Fundamentals of aerodynamic sound theory and flow duct acoustics

Journal of Sound and Vibration (1973)28(3), 527-561

FUNDAMENTALS OF AERODYNAMIC SOUND THEORY A N D FLOW DUCT ACOUSTICS P. E. DOAK

Institute of Solmd and Vibration Research, Unirersity of Southampton, Southampton S09 5NH, England

(Receired 16 February 1973)

Some fundamental aspects of the theory of internally generated sound (or "sound generated aerodynamically") are reviewed and discussed. Particular stress is laid on the functional relationships between the radiated sound field and the equivalent source distribution of Lighthili's "acoustic analogy" model, as exposed by multipole analysis. Recent theoretical and experimentalprogress in both turbulent mixingregion noise and flow duct acoustics is cited, and discussed in the context of its fundamental impIications for the future development of "aerodynamic noise" theory.

I. INTRODUCTION To experimenters and engineers, theories are rather like travel brochures are to tourists. Good ones should point out the places of interest, and indicate the best ways of getting to them. For vists to Rome, they should prepare one to do as the Romans do. In short, one should be able to return from the holiday, look at the brochure again, along with the holiday photographs, and say "it was all that, and more". Thus, to the tourist, the brochure and the photographs are, respectively, the means of selectively anticipating and remembering the real experience of the holiday. For experimenters and engineers, then, aerodynamic noise theories are naturally and inevitably valued in terms of their potentialities for guiding and codifying the realities of physical observations and machinery design. In this brief review; covering mainly developments in the last decade, progress in the fundamentals of aerodynamic noise theory and in the derivative subject of flow duct acoustics is described and discussed in this context: that of the needs of experimenters and engineers. As this review forms part of the University of Southampton Institute of Sound and Vibration Research Tenth Anniversary Report (19631973), it is also natural that Southampton contributions, and points of view, are emphasized. The review is not strictly chronological in form. This is because, during the past decade, development has proceeded along two parallel, but conceptually different, lines. For purposes of broad classification the two lines of approach may be called the "equivalent source" (or "acoustic analogy") approach, and the "true source" approach, respectively. In the "equivalent source" approach, the actual pressure (or mass density) fluctuations are regarded as occurring in a fictitious "acoustic" medium, otherwise at rest (or in uniform motion), rather than in the real fluid. When regarded in this way, of course, the actual pressure fluctuations appear to be, everywhere, "acoustic" fluctuations (i.e., isentropic and of small amplitude), generated by a certain volume distribution of "equivalent sources", which is known, to sufficient accuracy, if the actual mean and fluctuating flow of the fluid is known, everywhere, to sufficient accuracy. 527

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It is self-evident that this approach is so tautological as to be useless if the "equivalent source" distribution cannot be specified a priori, independently of the pressure fluctuations that it is regarded as causing. The ultimate success of "equivalent source" models, therefore, inherently depends on the extent to which, in any given physical situation, it happens to turn out that the "equivalent sources" can be identified, and described either experimentally or theoretically, independently of the pressure fluctuations that the model is supposed to predict. No intrinsic guarantee of separate specification of cause and effect is provided by this type of theoretical model. It is also self-evident that the user of an "equivalent source" theory is constrained to think of the pressure (or mass density) fluctuations, everywhere, as an "acoustic" field: i.e., as a superposition of waves, each propagating at the speed of sound in the fictitious, "acoustic" reference medium. Of course there is nothing basically illogical in this, from a fundamental, theoretical point of view. Many specific instances are well known in theoretical physics where sets of such "acoustic" functions exist which, as it were, form a complete set of eigenfunctions into which any function of space and time can be analysed. Nevertheless, the physically minded experimenter is understandably uncomfortable, and often confused, when he is forced, for the convenience of the theoretician, to think of, say, an essentially frozen, convected pressure pattern (such as that in a v o n Karman vortex street, for example) in terms of a lot of acoustic waves, going in all directions, perhaps with complex wavenumbers, and all of this going on in a fictitious medium and not in the real fluid whose motion he is trying to picture in his mind and ultimately understand. In the "true source" approach, the shle qua non is ab hdtio and hi conthnto adherence to explicit, independent identification of cause and effect. The fluctuations in pressure, or in any physical quantity, are to be seen in the model as occurring in the manner in which they actually occur in the real fluid under consideration, whether this manner is "acoustic" or not. This prescription, for example, explicitly prohibits an "equivalent source" representation of refraction of acoustic waves by mean temperature or mean velocity gradients, which, of course, is permissible in an "equivalent source" model. If the fluctuating pressure, say, is the dependent field variable to be predicted by the theoretical model, then the "true sources" of this fluctuating pressure field, as identified in the model, must be explicitly independent of the fluctuating pressure. Further, the model should provide means for positive theoretical or experimental identification of the "true source" distribution, independently of the fluctuating pressures. To the physically minded experimenter, then, a "true source" theoretical model of, say, the generation and propagation of a fluctuating pressure field provides a direct representation of the actual processes going on locally, in the real fluid, without reference to a fictitious "acoustic" medium. Broadly, and somewhat invidiously, speaking, the present state of affairs is that "equivalent source" theoretical models tend tobe physically uninterpretable and "true source" theoretical models tend to be mathematically insoluble. Because of the fundamentally different philosophies behind the "equivalent source" and "true source" approaches, the two types of model are initially described and discussed separately in this review, more or less chronologically under each heading to point up any possibly significant historical trends. In the final section of the review, some comparative evaluations of the present potentialities of models of both types are attempted. As much of the ground has recently received detailed coverage in another review [l], especially aspects relevant to jet noise, the main emphasis here is on brevity and clarity, on fundamental matters of consequence to the theory of aerodynamic noise in general, on applications to flow duct acoustics, with reference to experimental evidence, and on Southampton contributions to the subject.

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2. "EQUIVALENT SOURCE" MODELS 2,1. LIGHTHILL'S ACOUSTIC ANALOGY

In his paper entitled "On sound generated aerodynamically: 1. General theory" [2], Sir James Lighthili provided the first complete formulation of the concept of "sound generated aerodynamically" and established an exact mathematical framework for this concept, in the form of an "acoustic analogy". In his own words, the idea of Lighthill's acoustic analogy is as follows (see reference [2], p. 566): "Consider a fluctuating fluid flow occupying a limited part of a very large volume of fluid of which the remainder is at rest. Then the equations governing the fluctuations of densiO' in the real fluid will be compared with those which would be appropriate to a uniform acoustic medium at rest, which coincided with the real fluid outside the region of flow. The difference between the two sets of equations will be considered as if it were the effect of a fluctuating external force field, known if the flow is known, acting on the said uniform acoustic medium at rest, and hence radiating sound in it according to the ordinary laws of acoustics." The beauty of this concept is not so much its mathematical exactitude as its simplicity, and this simplicity makes it useful as well as beautiful. The only logical flaw in the concept (which for fluids of small viscosity and thermal conductivity does not appreciably impair its utility but only its conceptual beauty) is that Lighthilrs choice ofdensiO', rather than pressure, as the variable whose fluctuations in the real fluid and the reference "acoustic medium at rest" are to be compared, makes the concept inapplicable, strictly speaking, to viscous, thermally conducting fluids (see pp. 289-298 of reference [1]). The physical reason for this is that (as Lord Rayleigh showed in 1877 [3]), in the limit of small amplitude motion of a (weakly) viscous and thermally conducting material, it is only the pressure fluctuations which propagate exehtsively as acoustic waves; the density fluctuations are made up of two components, an acoustically propagating part adiabatically related to the pressure fluctuations and a thermal part which propagates, independently of the pressure fluctuations, in the form of thermal diffusion waves. Thus "a uniform acoustic medium at rest" is physically one in which pressure fluctuations (and those parts of the density fluctuations adiabatically related to these) propagate acoustically. For this reason, and to facilitate later comparisons among the various theoretical models, it is preferable here to pretend that Lighthill had used the word pressure instead of density in the above quotation and to express his acoustical analogy model in formal mathematical terms accordingly. The result is (see reference [1], p. 294), in Cartesian tensor notion, O' p Ox2

C2 - -

OZp Ot2

OZL~l 2 c~ Ox~ Oxj

0z Ot 2 (p -- c2 p)'

