A lifting surface theory has been developed for a helicopter rotor in forward flight for incompressible flow. The method utilized the concept of the linearized acceleration potential and make use of the vortex lattice procedures. Results in terms of lift coefficient slope for several forward flight conditions are given

Runyan, H. L.

Tai, H.

English

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https://ntrs.nasa.gov/search.jsp?R=19830011438 2020-03-21T03:52:12+00:00Z
JCINT INSTITUTE FOR ADVANCEMENT OF FLIGHT SCIENCES
A PROGRAM OF RESEARCH IN
	
c" 
HELICOPTER AERODYNAMIC THEORY
	 MpRtiQR3
Sn FN^Ill^DEPT.
r
NASA GRANT NAGI-13
	 l^lGr^ _ _ -r a5
(NASA-CP,- IESS97) 	LIFTING EUFFPCF TEEOFY FOF	 683 -19709
A HFLTCCFTFF FC'TOF. TN FCRWAFC FLIGF7 	 Final	 f
Report (Joint Inst. f or Advancement of
Fliqht Scie[,ces)	 19 p HC A02/MF AC 1	 Unclas
CSCL nIA G3/02 02995
FINAL REPORT
•
School of Engineering and Applied Science
The George Washingtcn University
Washington, D. C. 20052
'ORIGINAL PAGE 13
OF POOR QUAD Y
LIFTING SURFACE THEORY FOR A HFLIc.nPTF.R ROTOR IN FORWARD FLIGHT
Harry L. Runyan and Hsiang Tai
Abstract
A lifting surface theory has been developed for a helicopter rotor in
forward flight for incompressible flow. The method utilized the concept of the
linearized acceleration potential and makes use of the vortex lattice
procedures. Results in terms of lift coefficient slope for several forward
flight conditions are given.
A. 	 INTRODUCTION
Rotating lifting surfaces are an integral part of the propulsive unit of
every aeronautical and nautical vehicle, from the compressor and turbine blades
of jet engines, the pumpe for rocket engines, to propeller and helicopter
rotors. The aerodynamics of these rotatin g elements has been under extensive
study since the advent of the airplane and with a combination of experimental
and analytical approaches, succcessful designs have ' peen achieved.	 In many
cases, two-dimensional theory has been used, usually modified by an assumed
spanwise distribution, and inflow angles. A practical lifting surface theory
has not been developed for the compressihle rotating case for the general case
of a helicopter in forward flights.
	 It is the purpose of this report to present
a lifting surface method within the limits of linearized theory.
The method is based on the concept of the acceleration potential,
originally introdured by Hans Kussner (ref. 1). The method was first applied to
an oscillating wi , ig in uniform translatory motion by one of the authors
including compressible flow in 1957 (ref. 7). The acceleration potential
approach has now become standard in the determination of the unsteady
aerodynamics forces for flutter studies.
ORIGINAL PAGE 15
OF POOR QUALITY	 2
The first use of the acce l eration potential approat.r, for a rotating system
was made in a paper by Hanaok3 (ref. 3) for the incompressible case in a paper
Oirected at application to the m<rine propeller. The acceleration potential has
been used in the past in studying the oropeller noise problem, but in all of
these noise propagation cases the problem was specialized early in the
analytical development to the so-called far-field case usually with the
stationary observer, whereas the lifting surface theory is essentially concerned
with the details of the near-field case for a co-moving observer as well as the
satisfaction of certain boundary conditions. Dat, (ref. 4), has derived a
general expression for an acceleration doublet for any motion. His development
has been used as a basis for this investigation. The procedure developed here
involves the precise numerical integration over the surface of the rotor. It is
hoped that the method lays fundamental ground work for propeller and helicopter
unsteady, three dimensional compressible aerodynamic theory.
With regard to the contents of the report, the next section contains a
brief derivation of the fundamental equations, including a discussion of some
implications of the equations. The third section contains a description of the
method of solution. Finally, the results of some preliminary calculation for
the incompressible case are given.
?.	 RAs I f. Fnl IAT 1 ONS
The fundamental equation required for the solution of the lifting surface
problem relates the downwash velocity (velocity normal to the lifting surface)
to the unknown pressures on the rotor blade.
or
a^ +tom
- -(	 ax) (2.3)
OFtlutowAL PAGE li	 3
OF POOR QUALM
The a,-,-eieration potential method will he used in the analysis. Tl,e
acceleration potential, W, is defined as
p - pm = -P^	 (2.1)
Rernnulli's equation for pressure in terms of the velocity potential, m, is
defined as
	
