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' i L I FT I N G S U R FA C E T HEO R Y F O R A HE L ICO PT ER R OT hR I N • ! FOR WARD F LIG HT H. T ai _'i NASA Langley Research Center _i Hampton, Virginia 23665 w _ I Harry L. Run y an , _ C c llege o f William and Mary i NA S A Lan_ley Re s ear ch Center I Hampton, Virginia 23665 I l _' ABSTRACT _ .
_ : A li ft ing s urface theory ha s been deve l oped with a s tationa r y observer, whereas the lifting ; for a helicopter rotor in forwar d flight for surface theory is essentially concerned with the compre ss ible and incompre ss ible flow. The detail s of the near- f iel d case for a co-moving L ' i me t h od u ti l izes th e c onc e pt o f the li ne ari z e d o bser ve r a s well a s t h e s ati s f action o f certain _-! acceleration po t entia l an d m ak es us e o f th e of c ertain boun d ary conditions. Runyan (1973) I vo r tex l at t i c e p r ocedu re. C alcu lation s demo n - u t i l i z e d the a cc elera t ion p otentia l approach to _trating t h e ap p lication o f t he _ t h od are giv e n obtain a so lu t ion to the oscillating p ropeller ' : i n term s of t h e l i ft d i s tribution o n a s i n gle i n c omp r e ss ible flow. Dat, (1973), has derived s wept-forward an d sw e pt-back tips. In add i tion, for any motion. Pierce an d Vai d yanathan (1983) _I rotor, a two-bladed r o t o r, and a rotor with a general expression for an acceleration doublet " the lif% on a rotor which is vib r ating in a have treated the helicop t er ro t or in forward pitchin_ mode a t 4 / rev i s given. Compre ss ibi- f l ight using the method e f matched a s ymptotic two-blade d ro t or are discussed, proce d ure develo p e d h ere involve_ the preci s e lity ef f e c t s an d interference ef f ects fo r a ex p an s ion for _he _ncom p res_ihle case. The • numerical inte g ration over t he surface of the INTRODUCTION formulation of a fundamental three dimensiona l , compres s ible, un s tea d y aerodynamic then_y for Rotating lif t ing surfaces are an integra l p r ope ll ers an d helicopter rotors.
i! rotor in a time f rame. The method sets forth a par t of the propulsive unit of every aeron- a u tica l and nautical ve h icle, from t he The next section contain s a brief compressor and turbine blades of jet engines, derivation of t h e f undamenta l equations, i_clud- the pumps for rocket engine s , to propel l er an d ing a d iscussion o f some imp l ication s of t h e he l i co p t er ro to r s . The aero d ynamics of these equations. T h e third section contains a r o tating e l eme n ts h a s b ee n un d er ex ten s ive s t ud y d e scr ip t ion o f the metho d o f s ol ution. F i nal l y, , since the advent of the airplane and with a t he results of some calculations for the s everal combination of experimen t al an d analytical specific examples a e yiv_n.
approaches, s uccce s sful designs have been ac h ieved. In many case s , t wo- d imensional theory SYMBOLS ha s bee n used , usu ally mo d i f ied by an a s sume d / s panwise di st ribution, a n d inflow velocities. A' rotor bla d e area Thi s p a p er p resents a c ompressible, li ft ing Anm aero d ynamic inf l uence s u rface meth od f o r a h elicopter rotor in forwar d coetficients f l igh t within the limits of linearized theory. An,Bn Fourier coef t icients _.
c speed of s oun d _w_-c The method is based on the concept of the C chord of rotor a cc eleration p ote n t ia l , o riginally introduced by CT thrust coefficient per blade Kussner (1941). The method was first applied t o vector distance ( t hrust / w°_2R*4)from doublet " an o s cillating wing in uniform t ran s latory _ to m o tion including e ff ect s of c o mp r essi b le flow by d ownwash point R u n yan an d Wool s t o n (195 7 ). Th e a cc el e ration 1) ab sol ute v a l ue o f _ p ot en t ial appr o ac h ha s no w be come s tan d ar d f o r D = _ / D unit v ector of _ ( ; th e dete rmination o f the u ns teady aerodynamic ¢I for c e s for f l utter studies o f l ifting su rfaces I value o f singular integral I in re ct ilinear mo t ion. K kernel f unction I unit ve c t o r at downwash point, I The firs t u s e of the acceleration po t en t ia l norma l to wlocity vec t or paper by Hanaoka (Ig62) for the l o ading on a normal to velocity vector marine propeller in incompre ss ible flow. The iI appr o ach for a rotating s ystem was made in _ _o unit ve c tor at d oublet point, acce le ration p ot e ntia l h a s b e en u se d i n th e p ast £,m,n dire c tion c o s ine s of _ J i n s t ud ying the p r ope ll e r no i se prob lem, b u t i n _o, me, no dir e ction c o s ine s of _o I a 11 o f t he s e noi se p r o pa g a t ion c a s e s the prob l em p pre ss ure wa s spec ia l i z e d e ar l y i n t h e a n alytical deve l o p - _ o p os iti on v ec t o r of do ublet m e nt to t he so- ca ll e d far-fi e l d case u s ually from iner t ial f rame o rigin po s ition vector of do wnwash • J p o int from i n ertia l frame o rigin
I
"' _ q sour ceo r d o ublet strength liftingrotor is as sum e dto lie i n t h e skewed "_ Rt rotor tip radius helical path taken by the rotor blade. One _ : ; R s r o t o r r oo t ra d ius reason for adoptingthe acceleration potential r distanceof downwashpoint approachis that the pressur_discontinuity . '; al o ng the sp an o ccu r s on l y on th e s urfa c e of the blade and thus _ ; r o d istance o f d o ubl e t al on g th e t he b o u nd aryc o n d iti o n s need only be appliedon _ s pan the b la d e surface and not throughoutthe wake. 4 Th e b la d e i s t reateda s a v ery thin surface of _. ru upper limit o f sp anwl s epanel d i s c o ntinuityacrosswhich a pressurejump _ rL low e r limit of sp anwisepane l occurs. Th effect of cnmpre s sibiity i s taken into accountby utilizingthe complete _i ' i r 0 d istance of doubl e t along linearizedpotentialfor a liftingdoublet, span at singularpoint time _i t field time along with the effects_f retardedtime.
'_ U v el o city o f ro t o r s y st em, As shown in Fig. I, an inertialcoordinate parallelto x-axis, positivein system has been used in which the origin of Yl negativex- d i r ection c oordinatesis fixed to a point on the ground.
I ,, The heli cop t er rotor i s m o ving in the negative V velocityat downwashpoints x-directionwith velocityU, in the positive Vn velocityc o mponentof V at :ounterclockwisewith a con s tantangular the downwa s h point n o rmal to velocityft. A p o int of intereston the rotor iil + z - d ir e ction with ve l o cityW an d I s r o tating th e r o t o r l e adinge d ge V o vel o city o f d ou bl e t bla d e is d esignate d by t h e ra d i us v e c tor X o (1) • W velocityof rot o r s y s te m , f rom the origin of the ground ba s e d coordi n ate parallelto ; axi s s ystem.
wn downwashvelocity x a di s tancefrom p itch axi s to L et ¥ be t h e accelerationpot e ntial of a d ownwashpoint s ource (or doublet),the perturbation p re ss ure x,y, z Carte s iancoordinate s of is then given by downwashpoint xo,Yo,Z o Carte s iancoordinates o f p = - pY (1) doublet p osition a twist angle at do wnwash point Th i s ex p ressionre p resentsthe p ressure p at ok) twist angle at d oublet + ar angle of axis of rotation relativeto z-axis (or doublet)locatedat Xo. The potential Y a con s tant"q" w h ich r ep r esen tst_e B Vo / c strengthof th e s ou rce and thus the ma_ : ,itude of , p os iti o n p o in t X due t o a s ingle source B* oV th e p re ssure , in thi s fo rm, t h ere i_ n o i _ 1 bou n daryconditiona v ailableto d eter,,.i-_ the o / _ = c _ v alu e o f " q " a nd t he r esul ting p re ss ur e .
