theoretical analysis of compressor...

Received 5 November 1968 14.7

Theoretical Analysis of Compressor Noise*

M. V. Lowso•p

Wyle Laboratories, Huntsville, Alabama 35807

A theory is presented for the discrete-frequency sound radiated by axial-flow fans and compressors. The theory is based on the noise radiation from the fluctuating forces on either a rotor or a stator stage due to interactions with upstream components. The model used is a cascade of point forces, one on each blade, radiating into free air. It appears that, provided that the correct phase relations are retained, this model may be expected to give accurate predictions of farfield noise even for long, hard-wall duct cases. Both over-all power and directionality curves for the sound radiation are presented, and methods are given for calculating the fluctuating forces acting from the wake geometry. Once wake geometry is defined, the theory enables one to perform calculations of the noise observed at any point. Preliminary agreement with experiment is demonstrated. Virtually all significant compressor design parameters can be included in the theory, and therefore it appears that the theory could be used in tradeoff studies to minimize noise at the preliminary design stage of an engine.

INTRODUCTION

The sound produced by axial flow fans and compressors has been increasing in importance as a source of community noise. The projected introduction of very high bypass-ratio fans, possibly operating at supersonic tip speeds, will lead to the sound radiation from this source dominating the observed sound field of an aircraft.

Much important work has now been accomplished on compressor noise, 1-12 both theoretically and experi-

* This paper is an extension of work presented at the 74th Meeting of the Acoustical Society of America, November 1967, Miami [-J. Acoust. Soc. Amer. 42, 1150 (A) (1967)-].

• Present address: Dep. Transport Technol., Loughborough Univ., England.

• J. M. Tyler and T. G. Sofrin, "Axial Flow Compressor Noise Studies," Trans. Soc. Auto. Eng. 70, 309-332 (1962). [-The deriva- tion of the spinning-mode equation (6.2.2) in the original preprint version contains errors, which were corrected in the final published version.]

•' S. L. Bragg and R. Bridge, "Noise from Turbojet Compres- sors," Roy. Aeron. Soc. 68, 1-10 (Aug. 1964).

a C. L. Morfey, "How to Reduce the Noise of Jet Engines," Engineering 198, 782-783 (Dec. 1964).

4 C. L. Morfey, "Rotating Pressure Patterns in Ducts: Their Generation and Transmission," J. Sound Vibration 1, 60-87 (1964).

5 I. J. Shaftand, "Sources of Noise in Axial Flow Fans," J. Sound Vibration 1, 302-322 (1964).

6 R. Hetherington, "Compressor Noise Generated by Fluctuat- ing Lift Resulting from Rotor-Stator Interaction," J.AIAA 1, 473-474 (1963).

7 j. L. Crigler and W. L. Copeland, "Experimental Noise Studies of Inlet-Guide-Vane-Rotor Interaction of a Single-Stage Axial-Flow Compressor," Natl. Aeron. Space Admin. Tech. Note No. TND-2962 (Sept. 1965).

mentally. Most of the theoretical work has been oriented towards a prediction of effects of the surrounding duct on the noise, and only Hetherington 6 and Slutsky TM appear to have attempted actual predictions of compressor-noise levels. Experimental work has generally been aimed at studies of rotor-stator interaction, and syste- matic acoustic data for compressors are sparse. In a previous paper, TM the author gave a review of the available information on compressor noise.

In the present paper, an attempt is made to provide a theoretical description of the noise radiated by a fan or compressor. The model used for the analysis is to regard the fan as an isolated rotor, or stator, in free space, on which fluctuating forces are acting. The effect of upstream or downstream blade rows is included only via the fluctuating forces that result from their presence,

8 M. J. T. Smith and M. E. House, "Internally Generated Noise from Gas Turbine Engines, Measurement and Prediction," J. Eng. Power 191, 177-190 (Apr. 1967).

9 B. T. Hulse and J. B. Large, "The Mechanisms of Noise Gen- eration in a Compressor Model," J. Eng. Power 191, 191-198 (Apr. 1967).

•0 C. L. Morfey and H. Dawson, "Axial Compressor Noise, Some Results from Aero-Engine Research," paper presented at the 11th Gas Turbine Conf., Amer. Soc. Mech. Eng., Zurich, Switzerland (Mar. 1966).

n M. V. Lowson, "Reduction of Compressor Nosie Radiation," J. Acoust. Soc. Amer. 43, 37-50 (1968).

•' S. Slutsky, "Discrete Noise Generation and Propagation by a Fan Engine," paper presented at AFOSR-UTIAS Symp. Aerody- nam. Noise, Toronto (May 1968).

The Journal of the Acoustical Society of America 371

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00:11:32

M. V. LOWSON

STATOR

Fro. 1. Basic wake geometry.

and methods are given for calculating these fluctuating forces.

Several other sources of noise are feasible. Ffowcs

Williams and Hawkings la have discussed the possibility of the acoustic stresses existing in the flow around the blade, causing significant sound radiation. In an ex- tended report on the subject of this paper, 14 the author gives an estimate for a model of mass source (siren) radiation by the compressor. Further possibilities include the occurrence of fluctuating-thickness noise terms and the interaction between various components, such as the quadrupole and dipole fields, and the blade and its surrounding turbulence. Calculations of some of these latter effects will require fairly sophisticated analysis. Fortunately, the fluctuating-force model is found to be sufficient to account for the observed acoustic field in

the cases studied in this paper, and it therefore seems reasonable to suppose that the other sources listed above are secondary. Certainly the siren terms are found to be negligible. 14 Furthermore, the basic phase effects discussed in the paper apply to any form of acoustic source function. Distinctions between individual mono-

pole, dipole, and quadrupole fields on the blades are overshadowed by the effects of basic phase parameters, such as number of blades and rotational speed.

I. BASIC MECHANISMS

Figure 1 shows, diagrammatically, the aerodynamic field existing in a compressor. Wakes stream from an initial stator and are intercepted by the rotor. As shown by the velocity diagram, the basic effect of the velocity defect in the stator wake is to cause a fluctuating down- wash at the rotor disk, which will be reflected in fluc-

•a j. E. Ffowcs Williams and D. L. Hawkings, "Theory Relating to the Noise of Rotating Machinery," Aeron. Res. Council, London, Rep. ARC29, 821 (Jan. 1968).

•4 M. V. Lowson, "Theoretical Studies of Compressor Noise," Nat. Aeron. Space Admin. Contractors Rep. No. CR-1287 (Mar. 1969).

tuating forces on the rotor blades. Exactly equivalent effects occur at the stator, which experiences a fluctuating force on its vanes owing to the rotor wake. Because the number of rotor and stator blades are unequal, the forces on the rotor or stator blades vary in phase from blade to blade. These cascade phase effects are of extreme importance in determining the efficiency of the final radiation.