(2.?)np - Pm=
where the operation TT represents the substantial derivative. For a wing
translatin- in the x-direction with velocity 11, the equation can be written
p - p m = -PV =P(aO +I1)
For wings oscillating in harmonic motion such as in flutter, an harmonic
substitution can he made to obtain
_ -(iw ^ + l^)
Solving for
	 9iveS
-iwx	 iwX
U	 x
m =	 u	 I v( T ,Y,Z)e U da	 (?.a)
Since the motion of a rotating helicopter hlade is non -harmonic, this
prc,cedure is not practical, thus the relationship is
_ Tt.-	
(2.5)
Equation (?.5) can he ir,tegrated with respect to t to give
^ =	 jt ydt	 (2.6)
URIGINAL PAW. :y	 4
OF PL)OR QUALITY
4'inally, the downwash is derived
t
W 
= an - ail	 J ydt	 (2.7)
3. ARBITRARY MOTION Or AN OSCILLATING D01 I RLET IN COMPRESSIBLE FLOW
The expression for an oscillating doublet moving in an arbitrary manner in
co.-. ressihle flow was derived by Oat (ref. ')_ A cnncise derivation of the
doublet expression is given below.
The expression for an accelerating source is
01^oIT)
I:	
^0(T)•[^ - ^o(T)]
o
where ^ o (T) the vector position of the source at time T, P is the position
i
of the field point, %(T) is the velocity of the source point at time T,
"a"	 is the speed of sound and q is the strength of the source.
An auxiliary equation which relates the tim,- interval t - T hetween the
time the source emits a disturhance at T and the time at which it is received
at the field point. at t, to the distance hetween the two points is given below
1 ^ 
- 
^n(T)j
a
A douhlet may he formed by placing a source and a sink in close proxii..ity
and allowinq the distance to approach zero, while at the same time maintaining
^L s !I,Lj =
	wil ,. jw%X6 VALE IS
	
5
OF POOR QUALITY
the product of the distance and the source strength finite. This re! •.ult may
also be accomplished by performing a differentiation in the proper direction.
flat applies this secnnd technique by first defining a vector distance n
I)=P - Po -	 n o
	(3.3)
where ^ is a parameter and no is the unit vector which is in the desired
direction.	 Substitiiting D for P - Po in equation (3.1) gives
s
all
The douhlet is now obtained by applying the derivative with respect to E and
letting ^ + n or
3'
vn =(ay )	 (^.^)
to give
	 i
dV
	
aT	 no	 q(M	 )(^	 no)
^n	 4,ra(^2	
X0.7	
and	
3	
-^0.^3I I	 1	 a P	 a I C I [ 1	 a ^ I
	
g [( a 2 - Vo)^ ' no - 
aIDI(^o	 ^0 ) + (^^ '	 )(^0	n0)]
V	 3
4,ra 2 1f  3 [1 -	 0
77F 
1
1
	nRiGINAL PAGE IS	 6
OF POOR QUALITY
anoq D	
aT	 .
+	 (3.6)
2	 Vo	 D 2
4,^aIDI
	
