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STOL aircraft and ground effect machines
J. Sound F%. (1966) 3 (3j, 454-466
BASIC MECHANISMS HELICOPTERS, GROUND
OF NOISE GENERATION V/STOL AIRCRAFT EFFECT
M. v.
BY
AND
MACHINES
LOWSON
Wyle Laboratories, Huntsville, Alabama, U.S.A. (Receieted I June 196.5) The basic mechanisms of noise generation due to mass introduction, applied force and applied stress are discussed with reference to their implications for helicopters, V/STOL aircraft and ground effect machines. The significance of the dimensional dependence of monopole, dipole and quadrupole fields is shown. The results of a new theory giving the effects of system accelerations on noise are presented. This theory is especially relevant to noise problems for these machines because of the centrifugal accelerations associated with many of the noise generating components. System accelerations give rise to higher order poles in the sound field which become particularly important at high speeds. An expression for the sound field produced by fluctuating lift and drag forces in a rotating and convected system is given. As a further example of the application of the general theory the sound field radiated by a hovering helicopter is analyzed. It is shown how a previously unrecognized source of sound arises from the outward components of force induced by the effects of blade coning angle and lag. The source of sound has its maximum in the plane of the rotor disc. The importance of including the proper momentum terms in calculations of noise radiated by moving mass sources is demonstrated.
I. INTRODUCTION
The fundamentals of the theory of sound were laid down by Lord Rayleigh as far back as 1877, and his book (I) remains in general use today. Indeed, until recently, virtually all acoustic calculations were merely simple extensions of the classical theories. However, the increasing magnitudes of radiated noise, especially those associated with modern high speed transport, have shown the need for improved methods for noise calculation and prediction. In fact, most high intensity noise sources are in motion relative to the undisturbed air, so that there is a particular requirement for methods which can predict the sound fields generated by sources in motion. The classical methods for calculating these sound fields attempted to relate the sound heard at any given instant to the position of the sound source at that instant. In reality, sound travels at a finite speed a0 through the air so that the sound heard at any given instant t was actually generated at an earlier “retarded ” time t’= t-r/q,, where r is the distance of the source from the observer at that earlier time. Thus, in order to relate the sound radiated by a moving source to its current position it is necessary to specify its earlier motion in some detail. This requirement results in much analytical complication, and when the magnitude of the source, or its motion, is varying in some complex manner the analytical difficulties often become prohibitive. In 1952, Lighthill (2) attacked the problem of noise generation by turbulence, and one of his contributions was to show the simplification that resulted from referring all calculations to the condition of the source at the retarded time of emission of the sound. The 454
BASIC MECHANISMS OF NOISE GENERATION
455
writer has recently taken advantage of Lighthill’s ideas to derive expressions which give the sound fields for sources of arbitrarily varying strength in arbitrary motion (3). Classical methods required definition of the path of the sound source before any analysis could commence, but calculations referred to the retarded time can produce a number of general results which greatly simplify further study. The generation of noise by helicopters, V/STOL aircraft and ground effect machines (GEMS) is an extremely involved phenomenon. There is a multiplicity of sources of sound, and each of these is in itself a complicated process. There is little possibility of eliminating all the sources of noise on these machines, but if the noise could be accurately predicted then it would become possible to design from the start to meet any required noise limitation. There is, therefore, a definite requirement for improved methods of estimating the noise. The general results referred to above can often be applied to these problems and detailed predictions of the sound field made. However, in order to separate the effects of these sources, and to understand their relative significance under various conditions, it is very useful to return to first principles. General arguments based on these principles are often sufficient to define quite closely the important characteristics of complicated noise sources. Therefore, in this paper the first task is to define the fundamental causes of noise generation and their basic characteristics. The general results obtained in reference (3) are then presented. These results are of significance for the noise produced by rotating systems since the centrifugal accelerations of such systems give rise to additional radiated noise. Applications of these results to a number of noise problems on helicopters, V/STOL aircraft and ground effect machines are made, and it is shown how, in many cases, the noise radiation can be specified in detail.
z. THE
BASIC
RESULTS
First of all it is necessary to write down the fundamental equation describing the generation of noise. Written in tensor notation with the summation convention it is
azp z-agV2p
aQ aFi a2Tij
= ---+
at
axi
i3xiaxj’
(1)
A derivation of this equation was given in reference (3), and this followed the lines laid down by Lighthill (2). The left-hand side is the three-dimensional wave equation which is the equation of motion for sound in a uniform medium at rest. It has been written in terms of the fluctuating density p since this offers a slight simplification. However, for most practical purposes the variable p may be replaced by p/a& the pressure divided by the square of the undisturbed speed of sound. This point has been discussed by Doak (4). The right-hand side of equation (I) represents the various sources of sound present. It will be observed that there are three terms present, each of which corresponds to a different type of source. The first term, aQ/at, gives the sound due to a “ simple source “. This is the sound produced by mass introduction at a point. Here Q is the rate of introduction of mass per unit volume, and since Q may be a function of x this mass introduction may be distributed throughout the field. Note that the sound produced is proportional to the time rate of change of Q. The second term aF#xi, gives the sound produced by forces acting upon the air (or any other acoustic medium). Again it is possible for Fi to be a function of xi so that this term can represent the effect of a force distribution. Since it involves a space differentiation this force term is usually a less efficient producer of sound than the first “ simple source ” term. The final term on the right-hand side of equation (I)
456
M. V. LOWSON
is actually an amalgam of all the other acoustic effects which can possibly occur. The acoustic stress, Tij, includes the effects of temperature, refraction, diffraction and inhomogeneity. For most practical purposes the importance of this term is that it represents the noise produced by turbulence so that T,~ p~~l-‘i,where viand vjrepresent the turbulent velocity components. A full discussion of the importance of this last term will be found in Lighthill’s original paper (2). Thus, there are three separate physical effects which give rise to sound radiation : mass introduction, applied force, and applied stress. Each gives rise to a characteristic sound field, and much useful information may be gained merely from a knowledge of these characteristics. Because equation (I) is the wave equation, albeit with a complicated right-hand side, explicit solutions for the sound fields produced by various types of sound source can be found. It is convenient to consider first the sound fields produced by elementary point sources of each type. In reality, sound is produced by an extended distribution of these sources, which cannot always be approximated as a point singularity, but the results for the sound fields of these singularities are nevertheless of general value in the interpretation of more complex effects. For the case of uniform motion it is found that the sound fields for each type of source are given (putting p M afp) by
1’ -yJ(xi -YJ a2 Tij ’=[(xi 1 xi -_Yi
p = [ 4mor2(1-MT)*
aF. -2
at
4m~r3(1--IMJ3at2
’
(3) (4)
Equation (2) gives the sound field radiated by motion of a “ basic acoustic singularity” with strength q. This is one part of the sound field produced by a moving “ simple source ” Q with q = aQ/at. Equations (3) and (4) g ive the sound field produced by uniform motion of point forces, Fi, and point acoustic stresses, Tij, respectively. In each of these equations Xi (i= 1,2,3) and yi (;= I, 2,3) are the Cartesian coordinates of observer and source, respectively, and r is the distance from source to observer ; r* = lr* I= (xi-yJ*. M, is the component of the convection Mach number in the direction of the observer, equal to Mz(xi-yi)=M,(xi -Yr) +Mz(xz-_Yz) +M 3(x3 -y3). Lighthill (2) used the form M,cos 19 for M,. Now the sound heard at any time t was actually generated at time t’= t - r/a0 and has travelled a distance r at the speed of sound a0 in the intervening interval. In fact, proper account of the effect of retarded time is one of the crucial requirements in acoustics, and each of the equations (2), (3) and (4) h as b een enclosed in square brackets, a notation used to indicate that the contents of the brackets are to be evaluated at the retarded time. A dimensional analysis can be performed on the results of equations (z), (3) and (4). The magnitudes of q, Fi and Tij are all proportional to p. U*l*, where U is a typical velocity and E is a typical length. In addition, a/& can be put proportional to a typical frequency n, and then by invoking constancy of Strouhal number, d/U, we obtain the dimensional dependencies shown in the third column of Table I. Each source has a typical dimensional dependence and this fact is often of value in identifying an unknown source type or predicting simple characteristics of a known source. However, before discussing the further repercussions of these dimensional results it is desirable to consider the directionality patterns of the various sources.
