Acoustic interference of counter-rotation propellers

Journal of Sound and Vibration (1988) 124(2), 357-366

ACOUSTIC

INTERFERENCE

COUNTER-ROTATION

OF

PROPELLERS

C. K. W. TAM'[" Department of Mathematics, Florida State University, TaUahassee, Florida 32306, U.S.A.

M. SAL1KUDDIN Lockheed--Georgia Company, Marietta, Georgia 30063, U.S.A. AND D. B. HANSON

Hamilton Standard, Division of United Technologies Corporation, Windsor Locks, Connecticut, U.S.A. (Received 22 April 1987, and in revised form 11 November 1987)

The noise fields from the rotors of counter-rotating propellers having the same number of blades and angular frequency tend to cancel or reinforce each other depending on the relative phase of the two fields at the point of observation. Because of this, the total noise field at the blade passage frequency or harmonics forms a characteristic standing wave pattern. A general investigation of this acoustic interference phenomenon is carried out. Unlike previous works, the present analysis allows the front and the rear rotor to 'have different blade geometry and loading. Further, the effect of forward flight is included. Numerical results indicate that at high subsonic cruise Mach number the acoustic interference pattern differs substantially from that at static condition.

1. INTRODUCTION

Recently there has developed a renewed interest in the use o f counter-rotation propellers to power commercial and military aircraft: These propellers, however, generate very high intensity noise which could cause great discomfort to passengers inside the aircraft c a b i n . Because of this, the noise generation mechanisms and characteristics of this type of propulsive system have become the subject of numerous current investigations. In a recent paper, Hanson [1] carried out a comprehensive study of the noise of counter-rotation propellers. On noting that the noise field is very complicated he suggested separating the mutual interaction noise o]" the front and the rear rotors into two categories. He called the first type Of interaction effects acoustic interference and the second type aerodynamic interference. Acoustic interference refers to the constructive reinforcement and destructive cancellation of the noise radiated independently from the two rotors. Aerodynamic interference refers to the fluid-mechanical interaction of t h e r o t o r s which generally gives rise to the generation of additional noise and the change in noise radiation characteristics. In this paper only the phenomenon of acoustic interference will be considered. Hubbard [2] was, perhaps, th'e lirst to recognize that the noise from each rotor of a counter-rotation propeller system may reinforce or cancel the other if they have the same frequency. The constructive and destructive interference occur over the entire volume o f tConsultant to Lockheed-Georgia Company. 357 0022-460X/88/140357 + 10 $03.00/0

9 1988 Academic Press Limited

358

C. K. ~,V. TAM E T AL.

space so that the spatial distribution of the noise field may produce a standing wave pattern around the axis of rotation. In this early work, the effect of rotor separation on the acoustic interference pattern was not considered. The rotor separation effect was, however, included in the work of Hanson [1]. In addition to providing an analysis of this eftect, Hanson also offered a convincing demonstration of the accuracy of his results by showing a good fit of his calculated directivity to Block's measurements [3] at low flight velocity. Hanson's analysis, however, did not account for the convection effect of forward flight. This effect becomes important at high subsonic cruise mach number. Furthermore, to simplify the analysis, he assumed that the noise source strengths of the front and the rear rotor are the same. The main purpose of this paper is to extend Hanson's work to include acoustic interference phenomena at high subsonic fo~vard velocity and for rotors that do not necessarily have the same loading and blade geometry. The only requirement is that the acoustic field patterns of both the front and the rear rotor, considered independently, have the same number of lobes azimuthally (or same number of blades) and rotate opposite to each other with the same angular frequency. 2. THE SOUND FIELD OF TWO AERODYNAMICALLY NON-INTERACTING COUNTER ROTATING ROTORS Consider a counter-rotating propeller in a uniform flow of Mach number Moo as shown in Figure 1. Let the separation distance of the two rotors be S. It will be assumed that the rotors are aerodynamically non-interacting. This assumption allows one to consider the total sound field to be the superposition of two sound fields each generated by a rotor unrelated to the presence of the other. The rotors are assumed to have the same number of blades and rotate opposite to each other with the same angular frequency. The first portion of the analysis is carried out in a co-ordinate reference frame fixed to the center of the propeller, as in the previous work of Tam and Salikuddin [10] and others. Section 2 treats the general case in which the loading and geometry of the i'otors are not the same. Section 3 specializes these results to the case of nearly identical rotors with a simple formula giving the shape of the interference pattern. In Section 4 the identical rotor case is treated in Hanson's retarded co-ordinate reference frame. This verifies the results of Section 3 and demonstrates the equivalence of analyses in the fluid-fixed and propellerfixed co-ordinates. T o c a l c u l a t e the noise field generated by the front or the rear rotor alone it is easiest to perform the computation in a co-ordinate system fixed at the center of the rotor under Front