( . ! e angularp o siti o n o f b la de at Recour s ecan be ma d e to the velocity potential, time t s in ce th e sp atiald e ri v ative of a ve locity : - : e o an g ular po s itiono f blade at pot e ntialrepr ese nt s a v e locity. T h e time T relati o n s hi p betweenthe pre ss u r e an d vel oc ity 0 w bla de _n g l e o f attac k po t en tialf o r a n in e rtialc oo r d inate s y s tem is e B b l a de ang le r e lativ e to p lan e _ I of rotation _t - p a d vanceratio p - - (2) p air dens ity time where_-_+ is the sub s tantial derivative.
T,I0 A time at which integrandin Eq, Dro p ping o ut th e se c o n d o r de r t e rm s (2 4 ) become s singular and integrating with respect to field time ¢ velocitypot e ntial r es ult s in V s so ur ce a c c ele ration pot e ntial t _ Y D d o u bl e t acc e l e ration ¢(t) = _ Y (t') d t' (3) p o t e ntial " L i _ a z imuth a ngle - - • I _ rotation speed o f r o t o r _ m v ibrationfr e qu e ncyof rotor Th e acceleration p ot e ntial Ys sati s fi es the wave equation 1 _ 2YS = BASIC FORMULA T ION V2 y s - _ _ - 4 , f ( _,t) (4) T he f o rmulati o n o f the aer o dynamic e quat io n s I s ba sed o n t he ll ne art zed * a c c ele rati o np ote nti a lappr oach . The f l uid i s wh e r e f ( X, t) i s a sou rc e d i s tr i b u tion. Fu rt he r- c o n s i d er ed p e r fec t , with n o se parationan d th e m o r e , if the path of an isolated sou r ce i s a f o rmulati o ni s ba sed up o n t he a ss um p ti o n of f u ncti o n o f ti me va ri a bl e ,_ o( t ) , then s ma ll p e r tu rbation s . The wake cr e a ted by t he f( X ,t) • _(_ " _O) where 8 is the delt a functi o n .
9 0 ! ,
i '
acce l eration p o tentia l expre ss i o n f o r a moving th e flight path at the l o cation of the downwa s h source, Y s can be _.rittenas (Morse and point, as follows Fes hbac h , 19 7 8, p. 841 ) @¢D _ i awn = = ' ! . _ q(I o ,T) Is ( X ,t). . : tO obtain
L:!
; : , °o + C),) n 4 w cD2[1_6 • _3 "
_. (b) +(_o" g _" _ " - ' C o _';
# +
where Xo(_ ) designates the p osition o f the " ; s our c e at time T, X i s the po s ition of the ' ') + " _ 6 6 _ • 6 + _ 6 6 _ • _) / (1 - 6 • _) (9) C l field p o int at the tim e t, Vo (T ) i s the -(n o velocity of the sou rce point at time T, c is the speed of s ound and q i s the strength of the time interva l (t - T) to the di s tance 1 " - the source. An auxiliary equation which relates /To(ro) _ -_o 3_ o'- _" _ i + " _ _ " D[I'B2+ - " + ) I (I-6. _ ) }IO between the two _ints i s + _ -. q [ D 3 ] dT = Eq. (9) gives the downwash at a field point
1 J "
t c,> (x. y. z . t> due to adoublet p laced at apoint
• (xO,Yo,Zo,T) having a strength q. In which is usually referred to as the causality order to represent a lifting surface such as a condition. Eq. (5) expresse: the potential as rotor, it is necessary to distribute the an explicit function of T, and only through Eq. doublets over the lifting surface and integrate over the surface to obtain the downwash at a (6) as an implicit function of tans 3. From field point. If the downwash is known, the Eq. (3), the velocity potential due to a moving quantity "q" can be determined. Letting K be _ source is the expression on the RHS of Eq. (g), the final - equation is t t q(_,) dr' w : K dA' (10) Cs(t) = I _s(t') dt' = I 4 w LD _ • (7) n _ / ...... gJt' _ where A' is the area of the rotor surface.
The LHS, Wn, represents the known boundary , LI : _ dT', condition and is the velocity normal to the where _ = _ - 0 = _ the no flow condition for the velocity _. perpehdicular to the blade surface, the velocity component in the _ directi o n is V n tanOw or _] Wn(r,t) = Vn tan O w = K d A' (II) The quantities T', t' and t, _ satisfy Eq.
(6) where Vn is the velocity component of V _t the #_ . _ • downwash point and is normal to the rotor leading edge and 0w is the angle of attack.
By definition, the doublet velocity Thus the problem requires setting up a method of • can he written a_ the unknown doublet strength, can be determined potential _ of a _oublet aligned along _o solution of Eq. (II) from which a v alue of q, which satisfy the known velocity boundary "I : B w n.
i CD(t) _ Cs(t) : to • V_ o ¢s : - no • V_ ¢s This represents a rather formidable _ (R) computing ta s k and the history o f lifting ') _ • _ T _ , _ surface theory even for non-rotating wings has
;i :L o o
JT + y q--_._di ' centered on devising approximate methods to 4w c(0-_._) -- _ __ a cc omp l ish the integration in an economical Note that for incompressible flow, c + ®, the meth.d, has been very successfully applied to first term and the integral remains aircraft wings, and is probably the more manner. One method, termed the vortex lattice • u nch anged e x ce p t f o r th e up pe r limit where c - economical procedure of the many variants. This t. method wa s first demonstrated for the unsteady ' case by Runyan and _oolston (1957) and was later To obtain the final equation for downwash expanded by Albano and Rodden (196g). This is AWn, a second directional derivative is the method adopted in this paper and the a p plication will be disc u sse d later.