BASIC MECHANISMSOF NOISE GENERATION
437
Physical intuition suggests that a simple pulsating point mass source placed in the air has no preferred orientation so that its sound field is spherically symmetric. This type of sound radiation is known as a “ monopole ” field. Consider next the combination of two fluctuating mass sources of equal but opposite strength. It can be seen that there is zero total mass output from the combination, but that a fluctuating momentum transfer will occur. This fluctuating momentum may be interpreted as being the result of a fluctuating force. TABLE I
Dimensional dependence and directionality of various types of acoustic source
Source type
Typical cause
Monopole
Part of field of simple mass source
Dipole
Force in uniform motion
Quadrupole
Acoustic stress in uniform motion
Dimensional dependence
r
PO1
-
~
u2
I-M,
PO1 a,r (I
u3
PO1
u4
Directionality (when at rest)
0+ 8
&(ai@
Thus, the sound radiation of the fluctuating force can be regarded as the sum of two monopoles; i.e., a dipole radiation. The strength of the dipole is the product of the source strength and separation, and is equal to the magnitude of the force. Dipole radiation has a definite directionality associated with it, as shown in Table I, with the maximum sound radiation occurring in the direction of action of the force. In a similar fashion a quadrupole field may be constructed by combining two dipoles of equal but opposite strength. In this case a double directionality dependence occurs so that the radiation pattern of a quadrupole may either be a four-lobed “ lateral quadrupole ” or an extended two-lobe “ longitudinal quadrupole “. A quadrupole radiation field is appropriate to the acoustic stress. Because of the cancellation effects inherent in the higher order poles, the monopole or simple mass source is by far the most efficient producer of sound, at least at low speeds. However, Table I shows how an increase of velocity has a much larger proportional effect on the higher order poles. In fact, this velocity dependence can be traced directly to the mitigation of cancellation brought about by retarded time effects at high speeds. Thus, as speed increases the higher order poles become important sources of sound radiation. All of these results have applied to the case of uniform motion of the source through the air. In fact, many important noise producing components undergo accelerated motion due to their rotation. The effect of acceleration is to modify many of the conclusions reached above. Therefore, the sound fields due to the arbitrary motion of sources of sound must be considered. Practical application of the modified results leads to a number of interesting conclusions.
M. V. LOWSON
458 3. THE SOUND
PRODUCED
BY FORCES IN MOTION
It is convenient to consider first the sound produced by moving point forces. In reference (3) it was shown that the sound field for a point force in arbitrary motion is (xi-Yi) p =
4ra,,r*(r-M,)*
s+
Fi
at
I-M,
aMr
at
II ’
where the notation is as before. Note that an/r,/& = (xi-yi) (aMi/&). Extension of equation (5) to the case of a force distribution is easily made by putting Fi= PidS and integrating over the area of action S, taking appropriate account of retarded time. Comparison of this result with that given in equation (3) shows how generalization to an arbitrary motion causes an additional term to appear in the sound field. This second term in equation (5) is proportional to the acceleration (more strictly, to the component of the acceleration in the direction of the observer) so that for uniform motion equation (5) reduces to equation (3). However, the important and immediate conclusion from equation (5) is that accelerated motion causes additional sound generation. Performing a dimensional analysis on equation (5) on the lines followed before shows that
P
POJ - F(I
u3
Pal
-MJ*+&+
Here acceleration has been put proportional
to y,
u4
-MJ3’ as is usually the case. Comparison
of the result with those in Table I shows that the first term is the conventional dipole term as expected. However, the second term has the dimensional dependence of the quadrupole. By again referring to equation (5) it can also be observed that the first term has a single directional dependence, through (xi-yi) (aF,/at). The first term is thus dipole. The second term, however, has a double dependence, through both (xi-yi) Fi and aikf,/i3t, and is therefore quadrupole in type. Thus, equation (5) shows a quadrupole contribution of acceleration to the sound radiated. Identification of the acceleration term as quadrupole allows two immediate conclusions to be drawn. First, the effect of acceleration is relatively unimportant at low speeds, but is increasingly important at higher subsonic speeds. Second, typical radiation patterns for an accelerated motion can be expected to have directionality patterns similar to those shown in Table I. There are many applications of equation (5). There have been previous studies of the noise generated by accelerating systems; for instance, the case of propeller noise studied by Gutin (5) and Garrick and Watkins (6). However, previous studies have had to specify the motion in detail in order to commence the analysis. One advantage of the present results is that the result for any given motion may be obtained directly by substitution. Because the equation is completely general, the sound field can be predicted if the magnitudes and motions of the forces acting can be specified. It is thus possible to deal with any problem of the noise radiation by steady or fluctuating forces in any arbitrary motion. 4. APPLICATIONS
TO HELICOPTER
NOISE
An example of the application of the theory is the problem of noise production by a hovering helicopter. In principle, the rotor forces on a hovering helicopter should be constant. In practice, it is often found that these are fluctuating, but for the present purposes a steady rotor loading is assumed. The object of this analysis is to show how the coning angle and lag of a helicopter rotor blade lead to additional terms in the noise field. Nearly
459
BASIC MECHANISMS OF NOISE GENERATION
all helicopters have rotors which are hinged at the root and free to flap. In flight the rotor blade “cones ” at an angle fl (cf. Figure I), so that the hinge moment induced by the rotor thrust is balanced by that due to the centrifugal forces. Many helicopters also have drag hinges in addition, so that the rotor blade lags at angle y behind the perpendicular to the axis of rotation. The usual approximation for propeller noise theory can be made and the loading on the rotor thus reduced to a point force acting at some equivalent radius R from the hub. This force has the usual components of thrust T and drag D which act on the air in the opposite direction to that in which they act on the blade. However, due to the coning angle and blade lag there is an additional component of force which acts perpendicular to the axis of rotation. This component is denoted by C, and since both the coning angle, /3, and the lag angle, y, are small, C= /IT + ~0. The drag term yD is almost negligible but is included for completeness. Observer
Axis of rotation
Centrifugal
f
(b)
Figure I. Rotor forces on a hovering helicopter. (a) Blade “cones ” at an angle p so that hinge moment due to centrifugal forces cancels that due to thrust. (b) Loading imposed on the air is approximated by point force components.
Figure I shows the case assumed for the calculations. The sound field is axially symmetric so the analysis may be confined to the xy plane without loss of generality. Note that the thrust T and drag D act perpendicular to and in the plane of the rotor disc and are not necessarily equal to the actual thrust and drag experienced by a blade element. The components of the various parameters in equation (5) can be written down as
r = (xi-yi) = x,y-Rcos8, M = Mi = o, -MsinO, Fi = -T,
-RsinO,
Mcose,
-DsinB+CcosO,
DcosB+Csin9,
so that 8Milat = 0, - MS2 cos 0, - ML’sin 0, aFi/at = 0, - 06’ cos 8 - CQ sin 8, - DQ sin 8 + CQ cos and M, = (Xi-yi) Mi = -My sin B/r, aM,/al (~i-yi)Fi
= (Xi-yi) (aMi/at)
= -QM(y cos 8 - R)/r,
= -TX-DysinO+C(yc0~8-R),
(xi -yi) (aFi/at) = - DQ(y cos 8 - R) - Cy.Qsin 8.
8,
M. V. LOWSON
460
In these equations M is the rotational Mach number ( =QR/aO) and Q is the speed of rotation. Thus, substituting in equation (5) gives, after some cancellation i-2 s (ycod-R)(e-D) 4ra0 r’( 1+ My sin 19/y) (
P=
ysinO+(y2-ayRcosB+R2)r
. 111
Equation (7) gives the pressure as a function of time at the point (x,y). It is an approximation since the blade loadings have been reduced to equivalent point forces but, as mentioned previously, the effect of distributed loading may be calculated by replacing the point forces by pressure distributions and integrating over the area of action taking appropriate account of retarded time. In fact, it turns out that in the far field, for point forces, the frequency spectrum of the pressure pattern can be calculated. This is done in the Appendix, and the result obtained is that the magnitude of the sound pressure on the mth harmonic of a B blade rotor is l&l
= +g
(J,,+1 (+y
(yy)‘1”‘.
