holt ashley; eugene brunelle; herbert h. moser -- unsteady flow through helicopter rotors.pdf

Vol. IXb, 1958 57

Unsteady Flow Through Helicopter Rotors 1) By HOLT ASHLEY, EUGENE BRUNELLE, a n d HERBERT H . MOSER,

Cambridge, Mass., USA 2)

1. In troduc t ion

The related problems of aerodynamically-induced vibration and aeroelastic instability are a regular source of annoyance and danger to designers of rotary wing aircraft, but most past attempts to cope with them have involved rule- of-thumb modifications to the rotor and its linkages. I t has been found, for example, that mass-balancing the rotor blade with respect to its quarter-chord line, which is done to minimize control forces, normally assures that flutter speeds lie outside of the operating range. Theoretical analyses of vibration phenomena generally put more emphasis on accurately representing the me- chanical properties of the rotor than on finding reliable expressions for the aerodynamic loads. As a rule, some adaptation of steady-flow theory has been employed, often on a two-dimensional basis without regard for the several sources of aerodynamic induction present in the actual rotor flow field.

As in the case of fixed-wing aircraft, however, post-war efforts to push the performance of the lifting rotor closer to its inherent limits are creating, among many other requirements, a need for more rational, precise aerodynamic tools. The ultimate obiective is clearly a complete theory of three-dimensional, unsteady flow through the rotor in any flight condition. This objedtive is far from achievement today, for reasons that need not be elaborated to anyone familiar with unsteady lifting-surface theory. Nevertheless, a significant forward step was recently made by LOEWY [3] 8) (and independently by TIMMAN and VAN DE VOOREN [5]), who was first able to explain the large loss of aerodynamic damping experienced by a hovering rotor when it vibrates at or near frequency ratios m which are integral multiples of the number of blades. LOEWY'S theoretical model, which is discussed further in subsequent sections, is purely two-dimensional but accounts for the influence of the counter-vortices that are shed from the trailing edge and spiral back into close proximity with the blades because of the helical shape of the wake.

1) This investigation was performed under USAF Contract AF33(616)-3270 sponsored jointly by the US Army and Dynamics Branch of the 'Aircraft Laboratory, "~Vright Air Development Center.

3) Department of Aeronautical Engineering, Massachusetts Inst i tute of Technology. 3) Numbers in brackets refer to References, page 78.

5 8 HOLT ASHLEY, EUGENE BRUNELLE, and HERBERT H. MOSER ZAMI"

The success and mathematical simplicity of LoEwY's work give strong encouragement to other similar unsteady-flow investigations. The present paper undertakes to review two such, which were carried out incidentally to a broad program of research on helicopter vibrations at the Massachusetts Institute of Technology. One, a two-dimensional study of ground effect on an oscillating rotor blade, is described in section 3. The other a first and rather unwieldy attempt to predict unsteady three-dimensional loading of a blade in simple harmonic motion by analogy with REISSNER'S theory [12] of oscillating wings, is covered in section 4. By way of introduction to the important influence of unsteady effects, section 2 discusses the comparison between two-dimensional calculations and measurements of the forced response of a simplified rotor model.

a

b' bo C(k) = F + i G C ' = F ' + iG '

C F . D .

Cp.D. h' = 2 ~z u/Y2

n ' o H~'I i = ]/-- 1 In, ]n, K~ k = w b ' /V = m/r

t ko L , m = ~o/Y2 n

P r

r', ~f , r*, ~7" R, R~ S

t

U

V = ~ r

W , #

Principal Symbols

location of pitch axis, in semichords aft of midchord axis; blade semichord; semichord at midspan of blade; THEOOORSEN'S lift-deficiency function [2] ; LOEWY'S lift-deficiency function [3]; flap-damping coefficient; pitch-damping coefficient; spacing between successive rows of helical wake; distance between rotor plane and ground plane; Hankel function of the second kind and order n; imaginary unit; Bessel functions, as defined by WATSON [141 ; reduced frequency of oscillation; reduced frequency based on b'o; modified Struve function of order n; frequency ratio ; physical rotor revolution index; functions defined by equations (22), (23), and (24); ambient pressure, total number of shed wakes; radius of blade section; radial coordinates measured from blade midspan (Figure 9) ; radius of rotor; dimensionless semispan of blade; time coordinate; inflow velocity to rotor; upwash at blade surface; ' free-stream' velocity; wake weighting functions;

Vol. IXb, 1958

X', ~1

x,~,z,~ Y Zm

Unsteady Flow Through Helicopter Rotors

dimensional streamwise coordinates, positive aft; dimensionless streamwise coordinates; vertical displacement of blade section; dimensionless midchord sweep (Figure 9).

59

r

F /,,

0,~o ~,$

0)

(L~ z

{0 0

.c2

angle of attack; spanwise and chordwise components of running circulation of vortex sheet; total circulation around airfoil; circulation function defined by equation (18) ; angle variables defined by equation (32) ; dummy variables of integration in streamwise direction; density of incompressible fluid; circular frequency of simple harmonic oscillation; frequency of vertical displacement of rotor hub; frequency of collective pitch Variation; angular velocity of rotor.

a

n

( )("/ ( - )

Denotes a quantity on the blade surface; rotor revolution index; order of function; denotes a quantity in the shed wake; denotes a dimensional quantity; two-dimensional flow; complex amplitude of a quantity with simple harmonic time dependence.

2. Forced Response of a Hover ing Rotor Calculated by Aerodynamic

Strip Theory

In a recently issued report [11, one of the present authors and co-workers describe an extensive study of the dynamic response of a helicopter rotor sub- jected to several types of forcing. The determination and mathematical de- scription of the airloads, both steady and unsteady, was based on the well- known strip-theory approach, in which the flow at any given blade station is assumed unaffected by that at any other station. The application of this assumption to the case of a hovering rotor is, of course, equivalent to describing the flow at any blade station as two-dimensional at all azimuth locations.