(1)

where p is the thermodynamic pressure, p is the mass density, c~ is the (constant) speed of sound in the reference "acoustic medium at rest" and L~j is the Lighthili tensor (this terminology is the author's) Ltj = pvl vj + Po -- Pgij,

vi being the barycentric particle velocity, Pt~ the stress tensor and 61j the Kronecker delta (61j = 0, i ~ j ; 6tj = 1, i = j ) . This inhomogeneous, "acoustic analogy" wave equation for the pressure is formally exact, for aJO, classical continuum (i.e., non-relativistic, so that mass is locally conserved) in the

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absence of external body forces (i.e., gravitational or electromagnetic forces). All the quantities in the equation are the exact, total quantities: e.g., p is the complete thermodynamic pressure, including both mean and fluctuating parts. Also, apart from cz, which has been introduced to characterize the fictitious reference medium (the "uniform acoustic medium at rest!'), all the variables represent the actual respective physical quantities in the real continuum being considered. The right-hand side of equation (1) is, in accord with Lighthill's prescription, the difference between the respective equations for the actual pressure when viewed as occurring in the reference acoustic medium and when viewed as occurring in the real continuum. Thus equation (1) expresses the actual pressure field in the real continuum but viewed as an "acoustic" pressure field, occurring in the reference "acoustic" medium and caused by a certain "fluctuating external force field", which, according to this interpretation of the equation, makes its presence felt as an "acoustic source" distribution for the pressure in the reference medium, of strength a2 L u q(xk, t) =

1 az + -" at'--";'z~" (P -- c~ p)

Ceo

per unit volume. Hence, ~this source strength density, q(xk, t), is known, for anyparticular real fluctuating fluid flow, say, then the real fluctuating pressure associated with it (and indeed the total pressure) can be calculated from equation (1), regarded as an inhomogeneous scalar wave equation,

02p

ax~

I aZp i = - q ( x k , t),

c~ at2

subject, ofcourse, to suitable boundary conditions. In other words,in so far as q(Xk, t)is known for any arbitrary fluctuating fluid flow, the otherwise intractable problem of solving the full non-linear equations of transport of mass, linear momentum and energy can be reduced to the infinitely simpler and relatively well-understood problem of solution of the inhomogeneous scalar wave equation (at least as far as determining the pressure is concerned). This is the beautifully simple, and potentially powerful, mathematical essence of Lighthill's "acoustic analogy" concept. There are three absolute prerequisites for successful application of the acoustic analogy model to prediction of the actual sound pressure field generated aerodynamically in a real physical situation: (i) q(xk, t) must be known to sufficient accuracy for the particular situation; (ii) it must be possible to similarly specify suitable boundary conditions for the pressure; (iii) suitable mathematical techniques must be available for solving the resulting boundary value problem for the inhomogeneous scalar wave equation. Prerequisites (ii) and (iii) are largely mathematical matters. The inhomogeneous scalar wave equation has been known and assiduously studied for over 150 years (one might even say, with Pythagoras in mind, for over 2150 years). In connection with acoustical theory, however, at least before 1952, far more attention was devoted to study of the equation in its homogeneous form, since most problems of interest were concerned with acoustic fields produced by vibrating surfaces. The most concise and yet general formal solution of boundary value problems for the inhomogeneous scalar wave equation is probably that in terms of a Green function. Certain aspects of the nature of the dependence of the pressure, p, on the equivalent "acoustic" source strength density, q(x,,t), as revealed by this formal solution, are highly relevant to applications of Lighthill's "acoustic analogy" model to real problems of aerodynamic sound generation.

531

AERODYNAMIC SOUND A N D D U C T ACOUSTICS

2.2.

SOME ASPECTS OF THE FORMAL SOLUTION OF THE INttOMOGENEOUS SCALAR x,VAVE EQUATION

The essence of the Green function method is that, since the equation to be solved 02p

1 32p

ax]

2 0i 2 coo

-q(xk, t),

(2)

is linear in the pressure,p, the solution can be formed by superposing the separate contributions from each volume element of the source distribution, q(xk,t)dxl dx2dxa. Hence, instead of solving equation (2) directly, the first task is to solve the inhomogeneous scalar wave equation for the pressure, po(Xk, tlyk, 0 due to a point source (in space and time) of unit strength, located at any arbitrary point in space and time (Yg,0:

2 ~-2" p~(xa, tly~, ~) = -cS(xk -y~)cS(t - O,

(3)

Coo

where 6( ) is the Dirac delta function and the notation ~5(Xk-- 3'a) means 6(xl - y~) 6(x2 - Y2) 6(x3 -- Ya)The pressure due to this "point source" in space and time, p6, satisfying equation (3), is called a Green function for equation (2), for reasons to be made apparent in the next paragraph. Physically, it is thc "impulse" pressure response--the pressure due to a source that is "impulsive" in nature in both space and time. When such a Green function, or impulse response, satisfying equation (3), can be found (i.e., when it "exists"), it is a straightforward matter to prove, by simple manipulation of equations (2) and (3) and use of Green's Theorem relating certain volume and surface integrals (see, e.g., reference [5], volume l, pp. 834-837) that the desired pressure, p(x~,t), satisfying equation (2), can be expressed as

p(xk, t) = j" f q(Yk, "OPo(Yk, ZlXk, t) dyl dy2 dy3dr +

p , ~

- p

dS(y,) dr

TV

P6

C~o

-- P T ~ | dyt dy2 dy3 dr,

(4)

T I,"

where Vis any region in space and Tany interval in time in which both equations (2) and (3) are satisfied, Sis the boundary surface of Vand 0/0n is the component ofthe gradient operator in the direction outward from V. In the second and third integrals of equation (4), explicit indications of the arguments of the functions o f p and p~, which are P(Yk, r) and P~(Ya, ~[Xk, t), respectively, have been omitted for brevity. Physical interpretations of the three integrals on the right-hand side of equation (4) are as tbllows. The first integral, self-evidently, includes the anticipated superposition of the individual direct contributions to the pressure p(xk, t) from the source elements q(Yk, t)dyl dy2 d3'adz lying within the volume V and the time interval T. The second integral, involving the integration over the surface S, includes the contributions from sources outside V, and also can include the effects of reflection of waves from sources in V by any parts of the surface S. It is a straightforward matter to prove (and thus express) these interpretations of the first two integrals by demonstrating that the first integral is a particular solution of equation (2) whilst the second integral is a complementary solution: i.e., it satisfies equation (2) with the righthand side equal to zero (the homogeneous part of the inhomogeneous equation (2)). The last integral of equation (4) (i.e., the second of the volume integrals; see again, e.g., reference [5],

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P.E. DOAK

volume I, pp. 834-837 for a proof of this statement) does not represent any contributions to the pressure, p(xk, t), that are caused, either directly or via reflections, by the source distribution, q(x~, t), in the time interval T. Rather, it represents the initial value problem solution of the homogeneous equation: that is, it describes the redistribution, as time goes on, of any hlitial pressure field that may have been present at the beginning of the time interval T. For physical purposes, of course, it is only necessary to consider mathematical models in which the source distribution q(xk, t) and the pressure (x~, t) are both zero before some particular time, to, at which time the source is first "turned on". Thus, and again this argument basiizally depends upon application ofthe principle ofsuperposition (in this case superposition in time), it can be said that whatever the contribution of the third integral in equation (4) may be, it is wholly irrelevant to the problem of determining the pressure field caused by a source distribution q(xk, t) that can be regarded as turned on at some specified, finite time, to, and known thereafter as a function of position and time. For purposes of further discussion here, therefore, the third integral can be dropped. The physical significance of the first two integrals in equation (4) can now be looked at in a little more detail. Although, in any physical problem to which equation (4) is applied, suitable boundary conditions for the pressure will be prescribed by the physical situation, there are no such physically required boundary conditions on the Green function, p~. Equation (4) is formally valid as long as p~ satisfies the inhomogeneous partial differential equation (3), and thus the Green function P6, can be required to satisfy any convenient boundary conditions, provided only that these are appropriate to the scalar wave equation [6]. It is usual, on physical grounds, to require the Green function to satisfy a causality condition: i.e., reception of a signal by an observer cannot precede emission of the signal by the source. Subject to this condition, but without any further loss of generality, the Green function can be written as the sum ofthe"free space" Green function,p~.~, and a complementary function, P6,c, the latter satisfying the homogeneous scalar wave equation and any convenient specification of either its value or that of its normal derivative (specification of a linear combination of these values is also permitted) on the boundary S:

p~(x,, t l)'k, r) =

,~[r - (t - [x~ - y d l c ~ ) ]

4nix,

-3',1

F po. c(x,, t),

(5)

the first term being, of course, the free space Green function, which self-evidently displays its adherence to the well-known and important principle of reciprocity: pn. ~(x,, tlyk, 0 = pn. ~(y,, rlx~, t).