-J&?--1
(8)
Here JmB is a Bessel function of the first kind and of order mB, and r is the distance from the propeller center, (x’ +y2)‘j2. The term (JmB-l - J,,,B+1)1/2is typically about the same order of magnitude or slightly larger than the term Jma and hence the outward components of force, C, on a coned or lagging helicopter blade can be important. The first term in equation (8) is the term obtained by Gutin (5) and Garrick and Watkins (6) for the case of a propeller. The difference between quadrupole and dipole effects may be observed in both equations (7) and (8). The drag term is essentially dipole in type so that it produces a two lobe sound pattern and is important at lower speeds. The thrust term is essentially quadrupole and produces a four lobe sound pattern which is important at higher speeds. The combination of these two effects gives rise to the well-known phenomenon of maximum sound production slightly behind the disc, as shown in Figure 2.
+ + w
Rotor oxis
Rotor oxis
t
(a)
t
(b)
(cl
Figure 2. The cause of the sound directionality of propeller noise due to thrust and drag. (a) Dipole type contribution due to drag. (b) Quadrupole type contribution due to thrust. (c) Total sound directionality pattern.
The additional noise due to the outward components of force is also a quadrupole type field so that it is significant at high speeds. However, as shown in Figure 3, the directionality pattern is the extended double lobe of the longitudinal quadrupole rather than the four lobe lateral quadrupole case appropriate to the thrust noise. The maximum for the noise due to outward forces occurs at right angles to the propeller axis, which is just the position
BASIC MECHANISMS
OF NOISE GENERATION
461
where it is not desired on a helicopter. In most practical cases the effect of this new source of noise is of minor significance. In a typical case when the rotational Mach number of the effective point force is 0.5, T/D= 12, and 8=4”, sample calculations show the noise to be increased by about 2 dB in all harmonics. However, the adoption of rotors with higher tip speeds, or improved lift to drag ratios, could easily lead to important contributions from this source. The use of equation (5) has given, without much difficulty, the characteristics of a new source of sound from a hovering helicopter. In fact, the noise field generated by fluctuating forces in any specified motion can be written down with almost the same ease. A problem that is relevant to helicopters, VjSTOL Aircraft and GEMS is the case of fluctuating forces in a rotating and possibly convected system. The existence of fluctuating forces on a helicopter blade in forward flight is well known, but V/STOL aircraft also may often
Rotor
axis
t
Figure force.
3.
Approximate
directionality
pattern
for the sound
due to outward
components
of
have uneven forces on their rotating components due to uneven flow into the fan, etc. Some designs of hovercraft utilize rotatable propellers which often work in non-axial flow with consequent fluctuating forces imposed on the blades. An expression giving the noise radiated by fluctuating forces in a convected rotating system can be derived by substitution into equation (5). This was done in reference (3) with the result p =
{MTx - rD( I -Mor)}Q(y 4nao{r( I -lb&,)
+ My
cos 0 - R) _~@Dpt) sin 8 + @T/at) 4~a&( I -&lo,) + My sin ey
sin
ey
1’
(9)
where MO, is the component in the direction of the observer of the convection Mach number of the center of rotation. Equation (9) shows two terms contributing to the noise : one dependent on the instantaneous value of the forces and one dependent on their rate of change. Thus, once thrust and drag are given as functions of time or blade position the noise radiated can be calculated using equation (9). Equation (9) will also give the noise radiated by helicopter blade slap or wake and fuselage interference providing the loadings imposed by these phenomena can be predicted. Equation (9) assumes zero coning angle or blade lag although these could be incorporated if desired, with consequent lengthening of the expression. Another effect which could be important on helicopters is the flapping motions of the blades. It is apparent from the standpoint of the general theory of equation (5) that flapping accelerations could cause noise to be radiated, and again this effect could be incorporated if desired. The magnitude of this effect can be estimated by comparing the flapping accelerations to the centrifugal accelerations and would probably only be important for the higher harmonics. A further noise problem which can arise in rotating systems is the broad band or “vortex” noise caused by the random pressure patterns that exist on a rotating blade with a turbulent boundary layer. The most complete attempt to predict the sound radiated 31
M. V. LOWSON
462
by these random pressure patterns was made by Sharland (7). He did not include the effect of the acceleration terms in his calculation, but this would not affect the validity of the results in his range of interest with blade tip speeds less than M= 0.4. However, the effect of the second, acceleration, term in equation (5) could be important at higher Mach numbers. Unfortunately, theoretical study of this problem is difficult since specification of the random pressure patterns in a sufficiently detailed manner leads to a number of difficulties. One important problem arises from the possibility that these random pressure patterns do not, in fact, undergo centrifugal acceleration during their passage over the blade. Some additional problems were discussed by Lighthill (8). A valuable experiment would be to investigate the velocity dependence of the “vortex ” noise component at higher tip Mach numbers. This would show the general importance of the acceleration terms. However, resolution of these problems will require detailed study. 5. “SIMPLE
SOURCES”
IN MOTION
The analysis which gave the sound field for a point force in motion can be adapted to give the sound field for a point source in motion. This was done in reference (3) and the result obtained was
(10)
* =
The notation is as before. Again this result proves to be the sum of two terms. Since 4 = aQ/i3t, the first term in (IO) is identical to the result in equation (2). The second term is identical to the result for the arbitrarily moving force given in equation (5) except that Momentum transfer Mass0
_
Gin
Figure 4. Showing cause of momentum transfer on a convected “simple” mass source.
the term Fi is replaced by Qv~, which gives the rate of momentum introduction. The reason for this second term can be readily understood. The quantities in equation (IO) are evaluated relative to the moving system so that the introduction of mass in the moving system naturally implies an introduction of momentum relative to the free air. Even for the idealized case of a convected pulsating sphere the momentum is not instantaneously balanced. Figure 4 shows how the mass output of the system arises at a different location to the mass input and thus gives rise to a net momentum fluctuation. In fact the two terms in equation (IO) may be combined in a simpler form as
*=
I [ 4~(1-M,)~
aQ
Q
( at+I-M,?K
aMr
)I’
(11)
which therefore represents the general result for the sound field produced by a “simple source ” in arbitrary motion. Even for uniform motion this expression differs from that
BASIC MECHANISMS
OF NOISE GENERATION
463
given in equation (2), but this difference may now be readily understood as being the result of the momentum terms. In applying these results, for instance, to the case of a tip jet rotor, it is clearly important to include the effect of the momentum terms. In fact, a tip jet rotor represents an important source of momentum as well as of mass so that both of these variables require consideration in a complete study. The present results show that if the momentum output is calculated relative to the rotor, then equation (I I) should be used for the effect of the mass output. Alternatively the momentum output can be calculated relative to the still air and equation (2) used. In either case, the effect of the momentum terms would be calculated via equation (5). M ore d e t ai 1e d ca 1cu 1a t ions of the sound field could follow the lines of section 4. 6. ACOUSTIC
STRESSES
As might be expected, the analysis which applied applied to the case of acoustic stresses in motion. was that
(xi-yi)(xj-yj) * = [ 4mgr3(1
-M,)3
IN MOTION
to forces and sources in motion can be The result obtained in reference (3)
azTij I aT,3(aM,/at)+ -~ Tij a’M,+
1 at2
at
I-M,
I --al,
at2
$JJ 2 .
