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NASA Technical Memorandum 86315
NASA-TM-86315 19850002608
LIFTING SURFACE THEORY FOR A HELICOPTER ROTOR
IN FORWARD FLIGHT
H, TAl AND HARRY L, RUNYAN
SEPTEMBER 1984
It
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National Aeronautics and Space Administration Langley Research Center Hampton, Virginia 23665 LIFTING SURFACE THEORY FOR A HELICOPTER ROTOR IN FORWARO FLIGHT H. Tai NASA Langley Research Center Hampton, Virginia 23665 Harry L. Runyan College of William and Mary NASA Lan~ley Research Center Hampton, Virginia 23665 ABSTRACT A lifting surface theory has been developed with a stationary observer, whereas the lifting for a helicopter rotor in forward flight for surface theory is essenti ally concerned wi th the details of the near-field case for a co-moving compressible and incompressible flow. The observer as well as the satisfaction of certain method utilizes the concept of the linearized of certain boundary conditions. Runyan (1973) . acceleration potential and makes use of the utilized the acceleration potential approach to vortex lattice procedure. Calculations demon- obtain a solution to the oscillating propeller strating the application of the method are given in terms of the lift distribution on a single in compressible flow. Oat, (1973), has derived a general expression for an acceleration doublet rotor, a two-bladed rotor, and a rotor with for any motion. Pierce and Vaidyanathan (1983) swept-forward and swept-back tips. In addition, have treated the helicopter rotor in forward the lift on a rotor which is vibrating in a flight using the method of matched asymptotic pitching mode at 4/rev is given. Compressibi- ex~ansion for the incompressible case. The lity effects and interference effects for a procedure developed here involves the precise two-bladed rotor are discussed.
numerical inte~ration over the surface of the rotor in a time frame. The method sets forth a formulation of a fundamental three dimensional, I NTRODUCTI ON compressible, unsteady aerodynamic theory for propellers and helicopter rotors. Rotating lifting surfaces are an integral part of the ~ropulsive unit of every aeron- The next section contains a brief autical and nautical vehicle, from the derivation of the fundamental equations, includ- compressor and turbine blades of jet engines, ing a discussion of some implications of the the pumps for rocket engines, to pro~eller and equations. The third section contains a helicopter rotors. The aerodynamics of these description of the method of solution. Finally, rotating elements has been under extensive study the results of some calculations for the several since the advent of the airplane and with a specific examples are given. combination of experimental and analytical approaches, succcessful designs have been SYMBOLS achieved. In many cases, two-dimensional theory has been used, usually modified by an assumed spanwise distribution, and inflow velocities. A' rotor blade area This pa~er presents a compressible, lifting A aerodynamic influence nm surface method for a helicopter rotor in forward coefficients Fourier coefficients flight within the limits of linearized theory. An,B n c speed of sound The method is based on the concept of the chord of rotor C acceleration potential, originally introduced by thrust coeffic!ent per blade CT Kussner (1941). The method was first applied to (thrust/lIp n2R t )
an oscillating wing in uniform translatory 0 vector distance from doublet to
motion inclUding effects of compressible flow by downwash point
Runyan and Woolston (1957). The acceleration 0 absolute value of 0
A potential approach has now hecome standard for DID D = unit vector of ~ the determination of the unsteady aerodynamic I value of singular integral forces for flutter studies of lifting surfaces K kernel function in rectilinear motion.
-I- t n unit vector at downwash point, The first use of the acceleration potential normal to velocity vector -I- approach for a rotating system was made in a unit vector at doublet point, no paper by Hanaoka (1962) for the 1oadi ng on a normal to velocity vector marine propeller in incompressible flow. The direction coslnes of ~ .t,m,n acceleration potential has been used in the past in studying the propeller noise problem, but in mo, direction cosines of no .to, no all of these noise propa~ation cases the problem p pressure was specialized early in the analytical develop- position vector of doublet
X
o ment to the so-called far-field case usually from inertial frame origin X position vector of downwash point from inertial frame origin source or doublet strength q lifting rotor is assumed to lie in the skewed rotor tip radius Rt hel ical path taken by the rotor blade. One rotor root radius R reason for adopting the acceleration potential s r distance of downwash point approach is that the pressure discontinuity along the span occurs only on the surface of the blade and thus distance of doublet along the the boundary conditions need only be applied on soan the blade surface and not throughout the wake.
The blade is treated as a very thin surface of discontinuity across which a pressure jump upper limit of spanwlse panel occurs. The effect of compressibiity is taken lower limit of spanwise panel
•
into account by utilizing the complete distance of doublet along linearized potential for a lifting doublet, span at singular point time along with the effects of retarded time.
field time t
•
velocity of rotor system, U As shown in Fig. 1, an jnertial coordinate parallel to x-axis, positive in system has been used in which the origin of negative x-direction coordinates is fixed to a point on the ground.