A paper by Embleton and Thiessen 15 calculated the noise from a series of stationary monopole sources with these cascade effects. They found that the number of individual sources in the array was important only when there were less than about eight. For greater numbers of sources, only the phase parameter u, which describes the effective rotational speed of the phased array, was important. This is a significant result that may be expected to apply to the present case where there are more than about eight blades present. Furthermore, it suggests that detailed representation of the field on the blade is unnecessary, and, in the present paper, the forces on each blade are reduced to point forces acting at some effective radius. This approximation would, in any case, be expected to be accurate if the spatial extent of the fluctuating pressures were much less than a wavelength of the sound generated, and this is generally true for the chordwise distribution. However, Embleton's and Thiessen's result also suggests that the approximation is probably also adequate for the spanwise distribution, unless there is a substantial change in phase across the span.

Several important effects related to the frequency characteristics of the compressor radiation may also be deduced from a study of Fig. 1. Consider first of all the rear-most stator row. Each blade is stationary and undergoes a force fluctuation due to the velocity profile of the rotor wakes. The time variation of these fluctua-

tions is governed by the rotor speed. Thus, the frequency of the noise radiated by the stator is directly related to the rotor frequency. If the rotor has B blades and rotates at angular velocity ig, then the fundamental frequency of the stator radiation is Big.

The wake-velocity field behind the rotor can be analyzed into spatial Fourier components. Because of the rotation, each spatial harmonic of the rotor wake is transformed into a single temporal harmonic of the stator radiation. Now, immediately behind the rotor, the wake-velocity profile is very sharp and thus contains all spatial frequencies. As the wake expands downstream, however, it becomes smoother and less intense, so that at large rotor-spator separations only the first spatial harmonic of the wake may be significant. Thus, it may be predicted that the stator radiation for small rotor-stator separations will contain substantial noise at all multiples of the blade-passage frequency Big, and

•5 T. W. F. Embleton and G. J. Thiessen, "Efficiency of Circular Sources and Circular Arrays of Point Sources with Linear Phase Variation," J. Acoust. Soc. Amer. 34, 788-795 (1962).

372 Volume 47 Number 1 (Part 2) 1970


00:11:32

THEORETICAL ANALYSIS OF COMPRESSOR NOISE

that the effect of increasing rotor-stator separation will be to reduce preferentially the higher harmonics of the sound radiated by the stator.

The effects at the rotor, however, are rather different. The rotor field passes through what is essentially a stationary velocity field owing to the wakes from the first stator (see Fig. 1). The frequency of the fluctuating forces on, and thus the acoustic radiation by, the rotor is governed by the rotor speed. Thus, the angular velocity of the rotor governs temporal frequencies at both the rotor and the stator. Clearly, the stator wake can still be analyzed into spatial Fourier components, and their relative magnitude depends on stator-rotor separation in the same way as discussed above; that is, at large separations, only the first spatial harmonic of the stator wake is significant. Furthermore, successive spatial harmonics of the stator wake give rise to successive loading harmonics on the rotor. However, in contrast to the stator case, each loading harmonic on the rotor gives rise to more than one sound harmonic in the radiation field. This is due to the motion

of the rotor blades. As is well known, relative motion between source and observer gives rise to Doppler frequency shifts, with the observed frequency rising as the source approaches the observer and reducing as the source recedes. Thus, the rotation of a particular frequency source moving with the rotor causes a periodic variation of the frequency observed at a fixed point. The effects are very similar to those of FM radio sig- nals. It is found that the frequency modulation causes any single frequency input to be observed as a series of frequencies, each displaced from the input frequency by some multiple of the modulation frequency. Thus, fluctuating forces at one harmonic cause radiation in all harmonics. For this reason, increase of stator- rotor separation may be expected to reduce the noise basically in all the sound harmonics radiated by the rotor. This may be contrasted with the predominant reduction in the higher harmonics predicted above for the equivalent stator-radiation effect.

It is shown later in this paper that each loading harmonic (X) gives rise to a different model of radiation from the rotor, in a manner very similar to that described in the work of Tyler and Sofrin. • Each mode has different acoustic-radiation characteristics, so that the observed radiation level from the rotor is dependent on both the acoustic radiation efficiency of each mode and the relative magnitude of its contribution from the various loading harmonics X. The over-all radiation pattern is thus given by the sum of all the modes.

II. SOUND RADIATED BY THE ROTOR

We calculate first the sound radiated by the fluctuating forces on the rotor. The author has previously derived •6 a result for the sound pressure radiated by a

•0 M. V. Lowson, "The Sound Field for Singularities in Motion," Proc. Roy. Soc. (London) A286, 559-572 (1965).

point fluctuating force in arbitrary motion, which can be written

[ (xi--yi) 0 ( Fi )] • o P= (1--Mr)aor at 4•r(•--M,) (•) Note that Eq. 1 must be evaluated at retarded time r, but it is desired to calculate sound harmonics in the ob-

server's time t, where r--t--r/ao. This equation can be used to find an expression for

the sound from a point force in arbitrary harmonic motion. Suppose that the force and motion periodically repeat some otherwise arbitrary motion. Then,!i..defining the complex magnitude of the nth sound harmonic in the usual way gives

w/[ x•--yg O( Fg )] ß

cn=anq-ibn=- (1 M,)aor •X4•r(•--M,) Xexp(inwtd)t. (2)

The integral is over any period 2r/w. Changing variables back to retarded time r gives,

using dt = (1 -- Mr)dr,

•r,o aor Jt\4•rr(i-Mr) X exp[inw (rq-r/ao) ]dr,

and, integrating by parts,

4•r•'r ao 1--MrL r r •'

X exp[inw (r q-r/ao) ]dr,

where Fr=Fi(xi--yi)/r, the component of force in the direction of the observer. The second term is important only in the acoustic nearfield, because of the additional factors r in the denominator, so that the result for the farfield harmonics becomes simply

exp•inw(rq-r/ao)•dr (3) 4•r2r Jo \ ao /

To use this result, the fluctuating force field must be defined. Set up Cartesian coordinates as shown in Fig. 2 with x along the compressor axis and define fluctuating thrust and drag (torque) forces T and D, as shown. Then, Fig. 2 shows

F•=--T, --I) sin0, D cos0,

xi--yg=x, y--R cos0, --R sin0, (4)

from which

Fr= --xT/r-- (yD/r) sin0, (s)

The Journal of the Acoustical Society of America •7•


00:11:32

M. V. LOWSON

z

D T Y

x

Observer

(a)

z

2:/VN•, o o o o

•, •ot•, S___.qg!5 e•

x o Observer

(b)

FiG. 2. Coordinate systems for compressor-noise analysis. (a) Coordinates for rotor radiation. (b) Coordinates for stator radiation.

and

r= I xg--y, [ • rs-- (yR/r•) cos0, (6)

where rs is the distance from observer to hub. A geometric farfield approximation has been applied in Eq. 6, and terms of order (R/r•) 2 and higher have been ig- nored. Defining the fluctuating thrust and drag terms by a complex Fourier series

= exp(--iX•t) (7)

and substituting the other results above in Eq. 3 gives the expression for the harmonics of the farfield sound radiation from the rotor as

Cn •• in• •O2'r •(xTx yDx ) -}-• sin0 4•-•0/- X:--oo\ /'2 /'2

Xexp[i (n-- X)O--ina cosO']dO, (8)

where a=•Ry/aor•=My/r•, where M=•R/ao is the

rotational Mach number of the point of action of the force.