Rear pro )eller

propeller

u~

ii

-Sl 2..~q~-512-

IL

-

(b)

(o) Figure 1. Schematic diagram of a counter rotation propeller system showing the individual propeller-centered spherical co-ordinate systems and the propeller system centered spherical co-ordinate system. (a) side view of the propeller system; (b) front view of front propeller. ~"

ACOUSTIC

INTERFERENCE

OF CR PROPELLERS

359

consideration. But to find the acoustic interference pattern it is most straightforward to use a co-ordinate system fixed midway between the two rotors, as shown in Figure 1. In the following, therefore, the sound field associated with each rotor will first be written out in the respective propeller-centered co-ordinate systems. Then a co-ordinate transformation will be performed to shift the reference frame to a common origin centered midway between the two rotors. Let (R~, X~, 0s), (R2, X2, 0,) be the spherical coordinates (with the Z-axis in the opposite direction to that of flight as the polar axis) of two co-ordinates systems centered at the front and the rear rotor respectively. Let (R,x, 0s) be a similar spherical co-ordinate system centered in between the rotors. The radial co-ordinates RI, R2, R and the polar angles X~, X2, X are related geometrically by R~ =

[R 2 sin 2 X

+ (R cos X + S/2)2] t/2.

(2.1)

For R >>S, expression (2.1) reduces to R, = R +-~(S cos X) + 0($2/R2) 9

(2.2)

R2 = R --~(S cos X) + 0($2/R2) 9

(2.3)

Similarly,

Also to order ( S / R ) 2 it is easy to find cosxl=cosx+(S/2R)sin2x, sinxl=sinx-(S/2R)sinxcosx cosx2=cosx-(S/2R)sin2x,

(2.4,2.5)

s i n x 2 = s i n x + ( S / 2 R ) s i n x c o s x . (2.6,2.7)

The sound field associated with the thickness and loading of a propeller in forward flight with respect to a propeller-centered spherical co-ordinate system has been discussed by Tam and Salikuddin [10]. Figure 2 represents a blade of the front r o t o r which is intersected by the curved surface F of the circular cylinder of radius r. The thickness distribution of the blade is prescribed on the intersection of surface F and the blade and is shown in Figure 3. Suppose the front rotor has B blades o f length b~ and rotates with angular velocity w~; the acoustic field associated with blade thickness distribution h~(~:, r), blade sweep D~(r) (see Figure 3) and loading distribution LI(~:, r), where r is the radial

Pilch-chonge oxis

L

Figure 2. Schematic drawing showing the pitch-changeaxis of the ruth blade of the front propeller and the intersection of the blade and the cylindrical surface F at b~lius r.

360

C. K. W. T A M

E T AL.

A = Pilch 7ehonge

B

,~

r

Ull e z

Velocity

diogrom

Figure 3. Local co-ordinates (r, ~r and various dimensions of the ruth blade of the front propeller on the surface F.

distance of ihe blade section from the axis of rotation, then is 1

n~oo

p(R,,x,,Os, t)=--~ Y~ Rl>>b

I

Al(n, to, Xi)

tl =--o0

t

L.M~ocos XI + ~/1- M ~ sin z XI

t-O,-tot-Oi]}, (2.8)

where

A,(n,

w, XI) = - 2~'a=(1 - M ~

imoB2~r(n, to, XI) sin 2 Xl)(Moo cos X, +~/1 -

inwB21~'(n,t~ + 47rltol,]l -~--~= fo" ~ /

kay~

M ~ sin 2 X~)