1 - 
aID
This expression checks an expression derived in a different manner in ref. 7.
4.	 INCOMPRESSIBLE F LOW. - The incompressible case may be obtained by
letting a	 W to get
_ qD	 no
wD4,rIDj3	 (4.1)
the downwash equation may now be written
aI 'D1
	 t	 n	 n o	3(n	 D) (n o ' D_)
w = an
	 4n	
f q(ro ,T)	 3 -	 5	 dT	 (4.2)
SDI
The integral equation relating the known downwash, w(P,t) to the unknown
pressures, q(r o ,T), is given by the following equation for incompressible
flow
y	 1	 t	 no.n
W (P ' t) - 47Ta Chord f Spanf _f q(r o' T)	 IDI3
do dro
D	
dt	 (4.3)
^I5
In eq. (4.3), D = P(r,t) - P 0 (ro ,T) denotes the distance between the position
vectors of the downwash and doublet points relative to a fixed coordinate
system, where r, r  are measured rad i ally along the span; t,T are the
independent time %l ariables of the dowrwash and doublet point respectively.
q (r
0 97) is the lift strength per unit length of span; a is the density of
the fluid. n and no are the unit normal vectors perpendicular to the
surface at the downwash and kA- blet point, respectively.
-3 P ' n P	 n
0	 0
ORIGINAL PAGE IM
OF POOR QUALITY	 7
The fin?l downwash equation involves a double integration over the surface
of the blade which has proved to be a stumbling block to the efficient
application of lifting surface theories. Most of the research directed towards
the application of the kernel function for the planar wing in uniform motion has
been aimed at devising feasible and adequate means for accomplishing this
integration. Of the several techniques availahle, the one selected for
application to the rotor case is termed the doutlet-lattice method, a princ-'Ipal
assumption in the method is the placement of line loads at the 1/4 chord of the
wing and the satisfaction of the downwash condition at the 3/4 chord location of
each element. This would make eq. (4.3) tractahle, namely by yielding a set of
linear equations for qi . The total lift of the blade at the time t, or
equivalently at certain azimuthal position will he jgi(t).
n . ^' W
iA
at i
W t 
I ^z^
\	 I
1Lt
of t =
Y
n
2 ^
I	 ORIGINAL PAGE Is	
a
OF POOR QUALITY
Integration over T, it is tantamount to considering ?.11 positions successively
occupied by each lifting element in the past, which amounts to scanning the
whole wake. The blade has the cord c ?nd length Rt - Rs, Rs being the
root of the blade, R t
	is the maximum length of the blade. Suppose at the
time origin, the bladQ momentarily coincide with the coordinate system along the
x-axis as shown in the sketch.
OF POOR QUALV
I
The blade executes a courterclockwise rotation with angular velocity
n while moving with velocity U along the negai,ive x direction and W along
the positive z direction. Since the quarter line method has been adopted, the
doublet point lies 1/4 c ahead and downwash point lies 1/4 c aft of the
midchord. The positions of the douhlet point as well as the downwash point can
he established as follows. The Cartesian components of the douhlet position are
xo = -UT + ro COS (QT ) - c/ a sin(i2T)cosaO
y o	ro sin(QT)+ c/4 cos(SZT)cosao
zo = WT + ( c / 4 ) sin ao	 (4,4)
With the suhstitution of c + -c, ro ; r, T + t the position of the downwash
point is given by
x = -Ut + r cos(ot)+ c/4 sin(sit)cos a
y = r sin(sit)- c/4 cos(Qt) cos n
	 (4.5)
z = Wt - c/4 sin a
In egs (4.4) and (4.5), the angle a,ao are the twist angle of the blade
defined below
OR,ajr4AL PAGE If	 10
OF POOR QUALITY
tan a =	 Ws n Qg -T+-	 l
tan a -	 W	 (4'6)o	 11 sin , S2t + r0Sl
This means that the twist argle is determined by the velocity vector at the time
t	 (i.e., the final time) which implies that the angle of twist is constant in
time, and is only a function c° radial distance. Therefore, the reference plane
(defined by the doublets and downwash points) is a twisted surface. Only when
the douhlet (hereafter denoted as d.h.) and the downwash point (hereafter
denoted as d.w.) possess the same radial distance, r = ro, then the d.h. and
d.w. lie :,n a plane (i.e., n II no at	 T = t).
From eq. (4.4) the do-iblet velocity can he computed, namely the time
derivative of the position vectors.
+	 . +
	