The integrals in Eq. 8 can be identified as Bessel functions, and, using the expressions •7

o exp[i(nO-z cosO)']dO=2•ri-nJn(z), (9)

•0 2 • • exp[i(nO--z cos0)• sinOdO=--2•i-•-J•(z), (10)

Eq. 8 can be evaluated directly to give the sound radiation from a single rotor blade as

Cn • ' , • (__i) n-X 2waor•

xTx n--X Dx \ /nMy\ X,,-• n M')Jn-x•-•-• ) ' (11) If B rotor blades are present, harmonics that are not integral multiples of the number of blades will cancel; if V stator vanes are present, we may also identify X-kV, where k is integral. Thus, the final result for the complex magnitude of the mth harmonic of the noise radiated by the fluctuating forces on the rotor is

imB•

era:-- • (--i) mB--kV 2 •r aor •

xTk mB-k V \ /mBMy\ •'D•)JmB-•v• rs )' (12) Xx'• mBM Before discussing this equation, the result for the stator radiation is derived.

III. NOISE RADIATION FROM THE STATOR

The sound radiated by a point fluctuating force on a single stationary vane is given by a well known result---

(xi--yi) OFi 1 p= (13) 4•raor • • 'j

--which can also be derived directly from Eq. 1 by putting Mr=0. The key problem in the stator case is to define the retarded phase effects from vane to vane. Now, the time variation of the fluctuating forces on each vane is directly related to the spatial variation of the velocity in the rotor wakes:

Os=O•+•t, (t4)

where Os and 0R are fixed in the rotor and stator, respectively. At the vth stator vane, Os=•q-2rvV, where

•7 N. W. McLachlan, Bessel Functions for t•ngineers (Oxford University Press, London, 1955).



00:11:32


½ is the angular position of zeroth stator vane so that written as

O•=•+2rv/V-•t (15)

at the vth vane. Thus, because of the direct relation between wake velocity and fluctuating force, the wake- velocity profile

w = • A x exp(--iX0•) (16)

is transformed into a fluctuating thrust and drag variation'

exp 2•rv '• x=-•XDx/ V--•2t/'

(17)

The relation between the force coefficients Tx and Dx and the wake coefficients A x is discussed in Sec. VI.

The over-all sound radiation from the stator is found

by the sum of the individual sound fields for each vane given by Eq. 3, evaluated at appropriate retarded time r-t--r/ao. Using the result of Eq. 5 together with the expressions derived above gives

v-• +• -

p=z z v=0 X---o• 4wt/0fl 2

•[xTx +yDx sin (rk-nt-2•rv/ V) ']

exp iXr 2•rv a(t--r•l (18)

V--1

• exp.• v=0

• cosu •+ sin V k•._oO

V =-- • exp--i(kV--X)

4i

(k V-x+ V-x-

Substituting into Eq. 21, and using Eq. 17 also---

2/$

(z) J_,• (z) = (- 1)'•J,• (z)

--yields, after some algebra, the final result for the complex magnitude of the mth sound harmonic radiated by the stator forces owing to rotor-stator interaction. This is

+•o --imB V•2[xTm (mB--k V).Dm 1 ½m • E •_--o• 2•raor• c. r• mBM

(m7Y) XimB--•VJmB_•V . (22)

The geometric farfield approximation for r, in this case, can be written

r,•r•-- (yR/r•) cos(rk-k2•rv/V), (19)

and the resulting exponential cosine term can be ex- panded in Bessel functions 17 by

expiz cos0 = • "• z J• (z) cosn0, (20) n•---oo

which gives

p= • • 4•r-•or•2LxTx+yDx sin +

XexpIiX½ t •2r•l - 2•iX•) +oo /XMy\ / 2•rv\

x /z------oo \ F1 / (21)

Now, in summations of Fourier series, orthogonality relations apply, and two convenient expressions can be

IV. COMMENTS ON THE RESULTS

Equations 12 and 22 are observed to be remarkably similar. The Bessel-function term occurs to the same order and argument, and the multiplication terms out- side the Bessel function are quite closely the same. In other words, the principal features of the sound radiation from the fluctuating forces in a compressor are essentially the same for both stator-rotor and rotor- stator interactions. This result is somewhat unexpected and of clear significance.

In fact, there are some detailed variations between the two results. Equation 22 for the rotor-stator interaction contains coefficients in m as multipliers, whereas Eq. 12 for the stator-rotor case has coefficients in k. In the stator case, the radiated frequencies are determined by the spatial-harmonic content of the rotor wake, as discussed in Sec. I. Equation 22 shows that each sound harmonic from the stator is composed of the total of all k values in the summation, with the same coefficients Tin, Din. The rotor-radiation case is rather different. Here, increasing stator-rotor separation affects the coefficients Tk, Dk, and, in the case of extreme separation, only one value of T•, D• may be significant. How- ever, this radiates in all harmonics with an efficiency depending on the value of the Bessel function. Thus, as



00:11:32

M. V. LOWSON

discussed in Sec. I, it appears that increasing separation for the rotor-stator interaction tends primarily to reduce the higher harmonics of the noise, whereas increasing separation for the stator-rotor case reduces all harmonics, although not necessarily equally because of the effects of the Bessel function. Each value of k may be considered to give rise to a single "mode" of radiation in the same way as discussed by Tyler and Sofrin. t

Equations 12 and 22 give the results in terms of complex force coefficients T• and Tk. The results can be rewritten in terms of ordinary Fourier coefficients. Equation 12 gives the result

• (__i)n--X--1 [Jn-x+ (-- 1)XJn+x• 4•raor• x=0 t rt

inxbxT + "[Jn_x--(--1)XJn+x• ....

OZXD

M

ibxz)

M (23)

where the argument of all the Bessel functions is nMy/r•; axe, bxT, axz), and bxz) are ordinary Fourier coefficients of the fluctuating thrust and drag directions related to the complex coefficients Tx and Dx, respectively.