A possible choice of p6, c(xk, t) is zero; when this choice is made equation (4) becomes the well-known Kirchoff-type solution of the inhomogeneous scalar wave equation:

p(x,,,)= f q(),k,t[xk-),~[/c~) ~-x--~ --5,~ dy, dyz dy3 + v

f f( P ' ~ ' ~ o,

Onty,)]

TS

(6)1 The volume integral in this expression represents the superposition of the waves propagating directly to the observation point from each of the volume elements of the source distribution. In the surface integral, the first term represents contributions from a surface source distribution of strength Op/On per unit area and the second term represents contributions of a surface 1 in equation (6) the volume integral is given in a form that is strictly valid only when the volume Vis not a function of time, in order to make its physical interpretation more clearly apparent. The restriction is not necessary, however, as the basic equation (4) can include cases where Vis time-varying. In such more general cases the physical interpretation of the volume integral is in no way essentially different.

AERODYNAMIC SOUND AND DUCT ACOUSTICS

533

dipole distribution of strength p per unit area, the elementary dipoles being oriented normal to the surface, in the outward direction. Since the pressure is a scalar field which is regular and continuous in V (including S), p and Op/Oncannot be specified independently on S; the existence ofp(xk, t) in Vimplies compatible values ofp and Op/Onon 5'. Further, because of the surface integral, equation (6) is really an integral equation for p(xk, t), which, however, permits calculation ofp(xk, t) everywhere in the volume V if the compatible values o f p and Op/Onare known only on the boundary surface S. When, say, measured (or theoretically known) values of Op/On on S are available, the Green function boundary condition can be chosen as

ap, lOn= 0 on S. Then, where this Green function is denoted asp~+,t equation (4) for the pressure becomes m,,,) =

ap dS(y,) dz. f f q(y,,z)p,+(x,,tly,,z)dY, dy2 dy3dr + f f p,+ an(),,--~ T$

(7)

TS

Here the volume integral includes not only all the waves radiating directly from the volume elements of the source distribution (as explicitly shown in the volume integral of equation (6)) but also all the reflections, and multiple reflections, of these waves from the boundary S as an acoustically rigid surface. Similarly, if the pressure on S is known the choice p~ = 0 on S can be made. The volume integral then represents the waves propagating directly from the various volume elements of the source distribution together with all the reflections and re-reflections of these waves from S as a pressure-release surface. Again, in situations where an acoustic impedance is known on S, the Green function can be chosen to satisfy this impedance condition and the volume integral then represents the directly radiated waves from the source distribution together with the reflections and rereflections of these waves from the impedance boundary S. All these three cases are special cases of a more general impedance boundary condition, Aapdan + Bp6 = 0, on S. The "hard" wall case, apdOn = 0, is that where B = 0 and the "soft" wall case, p~ = 0, is that where A = 0. Now if the problem were really only an acoustic scattering problem, instead of an "acoustic analogy" ofaerodynamic sound generation and propagation, then, say, ifSwere the composite surface of rigid, fixed bodies immersed in the real fluid, ~p[Onwould, on physical grounds, have to be zero on S. In this case, therefore, choice of apdan = 0 on S would make the surface integral in equation (7) (or in equation (6)) vanish completely, because both the pressure, p(xk, t), and the Green function, p~(xk,t]yk,z) then satisfy the same boundary conditions. More generally, in acoustic scattering problems, it can thus be arranged for the surface integral to vanish whenever the pressure, p(x~,t), satisfies an impedance condition of the type A Op/an + Bp = 0, simply by requiring the Green function to satisfy the same impedance boundary condition. In the aerodynamic noise problem, however, the surface integral cannot generally be made to vanish by such choices of special boundary conditions for the Green function. Thus, in aerodynamic noise problems, it is necessary to visualize not only the "equivalent acoustic" pressure waves propagating out from the volume elements of the "equivalent acoustic" source distribution, and being reflected and re-reflected "acoustically" at the boundaries S of the real fluid according to some physically and/or mathematically convenient choice of acoustic impedance type boundary condition (again in the "equivalent acoustic" medium), but in addition these real physical boundaries S must be visualized'as covered with "equivalent" t Likep6.~, this Green function is known to satisfy the reciprocity condition [5].

534

P.E. DOAK

surface source and[or surface dipole source distributions, from which other "equivalent acoustic" pressure waves are being sent into the volume of fluid under consideration. It is clear from this, too, that in general the real physical boundaries, in aerodynamic noise problems, do not act only as "acoustic reflectors" of the "equivalent acoustic" pressure waves generated by the "equivalent acoustic" volume source distribution. In this connection, the problem first considered by Curie [7] is of interest. Curie's problem is that of aerodynamic sound in a fluid in which solid, fixed bodies are immersed. Application of the "hard" wall Green function formula, equation (7), to this problem is almost trivially simple, and yields exact results having, for physical purposes, the same interpretation as those of Curie (who used the inhomogeneous scalar wave equation for the mass density) and which in some respects are more enlightening. From the equation of linear momentum transport for a Stokesian fluid, the pressure gradient at a rigid surface (where all components of the particle velocity must be zero, the fluid being viscous) is 8p[axl = asu/ax j, where S u is the viscous stress tensor

p being the coefficient of shear viscosity and ( that of bulk viscosity. Thus equation (7) for the pressure is

p(Xk, t) = f f q(y,,

t ly,, 0 dYl dY2 d,Y3d~

TV

+

ff

dS,(yk) dr,

(8)

TS

where dS~ is the vector representation ofthe elemental surface of area dS. The surface integral in equation (8) clearly represents a contribution to the 'pressure originating in the reaction of the surface to the normal viscous force per unit volume, -OSnflaxj, exerted on it by the fluid. The contributions to the pressure originating from the reaction of the surface to the real pressure forces exerted on it by the fluid are all contained in the volume integral. As has been pointed out in the preceding paragraph,this volume integral consists solely of the "equivalent acoustic" pressure waves in the ideal, inviscid, non:heat-conducting reference medium that would be caused by the given "equivalent acoustic" sources in the presence of the same boundaries, S, acting as "hard" acoustic reflecting surfaces. In other words, the volume integral is simply the solution of a purely acoustic scattering problem: namely, that of a given acoustic source distribution q(xk, t), in an ideal (inviscid and non-heat conducting) t acoustic medium at rest subject to scattering by the acoustically (and also in reality) rigid surfaces S. It can (and has often been) argued, in connection with aerodynamic sound generation in the presence of rigid boundaries, that the reaction of the bodies to the viscous stresses on it are probably an order of magnitude less important than the reaction to the normal pressure stresses. In physical situations where this is a reasonable assumption to make, equation (8) reduces simply to the "equivalent acoustical scattering problem" result:

p(xk, t) ~- f f q()'k, ~)P~+(Xk, t] Yk, 0 dy, d.v2dya dr.

(9)

TV

The remarkable theoretical simplicity of this result (which to the author's knowledge is, in the context of its general interpretation given here, a new result, theoretically, although it is t And thus characterized, for. the purposes of calculating the pressure p(xk, t) from equation (8), only by the reference speed of sound, c~.