]

M~s--in2----X,d '

(2.9)

exp[-iks(n,w),h(r) iwnB(Dl-xlc)

-I a ~ d l - M ~ sin 2 x,(Moo cos X, +~/1 - M ~ sin 2 X,)~/M~+

/ xf.(r, xi,to)e- i.~=-2 ' J.~["

-wnB ~ - - sin Xl

\a~x/1 - Moo sin 2 XI

] (wr/a~o)2

r}'~ rdr,

(2.10)

f,(r, X,, to) f< Ohl(s~'r)

=Jo

[ xexp

a~

itonB~

]

a~ox/l_MLsin2xt(Moocosx,+~/lgMLsin2x,)4ML+(tor/a~) 2

d~:,. (2.11)

ACOUSTIC

INTERFERENCE

to 2

OF CR

Moo~+M~] exp[-ik~(n, to)rh(r) ] r2j

cosxi

15L(n'w'Xt)----I;'[aoo(l_M~) -- 2

361

PROPELLERS

+

2 ioJnB(O,-x,~) i]J a~x/Moo+ (wr/a~)2x/l - M~ sin2x~(Moo cos Xt +x/1 - ML sin 2 X~ { wnB sin Xl '~ x/3.(r, xt, u,j ~ a.Bt ;-----72 -- : r] rdr, (2.12) \aoo41 - Moo sin Xt .

FoCr, X,, w) = x exp

"~ ^ - - i n B r r

Io

L,(~r r)

[ .iwnB~ ] d~: (2.13) , 2 2 I-aoo41- Moo sin 2 xt(M~ cos Xl+----~1- Moo sin 2 Xt)vIM2+ (tor/a~) ~ ks

wnB

(4

a~(1- m~)

cos x, -Moo). 1-M~sin2x,

(2.14)

0~ characterizes the initial azimuthal position of the blades. Moo and aoo are the cruise Mach number and ambient speed of sound respectively. It is straightforward to show from equations (2.10) and (2.12) that the functions ~r(n, w, X) and /iL(n, w, X) possess the following properties:

t~L(-n,w,X)=~*(n,w,X),

I~L(n,--w,X)=lS*(n,w,X),

(2.15) where * denotes the complex conjugate. By means of expressions (2.15) it can be deduced that

A,(-n, ~, x l ) = a*~(n, o~,x,), A,(n, -w, X,) = a*(n, to, X1). '" (2.16) "For the rear rotor with angular velocity -oJ~, the sound field with respect to the coordinate system (R2, XI, 0s) is given by an equation similar to equation (2.8): i.e.,

1 R~>>b~

E

A (n, ,,x,)exp[inB[

R2 . . . .

k

-o,R:/,,oo

L M~ cos X2+ x/1 - M ~ sin 2 X2

+O,+oJt-02]}.

(2.17)

The functions A2(n, % X2) in equation (2.17) also satisfy property (2.16). The total acoustic field from the two rotors is the direct sum of the two non-interacting sound fields provided in expressions (2.8) and (2.17). To facilitate the process of adding the two complex quantities it is advantageous to transform both sound fields so that they are measured from a common co-ordinate system. For convenience the co-ordinate system (R,x, 0s) centered midway between the two rotors will be used (see Figure 1). Here it will be assumed that the separation distance S is smaller than the length of the blades so that in the far field R >>bz> S. It is simple to show that if terms to order S]R is neglected then only the change in the phase function of equations (2.8) need to be carried out in changing the co-ordinate system from (R~,xt, Os) to (R, X, 0s). Upon using equations (2.2) to (2.7) it is easy to find the following formulas for the change in the phase functions of equations (2.8) and (2.17):

wRt/ aoo,"

-

~+~+O(S/R),

Moo cos Xi +x/1 - M 2 sin 2 Xt

wRi/aoo

=-q~+~+O(S/R),

Moo cos X2+x/1 - M ~ sin 2 Xa-

(2.18,2.19).