For the great majority of practical cases, the Jn-x terms are very much greater than the Jn+x terms, provided that neither n or X is small. This may be attributed to the fact that the n-X terms correspond to a rapidly rotating source pattern, whereas the nnLX terms correspond to a slowly moving pattern. This latter is usually an extremely inefficient noise radiator. This also corresponds to the finding of Embleton and Thiessen •5 that the number of sources present was significant only if it were less than about eight. Ignoring the nnLX terms, Eq. 23 can be written as

mB2• Cm:------ • (--i) mB--kV--1

4 •r aor •

X .•(axzq-ibxz)--• mB -- k V ( axz) +ibxz) ) 1 mBM

mBMy•. XJmB-k (24) V\ Y1 '/

It is also of interest to consider the special case when only steady forces exist on the rotor. This corresponds to putting X-0 in Eq. 23, with the result

(--i)n-l½½Q(xTo Do\ /nMy\ Cn= •a•rl x • •)Jn[-•--• ). (25)

This is the result obtained by Gutin 18 for the case of propeller-noise radiation (see also Ref. 16). Reduction to this classical solution provides a useful test of the result. The same result for the rotor-noise radiation also

was derived by other methods, 10and it agrees with that given in Eq. 11.3.20 of Morse and Ingards' recent book. 'ø A report by Arnold et al. '1 gives an analysis of a similar case, and here frequencies other than harmonics of the rotational speed were allowed to occur. As expected from the discussion of Sec. I, they found that single-frequency input could give multiple-frequency output. This is the result of the modulated Doppler-frequency shift due to rotor-source motion.

V. POWER RADIATED

One of the single most important parameters in the description of an acoustic source is the over-all acoustic power radiated. Knowledge of over-all power enables pressure at any point to be estimated by using a spherical spreading law. It is thus of considerable interest to be able to predict the acoustic-power radiation in the present case. If the sound pressure at any field point in the nth harmonic is given by p•, then the mean power W• in that harmonic is given by a space-time integral around the source:

W n =- --dA dt T poao

=- r• 2 sing/rig/de. (26) 2 o poao

The factor of « is the result of the time integration and corresponds to the rms value of pressure being l/x/2 times the amplitude for sinusoidal fluctuations.

In Eq. 26, ½ is the angle around the source (½=0 on the y axis), and ,I• is an angle over the top of the source, with ,I•=0 along the compressor (x) axis. The ½ integration may be performed directly. It will be observed that in both Eqs. 12 and 22 there is no effect of ½ on the amplitude of the sound radiation, although it does enter the phase terms. Since ½ is only a phase variable, the mean-square value of the fluctuating pressure field for a single mode is the same at any point, and the ½ integration is trivial. Thus,

71'/'12 fO r mn---- I Cn 1 2 sing/rig/. (27) poao

ts L. Gutin, "On the Sound Field of a Rotating Propeller," Phys. Z. Sowjetunion A1, 57-71 (1936). EEnglish transl.: Nat. Aeron. Space Admin. Tech. Mere. No. TMl195 (Oct. 1948)•.

t9 M. V. Lowson and J. B. Oilerhead, "Studies of Helicopter Rotor Noise," USAAVLABS Tech. Rep. 68-60 (1968).

•.0p. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw-Hill Book Co., New York, 1968).

•'• L. Arnold, F. Lane, and S. Slutsky, "Propeller Singing Analy- sis," Report 221, Gen. Appl. Sci. Lab. Rep. 221 (1961); AD 257 424.



00:11:32


Direct analytical evaluation of the over-all power, using Eqs. 12 and 22, does not give results in a convenient form, although the necessary integration can be performed on a computer. However, a fairly simple analytic result can be obtained after some approximation. We ignore the nq-X terms compared to the n--X terms, as discussed above, and calculate the over-all power radiated by a single term in the summations of Eqs. 12 and 22, that is, for a single mode. For the rotor-radiation case, this corresponds to the power radiated by a single spatial harmonic k, and it corresponds closely to the actual power radiated at large stator-rotor separations. For the stator radiation case, the calculation is simply an analytic convenience. The analysis below is given in terms of the rotor parameters. Exactly equivalent results also apply to the stator. The power radiated by a single mode is therefore, using Eq. 24,

W•x = -- n•cx• • cosyq --- 1 16•ra0 øp0 M a _1

XJn_x2(nM sin•) sin•d•, (28)

where CXT2=axT9'q-bxT 9', cxD•=axz>•q-bxz) •. The cross terms between the thrust and drag vanish identically on integration. To evaluate the integral, we note its symmetry in xI, and use the results la.a•

t Jn•(Z sin0) sinOdO do

(-- 1)

r=0 r!(2n +r)! (2n q- 2r+ 1)'

I J,?(z sin0) cos•0 sinOdO .! 0

•=0 r! (2n +r) ! (2n-+- 2r-+- 1) (2n4- 2r-+-3)'

(29)

where r is simply a summation parameter. Thus, the expression

1 (-- 1) '(nM)2. +2r z

8•-aoooR • • r!(2tz+r)!(2tz+2r+l)

[ (Mn)•cxT2 Fl•2cxz>• 1 X k2tz-k2r-+-3 (30)

gives an approximation to the acoustic power radiated by a specific rotor-loading harmonic. As before,

•.2 T. W. F. Embleton, "Relation of Mechanical Power of a Pro- peller to Radiated Power of the Resulting Acoustic Sources," J. Acoust. Soc. Amer. 34, 862-86,3 (1962).

this approximation is most accurate when ]n--X I <<In+X I, i.e., when 2,>>0, n>>0, which usually is the case for a compressor.

VI. DEFINITION OF THE FLUCTUATING FORCES

Before the sound radiation can be calculated, it is necessary to define the forces at the rotor- and stator- blade rows. Fortunately, the prediction of the fluctuating-force levels on the blades has been the subject of studies by Kemp and Sears? 3.• They showed that two effects should be considered: the potential flow interac- actions between the blades and the effect of the viscous

wakes from an upstream row impinging on a downstream row. Solution of the potential interaction problem is not straightforward. Account must be taken of the interference of the fluctuating trailing vortex sheet from the first blade with the downstream row, and also of the mutual effects of the bound vorticity on each blade. The viscous-wake problem, illustrated in Fig. 1, is more straightforward, as there is only a single effect to consider: the passage of the fluctuating-velocity field of the wake through the downstream row.

Now, the velocity field of a vortex is proportional to the inverse of the distance, whereas the velocity decrement in wake is approximately proportional to the inverse square root of the distance. Thus, at large blade separations, only the viscous-interference effects may be expected to be of significance and, conversely, at small separations, only the potential interactions are probably significant. Kemp and Sears 2• found that, in a typical case, the viscous-interference effects were of the same order as the potential interactions for a rotor- stator separation equal to 0.! of the rotor chord. To reduce noise and vibration, most modern fans and compressors operate at a rotor-stator separation of more than 1.0 chord lengths. Thus, it seems unlikely that the potential flow interactions play any significant role in determining the fluctuating forces on the rotor and stator. Therefore, attention may be confined solely to the viscous interactions. An added advantage of this is that this interaction is far easier to analyze. Some additional comments on this point are given in the published discussion to Ref. 9.