+ The helicopter rotor is moving in the negative
velocity at downwash points v
x-direction with velocity U, in the positive + z-direction with velocity Wand is rotating velocity component of V at counter clockwise with a constant angular the downwash point normal to velocity n. A point of interest on the rotor the rotor leading edge + + blade is designated by the radius vector XO(T) velocity of doublet V o from the origin of the ground based coordinate velocity of rotor system, W system.
parallel to z axis downwash velocity Let ~ be the acceleration potential of a distance from pitch axis to source (or doublet), the perturbation pressure downwash point is then' !Jiven by Cartesian coordinates of x ,y,z downwash point
(1) p = -p~
Cartesian coordinates of doublet position This expression represents the pressure p at twist angle at downwash point + twist angle at doublet point X due to a single source position + angle of axis of rotation (or doublet) located at X The potential ~ relative to z-axis o • contains a constant "q" which represents the + + strength of the source and thus the magnitude of Volc the pressure. In this form, there is no + + 1 aiJ"o boundary condition available to determine the Volc
a
value of "q" and the resulting pressure.
caT
Recourse can be made to the velocity potential, of blade at position angular e since the spatial derivative of a velocity time t potential represents a velocity. The angular position of blade at relationship between the pressure and velocity time T potential for an inertial coordinate system is blade angle of attack blade angle relative to plane of rotation (2) p =-~ advance ratio \.I air density P where %t is the substantial derivative.
time T, TO Dropping out the second order terms time at which integrand in Eq.
T and integrating with respect to field time (24) becomes singular results in velocity potential t source acceleration potential doublet acceleration (3)
f ~(t') dt' 4'(t)
potential azimuth angle rotation speed of rotor The acceleration potential ~s satisfies vibration frequency of rotor the wave equation 2 1 a ~ BASIC FORMULATION
'V ~ - 2" -r = -411 f(~,t) (4)
Scat The formulation of the aerodynamic + equations is based on the linearized where f(X,t) is a source distribution.
acceleration potential. approach. The fluid is Furthermore, if an isolated source is considered perfect, with no separation and the + + + formulation is based upon the assumption of moving with velocity V f(X,t) O(X o, then sma 11 perturbations. The wake created by the Vot) where 0 is the delta function.
Usiny the Green's function formulation the required. This derivative is taken normal to acceleration potential expression for a moviny the flight path at the location of the downwash source, 's can be written as (Morse and point, as follows Feshbach, 1978, p. 841) q(~O,T) to obtain
"s (~, t) = -----'------.:::....------
_ ~O(T).[~ - ~o(T)]1 A A + + + Aw - q
n • 0 n • n
tn •
no + n - 2 A
cit - to (T) I
411cO [l-O • aJ + A (!» 0
-t(n .! it • 0
D -c'
+ " n· 0 0
where XO(T) designates the position of the A + A + A + A + +
. !) o + n • 0 n • 8)/(1 (9)
+ noD n • - 0 source at time T, X is the position of the +
A + A 2 0 -:-!J A
field point at the time t, VO(T) is the -( no . D n • 0[1-8 + ~ • 8 + ~2 J)/(l - 0
c c 0 velocity of the source point at time T, c is the speed of sound and q is the strength of the source. An aUXiliary equation which relates + + 3n+. OA +. OA () n 1 TO r 0 n' no - 0 dT the time interval (t - T) to the distance
+ 4l;"" J -co q [ 03 ]
between the two points is Eq. (9) gives the downwash at a field point (6) (x, y, z, t) due to a doublet placed at a point (xo,yo,ZO,T) having a strength q. In order to represent a lifting surface such as a which is usually referred to as the causality rotor, it is necessary to distribute the condition. Eq. (5) expresses the potential as doublets over the lifting surface and integrate an explicit function of T, and only through Eq.
over the surface to obtain the downwash at a (6) as an implicit function of t ans~. From field point. If the downwash is known, the Eq. (3), the velocity potential due to a moving quantity "q" can be determine~. Letting K b~ source is the expression on the RHS of Eq. (9), the final t t equation is q( T') (10)
Jf K dA' w =
~s(t) = J .,s(t') dt'
n A -co where A' is the area of the rotor surface.
T The LHS, wn, represents the known boundary
= J !tLT~J dT'
condition and is the velocity normal to the
D17T '
-co velocity vector at the downwash point. By using the no flow condition for the velocity
where 0 = ~ - ~o' 0 = 1 1
perpendicular to the blade surface, the velocity
component in the n direction is V
n tan6w or
and dt'= [l - D~a] dT'
w (r,t) = V tan6 = If K dA' (11)
w n n A The quantities T', t' and t, T satisfy Eq.
where V is the velocity component of V tt the n (6) • downwash point and is normal to the rotor leading edge and 6w is the angle of attack.
