J. AIRCRAFT VOL. 16, NO. 1

Local Momentum Theory and ItsApplication to the Rotary Wing

Akira Azuma*University of Tokyo, Tokyo, Japan

andKeiji Kawachit

National Aerospace Laboratory, Tokyo, Japan

A new momentum theory, named the local momentum theory, has been developed and applied to study rotarywing aerodynamics. The theory is based on the instantaneous momentum balance of the fluid with the bladeelemental lift at a local station in the rotar rotational plane. A rotor blade is considered to be decomposed into aseries of wings, each of which has an elliptical circulation distribution. The elliptical wings are so arranged that atip of each wing is aligned to the blade tip. By neglecting the upwash flow outside the wings and by introducingan attenuation coefficient to represent the timewise variation of the local induced velocity following an impact ofblade passage, the induced velocity distribution and the spanwise aerodynamic loading along the blade span canbe obtained easily. Applying the proposed theory to both steady and unsteady aerodynamic problems leads tofruitful results with much less computational time than that required in the vortex theory, in which complexity ofcalculation and difficulty of convergence usually are unavoidable.

Nomenclaturea = lift curve slopeb = wing span and blade numberC, C\m = attenuation coefficientsCT = thrust coefficient = T/pnR2 (#Q)2

c = wing chordIp = moment of inertia of a blade about flapping

hinge = i^(r-r / 3)2dm/ ^inclination angle of tip path plane; any number

of running indexj = imaginary = \T^\\ any number of running indexkp = spring stiffness at flapping hingeL =lift/ = airloading and coordinate of rotor wake planeM0 = apparant mass of rotor disk = (8/3)p/?3

Mp =mass moment of blade about flappinghinge = {j?0(r-r0)rd/n

m = mass of air and coordinate of rotor wake planerrij = mass of air of /th elliptical wing given by Eq. (7)m = mass of air associated with the local momentum

given by Eq. (6)n = number of spanwise partitionp = rolling angular velocity of rotorq = pitching angular velocity of rotorR = rotor radiusr = radius70 = radius at flapping hingeS = sectional area of related mass flowT = thrust/ =timeUT> Up = tangential and normal components of velocity at

a blade elementV * = forward velocity

Presented as Paper 75-865 at the AIAA 8th'Fluid and PlasmaDynamics Conference, Hartford, Conn., June 16-18, 1975; submittedJune 30, 1975; revision received Feb. 9, 1978. Copyright © AmericanInstitute of Aeronautics and Astronautics, Inc., 1975. All rightsreserved.

Index categories: Helicopters; Aerodynamics; ComputationalMethods.

* Professor, Institute of Space and Aeronautical Science. MemberAIAA.

tResearch Engineer.

= local horizontal inflow given by Eq. (10)= mean horizontal inflow of the /th wing = V

• VN = normal component of inflow velocityVtip, Vroot - tip and root speeds, respectively, of a bladev = induced velocityv\m = induced velocity given by Eq . ( 1 1 )(X,Y,Z) = coordinate system shown in Fig. 1x = nondimensional radial position = r/RXQ = nondimensional radius at flapping

hinge = r$/Ry = spanwise coordinate of fixed wingz ' = distance from the rotor rotational plane,

positive downwarda. — angle of attack]8 = flapping angle = /30 + 0lccos\l/ + j8/ssin^ -I- .../30 = preconing angle7 = lock number = pacR 4 /I&A = small increment5tm =5 functionrj = nondimensional spanwise coordinate of fixed

wing6 = blade pitch angle = 60 + B,x + 8lccos\l/

+ 0/5sim/' + ...0, = blade twist rateX = inflow ratio = (Ksin/+i;)//H)//, = advance ratio = V cos/7 /?Q£ = nondimensional spanwise coordinate of every

elliptical wingp = air density£ = summationa = solidity = bc/irR<t> = inflow angleX = skewed angle^ = azimuth angleft = rotor speed

Subscripts= spanwise partition and the quantity of /th

elliptical wing=azimuthal or timewise partition, spanwise

position, and the quantity of ./th elliptical wing= blade index

JANUARY 1979 LOCAL MOMENTUM THEORY APPLIED TO THE ROTARY WING

0 = constant or initial valueIc, 15- = first harmonic contents of Fourier cosine and

sine series0.75R = three-quarter radial positionSuperscript

( ') = d( )/d/

Introduction

THE momentum theory has been a useful tool for in-vestigating general flow behaviors of airplane wings,1

helicopter rotors,2 ducted fans, and so on, without knowingany local pressure distribution on the machine. Whenever themomentum theory is applied, the wing or the rotor sometimesis approximated by an elliptical wing so that the inducedvelocity distribution is constant in certain circumstancesthroughout the machine. Experimental studies have revealed,however, that the induced velocity distribution over a rotordisk is not uniform. To assume that the induced velocitydistribution is uniform all over the rotor disk leads not only toinaccurate estimation of the rotor static characteristics, suchas performance, but also to erroneous determination of therotor dynamic characteristics, such as the blade flappingmotion and the transient airload variation to a control input.

