jJftcftL- 9^ 
 
 ARE No. 3L09 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 WARTIME REPORT 
 
 i 
 
 ORIGINALLY ISSUED 
 
 December 19^3 as 
 Advance Restricted Report 3L09 
 
 PROPELLERS IN TAW 
 By Herbert S. Ribner 
 
 Langley Memorial Aeronautical Laboratory 
 Langley Field, Va. 
 
 NACA 
 
 I 
 
 WASHINGTON 
 
 NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of 
 advance research results to an authorized group requiring them for the war effort. They were pre- 
 viously held under a security status but are now unclassified. Some of these reports were not tech- 
 nically edited. All have been reproduced without change in order to expedite general distribution. 
 
 L - 219 
 
Digitized by the Internet Archive 
 
 in 2011 with funding from 
 
 University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation 
 
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 http://www.archive.org/detadls/propellersinyawOOIang 
 
n 
 
 NATIONAL. ADVISORY COLL ITIES FOR AERONAUTIC* 
 
 ADVANCE RESTRICTED REPORT 
 
 PROPELLERS IN YAW 
 By Herbert S. Ribner 
 
 ARY 
 
 It was realized as early as 1909 that a propeller in 
 yaw develops a side force like that of a fin. In 1917, 
 R. Or. Harris expressed t n ' >f " 
 
 coefficient for t] unyav slier. Of several 
 tempts to express :. fee directly in terms of the 
 shape of the bladei none has been cc ly 
 
 satisfactory, An anal 
 
 effects not adeq nd 1 
 
 giv&s good a . ' . Lt riment ov< r wld - of 
 operating condition, r In. The present 
 
 analysis shows that fir ale d to 
 
 the form of the e orce e and that th . . sc- 
 
 tive fin area may be taken as th< projected side area of 
 the propeller. The e tio i: 
 
 order of 2 end the appropriate iynamj -.ly 
 
 that at the propeller dish at; augmented by the inflow. 
 The variation of the inflow velocity, for pitch 
 propeller, accounts for most of bho of s.i 
 
 force with adv . -diameter ratio V/nD. 
 
 - e propeller forces due to an a r velocity of 
 pitch are also analyze. I ^nd are \ ' all 
 
 for the pitc velocities that Ly be realized 
 
 in maneuvers, with fc] ion of the spin. 
 
 Further conclusions are! A iual-rotating propeller 
 in yaw develops up to one-" dj sree than a 
 single-rotatin< propeller. .--ting 
 
 propeller experiences a pitching moment in addition to 
 the side force. The pitchi] moment is of the order of 
 the moment produced by force squal to ide force, 
 acting at the end of a lever arm equal to the propeller 
 radius. This cross-cc - betwe n pitch yaw is 
 small, but possibly not n >le . 
 
A correction to the side force for compressibility 
 is included. 
 
 INTRODUCTION 
 
 r ke effect of power on the stability and control of 
 aircraft is becoming of greater importance with increase 
 in engine output, and propeller solidity. An important 
 part of this effect is due to the aerodynamic forces 
 experienced by the propeller under any deviation from 
 uniform flight parallel to the thrust axis. The remaining 
 part is due to the interference between the propeller 
 slipstream and the other parts of the airplane structure. 
 
 A number of workers have considered the forces ex- 
 perienced by the propeller. It was pointed out in 1909 
 (reference 1), apparently by Manchester, that a proueller 
 in yaw develop:., a considerable side force. The basic 
 analysis was published In 19l8 by R. G. Harris ( reference 2), 
 who showed that a pitching moment arises as well. Glauert 
 (references J an< 3 4) extended the method to derive the 
 other stability derivatives of a oropeller. 
 
 Harris and Glauert expressed the forces and moments 
 -in terms of the thrust and torque coefficients for the 
 unyawed oropeller, which were presumably to be obtained 
 experimentally. The analyses did net take into account 
 certain induction ^fleets analogous to the downwash as- 
 sociated with a finite wing. It is noteworthy tnat with 
 a semi empirical factor the Harris equation for side force 
 does give-- good agreement with experiment (see reference 5) • 
 Pistoles! (reference 6) In 1T2C considered the induction 
 effects but his treatment was restricted to an Idealizsd 
 particular case. Klingemann and Weinig (reference 7) 
 in I93G published an analysis neglecting the induction 
 effects; the treatment appears almost identical with the 
 account given in 1935 °7 Glauert in reference I4.. 
 
 There have been several notable attempts to express 
 the side force- directly in terms of the shape of the 
 blades. Bairstow (reference 8) presented a detailed 
 analysis in 1919 that neglected the ineluction effects, 
 "isztal (reference 9) published an investigation in 1932 
 that did not have this limitation and that is probably 
 the most accurst:- up to the present. ? T isztal's result, 
 however, is in a very complex form from the point of view 
 of both practical comoutati on and physical interpretation; 
 
there is, in addition, an inaccurac; in J te jmission of 
 
 the effects of the additional apparent of the air 
 
 disturbed by the sidewash of the slipstre 
 
 " ry recently Humph, White, and rirumman ( r r ace 3D) 
 
 published an analysis that relates the side force d r itly 
 to the plan fo] a very simple manner. nee 10, 
 
 however, (1) docs not include the o i th 
 
 analysis and (2) applies ui y-1 I : >ry in an im- 
 
 proper mi • H 3 acc I ' 3tion eff - As 
 
 a consequence 3-f (1), th ' 'y in srror it 
 
 high slipstrea velocities. of (2), 
 
 i . ■ fai] c - T I ttial incr 
 
 in sic ;s tha : al 
 
 rotal i --.. ■ -lift 1 
 
 si s ted in using - ?ly to 1 sa.se of i 
 
 finite airfoil with : wake, 
 
 vortex loo-os s] L3 i 
 
 the 12 flo 1 ", ar recti- 
 
 11 n :e. - or.di::. 
 
 pro,' Her blad w, howe 1 . helica] 
 
 path traverse e. is 
 
 quite di ! flow for t recti- 
 
 linear wake. .- ct, Lt can tex 
 
 loops shed luring the uns + :- 
 
 such & wa ■ to produce i ow ani try. 
 
 antisymmetry L; •■ of t 'ts t3 
 
 will b id in the t an fr iomen.1 
 
 cons.' derati ons . 
 
 To suir up, there are av lable n analyses ■ on 
 
 the blade £ icural r f he 
 
 whole range of ipelle] tions and t: 
 
 dysis that is the most accurate is not 1. satis- 
 factorily simnle fori . F'or this r n a new me1 >d of 
 ana] - presented fc] ' ter 
 
 ■ n i accuracy, 
 
 that the fin malog ;. 1 to th form of t] 
 
 side-fore 3 , itive f in are 
 
 i ren as the orojected si< - of 1 
 
 The proj id 3d I - I rsa jjected by 1 
 
 blades on t lai througl the xis ition. 
 
 one or I - is arc - i u ut 
 
 the average va] ae is of tone 
 
 jjected side i Lis u ' : - xj mat Ion 
 
 by one-half '■. nu : r Les ti - : i r<= pro- 
 
 se ted by e sing] e blade on a plan the 
 
 blade center line and the axis of r I in. 
 
ths effective aspect ratio i s of the order of S. This 
 equivalent fin area nay, with small error, be regarded 
 as situated in the inflow at the propeller disk and 
 subject to the corresponding augmented dynamic pressure. 
 The variation with V/nD of the dynamic pressure at the 
 propeller disk, for a fixed-pitch propeller, therefore 
 accounts for most of the variation of side force with 
 V/nD. 
 
 SYMBOLS 
 
 The formulas of the present report refer to a system 
 of body axes. For single-rotating propellers, the origin 
 is at the intersection of the axis of rotation and the 
 plane of rotation; for dual-rotating propellers, the 
 origin is on the axis of rotation halfway between the 
 planes of rotation of the front and rear propellers. The 
 X-axis is coincident with the axis of rotation and directed 
 forward; the Y-axis is directed to the right and the 
 Z-axis is directed downward. The synools are defined as 
 follows: 
 
 D propeller diameter 
 
 S' disk area ^rrD-/!j.) 
 
 S wing area 
 
 R tip radius 
 
 r radius to any blade element 
 
 r Q minimum, radius at which shank olade sections 
 devel oo lift 'Taken as 0.2R) 
 
 :: fraction of tip radius (r/p) 
 
 x Q value of x corresponding to r ' . . i't q /R) 
 
 % s ratio of spinner radius to tip radius 
 
 5 number of blades 
 
 b blade section chord 
 
 c wing reference chord 
 
M- 
 
 : ] ttive blade section chord 
 
 or 
 
 b/D 
 
 b o.7|jR (b//D) 0.75R 
 
 a' = b - 
 
 V 
 
 a 
 
 S 
 V, 
 
 . 
 
 V 
 
 ■) 
 
 f x (a) 
 
 n 
 
 '0.75R 
 
 U ty at 0.1 
 
 L 
 
 \ 
 
 7 ' I 
 
 ' 
 
 =- stream vi 
 
 111 —.! 
 
 / 
 
 gp ? 1 Of 
 
 L ac c J 
 
 )n of gra by 
 exi s ] v€ ] : t prooe] L + a) 
 
 ■■ - 9 
 
 ! ! 
 
 I 
 
 S 1 ' ] 
 
 orac bice, 1 di a i. or More) ) ' 
 
 - - 1-'~ I amic urs 
 
 foncti sn 
 
 
 J 
 
 , ilar 
 
 b o r 
 
 
 '1 + a) 
 
 (1 + e (1 + 2s 
 
 L + (1 + 
 
 :a) 
 
 =i 
 
 h (1 +■ ^a)^ 
 
 advar.ce-u - 
 
 bib 1 = to refe - 
 

 
 a 
 e 
 
 v. 
 