3Tij (I -M,)2
(
at
c12)
HI
The first term in equation (12) is identical to the result for uniform motion given in equation (4), but again the effect of acceleration is to introduce additional terms into the sound radiation field. Equation (12) contains terms dependent on acceleration, rate of change of acceleration and acceleration squared. Interpretation of Tij is always difficult, particularly since the approximation of putting p w a& often requires careful consideration. When these difficulties are combined with the complexities of the equation above it becomes extremely difficult to make any detailed prediction. All that can be achieved at present is a broad statement of the possible importance of the acceleration terms. In general, it appears that accelerated turbulent eddies produce more noise than those moving at constant velocity. This could be very important in such cases as shockturbulence interaction, and in the general production of noise from turbulence at supersonic speeds. However, the applications of immediate interest in this case are to tip jet rotors and ground effect machines. The sound field produced by tip jet rotor turbulence is a major source of noise, and equation (I 2) indicates that centrifugal accelerations increase this noise. Unfortunately, it is difficult to be certain that tip jet rotor turbulence actually undergoes centrifugal acceleration so that it is possible that these remarks will not be relevant. However, it is certain that the turbulent eddies undergo acceleration in the peripheral jets of ground effect machines since these vehicles derive their lift from jet deflection. At present, even large hovercraft operate at relatively low jet speeds so that jet noise is not generally significant. However, successful development of these vehicles will inevitably lead to higher jet exit velocities when the effects of jet noise will be important. The acceleration terms may well require consideration in this case. The noise arising from the deflected jet flows of STOL aircraft is another case where the effects of acceleration could sometimes be important. 7. DISCUSSION
AND
CONCLUSIONS
Because the sources of noise on all these vehicles are so complicated it is often very valuable to return to basic principles. Classification of a source as monopole, dipole or quadrupole usually reveals the speeds at which it can be expected to predominate, its probable significance, and sometimes its directionality pattern. An interesting example of
464
M. V. LOWSON
this was furnished in a recent paper by Ffowcs Williams and Gordon (9). They showed, as expected, that the quadrupole effects of the acoustic stresses in the turbulence are responsible for jet noise at high subsonic Mach numbers, but, they also showed how, at low speeds, say M < 0.4, the monopole effects of the jet mass fluctuations induced by jet pipe boundary layer turbulence are predominant. At an intermediate range of Mach number the dipole contribution of the fluctuating forces induced at the end of the jet pipe is significant . Thus, noise which apparently arises from a single phenomenon (in the case above, jet turbulence) can often be found to be quite a complicated effect. For instance, the possible importance of acceleration on the turbulent noise produced by tip jet rotors, STOL aircraft and GEMS was pointed out in the last section. This conclusion is valid only at high speeds. Both tip jet rotors and GEMS can be expected to generate monopole type noise by mass fluctuation at lower speeds. In particular the thin, two-dimensional, low speed, peripheral jets currently favoured by hovercraft designers may be expected to be quite efficient generators of monopole noise. Again, the noise radiated by the deflected jet streams of STOL aircraft would be expected to arise mainly as a dipole source from the random pressure patterns occurring over the deflecting surfaces. Noise radiation from such sources was considered by Sharland (7), and his work may be extended to cover this case. In general, acceleration causes higher order poles to occur in the sound field. For a force in arbitrary motion a quadrupole acceleration term is added to the conventional dipole field. For the acoustic stress, acceleration effects cause the addition of two higher orders of pole. A moving source of mass is necessarily associated with a source of momentum. The simple monopole field of the mass source is reinforced with a dipole field due to the momentum source, and acceleration effects on the momentum source cause a further quadrupole field to be added. All these acceleration effects generally become increasingly important at higher speeds. The general expressions found for the acceleration effects are convenient for practical applications. For example, the analysis shows directly that the outward components of force induced by blade coning angle and lag on a hovering helicopter give rise to a new type of sound field with a maximum in the plane of the rotor disc. A general formula for the calculation of noise radiation from fluctuating forces in a rotating and convected system can be applied to direct calculation of the noise radiation from yawed propellers, helicopters in forward flight, and fans with uneven velocity distribution, providing the appropriate loadings can be specified. The results are particularly useful when the acoustic sources can be reduced to point singularities. For the more frequent practical case of an extended distribution of noise sources calculation of the sound field requires evaluation of the integral forms of the equations, taking appropriate account of retarded time. However, this analysis can usually be made on a computer. ACKNOWLEDGMENTS The general results described in this paper originated from work at the Institute of Sound and Vibration Research, Southampton University, and the writer would like to thank Professor E. J. Richards for initiating his interest in the problems. The applications of the results discussed herein were made under Wyle Laboratories Independent Research Program with the guidance and encouragement of K. McK. Eldred. REFERENCES volumes) New York: Dover Publications. Second
I. LORD RAYLEIGH 1877 Theory of Sound (two
edition, 1945 re-issue. a. M. J. LIGHTHILL 1952 Proc. R. Sot. A 211, 564-587. On sound generated aerodynamically. I. General theory.
BASIC
MECHANISMS
OF NOISE
465
GENERATION
M. V. LOWSON1965 Proc. R. Sot. A 286, 559-572. The sound field for singularities in motion. P. E. DOAK 1960 Proc. R. sot. A 254, 129-145. Acoustic radiation from a turbulent fluid containing foreign bodies. Tech. Memo 1195 (1948). On the sound field of a rotating propeller. 5. I,. GUTIN 1936 N.A.C.A. (Translation.) Rep. I 198. A theoretical study of the effect 6. I. E. GARRICKand C. E. WATKINS 1954 N.A.C.A. of forward speed on the free space sound pressure field around propellers. I. J. SHARLAND1964r. Sound l/‘;b. I, JOZ-pz. Sources of noise in axial flow fans. ii: M. J. LIGHTHILL1962 Proc. R. Sot. A 267, 147-182. Sound generated aerodynamically. ourna~3, 791-793. Noise of highly 9. J. E. FF~WCSWILLIAMSand C. G. GORDON1965 A.I.A.A.J turbulent jets at low exhaust speeds. IO. N. W. MCLACHLAN1955 Bessel ~unctionsfor &zgineers. Oxford University Press. 3.
4.
APPENDIX ANALYSIS
OF THE FAR FIELD
FREQUENCY
SPECTRUM
FOR A HOVERING
HELICOPTER
Equation (7) gave the sound as
.
(
-C
~sinBf~(y2-zyRcosB+R2)
Now
r2 = lrl2 = ~~+y~+R~-zyRcosO,
(AI)
so that in the far field, where r 9 R, r z rl - (yR/ r 1)cos 8 where rr is the distance from the rotor centre, (x2+y 2) 1/2. The effect of retarded time (t-r/aO) need only be retained on 8, SOthat
[e] = T+$GO~[ej, where T=Q(~-rI/ao). With these approximations be written down as 2n+a (;I)=:
1
the component
QV
642)
of far field sound in the nth harmonic may
e - Crl(~ + sin (I +crsin8)3
(MTzc - DY t) cos
a(
OL
cos
e) )(
127
d7
sinn~ 1
’
(A3)
where a = My/rl. The integral can cover any interval 2~. Equation (A3) is evaluated by changing the variable from T to 8, using the form indicated by equation (Az), viz : 7
=
(A4)
8-cCCOS8.
Thus
a,
0 bn
s
QY
=4= 0
cos 8
--I and 0c(r+crsinO) s equation (As) may be integrated by parts to give Now
(r +ccsin8)2
d2y
a,
0 b,
de=
=4-
2nMTx-
Jl 0
a
Drl r
1
so that
-sin{n(O-cccose))
I(
cos 8
cos
(f2(e -a
cos e))
de 0w 1
??
466
M. V. LOWSON
The integrals in equation (A6) are all various forms of Bessel functions. McLachlan (IO) gives (formula 42) Jn(z) = ‘G jexp{i(zcosB+nB)jdB.
(A7)
0
is a Bessel function of the first kind and of order n. In the present case, x is a real quantity, so that expanding equation (A7) and taking real and imaginary parts gives
J&z)
297 s 0
cos(zcosO+nO)dO
=
-
I)“‘2J,(z),
n
n
,
2n I 0
24
o
sin (Z cos 8 + ne) df9 =
27r(-
I)'"-'"~J,&),
0,
AIs0 cos ecOs(zcos e+ne) = &{COS[ZCOS e+ (n + identities (A8) gives
even odd 1
odd n even
(AS) '
n
I
I) e]+ COS[ZCOS e+ (~2 - I)e]).Thus,
using
2n I 0
cosec~~(~~o~e+~e)
de =
4-I)
(n+1)'2{Jn+i(4 -J+&))>
n odd 71even
(0,
and similarly 27 c0sesin(xc0s8+d)
I
de =
0
n
r( - I)“‘2url+1(d -Jn-1(z)], o 3
even
n odd
Using (A8) and (Ag) in (A6) then gives a, = -2:
4 =
‘3
(MT’~Drl]
2( - I)(n-1)12Jn( -ntL), n odd,
(- I>“‘~{.L+~( -4 -.L(
-4,
n even,
and n
b, = -f&~(
even,
- Ip+l)'2{Jn+l(-n0~)-_~-1(-n~)}, n odd.
Now Jn(- 4 = ( - I)“_L(+ so that the magnitude of the sound pressure in the nth harmonic finally becomes lAn[ = (af+bi)‘i2
=
where CLhas been replaced by its equivalent, My/r,. For a B bladed propeller, harmonics which are not multiples of the number of blades cancel, leaving the result as given in equation (8). This paper was first presented at the Symposium on the Noise and Loading Actions of Helicopters, V/Stol Aircraft and Ground Effect Machines, September x965 at the Institute of Sound & Vibration Research of the University of Southampton.