AERODYNAMICSOUND AND DUCT ACOUSTICS

535

one which in effect has nearly always been used in practice since viscous stresses have nearly always been neglected) is very striking indeed. Curie's result [7] shows that the solid bodies radiate sound as if they were forcing the equivalent acoustic medium in exact reaction to the real fluid forces exerted on them. The exact result obtained here, equation (8), corroborates this picture physically but produces an even simpler analysis of the process. The "equivalent acoustic" waves making up the real physical pressure are simply scattered by the real rigid bodies, as "acoustically rigid" bodies in the "equivalent acoustic" medium. The additional effect of the rigid surfaces is that of small "acoustic pistons" on the surfaces, each having an acceleration proportional to the local reaction of the actual surface element to the actual normal viscous force exerted by the real fluid on it. (It is a straightforward exercise in order-ofmagnitude dimensional analysis, incidentally, to show that, in similar physical situations, the "monopole" surface source distribution of the surface integral in equation (8), together with the volume source distribution contribution, lead to the same prediction of a radiated sound power proportional:to velocity to the sixth power as does Curie's "dipole" surface source distribution.) ~: These exact formal solutions, of the type of equations (8) and (9) have interesting and relatively unexplored potentialapplications to a number of problems of aerodynamic sound generation and propagation in flows when reactions at boundaries are involved, but it is not appropriate to pursue such nev~ possibilities in a review article. It is relevant to note, however, that this type of formulation is both more convenient mathematically and more satisfying conceptually as a basis for some Of the recent "diffraction theory" type of studies of certain aerodynamic sound problems (see, e.g., references [8] and [9]) than is the Lighthill/Curle description in terms of "equivalent acoustic" mass density fluctuations and equivalent dipole source surface distributions. The remarkable rcduction of the generally non-linear aerodynamic noise problem when boundaries are present to a purely acoustic scattering problem when viscous surface forces are relatively unimportant (equation (9)) shows once again the enormous potential for mathematical simplification inherent in the acoustic analogy approach initiated by Lighthill. 2.3. TIlE NATURE OF TIlE FUNCTIONAL DEPENDENCE OF TIlE PRESSURE ON TIlE EQUIVALENT SOURCE STRENGTH DENSITY

It has been shown in the previous section that, when the equivalent acoustic source distribution is known, the "acoustic analogy" model permits an exact formulation of the general problem of aerodynamic noise generation and propagation in the presence of boundaries in terms of a purely acoustic radiation and scattering problem. This acoustic radiation and scattering problem has an exact formal solution. Also, for a very wide range of boundary geometries, methods are available for giving analytical and/or numerical expression to this exact formal solution (see, e.g., reference [5]). As the first step towards establishing, within this framework, any rules of general validity governing the functional relationship between the pressure and the equivalent source strength density, the functional nature of the Green function must be established. As was made plain in the preceding section, many choices (infinitely many, in fact) of Green function are possible in any given problem. Of these, the free space, "hard" wall and "soft" wall choices, respectively, are the most popular, as being usually the most convenient for the various physical problems of most practical interest. Of these three, the hard wall Green function is probably the most generally convenient for aerodynamic noise problems, as has been implied in the discussion of the preceding section. For the hard wall Green function, the complementary function part, P6, c(Xk,t) (see equation (5)) can be thought of as generated by a "mirror image" of the point source generating the

536

P.E. DOAK

free space part (see again equation (5)), in the surface S. Naturally, if the surface S is irregular this "mirror image" will be a "crazy mirror" image. In fact, only when the surface S is a single infinite plane will the image source be a single, identical replica of the point source generating the free space part of the Green function, located at the geometric mirror image point. If the surface S is two parallel planes, then (as in the optical case of two parallel plane mirrors) there will be a doubly infinite sequence of images of the free space source. If the surface S is curved, the image source, in general, will be extended in space and/or time and thus not be confined to a point or to a denumerable sequence of points. Despite all these varied possibilities, nevertheless, because when the surface radius of curvature is small compared to the acoustic wavelength the image is weak and when it is large compared to the wavelength the image tends fairly rapidly to the single, infinite plane image, the image distribution for the complementary function part of the hard wall Green function can usefully be thought of, for physical purposes, as approximating to a sequence of one or more such "plane mirror" images of the point source generating the free space part. When the surface has dimensions small compared to the acoustic wavelength the strength of the approximate plane mirror image can be thought of as appropriately reduced. It then follows (see equation (8) and/or equation (9)) that the pressure caused by a certain equivalent acoustic volume source distribution, q(xg, t), in the presence of boundaries can be thought ofas the pressure that would be caused in free space by the same source distribution plus the pressure that would be caused in free space by the mirror image(s) of this source distribution in the surface S. Since the pressures from the actual source distribution and its image(s) are thus being superposed (and not the mean acoustic intensities) the phenomenon of interference between the two wave patterns will, in general, be in evidence, and the resulting acoustic power radiated will not, in general, be the sum of the separate powers radiated by the source distribution and its image(s), respectively. However, in any case it is clear that, in the presence of boundaries, the total pressure can be thought of as caused by a generalized volume source distribution, including both the actual source distribution and its image(s) in the boundary surfaces S (and also, for the normal viscous reactions--see equation (8)--an appropriate surface source distribution). Therefore, as the basic problem essential to establishing the nature of the functional dependence of the pressure on the equivalent source strength densit3, it is sufficient to consider the case of an arbitrary volume source distribution, q(xk, t), in an ideal "acoustic" medium of infinite extent, characterized by the sound speed c~. The exact formal solution of this problem is that given by equation (6), with the surface integral absent: namely,

P(Xk, t) = f q(Yk, t -- IX~ -v

ykllc~)

4nlx~ - Ykl

dYl dy2 dy3,

(10)

which, of course, is just the well-known Kirchoff-type solution displaying the retarded time relationship between events at the source points and the corresponding events at the observation point. Inspection of equation (10) shows that two kinds of "interference" can influence the pressure at the observation point: (i) contributions from two, separate, in-phase elements of the source distribution can interfere at the observation point because ofpath length differences; (ii) contributions from two, separate elements of the source distribution that are equidistant from the observation point can interfere because they are inherently out of phase. The multipole expansion scheme for equation (10) [i 1, 12] was devised to provide, in so far as possible, separate exhibition of the effects of these two types of interference.

537

A E R O D Y N A M I C S O U N D A N D D U C T ACOUSTICS

The multipole expansion of equation (10) is obtained by expanding the integrand in a triple Taylor's series in the variables Yl/[xkl, y2/Ix~l and Y3/[Xkl and integrating term by term. The formal, exact result is

where the multipole operators, Dr.,., are defined by

ax~ k4rcl'xkl 1 "") ' D,,.,. = ax~(-a)'+"+"

(12)

and the overall, instantaneous multipole moments of the source distribution, Mo.., are defined by

[

M,m.(t) =-- .)'IYr'~2YJq()'k, t) dy, dy2 dya. llm!n!

(13)

r

V

The normalization in these definitions is done to ensure that each multipole term is also the leading term at large distances of the radiation from an appropriate cubical lattice of equal positive and negative sources and of unit lattice spacing. For source distributions that vanish everywhere outside a volume V of finite extent, and which also may be functionally of the form of an Nth order divergence of an Nth order tensor the following result can be easily established by application of the divergence theorem to equation (lO). The multipole tensor source strength density equivalence theorem. If q(Xk, t) is a regular bounded function on a bounded region V, which vanishes with all its space derivatives on and outside of the surface S of V, and ifq(xk, t) is of the functional form ( - a ) " m . . t~..... Mxk, t)

q(x~,t)=

axn axl2... OxiN

,

(14)

where mjl. ~2..... ~,v(xk,t) is called the equivalent Nth order multipole density of q(Xk, t), then

(-0) ~ p(xk, t)

f , , , , , . , , ..... , N ( Y , , t - I x , - y d l c ~ ) 4rtlxk - Ykl

Ox, Ox, z...Ox, N a

dyldy2dya

(15)

V

and all terms of order less than N in the multipole expansion (l 1) are identically zero, the leading term being D,,. ,~..... ,N Ixkl,

M,l. ,2..... ,N(t-- Ixdlc&

(16)

where D

~

)

1

D, -= 7x, 4rcVxkl . . . .

4r~lxd'

D,j = Ox, Oxj \ 4~lxkl . . . . . . .

07)

and

M(t) = I q(yk, t)dytdy2dya,

M , ( t ) = I y,q(yk, t)dyldy2dy3,

V

V

M~j(t) - 89f YiYJq(Yk, t)dyl dy2 dk'a. . . . v

(18)

538

P.E. DOAK

are, respectively, the tensor multipole operators and overall moments, in terms of which the multipole expansion (1 I) can be expressed, in the general case, as

p(Xk, t)= D QXkI, o~k) M(t-- lX~[[C=)+ D, ( ' X , ' , ~ ) M,(t-- IXkI[C=) + D,j(lxkl,~)M,j(t--,x,,/c=)+

....