 'L 
 
 blade angle to zero-lift chord 
 
 angle of bl9.de relative to Y-axis measured in 
 direction of rotation 
 
 effective helix angle including inflow and 
 
 rotation (tan -1 
 
 Vr 
 
 'a 
 dP 
 
 T 
 T c 
 
 c T 
 Q 
 
 <^c 
 
 w 
 
 X,Y,Z 
 
 angle of yaw, radians 
 
 effective angle of attack of blade element ( p - $) 
 
 angle of sidewash in slipstream far behind propeller 
 
 nominal induced angle of sidewash at propeller 
 disk 
 
 effective average induced angle of sidewash at 
 propeller disk 
 
 sidewash velocity far behind propeller 
 
 a 1 r p 1 an e lift c o e f f 1 c lent 
 
 blade section lift coefficient 
 
 blade section profile-drag coefficient 
 
 slope of blade section lift ciirve, per radian 
 (dc,/da; average value taken as 0.95 x 2tt) 
 
 force component en a blade element in direction 
 of decreasing 9 (See fig. 1.) 
 
 thrus t 
 
 thrust coefficient (t/PV 2 D 2 ) 
 
 thrust coefficient \T/pn d m) 
 
 t orque 
 
 torque coeff ic3 ent (fy/pV^D*) 
 
 weight of airplane 
 
 forces directed along positive directions of X- , 
 Y-, and Z-axes, respectively 
 
L>M,N momenta about X-, Y-, and Z-axes, respectively, 
 n.se of right-handed screw; in appendix B 
 
 id ." ire 9 ? refers to the fr e-s1 • ;h 
 
 nurrber 
 
 •: J ach number for pre Ler side force 
 
 ( S < • ) 
 
 A', ; : , ^ ; , D' defined in ions ( '. ) 
 
 a', b', c', ; rals del' : ; and 
 
 0) 
 
 b,, b 5 j lis : ' ■ 
 
 I, sic"--- ' tion (I4.I) ( -— 9 r 
 
 1 \V J 
 
 3 
 L 1 
 
 m 
 
 v a 
 
 J Y 
 
 C ' 
 
 ir 
 
 J Y 
 
 • ' ' 
 
 ) 
 
 A Tf 
 
 
 d ' 
 
 ] 
 
 ■ 
 
 defined 1 ('hk) (2 dual-rotatii 
 
 ] lers) 
 
 cor fi Dr del' Lned 
 
 sic 1 . 1 factor I ation 
 
 spinner facto 
 
 . 
 i • Le-forci : ffici< 
 
 pitc] -nc I 3 Deff j - 
 
 de-force t bh r s 1 ect tc yai foC v '/;^ 
 
8 
 
 c ' 
 
 m 
 
 \y 
 
 V 
 
 Ditching-moment derivative with respect to yaw 
 
 side-force derivative with respect to Pitching 
 6Cy r 
 
 c m\ 
 
 V 2vy 
 
 ling-i 
 pitching 
 
 oi tching-moment derivative with resoect to 
 00 m 
 
 o 
 
 »rojected side area of propeller (See footnote 1.) 
 A aspect ratio 
 
 Subscript-, ; 
 O.75B measured at O.75R station <x = O.75) 
 
 - T 2^ 
 
 p * 
 
 e 
 
 k 
 
 max 
 stall 
 
 divided by PV^D'" if a force, by pV^D- i if a moment; 
 designates quantities corrected for compres- 
 sibility in appendix B and figure 9 
 
 effective 
 
 inde:: .' a.t takes the valuer, 1 to B to designate 
 a :. . .ilar propeller blade 
 
 rax! mi 
 
 at stall 
 
 A bar over a symbol denotes effective average value. 
 
 ANALYSTS 
 
 Proneller in Steady Axial Plight 
 
 The section shown in figure ] is part of a right- 
 hand propel 1 ' - blade moving to the right and advancing 
 upward. I lonents or the relative wind are V„ 
 
 and Vq, where V* a is the axial velocity including the 
 Inflow and Vg is the rotatiorial velocity including the 
 
slipstream rotation. The force component in bhe direc- 
 tion of d>cr c ising is: 
 
 IP = dTj sin 0f + dD cos $ 
 
 t /c 7 S'h if + Cj cos jfl 
 = |v.Adi p ■ ^2 
 
 = |v 2 bdr ff. (pf)] (1) 
 
 and th( sontrlbution to the thru; 
 
 dT = d r , ccs / - 
 
 , /c 7 / - Cj r" .1 A 
 
 i... , \ / ) 
 
 _ 2 , ^ __. .. 
 
 = " (2) 
 
 The equati y be div'.i . | 
 
 the terms to nondim< ] T na r V a =V(l + a), 
 
 there res ilts 
 
 - — . {__ _ f ( /) 
 
 4 D 
 
 where 
 
 and 
 
 dP 
 c 
 
 dT £ __i. . 
 
 r : ' '• - r - - 
 
 r 
 R 
 
10 
 
 Propeller under Altered Flight Conditions 
 
 r orce com pon e nts on bl ad e element .- In equations (1) 
 and (2) for dF and dT, V a occurs explicitly in the 
 factor V a and implicitly in and in terms depending 
 on 0; Vq occurs only implicitly in and in terms 
 
 deoending on . The relationship is - tan 
 
 • 1'- 
 
 V 
 
 e 
 
 which can be seen in figure 1. 3y partial differentia- 
 tion, therefore, the increments in dF and dT due to 
 any small changes whatsoever in V fi and V & are, for 
 fixed blade angle, 
 
 6(d 
 
 „, = m± m. dVo + 
 
 6v 9 
 
 6(dF) 6(dF) k# 
 
 dV 
 
 a 
 
 and a similar expression for 5(dT). The substitution 
 of equations (1) and (2) gives, when put in nondimensional 
 form, -. 
 
 B/ . , (1 + a) 2 b 
 B(dF o> = k D dX 
 
 = /J » (1+a) b , 
 
 o(dT c ) = --^ 5 dx 
 
 dV c 
 
 W 6f l 
 
 + dv 
 
 /4£i + M 
 
 6f, 
 
 ; w e av 9 6? TQV a \j a + v a o^ 
 
 dV r 
 
 A.-v 6t, 
 
 °y 1 
 
 6V 9 d?T 
 
 /2t 60 6t\ 
 
 + dv a k + T- £ - T3*J 
 
 \v a 6v a djj/ 
 
 The following abbreviations are helpful: 
 
 >(3) 
 
 y 
 
 A' 
 
 - d ? 
 
 Ui l 
 
 
 V a 
 
 6V 8 
 
 60 
 
 
 P T 
 
 " a 
 
 _ 2f l 
 V a 
 
 6$ 
 
 Of, 
 
 £1 
 
 V 
 a 
 
 6T< 
 
 6 tj 
 
 6? 
 
 
 D' 
 
 v a 
 
 . 2t l 
 
 Va 
 
 
 dt, 
 
 " = @/(5)o. 75R 
 
 
 > 
 
 (1+) 
 
11 
 
 f, I t. a: Lned in equations (1) end (2), 
 
 1 X 
 
 : f :v:i , equations (3) bocoire 
 
 v / \ v v / 
 
 ~\ 
 
 [idx 
 
 \ 
 
 / (3 
 
 I = ; ) [C + 
 
 •' 11 1 letors ere nondimensional. 
 
 D ! 
 
 /(5) 
 
 ti dx 
 
 Fore I ite proiel ler . - 
 
 onent-fo nt r due to 
 
 red f] Lc n an element of & ~ . 1 ■ . divided 
 
 ty pV^D b incri ts experienced 
 
 ' of 3 blades, ,,r ith respect to 
 
 the body axes ire 2, may be written as 
 
 Pore: s: 
 
 X - 
 
 > / 5(dT) k 
 
 1 ' 
 
 Y = 
 
 
 5(dP) n sin 8, 
 k k 
 
 > / 5(dF). cos 9. 
 
 (6) 
 
 (7) 
 
 
 Moments : 
 
 L - '-•:) 
 
 
 Z 1 
 
 J 
 
 k=I w r o 
 
 / ro(dT) v sin S 
 
 k-1 o 
 
 k 
 
 (9) 
 
 (10) 
 
12 
 
 k=3 
 
 I = - "S / ro(dT) v cos 9, (11) 
 
 where the subscript k refers to the kth propeller 
 blade. In order to obtain th<= nondimensional form X, 
 
 Y, Z, and T are divided by PV 2 D 2 to give X Q , Y„, Z c , 
 
 and T and L, M, M, and Q are divided by PV 2 D* to 
 
 give L , M , N , and . Tn the equations (6) to (11), 
 ceo 
 
 6(dP), becomes 6(dF„), , 6(dT). becomes 6(dT_). , and 
 
 r becomes — = — . The limits of integration become 
 x to 1, where x = — ~ . 
 
 O O h 
 
 stability derivatives of propeller .- The analysis 
 up to this point has been of a general nature in that the 
 formulas are applicable, for a fixed-pitch propeller, to 
 any type of deviation from steady axial advance - that 
 is, the formulas may be used to calculate all the stability 
 derivatives of a fixed-pitch propeller. In addition, the 
 formulas are applicable to those stability derivatives of 
 a constants-speed propeller that arc not associated with 
 changes in blade angle. This restriction could be re- 
 moved, however, by extending the analysis at the outset 
 to include a term in dft n . 
 
 A particular stability derivative can be obtained 
 by determining and substituting in equations (5) the 
 values of dV Q /v a and dV a /V a appropriate to the 
 
 motion under consideration. For dual-rotating propellers 
 equations (S) mast be set up independently for both propeller 
 sections with signs appropriate to the respective direc- 
 tions of rotation. Values of dVg /V a and clV /V that 
 
 are average for both sections are used for each section. 
 Note that dVq is the change in the component of the 
 
 effective relative wind acting on a blade in its plane 
 of rotation and dVq must therefore include the effect 
 
 of any changes induced by the motion In the rotational 
 speed of the propeller relative to the airplane. 
 
13 
 
 The possible i ! scelerated motions of a propeller 
 
 prise ■ it (1) at a steady angle of yaw, (2) at a 
 
 dy i , (3) ^ ' th un angular velocity of 
 
 v., 'I ) liar 'City of pitch, (5) with an 
 
 angul of roll, [6) with an increment in f or- 
 
 igination of these. It is clear 
 from the symmetry of the propeller that motions (1) and 
 (k) are similar ar.d motions (3) and (l±) arc similar. 
 Accordingly, of bhe six possible deviations of a pro- 
 peller from u *iven mode of steady axial advance, only 
 four an listinct. These fear may be taken as 
 of '•-< y velocitj of pitch q, angular veloc- 
 
 ity of , ■ ' ■ in forward velocity. 
 
 Glauert La- n in reference 3 J Lther yawed 
 
 flight nor flight - Locity of ' ih, 
 
 when these dj nail, -s the torque on 
 
 the prep . ■■'■■, )de will tend to 
 
 : tational v<= nd derivatives with re- 
 
 ;t t( t ar velocii ~; Ltch are independent 
 
 of th ra e of change of engJ r< volu- 
 
 tions. ur1 . ■ 1 I fivatives ob- 
 
 i ■ ied for a f! - tcl r [ually applicable 
 
 to a constRnt-speed oropeller becaua I sonstant-speed 
 
 jhanism is not br t ition. 
 
 r velocity of roll id ' snt in for- 
 
 ward velocity clearly affect the torque of the propeller. 
 
 en wil] attempt to titer its revolutions to attain 
 
 an equil im value. I Ller has fixed pitch, 
 
 the ad, nt will take pla ■ and its amount will depend 
 
 m the lion of engine torque with eng 
 
 revolutions for th articular engine used. er- 
 
 ence 3«) ' '- propeller is th constant-speed type, 
 
 pitch-change m< : tism will atte: \ t to alter th: blade 
 pitch; the resulting change in aerodynamic torque opposes 
 the change in . volutions. fluctuations in rotational 
 
 and tl v riations in aerodynamic torque 
 
 it of the propeller ar th n functionally related 
 to bhe lav of control of th-: pitch- •. mechanism ard 
 
 the dynamics ' its operation. ! > 1 ference 11.) 
 
 Th I 111 be lii Lted to a study of the 
 
 effects of yaw and of • ' ar velocity of oitch. In the 
 
 following sections dVp/V and dV /V a are evaluated 
 for yav jtion. 
 
11+ 
 
 Propeller in Yaw 
 
 Ratio dVn/V a for yawed motion.- The increment dVg 
 
 is the component parallel to Vg of a side-wind velocity 
 computed as follows: The velocity V a is regarded by 
 analogy with wing theory as passing through the propeller 
 disk at an angle \|/ - e* to the axis, where \|/ is the 
 angle of yaw and e' may be termed the "induced sidewash 
 
 le" (fig. 2). The side-wind velocity, for small 
 values of both \|/ and €', is accordingly V a ( >\> - e' ) . 
 