)
. S pecification o f Coor d inate S y s t em W(U + roO sln(Q T) + (C / 4 ) _CO S( O T) COS_0) L0 l T he b la de ha s the co r d C a nd l en gt h R t - ' / W 2 + V _2 _i R s ' R s being d i st anceto the the r o ot of V o the b lade , Rt i s t he d istancet o the tip of 1 the blade. Let the b l ade m o me n tarily co i nc ide W (ro_ c o s (_t) - ( C / 4)_ s in ( _T )COSao) ",! with the co o r d inate s ystem along the positive = (16) x-axl s at t = 0 and executea counterc l ockwi s e mo - { I r ot ationwith angu l a r velo c ity_ w h ile moving V . _ with ve l ocity U a l on g t he negativex direct i on o o i_t and ve l ocity W a l ong the positivez V ' directidn. Since th e v o rtex l a t ti c emethod ha s = o
;4
_ been a d o p te d ,t h e do ublet po i n t lie s C / 4 ahead n o / W; '2 an d th e d ownwa s hpoint l ies C / 4 af t of the + Vo •_ s ec t ionmidc h ord. The po si t ionof the d o ublet po i n t a s _II a s the do w n washpoint can be • establi s hed as fo l lows. The Car t e s ian where com p one nts of the d ouble t p os i t ionare
\
V O '2 = ( U + ro_ s in ( gt ) + {C / 4 ) _ c o s( Qt )c os a o ) 2 xo = -Ut + r0 cos(_t)- (C / 4) s in(Qt)cos Ok) ' YO = ro si n (_t)+(C / 4) cosSeT)cos o o z0 = Wt + (C / 4) sin % (12) + (togcos(gt)- (C / 4)_ sin(_t)cos a o)2 ( 17 ) - . wh e r e r o i s t he r adi_l d istanceof the doublet _ " al o n g t h e s p a n. With th e su b s tit u tion of + + C + -C, ro t + t t h e po s i t i o nof t h e By th e s ame procedure,n = _i + mj + nk, downwashpoint is given by where 4 x = - Ut + r cos(_t)+(C / 4 ) s in(_t)co sa t = W(U + r_ sin(_t)-(C / 4)_ cos(_t)cosa) Y = r si,1(_t) - ( C / 4 ) c os (_ t)co s _ (1 3) z = Wt - C / 4 sin a V ' / W 2 + V '2 (1 8) In E qs . ( 12 ) and (1 3) , the a n gles a ,_o a r e the -W(r_ c os(Qt)+(C /4 )_ sin(_t)cos_) twist angles of the velocityvector s _ and _o, m = ° / , 2 re s pectively, d e f ine d by V W2 + V W ; tan _ = U si n( _t)+r_ V ' i n - W (1 4 ) _ :an a o = U sin(_t)+ro_ and Th e re fe rence p l_)_ d efi ned by the doub l ets a nd V' 2 ( U + r_ s in(_t) - (C / 4 )_ cos(£t)cos a )2 d o wnwashp o ints i s a twisted s urface. From = . :_ : ,L Eq . (12) th e d ou blet vel oc ity c a n b e c o mp ute d, + ( r_ c o s( _t)+ ( C / 4)_ s in ( _t )co s¢)2 (Ig) / _ namely the time d erivativeo f the p o s ition ve ct o r s. + + t h e ve c t o r D = X-X o de fined i n Eq . ( 7) c a n be - % + expr ess e d a s
• "T ;; " Vo = Xo + 0 + Zok
" D " { [U ( t - t ) + r c os (_t)- r 0 cos (_t) The unit vect o r _o i s c hos en to be + ( C / 4 )( sln ( _t ) co s _ + sin(_t ) cos_o ) ] 2 _" perpendicular t o the twi s ted s urfa c ecreated • by t he v eloc ity vec t o r Vo w h ich i s a f uncti on + [r si n (_t )- r o s i n (_t ) (2 0 ) of ro, through Eq. (1 4 ).
Exp r ess n o as - ( C / 4)( C OS( gt ) C OS _ + c os(n t ) C OS _ 0) ] 2
+
m n o = _ 0 1 + m o J + n o r (15 ) + [W(t-t ) - ( C / 4) ( s ln_ o + s in a " ' i With t he su b s tit u ti o n o f t h e q uantities,th e : _ i n t eg ral Eq . ( 11 ) wa s so lv ed f o r the unkn o wn where _ o 'm o 'n o ar e th e dire c tional cos in eso f q(r o ,t ) by us in g a c ollo cationpr oc e ss ba sed ¢ the unit vect o r n o . It can be s h o wn that 92 _ ' _ ®.
o n t h e v ortex lattic e a ssumpt i on. Th e kern e l Is s ingularity. Th e Integration domain was d ivid e d ) C _ stngula r w hen C • O, and t h i s w as hand l ed b y use tn t o a r ea _ a s show n i n F ig. 2 . Ar eas 1-4 i i I of t he f i n ite part t ec hn ique. (ha t ched) w e r e c ompu t ed n ume r i c a lly usi n g a "_I t v_ -dl monsi o na l R_b e rg Inte g ration (D avis an d "@ P a b ln o wlt z ,19 6 7 ) a nd t he c o ntrlb u tl o _ o f t he _ / . SOL U T IO N OF INTEGR A L E QU AT I ON s in g ular re g ion (u nh a tc he d ) w a s obtaine d i n '_,_ clos e d f o r m by consid e ration o f th e finite part I n f ollo wingt he vortex l attlc e t ec h n i que a s sh o wn i n t h e nex t s e ction . 4 _,T th e r o t or is divide d int o a nu mb e r o f J _ p r ede t er min ed p anels, both s p a nwl se a nd Tr e at m ent o f S i ngul arT e rm it , Int eg ral- T h e ,;j c ho r d wi se , In e a ch chord wl s epa ne l, a l i ne o f integralin t he downw as he qu ati o n , Eq. (11) , i s dou b le t so f u n kno wnstr eng th q l I s loc ated at s in gula rw hen D+O a nd produce s a com pl ication - _ th e 25 % cho rdwi se l o cati o n of th e particular which m us t b e tr e at e d pr o perly. It sh o uld be po i n t lo c a t e dat 7 5 % c h o r d wl seloc ati on of th e is the p ath t h e do u b l et ha s tak e n in arrivingat _ pa n el , a nd th e d ownw ash i s ev a lu a t eda t th e r e member e dthat th e integration path alo n g "T" pan e l. T he r e f o re, a collocati n npr o ce d ur e is the final do ub le t point at (c / 4, r ob mea su red use d t o o bt a i n a se t of equa tion s i n t erms o f i n the l oca l b l a d e c oo r d inat es and ca n be that t he spa nwl se l oa dl n9q l i s con s tantalon g place al o ng th e path fr o m - - to the final , : _ each o f t he pa n e l s. A s e t o f equ a ti ons i s thu s do u bl e t p os iti on at T o • The distanc e O is t he • o bt a i nedas sho wn be lo w , di s tanc e fr o m t he int eg ration po i n t at time T t o i_t t h e u n kno w n l o a din g s qi • It is a l s o ass u m ed c o ns id er ed asth e wake. The in t e grat i on ta ke s _] W n' _ Anmq m (21 ) t he downw as hpoint at _ • r T h er e is a particularset of val u e s o f r o . / u dr o a n o wh e re n r efe r s t o an d T fo r which t he d enominat o rD ap p ro a che s wh e re Anm r_ Knm z ero, th us re s u l tingin an infiniteintegrand.
Th e s ing u larpart of the Eq. ( 1 1) i s | ._ r t he downwa s hp o int a nd m r e f ers to th e vortex ru z2 _ '_o " + + ;_ " I whic hla tti ce 'i nvolvesTh e ke r nel an int e gr a tion K I s a co mp l i c a t edove r T . f unc tion I = J rL J x l - 3( D .n)( D • no) dT dr o ( 2 4)
:4 u3
Th e term q ( ro,i ) repres e nt s th e s tren g th A s _ at the downwa s hpoint, D beco m e s o f th e doubl e t l o cat e d at r o and a t time T, ° j and is proportional to th e u n k n o wnl o ading. In p e rpendicular t o _ , theref o re,at the s ingular or d e r to account for un s teadine ss , a s olu_lon p o int , th e _ e c o nd term i s zero and will be w as f orm ulat e dt o t a k e into accountth e time negl e ctedin th e treatment o f the s ingula r ity _ v ariation of the s tren g tho f t he w a ke • Th i s wa s H o w eve r,t h i s se cond ter m i s retainedin al l o f , done by a ssum i nga Fou ri e r se ri es o f the f o rm t he nu m e ricalint e grations in vo l v ing A rea s I-4 _ _' m since it represent s an importantcontribution _ particularly wh e n the b la de i s pa ss ing o v e r a _ ":_ q ( r o ,T ) " A o + _ ( AnC OS (nOi ) + Bn s in ( n_T ))( 22 ) tr a ilingwake.