. +
	
. +
V o = x 0 i + y 0j y z 0 k	 (4.7)
where
X0 = -ll - r oll sin(QT)- c /AQ cos(QT)coS a0
y 0 = r 	 COS QT - CSl /4 sln(2T)COS a0
z - IJ	
.%	 +
To deterriine n o as a function of T, write V 0 = V o + W, where V o '	 is the
projection of doublet velocity on the horizont-il plane. Write
n rl = JC 0 + m n .j + n 0 k	 (4.R)
where Z 0 m 0 no are the directional cosine of the -grit vector n o . Form the
vector quantity S by demanding S= V 	 X I-I
ORIGINAL PAGE lb	 11
OF POOP QVAL"
S	 r 	 sin(AT)- C/ 4 ,1 COS( SI T)COS a0 )1 + (r an cos QT
- csl/4 sin(SIT)Cos a)jl X A	 (4.9)
= W(II + r Q sin(RT)+ r_/40 C,OS(QT)coS a )j + W( r n COS(QT)
0	 0	 0
- CSl / 4 sin(QT)cos a)i
It is now required that
	 nu	 l S
and	 nO 1 v 
In o l = 1
From (7), (R) and (9), there is Ohi.ained
W(II + r 00 sin(QT)+ C/4Sl C0s(QT)C0s a0)
O
--- ,^T
V	 + VO	 O
W(r 00 COs OT - cQ/4 sin OT COs a0
m = -
0	 /^
lJ + V 0
V
r
0
=	 —
0
VW 2 + Vo2
i
where	 VO' is Vie magnitude of VO'
(4.1(1)
(4.11)
ORIGINAL PAGE :g	 12	 •
OF POOR QUALITY
Vol? = (U + r 
0 
Q sin( QT)+ c/4ri CO3(QT)COS a0 ) 2
 , (r 0Q COS( QT)
- CQ/4 sin(QT)COS a 0 )
	
(4.1?)
Ry the sane procedure, n is c rived by making the appropriate suhst.itutinis,
C + -c, a 0 + a, T + t, etc. (i.e., n = xi + m j + nk) to get
= W ,j + rr:_s'n(Qt)- c,'40 cos(Q+.)cn s al
I/W ? v	 v
m = -W(rs! cos(r'it)+ CQ / n sin(Qt)cos a)
ld ? V	 +
V^
n=	 —
W? + V
where
V 02 = (u + rQ sin(Qt) - c/4Q cos( Qt)Cos a)'' + (ro r_os(Qt)+ c:Q/4 sin(ot)cosa)2
(4.14)
sirce
+	 +	 +	 +
xi + yj + zk,	 P= x i+ y j+ z k
	
(4.1^)
O	 o	 O	 o
+	 +	 +	 +	 a
a vector D is defined as n : P - Po; then the magn i tude f p l in eq. (1) can
he expressed as
UKIUII I*AL rh%aC *'S
OF POOR QUALITY
13
't
I CI = {[II(T-t) + r cos(slt)- r0 cos(91T)+ c / 4(sin(Qt)cos a + sin( SIT)COS 00)12
+ [r sin(s2t)- r0 sin( s1T)-c/4(cos(Qt )cos a + cos( nT )cos ao)]2
+ [W(t-T) -c/4(sin an + sin a )i 1 } 1/2
	
