1、REPORT No.PROPELLERS m820YAWBY EIEIWEETS.RIMSLESUMMARYIt wws realized aa early m 1909 that a propeller in yawderelops a side force like that of a$m In IW7, R. Q. Harri8expressed fhiu force in term of the torguu co+nt for theunyawed propeller. Of wreral attempts to exprm the sideforce directly in ter
2、nu of the 8hape of the bhI the treatmentappears almost identical with the account given in 193k byG1auert in reference 4.There have been aeveraI notable attempts to express theaide force directly in terms of the shape of thebladea. Baimtow(reference 8) prwmted a detailed amdysia in 1919 thatnegleete
3、d the induction effects. Mid al (referenee 9)”pub-lished an investigation in 1932 that did not have this limita-tion and that is probably the most accurate up to the present.Miszt.sls redt, however, is in a very complex form from thepoint of view of both practicaI computation and physicalinterpretat
4、ion; there is, in addition, an inaccuracy in theomission of the effects of the additional apparent mass of theair disturbed by the sidewash of the slipstream.Tery recentIy Rumph,White,and Grumman (reference 10)published an amdysis that reIates the side force directly tothe plan form in a very aimple
5、 manner. Reference 10, how-ever, (1) does not include the ordinary inflow in the amdysiaand (2) applies mat eady-lift theory in an improper manneT toaccount for the induction effects. As a consequence of (l),the equations are badly m error at high slipstream velocities.A a consequence of (2), the eq
6、uations fail to predict thesubstantial increase in aide foroe that experiment ahowe isprovided by dual rotation. The improper use of qnateady-lift theory consisted in using formulas that apply to the caseof a fite airfoiI with an essentially rectilinear wake. Thevortex 100PSshed by the finite airfot
7、i, which produce theinterfermce flow, are distributed aIong this rectilinear wake. .192Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-194 REPOET NO.“82ONATIONAL ADV180RY COMMITTEE FOR AERONAUTKMThe corresponding vortex loops shed by a propeller blad
8、e inyaw, howevm, lie along the heli path traversed by theblade. The interference flow iE quite d for dual-rotating propellers, the origin is on theaxis of rohstion halfway beLweenthe planes of rotation of thefront and rear propellers. The X-axis is coincident with theaxis of rotation and directed fo
9、rward; the Y-axis is directedto the right and the z-axis is directed downward. Thesymbols are defined aa folIowe:DssRr90x%!x,B,bcPpropeller diameterdisk area (d7/4)Wingareatip radiusradius to any blade element ,.minimum radius at which shank blade sectionsdevelop lift (taken as 0.22?)fraction of tip
10、 radius (r/l?) -value of z corresponding to r. (rJR)ratio of spinner radius to tip radiusnumber of bladcablade section chordti reference chordivebade-oord( aleo, angu-Tfretream dynamic pressure pllar velocity of pitchingM#;” function defied in equation (1)fu.pction defined in equation (2)f(a) - q-fa
11、ctor ( (1+a)(1+a)+(l+2a1+(1 +2fz) q)-nJB/%6revolutions per secondadvance-diameter ratio (1/nD)blade angle ta reference chordblade angle to zerdift chord.-.,angle of- blade relative LO Y-axis mmsured indirection of rotationeffective helix anglo including inflow and” rotnlion(tin-*)angle of yaw, radia
12、nseffective M of attack of Made clcmcnt (A-#)angle of qidewcsh in slipstream far behhdpropellernominal induced angle of eidcwaeb at.propeller diskeffectivo average induced angle of sidcwash atpropeller diskeidewaah velocity far behind propcllwairplane lift coefEcicntblade section lift cocfiicicnt .b
13、lade section profikhng coefficienteIope of blade section lift curve, pm radian (dkx;average value t,nken as 0.95X2r)force component on a Mndo clcrncnt in directiondFof decre-ming 8 (k=k,KCrc=cr+cor;c.;s,Amomente about X-, K, and Zkes, wpeotiveIy,in sense of right-handed screw; in appendix Band figur
14、e 9, Al refers to the freestream Machnumbereffective Mach number for propeIIer side force (Seeappeudix B.)functions defied in equations (4integrels defied by equations (21) and (30)integde defined by equations (31) and (32)()sida index defked by equation (41) “aintegraI deilned by equation (42) hl)i
15、nkgrel deilned by equation (43) d)defied by equation (24) (zero for dud-rotatingpropellem)detied by equation (44) (zero for dud-rotatingpropeIIers)defid by equation (45)correction factor defuled by equation (34)eidewash faotor detied by equation (35)spinner factor defined by equation (36)constant in
16、 equation for k,ide-fOmem*cient(*Or:p)(M 8Mpitohing-moment coefficient;PPDS r-)eside-force derivative with respect to raw (bCr/b#)pitahing-moment derivative with respect to yaw(ac./a*)side-force derivative with respect to pitibing(4%) acr2pitchimoment derivative with respect toac=fpitching(w) h“proj