I The time and di s tanceat which the integralI M M If q (r o ,T) I s ass ume d t u be a functi o n o f r o becom ess in g ularare de s ignatedby T and r o.
a l o n e , w h ic h mans t h a t t he w_ ke st r en gth does T h e do , r ai n o f t he integrationin E q . ( 2 4) _ not vary w ith t ime . t he Fou rier se ri es r e du ces c ons i s t so f a re c tanglei n which th e d u rati o n . r . : _ to q (r o ) - A o . A so l u ti o n o btainedwlth 12-11 I s kept e x tremelysmall. In other _, th l s ap p rox im a tion i s t erm e d t h e q u as i- word s , t he int eg rationI s p erf o rmedal o ng a s lit _ __ st ead y solu ti o n , i n ro . ove r whi c h t he 2n d term in Eq. ( 2 4) i s ) n egl i g ible. Th e reforet he Int e gralI can b e ., -" L i Th i s ser i es w a s in se rt ed t n t he b as i c app r o xi ma t ed by e q uati o n a n d int eg r a ted with re s p e c t to _ .
; H ow eve r , th e r e w e r e m o r e u n kno wn s th a n j ru ix 2 _ " _ o ?i s im u l ta n eous equa ti ons to solve f o r t h e I - _ dr o (25) unkno wns. T h e a dd i t i o n a l requ ir e d equa ti o n s r[ _ 1 d_ !
w e r e obta i ned b y e v aluat i n g E q. (11) a t a number of az i muth l oca ti ons. For in s t a n ce I f m • 1 , F urth er m o r e , n ot i c in g t h at D 2 i s q uadraticin then ro, if _ is ind e p e nd e ntof ro, th e n th e q(r o, l ) • A o + A 1 cosn l + Bls tn O _. (23) int e gr a tion o n r o ca n b e p e rf o rm ed an a lyti ca lly. Th ls ca n b e a c hi e v ed by The azimuth w as d i vided int o equal segmen t s o f reco gni z in g t hat tn th e v o rt ex l a ttice m e thod, 120o a nd t h e prope r bounda ry cond it ion s t he r otor i s d iv ided in t o sp an w i se pa n e l s fr o m prov i di n g t he necessa ry a ddi t ional e q ua ti ons, s m all t hen t he va r i at ion i n ao t s s ma ll.
_! appl i ed a t + • 0 O, 1200 , an d 2400 t h u s r_ t o r u . If th ese sp a nwt se p a n els a r e
d % . _ n
4 N um rl c a l " Int e gr at i o n o f K e rn e l _ " ( U s ine o+ r on) 2 + W 2 (26) • Th e in tegratio n w a s performed by numer i ca l i ntegrat i on, except for the area s urr o u ndi n g t he If th e va l ue o f i s ap pr ox i ma ted b y i ts mt d p a n e l va l ue, it _ possi bl e t o Int e grat e Eq.
(24), i n c l osed form in t he r o di r ec ti on.
Th t s t s quite acc e pta ble tn t he he l icop t e r mode m be c a use d e e / dr o i s i n the or de r o f as At i s kep t la rg e because th e ve r y la rg e v a lues of t he i n t e gr and nea r t he s i ngula rity ar e m a g n it ude 10 - 3 o r s m a l le r . The value _ o i s avo i ded . On t he o t he r hand, r e g a r d i n g t he al so a func t io n of r o, h u t i n the r e glo n of fin4te p art in tegr a ti on, the d en o mi n at o rwa s th e singu l arit y it ha s a ve r y s ma ll v ari a ti on exp anded in a Ta yl or sertes a b o ut th e a n d is evalu a ted at t he si n g u lar po siti on, - s in gu lar po int , ¢ . The r efo r e, tt is d es ir ab l e ( _ , _ o) . Pe r fo rmi n g t he r o i n t eg r a ti o n t o mat n t a i a A_ as sma ll as,poss i ble t o k ee p resul t s i n t he fo rm wtt h t n t he ltmtt s of t he ap p l i cab i l ity of t he T 2 _ s e r i es exp an sion . N u me rouscalcu l a ti o n s w e r e I = J T 1 .k_l dz (27) m ade, v a r yin g A z un ti l a re a so n a b l e conv e rgence w as found . Th i s v a lue w as found t o A w here _j ( T ) i s a func ti onc o n t a ini n g al l t he b e .O l (t - z), i. e . I% o f , t he time d iff ere n ce .
non-singularpart after p e rf o rmingth e re Actually , ther e is v e ry littledifferenc e integration an d t (T) - O, at betweenI% or 10% of th e ti me diff e renc e and the - - c o m pu ti ng tim e a nd co st is c on si d e r a b l y reduced T = T ( Z 1 < Z < _ ) . It c an be a r g ue d b y usi n g 10 %. For t r en d s t udies 1 0 % i s , I phy s ically that s i n c e t he q u antity D ( T ,Po; r ecommen d e dp r i n cipally t o reduce co m pu ter t ,r) as w el l a s it s mod i f i ed for m f( T ) (af t er cos t s. F lo we ve r . for f i n al des i gn ty pe ana ly s i s , t egr att o n ove r r o) represents t he d i s t ance a s m a ll e r v a l u e of tim e d i fference & T i s mo re betwe e ntw o p o int s in sp a ce it m u st b e positive a pp r opr iate.
and real for all its arguments , and ne_er beco m e n eg ati ve . De n o te t h e v al ue, o f ,r o an d T a t For t he sp a n wi s e d i rec ti o n , Ar o i s a lso w h i c h D bec o m es z er o a s r o an d T o. Thu s, in a n i n tegrati o nlimit v ariable. The finite p art i n tegralwa s o btaine d b y a pp r ox i r a atin g th e a ng le the neighborhoo do f T the functi o nf(T) behave s o f twi s t of t h e ve lo cityvect or acr oss a s egment like a p a rab o li c functionand ha s a se c o nd order by a ss uming it c o n s tantacr oss the s egment, zer o . having a value a s d etermine d at the ce n ter o f s e g ment. Numerica l experime n tation in d icate s Expan d i n gf(_) in a Taylor s erie s ab o ut the that f or a helic o pter,Ar o = 0 i s sati s - A s in gul a r point T resu lt s in f act o ry.
f(,) = f (;) + f' (;)(, - ;) + f '( ;) (z - ;)2 / 2 + ...
