HELICOPTER MAIN AND TAIL ROTORS IN HOVER(U)I AERONAUTICAL RESEARCH LABS MELBOURNE (AUSTRALIA)

USI FE K J HAYMAN ET AL. JUL 84 ARL-RERO-TM-366 F/G 20/4 N

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DEPARTMENT OF DEFENCE

I DEFENCE SCIENCE AND TECHNOLOGY ORGANISATION

AERONAUTICAL RESEARCH LABORATORIESMELBOURNE. VICTORIA

Aerodynamics Technical Mlmrandun, 366

CALCULATION OF THE VELOCITY FIELD GEN1ERATED BY A HELICOPTERMAIN AND TAIL ROTORS IN HOVER

K.J. HAYIANand

K. R. REMDY

DTIC.FELECTENOV 2 1 Q843

BApproved for public release.

Uj THE UNITED STATES NATIONAL

-~~~ TI TCAC I INFORMATION SERVICE

1b. is %ur140RISED TO

IREPRODUCE AND SELL THIS REPORT

(C) COF1W4'JEALTH OF AUSTRALIA 1984

COPY No JULY 1984

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AR-003-935

DEPARTMENT OF DEFENCE 0V DEFLCE SCIENCE AND TECHNOLOGY ORGANISATION

AERONAUTICAL RESEARCH LABORATORIES -7.

Aerodynamics Technical Memorandum 366

CALCULATION OF THE VELOCITY FIELD GENERATED BY A HELICOPTERMAIN AND TAIL ROTORS IN HOVER

by DTICK.J. HAYMAN ~ L C E

K.R. REDDY1

SLIARY

- A theory is presented to calculate the flow field due to mainand tail rotors of a helicopter in hovering flight. To solve thiscomplex flow problem, simple vortex rings are used to model main andtall rotor wakes. Preliminary numerical results are found to beencouraging.

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©COIONwEALTH OF AUSTRALIA 1984

POSTAL ADDRESS: Director, Aeronautical Research Laboratories,P.O. box 4331, Melbourne, Victoria, 3001, Australia. .. .

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CONTENTS

PAGE NO.

NOTATION

1. INTRODUCTION1

2. DEVELOPMENT OF THE WAKE MODEL I

3. DESCRIPTION OF THE COMPUTER PROGRAM 2

4. RESULTS AND CONCLUDING REMARKS 3

REFERENCES

APPENDIX

FIGURES

DISTRIBUTION

U DOCUMENT CONTROL DATA

Accession For

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Ava ilabilitY CodeSD.

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NOTATION 0

a,b,c co-ordinates of the centre of the tail rotor

CT rotor thrust coefficient i.. .

h vertical distance between field point Q and thevortex ring

k vortex strength of a ring vortex

r,e radial and angular co-ordinates (see fig. 1)

R radius of vortex ring

s ring vortex circumferencial parameter

local tangent vector

velocity vector in cylindrical polar co-ordinates

UrUz radial and axial velocity components

vi induced velocity in the plane of the rotor

xposition vector of field point Q (see fig. 1)

i,,j,i unit vector in the Cartesian co-ordinate systemx,y,z (see fig. 1)

azimuthal position (see fig. 1)

o.~ -' .'o

. ,. -.... -.

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1. INTRODUCTION

In certain helicopter flight conditions, the fuselage,and other components such as the tail rotor, fin and hub, arepartially or wholly immersed in the main rotor slipstream. This --.leads to interactional aerodynamic phenomena which have been observedin many helicopters [1]. A clear knowledge of these phenomena isimportant not only because they could be a serious source of vibrationand fatigue [2], but also because they may affect the aerodynamicefficiency of various components. The present trend towards greatercompactness necessitates the various aerodynamic elements being closeto each other. This means that the aerodynamic interaction problemwill become even more acute in the future.