17、ected side area of propeIkr (See footnote I.)aspect ratioSUbscrilltxcmessured at 0.75R station (3=0.75chided by pVW if a force, by pPD if a moment;designates quantities corrected for comi-btity in appendix B and JIW 9effectiveindex that takes the values 1 to 1? to designate aparticular propelIer bla
18、demaximumat stallA bar over a symbol denotes effective average value.IN YAW 195ANALYSISPEOPWEE INSTEADYAXIALPLIGHTThe seotion shown in figure 1 is part of a right-hand pro-peller blade moving to the right and. advancing upward.The oomponente of the relative wind are V. md VA whereV is th axial veloc
19、ity inc.ludiqg the inflo and Ii is therotational veloaty includiqg tie slipstream rotaticm Theforce component in the direotion of decreasing 8 h:dF=dL sin #+iD cos +=;=b dr ()cl sin #+cq 00s am: =$% dil() (1)and the contribution. to the thrust iadT=dL COS+dD ain #=;V;b dftl(+) (2)The equations may b
20、e divided by PPD to reduoe the termsb nondimensional form. Inasmuch as J.= V(1+a), thereresultswherea -FmuML-VachrreMmsstsbImdaekment.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-196 REPORTN.-82NAmONfi “i)VISORYfXMmEE FOR AEH.ONAUTICS PEOPELLEEUND
21、ER ALTERED#IKHT CONDITIONSForoe components on bhde eIement.In equations (1)and (2) for dF and dfl, V. occurs explicitly in the factor V;and implicitly in d and in tmns depending on #; Vooccursonly implicitly in # and in terms depanding on #. Therelationship ie =tan-, which can .be sem in ure 1.By pa
22、rtial dilh.mntiation, therefore, the increments in dFand dZ due to any small changes whatsoever in Vg mdare, for fied bIade angI 1udfl= $ W,+ b(dn+ a( a+ dvex Tmand tt similar expression for 6(dT). The substitution7*ofequations (I) and (2) gives, whm” put in nondimensionalThe following abbreviations
23、 are helpful:+bD-(6 O.nw(a)(4)where fl and tl are defined in equations (1) and (2), respec-tively. Equations (3) becomewhere all the factors are nondimensional.Foroesand moments experienced by oomplete propeUer,Equations (5) give the component-force increments due toah%red flight conditions on an el
24、ement of a single blade,divided by pVW. The force and moment incrementsexperienced by the complete propehr of B bkulea, withrespect to the body axea shown in figure 2, maybe written asForces:(6)(7)(8)x1Av?“f”(=+d” -/4=Y9!iiAhfomenta:(9)(lo)(11)where the subscript k refers to the M propdlcr M the res
25、dting change in aero-dynamic torque opposea the change in revolutions. The fluc-tuations in rotational speed and the associated variations inaerodynamic torque and thrust of the propeller are then func-tionally rdated to the law of control of the pitcldmnge mech-anism and the dynamics of its operati
26、on. (See reference 11.The present report wiU be limited to a study of the effectsof yaw and of angular velocity of pitch. In the folIowingsections dT;/T”, and dT”J= are ewdnated for yawed motion.PEOPELLERLVYAWkti d17JV. for yawed motion,-The increment dt isthe component parallel ta T“tof a side-wind
27、 velocity computedas foIIowa: The docity T=is regarded by analogy withwing theory as passing through the propeller disk at anangle #-dtothe sxis, whwe#iatheme ofyavranddmay be termed the “induced sidewash angle” (. 2). Theside-wind velocity, for small value of both # and c, isaccordiiy T“”=(#e).The
28、aidewaeh arises horn the am-wind forces. Theseforces are the cross-wind component of the thrust 2 sin #anti of the side force Imown to be produced by yaw 1?cos #.(See . 3.) The analysis is restricted to small #; thesecomponents are then approximately and KIf the sidewash veloaty far behind the prope
29、ller is Unthe induced sidewssh at the propellm may be taken as uJ2by analogy with the relation between the induced down-wash at a finite wing and the downwash far behind the wing.Note that 1 diame may be consider “far” behind thepropeller es regards the axial dipstream velocity; 95 percentof the fin
30、al inflow velocity is attained at this distance.M810714IN YAW 197AEa first approximation, thrust and side force are assumedto be uniformIy distributed over the propelIer dwk; cor-rections due to the actual distributions are investigated inappendix A. Under this assumption the momentum theory,support
31、ed qualitatively by vort- considerations, showsthat the slipstream is deflected sidewise as a rigid cylinder.The sidewise motion induces a flow of air around the slip-stream as in ure 4. The transverse momentum of thisflow is, accordii to Mu that is,dT”=V=($-d) sin(l (13)The value of c from equation
32、 (12) may be introduced andthe reIation T,=a(l +a), from simple momentum theory,may be used to eliminata T There resultsEado da/c for yawed motion.-k V.= 7(1+a) for uh-yawed motion, the changes produced by yaw aredla dV daT.=T+l+arnlYa (16)8if dV/V, which is cos $1-$ is neglected as being of theseco
33、nd order in t.In order to evaluate da, Ilgure 2 is fit considered. Thecomponent of the effective tide wind in the direction oppositeto the blade rotation is dr= VJx-c) sin 6. This componentacta to increase the relative wind at the blade, and thereforethe thrust, in quadrants 1 and 2; it acts to decr