BASIC MECHANISMS HELICOPTERS, GROUND
OF NOISE GENERATION V/STOL AIRCRAFT EFFECT
M. v.
BY
AND
MACHINES
LOWSON
Wyle Laboratories, Huntsville, Alabama, U.S.A. (Receieted I June 196.5) The basic mechanisms of noise generation due to mass introduction, applied force and applied stress are discussed with reference to their implications for helicopters, V/STOL aircraft and ground effect machines. The significance of the dimensional dependence of monopole, dipole and quadrupole fields is shown. The results of a new theory giving the effects of system accelerations on noise are presented. This theory is especially relevant to noise problems for these machines because of the centrifugal accelerations associated with many of the noise generating components. System accelerations give rise to higher order poles in the sound field which become particularly important at high speeds. An expression for the sound field produced by fluctuating lift and drag forces in a rotating and convected system is given. As a further example of the application of the general theory the sound field radiated by a hovering helicopter is analyzed. It is shown how a previously unrecognized source of sound arises from the outward components of force induced by the effects of blade coning angle and lag. The source of sound has its maximum in the plane of the rotor disc. The importance of including the proper momentum terms in calculations of noise radiated by moving mass sources is demonstrated.
I. INTRODUCTION
The fundamentals of the theory of sound were laid down by Lord Rayleigh as far back as 1877, and his book (I) remains in general use today. Indeed, until recently, virtually all acoustic calculations were merely simple extensions of the classical theories. However, the increasing magnitudes of radiated noise, especially those associated with modern high speed transport, have shown the need for improved methods for noise calculation and prediction. In fact, most high intensity noise sources are in motion relative to the undisturbed air, so that there is a particular requirement for methods which can predict the sound fields generated by sources in motion. The classical methods for calculating these sound fields attempted to relate the sound heard at any given instant to the position of the sound source at that instant. In reality, sound travels at a finite speed a0 through the air so that the sound heard at any given instant t was actually generated at an earlier “retarded ” time t’= t-r/q,, where r is the distance of the source from the observer at that earlier time. Thus, in order to relate the sound radiated by a moving source to its current position it is necessary to specify its earlier motion in some detail. This requirement results in much analytical complication, and when the magnitude of the source, or its motion, is varying in some complex manner the analytical difficulties often become prohibitive. In 1952, Lighthill (2) attacked the problem of noise generation by turbulence, and one of his contributions was to show the simplification that resulted from referring all calculations to the condition of the source at the retarded time of emission of the sound. The 454
BASIC MECHANISMS OF NOISE GENERATION
455
writer has recently taken advantage of Lighthill’s ideas to derive expressions which give the sound fields for sources of arbitrarily varying strength in arbitrary motion (3). Classical methods required definition of the path of the sound source before any analysis could commence, but calculations referred to the retarded time can produce a number of general results which greatly simplify further study. The generation of noise by helicopters, V/STOL aircraft and ground effect machines (GEMS) is an extremely involved phenomenon. There is a multiplicity of sources of sound, and each of these is in itself a complicated process. There is little possibility of eliminating all the sources of noise on these machines, but if the noise could be accurately predicted then it would become possible to design from the start to meet any required noise limitation. There is, therefore, a definite requirement for improved methods of estimating the noise. The general results referred to above can often be applied to these problems and detailed predictions of the sound field made. However, in order to separate the effects of these sources, and to understand their relative significance under various conditions, it is very useful to return to first principles. General arguments based on these principles are often sufficient to define quite closely the important characteristics of complicated noise sources. Therefore, in this paper the first task is to define the fundamental causes of noise generation and their basic characteristics. The general results obtained in reference (3) are then presented. These results are of significance for the noise produced by rotating systems since the centrifugal accelerations of such systems give rise to additional radiated noise. Applications of these results to a number of noise problems on helicopters, V/STOL aircraft and ground effect machines are made, and it is shown how, in many cases, the noise radiation can be specified in detail.
z. THE
BASIC
RESULTS
First of all it is necessary to write down the fundamental equation describing the generation of noise. Written in tensor notation with the summation convention it is
azp z-agV2p
aQ aFi a2Tij
= ---+
at
axi
i3xiaxj’
(1)
A derivation of this equation was given in reference (3), and this followed the lines laid down by Lighthill (2). The left-hand side is the three-dimensional wave equation which is the equation of motion for sound in a uniform medium at rest. It has been written in terms of the fluctuating density p since this offers a slight simplification. However, for most practical purposes the variable p may be replaced by p/a& the pressure divided by the square of the undisturbed speed of sound. This point has been discussed by Doak (4). The right-hand side of equation (I) represents the various sources of sound present. It will be observed that there are three terms present, each of which corresponds to a different type of source. The first term, aQ/at, gives the sound due to a “ simple source “. This is the sound produced by mass introduction at a point. Here Q is the rate of introduction of mass per unit volume, and since Q may be a function of x this mass introduction may be distributed throughout the field. Note that the sound produced is proportional to the time rate of change of Q. The second term aF#xi, gives the sound produced by forces acting upon the air (or any other acoustic medium). Again it is possible for Fi to be a function of xi so that this term can represent the effect of a force distribution. Since it involves a space differentiation this force term is usually a less efficient producer of sound than the first “ simple source ” term. The final term on the right-hand side of equation (I)
456
M. V. LOWSON
is actually an amalgam of all the other acoustic effects which can possibly occur. The acoustic stress, Tij, includes the effects of temperature, refraction, diffraction and inhomogeneity. For most practical purposes the importance of this term is that it represents the noise produced by turbulence so that T,~ p~~l-‘i,where viand vjrepresent the turbulent velocity components. A full discussion of the importance of this last term will be found in Lighthill’s original paper (2). Thus, there are three separate physical effects which give rise to sound radiation : mass introduction, applied force, and applied stress. Each gives rise to a characteristic sound field, and much useful information may be gained merely from a knowledge of these characteristics. Because equation (I) is the wave equation, albeit with a complicated right-hand side, explicit solutions for the sound fields produced by various types of sound source can be found. It is convenient to consider first the sound fields produced by elementary point sources of each type. In reality, sound is produced by an extended distribution of these sources, which cannot always be approximated as a point singularity, but the results for the sound fields of these singularities are nevertheless of general value in the interpretation of more complex effects. For the case of uniform motion it is found that the sound fields for each type of source are given (putting p M afp) by
1’ -yJ(xi -YJ a2 Tij ’=[(xi 1 xi -_Yi
p = [ 4mor2(1-MT)*
aF. -2
at
4m~r3(1--IMJ3at2
’
(3) (4)
Equation (2) gives the sound field radiated by motion of a “ basic acoustic singularity” with strength q. This is one part of the sound field produced by a moving “ simple source ” Q with q = aQ/at. Equations (3) and (4) g ive the sound field produced by uniform motion of point forces, Fi, and point acoustic stresses, Tij, respectively. In each of these equations Xi (i= 1,2,3) and yi (;= I, 2,3) are the Cartesian coordinates of observer and source, respectively, and r is the distance from source to observer ; r* = lr* I= (xi-yJ*. M, is the component of the convection Mach number in the direction of the observer, equal to Mz(xi-yi)=M,(xi -Yr) +Mz(xz-_Yz) +M 3(x3 -y3). Lighthill (2) used the form M,cos 19 for M,. Now the sound heard at any time t was actually generated at time t’= t - r/a0 and has travelled a distance r at the speed of sound a0 in the intervening interval. In fact, proper account of the effect of retarded time is one of the crucial requirements in acoustics, and each of the equations (2), (3) and (4) h as b een enclosed in square brackets, a notation used to indicate that the contents of the brackets are to be evaluated at the retarded time. A dimensional analysis can be performed on the results of equations (z), (3) and (4). The magnitudes of q, Fi and Tij are all proportional to p. U*l*, where U is a typical velocity and E is a typical length. In addition, a/& can be put proportional to a typical frequency n, and then by invoking constancy of Strouhal number, d/U, we obtain the dimensional dependencies shown in the third column of Table I. Each source has a typical dimensional dependence and this fact is often of value in identifying an unknown source type or predicting simple characteristics of a known source. However, before discussing the further repercussions of these dimensional results it is desirable to consider the directionality patterns of the various sources.