Because of the large number of helicopter aerodynamiccomponents involved, a comprehensive mathematical analysis of theinteraction phenomena is extremely difficult. Mutual interaction ofthe vortex elements generated by the separate components will deformthe vortices which, because of viscosity, will eventually dissipate.To solve this complex interactive flow problem, various simplifyingassumptions are necessary. In particular, these include the numberand geometry of the aerodynamic elements, and the flow field generatedby each element. In this document only the two important aerodynamicelements, the main rotor and tail rotor, are considered.

The wake model to represent a helicopter rotor bladevaries from very complex lifting line theory to simple blade tipvortex theory. As an initial step, a simple tip vortex wake model isused here to represent the wake of a rotor blade. To simplify thenumerical process further, the trailing vortex pattern is assumed toform a series of vortex rings. Similar wake models have been usedpreviously by Cook [3] to calculate the helicopter main rotorperformance, and recently Rayner [4] used a ring vortex wake model tostudy hovering flight of animals. Other forms of representing thewake such as the skewed spiral, necessitate numerical integration andare therefore time consuming.

In developing the theory, consideration has been limited Sto aircraft having two rotors, although no restriction has beenplaced on the number of rotor blades. A simple and efficientprocedure, suitable for flight simulation studies, for the calculationof the flow field due to main and tail rotors is described. For thetest case, the induced flow in the plane of the main rotor in steadyhovering flight has been calculated and compared with results from a -more complex model. The theory can be readily extended to cope withforward flight. Throughout, it is assumed that the fluid is inviscidand incompressible.

2. DEVELOPMENT OF THE WAKE MODEL

To estimate the combined main and tail rotor velocityfield, the wake of each rotor is modelled as a stack of circularcoaxial vortex rings beneath the rotor. The rings are equally spaced,

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the first one being half the spacing distance below the rotor. Themomentum theory is used to calculate the spacing between the rings.

Initially, all the rings were the same radius as therotor so as to simplify the situation but a comparison with resultsfrom other models (Reference 5) showed a large variation in results.The radii of the rings are now given by -

R = (0.78 + 0.22 e-K*) R'

(see Reference 6) where K is a wake contraction parameter (=4(CT)),.= 2w(i-l) for the ith vortex ring and R' is the radius of the

rotor. This gives much improved results for induced velocity in theplane of the rotor, except for riR' > 0.975 - the'very edge of therotor disc.

The vortex rings are assumed to be sufficiently farapart so that they have no effect on one another. Thus, the totalvelocity induced by the entire "stack" of rings is Just the vectorsum of the velocities induced by the individual rings,

The tail rotor is modelled in exactly the same way asthe main rotor, although a different co-ordinate system is used, andthe results can then be combined with those from the main rotor togive the total induced velocity. For the data used, however, thecontribution to the induced velocity from the tall rotor is smallexcept near the rear edge of the main rotor disc.!.-

3. DESCRIPTION OF THE COMPUTER PROGRAM

The program itself is a straight-forward implementationof the formulae developed in the Appendix. The required inputs are:-

- the radius of the rotor

- the separation between consecutive vortex rings -- the number of vortex rings to consider- the circulation of each ring.

If the effects of a tail rotor are to be included then this informa-tion is also needed for the tail rotor, as well as the position ofthe centre of the tail rotor relative to the main rotor.

The induced velocity at the rotor is then found usingthe information provided, by application of the formulae in theAppendix. If the tail rotor is included, the induced velocity dueto each stack of rings is found and the results combined to give thetotal effect. In this case, the effect of each rotor is output, aswell as the total effect.

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The units of the output are determined by those of the5input data. If distances (i.e. radii and separations) are given in 0metres and circulations in metres2/second, then the results are in-/s" but if other output units are desired then the input data mustbe modified appropriately.