In order to estimate a more reasonable distribution of theinduced flow over a rotor or rotors and to obtain a moreprecise variation of airloading on the rotor blade, it is un-doubtedly necessary to rely on the vortex theory in whichBiot-Savart's law connects the induced velocity at an arbitrarypoint to the vorticity at any influential point in the wake. Byassuming a rigid and cylindrical or helical wake system, manyinvestigators have analyzed the flowfields of helicopterrotors. By the recent developments of computer techniques,studies have been further extended to include a free wakeanalysis in which the mutual interference among wake vor-tices was taken into account.3 The latter analysis, however,inevitably requires laborious and lengthy calculations subjectto the tendency of computational divergence.

In order to improve the preceding essential shortcomingsresulting from the constant-induced-velocity distribution, thelocal momentum balance on a pie-shaped area has beenconsidered,4 and the nonuniformity of the induced velocitydistribution has been dealt with analytically in the rotordynamics.5 As a further extension along this line, the presentpaper provides a new theory to calculate easily and preciselythe load distribution on a rotor blade in transient motion aswell as in steady motion. It is based on the aerodynamicbalance between the fluid momentum and the force acting ona wing or blade element at a local station. Therefore, thetheory may be called "local momentum theory."

Fundamentals of Local Momentum Theory

Fixed WingIt is a well-known result from the "lifting line theory" that,

when an elliptical wing is flying with a forward velocity K, theinduced velocity is given by v0 on the wing and is developed to2v0 in the far downstream, and the induced velocity is con-stant over the wing span, as shown in Fig. 1. Since the upwashoutside the wing span is concentrated near the wing tips, theeffects on the flowfield at some distance from the tips aresmall.

A wing having an arbitrary planform or an uncertain liftdistribution may be considered to be a superposition of nelliptical wings, the /th of which has the lift L, and the con-stant induced velocity At;/ all over that wing, as shown in Fig.2. Then the total lift is given by

(1)

El l ip t i ca l L i f t Dis t r ibut ion

Fig. 1 Lift and induced velocity distribution of an elliptical wing.

a) Symmetric arrangement

L2 L3 L4 Ln

b) One-sided arrangmentFig. 2 Decomposition of a wing to n elliptical wings.

The elliptical wings can be arranged either symmetrically,as shown in Fig. 2a, or one-sidedly, as shown in Fig. 2b. If theupwash velocity induced by each elliptical wing is neglected,or AVi = Q for l £ l >1, then the lift distribution and the in-duced velocity distribution at any arbitrary point for sym-metric arrangment can be described by

(2)

__ Vi &

where

$=y/(bi/2)=ii(b/bi) (4)

It must be mentioned that £ should be restricted within ± 1 or

In order to determine the induced velocity Av} of the ythelliptical wing, the lift due to the strip theory at a localsegment spanned by (-T?/, -*?/+/) must be balanced with thelocal momentum change:

(5)

where

-n dr,

0j=9[i,= -(,, +r,J+ (6)

A. AZUMA AND K. KAWACHI J. AIRCRAFT

—•—"—•- I Local Momentum Theory (One-Sided)

— — — — I Local Momentum Theory (Symmetr ic )

-1.0

L i f t i n g Line Theory by Multhopp

-0.5

'-(*).a) Lift distribution

'-(I)b) Induced velocity distribution

Fig. 3 Lift and induced velocity distributions for a rectangular wing(aspect ratio = 6, n = 50).

Equation (5) gives solutions

and thus /y successively from j=l to j = n. It will be ap-preciated that a very similar treatment can be applied to theone-sided arrangement.