In their analysis, Kemp and Sears • assumed that each blade acts independently, so that mutual interference effects in the cascade are eliminated. They also assumed that the wake-velocity decrement is small, so that second-order perturbation terms can be removed. Both these assumptions are probably reasonable for the relatively large axial and circumferential blade spacings typical of current compressors.

The effect of these varying wake velocities convected relative to the blades is to cause a time-varying down-

2a N.H. Kemp and W. R. Sears, "Aerodynamic Interference between Moving Blade Rows," J. Aeron. Sci. 20, 585-598 (1953).

,.4 N.H. Kemp and W. R. Sears, "The Unsteady Forces due to Viscous Wakes in Turbomachines," J. Aeron. Sci. 22, 478-483 (1955).



00:11:32

M. V. LOWSON

-2

/'• • J

•1 • I I j i I

J' !

1/,i 0 2 14 16 Modal Order •11; I j8 t .= I•-xl -

I t

!1i , , ' i

i

-8

-10

where the lift-response function S is given by

S(•)={irr[K0(i•)+K•(i•)-]} -•, (r=wc/2U•,

L=Re[•rcpoU• •, --ax sin•S(•) expiXigt•. (31)

cos ((r-- •r/4) q-i sin ((r--•r/4) S = . (32)

(2•r•) •

where K is a modified Bessel function of the second

kind. 17 This particular form for S was given by Kemp, 26 and it gives the phase relative to the half-chord point. Thus, combining these equations above, the fluctuating lift per unit span at the rotor is given by

-12

-14

-16

-18

-20

FzG. 3. Contribution of typical compressor-force term to power.

wash at the blades, as shown in Fig. 1. Considering first the stator-rotor interaction, suppose that the wake- velocity profiles vary around the circumference in a way defined by

The sine terms here have been neglected, since the wake may be assumed to be symmetrical. This approximation 'was discussed by Kemp and Sears. 24 Also, since the rotor is moving at angular velocity 12, we can put 0 =lgt.

From Fig. 1, if the velocity at a point in the stator wake is w, then the upwash at the rotor is v=w sinfl. Sears 25 has analyzed the case of a wing entering a sinusoidal gust. He showed that, if a thin airfoil experiences a nonsteady upwash of the form

v=Re•v' expiw(t--x/U•)•,

then the lift per unit span acting at the quarter-chord point is given by

L = Re•rcpU•v'S ((r) expiwt•,

•s W. R. Sears, "Some Aspects of Non-Stationary Airfoil Theory and Its Practical Application," J. Aeron. Sci. 8, 104-188 (1941).

As discussed by Kemp and Sears, •'•,•'4 this equation can be applied to the compressor blading, provided that there is reasonable circumferential separation. In fact, for moderate nondimensional frequencies (•> •r), a sim- plified expression for S can be found from the standard asymptotic forms for Bessel functionsJ 4 It is

The approximate form of Eq. 32 is probably sufficiently accurate for nearly all practical applications. Equation 32 may be used in Eq. 31 to give an expression for the fluctuating lift on the blade as

I • ax sinfi )1 6 8 •0 •2 74 •6 •8 20 L=Re --•rcposUz • ---- expi(Xl2t+o---•-/4 . (33) h:l (271'0') «

Nondimensional Frequency k R = n M

Thus, if the spatial Fourier coefficients ax of the wake- velocity profile can be established, the fluctuating lift on the blade can be calculated. Note that the lift is proportional to the inverse square root of •r, so that the blade operates to some extent as a damper of the higher- frequency input. Equation 33 also shows how the phase of L is basically dependent on the phase of the fluc-

• tuating velocity on the blade. The additional phase W=ao-k • ax cosX0. effects due to the imaginary exponential are dependent

x=• on blade chord and frequency but do not vary from blade to blade. This feature has already been assumed in the application of the fluctuating-force phasing rela- tionships in Sec. III. To complete the specification of the noise radiation, we need only to define the coefficients ax. This is the most difficult task of the study.

Kemp and Sears •'4 gave numerical estimates for the wake parameters in their paper, and these can be used if desired. However, review of the characteristics of wakes in practical compressors suggests that these estimates may easily be grossly misleading. Kemp and Sears •'4 applied the results of Silverstein et al? on the wakes behind an isolated airfoil. Indeed, these appear to be the only comprehensive airfoil-wake data avail-

•.0 N.H. Kemp, "On the Lift and Circulation of Airfoils in Some Unsteady Flow Problems," J. Aeron. Sci. 19, 713 (1952).

•.7 A. Silverstein, S. Katzoff, and W. K. Bullivant, "Down Wash and Wake Behind Plain and Flapped Aerofoils," NASA Rep. 651 (1939).



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-10


I •. • ,. :.&,,- "¾,,

' / ß

I I i -m=l k:l I j I / m:l k:2 I [ 'm=2 k =1 --

I It/ .... m=2 k=2 I I B: 16 I • M= 0.5

, / • I

J , ,

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 Number of Stator Vanes V

Fro. 4. Acoustic power radiation by a compressor variation with stator-vane number.

able. However, it seems unlikely that the isolated airfoil, particularly that investigated by Silverstein et al., is representative of the conditions inside a compressor, which is complicated by at least four factors: axial pressure gradient, radial pressure gradient, varying swirl angle, and compressibility effects. These phenomena, discussed in more detail in Ref. 14, may be expected to cause significant departures from any simple wake model, and a detailed account of their effects will require much additional study.

VII. RESULTS AND DISCUSSION

There are two features of the sound output that are important: the directionality and the acoustic power output, both of which are predicted by the present theories. Acoustic power is the principal gross feature of the sound and is affected by the gross features of engine design, such as mass flow and revolution rate. However, directionality can also play a significant r61e, especially since it is more readily modified by detail design and because it is generally the sound radiated radially outwards that has the most significant effect on community noise. This fact is often overlooked in discussions of compressor noise. These two features are discussed separately below.

The two key parameters are the modal order •=(n--X)=mB--kV and the frequency parameter nM(=mBM). Essentially, the frequency parameter in- cludes the effects of rotational speed and blade number whereas the modal order is governed by the difference in blade numbers between rotor and stator. Note that

nM--kR, where k is the wavenumber and R a typical radius and in this form may be more familiar to acous- ticians. The form nM is used here because of its more

direct relevance to compressor parameters. Equation 30 gave the over-all acoustic power radiated

by the compressor as a result of the stator-rotor interaction. It may be noted that the drag term in this equation is identical in form to that associated with the mass

source terms discussed by Embleton and Thiessen. •5 The over-all power result is presented in Fig. 3. This is the value of the summation in Equation 30, with CxD=CxT= 1, which may thus be used for direct calculations of power level by multiplying by the factor out- side the summation in Eq. 30, once the magnitude of the fluctuating force coefficients is known. Although thrust and drag (torque force) have been assumed equal, it is found that the result is dominated by the thrust terms, so that minor variations in the thrust/drag ratio are not expected to be significant. Values of power could not be calculated for nM> 20 in the present work because of computational difficulties. Additional acoustic power curves can be found in Ref. 14.