By definition, the doublet velocity Thus the problem requires setting up a method of solution of Eq.' (11) from which a value of 4, potential 40 of a doublet aligned along 00 the unknown doublet strength, can be determined can be written as which satisfy the known velocity boundary conditions wn' +
n • V+ '" no • V+ ~
= -
o X 't's X s o This represents a rather formidable (8) computing task and the history of lifting surface theory even for non-rotating wings has
n.O T n.D
centered on devising approximate methods to
= l 0 .9. J T + I q 0 d T' Of
0 3 accomp1 ish the integration in an economical 411c(0-a.o) -co 411D manner. One method, termed the vortex lattice method, has been very successfully applied to Note that for incompressible flow, c + co, the aircraft wings, and is probably the more first term + 0 and the integral remains economical procedure of the many variants. This
unchanged except for the upper limit where T =
method was first demonstrated for the unsteady t.
case by Runyan and Uoo1ston (1957) and was later expanded by Albano and Rodden (1969). This is To obtain the final equation for downwash the method adopted in this paper and the AWn. a second directional derivative is application will be discussed later.
Specification of Coordinate System . The blade has the cord C and length Rt- R being distance to the the root of s • R s the blade. Rt is the distance to the tip of the blade. Let the blade momentarily coincide W(roo cos (OT) - (C/4)O sin(oT)cos a ) o with the coordinate system along the positive (16) m =- o x-axis at t = 0 and execute a counterclockwise
V ',lw2 + V'2
rotation with angular velocity O· while moving o 0 with velocity U along the negative x direction and velocity W along the positive z V ' direction. Since the vortex lattice method has n = 0 been adopteQ, the doublet point lies C/4 ahead 2 and the downwash point lies C/4 aft of the
o /w + V'2
o section midchord. The position of the doublet poi nt as well as the downwash poi nt can be where established as follows. The Cartesian components of the doublet position are V ,2 ='(U + roo sin(OT)+ (C/4)n COS(OT)COS a )2 O o Xo = -UT + ro COS(OT) - (C/4)sin(OT)cos aa Yo = ro sin(OT)+ (C/4) COS(OT)COS aa + (roo COS(OT) - (C/4)n sin(Or)cos a )2 (17) Zo = WT + (C/4) sin aa (12) o where ro is the radial distance of the doublet ... ... ... ...
along the span. With the substitution of By the same procedure, n = ti + mj + nk, C + -C, ro ... r, T ... t the position of the where downwash point is given by t = W(U + rO sin(ot)- (C/4)o cos(ot)cos a) x = -Ut + r cos(ot)+ (C/4) sin(ot)cos a (13) y = r sin(ot) - (C/4) cos(ot) cos a V' / W + V '2 z = Wt - C/4 sin a (18) In E4S. (12) and (13), the angles a,aa are the m = -W(rn cos(ot)+ (C/4)O sin(nt)cos a) twist anyles of the velocity vectors ~ and ~o,
'I 2 ,2
respectively, defined by V W + V W tan a sin(ot)+ rO U V n = (14 ) W tan a o U sin(OT'+ roo and The reference plane defined by the doublets and V,2 = (U + rO sin(ot) - (C/4)O cos(ot)cos a)2 downwash points is a twisted surface. From Eq. (12) the doublet velocity can be computed, + (ro cos(ot)+ (C/4)O sin(ot)cos a)2 (19) namely the time derivative of the position ... ... ...
vectors.
the vector 0 = X-Xo defined in Eq. (7) can be expressed as o = {[U(t-T) + r cos(ot)- r COS(OT) o The unit vector no is chosen to be + (C/4)(sin(ot)cos a + sin(Or)cos a )]2 o perpendicular to the twisted surface created ...
(20) + [r sin(ot)- r sin(oT) by the velocity vector V which is a function o o of ro, through Eq. (14).
...
-(C/4)(cos(ot)cos a + COS(OT)COS a )]2 o Express no as ..