An example of application of the present method forcalculating lift and induced velocity distributions for a rec-tangular wing with aspect ratio (AR) = 6 is shown in Fig. 3.The number of partitions is n = 5Q. Multhopp's solution6 ofthe exemplified wing obtained from the lifting line theory isshown by chain lines for both the lift and the induced velocity.The discrepancy in the results between the present theory andthe vortex theory can be eliminated completely by introducingthe up washes that have been neglected in the presentcalculation as the first approximation. The upwash generatedby an elliptical wing can be calculated as an iterativeprocedure after having decided the loading share of therespective wing.Rotary Wing

As stated in the preceding subsection, a rotor blade isdecomposed into a series of wings, each of which has anelliptical circulation distribution.! In this case, however,wings are arranged one-sidedly as shown in Fig. 4 becausemost airloading acts near the blade tip, and, therefore, somepossible error due to neglecting the upwash might be reduced.Since the dynamic pressure is not constant along the bladespan, the lift distribution is not elliptical even if its circulationhas an elliptical distribution.

The computational procedure is very similar to that forfixed wings, except that some local station in the flowfield

JSince the rotor blade is operated in a sheared flow in rotationalplane, a wing having elliptical circulation distribution does not haveelliptical planform. For convenience, however, such a wing is calledan elliptical wing.

^Induced Velocity

Fig. 4 Decomposition of a rotary wing.

Fig. 5 Representation of the successive impulse of a local station inthe rotor plane of an advancing rotor.

may be influenced directly by the passage of several blades.The situation is as follows. Assume that a single rotor istraveling in a horizontal plane. Let us divide the flqwfield ofthe rotor rotational plane into a number of small squareelements with the coordinate system (l,m) as shown in Fig. 5.A square element (l,m) is shown to be influenced directly bytwo blade passages, i.e. by the /th blade element of the Athblade at timey or (i,j,k) and by the / ' th blade element of theA:'th blade at timey' or (//yY £').§ For a multirotor system,e.g., a tandem or a side-by-side rotor system, the squareelements in the flowfield also might be influenced by someblade elements of the other rotor or rotors.

A blade element is assumed to proceed intermittently in thefield within a small time interval. Let us assume that at timet—j—\ the blade element is located at a position (// m') withforward speed of Vitjtk. As seen in Fig. 6, the normal com-ponent of the velocity at a point occupied by a blade elementcan be given by the uniform inflow due to the inclination ofthe rotor plane, VN, and the sum of the induced velocities dueto the blade element itself, y / y _ / A : , and one due to thepreceding blade elements having passed over the point prior tothat time, vjr^, . At two other points, (l,m) and (//'w"), whichwill just be reached by the blade element at times t=j andt=j+l, respectively, there may also exist the induced velocitycreated by the preceding blade elements.

In Fig. 6, the coefficients C\r^, , C\m and C\.m. are "at-tenuation coefficients" expressing the decay or degenerationof the induced velocity with the time elapsed. The inclusion ofthe attenuation coefficient is necessary for the calculation ofthe induced velocity, because the disturbed air goes down-ward, and the field in the rotor rotational plane will bepartially filled with fresh air. A more detailed explanation forthe attenuation coefficient will be presented later.

Now, it may be postulated that the air mass related to themomentum change due to the /th elliptical wing spanning theblade segment between rt =xtR and R is given by

(7)

Referring to Fig. 7, any station x along a blade composed ofa series of elliptical wings is represented by each coordinate ofevery elliptical wing £ as follows:

or =_____ (8)

§Generally, the first subscript of any quantity indicates the /thradial segment of the blade, the second subscript y — 1 indicates thetime or azimuthal location of the blade element, and the third sub-script k indicates any quantity that is related to the kth blade of a b-bladed rotor.

JANUARY 1979 LOCAL MOMENTUM THEORY APPLIED TO THE ROTARY WING 9

b) f = j

H-,-u)

c) *=;+!

Fig. 6 Successive change of the induced velocity in the rotationalplane of a hovering rotor.

The load and induced velocity distributions of the blade atthe station x can be obtained as a summation of those of theelliptical wings, whose spans include that station. The sub-sequent calculation process is very similar to that of the fixedwing. That is to say, for the /th element,the strip theory givesthe following relation!:

dx

4L>RQx+

(9)

where

(10)

and where \l/k>0 and A^ are the initial azimuth angle of the Athblade and azimuthal step, respectively. It will be apparent thatthe horizontal velocity K,, the blade pitch angle 0,, and theinduced flow angle </>, are functions of azimuth angle of theblade and, therefore, are dependent on the subscripts j and k.

1A more detailed explanation has been presented in Appendix A ofRef. 7.

Fig. 7 Piling up of the induced velocity distribution.