In Fig. 3, it can be seen that, for any given modal order u, sound radiation is very inefficient for low frequencies but becomes significant for values of the frequency parameter nM roughly greater than the modal order u. The efficient and inefficient radiation conditions in Fig. 3 correspond, respectively, to supersonic and subsonic phase velocities for the fluctuating forces. It can also be seen that the radiation efficiencies of all

modes are equivalent, to within 2 dB, if the frequency is sufficiently high. The asymptotic value of the series

The Journal of the Acoustical Society of America


00:11:32

2

-10

-12

M. V. LOWSON

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

Number of Stator Vanes V

Fro. 5. Effect of rotational mach number on acoustic power.

in Eq. 30 is given quite accurately by nM/4; this curve is also shown in Fig. 3. This asymptotic curve may be used in estimates at large values of nM.

To evaluate the significance of the results further, it is helpful to consider a particular case. For the purposes of discussion, suppose that a 16-blade rotor (B-16) operating at M =0.5 is chosen. Figure 4 shows the result for the effect on the first- and second-harmonic noise (m-l,2) of the first- and second-mode radiation (k--1,2) due to the rotor force terms. As pointed out previously (Sec. I), the effect of large stator-rotor separation is to emphasize the contribution of the k= 1 term, so that the case k= 1, m= 1 is of principal interest. Recalling that •=mB--kV, Fig. 4 can be seen to be consistent with Fig. 3, but the effect of variation of vane number V is now made explicit. For the first-mode, first-harmonic case (k =m = 1), the curves are symmetrical about V= 16, the u =0 case. For k=m= 1, Fig. 4 also shows how, for small numbers of stator blades, efficiency is very low. At V = 8, we start to move into an efficient radiation region, reaching a maximum at V = 10; then there is a slight lowering of efficiency as stator blade numbers are increased up to 16. Past this point, efficiency again increases up to a maximum of about 2.6 dB additional at V = 22. It may also be noted that V-10, 22 corresponds to the most unfavorable sideline directivity pattern as well as the highest power, as is shown below. For V-10, the peak would be in front of the compressor and for V= 22 behind. For further increases in blade number, efficiency then drops off rapidly. The effect of second-harmonic loading on


first-harmonic noise (k-2,m= 1) may also be seen in Fig. 4. Substantially the same effects occur, except that the range of efficient radiation is for 4< V<12. Like- wise, the effects of first-harmonic loading on second- harmonic noise (k-1,m-2) is seen to be greatest for 16< V< 48 and for second-harmonic loading on second- harmonic noise (k= 2,m= 2)8< V< 24. Now, the actual fluctuating loadings occurring for k = 2 and higher can be reduced to a minimum by choice of large rotor-stator separations. Thus, it can be concluded that increase in rotor-stator separation is particularly effective at reducing first-harmonic noise when there are fewer sta- tors than rotors. A second interesting conclusion is that choice of many more starors than rotors tends to increase second-harmonic noise. These general conclusions should apply to any compressor.

Clearly, the optimum way to utilize the results shown in Fig. 4 is to choose the stator vane number such that radiation is in the inefficient region of the curves. Un- fortunately, this is difficult on practical compressors. Apart from the effects of multimode input and higher harmonics, the high idling Mach number of modern compressors is very restrictive. Figure 5 shows the effect of rotational Mach number. It can be seen that the

region of efficient radiation spreads out considerably and reaches down even to zero vane numbers for the

M=i case. (Note that the V-0 cases are underesti- mated by 6 dB in Fig. 5 because of the neglect of the n-kX terms, as discussed in Sec. IV). Thus, excessive numbers of stator vanes are necessary to achieve low-


00:11:32


Fro. 6. Directivity patterns for fluctuating force terms in stator-rotor interactions. Up is forward for X <n.

nM= 4 nM= 8 nM= 16 nM= 32

!..1 16

radiation efficiency, unless the engine has been specifi- cally designed to run at low rotational Mach number.

A further inference from Fig. 4 is that at supersonic speeds the steady loadings existing on the rotor radiate particularly efficiently. These correspond to the V-0 case. Because the steady loadings are so much greater in magnitude than the fluctuating loading, their direct radiation at the blade-passage frequency swamps any contribution due to the fluctuating forces. Thus, the niceties of choice of blade/vane number, rotor- stator, separation, and so on, may be expected to show little over-all effect on a supersonic rotor.

The basic similarities between the present results and those of Tyler and Sofrin • for acoustic propagation in an open circular duct are apparent. Tyler and Sofrin showed how the radiation of any given mode down a circular duct was dependent on its frequency. Above a critical cutoff frequency, the modes propagated directly with unit efficiency, but below their cutoff frequency the modes decayed exponentially within the duct. In fact, the theoretical cutoff frequency corresponds closely to the changeover between efficient and inefficient radiation for the present case. Morfey 4 has also shown the equivalence of the propagation and radiation cutoff for a duct. Furthermore, it is shown below how the broad t•eatures of the directionality of the source-radiation

terms are similar to those of the duct radiation given by Tyler and Sofrin. 1

The directivity pattern of compressor-noise radiation is important because it is generally the sound radiated radially outwards from the compressor that causes the principal community-noise problem. To evaluate the directionality effects, a matrix of directivity patterns has been prepared (Fig. 6), giving the directivity factor in decibels as a function of direction from the compressor hub. The dotted lines on the Figures give lines of equal sideline-noise level. Because the sound radiated at small

angles to the axis travels farther before reaching the ground than that traveling sideways, it undergoes additional inverse-square loss due to spherical spreading. The dotted lines are equal sideline-noise contours calculated assuming this effect. The sound level relative to these contours generally is the most significant parameter from the community-noise point of view.