The above-mentioned report considers, in particular, two cases. In the first case, the response of the first two natural modes, rigid flapping and first bending, of a fuUy-articulated, two-bladed rotor to a sinusoidal variation in collective pitch was investigated. Equations of motion were derived by an energy approach and solved for a range of frequency ratios ~o0/~Q from 0 to 4,

60 HOLT ASHLEY', EUGENE BRUNELLE~ and HERBERT H. MOSER ZAMP

and for a rotor corresponding to that used in the experimental phase. In defining the natural modes, the rigid flapping mode shape was, of course, known and the first bending mode was described by the common approximation

y(~) _~ 4 ~ - 3 7 ,

where ~ is the station radius measured from the hub, y(~) is the displacement from its equilibrium position of the blade section located by ~/, and R is the radius of the rotor.

The second case considered was the response of the first two modes to a sinusoidal vertical displacement of the rotor hub. These equations were derived by GALERKIN'S method, and the solution was obtained for the frequency ratios c%/f2 from 0 to 4, and for the same rotor as in the first case.

The primary purpose of the investigation was to measure the effects of the unsteady aerodynamic forces on the rotor. Accordingly, the rotor response to each type of forcing was computed by each of three existing unsteady aerodynamic theories, and the predicted response corresponding to each theory was compared with the measured response obtained in the experimental program.

The first unsteady aerodynamic theory used was the so-called' quasi-static' theory in which the effect of the wake upon the aerodynamic forces acting on the airfoil is neglected entirely. The second theory consisted of assuming that the shed vortices arising from the varying circulation about the airfoil trail off along the chord-line of the airfoil to infinity. This assumption gives rise to the 'lift-deficiency function' C(k) described by THEODORSEN [2].

The third theory considered, one applicable to helicopter rotors in particular, consists of assuming, and accounting for, the presence of not only the shed vortices trailing off behind the airfoil, but also those lying in planes (approximately) in or beneath the plane of rotation due to preceding blades and/or revolutions. This is the previously mentioned method of LOEWY [3], which was first verified experimentally by DAUGHDAY, DUWALT, and GATES [4]. A similar theoretical model was used by TIMMAN and VAN DE VOOREN [5], but these authors considered only the particular case of the vortex pattern in the plane of rotation (corresponding to no inflow through the rotor). Applying this theory in the expressions for lift in the equations of motion consists of simply replacing C(k) by a 'modified lift-deficiency' function C'(k, m, h') which depends not only on the reduced frequency k, but also on the inflow and number of blades.

To provide background for the developments that follow, some results of the above work are reproduced here in Figures 1, 2, and 3. Figures 1 and 2 present the magnitude and phase, respectively, of the flapping response to sinusoidal variation in collective pitch; Figure 3 shows the magnitude of the

Vol. I X b , 1958

L

U n s t e a d y Flow T h r o u g h He l i cop te r Ro to r s 61

1"2

1'0

0'8

0"6

0"~.

0"2

o

i i

0:5 1.0 3.5 1"5 to o 2"0 2"5 3"0 T

Figure 1

F l a p p i n g response to p i t ch i n p u t - a m p l i t u d e . Theore t i ca l curves h = 0 '25; expe r imen t a l poin ts h = 0.

quas i - s t a t i c ; - - C(k) = F + i G; - - - - C ' (k , h, m) =: F" + i G'.

220 ~

200 ~

180 ~

120 ~ -q~13

80 ~

4 0 ~

/ j ~

oK

/ o

0.,5 1.0 1.5 2.0 to 0 2.5 3.0 3.5 4,0 /z.5

Figure 2

F l a p p i n g response to p i t ch i n p u t - p h a s G Theore t i ca l curves h = 0"25; expe r imen t a l curves h = 0.

- - - - - - q u a s i - s t a t i c ; C(k) = F + i G ; - - C'(k, h, m) = F ' + i G .

62 HOLT ASHLEY, EUGENE BRUNELLE, and HERBERT H. MOSER

1"8 o

ZAMP

" ] '6 o

1-t~

12

1'0

0-6,

I I t I I I I I l l l \ J \ / /

/ /

04 r t

0 l.O 2'0 3 0 co z ~..0 5.0 6.0 7.0

2

Figure 3

Flapping response to displacement input at hub-amplitude. Theoretical curves h = 0-25; experimental points h = 0.

quasi-static; C(k) = F + i G ; - - o experimental points.

c ' (k , h, m) = F" + i 6";

flapping response due to sinusoidal variation in hub displacement. In each figure, the response predicted by each of the three aerodynamic theories is plotted together with some experimental points found in the experimental part of that study*).

The flapping responses calculated on the basis of quasi-static aerodynamics will be recognized as essentially the responses of a second-order system with a finite amount of damping to the different types of inputs. The effect of the wake, as predicted by THEODORSEN [21, is to decrease the magnitudes of both the flap-damping and the unsteady lifts, with the result that the response so

4) The authors are indebted to Messrs. JOHN ZVARA and NORXAN D. HAM, co-authors of the report [1], which this section of the present paper summarizes in part. Figures 1, 2, 3 are reproduced with their kind permission.

Vol. IXb, 1958 Unsteady Flow Through Helicopter Rotors 63

calculated is not very much different from that predicted by quasi-static theory at the frequency ratios considered (although tile difference becomes more significant as the reduced frequency increases).

However, the unsteady effects assume a new significance in vibration work when the influence of the returning wake is taken into account. In this case (Figures 1 and 3) the flap-damping coefficient falls off markedly as the flapping frequency ratio approaches multiples of the number of blades. It is seen from the pitch forcing case (aerodynamic forcing) that the magnitude of the unsteady lift also falls off at these values of frequency ratio, so that there are 'dips ' in the magnitude of the flapping response to an aerodynamic input. On the other hand, the flapping response of the articulated rotor to a displacement input at the hub is highly sensitive to the returning wake, in that while the driving forces are maintained essentially constant by the nature of the excita- tion, there is a loss of damping when the frequency ratio approaches multiple values of the number of blades. An increased amplification ratio at these fre- quencies results.