Here, however, more abbreviated expressions have been usedVijk, 6ijk, and <t>ijk unlessfor these variables instead of

other wise stated.It is very important to remember that the induced velocity

at the time t=j and at the station (l,m), vjm, can be deter-mined successively in a time sequence using known bladesituations in the past. Thus the vj

!m can generally be given bythe following recurrent form:

(ID

where the attenuation coefficient C\~l should be a function ofthe normal component of the velocity passing through thestation (l,m) at the time t=j—l, as will be stated in thefollowing subsection, and where 6/w should be one if anyblade element hits the station (l,m) at t=j — l and otherwisezero. Thus, if the coefficient Cj~' is known, the presentproblem of finding a solution for a given initial condition andfor a specified blade pitch input can be solved by the com-bination of Eqs. (7-11).

The present process can proceed successively from a start ofthe blade motion or from a given steady state to another stateand from the blade root to the blade tip. When the rotor hubis rolling or pitching and the blade is also flapping, the normalvelocity component VN can be modified as

VN= Ksin/4-

-I- R ( x — — Rx ( qcos \l/ + psin \l/ ) (12)

Similarly, the blade deformation may also be introducedwithout any essential change of the formulation.

The equation of flapping motion of a blade having a springat the blade root, the stiffness of which is given by kp, may begiven by

(13)where /30 is the preconing or neutral angle.

10 A. AZUMA AND K. KAWACHI J. AIRCRAFT

A Part of Blade Operatingin Upwash Field

Following Blade

Preceding Blade

''Tip Vortex

Fig. 8 Upwash effect on the succeeding blade outside of the tipvortex.

We have neglected the effects of the upwash flow outside ofthe opposite ends of the elliptical wings. In a hovering rotor,this omission will introduce only a small error for the loaddistribution near the blade root, as stated before. In an ad-vancing rotor, however, the upwash flow left by the precedingblades will not always be small outboard of the blade tip, asshown in Fig. 8, and it is necessary to take it into account forestimating the load distribution of a following blade operatingoutside of the tip vortices of every preceding blade. Thus,only the upwash velocity outboard of the blade tip (or tipsideupwash) will be included in the following analysis, whereasthe upwash velocity inboard of the blade tip (or rootsideupwash) will still be neglected.

Since the upwash velocity induced outside an elliptical wingcan be determined readily by At;/ ( 1 - l£ I A/£2 — 1) , as shownin Fig. 7, the total upwash velocity given by a blade, whichcomprises a series of elliptical wings, will be obtained as asummation of the upwash velocities induced by all ellipticalwings. Thus the tipside upwash at the station x can be given by

2x-l-xi

2V(*-7) (*-*,•) (14)

It is very important to mention that the inclusion of thetipside upwash of every elliptical wing does not bring anycomplex computational difficulty in the calculation of bladeload distribution. This is due to the one-sided arrangement ofelliptical wings by which the angle of attack at any spanwisestation of the blade is independent of the tipside upwash,unlike the rootside upwash.

The induced velocity vjlm must be calculated for the station

(l,m) influenced by a hypothetical blade element on an ex-tended span (*> 1) of every preceding blade, as well as by thereal blade element on the actual span (x< 1). The induced flowdegeneration is quite similar to that of the downwash velocity.The attenuation coefficient can be determined by methodsdescribed in the next subsection.

The lift of the kth blade at the time t—j can be given by asum of the load distribution // = lijk:

n

~ LJ (15)

and the total thrust of the rotor at t =j can also be given by

Tj = LJ LJk = T Ti»^ LJk=l / = /

(16)

Attenuation CoefficientsThe instantaneous variation of induced velocity, which is

induced by a blade element passing through a specifiedstation, could be determined by the integrated contributionsfrom the total vortex system. To apply Biot-Savart's law tothis problem, the relative position vector of a vortex elementof the wake system with respect to the specified station must

be available with enough accuracy at every moment. As statedbefore, however, this procedure is almost impractical becauseof the complexity of distorted wake system.

The concept of attenuation coefficients has been introducedto retain the essential feature of the flow unsteadiness and toavoid the preceding complexity. To determine the attenuationcoefficients, two assumptions are used: 1) the wake systemconsists of a semi-infinite and skewed-vortex cylinder ofuniform disk loading presented by Castles et al.8; and 2) theupper end of the vortex cylinder is located at a positiondetermined by the time integration of a resultant velocity atthe center of rotor given by the vortex system. That is to say,after a time At has elapsed, the upper end of the vortexcylinder is located at a distance of (Vsini + v0)At in thedownward direction and (Kcos/) At in the backward direction,where v0 is a mean induced velocity. The upper end of thecylinder is assumed to move instantaneously back to the rotorplane when the next blade hits the same station, and then theprocess repeats. Each station has, therefore, its own vortexcylinder, which is reciprocating for every blade passage.