(28) APPLI C AT I O N T O SPECIFICE XA MPLES S i n c e T i s a s e cond o rd er ze r o 4 T h e f o r e g o inga n alysishas bee n a pp lie d t o " s everal sp e c ifice x am p le s which ar e given in a n d f( _ ) = f ' ( _ ) - 0 (2 g) F i gs . (3) an d (4) . The fo ll o wing sec ti on pre s entsre s ult s for s everal paneling . ; Eq. (2 9 ) h as b ee n ve rifie d n u m e rically. I f o nly c onf iguration s ; e .g. 5 s pa n wi se an d I chordwise t he squ are t e rm i s k ep t i n E q . (28) , E q. (2 7 ) p an e l s (des ig n ate d (5 - i)) an d 7 sp a n wi se a nd 3 , can be written as c h o rdwl s e(de s i g nated(7- 3 )). The r o t o r blade wa s mai n tain e dat a c o nstantpitch settin gof j _ 2 + g'(z) + J d T (3 0 ) I " _ f "(_) [ (T-T ) ( T-r ) B B = . I r ad ian sfo r all th e c a lc u l a ti o n s .
I n Eq. ( 30 ) , i f T 2 and TI ar e c hos en Sln_l e Blade ) A s ymmetrica l ly ab ou t T, then the odd d eri v ative I n o rd e r t o in ves ti g ateth e co n v er ge nce o f i t e rm s int eg ratet o ze r o. Fu t he rm o r e ,th e thir d th e method w he n us in g th e vo rt ex lattice I _ ' . : .
p r oced ure,th e p r o gramwa s run fo r seve ral 1 te*m can be neglected s ince g"(_) i s s mall. The ch or dwiseand s panwi s eelement sfo r the incom- _ maj o r con tribution co m es fr o m the fir s t t e rm.
pr ess lbl e ca s e. T h e thru s t co e ffici e nt CT T h e n u s ing the s tandardint e grationtechniqu e v s . th e a z imuth angle i s s hown in fig. (5), (In {Mangl e r,1952) th e final r es ult for th e all of th e followingplot s for thru s t co ef fl- int egr a l i s c l e nt vs. a z im u t h a ng le, t he t h ru s t wa s ) I • - g{_) 4 _ _ 4_ ... (31) ca lcul atedf o r 1 6 unif or m l y spaced a z im u th i f" ( T) & _ angles a nd eac h cur v e was f a ir ed us in g a c u bic sp l ln e ). T he r o t o r was f ir s t divi de d int o . 5 s panwi se an d one cho r d wi se ([.I) p an e l and t he : := w he r e 2AT • T 2 - T I an d T I < T < _ Z , r e s u lt s are sho wn b y t he sol i d l in e. T h e chor d w t s e d i v i s i on w a s in c r ea s ed t o ( 5 - 2) an d A nu m e ri c al p r oble m a r i ses because t he th e r esul t s a re sho wn b y t he lon g dashed l in e.
fi n it e part int e grati o n re s u lt s in a nega tiv e It ca n be see n th at v e ry littl e c h a ng e ha s ta ke n / nu m be r whi ch i s close t o t he t o t a l of th e p lace. The sp a nwt se d i v i s i o n s w ere i n c r e a sed t o I s ur r ounding nu m e ri c al integrati o n a r eas w h i c h (7-1) a nd t he la rges t ch an ge o c cu rr ed at a r e pos iti v e . Th u s, it i s necessa ry t o t ake t he $ • O O w he r e t he d if ference i n C T t s abou t d iff e r ence be _w ee h la r ge n u m be rs , a nd t he fi n a l 11 % . Increa s i n g t he c h ord wts e d i v i s i ons t o 3 i ! i nte grat io n a cc u rac y t s de p e n d _t on t he (7- 3) sho w sconve r gence of th e (7-1) case to b e ac cu r ac yof t he tw o i n t eg r a ti o n s. On th e o n e ve ry g ood.
ha n d , t h e num e ricalintegr a tioni s more accur a t e An Interc : ting ph e nom e naoccur s in th e _ i r e gi o n of sma ll a z im u th an gles . Fo r SmOt O
1 "
_ _ ,.
_ ! 3 7o, t he l i ft i n creases t o a local maxim u m a t B lade Os cillati n_ i n Pitch " ; _ ' , , , i )- 37o th e n t he lt ft a b ruptl y f alls t o a loc a l v ; _ i m tnt m u mf or _ .60o an d t h en r apidly increa s ed An ex a mple o f u ns t eady lo a d s on a r o t or ' _" t o a ma x im u m a t _ -100o. t A s im tlar p he n o m e n o n bl ad e w l th ( 5-1 2 p a ne l i n g whi c h ts osc il l ati ng _, , i s s ho w n an al yt ic a ll y by E glof a n d Lan d greb e in a p i t c hi ng mode ab ou t th e mid-chor d at a . _ (1983) I n Fi g . 60 o f th a t r e p o r t wh e r e a f r e quen c y o f 4 per re vol u t l o n( 1 20 cyc les / s ec ) loc a l m i n im um a n d a loc a l ma x im u m occ ur i n the i s given on f ig. 11. F or thls ca s e a 17 ter_ 4 s ame r a n ge o f az im u than gles,eve n t hough th e F ou ri e r se ri es (m =8) wa s us ed t o s imu la tethe t g e o metry o f t h e tw o b l a d e s an d the fl ight o s ci ll atlng l oad, which was compri s edof one cond i t i o n s a r e d i ffe r en t. Also, i n F i g . 93 o f c ons t an t te rm , 8 c os ine ter ms , a n d 8 s ine the sa m e repo rt so me te s t d at a sho w s a s i m i l ar term s , T h e s t e a d y and un s t ead yr o t o r blad e variati o n o f l o a d ingin the s ame a z imuth range, l o a d ing is given for one rev o lution. The blade was o sci l lat ed thr ou gha n an gl e of . 1 rid. ) The ch o rdwisepre s suredistributions f o r ab o ut a mean ang l e o f .1 rad. The effe c t o f the the (7-32 c ase a r e pr e sen te d i n figure 6. It os ci ll ationi s r e adi l yappare n ta s c o m p aredt o shou l d be remem be r ed t h at in us i n g t he vo rtex ;he s tea d y c a se. With t he ha rm o nic la tti ce meth od ,th e lo a d i ng is co nc en trate d at r e pre s e n tation o f the loadi ng,the m agnitudeand th e l o cati on of t he vo rt ex whi c h fo r t h e (7 - 3 ) pha s e o f the s e ve ra l harm o nic lo ad s ar e ea s i l y !
c a se i s locat e d a t .0833C , . 416C , and .75C . The dete rmi n ed . The magnit u de sa r e p l otted in pr ess urewa s f a ire d u s i n g a c u bic s p l lne thr o ugh F ig. 12. The o n l y harm o nic l oad s that were t he thr ee vo rtex loc at ions an d t he k n o wn v a l ue sig n ificantly c ha ng ed fr om th e steady case w e re !, of ze r o at t he tr ail in g e dge. Th e d i s tri b uti o n s t h e 3rd, 4 th and 5th . B o t h th e 3rd and 5th Lr are gi v e n f o r 7 s pa n wi s e pos iti o n s . In general, harm o nicswere inc ,_;_d and the 4 th harmonic th e c u rv es exhibitthe ex p ect e d s hape, having wa s dramatica l lyincr.,sed. Another ca l culation • th e lar g e st v al u e s as t he le adlng ed g e is wa s ma d e f o r t h e non-os ci ll a to ry u n s t ea dy ca se ....
ap proa c hed. Fo r t h e sp an d istrib ut i on t he an d co mpare d t o the q u a s i -st ea d y ca s e . _ .