362

C. K. W. T A M

ET

AL.

where toR/aoo

1/,=

05_

coS(cos X - M~ sin 2 X/x/1 - M 2 sin 2 X)

Moo cos X +x/1 - M 2 sin 2 X'

2aoo(Moo cos X +,]1 - M E sin 2 X) (2.20, 2.21)

Now by adding the sound fields of the two counter rotating propellers together the total acoustic field is given by 1 p ( R , x , Os, t)='--~ R ~ OO

~

A l ( n , to, X)

einn['t'+e~+~176176

n ~ --oo

1

Oo .~_o n , Z2( _

to, X)

ei"nt-~'+*+ ~176176 + O ( S / R ) (2.22)

Consider now the directivity of the sound field associated with the nth harmonic of blade passage frequency. Let the pressure of this field be denoted by p,. For brevity, the absolute values and arguments of A~(n, to, X) and A2(n , to, X) will henceforth be denoted by .3,~., A2., fl~. and -f12. respectively: i.e.,

Ia,(n, to, x)l = ,~.,., arg ( A l ( n , to, X)) = fl,.,

Ia2(n, to, X)]=.3,2., arg (A2(n, to, X)) = -f12..

(2.23)

From equation (2.22) one finds Rp,, = .,{1. [e i"B(q'+~+~ . . . . ~

4-e -inD('P'+c'+O,-~t-O,)-il3,.]

R~OO

4- "~2,~[ e lnB(-'t" +q'+~176

4- e-;"m- ~'+*+ ~176176

= [,~.1. e i~" +A2. e -i~" ] e i"B['t'-'~176176176

]

)

+ complex conjugate,

(2.24)

where or. = nB[ Os+ 05 - (0, - 02)/2] + (fl~. - fl2.)/2.

(2.25)

By means of expression (2.24) the angular distribution o f the noise intensity of the nth blade passage frequency or the r.m.s, value of p. can easily be found: p ~ ( R , x , O,)'/2=(2'/2/R)[,42,+A2.+2.4i..42, cos (2a,)] '/2

(2.26)

R ~oo

(the overbar denotes the time average). Equation (2.26) gives the most general acoustic interference pattern of two counterrotating propellers (with the same number of blades and rotational frequency) at subsonic cruise Mach number Moo. 3. NEARLY IDENTICAL PROPELLERS In most counter-rotation propeller'systems it is expected that the blade geometry and loading of the front a n d rear propellers are nearly the same. In these cases the general result (2.26) may be somewhat simplified. Let .3.2. = A,,, + e,

..fl~. = fl~. + 6,

(3.1, 3.2)

ACOUSTIC

INTERFERENCE

OF CR PROPELLERS

363

where le/,'~,l<< 1 and 18~<< 1. Substitution of expressions (3.1) and (3.2) into equation (2.25) gives to order le/A=,l 2 the following formula for the r.m.s, value of p,:

p~( g, X, 0,) ~/2= (2x/2/ R )('2II,A2,)I/21COS ct, I.

(3.3)

R~co

On the conical surface X = constant the azimuthal distribution of the sound intensity is, therefore, described by the factor Icos ot, I. Writing out in full, one obtains, to the lowest order, ( /~2 ~/2 COS

[

toS(cos X-Moo sin 2 X / ' f l - M 2 sin2 X )

nB Osq

2ao~(Moocosx+x/l_M2 sin2x )

0=202])[.

(3.4)

In this expression, clearly the second term of the argument of cosine is the one which accounts for the effect o f forward flight on the acoustic interference pattern. If Moo is set equal to zero, equation (3.4) reduces to the result found previously by Hanson [1]. Hanson [1] earlier demonstrated the accuracy of his acoustic interference prediction formula by comparing the calculated pressure amplitude distribution with the measurements of Block [3]. Hanson's formula includes the effects of azimuthal acoustic interference and rotor separation. The forward flight effect was, however, not considered. In the experiment o f Block the data were obtained in an open wind tunnel at very small simulated forward flight Mach number (Moo = 0.0825). At this forward flight Mach number equation (3.4) predicts a directivity pattern which differs negligibly from that at the static condition. Figure 4 shows a comparison of the calculated pressure intensity pattern at blade passage frequency (n = 1) and the measurements of Block [3]. The agreement is excellent (see also the paper by Hanson [1]). In order to provide a measure o f the importance of finite forward flight Mach number the calculated pattern at Moo = 0.8 according to expression (3.4) is plotted in Figure 5 (dotted curve). On comparing this with the distribution at /~//~o= 0.0925 (solid curve) it is clear that the locations o f the peaks and valleys of the