The induced velocity at any station around the rotor havinguniform disk loading can be determined easily either by usingcharts of Ref. 8 or by a direct computation based on the samemathematical model as that of Ref. 8. Then the attenuationcoefficient is given by C\~l =vlv0 at a specified local station(l,m) occupied by a blade element (i,j,k) as a function ofrelative position of the local station from the wake cylinder attime t=j. It will be appreciated that the procedure fordetermining the attenuation coefficient does not include anyiterative procedure and is free from any convergence problem.

In the simplest case, we may introduce a constant at-tenuation coefficient that is, in hovering flight, determined bythe value of C\m at x= 3/4 or**

In forward flight, the elapsed time for a given local point(l,m) between two consecutive blade passages is a function ofazimuth angle as well as radial position. For simplicity,however, the elapsed time can be assumed constant,,i.e.,2ir/bQ,. and a constant attenuation coefficient C can beevaluated by reading v/v0 at x= 3/$ and ^ = 90 or 270 deg forgiven z/R=2-K\/b and x = tan~ 7 (>t/X). As will be seen innumerical examples in the next section, these assumptionsgive a sufficiently good estimate of the degeneration of theinduced velocity in forward flight except at very low speeds(jn<0.1), in which the attenuation coefficient must be givenas a function of radius and azimuth angle.

Applications of the TheoryHovering Rotor

The aerodynamic load distribution along a blade span of ahovering rotor was calculated by the theory given in theprevious section. The rotor was assumed to have articulatedinfinitely rigid blades. The number of blade spanwise par-titions was /z = 20, and the azimuthal increment wasA\l/=2v/b. The latter may seem too coarse, but it is allowedbecause only the steady state is concerned.

An example of the results is shown in Fig. 9 and is com-pared with the results obtained from a vortex theory9 andwith wind-tunnel test results of a model rotor.10 It can be seenthat the results of the present theory using variable at-tenuation coefficients are very close to those of the vortextheory and are reasonably close to the experimental results.Even with the constant attenuation coefficient, the presenttheory gives a good estimate for the blade load distribution ofa hovering rotor. It is important to say that the computationaltime of the present theory is at least about 0.1 of that of thevortex theory.

**A more detailed explanation on the determination of C can befound in Ref. 7.

JANUARY 1979 LOCAL MOMENTUM THEORY APPLIED TO THE ROTARY WING 11

(kg/m) ( Ib / in )10r

0.5-

=0.74 Present Theon

—— I Present TheoryIPresent Theory wi th Constant C*=0.80

— I Vortex Theory (Ichikawa 8 ' ) __I Experimental Results

(NACA TN29539 ')

Fig. 9 Spanwsie airloading of hovering rotor (/* = 0, b = 2, 0, =0deg).

(kg/m) ( I b / i n )500

J 300-

<S 200

(kg/m) (Ib/ in)500

_ 400

IJ 300-"^<.2 200scCO

^100

A : 0 = 185° Experiment (NASA TN D-16371")

Fig. 10 Spanwise airloading of advancing rotor 0* = 0.18,0,= -8.3 deg).

= 4,

Advancing RotorWhenever a rotor is operating in forward flight, the effect

of the upwash velocity observed outside the blade tip must, asstated before, be included in determining the angle of attackof every succeeding blade. A typical example of the analysis isshown in Figs. 10-12. In this example, the number of bladespanwise partitions was n = 2Q, and the azimuthal incrementwas A^ = 5 deg. The field was divided into a net of /?/80squared meshes, the number of which was lxm = 160x320.The lift coefficient in the reversed flow region was assumed tobe zero. The calculation was performed by a computer,FACOM 230-75, which is probably equivalent to the IBM.360-165. The computational time was (50 s/rev) x 6 rev= 300s.

It is well recognized that the results of the present theorygive very good coincidence with the experimental results,11

even though the constant attenuation coefficient has beenadopted for simplicity of calculation. The reason for this canbe explained as follows. Since, in contrast with hovering case,most of the blade elements of an advancing rotor operate in arelatively undisturbed or fresh region of the flowfield and anylocal station does not experience so many blade passages, thevariation of C along the blade span may be neglected.