The curves in Fig. 6 for the sound radiation due to point forces are not symmetrical about the compressor disk. This is because for small values of X, the thrust and drag terms cancel in the forward quadrants but are additive in the rear quadrant, so that more sound is heard behind the compressor disk than in front. This effect is well known in propeller-noise theory. •9 How- ever, as can be seen from Eq. 22, the situation is re-



00:11:32

M. V. LOWSON

0 ø 10 ø 20 ø

Basic

Guide Modal

Vanes Order None 53

31 9 • o 62 9

30 ø

40 ø

Acoustic Power re: No Guide

Vane Case

0

12.3

4.3

11.0

B = 53

M. !. = 0.346

50 ø

60 ø F•c. 7. Measured «-oct. (at blade-passage fre-

quency) radiation patterns for various rotor-Guide- Vane configurations. 7

70 ø

Airflow -•l 70 80 90 100

Sound Pressure Level, dB (re: 0.0002 dynesfcm 2)

110 90 ø

versed when X>n; i.e., kV>mB. For this case, more sound is radiated forward out of the compressor inlet than rearwards. The effect of the X> n case is given quite accurately by simply inverting the directionality patterns in Fig. 6. The effect could have practical application, since it gives a method by which the designer can choose the more intense sound to radiate forwards or

rearwards into an available sound-absorbing device in the duct.

It can be seen that the sound radiation is predomi- nantly sideways for small values of the frequency parameter nM and large values of the modal order whereas for large nM and small t• the radiation is greater along the axis. Figure 6 may be interpreted in terms of several parameters. Increase in tip Mach number corresponds to a left-to-right movement in the directivity matrix. Also n=mB, where B is the number of rotor blades and m is the harmonic number. Succes-

sive harmonics therefore occupy positions moving from left to right in the matrix. Assuming that there are more rotor blades than stator vanes, increase in stator number corresponds to movement upwards in the matrix. On the other hand, rotor-blade number effects both axes. Increase in rotor-blade number implies a movement downwards and to the right in the matrices.

The various directivity patterns may also be related to the over-all power radiation for the same case. It

will be recalled that efficient radiation occurred roughly for nM>u. Thus, the lobed patterns towards the top right-hand corner of Fig. 6 correspond to an e•cient radiation, whereas the sideways radiation patterns towards the bottom left-hand corner correspond to inefficient radiation conditions. •hus, the least efficient radiation cases from the acoustic power point of view have the worst directivity pattern from the corrrr. unity- noise point of view. Fortunately, the effects of the de- creased efficiency for nM<u generally overcome any directivity effects. However, there are other significant features. As can be seen in Fig. 3 or Fig. 4, there is a maximum in the radiation e•ciency close to the condition u=nM. Figure 6 shows how this case corresponds to strong sideline radiation. Thus, it is desirable from the point of both directivity and power to design compressors away from the u=nM point. Con- versely, Figs. 3 and 4 show how it is desirable to go to low-order modes (u--• 0) to reduce sound power. Fig- ure 6 shows this also has favorable directivity effects with minimal sideline radiation.

The effects discussed above correspond to the varying acoustic efficiency of the drag (torque) terms. As can be seen from Eq. 30, the drag terms contribute nothing at t• =0. This corresponds to the dipole being normal to the direction of propagation of the wave. The drag terms reach their maximum efficiency near the maxi-



00:11:32

THEORETIC'AL ANALYSIS OF COMPRESSOR NOISE

._ 8O

o

o o

FIG. 8. Comparison of theorv and experi- • c; ment. Maximum fan-discharge noise in octave 7 band containing fundamental blade-passage • frequency.

õo

• • 50

o

z

40

ß Rolls Royce LtS.

ß Pratt and Whitney Aircraft

ß General Dynamics (Convair)

ß The Boeing Company

ß Douglas Aircraft Co., Inc. ß Research Compressor

I " d "Theory • '

Fluctuating Forces

/ .

/-

300 500 1000 1500 2000

V t - Mechanical Tip Speed, ft /sac

mum radiation condition observable on Fig. 4. An equivalent effect inside a duct was found by Morfef t for a skew dipole case.

It is also of interest to compare these ring-source directivity approximations to the radiation from a complete duct, as given in Refs. 11 and 14. It is found that the same broad effects occur for the open-duct case as in the ring source. The principal difference is at high- frequency parameters (nM), where the radiation patterns for the complete duct are more substantially forward. This effect gives a general idea of what may be expected if results were found by integration over a complete compressor annulus, assuming that the sources were in phase, rather than in the present effective ring approximation.

Clearly, any arbitrary spanwise distribution of fluctuating forces could be represented by a series of Bessel- function radial modes and the resultant sound calcu-

lated. However, the basic physics of the problem and the dependence on source parameters would remain essentially the same. Thus, this approach was not pursued during the present work. In principle, the spanwise distribution violates the compact source assumption of this paper, because the span is not small as compared to the wavelength. In practice, this problem is not thought to be too serious. First, the fluctuating forces are much greater at the tip that at the root. Second, the noise may well be generated by local interactions with secondary flow, rather than by distributed wake interactions. Third, the center part does not radiate because its effective phase velocity is subsonic, and, fourth, even given a distributive effect, it is not expected to be larger

than the small difference between open and annular duct radiation discussed in the previous paragraph.

The similarity between the radiation patterns for the duct and the present calculations with no duct is significant. It shows again how many of the phenomena that are sometimes considered to be due to the duct are

simply the result of the source parameters. The similarity of the duct "cutoff" effect to the present case with no duct was discussed above. It is concluded that many of the features of compressor noise sometimes thought to be associated with the duct are, in reality, reflections of the source input characteristics. Conversely, the general results of Tyler and Sofrin • and Morfey 4 have applicability even in the case where no duct is present. Above cutoff, sound propagates in the duct with unit efficiency, and it can be seen that a source radiating at conditions well below the duct cutoff frequency would project so little sound to the farfield that the effects of decay within the duct would be of small practical significance. Clearly, if the duct is treated with acoustical attenuating treatments, it must be considered in detail, but for the hard-wall-duct case, the arguments above suggest that the present analysis is usually applicable to both long- and short-duct radiation problems.

It is also of interest to note that recent work by Lowson and Oilerhead •8.28 and Wright 20 applied results equivalent to Eq. 12 to the case of sound radiation by

•.a M. V. Lowson and J. B. Oilerhead, "A Theoretical Study of Helicopter Rotor Noise," J. Sound Vibration 9, 197-222 (1969).

•.9 S. E. Wright, "Sound Radiation from a Lifting Rotor Gen- erated by Asymmetric Disc Loading," J. Sound Vibration 9, 223- 240 (1969).

The Journal of the Acoustical Society of America •8•


00:11:32

M. V. LOWSON

a helicopter rotor. Here, all possible modes of radiation can be present so that theoretical developments take a slightly different line. However, good agreement between experimental and theoretical trends, spectrum shape, and absolute levels was achieved.