The induced velocities are calculated using two right-S handed Cartesian coordinate systems. The main one, to which all

input and output is referenced, is centred at the centre of the mainrotor, has the x-axis directed horizontally backwards, the y-axishorizontal and the z-axis vertically upwards. The second system,which is used for convenience in calculating induced velocities dueto the tail rotor, is parallel to but rotated from the main system -as shown in Figure 2. l

If the centre of the second co-ordinate system is(a,b,c) relative to the main system, then the point P(x,y,z) in themain system is given by P'(x-a,z-c, b-y) relative to the secondsystem.

The program was written in modular form, so that themain program controls the flow of execution but does very little ofthe actual work; most of the work is done in subroutines to readin the input values and calculate the induced velocities by applica-tion of the formulae. Note that the calculations require the use ofa numerical integration routine from the library DSKB.NAGF (7,7],which must be included at link time for the program to run correctly.

4. RESULTS AND CONCLUDING REMARKS

The wake in the hovering flight of a rotor is modelledby a stack of circular blade tip vortex rings. Experimentalevidence suggests that blade tip vortex is the most dominent elementin the rotor wake. Experimental results also show that this bladetip vortex though highly distorted farther away from the rotor,follows a path quite close to a circle in the immediate vicinity ofthe rotor. If calculation of the velocity field close to therotor is of main concern then representation of the rotor vortex Swake by a stack of vortex rings is physically reasonable. This alsosimplifies the mathematical analysis, the computer time is reducedwhich make the model very suitable for flight simulation studies.

Induced velocities in the rotor plane calculated usingthe ring vortex wake modelare presented in Figure 3 for a typical 9four bladed helicopter rotor in hover. A comparison of results fromthis model with those in Reference 5 shows reasonable agreementbetween the theories except at the edge of the blade, where some ofthe assumptions made deriving the current formulae may not hold.Methods to improve the present mathematical model are being investi-gated. The theory can be readily extended to cover forward flightof any two rotor helicopter.

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REFERENCES

(1] P.F. Sheridan and R.P. Smith, Interactional Aerodynamics-A New Challenge to Helicopter Technology, Journal ofAmerican Helicopter Society, January 1980.

[2] J.W. Leverton, J.S. Pollard and C.R. Wills, Main Rotor Wake/Tail Rotor Interaction, VERTICA, Vol. 1, 1977, pp 213-221.

[3) C.V. Cook, Induced Flow Through a Helicopter Rotor in ForwardFlight, Research Paper 374, Westland Helicopters Ltd.,Somerset, 1970.

[4) J.MV. Rayner, A Vortex Theory of Animal Flight. Part 1.The Vortex Wake of a Hovering Animal, J. Fluid Mech., Vol. 91,1979, pp 697-730.

[5] K.R. Reddy, Prediction of Helicopter Rotor Downwash in Hoverand Vertical Flight, ARi Aerodynamics Report 150, January1979.

( 6] J.D. Kocurek and J.L. Tangler, A Prescribed Wake Lifting

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APPENDIX

MATHEMATICAL DERIVATION OF THE VELOCITY FIELD INDUCED BY

A VORTEX RING ELEMENT (from reference 4)

Consider the effect of a single vortex ring, centred at theorigin and lying in the x-y plane, on a point in the x-z plane(fig. la). No generality is lost by this positioning of the pointas any point can be so located by a rotation of the co-ordinatesystem about the z-axis. 4.

As long as the point is not near the core of a vortex ring .the vortex core can be approximated by a line vortex, and the inducedvelocity field is given by the Biot-Savart law.

-kI ( i (s))Air's) d-= Ilx-X(s)ll9 ds (1) -

where s is a parameter of the circumference of the vortex ring.

In the case of figure la with points P and Q (assumed to be S

in the x-z plane),

X = R(cose, sine, 0)i = (r,o,h) (2)s = Re

S=d = (-sine, cose, 0)ds

so from (1) it follows that

2w SkR (hcose, hsine, R-rcose)

4w [r2-2rRcose + R2 + h2] 3/ 2 de (3)

11

Denote the unit vectors in the x,y and z directions by x,y,z " ..respectively.