 The sidewash arises from the cross-wind forces. 
 These forces are the cross-wind component of the thrust 
 T sin i]/ and of the side force known to be produced by 
 yaw Y cos \|/. (See fig. J.) The analysis is restricted 
 to small \J/; these components are then approximately T\|/ 
 and Y . 
 
 If the sidewash velocity far behind the propeller 
 is v , the induced sidewash at the propeller may be 
 
 ta^en as v. /2 bj analogy with the relation between the 
 
 induced downwash at a finite wing and the downwash far 
 behind the wing. Note that 1 diameter may be considered 
 "far" behind the propeller as regards the axial slip- 
 stream velocity; 95 percent of the final inflow velocity 
 is attained at this distance. 
 
 As a first approximation, thrust and side force are 
 assumed to be uniformly distributed over the propeller 
 disk; corrections due- to the actaal distributions are 
 investigated in appendix A. Under this assumption the 
 momentum theory, supported qualitatively by vortex con- 
 siderations, shows that the slipstream is deflected side- 
 wise as a rigid cylinder. The sldewise motion induces 
 a flow of air around the slipstream as in figure L.. The 
 transverse momentum of this flow is, according to Munk 
 (reference 12), equal to the transverse momentum of an- 
 other cylinder of cir having the same diameter as the 
 slipstream at ail points and moving sldewise with the 
 same velocity as the slipstream boundary. Mote that the 
 air within the slipstream has a greater sldewise com- 
 ponent of velocity than does the slipstream boundary. 
 
 Far back of the propeller the ratio is — a = 1 + 2a. 
 
 V 
 
 The time rate of change of the transverse momentum of 
 
 the air flowing at free-stream velocity through this 
 second cylinder should be included in setting up the 
 momentum relations for the sidewash. 
 
l r ; 
 
 By equating 1 oss-wind force to the total t' 
 
 be of cha. ige of i - I 
 
 T\|/ + 
 
 v - - 
 
 
 2 
 
 v + 
 
 i v/-S 25? 
 
 i: 
 
 to ' Lrst ordei in v|f, where the first term on the 
 
 e contribution of the slipstream and t 
 
 second term is the contribution of the air displaced by 
 the Lipstrei A " ^,h ,***,„ k,, r7r2^2 
 
 a 
 
 A n dividing by PV~D and us ' : 
 rel I is V n = V(l + a) - = V(1 + 2a), 
 
 v./2 
 
 1 1 
 
 
 + " ) 
 
 2 r 
 
 (i + ; ) 
 
 (12) 
 
 
 
 Her. 
 
 va 
 
 
 of 
 
 
 
 
 in 
 
 
 is 
 
 the 
 
 e' is t] ied angle of 3idewash at the prc- 
 
 Glauert (reference ':) deduces almost twice this 
 
 ■ all values of a by negl< sting the reaction 
 by the slipstrea . 
 
 shown earlier that the effective s:de wind 
 Le of the iropeller is V_(\J< - «' ) and 
 component parallel to Vq j that is, 
 
 - V. M - c ) sin 9 
 
 ( 13 ) 
 
 val 
 tl sry, 
 
 ue of e' from equal fl_ , > may be introduced 
 
 rel P_ = a(l + a), from le momentum 
 
 - be used to ' <te 
 
 T 
 
 rs results 
 
 r 
 
 aH - f x (a)^ 
 
 (] + a) 2 ' U3a) 
 
 ■re 
 
 ) = 
 
 a) 1.(1 + a) + (1 + 2a) 2 ] 
 
 1 + (1 + 2a) 
 
 (111) 
 
16 
 
 and 
 
 f ( .)= ^ + 2 »> 2 2 (15) 
 
 1 1 + (1 + 2a) ^ 
 
 Ratio dV /V a for yawed motion.- As V a = V(l + a) 
 for unyawed motion, the changes produced by yaw are 
 
 £Xa = ^/ + _da_. s _da_ 
 V, V 1 + a 1 + a l±u; 
 
 2 
 if dV/V, which is cos v|/ - 1 c ~— , is neglected as 
 
 being of the second order in ty 
 
 o 
 
 In order to evaluate da, figure 2 is first con- 
 sidered. The component of the effective side wind in 
 the direction opposite to the blade rotation is 
 dVo = V a (vi/ - e' ) sin 9. This comoonent acts to increase 
 
 the relative wind at the blade, and therefore the thrust, 
 in quadrants 1 and 2; it acts to decrease the relative 
 wind, and therefore the thrust, in quadrants 3 and ij.. 
 Fore exactly, the change in thrust due to the side wind 
 is distributed sinusoidally in 9 . It is clear that 
 this incremental thrust distribution by its antisymmetry 
 produces a pitching moment. 
 
 ''omentum considerations require an increase in in- 
 flow in quadrants 1 and 2, where the thrust is increased, 
 and a decrease in inflow in quadrants 3 and ii, where the 
 thrust is decreased. The variation should be sinusoidal 
 in 9, and the assumption that the variation is directly 
 proportional to the radius is sufficiently accurate for 
 computing the effect on the side force. Such a represen- 
 tation is illustrated in figure 5« The analytical ex- 
 pression is 
 
 dv = Vda 
 
 = Vr sin 9 (17) 
 
 where k is a constant to be determined. Applying the 
 momentum theory to evaluate the pitching moment M in 
 terms of the inflow modifications produced by the pitching 
 
 moment gives 
 
l? 
 
 M = 
 
 ^TT 
 
 pV r dS dr r sin 9 (2 dv) 
 
 <s o i/o 
 By substitution of the relation for dv, 
 
 p 2^ 
 
 M = 2k p 
 
 T Ipon integration, 
 
 1 J i/o 
 
 r- - si ^9 d9 dr 
 
 " 
 
 <•: = 
 
 where 
 
 (1 + a)rrR 
 
 
 r,5 
 
 but, by [uations (l6) and (17) > 
 
 or 
 
 where 
 
 dV. 
 
 da 
 
 1 + a 
 
 kr sin 9 
 
 v / 
 
 A 
 
 1 + a) 
 
 dV 
 
 a 
 
 a 
 
 x = 
 
 " , x s i n 9 
 
 (1 + a) 2 rr 
 
 i 
 
 (18) 
 
 S urnr. a t i o n o v e r blade index k . - The z orip o n e n t - v e 1 o c i t y 
 increments due to yaw have been obtained in the preceding two 
 sections as 
 
 dV Q 
 
 Q + a)* 
 
 f(a) 
 
 Y 
 
 f,(a) -a- 
 
 1 rr 
 
 sm 
 
 (13a) 
 
18 
 
 dv a 
 
 v ~ ( 
 
 a 
 
 1 + a)2 ^ " 
 
 (18) 
 
 where the subscript k has been added to refer to con- 
 dition? at the kth propeller blade. These values of 
 dV /7 g and dV a /V a May be substituted in equations (5) 
 
 to field values of 5'dF.) and 6(dT_). The values of 
 6(dF c ) and 5(d? c ) thus found may be inserted in equa- 
 tions (6) to (11), which give the several forces and mo- 
 ments the propeller might conceivably experience. 
 
 The summations over k indicated in equations (6) 
 to (11) affect only the factors involving sin 9^_ and 
 cos 9 V . The several factors arc, upon evaluation, 
 
 k=3 k=3 
 
 / sin 0, = y sin 6, cos B = 
 k=l " k=l 
 
 > 
 
 ft— J3 
 
 > sm^ 8, = — 
 
 If B = 2 or 1, 
 
 k=B 
 
 k=l 
 
 i B, 
 
 sin 4 - 9 k = — (1 - cos <^9 ) 
 
 but the average over 9 is B/2. 
 
 k*B p 
 The nonvanishing factor XT sin'- 9, occurs only 
 
 k=l 
 in equation (7) for the side force Y and inequation (10) 
 for the pitching moment V . n lhe other hypothetical forces 
 and moments that might be produced by yaw are, accordingly, 
 a] I zero. 
 
19 
 
 n the re li + .ion 
 
 k=3 
 
 
 X ; sin 2 6,. 
 
 
 — — 
 
 Z. * 
 
 2. 
 
 k=l 
 
 
 is used, equatj ) and (10) become in nondimensional 
 
 form 
 
 ^ 8 vVo.75* *4U 
 
 v 
 
 a)* - f.(a) — 
 
 1 ~ 
 
 L 
 
 L6. 1 
 
 A' + — M c 3'x>ixdx (19) 
 
 ! »o = ^) f% »♦-',<■>-* 
 
 i/: 
 
 I: 
 
 ^c 
 
 i • rr 
 
 *__.! 
 
 C'+ — P ! c D'x^xdx (20) 
 
 1 
 
 For simplicity the following additional abbreviations 
 are Introduced: 
 
 O' = 3 
 
 D / 0.75R 
 
 a ' = - / .: •' ' dx 
 
 n ' — - 
 
 It 
 
 l. 
 
 ^ 
 
 (21) 
 
 = i /" 
 
 u-. 
 
 [J.C-B 1 )a dx 
 
 iiC'x dx 
 
 a 
 
 ^ i 
 
 - -;- / n(-D«)x 2 dx 
 x. 
 
 > 
 
 ■ ' is have been chosen to make a' , b' , c' , 
 
 and d' positive quantities. 
 
20 
 
 Solution for Y, and M for single-rotating 
 
 propel lers . - with the preceding substitutions, equations 
 (19) and (20) become 
 
 Y. 
 
 ^o 3 
 
 r 
 
 L 
 
 1J f(a)+ - f.(a) I« 
 
 8 \_l 1 n. 
 
 a ' - — F-, b ' 
 rr ° 
 
 (22) 
 
 f(a)M/ - f^a) -^ 
 
 16 .. . A 
 
 M„d' 
 
 TT C 
 
 m 
 
 These are simultaneous linear algebraic equations 
 : and M . The solution for Y is, after 
 
 simplification, 
 
 Y 
 
 TT 
 
 f(a) a. 4. - , g '^ c ') 
 
 \ 1 + a'd'/ 
 
 fi ( a) 
 1 + a' _i 
 
 8 
 
 4, . Q'b'cjN 
 \ 1 + o'dy 
 
 which may be written in the form 
 
 Y 
 
 TT 
 
 8* 
 
 f(a) o'a' 
 
 A' 
 
 f'l(a) 
 
 — a' a' 
 
 8 
 
 where 
 
 A' = 
 
 o'b' c ' 
 1 + a'd 
 
 (23) 
 
 (21*.) 
 
 Numerical evaluation shows that the denominator of equa- 
 tion (25) does not differ greatly from unity; therefore, 
 Y c is roughly proportional to a'. 
 
 Similarly, the solution for M_ is 
 
 TT 
 
 2 
 
 M„ = -\{/ 
 
 f(a) a'c» 
 
 [ 3 
 
 + a' f (a) 
 
 (1 + a'd' ) - o» f x (a) b'c> 
 
21 
 
 - be put in the form 
 
 rr [a) m 
 M c = p fTTa) (25) 
 
 1 + -— — a' ( a' - t 1 ) 
 
 ':Ve 
 
 a'c' 
 
 m = (26) 
 
 2(1 + o»d') 
 
 relative LI les of the quantities are such that 
 
 |'-' is roarhly 03 '. ional to ;'. 
 