(19)

DM being the sum ofall terms in the multipole expansion for which N = I + m + n = 0, DtMt that for which N = l + m + n = 1, DIjM~j that for which N = l + m + n = 2 , and so on. DM can properly be called the overall monopole field & t h e source distribution, D being the monopole operator, M the overall monopole moment and q(xk, t) the monopole source strength density. Similarly, DiMI can properly be called the overall dipole field of the source distribution, Dt being the dipole operator, M~ the overall dipole moment and, if q(xk, t) = Omi(xk, t)/Ox, m~(Xk,t) the equivalent dipole source strength density; also D~jM~j is the over'all quadrupole field of the source distribution, D~ being the quadrupole operator, M~j the overall quadrupole moment and, if q(xk, t)= a2mij(Xk, t)/OX~Oxj,mIj(Xk, t) the equivalent quadrupole source strength densitY,. A source distribution of the functional form (14) can properly be called a distribution of multipole order N. Physically, these results from the multipole expansion and the " m u l t i p o l e . . . equivalence theorem" can be interpreted as follows. Each volume element, d V, of the source distribution can be thought of, ultimately, as contributing to the total pressure field like a "point" monopole (i.e., a small pulsating sphere) of strength q(Xk, t) d V. Thus ifq(xk, t) is zero in a particular volume element, that element makes no contribution. If, however, q(xk, t) is functionally a multipole divergence of some tensor field, q(xk, t) =

(_0)~' OXll OX12''" O'~lN

re, l, 12..... ,N(Xk,t)

then, equivalently, each volume element can be thought of as contributing to the total field like a point multipole oforder Nand ofstrength and orientation given bym~L ~2..... in'(Xk,t)d V. In the case of the equivalent multipole moment density it is quite possible, of course, that although q(Xk, t) may be zero in a given volume element and so, ultimately, the element can be regarded as making zero contribution to the total field, nevertheless the equivalent multipole source strength density, mi L ~2..... ~N(Xk,t), may not be zero in the volume element and thus contributes to the integral in equation (15). This is just one example of the fairly selfevident redundancies in the equivalent multipole source strength density description of what is, ultimately, a source distribution of monopole source strength density, q(xk, t). Another example is to say that 02Lij/O.'clOxj in the basic acoustic analogy equation (1) can be regarded, equally well, as a distribution of monopole source strength density OZLij/Ox~Oxj, or as a 9 distribution of dipole moment density -OL~flOxj, or as a distribution of quadrupole moment density L~j. Thus the disadvantage of the equivalent multipole moment density description is its redundancy: when the multipole order of the distribution is N, then specification of the equivalent multipole moment density requires knowledge of all the scalar fields making up the tensor mt~. ~2..... iN(x~,t) (three for a dipole order field, six for a quadrupole order field, and so on) instead of just knowledge of the one ultimate scalar field, q(xk, t), itself. On the other hand, the equivalent multipole moment density description gives a form of result which inherently displays the leading term in the multipole expansion (see equations (11), (12) and (I 5)-(19)). When the source distribution is acoustically compact (i.e., when the

AERODYNAMICSOUND AND DUCT ACOUSTICS

539

behaviour of the source distribution at a given point is uncorrelated with its behaviour at points more than half an acoustic wavelength distant) this leading term is the dominant termt and then good estimates of the radiated pressure, field can be made from knowledge only of the order of magnitude, length scale, orientation and frequency spectrum of the equivalent multipole moment density. Exploitation of this possibility has been an essential ingredient in the techniques originated by Lighthill [2] for providing estimates of the aerodynamic noise generated by turbulent flows. In such cases, estimates on the basis of the order &magnitude, length scales and frequency spectrum of the source strength density itself, q(xk, t), do not automatically include the radiation inefficiency inherent in an acoustically compact source distribution of higher multipole order. Bq considering the form of the multipole expansion in the radiation field of the source distribution the fundamental result displaying the essential nature of the dependence of the pressure field on the source distribution can be easily obtained. The radiation field occupies the region where the distance, Ixkl, from the source distribution is large compared both with the representative acoustic wavelength, ). = 2nk = 2nc=[og, where/," is the acoustic wavenumber and o9 is a radian frequency representative of the time rates of change in the source distribution, and with the representative linear dimension of the source region V. In this radiation field region, terms in Otmnl~lm n oforder higher than the first in 1/Ix~l can be neglected and consequently the radiation field pressure is, from equation (11),

Prad(Xk,t)=

~

(xtIlxkl)'(X2/IX~I)m(X3/IXkl)

(1

0 ]]'+"+" M,,,.

( t - Ix,

(20)

I, m, n = O

It is not difficult to see, from the definition ofthe moments M~,.., that this expression, equation (20), is just the series expansion of a fourfold inverse Fourier transform, with respect to frequency tn and source position )'k: namely, eo

' ffs

praa(xj, t)=4rclxj---~]

{(ot - k [ x j l + k F ~xj [ y ,))dogdyxdy2dy3,

q(yk, og)exp i

(21)

-0o

where q(y~, 09) is the Fourier transform on time of the source distribution,

q(xk, og) - ~

q(xk, t)exp{--iogt}dt.

(22)

By virtue of this result, equation (21), and the well-known, unique inverse relationships between Fourier transform function pairs, it can be said that the pressure radiation field is a unique (fourfold) Fourier transform of the source distribution, and vice versa.~. The strong, and essentially simple, relationship between the behaviour of a function and that of its partner in a Fourier transform pair is of course well known to students of signal processing and operational calculus generally. One example is sufficient to indicate all the principal features of the way the source distribution determines the radiation field pressure, and vice versa. Consider a single frequency, uniform source distribution, of equivalent multipole moment density m~m,(og)(when Fourier transformed with respect to time), the distribution being confined to a volume in the form of a rectangular parallelepiped of edge lengths a, b and d, respectively, in the xl, x2 and x3 directions. The overall (time transformed) multipole moment of the distribution is then t This will be explicitlydemonstrated in a subsequent example. ~tStrictly speaking, quantitatively, this statement applies to 4~lxk[ times the pressure radiation field, of course.

540

P.E. DOAK

Mt,~.(o~) = abdmo~.(oJ). For this source distribution the radiation field pressure is given by, exactly,

4n]x~l P,aa(xj, t) = [(abd) m~,..(w) exp {ioJ(t - Ix~]lc~)}l x 9

Xl

I

.

X2

m

.

X3

n

[sin (kaxdlxll) sin (kbx~llx~l) sin (kdx~llx~l)] x [ (kax,llxjl)

(kbx2/}xj})

(kdx31lxjD J"

(23)

The first square bracket on the right-hand side ofequation (23) is just the time retarded overall multipole moment of the distribution, M~,..(t- Ix~[[co~). The second square bracket on the right-hand side of equation (23) is a complex "transfer function" factor influencing both radiation efficiency and directivity, and depending only on the multipole order and frequency of the distribution (not on the shape or size of the volume containing it--which can conveniently and appropriately be called the "correlation volume"). The third square bracket on the right-hand side of equation (23), involving the (sinz)/z functions so familiar in the theory of wave radiators, is a real dimensionless "transfer function" factor depending only on the three dimensionless wavenumbers, kaxt/lxjl, kbx2/Ixjl and kdx~/lxil, and thus this factor, which also influences both the radiation efficiency and the directivity, is completely hldependent of the multipole order of the source distribution. For an acoustically compact correlation volume (ka, kb and kd all appreciably less than unity), the radiation field pressure, from equation (23), reduces to 4nixjI p,~d(xl, t)~-[(abd),,h,..(o3)exp{io~(t-

Ixjl/c~o)}] -ik ~-~j[]

~jI) ~-,k ~-x-~l}J'

(24) the "point" multipole result9 For an acoustically extensive correlation volume, however (ka, kb and kd all appreciably greater than unity), the result is

4~zlx~l p,ad(xj, t) "" [m,m.(co)exp {io3(t -- Ixjllc~)}] • 9

trl--I

x [sin (kaxlllxj[) sin (kbx21lxjl) sin

(kdx31lxjl)].

(25)

In this case the result is proportional to mo..(og), rather than to Mt,..(m) as in the low frequency (acoustically compact) case, and also the dependence on the wavenumber, k (and hence on the frequency ~ = kc=) has changed from M +'+" in the low frequency case to k ~+'+"-3 times a spatially oscillatory factor of maximum modulus unity in the high frequency case. Note, too, that the angular dependence of the radiation field pressure has changed from (for the x,/Ix~l dependence) from (xl/lx~l) in the acoustically compact case to (x,/lxjl)'-' sin(kaXl/ Ixjl) in the acoustically extensive case, and similarly for the dependence on x=llx~l and

xdl-','~lSimilar results can be relatively easily obtained for correlation volumes of cylindrical, spherical and other shapes. For the spherical shape, for example, the result is identical to equation (23) except that the third square bracket is to be replaced by 3(sin ka - kacos ka)l(ka) 3, where a is the radius of the sphere.