BASIC MECHANISMSOF NOISE GENERATION
437
Physical intuition suggests that a simple pulsating point mass source placed in the air has no preferred orientation so that its sound field is spherically symmetric. This type of sound radiation is known as a “ monopole ” field. Consider next the combination of two fluctuating mass sources of equal but opposite strength. It can be seen that there is zero total mass output from the combination, but that a fluctuating momentum transfer will occur. This fluctuating momentum may be interpreted as being the result of a fluctuating force. TABLE I
Dimensional dependence and directionality of various types of acoustic source
Source type
Typical cause
Monopole
Part of field of simple mass source
Dipole
Force in uniform motion
Quadrupole
Acoustic stress in uniform motion
Dimensional dependence
r
PO1
-
~
u2
I-M,
PO1 a,r (I
u3
PO1
u4
Directionality (when at rest)
0+ 8
&(ai@
Thus, the sound radiation of the fluctuating force can be regarded as the sum of two monopoles; i.e., a dipole radiation. The strength of the dipole is the product of the source strength and separation, and is equal to the magnitude of the force. Dipole radiation has a definite directionality associated with it, as shown in Table I, with the maximum sound radiation occurring in the direction of action of the force. In a similar fashion a quadrupole field may be constructed by combining two dipoles of equal but opposite strength. In this case a double directionality dependence occurs so that the radiation pattern of a quadrupole may either be a four-lobed “ lateral quadrupole ” or an extended two-lobe “ longitudinal quadrupole “. A quadrupole radiation field is appropriate to the acoustic stress. Because of the cancellation effects inherent in the higher order poles, the monopole or simple mass source is by far the most efficient producer of sound, at least at low speeds. However, Table I shows how an increase of velocity has a much larger proportional effect on the higher order poles. In fact, this velocity dependence can be traced directly to the mitigation of cancellation brought about by retarded time effects at high speeds. Thus, as speed increases the higher order poles become important sources of sound radiation. All of these results have applied to the case of uniform motion of the source through the air. In fact, many important noise producing components undergo accelerated motion due to their rotation. The effect of acceleration is to modify many of the conclusions reached above. Therefore, the sound fields due to the arbitrary motion of sources of sound must be considered. Practical application of the modified results leads to a number of interesting conclusions.
M. V. LOWSON
458 3. THE SOUND
PRODUCED
BY FORCES IN MOTION
It is convenient to consider first the sound produced by moving point forces. In reference (3) it was shown that the sound field for a point force in arbitrary motion is (xi-Yi) p =
4ra,,r*(r-M,)*
s+
Fi
at
I-M,
aMr
at
II ’
where the notation is as before. Note that an/r,/& = (xi-yi) (aMi/&). Extension of equation (5) to the case of a force distribution is easily made by putting Fi= PidS and integrating over the area of action S, taking appropriate account of retarded time. Comparison of this result with that given in equation (3) shows how generalization to an arbitrary motion causes an additional term to appear in the sound field. This second term in equation (5) is proportional to the acceleration (more strictly, to the component of the acceleration in the direction of the observer) so that for uniform motion equation (5) reduces to equation (3). However, the important and immediate conclusion from equation (5) is that accelerated motion causes additional sound generation. Performing a dimensional analysis on equation (5) on the lines followed before shows that
P
POJ - F(I
u3
Pal
-MJ*+&+
Here acceleration has been put proportional
to y,
u4
-MJ3’ as is usually the case. Comparison
of the result with those in Table I shows that the first term is the conventional dipole term as expected. However, the second term has the dimensional dependence of the quadrupole. By again referring to equation (5) it can also be observed that the first term has a single directional dependence, through (xi-yi) (aF,/at). The first term is thus dipole. The second term, however, has a double dependence, through both (xi-yi) Fi and aikf,/i3t, and is therefore quadrupole in type. Thus, equation (5) shows a quadrupole contribution of acceleration to the sound radiated. Identification of the acceleration term as quadrupole allows two immediate conclusions to be drawn. First, the effect of acceleration is relatively unimportant at low speeds, but is increasingly important at higher subsonic speeds. Second, typical radiation patterns for an accelerated motion can be expected to have directionality patterns similar to those shown in Table I. There are many applications of equation (5). There have been previous studies of the noise generated by accelerating systems; for instance, the case of propeller noise studied by Gutin (5) and Garrick and Watkins (6). However, previous studies have had to specify the motion in detail in order to commence the analysis. One advantage of the present results is that the result for any given motion may be obtained directly by substitution. Because the equation is completely general, the sound field can be predicted if the magnitudes and motions of the forces acting can be specified. It is thus possible to deal with any problem of the noise radiation by steady or fluctuating forces in any arbitrary motion. 4. APPLICATIONS
TO HELICOPTER
NOISE
An example of the application of the theory is the problem of noise production by a hovering helicopter. In principle, the rotor forces on a hovering helicopter should be constant. In practice, it is often found that these are fluctuating, but for the present purposes a steady rotor loading is assumed. The object of this analysis is to show how the coning angle and lag of a helicopter rotor blade lead to additional terms in the noise field. Nearly
459
BASIC MECHANISMS OF NOISE GENERATION
all helicopters have rotors which are hinged at the root and free to flap. In flight the rotor blade “cones ” at an angle fl (cf. Figure I), so that the hinge moment induced by the rotor thrust is balanced by that due to the centrifugal forces. Many helicopters also have drag hinges in addition, so that the rotor blade lags at angle y behind the perpendicular to the axis of rotation. The usual approximation for propeller noise theory can be made and the loading on the rotor thus reduced to a point force acting at some equivalent radius R from the hub. This force has the usual components of thrust T and drag D which act on the air in the opposite direction to that in which they act on the blade. However, due to the coning angle and blade lag there is an additional component of force which acts perpendicular to the axis of rotation. This component is denoted by C, and since both the coning angle, /3, and the lag angle, y, are small, C= /IT + ~0. The drag term yD is almost negligible but is included for completeness. Observer
Axis of rotation
Centrifugal
f
(b)
Figure I. Rotor forces on a hovering helicopter. (a) Blade “cones ” at an angle p so that hinge moment due to centrifugal forces cancels that due to thrust. (b) Loading imposed on the air is approximated by point force components.
Figure I shows the case assumed for the calculations. The sound field is axially symmetric so the analysis may be confined to the xy plane without loss of generality. Note that the thrust T and drag D act perpendicular to and in the plane of the rotor disc and are not necessarily equal to the actual thrust and drag experienced by a blade element. The components of the various parameters in equation (5) can be written down as
r = (xi-yi) = x,y-Rcos8, M = Mi = o, -MsinO, Fi = -T,
-RsinO,
Mcose,
-DsinB+CcosO,
DcosB+Csin9,
so that 8Milat = 0, - MS2 cos 0, - ML’sin 0, aFi/at = 0, - 06’ cos 8 - CQ sin 8, - DQ sin 8 + CQ cos and M, = (Xi-yi) Mi = -My sin B/r, aM,/al (~i-yi)Fi
= (Xi-yi) (aMi/at)
= -QM(y cos 8 - R)/r,
= -TX-DysinO+C(yc0~8-R),
(xi -yi) (aFi/at) = - DQ(y cos 8 - R) - Cy.Qsin 8.
8,
M. V. LOWSON
460
In these equations M is the rotational Mach number ( =QR/aO) and Q is the speed of rotation. Thus, substituting in equation (5) gives, after some cancellation i-2 s (ycod-R)(e-D) 4ra0 r’( 1+ My sin 19/y) (
P=
ysinO+(y2-ayRcosB+R2)r
. 111
Equation (7) gives the pressure as a function of time at the point (x,y). It is an approximation since the blade loadings have been reduced to equivalent point forces but, as mentioned previously, the effect of distributed loading may be calculated by replacing the point forces by pressure distributions and integrating over the area of action taking appropriate account of retarded time. In fact, it turns out that in the far field, for point forces, the frequency spectrum of the pressure pattern can be calculated. This is done in the Appendix, and the result obtained is that the magnitude of the sound pressure on the mth harmonic of a B blade rotor is l&l
= +g
(J,,+1 (+y
(yy)‘1”‘.