U ~

Microphone number 2 5 4 I I I I

Propeller

PX3 5

I

6

7

i

8 I

9

i

/

10

I

j r

- ~ l~dB 60

I 40

20

~"

I

I

0 -20 (Sstance, x2(inches)

-40

-60

Figure 4. Comparison of Block's azimuthal directivity data with calculated results based on equation (3.4) (see also the paper by Hanson [1])) at the blade passage frequency. Four bladed rotor; M, =0-632. Calculated curves were shifted vertically for good fit. 0, Data from Bl~k [3]; , equation (3.4) M~=0.0925.

364

C. K. %V. TAM E T A L .

.~....._.~

Propeller

YX 3 i

I

I

I!

r /I

pB/I 60

40

I

20 0 -20 Distance, x2 (inches)

t

-40

Figure 5. ElIect of forward flight at the same conditions as Figure 4.. , equation (3.4), M~ = 0-8.

-60

9 Equation (3.4), M~=0.0925;

. two patterns have more or less been interchanged. This indicates that a t h i g h subsonic cruise Mach number failure to properly accounting for the flight effect could result in a totally erroneous prediction of the noise directivity pattern. %

4. IDENTICAL ROTORS ANALYZED IN RETARDED Co-oRDINATES If it is assumed from the outset that the blade load distributions on the two rotors are the same, then the directivity pattern of expression (3.4) can be derived more easily. The results are given below as equation (4.1) in Hanson's retarded co-ordinate notation for the purpose of verifying expression (3.4) and also to illustrate the equivalence of the retarded and visual (or nacelle-fixed) co-ordinate systems. The analysis in reference [1] is valid only for the static condition (Moo = 0). As explained at the end of this section, the complete formula including forward flight effects is

H=cos[nB(q~q SMr D

cos0 1-Moocos 0

)] ~bc

(4.1)

in the notation of reference [1], in which the Doppler factor 1 - M~o cos 0 was omitted. To show the equivalence of expressions (4.1) and (3.4), differences in notation must be resolved. First, 0 t and 02 were equal so that (0t-l-02)/2=q~ c. Second, the observer circumferential angle ~b is measured in the direction of rotation of the rear rotor rather than the front so that ~b = - 0 , and ~bc--* -4'c. Third, S M r / D can be recognized as toS/2aoo so that expression (4.1) can be rewritten as [ H=cos

(

toS

cos0

nB 0~" 2aoo l - Mo~ cos O

0,+ 02.)] 2 "

(4.2)

Now it only remains to show the equivalence of ,;bu = cos 0/(L*- Moo cos 0)

(4.3)

ACOUSTIC INTERFERENCE

OF CR PROPELLERS

365

from expression (4.1) and ~brs =

cos X - Moo sin EX/x/1 - M ~ sin 2 X

(4.4)

M~ cos X +x/1 - M ~ sin EX

from expression (3.4). To accomplish this, one can define a visual angle measured from the propeller forward axis,

Ov = ~r-X,

(4.5)

which changes cos X to - c o s Ov and sinE x to sin 20v with the result qSrs -

cos Ov+M~sinEx/x/1

-

-

2 sin 20v Moo

(4.6)

-Moo cos Ov +x/1 - M ~ sin E Ov

Multiplying numerator and denominator by x/1 - M 2 sin 20v and rearranging gives [cos Ovv/l- M 2 sin 2 0 v + M~ sin 2 Or] q~rs - 1 - M~[cos Ovx/1 - M 2 sin 20v + M ~ sin 2 Or]"

(4.7)

But the expression in square brackets is just cos O: that is, cos 0 = cos Ov,/1 - M ~ sin 20v + M~ sin 20v