(4.16)
The final equation may now be written as
	 j
n • n -3(n • n (n •n )
W = 4np P f_m q(r0 ,T)[	 0	
3n 
o
.
 (4.11)
o	 II
where nn is a unit vector in the ^ direction, namely, nn
ICI
lnis integral equation was solved for the unknown q(ro,T) by using a
collocation process based on the vortex lattice assumption (ref. 6). The kernel
is singular when n _ n, and this was handled by use of the finite part
technique (ref. 5). The remainder of the integrations was accomplished
r-merally.
5.	 RFSIILTS ANn nISCIISsioN
The foregoing procedure was applied to a rotor having the dimtosion shown
in fiq. 1. The rotor was divil p d into 5 equal spanwise sections. The vortex
lattice approximation was used which resulted in the placement of a vortex at
the 1/4 thud positon on each section and a downward point was located at the
center of each span section and was placed at the 3/4 chord position as
indicated on the fi.;ure.
OftIGiN R rUALITrOF POOR Q 14
The Psults of calculations are given in fig. 2-4 for three cases. For all
calculation the rotor speed was held constant at 30 rad/sec and the vertical
velocity was 2n ft/sec. On fi gure 2 is shown the overall lift curve slope,
C La , for inn ft/sec forward velocity plotted against azimuthal angle. For
the range of angles from approximately 90 degrees to 2.7n deg, •ees, the lift curve
slope is nearly constant. However, for the position of the angular position
between 315 degrees to 67.5 0 the lift curve slope is considerabl ")wer. It
could be sti •mised that with the rotor in a trailing position, that the effective
aspect ratio of the rotor is lower than at other azimuthal positions and would
accordingly have a lower left curve slope.
On figure 3, the forward velocity has heen increased to 200 ft/sec. Note
that a large increase in CL., is found at 225 degrees. This one point is
suspect, in as much as the ,tall region is being approached and the theory is
based on linear approximations. Note that the region close to n deg. azimuth
C La , is again low, where the mininum CL(, is 3.4 as com[ared to 3.9 for
the inn ft/sec case.
On figure 4, the case for a forward velocity of NO ft/sec is shown. For
this case, the rotor is definitely stalled in the azimuthal region e = 27nO
to 3150.
F. CONCLUDING REMARKS
A lifting surface theory has been developed for a helicopter rotor in
forward flight for incompressible flow. The method utilizes the concept of the
linearized acceleration potential, and makes use of the vortex lattice procedure
which has been successfull,- used for the non-rotating case.
Results are shown for three forward velocities, li = inn, 200 and 3nn
ft/sec. The vertical velocity was held constant at W = 2.O ft/sec as well as the
V
IS^^ '^ECM	 15
rotation speed, n = 3n rad/sec. For regions when the rotor blade was in a
stalled condition the analytical results are not applicable and these regions
are easily discernible in the plots. A significant finding in the results is
the degradation in the lift curve slope, C L . , in the neighborhood of 6 =
0 deg. It should be pointed out that if two-dimensional coeffic*ents were used
in an analysis that this degradation in CL would not be found.
CL
nrrrnrw,rr
1. Kussner, H. G.: General Airplane Theory. NACA TM 97Q, 1Q41.
2. Runyan, H. L.; and Woolston, D. S.: Method for Calculating the Aerodynamic
Loading on an oscillating Finite Wing in Subsonic and Sonic Flow. NACA TR
1322, 1957.
3. Hanaoka, T.: Hydrodynamics of an oscillating Screw Propeller. Office of
Naval Research. ACR 92, 1962.
n, flat, R.: Representation of a Lifting 'Line in an Arbitrary Motion by a Line
of Acceleration doublets, NASA TT, F-12, 952, 1970.
5. Mangler, K. W.:	 Improper Integrals in Theoretical Aerodynamics, A.R.C.
Technical Report C.P. No. 04, 1952.
6. Aloano, E. and Rodden, W. P,:	 A double-Lattice Method for Calculating Lift
nistrihutions on Osc'llating Surface in Subsonic Flows. AIAA Journal, Vol.
7, No. 2, 1969, pp. 27Q-285, anti No, 11, 1QFQ, p. 2192.
7. Runyan, H. 1_.: ilnsteady Lifting Surface Theory Applied to a Propeller and
Helicopter Rotor," Ph.O Thesis, Loughborough Univ. of Technology, great
Rritian, Jul y 1973.
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OF POOR QUALITY
^	 I	 1	 r	 N
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Fig. 1 
Rotor 81ade Dimension
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OF POUR QUALITY
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Lifting surface theory for a helicopter rotor in forward flight

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