Discrepancies observed 'between the theoretical and ex-perimental results, specifically at \l/ = 240 deg in Fig. 1 la, must

25-

A >//?= 0.75 Experiment (NASA TN D-163710)———— '.r/R= 0.75 Vortex Theory Based on Rigid Wake11

(kg/m) ( I b / i n ) — — — — Ir/fl =0.75Vortex Theory Based on Distorted Wake"30r

500

£400

oI 300-<

1 200

100

Azimuth Angle, </> (deg)a) * = 0.74 and 0.75

A \r/R =0.55 Experiment(NASA TN D-1637101)

QL 0

(kg/m) ( Ib / in )500

-300<

| 200c

CO100-

Azimuth Angle, <p (deg)b) x -0.56 and 0.55

Fig. 11 Azimuthal variation of airloading (/* = 0.18, b = 4, Ot = - 8.3deg).

be due to the following reason. Since the wake contraction inthe radial direction has not been considered in the presenttheory, a point where the tip vortex of the preceding bladeintersects with the following blade must have beenerroneously estimated in azimuthal direction. The effect ofthe wake distortion on the airloading of the same blade hasbeen presented by Johnson et al.12 and their results are alsoshown in Fig. lla. Figures 12a and 12b show the radialdistribution of mean induced velocity at \l/ = 0 and 180 deg and^ = 90 and 270 deg, respectively, which is the time averageover one rotor revolution.Rotor Response Due to Rapid Increase of Collective Pitch

When a rapid change of collective pitch is applied to ahovering rotor, the rotor thrust and the blade flapping anglechange rapidly. Such transient response of the rotor thrustand the related induced flow variation have been observed inexperimental tests. Analyses also have been conducted byintroducing an inertia term due to the added mass associatedwith the rotor disk which is equivalent to that of an im-pervious disk in unsteady translation perpendicular to itsplane.13'16

The approach with the vortex theory to this problem isorthodox but needs complex computational technique, eventhough the rigid wake has been assumed.17'18 The presentmethod of analysis is very simple in the application to thisproblem and yet gives a very good result, as shown in Fig. 13.In the calculation, the mean induced velocity v0 was deter-mined by two ways, one of which was to take a mean value onthe rotor disk at each instant (shown by a solid line), and theother of which was to use the one assumed by Segel17'18

(shown by a chain line). It will be apparent that the firstdecrement following the initial peak of the aerodynamicthrust coefficient results from the blade flapping motion,which reduces the blade angle of attack, and that the seconddecrement following the maximum thrust is due to thedevelopment of the wake. In either way for the evaluation v0,

12 A. AZUMA AND K. KAWACHI J. AIRCRAFT

0.5r/R

v/QR

0.03

10

^=180°

0.02

0,01

0.5r/R

1.0

a) ^ = 0° and 180°

————— Present Theory

— — — —Vortex TheoryBased on RigidWake System7*

= 90°

b) ^ = 90° and 270°

Fig. 12 Mean induced velocity along the blade span (/* = 0.18, 6 = 4,0,= - 8.3 deg).

2x

a) Cr Total ( Thrust

I Present Theory

I Present Theory w i t h Assumed

I Vortex Theory16-17

360( d e g )

540 720

0.16

0.14

- 0.12

0.10-

0.08360

> (deg)720

c)

3.6°-13.3°

-9.7°

the transient thrust change by the present theory coincidesalmost perfectly with the result of the more sophisticatedvortex theory. The total thrust is observed to be quite dif-ferent from the aerodynamic thrust because of the bladeflapping inertia.

It must be mentioned that in the present theory no con-sideration has been given to effects caused by additionalapparent mass of the air associated with the blade pitchingand flapping motion. Therefore, the result cannot, as itstands, be applied to phenomena related to very quick motionsuch as blade flutter, including the blade pitching oscillation.The disregard of the apparent mass is nearly equivalent to theneglect of the shed vortices in the vortex wake system.

Figure 14 shows a comparison between the results of thepresent theory and the classical momentum theory in whichthe apparent mass, M0 = (S/3)pR3, of the entire rotor disk hasbeen introduced. Although the flapping motion has beenrigidly constrained in this calculation, the irregular curves ofthe thrust and the inflow have resulted from the mutual in-terference among the induced velocities of all (three) blades.In the numerical calculation, the blade .was divided into n = 20elements along the radius, and the azimuthal increment wasA\^ = 10 deg. The computation time was 2.7 s/rev. The presentcalculation supports Carpenter's result13 that the added orapparent mass of the rotor due to a rapid change of collectivepitch is equal to that of the solid circular disk in normaltranslation.