I t may, therefore, be concluded that the effect of the returning wake is indeed appreciable in vibration analysis, as predicted by LOEWY and TIMMAN- VAN DE VOOREN, and already confirmed experimentally by DAUGHADAY,

DuWALT, and GATES. It is, in addition, noted that the two-degree-of-freedom analyses not only predict with reasonable accuracy the effects of the returning wake on the rotor response, but also show satisfactory correlation with the measured responses throughout the entire range of frequency ratios considered. Thus the importance of unsteady flow is established, and justification is provided for advancing the theory beyond the fairly severe restrictions to which it is still subject. I t would be unreasonable to infer from the above that in all rotor vibration studies the Loewy airloads will be adequate. Both the data in Figures 1-3 and those of [41 were taken outside of appreciable ground effect. Furthermore, the reductions in amplitudes of airloads due to three-dimensional flow occur in all aerodynamic terms of the equations of motion, and there is reason to believe in the example presented here that the accuracy of two- dimensional predictions of pitching and flapping response may result partly from self-cancelling errors. One cannot assume that three-dimensional effects will be equally secondary in every application.

3. Uns t eady Load ing of a Ro to r in the Presence of a G r o u n d Plane

The theory of LOEWY E3] was of course derived for rotary wings operating in an unbounded medium. For practical reasons, however, it appears desirable to re-cast tile problem so that the effects of a solid boundary may be considered. This re-casting yields (1) an expression for the aerodynamic loads and damping coefficients as a function of the distance above the ground plane,

64 HOLT ASHLEY, EUGENE BRUNELLE, and HERBERT H. MOSER ZAMP

(2) a more accurate interpretation of rotary wing test data taken on a test pylon in the proximity of a ground plane, and (3) a rough approximation to the aerodynamic pulse loading experienced by a rotor as it passes over a fuselage.

The basic aerodynamic model postulated in E3] is retained, hence no attempt is made in this paper to reiterate that complete development. Further- more, since the main consideration is a study of the ground plane effects, a single-bladed rotor has been considered, so as to separate clearly the ground effects from multi-blade considerations. This step promotes the interests of clarity and simplicity, the extension to a multi-bladed rotor being straight- forward.

When the basic aerodynamic model postulated by LOEWY E3] is placed in the proximity of a ground plane, additional conditions must be fulfilled. One obvious condition is that the velocity component normal to the ground plane is zero along the ground plane. Also it is assumed that the shed wake does not rebound from the ground plane. The wake is collected, held captive at the ground level and quickly dissipated by vortex-ground plane interaction (Figure 4). This behavior is observed, for example, when a smoke ring is blown against a flat surface.

-CO. Cu

CO

Shed vopticity coptuped end annihiI~ted by vortex - g round inteructton

Ground plane

Figure 4

Schematic of vortex cancellation.

The assumption of quick dissipation of the vorticity at the ground level implies that the instant a row of shed wake touches the ground it can no longer contribute to the downwash distribution over the airfoil. With this simple physical picture in mind, it is easy to construct a vortex image system that provides (1) the required normal velocity at the ground plane, (2) the condition of no wake reflection, and (3) the annihilation of all vorticity at the ground level (Figure 5). If vortices first begin to be shed at time t = 0, the aerodynamic model depicted in Figure 5 is valid for all times t ~ Ho/u; that is, when at least one row of shed vorticity has contacted the ground plane. The time history of the aerodynamic loads in the interval may be obtained by deleting successive rows of vorticity (p, p - 1 . . . . . etc.).

Vol. I X b , 1958 U n s t e a d y Flow Through Helicopter Rotors

v , alto(x) due to image vortex

" , "/"~- d ' ~ - ( l ' l ' t l ~ y S i C (lt vortex n= 1 - _ . / , e , ,,_ . , 2 ~ r u

n : 2 - ( ~ - / . . . . - . . . . ' ' g2 Ho,

n*=p ~ " 7 ~ I Ground plane

n ; 2 . . . . ~/ - ~ _ _ [ Image vortex

n ' l - " I wake system

J

Figure 5

Pic ture of theoretical model with ground plane.

65

Using the configurat ion i l lustrated in Figure 5 and applying the law of BIOT and SAVART, the induced upwash (i.e., neglecting sidewash and wake- wake interact ion effects) on the physical airfoil resulting from the complete vor tex sys tem is:

1 ra(~', t) " ~o(x', t) 2 .-, -~'- ix ~ --~')~ (/i~')"- "

~ , d.~' - ~,~(r t) (.~ - r ) d~

--b' --b

oo oo

+ i ~,o!?'~-!., ~, d ~ ' - / rA , , t ) (~ - r 1 6 2 (1) J ,~ - ~ d W ~-~')~ + (/-h') ~" -

b' b" ( oo 0,3 /q

P

+ L (.;'= (.,.'- + h' :ll'

where pr imes (') denote dimensional quant i t ies and

n * - - 2 p + e - n = / - n , (2)

/ = 2 p + e . (3)

Le t t i ngyw - = 7w e'~*, and since y~o~ is the vor t ic i ty shed at some earlier t ime,

Now using the bounda ry condition of zero pressure difference across the wake pa t te rn , defining a circulation function

b'

b' / '~a d~' Zb

ZAMP IXb/5

66 HOLT ASHLEY', t~UGENE BRUNELLE~ a n d HERBERT H. ~[OSER ZAMP

and nondimensionalizing the upwash equation with respect to the semichord b'; the upwash equation becomes

I 1 ~ " e CO . e

~o(~) = - _ ~ - [ ~ ~-_-~ d~: - ~ k r ' / - ~ = d~ + ik_~ j x - r J ( x - ~ ) ~ + ( / h ) - - 1 1

P ~e - - i2nmn- - i k .~ -ik?' ZI ; (5)

I ;

+ ( ~ = ~)~ 4c- (;;~, ;;? �9

Resorting to integrations in the

x - - ~ = ~ : L. = i ~ e -kq'- e - i ( k x+2 . . . . ),

complex plane

Ia

and a change of variable

(6)

where

i:; -.= - - i ~ e - kqs e - i(kx +2nm~,)

q.,.= n h , q 3 = n* h .