-J&?--1
(8)
Here JmB is a Bessel function of the first kind and of order mB, and r is the distance from the propeller center, (x’ +y2)‘j2. The term (JmB-l - J,,,B+1)1/2is typically about the same order of magnitude or slightly larger than the term Jma and hence the outward components of force, C, on a coned or lagging helicopter blade can be important. The first term in equation (8) is the term obtained by Gutin (5) and Garrick and Watkins (6) for the case of a propeller. The difference between quadrupole and dipole effects may be observed in both equations (7) and (8). The drag term is essentially dipole in type so that it produces a two lobe sound pattern and is important at lower speeds. The thrust term is essentially quadrupole and produces a four lobe sound pattern which is important at higher speeds. The combination of these two effects gives rise to the well-known phenomenon of maximum sound production slightly behind the disc, as shown in Figure 2.
+ + w
Rotor oxis
Rotor oxis
t
(a)
t
(b)
(cl
Figure 2. The cause of the sound directionality of propeller noise due to thrust and drag. (a) Dipole type contribution due to drag. (b) Quadrupole type contribution due to thrust. (c) Total sound directionality pattern.
The additional noise due to the outward components of force is also a quadrupole type field so that it is significant at high speeds. However, as shown in Figure 3, the directionality pattern is the extended double lobe of the longitudinal quadrupole rather than the four lobe lateral quadrupole case appropriate to the thrust noise. The maximum for the noise due to outward forces occurs at right angles to the propeller axis, which is just the position
BASIC MECHANISMS
OF NOISE GENERATION
461
where it is not desired on a helicopter. In most practical cases the effect of this new source of noise is of minor significance. In a typical case when the rotational Mach number of the effective point force is 0.5, T/D= 12, and 8=4”, sample calculations show the noise to be increased by about 2 dB in all harmonics. However, the adoption of rotors with higher tip speeds, or improved lift to drag ratios, could easily lead to important contributions from this source. The use of equation (5) has given, without much difficulty, the characteristics of a new source of sound from a hovering helicopter. In fact, the noise field generated by fluctuating forces in any specified motion can be written down with almost the same ease. A problem that is relevant to helicopters, VjSTOL Aircraft and GEMS is the case of fluctuating forces in a rotating and possibly convected system. The existence of fluctuating forces on a helicopter blade in forward flight is well known, but V/STOL aircraft also may often
Rotor
axis
t
Figure force.
3.
Approximate
directionality
pattern
for the sound
due to outward
components
of
have uneven forces on their rotating components due to uneven flow into the fan, etc. Some designs of hovercraft utilize rotatable propellers which often work in non-axial flow with consequent fluctuating forces imposed on the blades. An expression giving the noise radiated by fluctuating forces in a convected rotating system can be derived by substitution into equation (5). This was done in reference (3) with the result p =
{MTx - rD( I -Mor)}Q(y 4nao{r( I -lb&,)
+ My
cos 0 - R) _~@Dpt) sin 8 + @T/at) 4~a&( I -&lo,) + My sin ey
sin
ey
1’
(9)
where MO, is the component in the direction of the observer of the convection Mach number of the center of rotation. Equation (9) shows two terms contributing to the noise : one dependent on the instantaneous value of the forces and one dependent on their rate of change. Thus, once thrust and drag are given as functions of time or blade position the noise radiated can be calculated using equation (9). Equation (9) will also give the noise radiated by helicopter blade slap or wake and fuselage interference providing the loadings imposed by these phenomena can be predicted. Equation (9) assumes zero coning angle or blade lag although these could be incorporated if desired, with consequent lengthening of the expression. Another effect which could be important on helicopters is the flapping motions of the blades. It is apparent from the standpoint of the general theory of equation (5) that flapping accelerations could cause noise to be radiated, and again this effect could be incorporated if desired. The magnitude of this effect can be estimated by comparing the flapping accelerations to the centrifugal accelerations and would probably only be important for the higher harmonics. A further noise problem which can arise in rotating systems is the broad band or “vortex” noise caused by the random pressure patterns that exist on a rotating blade with a turbulent boundary layer. The most complete attempt to predict the sound radiated 31
M. V. LOWSON
462
by these random pressure patterns was made by Sharland (7). He did not include the effect of the acceleration terms in his calculation, but this would not affect the validity of the results in his range of interest with blade tip speeds less than M= 0.4. However, the effect of the second, acceleration, term in equation (5) could be important at higher Mach numbers. Unfortunately, theoretical study of this problem is difficult since specification of the random pressure patterns in a sufficiently detailed manner leads to a number of difficulties. One important problem arises from the possibility that these random pressure patterns do not, in fact, undergo centrifugal acceleration during their passage over the blade. Some additional problems were discussed by Lighthill (8). A valuable experiment would be to investigate the velocity dependence of the “vortex ” noise component at higher tip Mach numbers. This would show the general importance of the acceleration terms. However, resolution of these problems will require detailed study. 5. “SIMPLE
SOURCES”
IN MOTION
The analysis which gave the sound field for a point force in motion can be adapted to give the sound field for a point source in motion. This was done in reference (3) and the result obtained was
(10)
* =
The notation is as before. Again this result proves to be the sum of two terms. Since 4 = aQ/i3t, the first term in (IO) is identical to the result in equation (2). The second term is identical to the result for the arbitrarily moving force given in equation (5) except that Momentum transfer Mass0
_
Gin
Figure 4. Showing cause of momentum transfer on a convected “simple” mass source.
the term Fi is replaced by Qv~, which gives the rate of momentum introduction. The reason for this second term can be readily understood. The quantities in equation (IO) are evaluated relative to the moving system so that the introduction of mass in the moving system naturally implies an introduction of momentum relative to the free air. Even for the idealized case of a convected pulsating sphere the momentum is not instantaneously balanced. Figure 4 shows how the mass output of the system arises at a different location to the mass input and thus gives rise to a net momentum fluctuation. In fact the two terms in equation (IO) may be combined in a simpler form as
*=
I [ 4~(1-M,)~
aQ
Q
( at+I-M,?K
aMr
)I’
(11)
which therefore represents the general result for the sound field produced by a “simple source ” in arbitrary motion. Even for uniform motion this expression differs from that
BASIC MECHANISMS
OF NOISE GENERATION
463
given in equation (2), but this difference may now be readily understood as being the result of the momentum terms. In applying these results, for instance, to the case of a tip jet rotor, it is clearly important to include the effect of the momentum terms. In fact, a tip jet rotor represents an important source of momentum as well as of mass so that both of these variables require consideration in a complete study. The present results show that if the momentum output is calculated relative to the rotor, then equation (I I) should be used for the effect of the mass output. Alternatively the momentum output can be calculated relative to the still air and equation (2) used. In either case, the effect of the momentum terms would be calculated via equation (5). M ore d e t ai 1e d ca 1cu 1a t ions of the sound field could follow the lines of section 4. 6. ACOUSTIC
STRESSES
As might be expected, the analysis which applied applied to the case of acoustic stresses in motion. was that
(xi-yi)(xj-yj) * = [ 4mgr3(1
-M,)3
IN MOTION
to forces and sources in motion can be The result obtained in reference (3)
azTij I aT,3(aM,/at)+ -~ Tij a’M,+
1 at2
at
I-M,
I --al,
at2
$JJ 2 .