(4.8)

is the standard relationship between visual and retarded angles Ov and O. (See reference [11] for further discussion.) It follows that qSrs = cos 0/(1 - m o o cos 0) = qSn,

(4.9)

wlfich was to be shown and demonstrates the equivalence o f derivations in ifie retarded and visual co-ordinate systems. The revised derivation of expression (4.1) proceeds as follows. In reference [1] an expression was needed for r+Ar, the radiation distance from the rear rotor location at t = 0 to the observer location, r is the corresponding distance for a point midway between the rotors. To find this correctly, r must be expressed in propeller-fixed (visual) coordinates because these are used to specify the s e p a r a t i o n d i s t a n c e S. From formulas in the Appendix of reference [6], it can be deduced that

r = (l/flE)CM~x~ + So),

(4.10)

So=x]xE + flEy 2,

(4.11)

where

y is the sideline distance to the observer and xl is his distance forward of the propeller reference plane. I f t h e distance xl is increased to x~ + S/2, then the corresponding radiation distance is r+ar. By series expansion Ar can be found to first order: Ar = [(x, + Mo~So)/flESo](S/2).

(4.12)

The formulas of reference [6] can be used to return to retarded co-ordinates with the result

Ar='(S/2) c~s 0 / ( 1 - M ~ cos 0)

(4.13)

This was given as ( S / 2 ) cos 0 in reference [1], which applies only for the static case. Thus, including the D o p p l e r factor in equation (21) of reference [1] extends that work to account for the forward flight effect.

366

c . K . w . TAM ET AL 5. CONCLUSION

In this paper the far-field acoustic interference pattern o f a counter-rotation propeller is derived in propeller-fixed co-ordinates for the general case where the front and the rear rotor geometries and loadings need not be the same. The only constraint is that the rotors must have the same number of blades and the same rotational frequency. A simplified formula is given for the case of nearly identical rotors. Finally, a derivation is given in fluid-fixed co-ordinates showing the equivalence of the propeller- and fluid-fixed viewpoints. A sample calculation shows that the interference pattern is strongly affected by forward flight at cruise Mach numbers.

ACKNOWLEDGMENTS This work was supported by the Independent Research and Development Program of the Lockheed-Georgia C o m p a n y and Hamilton Standard.

REFERENCES 1. D. B. HANSON 1985 Journal of Aircraft 22, 609-617. Noise of counter-rotation propellers. 2. H. H. HUBBARD 1948 NACA TN 1654. Sound from dual-rotating and multiple single-rotating propellers. 3. P.J.W. BLOCK 1984 NASA TM 85790. Installation noise measurements of model SR and CR propellers. 4. J. E. FFOWCS WILLIAMSand D. L. HAWKINGS 1969 Philosophical Transactions Royal Society (London), Series A 264, 321-342. Sound generation by turbulence and surfaces in arbitrary motion. 5. D. B. HANSON 1980 American Institute of Aeronautics and Astronautics Journal 18, 1213-1220. Helicoidal surface theory for harmonic noise of propellers in the far field. 6. D. B. HANSON 1980 American Institute of Aeronautics and Astronautics Journal 18, 1313-1319. Influence of propeller design parameters on far field harmonic noise in forward flight. 7. D. B. HANSON 1983 American Institute of Aeronautics and Astronatiiics Journal 21,881-889. Compressible helicoidal surface theory for propeller aerodynamics and noise. 8. F. FARASSATand G. P. SUCCI 1980 Journal of Sound and Vibration 71,399-419. A review of propeller discrete frequency noise prediction technology with emphasis on two current methods for time domain calculations. 9. F. FARASSAT1981 American Institute of Aeronautics and Astronautics Journal 19, 1122-1130. Linear acoustic formulas for calculation of rotating blade noise. 10. C. K. W. TAM and M. SALIKUDDIN 1986 Journal of Fluid Mechanics 164, 127-154. Weakly nonlinear acoustic and shock wave theory of the noise of advanced high-speed turbopropellers. 11. D. B. HANSON 1988 Journal of Sound and Vibration. Shielding of prop-fan cabin noise by the fuselage boundary layer (to appear).