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By symmietry about the z-axis n

U.; 0 (4)

p Define .- =kRh

u~x r- 2(r.R~h)-~kR

;.z = (Rll(r,R,h) -r1 2(r,Rh)) (5)PA

whee rIAand 12 can be evaluated in terms of the complete elliptic

K(e) (1f-e 2 di

(6)

and E(e) (1de

where the eccentricity e is given by

e 4rR[(r+R) + h2J (7)

For K(e) to be real, e2 < I which is always true as follows

(r-R )2 + h 2 > 0 V rR,h

S r+2+Rhz 4rR

*4rR e e2 :gC(r.R)2 + 2]

Comparing (3) and (5) gives

u.* kRh 12w cose de4w (r R)2h2-2rR-2rb1.DIl

.4 - (8)

=kh1 2(r,Rh)

A. 3

~ 2 =f coseUr+R)2+h'-2rR(1+cose)j3 de (9)

* 12 = (-+R)+h2 cose [i-ea(+oeJ de (10)

But COS2e a 1+cos2e

K' 2: 2= 2[(r+R)2+h2J1 cos2gE1-eco2y/ di (11)

Also notedoS2]-/ 2C /2 -3/2

coEcos2 os~eJ3/ d J cos2e[1-e~cse d5

(12)

Thus

12 4[(r+R)2+h2Y/ J/cos2i[I-e2co2J/ di (13)

=4H(e)[(r+R)2+h2Y/ (14)

from tables of integrals, where H(e) is defined by

H(e) = L[2-e' E(e) -2K(el (15)e2 I -e2

Substituting (9) back into (5) and comparing the result to (3)".

and proceeding similarly to the above gives* .%

I1 4[()2.2-3/2 'P/2 (1 e 2CO~)-3/2 di (16)

=4G(e) [frR)2 +h 2]-3 / 2 (17)

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where G(e)= E(e) (18)(1-e2)

From (14) and (17) it can be seen that

ur x + uz z (19) -

Ur = kRh H(e) (20) 44wT [(r+R Ah ] 1 ,, -.' '

kR 4.

i u 4 (RG(e)-rH(e)) (21)uz =4w [(r+R )2+h 2]312 ;_ -: .

with e given as in (7).

If the point Q does not lie in the x-z plane then the quantityUr is a radial velocity directed outwards from the z-axis, which -must be split into true x and y components.

Then !:::::

u x= ur cos (22)

U y = Ur sin .

where * is the angle shown in figure lb.0

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DOCUMENT CONTROL DATA1.e0. ARNo 1. b. Es 4Iihmsa No 2. Daont Dmt 3. TmgIh NoAR-003-935 ARL-AERO-TM-366 JULY 1984 83/004

CALCULATION OF THE VELOCITY FIELD t ~GENERATED BY A HELICOPTER MAIN AND TAI NC3SFEROTORS IN HOVER b. titl f. uraot 7. "1U U 6R@

K.J. HAYMANand

-K.R. REDDY

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AERONAUTICAL RESEARCH LABORATORIESb.am G.PO BOX 4331, MELBOURNE. VICTORIA, 3001

12. becoramy Diwb~ofl (of dWi doumijt

Approved for public release.

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U.. iDorwa 16. COSATI GrowRotary-wing aircraft010Rotor wakes010AerodynamicsHelicopter wakes

W. A6W A teory is presented to calculate the flow field dueto main and tail rotors of a helicopter in hovering flight. Tosolve this complex flow problem, simple vortex rings are used tomodel main and tail rotor wakes. Preliminary numerical results arefound to be encouraging.

This pop is to be ud to rewrd inforieton whidh is requiid by the Eusbignm for Its own e but

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Aeronautical Research Laboratories, Melbourne

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* MEMORANDUM 366

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