+ [W(t-r) -(C/4)(sin a + sin a )]2}1/2 o (15) With the substitution of the quantities, the integra 1 E4. (11) was solved for the unknown q(ro,T) by using a collocation process based where to' mo' no are the directional cosines of ...
the unit vector It can be shown that "0.
on the vortex lattice assumption. The kernel is singularity. The integration domain was divided
singular when 0 = 0, and this was handled by use into areas as shown in Fig. 2. Areas 1-4
of the finite part technique. (hatched) were computed numerically using a two-dimensional Romberg integration (Davis and Rabinowitz, 1967) and the contribution of the SOLUTION OF INTEGRAL EQUATION singular region (unhatched) was obtained in closed form by consideration of the finite part In following the vortex lattice technique as shown in the next section.
the rotor is divided into a number of predetermined panels, both spanwise and Treatment of Singular Term in Integral - The chordwise, In each chordwise panel, a line of integral in the downwash equation, Eq. (11)~ is singular when 0+0 and produces a complication doub 1 ets of unknown strength qi is located at the 25% chordwise location of the particular which must be treated properly. It should be panel, and the downwash is evaluated at the remembered that the integration path alony "T" point located at 75% chordwise location of the is the path the doublet has taken in arriving at panel. Therefore, a collocation procedure is the final dOUblet point at (c/4, ro) measured used to obtain a set of equations in terms of in the local blade coordinates and can be the unknown loadings qi. It is also assumed considered as the wake. The integration takes that the spanwise loading qi is constant along place along the path from -~ to the final each of the panels. A set of equations is thus doublet position at TO. The distance 0 is the obtained as shown below. distance from the integration point at time t to the downwash point at X.
(21) There is a particular set of values of ro r and t for which the denominator D approaches
where A = I u K dr and where n refers to
nm nm zero, thus resulting in an infinite integrand.
o r The singular part of the Eq. (11) is t the downwash point and m refers to the vortex
r T n.o - 3(0.0)(0 • 00)
lattice. The kernel K is a complicated function 2 o u I = J IT 0 dT dr (24) which involves an integration over t. r 3 o R. I The term q(ro,t) represents the strenyth
As 0+0 at the downwash point, D becomes
of the doublet located at ro and at time T, and is proportional to the unknown loading. In perpendicular to ~, therefore, at the singular order to account for unsteadiness, a solution point. the second term is zero and will be was formulated to take into account the time neglected in the treatment of the sinyularity.
However, this second term is retained in all of variation of the strength of the wake. This was done by assuming a Fourier series of the form the numerical integrations involVing Areas 1-4 m since it represents an important contribution particularly when the blade is passing over a
q(ro,t) = A + L (A cos(nGT) + B sin(nGT)) (22)
trailing wake.
o 1 n n The time and distance at which the integral I A A If q(ro.t) is assumed to be a function of ro becomes singular are designated by t and roo alone, which means that the wake strength does The domain of the integration in Eq. (24) not vary with time, the Fourier series reduces consists of a rectangle in which the duration
to q(ro) = A A solution obtained with T2-t1 is kept extremely small. In other
o• this approximation is termed the quasi- words. the integration is performed along a slit steady solution. in roo over which the 2nd term in Eq. (24) is negligible. Therefore the integral I can be This series was inserted in the basic apprOXimated by + + equation and integrated with respect to T.
r T n • no However, there were more unknowns than 2
I = I u J -3- dt dr (25)
simultaneous equations to solve for the o rR, T D unknowns. The additional required equations 1 were obtained by evaluating Eq. (11) at a number Furthermore. noticing that 02 is quadratic in of azimuth locations. For instance if m = 1, ..
thpn ro, if ao is independent of roo then the integration on ro can be performed q(ro,t) = A + Al COSOt + BlsinGt. (23) o analytically. This can be achieved by recognizing that in the vortex lattice method, The azimuth was divided into equal segments of 120 and the proper boundary conditions the rotor is divided into spanwise panels from 0 0 0 applied at ~ = 0 , 120 , and 240 thus rR. to rue If these spanwise panels are small then the variation in ao is small.
prQ\'idimLt!le necessary additional equations.
d<x ~IG o Numerical Integration of Kernel dr .. - (U sine + r G)2 + ~12 (26) o o a The integration was performed by numerical i ntegrati on, except for the area surround~ ng the If the value of ao is approximated by its mid panel value, it is possible to integrate Eq.
(24), in closed form in the ro direction.
This is quite acceptable in the hel icopter morie, because dao/dro is in the order of as liT is kept large because the very large values of the inte9rand near the singularity are magnitude 10-3 or smaller. The value ~o is avoided. On the other hand, regarding the also a function of ro, but in the region of finite part integration, the denominator was the singularity it has a very small variation expanded in a Taylor series about the and is evaluated at the singular position, A " " singular point, T. Therefore, it is desirable (T,ro)' Performiny the ro integration to maintain lit as small as'possible to keep results in the form within the limits of the applicability of the series expansion. Numerous calculations were
J = /2 .9l!l. dT (27)
made, varying At until a reasonable T lTtT convergence was found. This value was found to A where g(T) is a function containing all the be .01(t-t), i.e. 1% of the time difference.