 Solut ' and M for dual -rotating pro- 
 
 3 
 
 Here . - The foj equations . Lj only to sj l Le- 
 
 rotating rs. dual-rotating propellers the 
 
 .~try of + ' loading, which for a single- 
 
 >duces the pitching moment c'ue to 
 , is iposl ! over the front and rear sec- 
 
 tions. "The resultant over-all disk loading, therefore , 
 
 and gj ves rise to a negligible 
 pitching moment - that is, 
 
 M c « (27) 
 
 The induction ef: its associated with the respective 
 disk-loading asymmetries of the front and rear- ieller 
 
 sections very nearly cancel even though there is a finite 
 separation tv en the two sen ' 3. This fact, which 
 
 '■'■ regarded as a con° :. ie of the relation (27), 
 is represented by cutting V = in equation (22). 
 The result is 
 
 f(a) o'a' 
 
 v 
 
 c G 1 (26) 
 
 1 + - f (a) o'a' 
 
 This equation differs from equation (<^3)» which 
 applies to single rotation, in that unity reolaces the 
 
 larger quantity — in the denominator. The 
 
 a' - &« 
 
 side-force coefficient v r is therefore larger in the 
 case of dual rotation. With data for conventional 
 
22 
 
 propellers, the increase averages about 18 percent and 
 reaches ~^<± percent at low blade angles. 
 
 The increase in side force ic due to the lack in the 
 dual-rotating propeller of the asymmetric distribution 
 of inflow velocity across the disk which, for the single- 
 rotating propeller, is induced by the asymmetric disk 
 loading. The inflow asymmetry is so disposed as to reduce 
 the change in angle of attack due to yaw on all blade 
 elements. e behavior is analogous to that of downwash 
 in reducing the effective angle of attack of a finite 
 wing. 
 
 The inflow asymmetry is not the only effect analogous 
 to downwash in wing theory; the sidewash of the inflow is 
 another such effect and serves to reduce the side force 
 still further. Sidewash is, however, common to singie- 
 and dual-rotating propellers and affects both in the same 
 way. An examination of the steps in the derivation shows 
 
 that the term ^ f ( a) a'a' in the denominator, the 
 8 i - 
 
 absence of which would increase the value of Y , is due 
 
 to the sidewash. 
 
 Equations ( 23) to (28) give the stability derivatives 
 of single- and dual-rotating propellers with respect to 
 yaw, but the results are not yet in final form. There 
 remain the evaluation of a', b' , c', and d' and the 
 introduction of a factor to account for the effect of a 
 spinner and another factor to correct for the assumption 
 of uniform loading of the side force over the propeller 
 disk. 
 
 Exolicit representation of 
 
 b ' , c ' , and d ' . - 
 
 Equations (21) show a', b', c', and d' to be integrals 
 involving the functions A', 3', C ' , and D' , respectively, 
 which are defined in equations (h.) in conjunction with 
 equations (1) and (2). The quantities A', 3', C, and 
 D' . are, upon evaluation, 
 
 3' 
 
 sin + c cos 
 a 
 
 (29) 
 
 C C O 5 ,0 - C ( S i n fi + C3C 0) 
 
 L a l 
 
23 
 
 C = c 7 cos - c (sin 0-2 cso 0) 
 L a l 
 
 f cos 2 
 D' - - ic, 7~ - c cos 
 
 if terms in the coefficient of orofile drag c, are 
 
 d o 
 neglected as being small in comparison with the terms 
 in c . The neglect of c - is valid onlj for values 
 
 'a o 
 
 of not too near 0° or 90°. 
 
 Prom figure 1 , p = + a. Then, for a in the 
 unstalled range, 
 
 sin p r = sin $ cos a + sin a cos 
 
 ~ sin + a cos 
 and 
 
 : sin p ~ c sin + c a cos 
 
 c. 
 
 a 
 
 = c sin 0" + c cos 
 a 
 
 the right-hand member of which is just A' in equation (29) 
 _s relat'on provides tne important-: that, although 
 both and c 7 depend on the inflow, the slipstream 
 
 rotation, and trie value of V/nD, the function A 1 is 
 independent of these quantities and uepenus solely on the 
 geometrical blade angle (3 . Is relationship leads 
 directly to the interpretation, to oe established eresently,. 
 
 ; : the effective fin ares of a.oropeller is essentially 
 the projected side area. 
 
 The introduction of (3 does not succeed in similarly 
 eliminating and c 7 from B', C ' , and D' but does 
 result in a simplification in E 1 and C. The summarized 
 results are, to the first order in c, with D' left 
 unchanged: 
 
2k 
 
 A' = c 7 sin p 
 a 
 
 B' = -10 
 
 cos (3 - c esc ^ 
 
 o "J 
 
 C = c cos (3 + 2c esc 
 
 D 1 = -(c 
 
 :0 S^ (# 
 
 - c 7 cos 0, 
 
 The integrals (21), in which A', 3', C ' , and 
 occur, must now be evaluated. Upon substitution, 
 
 D< 
 
 a' 
 
 = — / ix si 
 
 n p dz 
 
 "O 
 
 il 
 
 C 7 1 
 
 7 a P 1 / )J - 
 
 : ~~ J \ix cos p dx - — / uxc csc ,25 dx 
 
 -o x o 
 
 IT 
 
 c' = / U.X cos S, dx +T? / LAXC esc dx 
 
 ) (30) 
 
 d' = 
 
 a 
 
 .«2 
 
 X 1 " CO-' 
 
 1 
 
 sir 
 
 -i 
 
 dx 
 
 Z' 1 
 
 ji.x^a cos dx 
 
 x. 
 
 where 
 
 b 
 
 H 
 
 0.75R 
 
 and b is the blade width. 
 
 E valuation of a ' , b ' , and c ' . - The integral a' 
 is already in its simplest form in equations (JO), as is 
 the first integral of b' , which is identical with the 
 
25 
 
 first integral of c'. If the first integral of b 1 
 as b-, , 
 
 b» = b, - b. 
 
 C = b 1 + 2b 2 
 
 } 
 
 (3D 
 
 whe re 
 
 - _l£L 
 
 /» 
 
 b n —"' / ax cop 3 dx 
 
 o = — / u::o 7 C3C $# dx 
 
 (52) 
 
 In the attenrot to eva] b Lt i as found bhat, 
 
 if the blade section coefficient of profile drag and the 
 
 rotation of the si:. pstream are ne, 3d, the thru 
 coefficient is 
 
 ■ ' OTt2t,2 
 
 P 1 fi 4- a 
 
 O ro' 
 
 kJ 
 
 [j.xc cso dx 
 
 6 
 
 where 
 
 .T - 
 
 nD 
 
 If an average value of 1 + a over the disk is used, 
 i + a can be taken from under the integral sign, and 
 
 liT, 
 
 D- 
 
 o' rr^ (l + a) 
 But, by the simple momentum theory, 
 
 W r„ - aa + a^ 
 
26 
 Therefore, 
 
 , _ _J_ 2a 
 2 a ' rr 
 
 A graph of the variation of 2a/rr with T^ is given in 
 figure 6. 
 
 Appr oximate evaluat ion of d' .- The contribution of 
 
 d' - -^ f 1 ^ ^Jl dx . !ia / , x 2 ac os dx 
 
 tt </. ^ sin tt J * 
 
 to Y is small. It is found, by using the largest 
 
 value which a may have without causing stalling of 
 the blades (about i/k radian), that the second integral 
 can be neglected, with the result that 
 
 , , _ a / o cos^ , 
 
 d' = •——- / ax' -t- dx 
 
 n iX sin 
 
 J o 
 
 Note that involves the inflow velocity and the slip- 
 stream, rotational velocity. These velocities, if assumed 
 to be constant over the propeller disk, may easily be 
 related to T_ and 5,,, , respectively, from momentum 
 
 considerations. 
 
 Curves of d' have been computed for a typical plan form 
 (Hamilton-Standard propeller J155-&) an ^ are presented 
 in figure 7- This chart makes use of an altered notation 
 introduced later in the report: the ordinate is the 
 quantity 
 
 ij = t d ' 
 
 and a parameter is the solidity at O.75R, 
 
 1l8 /b 
 Trr Id 
 
 0.75R 
 
^7 
 
 Thp abscissa is V/nD. The error in computed side force 
 due to using this chart for plan forms other than the 
 Hamilton-Standard 5155-6 should be negligible. The 
 chart is not sufficiently accurate, however, for precise 
 computation of the pitching moment due to yaw. 
 
 Correction for nonuniform distribution of side force .- 
 
 The induce 
 
 foregoing 
 and the si 
 the pro 
 
 si dew ash d 
 tribution 
 thi-. assume 
 reciabl 
 side force 
 
 d sidewash angle c' 
 is based on the as sump 
 de force are each unif 
 ler disk. The error 
 ue to the assumption o 
 can be shown to be sma 
 t i on of uniform side-f 
 e. The effect of thi 
 is small, but not neg 
 
 as calculated in the 
 tions that the thrust 
 ormly distributed over 
 in effective aver- 
 f uniform thrust dis- 
 11: the error due to 
 orce distribution is 
 s error on the computed 
 Lbl . 
 
 The sid<~ force is actually distriouted over t. 
 
 pro r disk nearly as t . product of the integrand 
 
 of the„most important term in the side-force expression a 1 
 
 The integrand of a' is proportional to the 
 
 ea the sine of the 
 
 greatest toward the blade 
 2 
 
 snd sin d P. 
 blade width 
 tends to be 
 twist, and 
 
 blade e, which 
 
 roots dup to the 
 sin^- 9 has maximums at 6 = 90° and '^O . 
 The sid^ force is therefore concentrated near the blade 
 roots and along the Z-axis. In calculating an effective 
 averag of the part of e' due to the side force, this 
 distribution of the aide force is taken into account by 
 using in effect the integrand of a' , which is u. sin p, 
 times sin^ as a weight factor. The detailed treat- 
 ment is given in appendix A. There is obtained for the 
 effective average of the induced sidewash angle 
 
 i — 
 
 % (T * + k/j 
 
 (1 + a) 1 
 
 1 + 
 
 (1 + 2a) 
 
 (33) 
 
 where 
 
 ilk) 
 
28 
 
 is a correction factor derived in aooendix A. If e ' 
 is inserted for €' in the analysis for side force and 
 pitching moment, the corrected forms of equations (23) 
 and (25) are, respectively, 
 
 Y = Jty f(a)a'a< 
 
 a' 
 
 a' - &> 
 tt , f (a) m 
 
 v 
 
 , = 7^ 
 
 8 1 + fc.a 1 (a* - A 1 ) 
 
 where the abbreviation 
 
 f\'a) k. 
 
 /" (n sin 3-)" 
 
 <S* n 
 
 dx 
 
 V^-7— ? — ^ (55) 
 
 \l sin P Q dx 
 
 has been used. The factor k„ may be called the side- 
 wash factor. 
 
 Corr ection for augment! ve effect of spinner . - If the 
 
 spinner-nacelle or soinner-fusel age combination has a 
 fairly large fineness ratio, the circumferential component 
 of the side wind is speeded up in passing around the blade 
 shanks (fig. 8) by approximately the factor 
 
 <$' 
 
 1 + I 
 
 where 
 
 so inn er radius 
 
 X e- — — ~ ~ ~~ ■—- — — — — 
 
 R 
 
 constant (1 for fineness ratio 00 ; . 90 for fine- 
 ness ratio 6) 
 
. 
 