(7)

N o w l e t t i n g (~ - x) = 2, a l s o n o t i n g t h a t - 1 _< x ~ 1 a n d 0 _< ). < o 0 :

�9 "'~,~ e - 11 ~ - e - ' k ~ j - ~ , . d;t.

0

Two more integrations in the complex plane yield the result"

i l --"~- - - e - i k X [ B ( k l) - i 2 - e - - k 2 ] ,

where 1

B(k l) = -2- { E- E i ( - - k l)l e ~ - EEi(k Z)i ~ -~} ,

(8)

(9)

(10)

o o _

kl

Z = / h .

exponential integrals,

Collecting results, the upwash equation may be rewritten as

- - oo 1

~(x)- 2-~-jy2_-~-d~+ ~/~ 2~ j x - ~ 1 --1

(11)


where

n = l

(12)

Applying S6HNGEN'S [61 inversion relation to equation (11) and following the classical work of SCHWARTZ [15], the pressure distribution is expressible as

1

-AFa(*' -- 2 [ 1 - C ' ] i V ~ - x V 1 + ~ 77a(~)d~

1 - 1 (13)

2 i (V + W ~ 1 - ~ . - ~ - -1

where

A l ( x , ~ ) = f i n (14)

and

c ' = Hi;'(k) + 2 Jdk) ~'? (15) H~-~(k) + i H~"~(k) + 2 [A(k) + i Jo(k)] I,V "

The above pressure distribution equation, has the same form as that for a corresponding two-dimensional fixed wing, oscillating in incompressible flow. Therefore, as noted in [31, the usual fixed-wing expressions for lift and moment may be used when describing the simple harmonic unsteady aerodynamics of a rotating blade section in the presence of a ground plane, provided that the above definition of C' is substituted for the fixed-wing Theodorsen function C(k). Additionally, the flap-damping coefficient is given by CU.D. = 2 ~ F' , and the pitch-damping coefficient E7] is given by

1 I a~ 2 G'

where C' = F ' + i G' (a pitch axis location measured aft of the midchord in semichords).

Therefore, by calculating IY, and hence C', all the necessary aerodynamic information is then available to perform the standard dynamic stress analysis routines, the ' typical section' flutter analyses and the approximate three- dimensional (strip theory) flutter analyses.

Of most immediate interest, however, is an assessment of the ground-plane effects on the rotary wing. This assessment can be made by observing the func- tional dependence of C', Cy.D. and Cp.D., on frequency ratio, with the wake- spacing and the total rotor-ground spacing as parameters. Furthermore, since the dependence of C', Cr.D. and Cp.D. on h' is qualitatively the same as exhibited in [3}, only H o will be considered as a new parameter. These func- tional dependences are shown in Figures 6, 7, and 8. As to be expected, when

68

1'0

0'8

I 0.8

F' 0.4

0'2

0

0

HOLT ASHLEY, EUGENE BRUNELLE, and HERBERT H..~IOSER

r =16 h'l"O0

Ho, 21.33 ---, T

Ho, co (Loewy value)

0-5 10 1.5 2-0 2 5 3:0 FPequency patio m.co/f~

F i g u r e 6

F l a p d a m p i n g coeff ic ient .

3.~ 4.0

ZA.MP

500

~00

300

~00

I00

O - - O H o = 3 . 5 5 5 6 ;

,~, )~- I- '~ '~, - ~ - " ~

0~ 10 15 , 0 25 3o 3~ ,0 m

F i g u r e 7

P i t c h d a m p i n g coef f ic ien t (a = 0). A - - & H o = 7 . I l l l ; . . . . H o = ~

H o + oo, I~V uniformly approaches W, where W is the wake weighting function given in E3].

I t can be concluded from the figures that the presence of the ground plane provides additional positive aerodynamic damping to all blade modes of

Vol. I X b , 1958

80

60

U n s t e a d y F low T h r o u g h He l i cop te r Ro to r s

20

-~0 0"5 l'O 1.5 2"0 2'5 3.0

m

1;igure 8

P i t ch d a m p i n g coefficient (a = -- 1). C P, D. vPI'S|lS II$.

Y 3.5 4,0

69

motion, hence greatly reducing the possibility for one-degree-of-freedom flutter to occur when the helicopter rotor plane is less than one rotor diameter above the ground plane. Note, however, that this type of flutter is still possible under unfavorable circumstances. When the helicopter rotor plane is more than two rotor diameters above the ground plane, the ground effects are

completely negligible and t'V ~- W.

4. T h r e e - D i m e n s i o n a l U n s t e a d y F l o w Effects

Flow past a single helicopter rotor blade is three-dimensional because of the curvature of the streamlines, the presence of 'wingtips' and the spanwise variation of relative wind. Since the blade aspect ratio is usually quite large, the third effect may have equal or greater importance than the other two. As applied to rotors, two-dimensional strip theory contains an inconsistency, as- sociated with this nonuniform relative wind, which does not occur in the case of a conventional airplane wing: even in steady rotation at constant angle o f attack the bound circulation varies linearly with radial distance from the hub thus violating the law of continuity of vortex lines. For steady, vertical flight of a lightly-loaded rotor this and other objections are fully overcome by GOLD- STEIN'S vortex model [8]. Discussions of several other approximate, steady- state theories of the finite rotor will be found in NIKOLSKY [9~. An elaborate

70 I-IOLT ASHLEY, EUGENE BRUNELLE~ and HERBERT H. ~IOSER ZAMP

vortex representation of the hovering condition has been given recently by GRAY and NIKQLSKY [10].

Sections 1 and 2 above demonstrate the need for a theory that predicts loads due to unsteady, three-dimensional flow. Since such a theory would per- force be linearized and its results therefore capable of superposition, a restriction to simple harmonic motion is acceptable. It is not surprising, however, that no efforts along these lines appear to have been published; no fully rational theory exists even for the finite airplane wing in unsteady motion, and most of the suggested approximations involve severe mathematical complexi- ties. The paragraphs below describe a first a t tempt at analyzing the oscillating finite rotor. The authors freely admit that radical modifications of the true physical system have been made in constructing their vortex model. Even then, the computations remain lengthy, and only a few examples have been worked out. Two facts lent encouragement in carrying the development through to completion. First, the large aspect ratios of typical rotors insure that a theory of the lifting-line type will furnish the loads needed for nearly any practical dynamic calculation. Second, a modified lifting-line theory was found to yield satisfactory steady-flow results.