values a t r / R T - .8 t a re slig h t l y l arger tha n V irt u ally no diffe r encewa s o b s er ved ,i nd icating t he values at r / R T , % ,, in d icating a fal l ing that, at l ea s t f o r this ca s e, t he rate o f ch a nge ' off In the tlp r eg i on , o f lo a d ingin a re vol uti onof the b l a d e is small _ ; en o ugh so that the effect o f a variab l ewake i s F r o m t hese concen trat ed f o rc e s ,th e sec ti on n e g l i g ible.
pitchi ng m o ment c a n be ca l cu l ate d . F igure 7 p re s ent s the s e re sul t s fo r _= 9 0 degree s . T he s e c ti o n m o m en twa s t ake n ab ou t the I / 4 C a nd a Compressible Effects (5-i) " J : n ose d o wn m o me n t is taken as p o siti v e. Th e p i t chi n g m o ment sh o w s so me ratherdramatic F o r a o ne-b l adedr o tor, th e effect of _ chan g es a lo n g th e s pa n. T h e mome n t i s nose u p co mpressibility i s i ll u s tratedin Fig. 13 , in _ ne ar the tip (r / R T - .95), change s t o a sma l l which the CT i s pl o tted again s tazimuth n ose dO w n va lue , t hen b ecomes nose u p f o r mos t an gle. The inc o m p r ess ib l e re sult i s i n clu de d _" _' . _ .
of th e In b oardre g io n . I n tegr a ti o n of th e f o r c o mpari son. A s ex p ec ted,the c om pre ss ib le j .
m o m e n t w o u ld re sul t in a t o ta l pitch moment up l o ad i s l arger than the inc o n, pre ss ib l e a t _- g o o . thr o ug ho ut o ne rev o luti o n. The effect i s , greate s tin the regi on o f the advancingb l ade and s m a ll e s tin the retre a tingregion as w o u l d Swept TIp be expe c t e d. _ .
The segme n t s used fo r t he vo rt ex la ttic e for t he s w ep t t ip s t ud i es were( 5-1) ,wher e tw o Two-B 'a dedRotor in CompressibleFlow (5-1 per _" equal seg m en ts w e r e us e d i n th e ti p r eg i o n a nd _ @ " , t hree equ a l seg m en t s w e r e used t n the un s we p t _ i nb o ar d s ectton. I n F ig . 8 th e l i f t i s sho wn T h e m e t hod h a s been e x t end e d t o th e plot t ed ag a i n st a z i mu t h f o r th e tw o s w eep tw o - bl a de d r o t o r fo r th e co mpr ess ible c a se an d ' co n ditions a nd fo r ze r o s w eep. I n ge n e ra l , th e th e r es u l t s are s h o w n i n Fig. 14. Th e t h ru s t th r ee r esults sho w l i ttle dif f e r ence. T h e c oeffi ci e nt C T pe r o l a d e i s g iv en v s. az im u th i sweptbackc o n f i g urati o n has a l a r ge r llft fro m a n gle fo r a s ing l e b l a ded r o t o r an d fo r a - 3 000 t o 4 0 0. F or _ - 100o t o 2400 , tw o -b l a d e d r o t o r . Fo r a z im u th an g l es fr o m the s w ept f o rwar d co nf ig urati o n has a s l f yn t l y _ • 20 0 t o 1 2 0 0 th e sing le b l a de r o t o r ha s a : la r ge r lift. I t a pp e a r s tha t th e to t a l l ift fo r la r ge r C T . For _ , 1200 t o ? 600, t he C T _ o n e r o tati o n f or the sw e pt- back case a n d the on th e o n e a n d tw o -b l a ded r o t o r s a r e s wep t f o rwar_ c a se w ould g tve a bout t h e sa me lt ft appr oximat e l y the same . Ho w eve r , fo r _ • 2600 ( l_ as pro d ucN b y t he unswept r o t o r . In Fig . g t o _U o a d ra.atlc r educ t ion in 1 1ft occu r s t he lt f t d istr ib u t i o n a lo ng the _ o t o r s p an I s f o r t he tw o -b laded r n t o r a s co mp a red t o th e o n e { _'_ _ tv e n f or _ - 0o. The m a jo r e f fect o f s w ee p bladed r esul t s . T h e lo west l ift o c curs at ; t s conce n t r ated at t he ti p , wh e r e th e sw e pt-ba ck _ - 2 9 20 whi c h p l a ce s t he o ther bl a de o f th e _1 tt p lo a d Is g r e a te r th a n b oth t he unswept an d two- b l a ded r o t o r at _ - 1 12o, t he po int o f swep tb a c k cases . In f t g, 1U, _ '- 18 0 ° , m axi m u m li ft on t he o t he r b l a de. A pparently t he ) Co m p arin g t o ftg . g, the sw e pt- back tip load I s ht gh li f t on the bl a de at ) • 1120 c r e at e s a i la r ge r than bO t h t he u n s w e p t a nd t he ve ry u nfa vo ra ble in d uc ed vel n c tty o n the s econd ._ swept-f o rwar d t i ps , b lade at _, 2 9 20 w h i ch r e _ J t r es th e load in g t ,_ to go to Z er o tn o r de r t o satis f y the bou n da ry , c o n d it io ns at @ - 2g 2o .
)
9 5 + i.
H e licopt e r R ot o r L oad s U s ing - _ A llne ar l z e dli f tin g s ur f ac e th e ory D l s cr e t lzedMa tc hedAs ymptotic Exp an sio n s, ' includingth e e ff e ct s o f compr ess ibility ha s NASA CR 166092.
ii l C ON C LU DI NGRE MA RKS P i e r ce, G . A. ; a n d V a ldyanathan,A. R. 1983: b ee n d e v e lopedfor a h e licopt e rrot o r in forward acc e l e rationpot e ntial,an d m ak es u se of th e for Calcu l atingth e Aeno d ynamlc vort e x - lattlc e proc e dur e for p e rf orm ingth e Loading o n an O s cillating F init e Wing in r e quir e dint egr ation s . In a dd iti o n,the me th o d S u b so nicand S o nic F low. NACA TR 1322. 4 ha s been ex t ended to i n c lude t he effe ct s of , uns t ead y f l o w. Run y an, H. L. 1 97 3 : Uns t ead y Lif ti ng Su r fa c e Theo ry A p pl i ed t o a P r opelle r and f light . Th e m e th od utilizesthe conc e pt of t he R u nyan , H. L., an d Woolston , g. S. 195 7: M e thod seve ra l cases. These i nc l ude t he effec t o f Lou _ n b o r ough , Un i ve r si ty of Techno l og y .
Sample c al c u l a ti ons ha v e been done f or Hel ico p t er Ro t or, / h .D . Thes i s, swe p t-b a c kan d swe pt-for wa rd tip. T h e effec t o f + th e s e two tlp co n fig u ration s wa s m in ima l on th e ( total loadingfor on e r e volution. How e v e r,th e l o a d i ngd istri bu ti o n c hangedc o n s ide rably f or se ve r a l azi m u th p o s it ions. A c o m p a ri son of th e Z th rus t co e ff i c i en t , C T, of a on e blad e d roto r and a tw o b l ade d ro t o r w as m ade . I n t he I a z l muth a l r a n g e betwee n 2 0 o an d 12fl o , the o n e b l a ded r otor s howe d higher li ft. H o we v er
I
rotor indicat e da l ower CT. Co m pressibility was inv e stigat e d for on P config u r a tion.As . a dva n c i n g blade r eg io n ( $ = g o o ) a n d wa s i'" -- % expe ct ed , t he effe ct w a s g r ea t es t i n t he + X - O (T ) / j // ..
m i n ima l in t h e r e tr ea ti ng b l a de r e gi on . Th e / J _ / s " at 4 / r e v t s g i v en . T h e e f fec t on t h e total _ . .. ._. .