AERODYNAMIC SOUND AND DUCT ACOUSTICS

541

Thus it can be seen that in aerodynamic noise problems questions about the size and shape of the correlation volumes can be important. Only if the correlation vohones are known to be acoustically compact can the low frequency, point mtdtipole approxhnations (such as equation (24), which is independent of the shape of the correlation volume) be ttsedjustifiably. These results for the radiation field pressure demonstrate the unique, Fourier pair, relationship between the equivalent source distribution and the radiation field pressure. It is also easily possible to investigate the relationship between the pressure and the source distribution inside the volume occupied by the source distribution (see reference [1], pp. 303-313).t Details.of such an investigation have been provided very recently in reference [1 ] and it is only necessary here to describe the results briefly. The analysis shows that inside an isolated,:~ acoustically compact source distribution of aIo' multipole order the exact governing equation for the pressure, equation (2), reduces to the Poisson equation

02p

ax~

= -q(x,,t),

as is to be expected. Broadly speaking, then, the pressure fluctuations inside the correlation volume of such an isolated, acoustically compact source distribution are very nearly the same as the "incompressible" pressure fluctuations that the given flow would generate (i.e., the so-called "pseudo-sound" pressure fluctuations). Just outside the correlation volume, the pressure decreases from these interior "hydrodynamic" pressure values according to the same laws as regards variation with distancc and angle that govern the behaviour of the pressure field of a point monopole of the same order as the order of the distribution. Thus, in descriptive terms, one can say that inside the corc of an acoustically compact turbulent eddy, which according to equation (l) must be regarded as a source distribution of quadrupole order, the pressure fluctuations are of the order of the fluctuating part of (89 At the edge of the core the pressur~ will drop rapidly to a value which may be down to (ka)3/8 its value inside the core, where a is the eddy core radius. From the edge of the core outwards, the pressure will then decrease with distance, r, from the centre of the eddy, first in proportion to 1/r 3, then in proportion to l/r 2 and finally in proportion to l/r, being the acoustic radiation field behaviour. Thus, hlside the source distribution the pressure is, as it were, dominated by its non-radiating, "near field" values, which, as for any acoustically compact radiator, are very nearly completely reactive and thus generally unrepresentative of the active radiation field pressures. Acoustically, then, the eddy could be thought of as behaving like, as far as its radiation efficiency is concerned, a very small acoustically radiating surface characterized by relatively very large surface velocity fluctuations but a very small radiation impedance. The implication is clearly that for aerodynamic sound radiation generally, not only the order of magnitude of the source strength density of a given correlation volume of the source distribution must be known, but also the radiation impedance as seen locally by the correlation volume must be just as well known, or perhaps even more accurately known. When the source distribution is not acoustically compact, or isolated, then the analysis of the pressure field inside the source distribution [I, see again pp. 303-313] shows that the pressure inside a given correlation volume can easily be dominated by the (apparently) acoustically propagating contributions to the pressure from either acoustically remote parts of the correlation volume itself or other correlation volumes of the source distribution. In such cases, then, the pressure inside the source distribution is no longer approximately equal I The analysis in reference [1] is for the mass density acoustic analogy equation but it is evident from that analysis that similar results will apply for the pressure equation. J; i.e., far from other sources.

542

P.E. DOAK

to the local "hydrodynamic", or "incompressible", or "pseudo-sound" pressure. Furthermore, its characteristic length scale becomes the acoustic wavelength and thus is no longer that ofthe local eddy core radius. Very crude estimates given in reference [I ] (see pp. 303-313), indicate that in turbulent flows known to produce significant aerodynamic noise, such as high speed jets, what little evidence there is on whether the source distribution correlation volumes could be regarded as primarily either acoustically compact or acoustically extensive is wholly inconclusive. At present, then, the only sensible conclusion is that the correlation volumes of both types are probably present, and that each type is responsible for some of the significant effects which are observed. In brief, the main conclusions of this section are as follows. (i) The radiation field pressure and the equivalent source distribution of the acoustic analogy model are uniquely related, reciprocally. (ii) All the following characteristics of the source distribution may, according to circumstances, be of dominant, or at least relatively equal, importance in determining the radiation field pressure: (a) (b) (c) (d)

the order of magnitude of the source strength density, the frequency spectrum of the source strength density, the multipole order of the source distribution, the sizes, shapes and orientations of the various correlation volumes making up the source distribution (for a given frequency spectrum, and multipole order, of course, these will determine the respective radiation impedances for the correlation volumes).

The possible significance of the influencing factors mentioned in (lid) has been almost entirely overlooked, and (in the author's view) underestimated, in theoretical work on aerodynamic noise to date. 2.4. PRINCIPAL ACttlEVEMENTS AND LIMITATIONS OF "EQUIVALENT SOURCE" MODELS The inhomogeneous scalar wave equation is one of the best understood equations of mathematical physics. Because of this, and because of the mathematical exactness of the "equivalent acoustic source" model of aerodynamic sound generation and propagation, it has not proved too difficult to apply this model to the study of many important aspects ofthe subject. The ease of application is only relative, however, and indeed these applications have required, and produced, significant improvements in existing mathematical techniques, and have also resulted in development of some new techniques, particularly in connection with problems involving moving source distributions, and source distributions of random character in space or time or both. Perhaps coincidentally, these developments have also coincided with rapid development of machine computational methods of solution. As has been explained in the preceding sections, a formal, exact solution of any problem of aerodynamic noise generation and propagation is available. All that is required in principle is knowledge of the "equivalent acoustic source" distribution, and, in addition for practical purposes, the analytical and/or machine computational means to evaluate the formal result. To date, these practical problems of obtaining numerical or graphical answers have not proved an insuperable obstacle in any case where results for a particular assumed equivalent source distribution were really worth evaluating. Thus it can be said that the only really serious limitations of the equivalent acoustic source model arise from whatever limitations there are to one's ability to either measure or theoretically describe the equivalent source distribution sufficiently accurately for each particular situation of interest. It has turned out (and the reasons for this will be explained briefly in section 3) that there are certain limitations ofthis kind and that they appear to severely restrict the ability of the equivalent source models

AERODYNAMICSOUND AND DUCT ACOUSTICS

543

to describe the effects, on aerodynamic sound propagation and on the acoustic radiation efficiency of a given source, of gradients of the mean flow velocity and mean temperature (and/or mean pressure or mass density). Although the equivalent source model appears to have limitations of this kind in respect to describing accurately the effects on the far field acoustic radiation of the mean velocity and mean temperature distributions in the immediate neighbourhood of the sources of the radiation, nevertheless there are good reasons, both experimental and theoretical, for believing that it does provide accurate indications of certain effects of source distribution structure (e.g., correlation lengths and frequency spectrum of a turbulent flow) and of the motion of such a space-time structure with respect to the distant acoustic medium at rest. In the case of jet mixing region noise the nature of the discrepancies between the best "equivalent source" model predictions to date and experimental results have recently been definitively exposed and analysed [13, 14, 15]. Detailed discussion of these discrepancies is available elsewhere. In terms of the implications of the results for theory, however, it is appropriate to say here that it certainly appears from the results that only two things are missing from the equivalent source model predictions. These are the effects of the mean flow velocity and mean temperature profiles in causing (i) refraction and (ii) an altered radiation efficiency of sound generated by source correlation volumes that are "acoustically buried" in the jet mixing region. Here, of course, "acoustically buried" means that the source correla= tion volume communicates to the surrounding relatively quiescent acoustic medium only through a region ofinhomogeneous mean flow velocity and/or temperature having a thickness of the order of an acoustic wavelength or more. The idea that refraction is important and must be included is strongly reinforced by very recent results from Professor Ribner and his colleagues [16]. The idea that radiation efficiency is also highly relevant has been elegantly demonstrated in a theoretical example by Mani [17], again very recently. Taken iogether, the experimental and theoretical results that have been recently obtained [13-18] strongly imply that a much improved and quiet accurate formula for engineering predictions of both the frequency power spectrum and the directivity of turbulent mixing region noise ofjets is now a definite possibility. This formula would be essentially the Lighthill/ Ffowcs Williams formula modified by three semi-theoretical, semi-empirical correction factors, which could be deduced from the results of references [13-18]. The first of these correction factors would provide an adjustment for the observed dependence of the noise on jet mean mass density [18]. The second factor would correct for refraction [13-16]. The third factor would correct for the radiation inefficiency of the "acoustically buried" parts of the source distribution [13-18]. It must be emphasized, however, that the accuracy of such a semi-empirical formula could not be in any way guaranteed, in the present state offundamental knowledge, for all jets, as the presently available theoretical and experimental data is almost entirely restricted to round jets without appreciable initial annular layering of either mean temperature or mean velocity. Correction factors for jet cross section shapes of high aspect ratio, or with appreciable annular layering of either mean velocity or mean temperature, could well be significantly different from those that can be established for round jets from presently available data. Further fundamental work on aspects of the problem like refraction and radiation efficiency is required before generalIj' applicable formulae can be deduced, even for empirical prediction purposes (see section 3). Perhaps the most successful applications to date of"equivalent source" models have been those to noise generation and propagation of rotating machinery. For rotating machines in which the rotating elements are moving at subsonic speeds--as is the case for most fans, compressors, helicopter rotors, propellers, etc.--it is possible to identify some parts, at least, of the equivalent source distribution with fairly high accuracy. Correspondingly, for assumed equiv-alent source distributions having functional forms appropriate to such identifiable parts