non-singular part after performing the ro Actually, there is very little difference
integration and f(t) = 0, at
between 1% or 10% of the time difference and the computing time and cost is considerably reduced " " T=t (t1 < t < 12). It can be argued by using 10%. For trend studies 10% is physically that since the quantity D(t,ro; recommended principally to reduce computer t,r) as well as its modified form f(t) (after costs, However, for final design type analysis, integration over ro) represents the distance a smaller value of time difference liT is more between two points in space it must be positive appropriate.
and real for all its arguments, and never become negative. Denote the value of ro and t at For the spanwise direction, lir is also o " " whi ch D becomes zero as ro and to' Thus, in an integration limit variable. The finite part integral was obtained by approximating the angle " the neighborhood of t the function f(t) behaves of twist of the velocity vector across a segment like a parabolic function and has a second order by assuming it constant across the segment, zero.
having'a value as determined at the center of segment. Numerical experimentation indicates Expanding f(t) in a Taylor series about the
that for a helicopter, liro = 0 is satis-
factory.
singular point T results in A A A A A 2 f( t) = f( t) + f' (T)(T-t) + fll( t) (T - T) -/2 + ••• (28) APPLICATION TO SPECIFIC EXAMPLES Since f is a second order zero The foregoing analysis has been applied to several specific examples which are given in " " (29) f(t) = f'(t) = 0 and Figs. (3) and (4). The following section presents results for several paneling If only Eq. (29) has been verified numerically.
configurations; e.g. 5 spanwise and 1 chordwise (27) the square term is kept in Eq. (28), Eq.
panels (designated (5-1» and 7 spanwise and 3 can be written as chordwise (designated (7-3». The rotor blade A A t A was maintained at a constant pitch setting of
0' (t) 1Li!l
I = J 2 ---?. l g~ t)
+ ~ + 2 j df (30) 6B = .1 radians for all the calculations.
t 1 fll(T) (t_t)2 ( t-t) In Ell. (30), if t2 and t1 are chosen Single Blade A In order to investigate the convergence of symmetrically about t, then the odd derivative the method when usiny the vortex lattice terms integrate to zero. Futhermore, the third A procedure, the program was run for several term can be neglected since g"(T) is small. The chordwise and spanwise elements for the incom- major contribution comes from the first term.
pressible case. The thrust coefficient Cr
Then using the standard integration technique vs. the azimuth angle is shown in fig. (5). (In (Mangler, 1952) the final result for the all of the following plots for thrust coeffi- integral is cient vs. azimuth angle, the thrust was calculated for 16 uniformly spaced azimuth
I = - M_4_ (31)
angles and each curve was faired using a fll(;) lit cubic spline). The rotor was first divided into
•
5 spanwise and one chordwise (5-1) panel and the resu'lts are shown by the solid line. The where 2liT = T'~ - T1 and T1 < T < T2 • chordwise division was increased to (5-2) and A numerical problem arises because the the results are shown by the long dashed line.
It can be seen that very little change has taken finite part integration results in a negative number which is close to the total of the place. The spanwise divisions were increased to surrounding numerical integration areas which (7-1) and the largest change occurred at W = 0 where the difference in CT is about are positive. Thus, it is necessary to take the difference between large numbers, and the final 11%. Increasing the chordwise divisions to 3 (7-3) shows convergence af the (7-1) case ta be integration accuracy is dependent on the very \,load.
accuracy of the two integrations. On the one hand, the numerical integration is more accurate An interestiny phenomena occurs in the region of small azimuth angles. For w=0 to 37 , the lift increases to a local maximum at Hlade Oscillating in Pitch 0 then w=37 the lift abruptly falls to a local minimum for'v=60 and then rapidly increased An exam~le of unsteady loads on a rotor to a maximum at,1jI"100 .:, A similar phenomenon blade with (5-1) paneling which is oscillating is shown analyticaily by Eglof and Landgrebe in a pitching mode about the mid-chord at a frequency of 4 per revo 1ut ion (120 cy~l es/sec) (1983) in Fig. 60 of that report where a is given ,on fig. 11. For this case a 17 term, local minimum and a local maximum occur in the same range of azimuth angles, even, though tne Fourier series (m=8) 'was used to simulate the geometry of the two blades and the flight oscillating load, which was comprised of one constant term, 8 cosine terms; and 8 sine conditions are different. Also, in Fig. 93 ,of the same report some test data shows ,a similar, terms. The steady and unsteady rotor blade " variation of loading in the same azimuth range,' loading is yiven for one rev~lution. The blade was oscillated through an angle of .1 rad.