 This local increase in s ; Lnd is equivalent to an in- 
 
 crease in the an ] oJ raw 1/ 5 r. tiae same ratio. Thus 
 at rad.3 is xR the effective angle Ls 
 
 *o(x) - > 
 
 'y \ d 
 
 1 + ' 
 
 effective average yaw over the disk Ls ined 
 
 from the consideration that dY c is nearly proportional 
 
 to t! }i sin (3 of the dominant term a'. 
 
 ■' Ly * t] re , 
 
 Y = k* / 
 
 -*fe) 2 
 
 jj. sin 6 dx 
 
 = -1 P 
 
 u, 
 
 \i s - lx 
 
 wher k Ls a constant. jording 
 
 .1 . . 2 
 
 - 
 
 -- - ■ 
 
 r( x *Y ** 
 
 : V, 
 
 /" ,x 
 
 / a Sin [Ldx 
 
 i/jt 
 
 1 
 
 (36) 
 
 According to this result, if the prone Her is 
 equ with a sninner, ihc previously given expressions 
 for side force and pitching m ' * lied 
 
 by the 302 tant k , which may be termed the "spinner 
 factor. :| e value cf k is of the jrder of l.llj. and 
 varies slightly with bladi ;le. 
 
 1 definitions.- It is worth while to introduce 
 
 certain dtions at this point to put the : na] 
 
 ations in better form. The original definitions were 
 
30 
 
 chosen solely with a view toward clarity in presenting 
 the derivation. The principal change is the replacement 
 of 
 
 - (4. 
 
 75R 
 
 which is proportional t:> the solidity at 0.75^» by the 
 actual solidity at O.75R 
 
 = — C 
 
 5rr 
 
 _ Its (^ 
 
 3* \ D y 
 
 I (37) 
 
 0.75R 
 
 This change entails replacing all the integrals occurring 
 in the equation by Jt/Ij. times the iormer values. Thus 
 a' is replaced by I , b-, by J d' by I , and A' by A, 
 
 where 
 
 T 
 "1 
 
 3^r , 
 
 = 1r a ' 
 
 L 2 
 
 5H . 
 
 -r°i 
 
 T 
 
 
 A 
 
 3tt , 
 
 In addition, the following definitions are intro- 
 duced: 
 
 v 
 
 Y qs ' 
 
 o 
 
 n ■*■ 
 
 6C V 
 
 Cv' = ~ET7 
 
-r.d 
 
 
 
 "in 
 
 qDS ' 
 
 TT C 
 
 
 
 C ' 
 m v|/ 
 
 * a ty 
 
 where 
 
 1 
 
 propeller di sk a 
 
 re a 
 
 
 
 S' 
 
 
 The symbols Cy' and C have been so chosen in rela- 
 
 tion to the conventional side-force and pitching-moment 
 coefficients of an airplane C v and C m , that con- 
 
 version is obtained through the relations 
 
 IS 
 SJ_ 
 
 ■ 'AT- 
 
 -1 
 
 
 -I'-'V^ 
 
 - 
 
 ■ • d 
 
 S c G ^' 
 
 where S is bhe wing area and c is the wing refer- 
 ence chord. • ; . n all the foregoing v!/ is 
 
 raes n radians. 
 
 i on of a: <..e ' L lty . - It 
 
 is horn ■ ipendix I orrection for 
 
 bility is obtained by ' side fore 
 
 by \A - * ' /~ , "here M is related to the stream "ach 
 
 numb '' and v/nD :~v tl jurve of figure 1 . The 
 
 correction is valid Dnly below the critical Wach number 
 for the r ipi Her. 
 
Summ arized effects of yaw.- With the new definitions, 
 the side-force derivative for a single-rotating propeller 
 
 is 
 
 Y * qS ' 
 
 k e f(a) oL, 
 
 ' — (35) 
 
 It 
 -h' + k a*l 
 
 and the side-force derivative for a dual-rotating pro- 
 peller is 
 
 V, : 
 
 = 6Y/6Uf 
 qS' 
 
 
 k a f(a) c^ 
 
 (39) 
 1 + K a ol 1 
 
 For a single-rotating propeller the Ditching-moment 
 derivative is 
 
 r i - b¥ /^ J 
 ra \|/ qDS< 
 
 k„ f(a) m 
 
 1 + k o (i -a) Ur ; 
 
 and for a dual-rotating propeller the pitching-moment 
 derivative is negligibly small. 
 
 The side-force derivative may be corrected for com- 
 pressibility by dividing by y/l -■ M ^. The same cor- 
 rection may be applied to the pi t c hi ng -moment derivative 
 but with less accuracy. 
 
'i he i u a n 1 1 1 i e s 1 n v o 1 ved a r e : 
 Spinner factor 
 
 K/ (— ) a sin (3 
 </y _ V x I 
 
 Si cle wash f • c f or 
 
 Lr 
 
 ■ ■ ■ i 
 
 -u - ' **) 
 
 
 where 
 
 \ - 
 
 1 + (] 2a) 
 
 Inflo fac 
 
 -, / 1 + c 
 
 a 
 
 2 
 
 3 
 
 /:-N 
 
 3" Wo. 7 
 
 
 35 
 
 i? o dx 
 'x \ x / o 
 
 I 
 
 L y 
 
 , = f , a) „_ — (35) 
 
 ■ (1 + «) Ld * a) + a * 2«)^ (ll|) 
 
 1 4- ( 1 + 2a) ' 
 
 3olJ . t , at 1.75? 
 
 "=?:; (27) 
 
3k 
 
 T = f c / i sin (3 dx (I4.I) 
 
 I = f c, / it cos p q x dx (1+2) 
 
 2 - a t/. 
 
 1, =f e, / p, S2£-i x 2 dx (U3) 
 
 5 '- l a J sin 
 
 K - ^ k * «¥) 
 
 (W 
 
 o(l + al^) 
 
 ol, + 2 J 2 * 
 
 m = — 2 JL (| t5 ) 
 
 2(1 + oI_) 
 
 and in equation (56), for a nacelle fineness rstio of 6, 
 K ~ O.9O and, for a nacelle fineness ratio of co, K ~ 1.00. 
 
 The charts of figures 6, 7> 10 > an d 11 are provided 
 for determining 2a/rr, I 7 , f(a), and f-,(a)j respectively. 
 
 Required accuracy of k_ , k„ , and A.- To the degree 
 
 in which comparison with existing experiments establishes 
 the accuracy of the side-force formulas - about ±10 per- 
 cent average error - it is sufficiently accurate to use 
 the mean values O.L. for k and, for the usual-size spinner 
 
 fx s = 0.1b), I.1I4 for k . To the same accuracy, the 
 
 terms in J may be omitted from L, and I 7 may be set 
 
 equal to the average value 3, with the result that 
 
 1 + Jo 
 
Aval lability of c ^_a rt s o f s id e-f or ce d e ri_ vat i ve . - 
 
 In reference 1J is presented an extei : '" Les of charts 
 
 computed from equations (5^) to (ij.ii) for two convent ; i] 
 pro '-'s. e derivative G, ' is given as a function 
 
 of v/nD for blade an: Les ra - from 15° to 60° s d 
 
 for solidities from two blades to si , th sing] 
 
 rotation and dual rotation. In 1 rence lit is presented 
 a method of extra n where b ae1 of chart" i 
 
 be used for deteri ' ■' ' ' for conve I >nal oro- 
 
 pellers without resort to * Lnal eq ati ns ; to 
 
 ) 
 
 Pi tching-mom< I . - L evaluation 
 
 of 3V- ■ f a sing] -ro1 
 
 propeller in ys Is r of the momei 
 
 produce rce Li le for ■ acti] 
 
 of a level - L 1 th .1 is 
 moment is ' ted in air- 
 craft stabilit I ' : s. t the effect is a cross- 
 coupling be tv 
 
 El-rotat: ■ I 
 
 moment. 
 
 Pi )ler ' it to Lty of Pitc 
 
 tio d . ' - ■ ' ■ I .. tch. - The 
 
 31 1; ■ \ no di reel nation to 
 
 the rotationa] v- locit; >1; 1 I ' 1 eller 
 
 disk V . It is kn auert's wor (reference 3), 
 
 o 
 ver, that ■• Lves rise to a si and to 
 
 a nit- . j side J Lucei 
 
 that affects Vn , 03 in the of the yawed er. 
 
 The 1 e in V p is accord as the Lced 
 
 part of the tot l1 c )tion. 1 s j] 
 
 is obi by sett] = C ' . t ion (1J I : 
 
 s' n 9 
 
 fa) -^ (16) 
 
 V a (1 + 
 
 where 
 
 2(1 + 2 a) 2 
 
 (a) = — ^ — ' (15) 
 
 , p 
 
 1 + : 1 + 2 -: ) - 
 
36 
 
 P.atio dV„/V a for angular velocity of pitch.- The 
 
 direct increment, due to pitching, in the axial velocity 
 Y a Is qr sin 9. 
 
 The induced increment due to the nf ore-mentioned 
 pitching moment is, by equation (1&), 
 
 
 (1 + a) 
 
 C TT 
 
 sin 
 
 The total increment dV g 
 th e i nduc e d i no r° me n t s . 
 
 is the sum of the direct and 
 Therefore 
 
 dVc 
 
 V, 
 
 a 
 
 sin 9 
 (1 + a) 2 
 
 (i ♦ ,) as*. If 
 
 2V 
 
 TT 
 
 (hi) 
 
 Expressions for Y and M.,.- Upon introducing the 
 
 equations (lib) and (ij-7), the equations that result here 
 in place of equations (1§) and (20) for the propeller in 
 vav: are : 
 
 2 L 
 
 2M -i^ft 
 
 c 2 U \lJ 
 
 1 M z 11 
 
 fr 
 
 
 , Y c i 
 
 f (a) T A' + 
 
 loM c x 
 
 (1 + a ) -i — + — - 
 
 £ cb 
 
 f J 
 
 0.75V J . 
 
 -f\(a) -=■ C» + 
 1 n 
 
 -| A 8 ) 
 
 Solution for Y Q end M . - 3y using the abbrevia- 
 tions of equations (21), equations (li3) become 
 
 * =^f<!-Ma) 
 
 a' - 
 
 ft + a) S2 + L^2| b .l 
 
 2V TT 
 
2* 
 
 xr a 
 
 t ! 
 
 ["<! ♦ a) W + l^ 
 2V 
 
 TT 
 
 - — s - f ( a) — - c ' - f 
 
 6 j 'l [ - TT ' [ ( 
 
 which ar^ tneous linear algebraic equations in Y 
 
 1 
 
 7 
 
 ana 
 
 The .solution for Y and M e is, after 
 
 sirrplificati 
 
 Y c = 
 
 ji 
 
 2V 
 
 
 o 
 
 ,2 a 
 
 : 
 
 8 
 
 afl 1 - a) 
 
 (1 + al 3 ) 
 
 T,r - 
 
 • c 
 
 
 3I I £ 
 
 \ 1 + aTz, 
 
 f x (a) 
 -^75 — a A 
 
 1 6 
 
 fl(a) . 
 