Rotor Response Due to Rapid Increase of Cyclic PitchFigure 15 shows the rotor thrust variation and the blade

flapping motion of an advancing rotor caused by a rapid

0.065

: 0.055 -

0.0455 4 0 7 2 0360

<!> (deg)

Fig. 13 Rotor response caused by a rapid increase of collective pitch(,1 = 0, 6 = 4,0, = -8.3 deg).

increase of cyclic pitch from a steady trimmed condition.Shown in Fig. 15a are the total thrust coefficient, includingblade inertia term, the aerodynamic term, and the meanflapping angle or instantaneous coning angle. It will be ap-preciated that the almost periodic variation observed in thethe thrust coefficients resulted mainly from the nonunifor-mity of the induced velocity distribution. The phase dif-ference between the total and aerodynamic thrust coefficientscan be seen, the reason for which will be found by referring tothe discussion related to Fig. 13. The longitudinal and thelateral flapping components, &Ic and 0/5, are shown in thepolar coordinates of Fig. 15b. In this example, the spanwiseand azimuthal partitions were n = 20 and A^=10 deg,respectively, and the computation time was 300 s for 10 rotorrevolutions, in which 5 rev were used to obtain the steadystate, and the other 5 rev were spent for tracing the timeresponse following a step input.

ConclusionA new momentum theory, called the local momentum

theory, has been developed and applied to investigate both

JANUARY 1979 LOCAL MOMENTUM THEORY APPLIED TO THE ROTARY WING 13

1080

a) Thrust coefficient

———— ! Present Theory

—— ~^— ' Classic Momentum Theor

Mo-f PR*

360 720 ~~ 1080<l> (deg)

b) Inflow ratio at r/R = 0.75Fig. 14 Comparison of the results obtained by the present theory andthe classic momentum theory (/* = 0, b = 3,0, = 0 deg).

Total Cr Including Blade Inertia Term

Where /?'"is the Flapping Angle of / th Blade360 #(Deg . )

a) Time response

b) Polar locusFig. 15 Time response to a step cyclic pitch input0,= -8.3 deg).

* = 0.18, 6 = 4,

steady and unsteady rotor aerodynamics. The theory is basedon the instantaneous momentum balance of fluid with theblade elemental lift at a local station in the blade rotationalplane.

A rotor blade is considered to be composed of multiplewings, each of which has an elliptical circulation distribution

and has its outboard tip aligned to the blade tip. By neglectingthe upwash flow outside each elliptical wing, the inducedvelocity of the blade is simply given by a sum of the constantinduced velocities associated with each wing, and the span wiseaerodynamic loading can be readily obtained in a straight-forward manner.

By considering that a local station in the blade rotationalplane is hit many times by different blade elements, an at-tenuation coefficient has been introduced for taking intoaccount the timewise decay of the induced velocity duringsuccessive blade passages. Specifically, for a hovering rotor,the attenuation coefficient plays an important role in thetheory and is, therefore, given as a function of radial position.For the advancing rotor, however, the coefficient may beconsidered constant, but, on the other hand, the upwash flowoutside the preceding blades' tips must be taken into accountfor evaluating the angle of attack at a following bladeelement. Results obtained by applying the theory to hoveringand advancing rotors show very good agreement with ex-perimental results, as well as theoretical estimations based onthe vortex theory.

The theory has been applied to the unsteady aerodynamicproblems of rotors in the low-frequency range, such asresponses of rotor thrust and blade flapping motion due to arapid change of collective pitch or cyclic pitch. Fruitful resultshave been obtained without any laborious calculation or anycomputation difficulty resulting from the numericaldivergence. Those results indicate that the present theorycompletely eliminates various difficulties usually associatedwith the vortex theory.

AcknowledgmentThe authors wish to express their deepest appreciation to

W.Z. Stepniewski, Aeronautical Consultant, for his warmsupport and constant encouragement throughout thedevelopment of the present theory.

References^randtl, L. and Tietjens, O.G., Fundamentals of Hydro- and

Aeromechanics, McGraw-Hill, New York, 1934.2Glauert, H. "A General Theory of the Autogyro," British

Aeronautical Research Council, R&M 1111, 1926.3Landgrebe, A.J. and Cheney, M.C., Jr. "Rotor Wakes—Key to

Performance Prediction," AGARD Conference Proceedings onAerodynamics of Rotary Wings, Fluid Dynamics Panel Specialists'Meeting, Marseilles, France, AGARD CP-111, Sept. 13-15, 1972.

4Shupe, N.K., "A Study of the Dynamic Motions of HingelessRotored Helicopters," United States Army Electronics CommandECOM-3323(AD713402), 1970.

5Azuma, A. and Nakamura, Y., "Pitch Damping of HelicopterRotor with Nonuniform Inflow," Journal of Aircraft, Vol. 11, Oct.1974. pp. 639-646.