34、ease the rela-tive wind, and therefore the thrust, in quadrants 3 and 4.More exactly, the change in thrust due to the side wind isdistributed shmeoidally in (?. It is C.lemthat thk incrementalthrust distribution by its antieymmetry produces a pitchingmoment.Momentum considerations require an increas
35、e in Mow inquadranta 1 and 2, where the tbruet is increased, and adecrease. in inflow in quadrante 3 and 4, whrc k thrustk decreased. The variation should be sinusoidal in :/uidxa=- ks,11=r r, p(w)z therefore, Y. isroughly proportiomd to a.Similarly, the solution for M, ief(a)dtiv=! # 1+gjl(a) us(23
36、)This equation dif%e ftom equation (23), which appliee tosingle rotation, in that unity replaces the Iarger quantityaat _A, in the denominator. The sidforce coet%cient Ic istherefore Iarger in the case of dual rotation. TM datafor conventional propellem, the increase averages about18 percent and rea
37、ch= 32 percent at low blade angles.The increase in aide force is due to the Iack in the dual-rotding propeIIer of the asymmetric distribution of Mowvehcity across the disk which, for the single-rotating propel-ler, is induced by the asymmetric disk loading. The inflowasymmetry is so the eidewash of
38、the Mow isanother such efiect and serv to reduce the side force stillfurther. Sidewesh is, however, common to aingl and dual-rotating propellem and affecta both in the same way. Anexamination of the steps in the derivation chows that theterm f,(a)ua in the denominator, the absence of whichwotid incr
39、ease the value of ., is due to the aidewssh.Equations (23) to (28) give the stability derivatiws ofsingle- and duaLrotating propellers with -pect to yaw, butthe resuh.e are not yet in iintd form. There remain theevaluation of a, b, c, and d and the introduction of a factorto account for the effect o
40、f a spinner and another factor tocorrect for the assumption of uniform Ioading of the aideforce over the propeIIer disk.”Explicit representtion of a, b, c,and d.Equations (21)show a, b, c, and d to be integrals involving thefunctions A, l?, C“, and D, respectively, which are definedin equations (4)
41、in conjunction with equations (1) and (2).The quantities A, B, C, and D are, upon evaluation,.4= C,=Sin +ci cm #B= c,= Coe +c, (sin #+csc 1#)C=q= Cos 4C, (sin 42 o I (29)if terms in the dcient of profi drag c%me ne.gkcted asbeing small in comparison with b terms incg=. The neglectof e%is vaIid only
42、for vahm of # not too near 0 or 90.From figure 1, 2:A gmph of the ariation of ti)r with T. is ghren in ure 6.Approximate evaluation of d.The contribution ofto 1“.is small It is found, by using the largest alue whicha may have without causingstalling of the blades (about1%radian),hat the secmd integd
43、 canbe neglected, with theresult thatNote that # invohs the inflow vehcity and the dipstreamrotational velocity. These docities, if assumed to beconstant oer the propeIIer disk, may easily be reheal toT, and Q., respectively, from momentum considemtiormCurree of d have been computed for a typical ph
44、m form(Hamilton Standard propeIler 3155-6) and are presented infigure 7. This chart makes use of an shred notation intro-duced later iR the report; the ordinate is the quantityra=dand a pmameter is the solidity at 0.i5R,413b()-Fr D .The abscissa is l-/nD. The error in-computed side force dueto using
45、 this chart for plan forma other than the HamiltonStandard 3155-6 shotid be negligible. The duirt is notsnflioiently accurate, however, for precise computation of thepitching moment due to yaw.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-202 REPOR
46、TNO. 82*NAYIOKAL AWISORYCOMM1lrEEFORAERONAUTICSA?8G40 /m 3 4FIGWE i.hth of theerror due to the aesumpt ion of uniform side-force distributionis appreciable. The effect of this error on the computed sideforce is small, but not negligible.The side force is actually distributed orer the propellerdisk n
47、early as the product of the integrand of the most im-portant term in the side-force espresion a and sin% Theintegrand of a is proportional to the blade width times thesine of the blade angIe, which tends to be greatest towardthe blade roote due to the twist, and sin* 8 has maximumsat 8=90 and 270. T
48、he side force is therefore concentratednear the blade roots and along the Z-c 0.90 for fineness ratio 6)FIorM.a-EIYectdnMrm* mmponsncofthenowbltlmpheofthepr ordterm.This local increase in side wind is equivalent ta an increasein the angIe of yaw # in the same ratio. Thus at radius zI?the effective angle isProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-PEOPIZLLER8 YAW 203The effective awrage yaw over thePsinA /lain%l, “ )land the side-force derivative for a dual-rotating propeIIer isFor a single-rotating propeller thederivative