3Tij (I -M,)2
(
at
c12)
HI
The first term in equation (12) is identical to the result for uniform motion given in equation (4), but again the effect of acceleration is to introduce additional terms into the sound radiation field. Equation (12) contains terms dependent on acceleration, rate of change of acceleration and acceleration squared. Interpretation of Tij is always difficult, particularly since the approximation of putting p w a& often requires careful consideration. When these difficulties are combined with the complexities of the equation above it becomes extremely difficult to make any detailed prediction. All that can be achieved at present is a broad statement of the possible importance of the acceleration terms. In general, it appears that accelerated turbulent eddies produce more noise than those moving at constant velocity. This could be very important in such cases as shockturbulence interaction, and in the general production of noise from turbulence at supersonic speeds. However, the applications of immediate interest in this case are to tip jet rotors and ground effect machines. The sound field produced by tip jet rotor turbulence is a major source of noise, and equation (I 2) indicates that centrifugal accelerations increase this noise. Unfortunately, it is difficult to be certain that tip jet rotor turbulence actually undergoes centrifugal acceleration so that it is possible that these remarks will not be relevant. However, it is certain that the turbulent eddies undergo acceleration in the peripheral jets of ground effect machines since these vehicles derive their lift from jet deflection. At present, even large hovercraft operate at relatively low jet speeds so that jet noise is not generally significant. However, successful development of these vehicles will inevitably lead to higher jet exit velocities when the effects of jet noise will be important. The acceleration terms may well require consideration in this case. The noise arising from the deflected jet flows of STOL aircraft is another case where the effects of acceleration could sometimes be important. 7. DISCUSSION
AND
CONCLUSIONS
Because the sources of noise on all these vehicles are so complicated it is often very valuable to return to basic principles. Classification of a source as monopole, dipole or quadrupole usually reveals the speeds at which it can be expected to predominate, its probable significance, and sometimes its directionality pattern. An interesting example of
464
M. V. LOWSON
this was furnished in a recent paper by Ffowcs Williams and Gordon (9). They showed, as expected, that the quadrupole effects of the acoustic stresses in the turbulence are responsible for jet noise at high subsonic Mach numbers, but, they also showed how, at low speeds, say M < 0.4, the monopole effects of the jet mass fluctuations induced by jet pipe boundary layer turbulence are predominant. At an intermediate range of Mach number the dipole contribution of the fluctuating forces induced at the end of the jet pipe is significant . Thus, noise which apparently arises from a single phenomenon (in the case above, jet turbulence) can often be found to be quite a complicated effect. For instance, the possible importance of acceleration on the turbulent noise produced by tip jet rotors, STOL aircraft and GEMS was pointed out in the last section. This conclusion is valid only at high speeds. Both tip jet rotors and GEMS can be expected to generate monopole type noise by mass fluctuation at lower speeds. In particular the thin, two-dimensional, low speed, peripheral jets currently favoured by hovercraft designers may be expected to be quite efficient generators of monopole noise. Again, the noise radiated by the deflected jet streams of STOL aircraft would be expected to arise mainly as a dipole source from the random pressure patterns occurring over the deflecting surfaces. Noise radiation from such sources was considered by Sharland (7), and his work may be extended to cover this case. In general, acceleration causes higher order poles to occur in the sound field. For a force in arbitrary motion a quadrupole acceleration term is added to the conventional dipole field. For the acoustic stress, acceleration effects cause the addition of two higher orders of pole. A moving source of mass is necessarily associated with a source of momentum. The simple monopole field of the mass source is reinforced with a dipole field due to the momentum source, and acceleration effects on the momentum source cause a further quadrupole field to be added. All these acceleration effects generally become increasingly important at higher speeds. The general expressions found for the acceleration effects are convenient for practical applications. For example, the analysis shows directly that the outward components of force induced by blade coning angle and lag on a hovering helicopter give rise to a new type of sound field with a maximum in the plane of the rotor disc. A general formula for the calculation of noise radiation from fluctuating forces in a rotating and convected system can be applied to direct calculation of the noise radiation from yawed propellers, helicopters in forward flight, and fans with uneven velocity distribution, providing the appropriate loadings can be specified. The results are particularly useful when the acoustic sources can be reduced to point singularities. For the more frequent practical case of an extended distribution of noise sources calculation of the sound field requires evaluation of the integral forms of the equations, taking appropriate account of retarded time. However, this analysis can usually be made on a computer. ACKNOWLEDGMENTS The general results described in this paper originated from work at the Institute of Sound and Vibration Research, Southampton University, and the writer would like to thank Professor E. J. Richards for initiating his interest in the problems. The applications of the results discussed herein were made under Wyle Laboratories Independent Research Program with the guidance and encouragement of K. McK. Eldred. REFERENCES volumes) New York: Dover Publications. Second
I. LORD RAYLEIGH 1877 Theory of Sound (two
edition, 1945 re-issue. a. M. J. LIGHTHILL 1952 Proc. R. Sot. A 211, 564-587. On sound generated aerodynamically. I. General theory.
BASIC
MECHANISMS
OF NOISE
465
GENERATION
M. V. LOWSON1965 Proc. R. Sot. A 286, 559-572. The sound field for singularities in motion. P. E. DOAK 1960 Proc. R. sot. A 254, 129-145. Acoustic radiation from a turbulent fluid containing foreign bodies. Tech. Memo 1195 (1948). On the sound field of a rotating propeller. 5. I,. GUTIN 1936 N.A.C.A. (Translation.) Rep. I 198. A theoretical study of the effect 6. I. E. GARRICKand C. E. WATKINS 1954 N.A.C.A. of forward speed on the free space sound pressure field around propellers. I. J. SHARLAND1964r. Sound l/‘;b. I, JOZ-pz. Sources of noise in axial flow fans. ii: M. J. LIGHTHILL1962 Proc. R. Sot. A 267, 147-182. Sound generated aerodynamically. ourna~3, 791-793. Noise of highly 9. J. E. FF~WCSWILLIAMSand C. G. GORDON1965 A.I.A.A.J turbulent jets at low exhaust speeds. IO. N. W. MCLACHLAN1955 Bessel ~unctionsfor &zgineers. Oxford University Press. 3.
4.
APPENDIX ANALYSIS
OF THE FAR FIELD
FREQUENCY
SPECTRUM
FOR A HOVERING
HELICOPTER
Equation (7) gave the sound as
.
(
-C
~sinBf~(y2-zyRcosB+R2)
Now
r2 = lrl2 = ~~+y~+R~-zyRcosO,
(AI)
so that in the far field, where r 9 R, r z rl - (yR/ r 1)cos 8 where rr is the distance from the rotor centre, (x2+y 2) 1/2. The effect of retarded time (t-r/aO) need only be retained on 8, SOthat
[e] = T+$GO~[ej, where T=Q(~-rI/ao). With these approximations be written down as 2n+a (;I)=:
1
the component
QV
642)
of far field sound in the nth harmonic may
e - Crl(~ + sin (I +crsin8)3
(MTzc - DY t) cos
a(
OL
cos
e) )(
127
d7
sinn~ 1
’
(A3)
where a = My/rl. The integral can cover any interval 2~. Equation (A3) is evaluated by changing the variable from T to 8, using the form indicated by equation (Az), viz : 7
=
(A4)
8-cCCOS8.
Thus
a,
0 bn
s
QY
=4= 0
cos 8
--I and 0c(r+crsinO) s equation (As) may be integrated by parts to give Now
(r +ccsin8)2
d2y
a,
0 b,
de=
=4-
2nMTx-
Jl 0
a
Drl r
1
so that
-sin{n(O-cccose))
I(
cos 8
cos
(f2(e -a
cos e))
de 0w 1
??
466
M. V. LOWSON
The integrals in equation (A6) are all various forms of Bessel functions. McLachlan (IO) gives (formula 42) Jn(z) = ‘G jexp{i(zcosB+nB)jdB.
(A7)
0
is a Bessel function of the first kind and of order n. In the present case, x is a real quantity, so that expanding equation (A7) and taking real and imaginary parts gives
J&z)
297 s 0
cos(zcosO+nO)dO
=
-
I)“‘2J,(z),
n
n
,
2n I 0
24
o
sin (Z cos 8 + ne) df9 =
27r(-
I)'"-'"~J,&),
0,
AIs0 cos ecOs(zcos e+ne) = &{COS[ZCOS e+ (n + identities (A8) gives
even odd 1
odd n even
(AS) '
n
I
I) e]+ COS[ZCOS e+ (~2 - I)e]).Thus,
using
2n I 0
cosec~~(~~o~e+~e)
de =
4-I)
(n+1)'2{Jn+i(4 -J+&))>
n odd 71even
(0,
and similarly 27 c0sesin(xc0s8+d)
I
de =
0
n
r( - I)“‘2url+1(d -Jn-1(z)], o 3
even
n odd
Using (A8) and (Ag) in (A6) then gives a, = -2:
4 =
‘3
(MT’~Drl]
2( - I)(n-1)12Jn( -ntL), n odd,
(- I>“‘~{.L+~( -4 -.L(
-4,
n even,
and n
b, = -f&~(
even,
- Ip+l)'2{Jn+l(-n0~)-_~-1(-n~)}, n odd.
Now Jn(- 4 = ( - I)“_L(+ so that the magnitude of the sound pressure in the nth harmonic finally becomes lAn[ = (af+bi)‘i2
=
where CLhas been replaced by its equivalent, My/r,. For a B bladed propeller, harmonics which are not multiples of the number of blades cancel, leaving the result as given in equation (8). This paper was first presented at the Symposium on the Noise and Loading Actions of Helicopters, V/Stol Aircraft and Ground Effect Machines, September x965 at the Institute of Sound & Vibration Research of the University of Southampton.