'+' eff e ct °n C T °f a b ladu °sclllatingIn pltch t _ " " : _l b lade l i f t is sho w n a nd t he e f fe ct o f t he i l l .)_ c o n t e nt w as ca lcula t e d an d th e g r e at es t dif f ere n cebetwee n the o s cill a to r yand osc i ll ati on i S r e adi l y appa r en t . The h a rm o n i c _ y ha rmo n ic. n° n' °sc i ll at ° r Y c a ses wa s f o u nd i n t he 4 th _ j / l REFERE NC ES X + .
Alba no , E. ; Ro dd en , W. P. lg 6g: A D ou b le t L a tti c eM e thod f or Calcul a tingL lf t F i g. I I n e rt ia lC oo r di n a t e Sy s tem , Di s tribution on O s cillatingSurface s in Sub s onicFlow s . AIAA Journal, r U Vol. 7 , N O. 2 , p p. 27 g- 285 . _+' Oat , Roland 19 73 : Tn e Li f ting Su r f a ceT h eo ry + Y A p p l i ed t o Fi xed W in g s a nd P rop ellers O NRA TP N o . 1298 . _ T=T O' _ : _+ D a vi s , P . J . a n d Rabl n owl ti z. P . , 1967 : Singular Region .:i ,: E golf. T, A . an d L a n d g r e be, A. J . 1 983 : Hel i cop t e r Ro t or Wa k e Geo m e t r y and its I • Inf luen c e in Fo r wa r dFlly ht ,Vol, 1, NA S A _ : C R 3726.
_ " C o m p a ny . N u m e r i c a l Int egr ati o n , Bl a ls d ellPu b l l s hlng m ; + " H anaoka, T . 1 962: H y d r od y na mi cs of a n i _ O s c i ll a ti n g S c r ew P r opelle r . ONR , AC R ( ++_- (92) (19 6 2).
": _ ' d K u s sn e r, Hens G. 1941 : Ge n eral A irfoil Th eory.
'_ ,1 NAC A T M 979.
_ :1 Mo r se , P. M. a n d Fa sh h a ck, H . 1978: Me t hod s of ,u Th eo re, lcal Phy sic s , M cG raw- H i ll, In c .
" Ma n g ler,K. W . 19 52 : I m prope rInt e gral s in : i T he o reticalA e ro d ynamics. F ig . 2 Integrati o n Area s 96 " _ , f r , !
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[ " %% _ " _ . _ - ._h._ 0 45 90 135 180 225 270 315 36 0 0 0.2 0.4 0.5 0.8 1.0 ,.:, i Anqle,(leg C hord ,.. , Fi g. 5 Thr u s t C oeff i c i en t vs. Az i m ut hAng l e for F i g. 5 C hordw i seP r essure D i s tribution f or Four Panel Co nf i gu ratio ns for a Si ngle Sever a l S p a nw t se Loca ti o n s, _ • 90o, Ro t o r Bl a de, I n compress ib le, (u 0.17 e a = O.1 r e d, _ - . O F r e d , i ncomp r essible, (p .17, el _- 0.1 O _ " 30 ra d / s ec) ca d, _ - .05 r ed, ; 30 ta d / s ac) S_ , ) - , 0 4 000 Section Momnt _ ...... Smpt F orvmrd 22.8 ° ; fl-lb / ft -5 I _ _ - No S l , p ; , o \ - .... , - , -.- , .
. 25 , i z ) I000 n I I I I I___ ) 0,2 0, 4 0, 5 0.8 l .O 0 _ t 0 1_ 190 225 _ 3 15 _ S N nr l RT Angl o , d o g ) FIg. 7 Spanwlse S e c t ion Homent Di s tribution ) about 1 / 4 C - Po s i t iv e N nse Down , Fi g , B C ompari s on of Lift on a S _ ept-Beck Z ero t" 900. i ncom p r e ss ibl e . S weep a nd Swe pt-Forw a r d Blade. !
(p • 0.17 . e I • 0.1 red . _ r " .05 in com pr essi bl e . (, • 0.17. ee - 0.1 J tad , II 30 red / s ic) rid, or • ,05 r i d , I I 30 r l (] l lo {) .... . _ , , . _ • "_ lllr _ d V " El Swept B ade 22.8 ° NoSweep _ i 0 No Sweep A Swe , _it / _ S w ept For m rd 22.8 ° 200 m / n m / ft . .
,_ I00 0 0 I I I I • 0.2 0.4 0,6 0.8 t.O 0.2 0.4 0.6 0,8 l.O ,, ,: r / o_ Spa n r I RT _ ..
- " Spa n "_ . ;: _ - _ - _ F t g. 9 Spanw t se Sect i on Lt ft D i st ri b u t io n f or F t g. 10 Spa n wtse Sec t ion Ltft Distribu t ion fo r _ Swept-Tip Configu r ations, _ • 0o, Swept-Tip Con f igu r a t ions , _ - 180o, I n co m pressible, (u 0.17, e a - 0.1 I n co m pres si ble, (p 0.1 7 , e g - 0.1 r ad, =r " .0fi r ad, ; - 30 ra d / sec) r ad, a r " .05 tad, ; 3 0 raa l sec) ;.:' : .
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Angle,d_ _ _ } Ft9. II Comp a risonof Ltft on a Rotor Blade Ftg. 1 2 Ha rmonic Content for Non-Oscill a tory a nd I Ltft on a / :on-Olctllatorj _ Utade. tncoeprelst_ l e , (. • 0 ,1 7, 0. • 0.1 tncmpreaatbl e (u • 0.17. at • 0. 1 r ed, ar • , 06 rid, g - 30 red , le:) '1_ Oscilla t ing tn Pttch at 4 / Rnv. to the Os r .tlla t or ? Cotes- r / ; , r • " _ ' 1 ri d, • r • .05 r ed , n - _ rad / sec) " . : d_ ° _ .J _: f ' _ 4 _O n e Bla de i " 4 t _ .... TwoBlades J " _ _4 I / "%_ , - Comp r e s sibl e t .
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2 Llncompr e ss i bl e I - •• ' , t s 1 O- ! _ 2 • i , J , I I I I I I I --J -1 I I I I I I I I ; 0 45 g O 135 180 225 2 1 0 315 _ 0 0 45 90 135 180 225 270 31 . c 3 60 _ _ " / Angl e ,de9 A_ ,gle,de9 __ F tg . 13 ]n c ompresstble a nd Co m p r ess i ble L ift f o r Ft9. 14 L i f t on Two-B] a.t ed and One-Bl a de d R o t or' One-Bladed Roto r , ( _ = 0.17, e B = 0.1 vs. Az i mu th An9 _ e, co m p r ess i ble, tad, ar = .05 t ad, HT[P 0. 54) ( _ = 0. 17, e e = 0.1 t ad, ar = . 0 5 1 -' r ad , M l - i P 0. 54) _ __- ; . | 1 , | - !
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DISCUSSIO N • , _ Pape r go. 7 ;_+ ' .] Bob S o phermSlkorsk_ j Alrora t' t: Just a couple o f que s tions. O n e, you had a r i gid : otor In thi s _! analy s i s ?
_ Ta i : Ye s .