about a mean angle of ~1 rad. The effect of the The chordwise pressure dis~ributions for the (7-3) case are presented in figure 6. It oscillation is readily apparent as compa~ed to 'the steady case. With the harmonic should be remembered that in usinv the vortex lattice method, the loading is concentrated at representation of the ,loading, the magnitude and phase of the several harmonic loads are easily the location of the vortex ~Ihich for the (7-3) case is located at .Oa33C, .416G, and .75e. The determined. The magnitudes are plotted in Fig. 12. The only harmonic loads that were pressure was faired using a cubic spline through significan~ly changed from the steady case were the three vortex locations and the known value the 3rd, 4th and 5th. Both the 3rd and 5th of zero at the trailing edge. The distributions harmonics were increased and the 4th harmonic are given for 7 spanwise positions. In general, the curves exhibit the expected shape, having was dramatically increased. Another calculation the largest values as the leading edge is was made for the non-Qsc ill atory unsteady case and compared to the ~uasi-steady' case.
a~proached. For the span distribution the Virtually no difference was observed, indicating values at r/RT •• 85 are slightly larger than that, at 1 east' for thi s case, the 'rate of change the values at r/RT •• 9!:) .. indicating a falling off in the tip region. of loading ina revolution of the blade is small enough so that the effect of a variable wake is negligible.
From these concentrated forces, the section pitching moment can be calculated. Figure 7 presents these resul ts for 1/1 = 9U degrees. The Compressible Effects (5-1) section moment was taken about the 1/4 C and a nose down moment is taken as positive. The For a one-bladed rotor, the effect of ~itching moment shows some rather dramatic changes along the span. The moment is nose up compressibility is illustrated in Fig. 13, in which the CT is plotted against azimuth near the ti ~ (r/RT = .95), changes to a small anyle. The in,ompressible result is included nose down value, then becomes nose u~ for most for comparison. As expected, the compressible of the inboard region. Integration of the load is larger than the incompressible moment wou 1d res u It ina tota 1 pitch moment up throughout one revolution. The effect is at 1/1= 90°.
greatest in the region of the advancing blade and smallest in the retreating region as would be expected.
Swept Tip The segments used for the vortex lattice Two-Bladed Rotor in Compressible Flow (5-1 per for the swept tip studies were(5-1), where two blade) equal segments were used in the tip region and three equal segments were used in the unswept The method has been extended to the inboard section. In Fig. 8 the lift is shown two-b 1 aded rotor for the com~ress i hIe case and plotted against azimuth for the two sweep the results are shown in Fig. 14. The thrust conditions and for zero sweep. In general, the coefficient CT per blade is given vs. azimuth three results show little difference. The angle for a single bladed rotor and for a sweptback confi gurat i on has a 1arger 1.i ft from 0 • 0 0 , two-bladed rotor. For azimuth angles from 1/1 ~ 300 to 40 For 1/1 ~ 100 to 240 0 0 1jI'. 20 to 120 the single blade rotor has a the swept forward configuration has a gligntly
1arger CT. For 1/1 = 120 to 260°, the Cr
larger lift. It appears that the total lift for on the one and two-bladed rotors are one rotation for the swept-back case and the AI approx imate ly the same. However, for 1/1. 260 sweptforward case would give about the same lift to 34Uo a dramatic reduction in lift occurs as produced by the unswept rotor. In Fig. g for the two-bladed rotor as compared to the one the lift distribution along the rotor span is bladed results. The lowest lift occurs at !Ii ven for 1jI. UO. The major effect of sweelJ ..
1jI= 292 which 1J1aces the other blade of the is concentrated at the tip, Where the swept-back tWO-bladed rotor at 1jI= 1120, the point of tip load is greater than both the unswe~t and 0 maximum lift on the other blade. AlJparently the sweptback cases. In fig. 10, 1/1'= 180 • high lift on the blade at '1/1. 1120 creates a Comparing to fig. 9, the swept-back tip load is very unfavorable induced velocity on the second larger than both the unswept and the hlade at 1jI= 2920 which re~uires the loading swelJt-forward tips.
to go to zero in order to satisfy the boundary conditions at 1/1 = 292°.