9 544

P.E. DOAK

of the actual equivalent source distributions, it has been possible to develop generally applicable methods of theoretically calculating the resulting radiated acoustic fields (see, e.g., references [19-23]). The theoretical capabilities that have been developed in this connection are such that for a number of years now it has been apparent that these theoretical capabilities are considerably in advance of the knowledge of the local unsteady aerodynamic phenomena that is required to permit specification of the complete equivalent source distribution to a comparable degree of accuracy [19]. Thus, the equivalent source model of Lighthill has provided a mathematically simple, and hence very usable, theoretical basis for most of the advances in the subject of sound generated aerodynamically during the twenty years since 1952. These advances have been very considerable and very rapid, and there is no doubt that many opportunities for continued useful service by this model will occur in the years ahead. 3. "TRUE SOURCE" MODELS: PHILLIPS' EQUATION AND LILLEY'S FORMULATION OF THE TURBULENT MIXING REGION NOISE PROBLEM Certain apparently inherent limitations of "equivalent source" models, particularly in connection with refraction and radiation efficiency of "acoustically buried" sources, were mentioned in section 2.4. The fact that an "equivalent source" distribution would certainly contain terms representing the effects of refraction was emphasized by Lighthill in his first paper on the subject [2]. Neither in Lighthill's own work, however, nor in that of most people subsequently using his acoustic analogy model were any attempts made to unambiguously identify or evaluate these terms. Perhaps for this reason, the illogicalities that can arise when one does attempt to identify and evaluate these terms seem to have been largely overlooked in nearly all of the immense amount of work that has been published on the subject ofsound generated aerodynamically during the past twenty years. Nevertheless, a growing awareness of the potential importance of phenomena such as refraction was clearly evident from the beginning, especially in the early contributions by Powell, Ribner and Lilley. This growing awareness was first fully and publicly crystallized, as it were, in Phillips' paper of 1960 [24]. Phillips showed that, in terms of a pressure variable, r, defined as the logarithm of the pressure, r ~ lnp,

(26)

a - In ( P / P O ,

(27)

and an entropy variable, a, defined as

where ), is the ratio of the specific heats, the transport equations of an ideal Stokesian fluid could be combined to give an exact, inhomogeneous, "convected wave equation" for the pressure variable, in the form

if,x,\

ax, ]

-fffc =

- -ff7 §

(28)

where c is the local, instantaneous speed of sound, its square being c 2 = YPlP,

(29)

AERODYNAMIC

S O U N D A N D D U C T ACOUSTICS

545

p is the coefficient of shear viscosity, ( is the coefficient of bulk viscosity and D/Dt is the instantaneous, "hydrodynamic", or "material", derivative, D

0

0

Dt =-~ + vj -~xj"

(30)

Phillips' equation (28) is self-evidently exactly valid, mathematically, just as is Doak's version of Lighthill's acoustic analogy equation (1). Phillips argued that his equation was also a physically valid (i.e., meaningful) generalization, to arbitrary flow situations, of the "acoustic" scalar wave equation which is valid for media otherwise at rest. In this sense, r(= lnp) was the required generalization of the pressure variable, O(C2OF/OXI)/OXlthat of c202p]Oxi, and D'-r[Dt 2 that of OZp/Ot2. Also, in the same sense, the terms on the right-hand side of Phillips' equation (28) were to be regarded as "true source" terms, representing "the generation of pressure fluctuations by the velocity fluctuations...entropy fluctuations and fluid viscosity" [24]. Unfortunately, there is, ultimately, a fatal, fundamental flaw in this physical interpretation of Phillips' equation, which can most readily be made evident by attempting to apply the equation and this interpretation to the case of an inviscid, non-heat-conducting ideal fluid, in which case both the second and third terms on the right-hand side of the equation become identically zero. Phillips' equation is then (again exactly)

0 (c20r~ Ox, ~ Ox,]

D2r Dt'

Or, Ovj ? Oxj a x l

(31)

Also, from the exact equation of linear momentum transport for this fluid, namely,

Dvi - Dt

-t

cz Or ~ Oxi

it is evident that

O,

/

0Vk OV~

Hence, in view of the identity (32), it is evident that the material derivative of Phillips' equation can be written as (again exactly)

O 0 c2 ~ D-"t ~

-

Ot 2 ] - 2 -

Oxi Oxj

c'--

Oxi

=2~,

Oxj Oxk Oxl

.

(33)'t"

Now, for simplicity, consider the respective forms, and physical interpretations, of Phillips' equation (31) and its material derivative (33) in the special case where the mean flow velocity is in only one direction, the xt direction, say, where it and the mean temperature both depend on only one transverse space variable, x2, say, and where the mean pressure variable is a constant. Then, in terms &their mean and fluctuating parts, the pressure variable, r, and the particle velocity, vi, can be written as r - const. + r'(.,q, t),

v, -- (O,(xg, O,O) + v~(xk,t), the equations (31) and (33), respectively, can be written as (exactly)

0 [=-i

Or'\

D2r '

Ovl(x2)Ov2t-PC2'(xk,t)

i" T h e ratio of specific heats, ~,, has been assumed to be constant.

(31a)

P. E. DOAK

546 and

I)1 [ 0 [--

ar"~ _ E)[r']

Ogl(x2) O c2(x2)

=Lt2~(xk, t),

(33a)

where

DI 0 0 =- + f,~(x2) Dt Ot Ox~ and the functions Pt2~(xk, t ) and U2)(Xk, t) are both of quadratic (and/or higher) order in the fluctttathlg qttantities r', v~ and c 2' (since c 2 =-yp[p, fluctuations in c 2 are proportional to temperature fluctuations, of course). It is obvious that in the limit of small fluctuationst equation (33a) becomes homogeneous in r', given the mean velocity and temperature functions, vi and c 2, but Phillips' equation (31 a) remains explicitly inhomogeneous. The implications of these results are only too clear when it comes to the physical interpretations (31a) and (33a). According to Phillips' model, the problem of small amplitude pressure fluctuations in a transversely sheared mean flow is a forced motion problem: one of the fluctuating velocity components, v;(xk, t), as well as the mean velocity, must be known a priori to define the forcing function for the pressure fluctuations. The model provided by equation (33a), however, displays precisely the same physical situation as a free motion (i.e., a propagation) problem. In its physical interpretation, then, equation (33a) is much to be preferred to Phillips' equation (31a), as it provides a law, in the form of a third-order partial differential equation, which governs the propagation of small amplitude pressure fluctuations in a sheared mean flow and which is imlepemlent of the manner in which these fluctuations are produced. Knowledge of any other fluctuating quantity is not a prerequisite; only the mean velocity and mean temperature need to be known. Thus, in this special case of a transversely sheared mean flow, which is one of considerable practical importance, Phillips' interpretation of his left-hand side terms as proper generalized propagation terms for pressure fluctuations can be straightforwardly and conclusively shown to be both theoretically illogical and physically unsatisfilctory as a concept of any general applicability. The demonstration that Phillips' concept is unsatisfactory in this special case has shown, however, that, in the context of a successive approximation scheme, equation (33) is an eminently satisfactory generalization of the "acoustic" scalar wave equation for this special case, as its right-hand side automatically becomes at least quadratic in the fluctuating quantities (see equation (33a)). Thus, even inore than Phillips' equation, it merits a name. As a model for aerodynamic noise generation and propagation in turbulent mixing regions it has been derived, discussed and strongly advocated by Lilley [25]. Previously, however, it had been derived and applied, in appropriate forms, to problems of hydrodynamic stability and pressure fluctuations inside turbulent flows by Landahl and Lin, among others. Further, it was independently derived, in its homogeneous form, by Pridmore-Brown [26] for application to problems of propagation of sound in acoustically lined flow ducts and in this context has been re-derived and studied by Tack and Lambert [27], Mungur and Gladwell [28] and Mungur and Plumblee [29]. Prior to all these, however, Sir James Lighthill derived a special, but distinctly recognizable, form of the equation !++Rather than call it the Lighthill/Landahl/ Lin/Pridmore-Brown/Mungur/Lilley equation, it seems best to name it after the three persons who have been the first and/or strongest advocates of its virtues as a theoretical model for problems of turbulent pressure fluctuations, flow duct acoustics and aerodynamic 1" .e.,whenP ~2) and L{2) can be neglected. ~.The author is grateful to Dr C. L. Morley for this information.