6Multhopp, H., "Die Berechnung der Auftriebsverteilung vorTranglfugeln," Luftfahrt Forshung, Bd. 15, 1938, pp. 153-169;transl. as Aeronautical Research Council Rept.8516.

7Azuma, A. and Kawachi, K., "Local Momentum Theory and ItsyApplication to the Rotary Wing," AIAA Paper 75-865, AIAA 8thFluid and Plasmadynamic Conference, Hartford, Conn., June 16-18,1975.

8Castles, W., Jr. and De Leeuw, J.H., "The Normal Componentof the Induced Velocities in the Vicinity of a Lifting Rotor and SomeExamples of Its Application," NACA Rept. 1184, 1954 (supersedesNACATN2912).

9Ichikawa, T., "Linearized Aerodynamic Theory of Rotor Blades(III)," National Aerospace Lab. of Japan, NAL TR-100, 1966.

10Meyer, J.R., Jr. and Falabella, G., Jr., "An Investigation of theExperimental Aerodynamic Loading on a Model Helicopter RotorBlade," NACA TN 2953, 1953.

nScheiman, J. and Ludi, L.H., "Qualitative Evaluation of Effectof Helicopter Rotor-Blade Tip Vortex on Blade Airloads," NASA TND-1637, 1963.

12Johnson, W. and Scully, M.P., "Aerodynamic Problems in theCalculation of Helicopter Airloads," Symposium on Status of Testingand Model Techniques for V/STOL Aircraft, American HelicopterSociety, Oct. 1972.

14 A. AZUMA AND K. KAWACHI J. AIRCRAFT

13Carpenter, P.J. and Fridovich, B., "Effect of a Rapid Blade-Pitch Increase on the Thrust and Induced-Velocity Response of a Full-Scale Helicopter Rotor," NACA TN 3044,1953.

14Rebont, J., Valensi, J. and Soulez-Lariviere, J., "Wind-TunnelStudy of the Response in Lift of a Rotor to an Increase in CollectivePitch in the Case of Vertical Flight Near the Auto-Rotative Regime,"NASA TTF-17, 1960.

15Robont, J., Soulez-Lariviere, J., and Valensi, J., "Response ofRotor Lift to an Increase in Collective Pitch in the Case of Descending

Flight, the Regime of the Rotor Being Near Autorotation," NASA TTF-18, 1960.

16Rebont, J., Valensi, J., and Soulez-Lariviere, J., "Response of aHelicopter Rotor to an Increase in Collective Pitch for the Case ofVertical Flight," NASA TT F-55, 1961.

17Segel, L., "Air Loading on a Rotor Blade as Caused by TransientInputs of Collective Pitch," U.S. Army Aviation Labs., TR 65-65,1965.

18Segel, L., "A Method for Predicting Nonperiodic Air Loads on aRotary Wing," Journal of Aircraft, Vol. 3, Nov.-Dec. 1966, pp. 541-548.

From theAIAA Progress in Astronautics and Aeronautics Series

ALTERNATIVE HYDROCARBON FUELS:COMBUSTION AND CHEMICAL KINETICS—v. 62

A Project SQUID Workshop

Edited by Craig T. Bowman, Stanford Universityand J<t>rgen Birkeland, Department of Energy

The current generation of internal combustion engines is the result of an extended period of simultaneous evolution ofengines and fuels. During this period, the engine designer was relatively free to specify fuel properties to meet engine per-formance requirements, and the petroleum industry responded by producing fuels with the desired specifications. However,today's rising cost of petroleum, coupled with the realization that petroleum supplies will not be able to meet the long-termdemand, has stimulated an interest in alternative liquid fuels, particularly those that can be derived from coal. A widevariety of liquid fuels can be produced from coal, and from other hydrocarbon and carbohydrate sources as well, rangingfrom methanol to high molecular weight, low volatility oils. This volume is based on a set of original papers delivered at aspecial workshop called by the Department of Energy and the Department of Defense for the purpose of discussing theproblems of switching to fuels producible from such nonpetroleum sources for use in automotive engines, aircraft gasturbines, and stationary power plants. The authors were asked also to indicate how research in the areas of combustion, fuelchemistry, and chemical kinetics can be directed toward achieving a timely transition to such fuels, should it becomenecessary. Research scientists in those fields, as well as development engineers concerned with engines and power plants, willfind this volume a useful up-to-date analysis of the changing fuels picture.

463pp., 6x9Hlus., $20.00Mem., $35.00List

TO ORDER WRITE: Publications Dept., AIAA, 1290 Avenue of the Americas, New York, N. Y. 10019