Pierce, G. A.; and Vaidyanathan, A. R. 1983: CONCLUDING REMARKS Helicopter Rotor Loads Using A linearized lifting surface theory Discretized Matched Asymptotic Expansions, including the effects ot" compressibil ity has NASA CR 166092.
been developed for a helicopter rotor in forward flight. The method utilizes the concept of the Runyan, H. L., and Woolston, D. S. 1957: Method r " acceleration potential, and makes use of the for Calculating the Aerodynamic vortex-lattice procedure for performing the Loading on an Oscillating Finite Wing in required integrations. In addition, the method Stibsoni c and Soni c Flow. NACA TR 1322.
has been extended to include the effects of
•
unsteady flow. Runyan, H. L. 1973: Unsteady Lifting Surface Theory Applied to a Propeller and Sample calculations have been done for Helicopter Rotor, Ph.D. Thesis, several cases. These include the effect of Louyhborough, University of Technoloyy.
swept-back and swept-forward tip. The effect of these two tip configurations was minimal on the total loading for one revolution. However, the loading distribution changed considerably for several azimuth positions. A comparison of the
z
thrust coefficient, CT, of a one bladed rotor and a two bl aded rotor was made. In the 0 0, the azimuthal range between 20 and 120 one bladed rotor showed higher lift. However 0 to 0 between ~ = 260 340 the two bladed rotor indicated a lower CT. Compressibility was investi~ated for one configuration. As expected, the effect was greatest in the 0) advancing blade region (~= 90 and was
---
minimal in the retreating blade region. The
" I
effect on CT of a blade oscillating in pitch I at 4/rev is given. The effect on -the total
.... -
.... ----- --
blade lift is shown and the effect of the oscillation is readily apparent. The harmonic ..JC---,----- Y" content was calculated and the greatest I difference between the oscillatory and I non-oscillatory cases was found in the 4th / harmonic.
/
... " "
'-- --."".
REFERENCES x
Albano, E.; Rodden, W. P. 1969: A Doublet Fig. 1 Inertial Coordinate System Lattice Method for Calculating Lift Distribution on Oscillating Surfaces in Subsonic F10ws. AIAA Journal, o Vol. 7, No.2, pp. 279-285.
r ---:;===4 L _ oJ
l Oat, Roland 1973: The Lifting Surface Theory Applied to Fixed Wings and Propellers "'"''' ....-"'-" t' = TO' ONRA TP No. 1298.
~'d--Singular Region Davis, P. J. and Rabinowitiz, P., 1967:
c
Numerical Integration, Blaisdell Publishin~ Company.
E~olf, T. A. and Landgrebe, A. J. 1983: Helicopter Rotor Wake Geometry and its
•
Influence in Forward Flight, Vol. I, NASA CR 3726.
..
Hanaoka, T. 1962: Hydrodynamics of an Oscillatin~ Screw Propeller. ONR, ACR (92) (1962).
Kussner, Hans G. 1941: General Airfoil Theory.
NACA TM 979.
Morse, P. M. and Feshback, H. 1978: Methods of Theoretical Physics, McGraw-Hill, Inc.
Mangler, K. W. 1952: Improper Integrals in Fig. 2 Integration Areas Theoretical Aerodynamics.
i+-------20ft-----.
/
~---------20ft·------·1 " ~------_---oIiii22.!i~
6ft 6ft
m
1 l 1 r- 1 ~
-- __ ~ ---JL I . ~:E
U=l00ft/sec ar=O~05rad 1=2ft-l w= 5 ft/sec Q = 30 radl sec IJ = O. 17 Fig. 3 Unswept Configuration and Input Fig. 4 Swept Tip Configurations Parameters Chord Span r/R T 5 1 ----0.95 ---- 5 2 ----0.85 --------- 1 ----·----·0.75 7
-'-''''''''0.65 ----- 7
-"-"-0.55 -'---',-0.45 300 -----0.35 Li tt (bl ttl tt 45 90 135 180 225 270 315 360
o
0.2 o
0.8 0,4 0.6 1.0 Chord Anq!~. ~@g.
Fig. 5 Thrust Coefficient vs. Azimuth Angle for Chordwise Pressure Distribution for Fi g. 6 Four Panel Confiyurations for a Single Several Spanwise Locations, ~ = 90 , Rotor Blade, incompressible, (~ = 0.17,
incompressible, (~ = .17, 8a = 0.1
8a = 0.1 rad, ar = .05 rad, rad, ar = .05 rad, g = 30 rad/sec)
g = 30 rad/sec)
o
Section Moment -'-'- Swept Forward 22.8° fHb/ft -5 --- No Sweep ----- swept Back 22.8° -10 Total Lift, Ib -15
•
-20 1000 L-J_--.l._--l._-L_-L_..L-_.!.---..J -25 L __ L- __ .1- __ .1-_----J
•
o ~ ~ ill ~ m m ill ~
0.2 0,4 0.6 0.8 1.0 Angle, deg Span r/RT Spanwise Section Moment Distribution Fig. 7 Fig. 8 Comparison of Lift on a Swept-Back Zero about 1/4C - Positive Nose Down, Sweep and Swept-Forward Blade, ~ = 90 , incompressible, incompressible, (~ = 0.17, 8B = 0.1
(~ = 0.17, 8a = 0.1 rad, ar = .05
rad, ar = .05 rad, g = 30 rad/sec) rad, g = 30 rad/sec) 0 Swept Back 22.8 No Sweep Swept .Back 22.8 ...