To construct a steady lifting-line theory for vertical flight of a blade whose root and tip lie at r = R1 and r - - R 2, respectively, let the bound circulation / ' ( r ) be concentrated in a single straight vortex along the quarter-chord line. As seen from a coordinate system mounted on the blade, the oncoming stream has a velocity V(r) = .(2 r normal to the lifting line. The vortices trailing into the rotor wake, whose circulation is dr ' /dr per unit radial distance, are assumed to extend straight downstream to infinity. Hence one neglects both the helical shape of the actual wake and the centrifugal terms in the equations of fluid motion relative to the rotating coordinates, approximations which can be justified under certain circumstances but which are best assessed from the accuracy of the results. One is thus led to the problem of an ordinary airplane wing in a sheared main stream, and by direct analogy with conventional lifting-line theory (cf. GLAUERT [11], chaps. 10-11), the bound circulation satisfies the integrodifferential equation

R~

b s~ 4 ~ 9 r r - , l J Cla RI

Here cl~, b', and s o denote the local lift-curve slope, blade semichord, and incidence measured from zero-lift, all of which may be functions of r. ~Vhen equation (16) is solved by well-known methods, it yields spanwise load distributions and rotor thrust coefficients which fall among the values estimated by more commonly used theoretical methods Eg]. (Certain other quantities such


as induced inflow velocity do not compare so favorably, but the emphasis here is on airloads.)

The theory of the oscillating finite wing in incompressible flow which is a direct extension of lifting-line theory, and which reduces thereto when the frequency approaches zero, is REISSNER'S E12]. The authors have, in effect, modified REISSNER'S physical approximations and mathematical derivation to include the effect of a space-variable main stream V(r ) . I t is impossible to present the detailed steps here. Only the general outline is given, but the interested reader can find the full development in [13].

Figure 9

A rotor blade osciUating sinusoidally in a sheared flow, showing locatkms of coordinate systems. Definitions:

1 r' +7' (z', ~') - o- [zi + zl] z; + z~

, ' = , - - R o ; ~*"~*= b~' -hi-; (z,~-)= ~ ; ~ = ~b--~o" -~ [z; - zl]

Figure 9 illustrates the rotor blade in a sheared flow and defines the dimensional and dimensionless coordinates used in the development. The motion of the blade is specified in terms of the normal velocity component of fluid particles in contact with it (upwash)

va(x ' , r' , t) = -~,~ (x' , r ') e i ' ' t , (17)

which is assumed to be a known quantity. The flow over the blade and its wake is approximated by a vortex sheet in the (x' - r')-plane with circulation components (velocity discontinuities) 7 spanwise and ~ chordwise. The amplitude

of total circulation [ ' (r ' ) bound to the blade is replaced by a modified circu-

lation function/w, defined, following REISSXER [12!, as

]~(r ' ) = r ( 7 ) e.,,~,v (18) b~

App l ica t ion of the B io t -Savar t law, the K n t t a condi t ion of smooth f low-of f from the trailing edge and the requirement of zero pressure discontinuity

72 HOLT ASHLEY, EUGENE BRUNELLE, a n d HERBERT H . MOSER ZAMP

through the wake vortex sheet leads to the lifting-surface integral equation (factor e i~ has been cancelled)

sb~ x t _ _ , ~ ~,~(~,,~)[x ~ r - , / ] vo (x , r') = 4 ~ ( ~ - = ~ U # ~ = 3 ~ 3 ~ a~' d , /

--sb'. x t

/ ~ d [p,( , ) e_~e,,v (~'1] It" - ~'] - ~ ~ P ( , / ) e -~'~'zl"'~ Ix" - ~'~ b6 dn'

4 ~ {Ix' - ~']~ + fr' - V']~p: ~- --sb'. x t

Equation (19) involves only the approximations of ordinary lifting-surface theory and is the counterpart of equation (18), Part I, of REISSNER [12].

Reduction of equation (19) to a form suitable for convenient mathematical manipulation requires several modifications of the physical model, two of which have direct parallels in the development of lifting-line theory. These may be stated in words as follows: (1) The bound vortices are treated as if the blade were replaced by an airfoil

in two-dimensional flow, having the same chordwise toad distribution as the section at station r' ~i.e., 5n ~- 0 and 7-~ (~', 7') -~ ~ (~o, r') in the first integralJ.

(2) The trailing vortex pattern in the wake is projected forward from the trailing edge xt(~f) to a spanwise line passing through point (x', r ') , where tile vertical velocity is to be determined.

(3) In the pattern of shed 7,o-vortices, those portions which represent devi- ations from two-dimensional flow at any spanwise station are also projected up to a spanwise line through (x', r'). In this same pattern the wavelength factor e-~r is approximated by e- i,o~'i7(/) When the foregoing steps are carried out and the dimensionless notation

of Figure 9 is introduced into equation (19), one obtains 1 c'~

. . . . L. d:~(r162 + i ko T'(~*) ~-"~ .... 1" ~ - ~ ~o(z, y* ) dr --1 1

4 . y at/* k~ .TV B (ko(~*) Jr* -- ~/'3) --s

+ i ~-' t~~ :Vo (ko ('*) E'* -- ~*j)} d~*

"+ 4~'z ('rl*) ~ t V(t/*)] e- i[ /~ . . . . +/:zit'(,*)/v(,~*) --$

V(r*) • (N~ (ko(,~*/I~*-~,~)+ Eke+ kO~m? V--r N, (ko('~*/ E~*--~*l))

- ~-~ ..... +~J ~? (ko(r*)?* - ~*~)~ dr* . /

I d~' d~'.[ (19)

(2o)

Vol. IXb, 1958 U n s t e a d y F l o w T h r o u g h H e l i c o p t e r R o t o r s 73