 1 + — i*- — a(I. 
 
 A) 
 
 Side-force derivative ' and i itching -moment 
 
 - q 
 
 derivative C m ' .- Side-force sitching- moment ie- 
 
 rivatives may be del ollows: 
 
 C/ / 
 / \2V/ 
 V ~ 
 
 2L V^S' 
 
 2 
 
 fl + a) 
 
 f t - Ti2 
 
 C. TT 
 
 fl ♦ '- 
 
 Ma) 
 
 o 
 
 a(i ] - /J 
 
 (1 + ai ) 
 
.8 
 
 n ' 
 
 >»/(!) 
 
 q r V*D3 f 
 
 < + f ' i(a) A 
 
 T ( "T~ gI l fi(a) ^ 
 
 1 + a 0I 5\ 1 * QI ? 7 " ~U~ CTA 
 
 1 + -^ — a(i - a) 
 d J - 
 
 where 
 
 Hough approximations may be obtained by omitting the 
 induction terms - that is, the terms due to sidewash and 
 to inflow asymmetry. There result 
 
 C ' =* - (1 + a) crip (i;9) 
 
 ' q 
 
 (V ~ - (1 + a) — 1 
 
 q 2 
 
 ir 
 
 C ditto ■- r i 3 o w of _ar. ^;le of yav; wit h angular velocity of 
 pitch to p ro duc e s:ime side fo rce . - To the same rough 
 approximation as equation HIT) , 
 
 -' ,1, S 1 
 
 Die ratio of ^ to qD/2V to produce the same side 
 f o re e i i t h e re £ o r e 
 
59 
 
 
 c ' 
 
 Y i 
 
 qD/2V 
 
 f(a) k T, 
 
 3 1 
 
 ' dL 
 
 K ii 
 
 (50) 
 
 This ratio is of the order of unity. 
 
 _ s '-^ e ? orce & lf - to ° i t cn: i n 5 • " — "■- e 
 maximum side-force coefficient due to pitching occurs, 
 for Lade -an qD/2V is a 
 
 Mas ;.,, V in unstalled flight is de- 
 
 teri sceleration that the air- 
 
 oan develop, v r s determined by the maximum lift 
 
 coeffici int. 6 normal acceleration is 
 
 a n = qV 
 
 fro ^ - i 3h 
 
 qD _ a n D 
 
 -" 2V d 
 At a given speed the n am normal acceleration a 
 
 (3D 
 
 "max 
 
 could he realized at the top of an inside loop. The 
 rel at ion i e 
 
 \ - Do nward lift + Weight 
 
 => max 
 
 s ^ v 2 + - 
 
ho 
 
 or 
 
 !smz = fl^hm + -L^j g ( 52 ) 
 
 v V2W/S v y 
 
 The value of a /v 2 is greatest when V is least 
 
 max/ 
 
 If the discussion is United for the present to the 
 
 minimum soeed for level flight V .„,,. 
 
 stall* 
 
 _ ^rr.ax - 1 
 
 - 2 ^ V stall 
 From equation (52), 
 
 r ' n :nax _ __££_ 
 
 V 2 ' V 2 
 
 stall 
 
 and therefore, from equation (51), 
 
 2V/ " V 2 
 
 ' max stall 
 
 A oractj cal uooer limit to (qD/2V) at the 
 
 v n ' 'max 
 
 stalling speed would be afforded by a hypothetical fighter air- 
 plane having the following characteristics: 
 
 V stall = 75 mph 
 = 113 fos 
 D = 12 ft 
 - V 2 - 2 y 12 
 
 ^ v /ma.x (HO) 2 
 = 0.332 
 
ill 
 
 By . ' r /^ : ) ' b !. le of y^" ; - ■ iians, thi ' 
 
 provide the same side forci is tely 
 
 
 
 If a mini jtj ub blad« an Di a1 ; s c3 ,; s 
 
 assumed, tb i ' I-./^o"! ' ' 1.13 for the 
 
 ] L] ..:■•; : - , (q '2V) 
 
 I ' producing : >rce to an s 
 
 o f y aw 
 
 = -1.1J x 
 
 = -O.O36 tj an -2.] 
 
 The resul T ■ " . - ! i • all. 
 
 The preceding data, it 
 
 to pitching, t the max ' jiable 
 
 ^ i t ■-■ • 1 rde r o f thi ■ : J I 
 
 ameter and the ■.-.■• - side for 01 
 
 due t ■ . leral conclu 1 ' side 
 
 for: >lies that tb ' due t o pitch - 
 Ls small vei : ti aver, with bh xc tion 
 
 of the spin, ■ i all ordinal s is : : 11. .b] . 
 
 _ Z2 C C— L'_1_lr__ 111- \ ar veloc I _of aw . - il ar 
 
 velocities of yaw as 
 
 angular velc tj s • >itc] . .e forces on a ieller 
 
 due to ys - ng are, like + :'. >se due t Ltchi ..', n Lble 
 
 except in the s sin . 
 
h2 
 
 PHYSICAL INTERPRETATION OF PROPELLER IN YAW 
 
 Concept of p rojecte d side area .- The area projected 
 
 by a propeller blade on a Diane through the axis of ro- 
 tation and the axis of the blade is 
 
 j b pin h V dr 
 
 o 
 The average area projected, by all the blades of a rotating 
 nropeller on any plane through the axis of rotation is 
 the prelected side area 
 
 b sin p dr 
 
 fl 
 
 
 B 
 
 
 r 
 
 
 
 
 i 
 
 p 
 
 
 c. 
 
 
 
 r o 
 
 where B is the number of blades. From this relation, 
 it can be < 
 
 Dressed as 
 
 it can be established that the product oL rr.ay be ex- 
 
 where S' = -~— is the propeller disk area. Thus o~l , 
 
 h 
 
 which figures so prominently in the expressions for the 
 
 side-force derivative Cy' , is proportional to the 
 
 - y 
 
 projected side ar«a of the propeller. In reference 1J, 
 I, is termed the "side-area index." 
 
 Effective fin area and aspe ct ratio .- Inasmuch as 
 D /S is the aspect ratio A of the projected side 
 area S n , it is also true that 
 
 c 7 n 
 
 ^ t - a o ( r\ \ 
 
 oi^- — - (5k) 
 
 Substitution of equation (53) in the numerator and equa- 
 tion (5I+) in the denominator of equation (39) gives for 
 a dual-rotating prone Her 
 
k3 
 
 6 v /.:>>!/ 
 
 r(a) q £ 
 
 
 • ' 
 
 a) q S v 
 
 - k. 
 
 k 1- 
 
 3 i 1 
 
 1 + . 
 
 v. 
 
 (55) 
 
 k ~ D - 1; on the and 
 
 
 0.95 
 
 for cor . " ■ correspondinj resslon for an 
 
 actui 1. ar^a and asp tio, at which 
 
 the local mic pressure Is f(a)q, is 
 
 bY/h^s 
 
 f(a) 
 
 
 D 
 
 z 
 
 (56) 
 
 n the lifting-line form of aspect-rati correction is 
 uceci. btii k . , whi :h :-■■ rely - ints for the 
 
 favors ! Lnterfi : ■ een spinner and propeller, 
 
 equation (55) Gan nt v form of equation (5&) 
 
 t rodu iin an I : t ratio 
 
 h 
 
 c 7 Q 
 
 «* A 
 
 2 
 
 " + follows that a dual-rotating propeller in yaw 
 
 acts like a fin of which the area is the projected side 
 . of the propeller, the effective aspect ratio is 
 
kh 
 
 approximately two-thi rds the side-area aspect ratio, and 
 the local dynamic pressure is f(a) times the free- 
 stream value. A single-rotating propeller may be shown 
 to act similarly, but the effective aspect ratio is mark- 
 edly less and is not so sir/ply expressed. A mean effec- 
 tive aspect ratio for both single- and dual-rotating 
 propellers is about 8. 
 
 Effective dynamic prea sure . - By the definition of 
 a, the expression V'l + a) is the axial wind velocity 
 at the propeller disk. Accordingly, (1 + a)^q is the 
 dynamic pressure at the propeller disk. The pressure 
 
 (1 + a)^q is only slightly greater than _f(a)q, the 
 effective dynamic pressure of equation (jb). Thus the 
 equivalent fin described in the preceding paragraph may 
 with small error be regarded as situated in the inflow 
 at the propeller disk and subject to the corresponding 
 augmented dynamic pressure. 
 
 C_o mp orison o f side force of s ingle - and dual -rotating 
 propellers . - it has been pointed out in the discussion 
 accompanying the derivation of y c and M c for dual- 
 rotating propellers in yaw that the dual-rotating pro- 
 peller averages l'C percent more side force than the single- 
 rotating propeller and that the increase reaches 3^ per- 
 cent at low blade angles. The detailed explanation is 
 given in the same discussion. Tn brief, dual rotation 
 eliminates certain induction effects associated witn single- 
 rotation: the dual -rotating propeller acts as if it has 
 a considerably higher asoect ratio and therefore develops 
 more side force for the same solidity. 
 
 ?' <T agni tud e of o ltchi n g mo me n t . - It has been shown 
 that yew gives rise to zero pitching moment for a dual- 
 rotating propeller and to a finite pitching moment, given 
 by equation (h.0) , for a single-rotating propeller. The 
 numerical evaluation of equation (iiO) for typical cases 
 shows that the pitching moment is of the order of the 
 moment produced by a force equal to the side force, acting 
 at the end of a lever arm equal to the propeller radius. 
 This cross-couoling between pitch and yaw is smr.ll, but 
 possibly not negligible. 
 
U5 
 
 PPOPELLERS IN PITCH 
 
 The results for propellers in yaw may be applied, 
 to propellers in pitch from considerations of symmetry. 
 The normal-force derivative of a propeller with respect 
 to pitch is equal to the side-force derivative of the 
 same propeller with respect to yaw, and the yawing- 
 moment derivative of a propeller with respect to pitch 
 is equal to the negative of the pitching-moment derivative 
 of the same prooeller with resoect to yaw. These re- 
 lations are invalid when the propeller is in the upwash 
 or downwash of a wing. (See reference 15, p- 12.) 
 
 COMPARISON WITH EXPERIMENT 
 
 Expe riments of Bramwell, Relf, and 3ryant .- The 
 experiments of Bramwell, ?elf~J and Bryant in 19li+ with a -four- 
 
 blade model prooeller in yaw (reference 15) are worth 
 noting because the experimental arrangement was designed 
 specifically for the problem. The balance was arranged 
 to yaw with the propeller and to measure the side force 
 directly with respect to body axes. Tare readings were 
 inherently small in comparison with the forces being 
 measured. Tunnel speed was calibrated by comparison of 
 thrust curves for the same oropeller in the wind tunnel 
 and on a whirling arm. 
 
 A calculated curve of C v '/*!>> which is the same as 
 
 C ' for small values of \{/, is comoared in figure 12 
 
 with the experimental values of reference 15. There is 
 included for further comparison the theoretical curve 
 calculated by ; "isztal (reference 9). The curve calcu- 
 lated from the formula of the present report appears to 
 give somewhat better agreement than that of iVIisztal but 
 the improvement is not conclusive. The orincipal objec- 
 tion to Misztal's formula remains the labor of its appli- 
 cation rather than its defect in accuracy. 
 