A Swept Forward 22.8 No Sweep A Swept Forward 22.8 t Ib/tt Ib/tt 0'----..L...-----I....---l. __ ---1 OL....------L.---~--~-----' 0.8. 0.6 0.4 0.2
LO
1.0 0.8 0.6 0.4 0.2 Span r IRr Span r/Rr Fig. 9 Spanwise Section Lift Distribution for . Fiy. 10 Spanwise Section Lift Distribution for 0 •
Swept-Tip Confiyurations. ~ = 00. Swept-Tip Confiyurations. ~ = 180
incompressible. (~ = 0.17. Sa = 0.1 incompressible. (~ = 0.17. Sa = 0.1
rad. Qr = .05 rad. 11 = 30 rad/sec) rad. ar = .05 rad. 11 = 30 rad/sec) 80 '\
i \ ..... _Oscillating
, , Non-Oscillati ng Oscillatory x Cr 1000 Mag , I \ I ,,I 8 7 3 4 5 6 2 45 90 135 180 225 270 315 360
o
Harmonic Number Angle. de~ .
and Fig. 12 Harmonic Content for Non-Oscillatory Comparison of Lift on a Rotor Blade
• Fi g. 11
Oscillatory Cases - r/RT = .95.
Oscillating in Pitch at 4/Rev. to the
incompressible. (~ = 0.17. Sa = 0.1
Lift on a Non-Oscillatory Blade.
rad. Q = .05 rad. 0 = 30 rad/sec)
incompressible (~ = 0.17. Sa = 0.1
r
rad. Qr = .05 rad. ~ = 30 rad/sec)
--One Blade " ' , , 4 \ --~- Two Blades I I
\r Compressible
I \ 3 \ \
•
\ CTx 1000 2 ~ C X 1000 T I I I I
-'
\ / , ..
-1 '----'-L-..--I_--l_--'-_--l._--I..._...I-----I
o 45 90 135 180 225 270 315 360
90 135 180 225 270 315 360 o 45
Angle, deg Angle, deg .
Fiy. 13 Incompressible and Compressible Lift for Fiy. 14 Lift on Two-Bladed and One-Bladed Rotor
One-Bladed Rotor. (~ = 0.17. 6a = 0.1
vs. Azimuth Angle. compressible.
rad. Qr = .05 rad. Mrl? = 0.54)
(1/ = 0.17. e B = 0.1 rad. Qr = .05
rad. MrIP = 0.54)
•
1. Report No. 3. Recipient's Catalog No.
I 2. Government Accession No.
NASA TM-86315
5. Report Date 4. Title and Subtitle
LI FTING ,SURFACE THEORY FOR A HELICOPTER ROTOR IN
September 1984
FORWARD FLIGHT
6. Performing Organization Code
505-33-43-09
8. Performing Organization Report No.
7. Author(s}
H. Tai and Harry L. Runyan
, 1--------------------------"""1 10. Work Unit No.
9. Performing Organization Name and Address
NASA Langley Research Center
11. Contract or Grant No.
Hampton, VA 23665
1--------------------------"""1 13. Type of Report and Period Covered
12. Sponsoring Agency Name and Address
Technical Memorandum
National Aeronautics and Space Administration
14. Sponsoring Agency Code
Washington, DC 20546
15. Supplementary Notes
This paper will be presented at the 2nd Decennial Specialists' Meeting on
Rotorcraft Dynamics, Ames Research Center~ November 7-9, 1984.
16. Abstract
A Hfting surface theory has been developed for a helicopter rotor in forward
flight for compressible and incompressible flow. The method utilizes the
concept of the linearized acceleration potential and makes use of the vortex
lattice procedure. Calculations demonstrating the application of the method
are given in terms of the lift distribution on a single rotor, a two-bladed
rotor, and a rotor with swept-forward and swept-back tips. In addition, the
lift on a rotor which is vibrating in a pitching mode at 4/rev is given.
Compressibility effects and interference effects for a two-bladed rotor are
discussed.
17. Key Words ISuggested by Authorls" 18. Distribution Statement
Compressible
Lifting Surface
Unclassified - Unlimited
..
Helicopter Rotor
Subject Category - 02
Forward F1 i ght
Acceleration Potential
Unsteady
22. Price 19. Secutity Oaalf. (of this report) 20. Security Classlf. lof this page} 21. No. of Pages
A02
Unclassified Unclassified
For sale by the National Technical Information Service. Springfield. Virginia 22161 "·30S