Here the reduced f requency appears in four forms

k(~*) - o~ v ( ~ * ) v(~*) ' k(,~*) -

ko(r,)_ oJ b6 v(~*) ' k0(~*) -

o~ b '(n*) [

o~ bd ] y (n*) '

the a rgumen t r* being implied wherever it is not indicated. Three special functions arise in this development which are related to the

so-called Cicala function of uns teady wing theory El21. Unlike the latter, two of t hem can be eva lua ted in terms of integral representat ions of t abu la t ed functions, as given by WATSON" [14]. The definitions and evaluat ions are listed below :

oo . . ~q /" Ze-"d3. [ql [1 - - i q K 0 ([q])] + -2- [L~ -- I~ (22) NA (q) = q J [fi-+§ }~i5~-~ -- q

0

]" e - z ) " d ) . 1 2v)~(q) = q j -j~-~]_~-~ . . . . q- + AN~(q)

0 (23)

= 1 _ i [q/ + ~ [/l(q) - Ll(q)] + I ] Kl(!q] ) _ q q - q '

oo

Nc(q) = -)3." § q; q I -D-- d2 o l~p !q: (24)

[ j = l qL In 1 2 q ] + [L0(x ) - I o ( x ) ] d x - i Ko(X )dx q 2

0 0

Here I~ and Kn are modif ied Bessel functions, while L~ is the modified St ruve funct ion [14] of the first kind and order n. The pole s ingulari ty of N8 at q = 0, which is just the s teady-s ta te singulari ty of lifting-line theory, has been sepa- ra ted for computa t iona l convenience, leaving a remainder ANB which is everywhere finite. Na and Nc have no singularities for real, finite a rguments ex- cepting an integrable, logari thmic singulari ty of Nc at the origin. No par t icu- lar difficulty was encountered when construct ing tables of these functions over the ranges needed, one convenience being tha t they are all odd.

The integral equat ion (20) is readily solved for ~ b y well-known methods of thin airfoil theory (cf. SCH~NARZ [153). The result can be cast as the sum of

two-dimensional theory plus a three-dimensional correction c o n t a i n i n g / " and the N-funct ions, In view of the relation

x t 1

/ / -- "' ' - - e - i [ k + k ~ ? ' ( r * ) . r o (~ , y') d x ' = b to(z , ~*) d z = b'o x z --I

(25)

74 HOLT ASHLEY', EUGENE BRUNELLE, a n d HERBERT I-{. MOSER ' ZAMP

the expression for Ya can be integrated chordwise to produce an integrodiffer-

ential equation for _~', which is the principal unknown

F '(~) = F ' + ;r i [Ht~(k) + i H~J(k)] e-ik~ -- d~l* V01*)

x NB (ko(,*) ~r* - 7*]) + i f ' ( ~ * ) ~ , ~V~-*y! Na (ko(rl*) It* - 7*])]

V(~*) : -- i f : @*

s d

- r ~(k)-;I . (k)l , - '~ .... f - ~ , , [F ( : ) v(,.)] ~((~-,)-j N c (ko(,*) Jr* -- 7"1) d~* - - S

S

v( .*) , d iv( .*)] + i ko /~(~*) -P(~-~y d.,. t v( , : ) j N. (ko(~*) [,* - 7*]) ' - " : " v(,*) 7(7*) - - 8

_V(r*) ~ : V(r*) ,1 • (.o [fo (,, J,

[:o } Here/~,('-'1 denotes the circulation function as given by two-dimensional methods [2], 1

j ~/-i--~-o(r162 b'

~'(~) = 4 .... b6 eik~ ~-:i k[H~'-'~(k) + i H[;2~(k)] (27)

REISSNER'S linearized relation [12],

dp.(z , r*) - (z, r*) + i k / ~ (~, r*) d ; (28) - - - ~ V - . . . . . . ~ a a ,

- 1

between Ya and the complex amplitude of pressure difference between the upper and lower surfaces of the rotor blade can be carried over into the present theory. When equation (28) is applied to the inverted form of equation (20), one obtains after considerable manipulation

1

,fo(....) 2 . I: r( )l f ;.l:. ,.) --1

+Wf[~ZVI-r (z-r 2 z'~-~lT~'# T-fi/j (29, --I

V I~- z [Fo(r, ) + z F:(r*) + z ~ G(r*)] • -ff~(~, r*) d~" -- -S+~-

(26)

VoL IXb, 1958 Unsteady Flow Through Helicopter Rotors 75

Here the first two lines are the strip-theory result, and Fo, F~, F~ represent

rather elaborate integrals across the rotor span, containing ~ ' and the N-functions but independent of the chordwise coordinate. Suitable chordwise integrations of Ap-~ give the complex amplitudes of lift force (positive upward) per unit span and pitching moment (positive nose-up about a midchord axis) per unit span'

E ~ " [ Fo(r*) + 1 Fl(r* ) 1 F2(r*)] q V~b' -- qv"-b' + ~ -- - - V - - -2- " - - ( 1 - - -- ~ " V . '

-FI ~ l '2' ~ [ Fo(r* ) Fl(r* ) 3 F2(r*) ] e V2 b'2 -- e V2 b'2 + - 2 - -- ~ + V - - -4- " V ]"

(30)

(31)

The practical application of the foregoing theory is carried out in two steps. First, the circulation integral equation is solved by introducing the angle variables

r* = s cos( F, ~* = s cos 0, (32)

approximating the circulation function by the Fourier series

I?'(r *) ~= ~ K~ si)~n_ ~ (33) n

and requiring equation (26) to be satisfied at as many spanwise stations (tips excluded) as there are constants K,, to be determined. Second, the loadings are calculated by substituting equation (33) into the expressions for P0, /71, F 2, then applying equations (30) and (31). This process resembles closely that devised by REISSNER and STEVENS [12], Part II. In the first examples, however, it was found convenient to use series expansions of f0, J1 and the complex exponentials; these were carried up to terms of orders k 2 and k o, which appears adequate for the majority of helicopter applications. It proved necessary to evaluate numerically several hundred integrals containing the N-functions, sinn 0, cosn 0 and various powers of

V(r*) R,* + r* R*/s + cos ~ (34) v ( ~ ) - = -R~;-+ ~* = R*/~ + Co~0

as factors in their integrands. Although the first examples were laborious, the formulation is now systematized and well adapted to high-speed digital computation.