J Sop har : The r e i s n o aeroela s tl c ity and y ou w en t up to a , o f .177 Ta__ _ l: That ' s cor r e c t.
/ _ Sopher: First of all , I t hin k that as soon as you pu t aeroel astic lty In you are going to see _ di ff erent trends w hen you pu t sw eep on In co mpari s on w ithou t s w eep.
+ . Ta__l i : Oh, s ure. _ + - _ ; be a v ery g ood a pproximation to the ac t ual w a k e, whi ch w il l be substantially d isto r t ed--mo r e " _] li ke a wak e that you w ou l d g e t un d e r hover c ondit i o ns . So I ques t ion t he u ti li ty of ' as sumin g a _ J skew ed h _ £1 ca l wa k e .
++ t + is + _ o. + .n -- .----.--not
proce s se s . You can g o back and ca lculate the wake and p u t it back a g ain. Hopefully, t hat procedure w ould -4 re you a better result.
So,her: Th e third question i s wha t advance rat i os do you expect to apply the analysis up t o?
" ' -=. .
Tal: The answ er Is I don't k no w . Ho w eve r , I w ouJd t hink U hat the higher f orward s pe ed w ould ; ^+ -.
Sopher: [ que s tion that be ca u s e a s you go up to higher s peeds you are going t o f ind t ha t you _ _ . _ . _ * . _ ; _ l _ : run i n to s i _ , uatlons w here you g et tran s onic f lo w o n the a d v an ci ng blade a n d I do n o t believe _ +_+ _ .
tha t the linear analy s i s w ill ap p ly accurately under tho s e c o n di ti o ns . I_ - + i_I probably ha ve a better an s wer. Becau s e you don' _ depend on the w ake that much. _ Ta__ _ l: Perha ps you are righ t . : - _ _ " Sopher: A s a ma tter of " f a c t thi s re s earch center has developed tran s o n i c fl o _ analy s e s w h i ch _ ; _ =_ - apply t o three dimensional li fting b l ades s o I w ould s ay t ha t the primary utility t ha t I w ould i . '_ ..
s ee in this analy si s i s f or hover appli cati on s w here the li near ana l y s is is valid, but you w o u l d L_ have t o u s e a d is tort ed w ake. _ ', " " Ta_. _ l: F or hover ca s es you w ould really expe c t the w a k es to st ank up and the n you en d up w it h a _ ' _ ' - / • very dl t Ticul t mathemati ca l problem. Ho w ever, I gues s _ ' rommy past exper i en oa you probably can _ / _ . . g et the loading by s , _ e n um eri ca l pro c edure. For example, you can do ex t rapola t ion. Assu m ing a cer t a in W and then you extra po late f or _ : O. I don 't kn o w . _ e don't have a ol ma r u nd er- s tand i r, _ . I admit that .
i Jim HoCrosksy m U.$. Ar_ Aeromechuni ca Laboratory: You have ma de some nice pro g re ss on t h is ,..
,- approa c h since y ou talk ed w_ th u s a year or s o a g o. I t ' s in t ere st in g ; it' s nice to se e so m e i • I presume this blade i s un twis te d , i s tha _ right? " "' " result s bein g g enera ted % r so m e _ e _ ll s_ :lc oase s . I w an ted to a s k a couple oP minor questions.
I • H cCroskey: H owdid you tr e a t t he reverse t ] ,_w re g ion?
Ta..._ l: The twis t can be added on very easily b e caus e [ we only have to ] add on th e bo un dary co ndi- ti on. T _ avoid a reverse ph e no m enon w e del i berately u se very lar l e outo _ ' P . You can see [that i t is ]j i x r es t . W e tr y to avoid that re gi on.
In t a c t i n , _ _ : the sk et ch t h e bo ok is a little mi sl ea ding b eca us e t h e ou t put i s more "_ .
like 3 0 _ i n st ea d o _ t h e 10 o r 15 _ [ that appears i n t h e Pi_ ur e ]. 30 you Ju st avo i d e d I t
b y
havi ng a roo t o ut ou t . _ T a _.!i: Yes . , + , _ ,
; co ++ - :+
., _+, ;+_.<+ i w hat y ou pr edi c t I n Wl g ur e 13; Is that a l ug er e ff ect t ha n y ou w ould predict I t" you Just u s ed m ki nd o f P r andtl-Gl ausr t s caling on the I n c o m pre ssi ble s olution?
i NoCroskey: The fl ns l th ing I s o n the I n flu en ce o f com p ress i b ilit y . B o you ha ve s o m e I d ea o f r "4 Ta le W ell, T g ue s s w e sh ou ld be bla m ed fo r not m akin g t h e ab str act v e ry elmer . W e us ed a very, 'i _ ver y hone st way o f do i n g t ha t. W e didn't u s e , - n y a pprox im ation a t 4 11. In othe r w o rds , as I pointed out, you E l nd the • as t _ n otl on o f " r_. In othe r w o r d s , y ou g ive It the ra d i u s o r" the trivl al matter. T o an sw er your que s t i o n w e s ay t hat w e u s e true, ho n e s t compre s s ibility et ' f e ct .
HeCroskey: But the que s tion is ho w g oo _ w ould the P randtl-Glauert type approxi mat ion be to w hat you actually ca lcula ted ?
Ta.11: W ell, to hone s tly an sw er tha t que s t i on-- w e don't kno w . W e didn't check [ i t], but I th i nk i t Is not very ea s y to check It out.
_it d o ublet [ an d] y o u g o there a n d f ind the T wg l _ h s erves a s your upper li m it-- w h ic h i s not a 4 like you have m ade s ome pretty g ood pro gr e ss i n the la s t year o r s o. The method s w e are u s in g no w f o r ro u t i ne r otor load s analy s i s a r e u s ually ba s ed on s o m e f a ir ly p r i m itive a ss umpt i on s like s t r ip theory and 2-D a irf o i l o oe f E l o lent s an d s o f orth an d wha t w e ult im ately ha ve to get to Is very, very s ophi s ticated t m my be , 3-D C F D k i nd s o f " analy s e s . It look s like w ha t yo u 've g ot I s an l ntemed i ate ty pe o f analy sis w h i ch ,my be very pra c t ica l. M y que s tion I s do you thin k there I s a pract i cal w ay to g eneralize t he re s ult s yo u have g ot t en, s ay, to co m_ up w ith g eneraliz ed I Bob Om Is t _ n _ U.S. Ar my A eromeehanf us Laboratory: I w ant to c o m end yo u r re s ult s . It look s forcing fl a l o tion s f or speci f i c loading di s tributions, a f a m ily of " loading di s tribution s t ha t you might be able to calculate and then not ha ve to re pe at the integration proble m f or each parti c u- lar oo n f l s ura t ion t ha t you a r e analyzing? I s there a pra ct ical w a y t o do that?
Ta. I : W ell to ans w er your que s tion, the an s w er i s ye s . I did not mention that w hen we break the blade Into di t' f erent s e gm ent s . Apparen t ly the matr i x I s highly diagonal. In other w ord s o t'E _ _ di _ onal mat r ix you can us e les s accurate me thod s to generate. Beyond t ha t, t o an sw er your que s tl _ , l , I think in a practica l s en s e w e ca n generate tho s e matrix element s and s tore them and only change t he boundary condition s to do all the type s o r" ca lculation s . In o t her w ord s, the - ans w er I s indeed It can be very p rac t ical.
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