 E xperiments of Lesley, 'Vorl ey, and Moy. - In the ex- 
 periments of Lesley, V.'orloy, and T 'oy reported in 1937 
 (reference l6), the nacelle was shielded from the 
 air stream, with the result that only forces on the 
 propeller blades were communicated to the balances. A 
 
he 
 
 3-foot, two-blade -propeller was used. Measurements were 
 made of six components of the air forces on the propeller. 
 
 Calculated curves of C ',A are compared with the 
 
 experimental values of reference l6 for ty - 10° in 
 figure 13- Note that the original data of reference l6 
 were presented therein with respect to wind axes, and the 
 data have been converted to the body axes of this report 
 in the presentation of figure 1J. 
 
 Experiments of Runckel . - The most complete experi- 
 ments on yawed propellers - the only published experi- 
 ments on full-scale propellers -are those of P.unckel (ref- 
 erence 17) • Runckel tested single-rotating propellers 
 of two, three, four, and six blades and a six-blade 
 dual-rotating propeller. The diameter was 10 feet. An 
 attempt was made to correct for the wind forces on the 
 rather large unshielded nacelle by subtracting the forces 
 and moments measured with zero yaw from the corresponding 
 forces and moments measured with yaw at the same value 
 of v/nD. 
 
 Calculated curves of C '/fy , including a spinner 
 
 correction, are compared in figure 1)4 with the faired 
 experimental curves from reference 17 for 10° yaw. In 
 reference 17, as in reference 16, the original data were 
 presented with respect to wind axes and the curves have 
 been converted to the body axes of this report in the 
 presentation of figure la. In figure 15 the unpublished 
 experimental points for the single-rotating six-blade 
 propeller are presented for comoarison with the faired 
 published curves as converted to body axes. 
 
 A ccuracy . - From these several comparisons of the 
 theory witE experiment it appears that the average dis- 
 agreement is slightly less than ±10 percent. This 
 accuracy is of the order of that obtainable by the vortex 
 theory for the uninclined propeller when the number of 
 
 The propellers of reference 17 were actually tested in 
 pitch rather than in yaw out, inasmuch as pitch becomes 
 yaw upon a C0° rotation of the axes, this conversion was 
 made to keep the discussion consistent. In this con- 
 nection, a vertical force due to pitch has herein been 
 called a side force due to yaw. 
 
Itf 
 
 blades is tacitly assumed to be infinite bv the omission 
 of the Goldstein correction for finite number of blades. 
 The same assumption is made in the present analysis. 
 
 CONCLUSIONS 
 
 The foregoing analysis of propellers in yaw and 
 propellers subjected to an angular velocity of pitch 
 permits the following conclusions* 
 
 1. A propeller In yaw acts like a fin of which the 
 area is the projected side area of the propeller, the 
 effective aspect ratio is of the order of o, and the 
 effective dynamic pressure Is roughly that at the pro- 
 peller disk as augmented by the inflow. The variation 
 of the inflow velocity, for a fixed-pitch propeller, 
 accounts for most of the variation of side force with 
 advance-diameter ratio. 
 
 2. A dual-rotating propeller develops up to one- 
 third more side force than a single-rotating propeller. 
 
 3. A yawed single-rotating propeller experiences a 
 pitching moment as well as a side force. The pitching 
 moment is of the order of the moment produced by a force 
 equal to the side force, acting at the end of a lever 
 arm equal to the propeller radius. This cross-coupling 
 between pitch and yaw is small, but possibly not negligible 
 
 Ij.. Propeller forces due to an angular velocity of 
 pitch or yaw are negligibly small for the angular ve- 
 locities that may be realized in maneuvers, with the ex- 
 ception of the snin. 
 
 Langley Memorial Aeronautical Laboratory, 
 
 national Advisory Committee for Aeronautics, 
 Lang 1 ey Pi eld, Va . 
 
1+8 
 
 APPENDIX A 
 DERIVATION OF SIDEWASF FACTOR k Q 
 
 If the assumption that the side force is uniformly 
 distributed over the propeller disk is abandoned, it is 
 necessary to oroceed differently beyond equations (5) in 
 deriving Y c . For the purpose of obtaining an effective 
 
 average induced sidewash, It is nerrrii scible to neglect 
 
 >■ - '. ■: dv 
 
 the small term 3' - in oquation (b)> which uives 
 
 v 
 ■ 
 
 (1 + a) 2 / \ 
 
 ^ V/0.75P 
 
 An equivalent differential relation for the time average 
 
 side force, divided by PV D , on an element of disk 
 area x d6 dx may be substituted for the summation of 
 equation ( 7) , as 
 
 J~J-dx d9 =S5(di c ) sine §§. (A .2, 
 
 G 
 
 2rr 
 
 ciY., 
 The fraction — — has been shown to be given by 
 
 V a 
 
 dV Q 
 
 ' a 
 
 (\i/ - e ' } sin G (1.3) 
 
 where ?' is the local induced annie of side 1 :^ it the 
 orooeller disk. Combining equations (A-l), (A-2), and (1J), 
 
 using A' = c 7 sin p , and assuming that (1 + a) 2 is 
 
 ''a 
 
 constant over the disk gives 
 dY„ c l a' v 1 + a ) 2 sir. 2 8 Z' 1 
 
 — - — I (♦■«')Mln Po dx (A-3) 
 
from vhich 
 
 L9 
 
 c 7 a'O + *) 
 
 
 i2tT n l 
 
 e< sin 9 p, sin p n 
 
 V6.V7L. 
 
 An effective aver?-:'- le oi c' ' tained ,: iefininj 
 
 3 - c ' H + a ) ' 
 
 Y„ = 
 
 a 
 
 - *? ' ) 
 
 :.n (3 Q d; 
 
 x 
 
 
 frorr. which the effects ' wash is 
 
 n2ir ^1 
 
 • : . " d.8 
 
 t = 
 
 ( l-5) 
 
 *o 
 
 ,j. sin (? dx 
 
 In this appendix the Lhduc-ed sidewash angle e' is 
 
 the local value at the disk element x dx 18, aot J 
 
 average value used in the main text; c 1 composed of 
 
 one part due to bhe side force e' and one. -art cue to 
 
 v 
 
 the cross-wind 3omponen.t. of t u .e + " e, m- ef~ 
 
 fective averages are designated "h and e~* „ Then 
 
 equations of the form of equation (A-5)' hold between 
 T' and e' and between <F'_ and e' . 
 
 y 
 
 i 
 
50 
 
 The evaluation of e ' follows: The product 
 
 V„ e'_ sin 6, hereinafter called v , is a velocity 
 a v y y 
 
 component parallel to Vn but not necessarily in the 
 same sense. In order to evaluate v for use in 
 equation ( A-<y) , it is useful to define a quantity f 
 
 such that 
 
 6 2 f 
 
 dx d6 is the time average of the 
 
 x 6x 68 
 increment due to yaw of the peripheral force on an element 
 of disk area x dx dG . taking the simplifying assump- 
 tion that the peripheral force on an element of the ^re- 
 veller disk affects only the air flowing through that 
 element and equating the peripheral force to the rate of 
 change of peripheral momentum which this force produces 
 far behind the propeller leads to the relation 
 
 6 2 f 
 
 6*69 
 
 x d* d9 = pV a r dr cfi 2v q 
 
 (see derivation of equation (12)), ,_ 
 
 2„ 
 
 v e 
 
 T 
 
 2 
 
 x dy 
 
 x 
 
 J) 
 
 /v\ 
 
 2 
 
 L + ( — ■ \ 
 
 V Q 
 
 L 
 
 o 2 f c 
 
 x ex 6 8 
 
 (1 + a) 
 
 L + 
 
 \B)' 
 
 (A-6) 
 
whe re 
 
 f 
 
 V g = V(l + a) 
 V g = V(l + 2a) 
 
 or 
 
 lternatj ve form of luatj an ( ft-2) ii 
 
 d r = c -—- sin 8 : dx 
 
 ° x ex 06 
 
 s 1 n 3 
 
 x ox 69 x ox 6 9 
 
 In equation (A-3) the it c- , :. on 
 
 G, is small cor th '!/ alloi ipro 
 
 on 
 
 6Y, 
 
 ■-.'- 9 
 
 60 
 
 is 
 
 ; Of k as "'f,/ 71 "- 1 
 
 where k is a bant. Integration es1 Lshes the 
 
 v p sin 2 9 
 
 de 
 
 
 TT 
 
 &2y c 
 
 
 C X 
 
 ox 6 9 
 
52 
 
 By equation (A-.L), 
 
 OX rtl 
 
 a. si n 8 C 
 
 I 
 
 sin (3_ dx 
 
 Therefore 
 
 <fr, 
 
 — rr V 
 
 £ : n ,: ' 
 
 sin' 
 
 or 6 9 
 
 ° „1 
 
 y, sm p„ ax 
 
 &2f o __ sin 
 
 y 6x 66 tt 
 
 sin 8, 
 
 r 
 
 i 
 
 x > n sxn p dx 
 
 (A-?) 
 
 Equations (A-6) and (A-7) establish the value of 
 v Q ,/Vq, which can be substituted for e' sin 9 in 
 
 equation (A-5) as applied ^o T' in place of ~ ! . 
 
 This value is 
 
 v, 
 
 -V 
 
 1 + a)^ l + 
 
 L (1 + z ^~~j 
 
 s in 8_ y c ii sm jJq 
 
 " ' .1 
 
 x / ll sin p_ 
 
 "! TT 
 
 1 
 
 
 
 Therefore, substitution in (A~5) as applied to e' gives 
 
 r^ TT p 1 
 
 2Y C / / sin- - - 9 
 u n '•_ ■■-> 
 
 ( ii s i n 8 n ) c 
 
 > t - 
 
 dx d8 
 
 rr^ (l + a)^ ! 1 + 
 
 i 
 
 I 
 
 1 + 2a)2 J K 
 
 sin Pq ^ x 
 
 ^ 
 
v> 
 
 it = grat I or. v;:* t h re s : " : : t to 9 re :-: u 1 
 
 r -i 
 
 2 
 
 nr 
 
 /" (M- 3ln P.) . 
 