To obtain some idea of the importance of three-dimensional effects on vibrating rotors, the theory has been used to compute the loads due to pitching and flapping of a constant-chord blade with aspect ratio 8. The frequency is given by r = 1. (Since the relative wind is everywhere proportional to rotor speed s it is unnecessary to specify a reference value of reduced frequency k ; all dimensionless quantities are fixed by c~/Y2L Five spanwise stations were chosen, spaced equally in the variable 9.) Figure 10 plots, vs. radial distance,

76 HOLT ASHLEY, EUGENE BRUNELLE, and HERBERT H. 3,lOSER ZAMP

8 2~ ~ \ ' TnPee-Oimensionat /

5 ~,/ phase / 2 o ~

\ Two-rtimensionnt / r \ am0tituoe - - - 7 " 160

. . , ~ 3 ~ 12" ~,

2 \ " x , /...~--._Tm'ee-tlimenslonall 8" ==

phase

"1 0.2 0-t 0"6 0"8 1-0 4 ~ P

It 2 Figure l0

Dimensionless ampl i tude and phase angle of lift due to pi tching oscillation, about the quar te r -chord axis for a cons tant -chord ro tor with R 1 = 8 in., Ro = 40 in., 2 b = 5 in. Circular f requency ~o = .O.

the dimensionless amplitude and phase angle of lift per unit span due to nose-up pitching oscillation

c~ = ~ e i~,t (35)

about an axis along the quarter-chord line. Dimensions of the blade planform are listed in the caption. ~pr~ is the phase angle by which the force leads the angular displacement; on a quasi-steady basis it would equal zero. The curves marked ' two-dimensional' represent the unsteady strip theory of THEODORSEa" [2], which is discussed in previous sections. Other airloads due to pitching and flapping show the same general behavior as Figure 10, for the particular rotor and value of o)/.(2 chosen here.

I t can be concluded that in this example the phase shift produced by three- dimensionality of the flow is small enough to be negligible when one is analyzing forced motion. The reduction of amplitude is more significant, particularly in the tip region. The total lift drops roughly 2 5 0 , while the moment about the flapping hinge of the blade is reduced by well over 30%. Further remarks on the imphcations of this calculation are made in the next section.

The three-dimensional theory can be modified in an approximate way to account either for the returning shed vortices in the helical wake during vertical flight, or for the small fluctuations of relative wind due to forward flight at low tip-speed ratios. I t is unlikely that both of these effects would have to be introduced simultaneously.


Without going into detail, the shed-vortex modification is made on a two- dimensional basis by analogy with LOEWY'S work [31. The two-dimensional circulation function of equations (27) and (26) must be replaced by (midchord sweep zm is set equal to zero here for simplicity)

1

2 fl/ J V 1 - ~ ~(~' r*) d~ Ft(2) = --1 j% } . (36)

~i k t 2 [ni2}(k) + i H~-~(k)] + [ y # ) + iYo(k)] W

Here W is a wake-correction function [31, which approaches zero as tile spacing between successive helical vortex sheets becomes large. The circulation integral equation is then solved as before, but in the two-dimensional parts of the final pressure, force and moment formulas the function C(k) must everywhere be replaced with the function C'(k , m, h') tabulated by LOEWY. This procedure omits the potentially large influence of the strong helical vortices trailing from the neighborhood of the blade tips, something which certainly merits further study but will lead to serious analytical complications.

Forward speed can be accounted for two-dimensionally by reference to the work of GREENBERG [16i, who studies an airfoil operating in a stream whose speed is given by the real part of

V = V o [l + a ei~"t 2 . (37)

Some care must be observed when handling terms containing the combined factors e i~'t and e i ' , t , but for a particular motion like flapping or pitching,

the principal changes come in the function F'(") and the two-dimensional parts of the loading expressions. It is important to realize that GREENBER~'S theory contains the implicit restriction that a /k , is small compared with unity, where k, = co, b ' / V o. This may limit its applicability to typical rotor problems rather severely.

No numerical examples involving forward speed or shed vortices have been worked out as yet. From the general formulation, however, there is good reason to believe that three-dimensional loadings incorporating the latter will exhibit the same drops in magnitude and losses of aerodynamic damping near integral values of m that were discovered in LOEWY'S original investigation.

5. C o n c l u s i o n s

Although much remains to be done in the way of practical application of the theories described in sections 3 and 4, some tentative conclusions can now be drawn concerning rotor vibrations and unsteady flow. When calculating forced vibratory response during hovering or vertical flight, unsteady effects

78 HOLT ASHLEY, EUGENE BRUNELLE, and HERBERT H. MOSER ZAMZ'

must be accounted for, with particular reference to the shed vortices in the helical wake. If this is not done, potentially dangerous resonances near frequency ratios which are integral multiples of the number of blades wilt be over- looked. It can be inferred that aeroelastic stability is subject to the same in- fluences, and that rotor flutter speeds much lower than those computed without regard for the returning wake may occur whenever the flutter frequency is close to any critical multiple of Q.

In the presence of the ground, the dangers of resonant vibration and flutter are reduced, since the ground plane provides additional aerodynamic damping for all modes of blade motion. Single-degree-of-freedom flutter can still occur in an unfavorable combination of circumstances. When the rotor plane is more than two diameters above the surface, ground effects become negligible.

For values of the reduced oscillation frequency typical of helicopter prac- tice, the three-dimensionality of the flow does not cause significant changes in the phase angle between a given motion and the airloads produced thereby. There will be a large amplitude reduction, however, especially near the rotor tip. This suggests, incidentally, that finite-span effects can perhaps be estimated on a steady-state basis. It cannot be concluded that forced vibratory responses calculated by strip theory will always be larger than those actually encountered, because three-dimensional effects reduce both the forcing functions and the aerodynamic 'springs' and 'dampers ' opposing them. Since these reductions are not the same for all terms in the equations of motion, further study is needed to discover the conditions under which strip-theory predictions are unacceptable. Three-dimensional flow is likely to be more significant for flutter analyses, since flutter occurs at a stability boundary, whose location may be sensitive to small changes in the aerodynamic properties of the system.

