e on a CDC 6600 s e r i e s d i g i t a lcomputer. The p r o g r a mu s e r n s t r u c t i o n s a r e gi ve n n Appendix C.The f l o wd i a g ra mfo r t h e program i s shown in F ig ur e 7 and al i s t i n g o f the program i s pr es en te d n Appendix D . Theprogram w a s a cc om p li sh ed by a n e x t e n s i v e r e s t r u c t u r i n g a n den la rgement of the ba s i c power-offwing s t a l l a n a l y s i s p r o -gram con t a ine d n R e f e r e n c e 1.The program i s i n i t i a t e d by r e a d i n g n t h e bas icw i n g - f u s e l a g ec o n f i g u r a t i o np a r a m e t e r s . n t h i s i n p u tf o r m a t , p r o v i s i o n has been made to nc lud e an nc rem ent

    r e p r e s e n t i n g the d r a gc o e f f i c i e n to f the n a c e l l e s . I f thec a l c u l a t i o n s a r e t o be p e r fo rme d fo r t h e power-on case t h i si s i n d i c a t e d t o t he p ro gram by s e t t i n g t he paramete r NSLIPe q u a l o 1. If NSLIP=O, t he s l i p s t r e a mc a l c u l a t i o n o o p s a r ebypassedand t h e programonlycomputes the power-off charact-e r i s t i c s .The computerprogramarrays a r e dimens ioned to en ab lec a l c u l a t io n s o f t h e s p a n o a d i n g ob e made u s i n g 1 0 c o n t r o lp o i n t s p e r s e m i s p a n .For t w i n p r o p e l l e r a i r c r a f t computa t ions where t h ep r o p e l l e r s a r e s i t u a t e d n e a r the c e n t e ro f each w i n gp a n e lo r

    61

  • 7/22/2019 19760004918_1976004918

    78/209

  • 7/22/2019 19760004918_1976004918

    79/209

    a t the w i n g t i p s , th i s number ofw i n g c o n t r o ls t a t i o n s i sa d e q u a t e . However, fors i n g l ep r o p e l l e rc o n f i g u r a t i o n s , ab e t t e r d e f i n i t i o n o f the s p a n o a d i n g n t h e s l i p s t r e a mr e g i o n i s o b t a i n e d i f the number ofc o n t r o l s t a t i o n s i sd o ub le d t o 20 per emispan. This i s r e a d i l ya c h i e v e d b yred imens ion ing t h e r e q u i r e d a r r a y s .

    Hav ing inpu t t h e b a s i c da ta , the


Keyboard Shortcuts for the Opera Browser for Mac OS X

Keyboard Shortcuts for the Opera Browser for Mac OS X

Do you admire Leonardo da Vinci?

Do you admire Leonardo da Vinci?

Physical Modelling Synthesis Overview

Physical Modelling Synthesis Overview

Chapter 24

Chapter 24

Star Wars Prequel Trilogy Trivia (Episodes I-III)

Star Wars Prequel Trilogy Trivia (Episodes I-III)

Basic Buffer Overflows Explained

Basic Buffer Overflows Explained

Introduction to Six Sigma

Introduction to Six Sigma

Chapter 23

Chapter 23

Minne hävisivät soneran rahat

Minne hävisivät soneran rahat

Improve the Color Quality Of Your Monitor

Improve the Color Quality Of Your Monitor

Aesops Fables

Aesops Fables

Iron Mills Essay

Iron Mills Essay

1 시스템분석입문

1 시스템분석입문

Steve Jobs' Commencement Speech at Stanford

Steve Jobs' Commencement Speech at Stanford

Explore The Levels of Creation

Explore The Levels of Creation

Fortran

Fortran

расписание электричек Москва-Железка

расписание электричек Москва-Железка

18 Tricks to Teach Your Body

18 Tricks to Teach Your Body

Heidegger Kritik

Heidegger Kritik

Life Is Just A Dream - Or Is It?

Life Is Just A Dream - Or Is It?