 - 
 
 '-■'x 
 
 V 
 
 (1 + a) 2 1 + 
 
 
 • 
 
 v 
 
 / ■■ - I • / 
 
 4 " J 
 
 y 
 
 "- ) 
 
 Tl )f e' d bo Y p In ition (12), which 
 
 is urn t ion of uniform distribution 
 
 ust and >rc ■ the prop' , Lifers 
 
 for £"'. ' \ '/i (A- ) 
 
 I . absence " 
 
 
 *1 = 
 
 •- :■ 
 
 Vo - y 
 
 sh Ls luat Lo ) . lysis for ~' _ 
 
 to thai for F'v results In • not 
 
 irecii ' ff ir froi irt of C to T_ In 
 
 Lon (12) ; t is, 
 
 2 
 
 - T. ■ 
 
 1 + 
 
 L 
 
 (] 
 
 Accord Ly, the effect:! 1 i Le of de- 
 
 wash e 1 , equals f',. + t ; ',., 
 
 | (T n vj/ + k-,y c ) 
 
 (1 H ) 
 
 '1 + d a ) 
 
5U 
 
 which is equation (33). If e' is inserted for e' 
 in equation (12), the factor f (a)/8 in equations (2$) 
 
 f-, (a) 
 rnd (25) i? 1 reolaced by — k_ . This is the quantity 
 
 that has been called the sidewash factor k_ . iVith the 
 
 value of k Inserted, 
 
 k c - f, (a) 
 
 /, 
 
 / ° 1 (i* sin p n ) 2 
 
 ax 
 
 A, / 
 
 V: 
 
 JV 
 
 (35) 
 
55 
 
 APPENDIX P 
 CORRECTION POP COMPRESSIBILITY 
 
 The side-force derivative C v ' is very nearly 
 
 proportional to the integral 
 
 I-, - f c / sin p dx (Ll) 
 
 To a first i iximation the effect of compress!! LI I is 
 
 accounted for by re n ; c by c A/1 - M , i 
 
 x 
 M L; 1 te resul' le section at x 
 
 divided by the speed of 30"und in ; 'ree stre 
 
 the subscript c is \ sed to designai ■ . cted 
 
 for- compressibility ■' , 
 
 ~ 3 P l sin 
 
 I x = f"-Cj / ! = ' -1) 
 
 c 
 
 Z 1 ,-• 
 
 - / 
 
 effective -'ech number ' t i ~'" J ion 
 
 T 
 
 -2) 
 
 o I p 
 
 would, also approximately sati I Lon 
 
 C- ' 
 
 ,i 
 
 f, --Z.) 
 
 . h _ v. 
 V- 
 
 Equation (B-J) cons tutes stion of I 
 
 side-force derivative for compressibility offsets. 
 
5 6 
 
 The determination of ? T e proceeds as follows: 
 
 By equations (ill), (B-l), and (B-2), 
 
 J u sin p 3 dx 
 1 _ _^2_ 1 2_ 1 /l_j L F^fL 
 
 e / \i sin (3 Q dx 
 
 n 
 
 .2 
 
 For determining the ratio.': '•'\/ 1 ' 1 ' it is suf ficiently 
 accurate to put 
 
 i M 2 
 
 = 1 + — - (B-4) 
 
 ,A - I, 
 
 ? * 
 
 ana 
 
 
 i 
 
 
 r 
 
 ~ 
 
 1 
 
 + 
 
 
 
 
 
 
 A 
 
 - 
 
 M, 
 
 2 
 
 (3-5) 
 
 although approximation (B-I4.) will not be applied to the 
 final equation (B-J). Then 
 
 M = 
 
 /0 
 
 1 
 
 
 
 
 
 / 
 
 i 1 
 
 */ H a 
 
 in 
 
 Pn 
 
 clx 
 
 Q 
 
 2 
 
 
 
 
 
 
 ,1 
 
 
 
 
 
 / 
 
 1 
 
 
 
 
 
 Jc 
 
 ; y 
 
 ^i sin 
 
 % 
 
 dx 
 
 
 By reference to figures 1 and 2, jf inflow and rotation 
 are neglec bed , 
 
, 
 
 a 
 
 « 2 = 
 X 
 
 V 
 
 2 
 
 , ^ ,i ,,2 t( 
 
 ti si J 
 
 u2 
 
 cin- 
 
 r M 2 ll + f^) 
 
 speed of scur 
 
 free- stream T ' 
 
 J = 
 
 nD 
 
 r 
 x = — 
 
 - 
 
 The appro:'! Lr atioa i = cons r 
 
 likewise adequate for th ; - - : . .as 
 
 tin i 
 
 
 \-V 
 
 equations (B-6) and .'"-.■ 
 
 t- 
 (_ 
 
 \ 
 
 ^0.2 V 1 *\T) 
 
 r.±. 
 
 I • . 22 
 
 Uv 
 
 ( >-8) 
 
58 
 
 Upon Integration 
 
 o J 
 
 v i + x - 0.2V0.0I4 + x 1 + /1 + \ 2 
 
 X2 T iU 6e ~ ~ r-T^ v2 
 
 J± O.s + yO.Gu + hy 
 
 2 log- 
 
 -2 
 
 (B-9) 
 
 0.2 + v/O.O, 1 . + >v^ 
 
 where 
 
 \=I 
 
 rr 
 V/nD 
 
 TT 
 
 Equation (3-9) provides the desired relation between the 
 effective !'ach number M and the stream ? -Tach number M 
 for use in equation (B-J). A graph of the variation of 
 M e /M with V/nD, computed from equation (B-Q), is given 
 
 in figure 9. Note that, in spite of the rapid rise of 
 M e /M with decreasing V/nD, for constant-speed propeller 
 
 Operation I' decreases. 
 
 e 
 
 It may be noted that equations (3-Ii) and (B-5) are 
 parabolic a pproxi mations to the Glauert compressibility 
 
 factor l//l - M 2 . Equations (3-6) to (3-9) are, 
 
 however, independent of the constants of the parabolic 
 representation. Thus the validity of these equations 
 is not restricted to the case of a variation of c. 
 
 l a 
 
 with Mach number that follows the Hlauert relation; 
 the equations are valid for any variation that may be 
 approximated in the region of Interest by a parabola, 
 such as 
 
 Q, = (A + 3M 2 ) c 
 c a 
 
 where A and 3 are constants. 
 
 The compressibility correction ceases to apply at 
 .Mach numbers above tne critical Mach number for the 
 propeller. 
 
59 
 REFERENCES 
 
 1. Lanchester, F. W. : The Flying-liachine from an 
 
 Engineering Standpoint . Constable and Co., Ltd. 
 (London), 1917. 
 
 2. Harris, R. G-. : Forces on a Propeller Due to Sideslip. 
 
 R. & M. No. 4-27, British A.C.A., 191S. 
 
 3. C-lauert, K.: The Stability Derivatives of an Airscrew. 
 
 R. & M. No. 6^-2, British A.C.A., 1919. 
 
 4-. iert, H.: Airplane Propellers. Miscellaneous 
 
 Airscrew Problems. Vol. IV, div. L, sees. 5 and 6, 
 ch. XII of Aerodynamic Theory, W. F. Durand, ed., 
 Julius Springer (Berlin), 193?- PP« 351-359 • 
 
 5. Goett, Harry J., and Pass, H. R.: Effect of Propeller 
 Operation en the Pitching Moments of Single-Engine 
 Monoplanes. MCA, A.C.R., Lay 19^1. 
 
 . Pistolesi, E.: New Considerations Concerning the 
 problem of Propellers in Yaw, (Trans.)* 
 L' Aerotecnica, vol. S, no. 3, March 1923, pp. 177-192 
 
 7. Klingemann, G-. , and Weinlg, I\: Die Krafte "and Memento 
 djr Luftschraube bei Schr'aganblasung und Flugzeug- 
 drehung. Luftfahrtforschung, 3d.. l r i, Lfg. h, April 
 6, 1933, PP. 206-213. 
 
 £ . Bairstow, Leonard: Atrolied Aerodynamics. Longmans, 
 Green and Co., 2d ed., New York, N. Y., 1939- 
 
 9. Misztal, Franz: The Problem of the Propeller in Yaw 
 with Special Reference to Airplane Stability. T.M. 
 No. 696, NACA, 1933. 
 
 10. Humph, L. B., White, R. J., and Grumman, H. R. : 
 
 Propeller Forces Due to 'Yaw and Their Effect on 
 Airplane Stability. Jour. Aero. Sci., vol. 9, 
 no. 12, Oct. 19^2, pp. i; -65-4-70. 
 
 11. Weiss, Herbert K.: Dynamics of Constant-Speed Propel- 
 
 lers. Jour. Aero. Sci., vol. 10, no. 2, Feb. 19^3, 
 p. 5$- 
 
 ♦Available for reference or loan in Office of Aeronautical 
 Intelligence, National Advisory Committee for Aeronautics. 
 
6o 
 
 12. Fun 1 :, Max K. : Fundamentals of Fluid Dynamics fcr Air- 
 
 craft Designers. The Ronald Press Co., New York, 
 N. Y., 1929, pp. 15-23. 
 
 13. Ribner, Herbert S.: Formulas for Propellers in Yaw 
 
 and Charts of the Side-Fores Derivative. NACA 
 ARR No. 3E19, May 1943 . 
 
 14. Ribner, Herbert S.s Proposal for a Propeller Side- 
 
 Force Factor. NACA RB No. 3L02, Dec. 1943 , 
 
 15. Bramweil, F. H., Relf, S. F., and Bryant, L. W. : 
 
 Experiments on Model Propellers at the National 
 Physical Laboratory, (ii) Experiments to Determine 
 the Lateral Force on a Propeller in a Sidewind. 
 R. & M. No. 123, British A. JUG., 1914. 
 
 16. Lesley, E. P., Wcrley, George F., and Key, Stanley: 
 
 Air Propellers in Yaw. Rep. No. 597, NACA, 1937- 
 
 17. Runckel, Jack F.: The Effect of Pitch on Force and 
 
 Moment Characteristics of Full-Scale Propellers of 
 Five Solidities, NACA A.R.R. , June 1942. 
 
NACA 
 
 i(dO-' 
 
 tldFh 
 
 Figs. 1,2. 
 
 -Sic/T) 
 
 Figure /- Vector re/otians at a blade element. 
 
 l^-£-J / ^„- Resultant wind 
 
 a to rop e Her in clu ding 
 inflow and s/de wash ~V. 
 
 > V 
 
 xial velocity in dud 
 ing~ in flow, -V a + dV 9 
 
 Side-wind velo c ity in - 
 eluding s : dewash, 
 
 Component of side wind 
 normal to r ~ / a (]p -€') 
 sine =dV g 
 
 Figure 2..- Vector re/ations fcr propeller in yaw. 
 
NACA 
 
 Figs. 3,4 
 
 YcosWy 
 Ts/np_ 
 
 Ve/cc/ty 1/ outside 
 the slipstream 
 
 A/J air ye ioc rf/es measured 
 
 re /of i ire to propeller hub. 
 
 Y 
 
 Figure 3. - Vector reloticns pertaining to the 
 side wash of a propeller in yaw. 
 
 Figure *?.- Flow induced by the side, wise mot/on 
 of an infinite cylinder in a fluid initially at 
 rest. 
 
NACA 
 
 F'\qs. 5,8 
 
 Figure S.~ Perspective view of three -dimensional 
 groph of the, assumed incremental mfio//, 
 
 Figure 8.— Effect of spinner on the 
 component of the flow in the plane of ihe 
 propeller disk. 
 
NACA 
 
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 Figure 13.- Comparison of calculated and experimental values Df Cyi/o 
 two-blade raodel propeller. Curves are terminated, except 
 3 = 1^.6°, at point where obvious stallin if blades occurs. Sxperims 
 lata from reference 13 and converted from .■■•'■ axes to body 8xes.ii/ = 
 
 1.4 
 
 • for 
 for 
 ntal 
 10°. 
 
Fig. 14 
 
 7/ nD 
 
 -jtv 14.- Comparison of calculated curves of Cyi/^with the faired experimental 
 curves from reference 1? for 10° yavv. The original data with respect 
 to pitch, with wind axes, have been co:iv< rtcd to '. ta with respect to yaw, with 
 
 body a::,. . . 
 
MCA 
 
 Fig. 15 
 
 n) 
 
UNIVERSITY OF FLORIDA 
 
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