Tile present investigation concentrates on the case of vertical flight. I t is possible, however, to adjust both conventional strip theory and the three- dimensional approach of section 4 to account for forward motion at small tip- speed ratios. This can be done along the lines proposed by GREENBERG [t6]. Since the most severe forced vibrations are often encountered on helicopters in this transitional range, further study of the problem is very desirable.

REFERENCES

Eli j . ZVARA, N. D. HA.~I und H. H. MOSER, E[[ects of Unsteady Aerodynamics on Helicopter Rotors, Part I, to be issued as a "W.A.D.C. Technical Report (1958).

~2] T. THEOOORSEN, General Theory o/Aerodynamic Instability and the Mechanism o/Flutter, Report 496, National Advisory Committee for Aeronautics (1935).

[3] R. G. LoEwY, A Two-Dimensional Approximation to the Unsteady Aerodyna- mics o/Rotary Wings, Report 75, Cornell Aeronautical Laboratory (1955).


[4J H. DAUGHADAY, F. DUWALDT, and C. GATES, Investigation o/Helicopter Blade Flutter and Load Amplification Problems, Inst i tute of the Aeronautical Sciences, Preprint 705 (1957).

[SJ R. TIMMAN and A. I. VAN DE VOORE~r Flutter of a Helicopter Rotor Rotating in Its Own Wake, J. aeron. Sci. 24, No. 9 (1957).

[6j H. S6HI~GEI% Die L6sungen der Integralgleichung

a

d /(~) g(x) = ~ y ~ dk

und deren Anwendung in der Tragfli~geltheorie, Math. Z. 45, 245-264 (1939), E 7] H. L. RUNYAN, Single-Degree-o/-Freedom Flutter Calculations /or a Wing in

Subsonic Potential Flow and Comparison with an Experiment, Technical Note 2396, National Advisory Committee for Aeronautics (1951).

[8J S. GOLDSTEIN, On the Vortex Theory o/ Screw Propellers, Proc. roy. Soc. [AJ 723, 440-465 (1929).

[gJ A. A. NIKOLSKY, Helicopter Analysis (John Wiley & Sons, New York 1951). El0] A. A. NIKOLSKY and R. B. GRAY, On the Motion o/the Helical Vortex Shed/rom

a Single-Bladed Hovering Model Helicopter Rotor and Its Application to the Cal- culation o/the Spanwise Aerodynamic Loading, Princeton University Aeronau- tical Engineering Dept. Rept. No. 313 (Sept. 1955).

[11] H. GLAUERT, Aero/oil and Airscrew Theory, American edition (The Macmillan Co., New York 1943).

[12J E. REISSNER, E//ect o/ Finite Span on the Airload Distributions/or Oscillating Wings, Parts I and II, Technical Notes 1194 and 1195, National Advisory Committee for Aeronautics (1947).

[13J H. ASHLEY, J. ])UGUNDJI, and H.H. MOSER, E//ects o/ Unsteady Aerodynamics on Helicopter Rotors, Part I[I , to be issued as a W.A.D.C. Technical Report (1958).

[14] G. N. WATSON, Treatise on the Theory o/ Bessel Functions, Second edition (The MacMillan Co., New York 1945).

[15J L. SCHWARZ, Berechnung der Druckverteilung einer harmonisch sich ver]ormenden Tragfldche in ebener Str6mung, Luftfahrtforschung (17. Dez. 1940), p. 379-386.

I16] J. M. GREENBERG, Air/oil in Sinusoidal Motion in a Pulsating Stream, Tech- nical Note 1326, National Advisory Committee for Aeronautics (1947).

Zu sam m e n/ass vl ng

Es wird ein 13eitrag geliefert zur Stfitzung der schon frfiher ge~usserten ~)ber- zeugung, dass nichtstationXre Str6mungsanteile das Schwingungsverhalten und die aeroelastische Stabilit~t yon Helikopter-Drehfliigeln wesentlich beeinflussen k6n- nen. Die Untersuchung betont besonders den D~mpfungsverlust, welcher sich aus den entlang der Spannweite eines schwebenden Drehfliigels schraubenf6rmig ab- gehenden Wirbeln ergibt.

Gem~ss dem Bediirfnis ftir verbesserte Mitre1 zur Behandlung nichtstation~rer aerodynamischer Vorg~nge werden zwei neue theoretische Entwicklungen beschrie- ben und zum Berechnen yon Luftkr~ften auf typische schwingende Drehflfigel angewendet.

Die erste dieser Arbeiten erfasst auf zweidimensionaler Basis das Vorhandensein einer Bodenebene. Es wird angenommen, class die \Virbel im Abwinde des schwe-

80 HOLT ASHLEY, EUGENE BRUNELLE, and HERBERT H..~[OSER ZAMP

benden Drehfliigels durch die Bodenebene nicht zuriickgeworfen, sondern schnell zerstreut werden. Als Resul ta t zeigt sich, dass die einer spezifischen Schwingungs- form entsprechende aerodynamische D~impfung durch dell Einfluss des Bodens, der prakt isch etwa zwei Drehfliigel-Durchmesser hinaufreicht, im allgemeinen ver- gr6ssert wird.

Die zweite der vorgelegten Theorien betrifft die dreidilnensionale Str6mung /iber ein schwingendes Fliigelblatt , welches als tragende Linie und unter Vernach- liissigung der Kri immung seines Abwindes untersucht wird. Numerische Rechnun- gen sind ziemlich miihsam, aber mittels Rechenmaschinen ohne weiteres durchfiihr- bar. In einem herausgegriffenen typischen Beispiele zeigte sich, dass die Luftkriifte bedeutend kleiner sind als die yon der Streifentheorie angezeigten, dass aber die Phasendifferenz zwischen Anstellwinkel und Luftkr~ften fast unveriindert bleibt.

(Received: October 3, 1957.)

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