VIII. COMPARISON WITH EXPERIMENT

Because of the large number of unknowns in the problem, particularly in specifications of the aerodynamic-source characteristics, detailed correlation with experiment must be deferred. However, it is of interest to make some comparison here. Figures 4 and 6 suggested the rather surprising conclusion that, if the compressor had to operate in the efficient-radiation region, then it would be desirable to go to equal numbers of rotor and stator blades (t• =0) for minimum noise radiation. Figure 7, taken from a report by Crigler and Copeland, 7 provides a verification of this prediction. Figure 7 shows that the acoustic power radiated for the 53-guide-vane (t• =0) case is about 8 dB less than for the 31-guide-vane and 7 dB less than for the 62-guide-vane cases. Extrapolation of Fig. 3 suggests a maximum effect of only 4 dB so that there are probably other effects present in Crigler-Copeland results. Nevertheless, the major reduction in sound radiation occurring for the t• =0 case here, particularly in the radial direction, is obviously of practical significance. Note on Fig. 7 how the peak levels for the u =0 case are higher, but carry very little acoustic power, so that power levels are low.

A further comparison with experiment may be accomplished by the introduction of a very simple model, discussed in more detail in Ref. 14. If it is assumed that the wake is of pure sinusoidal form (that is, single mode only), with a velocity defect equal to the blade-profile drag, then numerical values may be assigned to the fluctuating-force coefficients via Eq. 31. If the asymptotic expressions for the lift response function (Eq. 32) and the Bessel-function integral of Fig. 3 are also taken, then a simple result for over-all acoustic power is obtained. Using typical values for the parameters and assuming spherical spreading gives the result for the sideline-noise level in the fundamental frequency at 200 ft as

SPL = 501og10 V•,-[- 201og10D-- 75, (34)

where V• is the compressor tip speed in feet/second and D the compressor diameter in inches. This curve is shown in Fig. 8, along with data points from several sources. In view of the large number of assumptions made, it is very encouraging that the theoretical curve even falls on the data points.

The V• 5 law given by Eq. 34 has also been found in several experiments, but it is of interest that in the basic formulation •4 a VR 5 law was derived, where VR is the relative speed of the blade, as opposed to V•, the mechanical tip speed. This dependence of noise of V• has also been suggested by several investigators. The

dependence of the sound on diameter squared is also eminently reasonable. However, it should be noted that the result of Eq. 34 is the end result of many approximations and cannot be applied indiscriminately. The un- certainties associated with any simple approach are indi- cated by the wide scatter of the experimental data on Fig. 8. It can be seen that variations of q-10 dB are probable. Note also that Eq. 34 applies only for subsonic rotors. As discussed in the previous Section, the radiation from supersonic rotors, at least at the blade- passage frequency, is governed by the steady forces acting so that the approximations made for the above results are no longer valid. Although simple estimates for the supersonic case still generate a V • law, this is unlikely to apply because of the partial choking effects that must be expected for supersonic rotors.

IX. CONCLUDING REMARKS

A theoretical description of the noise radiated by the fluctuating forces on the rotor and stator of a fan or compressor has been given. The theory enables most of the design features of the compressor to be accounted for. Parameters that can be included in the theory include•

ß rotational speed;

ß blade/vane numbers;

ß multistage radiation;

ß compressor hub velocity (see Ref. 14).

Effects that are not included in the theory include•

ß duct treatment,

ß passage of sound through blade rows;

ß effect of duct velocities on acoustic propagation;

ß noise radiation due to thickness or acoustic stress.

The theory also suggests that the following features do not have substantial effects on the sound-

ß hard-wall-duct phenomena;

ß detailed compressor geometry.

The key problem is in predicting the aerodynamic characteristics of the shed wakes. Once these can be

predicted, the theory allows calculation of the acoustic effects of engine parameters, such as-

ß separation between rotor and stator;

ß broad features of blade and vane geometry;

ß stage aerodynamic and performance parameters.

The theory enables, at least in principle, comprehensive predictions of the acoustic radiation to be made. Even off-design conditions, including such effects as rotating stall, are described by the general theory, al-



00:11:32


though in such cases the orthogonal summations no longer apply and sound can be radiated at any harmonic of the rotational speed, rather than only at harmonics of the blade-passage frequency. Thus, at the present time, the key requirement is for comprehensive information on the unsteady aerodynamics of the compressor.

The primary effect on the noise radiation not included in the theory is the acoustic effect of finite flow velocity. Clearly, choked flows would not be predicted correctly, using the present results. However, lesser, but still significant, effects are possible. For instance, the theory shows how rotor-stator or stator-rotor interactions are

equivalent. On a real engine, the rotor radiation field propagates through the comparatively slowly moving velocity field of the stator. The stator radiation field, however, propagates out of the inlet against the high- velocity field due to the motor. Thus, in practical engines, where high subsonic Mach numbers behind the rotor are possible, the stator radiates less efficiently than the rotor out of the inlet. Such effects are not

included in the present theory. The principal practical conclusions of the study are---

ß The stator and rotor are basically of equal significance in noise radiation.

ß Increasing rotor-stator separation basically reduces the higher-harmonic radiation by the stator, whereas increase of stator-rotor separation reduces all harmonics of noise radiation by the rotor.

e The predominant radiation mechanism is via the fluctuating forces on the blades. Mass source (siren) terms are insignificant in practical compressors.

• The gross features of the noise-radiation characteristics (over-all power and directionality) are governed basically by the source parameters, rotational speed, and blade/vane numbers and are not substantially dependent on hard duct effects.

. In particular, compressor noise at frequencies below cutoff for a duct would radiate very inefficiently even in the absence of a duct.

. To achieve inefficient acoustic radiation, it is desirable to reduce rotational speed below a critical

value and generally to maximize the difference between the number of rotor and stator blades. Several further detail effects are discussed in the text.

ß If rotational speed cannot be reduced below the critical value, then it is desirable to minimize the difference between the number of rotor and stator blades.

e In particular, equal numbers of rotor and stator blades give a minimum acoustic power radiation coupled with favorable directivity patterns.

• Sound radiation at conditions close to the critical

(cutoff) case is particularly unfavorable having increased acoustic power and undesirable directivity characteristics.

• Use of the theory with a crude model for the fluctuating-force levels gives a D•'V 5 law for the noise radiation, in good agreement with both the trends and absolute levels from experiment.

ß Above the cutoff condition, the relative (effective) velocity at the blade appears to be more significant in the noise radiation than the mechanical tip speed, and it should therefore be minimized in a quiet engine.

• On compressors operating at supersonic tip speeds, the steady loads radiate efficiently and dominate the blade-passage-frequency sound. Noise-control mea- sures suggested for subsonic rotors are unlikely to be of particular value at supersonic speeds.

The results given enable acoustic power and directionality patterns to be calculated from a knowledge of compressor geometry and aerodynamics. None of the approximations used in the theory is thought to be un- reasonable, and it seems that the most likely source of error in the application of the theory is in the specification of the aerodynamic-source terms, particularly the wake characteristics.

ACKNOWLEDGMENTS

This work was supported by the National Aeronautics and Space Administration.

The, Journal of the Acoustical Society of America 385


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theoretical analysis of compressor...

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