AERODYNAMIC SOUND AND DUCT ACOUSTICS

547

noise generation and propagation in turbulent mixing regions, respectively, and thus call it, in its inhomogeneous form, the Landahl Lilley equation, and, in its homogeneous form, the Pridmore-Brown equation. The method of derivation of equations (33) and (33a) as given here is that ofthe author, and it is a straightforward matter to generalize this method so as to obtain these equations in forms applicable to an ideal Stokesian fluid (see p. 287 of reference [1]). The importance of both Phillips' equation and the Landahl Lilley equation as very considerable steps towards an ultimate "true source" model for aerodynamic sound generation and propagation is very great indeed. It can be seen from a comparative inspection of equations (33a) and (31a) that the Landahl Lilley equation (33a) must effectively reduce to Phillips' equation (31a) when the "shear refraction term",

a [-~ ar'\ ax-----~axl ~c (x2) ~x2),

- 2 ? a~t(x2)

on the left-hand side of equation (33a) is small relative to, say, the "convective refraction terms" in the triple material derivative D3r'/Dt a. An order of magnitude estimate indicates that this will be the case at high frequencies: that is, when the thickness of the mean velocity shear layer is large compared with the acoustic wavelength. Thus, for such higher frequencies, Phillips' model and the Landahl Lilley model should produce similar predictions of refractive and radiation efficiency effects. At the same time, one must expect to have to "correct" predictions of the simpler (second-order equation) model of Phillips at the medium and lower frequencies by a factor, or factors, to be determined by solutions of the Landahl Lilley equation (33a). As exceptionally good progress is being made in the development of both analytical and machine computational methods for solving both equations (31a) and (33a) (see, e.g., references [25], [30] and [31]) one can expect considerable information on the questions at issue to be forthcoming in the fairly near future. It should be mentioned that, in the work in progress on the Landahl Lilley equation, theoretically predicted values of the turbulent 9 source terms, Lt2~(Xk, t), Of equation (33a) are being used [32] and that experimental checks are included on both these source terms and the propagation terms on the left-hand side of the equation [33]. It can be hoped, therefore, that we are now well on the way to a "true source" solution of problems of aerodynamic noise generation and propagation in flows that have, effectively, a unidirectional mean velocity, transversely sheared, and where the mean temperature may also vary with the transverse space coordinate. Results of the preliminary studies to date give every indication that the theoretical results will include good predictions of the radiation efficiency "correction factors" that now have to be empirically provided to make the predictions of Lighthili's "equivalent source" theory fit the experimental measurements of turbulent mixing region noise. It is also to be hoped and expected that selective studies of the behaviour predicted by the mathematically relatively complicated Landahl Lilley model will give guidance on the possibility of devising and/or validating simpler models, such as, perhaps, "plug" flow models like that of Mani [l 7], which can provide adequately accurate predictions of refractive and radiation efficiency effects in special cases of practical interest. It is worth noting in this connection that some simplified "plug" flow models of this kind have already been devised and validated for certain uses in flow duct acoustics [34]. 4. FLOW DUCT ACOUSTICS As has been mentioned in the preceding section, the homogeneous form of the Landahl Lilley equation is, in fact, the independently derived Pridmore-Brown equation of flow duct acoustics. As such, it has been studied and applied very extensively in recent years (in addition

548

e . E . OOAK

to references [26]-[29] and [34], already quoted, there are literally scores of important papers on the subject in the literature of the past several years--among the authors of these who have made important published contributions are Morfey, Shankar, Eversman, Mariano, Savkar, Ko, Snow, Bolleter, MOhring, Plumblee, Dean, Kurze, Hubert, Neise, Perulli, Harel, Kapur, Tester, Vaidya and Swinbanks). After the pioneering theoretical work of Pridmore-Brown [26] and of Tack and Lambert [27], who also provided evidence of agreement between theoretical predictions and experimental results, the subject snow-balled from 1969 following the development and exploitation by Mungur and his colleagues [28, 29] of machine computational methods for coping with the difficult mathematical problem of solving the Pridmore-Brown equation subject to wavenumber dependent normal acoustic impedance boundary conditions. I0 0

I

I

I

I

I

I

(a) Direct

JC\

,:l//

I

I

I

I

"x~

I-0 rn

...x "~ x ' ~ t j

.:<>

/

0.1

001

f 500

"\X~X (bl Computed

I 630

I 800

I Ik

I I I2.~O Ir

I 2k

I 2.5k

Frequency(Hz)

I 3.15

I 4k

l 5k

Figure 1. Axial attenuation vs. frequencyfor 22 70 perforate liner at 100 ft/s flow(a) from direct measurement and (b) computed vla direct in situ impedance measurement. (After reference [35],)

Dean [35] and Plumblee and Dean [36] have recently provided conclusive evidence (in addition to that of Tack and Lambert [27]) ofgood agreement between experimental observations and the predictions of Mungur's theory, thus firmly establishing a correct physical interpretation of Mungur's results. Savkar [37] and Eversman and his colleagues [38] have devised alternative computational procedures, using Galerkin's method, that can be simpler to use (in some cases, at least) than Mungur's fourth order Runge-Kutta integration methods. Tester has developed a very promising Green function method of solution and his studies have also provided both a method of generalizing Cremer's "attenuation optimization" procedures and, perhaps most important of all, an understanding of the physical and mathematical nature of the "unstable", or "strange", modes that can arise as solutions of the Pridmore-Brown equation in addition to the "acoustic" mode solutions that Mungur's method provides [34, 39--42]. An indication of the degree of accuracy of attenuation predictions that is possible in practical situations by using Mungur's computational methods, or derivatives thereof, together with the in situ impedance measurement technique of Dean is provided by Figures 1 and 2 (adapted from reference [35]). In Figure I the dashed curve is the measured sound attenuation, as a function of frequency, in an acoustically lined duct carrying a mean flow of nominal speed 100 ft/s. The solid curve is the attentuaion predicted by Mungur's theory, in which the hz situ measured values of the liner impedance and the measured values ofthe mean flow profile have been used. Agreement between observations and predictions of the kind shown in Figure 1 thus validates both the Mungur prediction method and the Dean in sittt impedance measurement technique.

549

AERODYNAMIC SOUND A N D D U C T ACOUSTICS

The relatively small apparent discrepancies between observed and predicted values, of'the type shown in Figure 1, have also been carefully investigated by Dean [35] and by Plumblee and Dean [36], and it has been shown that these can largely be attributed to actual spatial variations in the liner impedance, due to the inhomogeneities of liner geometry which are usually inevitable as the result of the fabrication techniques. This is illustrated in Figure 2, which shows the measured sound attenuation rate and the range of values of measured wall impedance (for a (3,0) mode, zero flow case) overlaid on a "level attenuation isogram" plot. The curves labelled 6 dB, 7 dB .... on this plot are the contours of constant attenuation for the (3,0) mode as functions of the real and imaginary parts of the wall impedance. The circle labelled "measured wall Z " encloses the various local values of the liner material impedance, as actually measured at different points. The dashed curve at 7.5 dB (labelled "measured decay rate") is the directly measured attenuation, as a function of the effective values of the real and imaginary parts of the wall impedance. It has been shown [35, 36] that in general, as in this case, such an average measured decay rate does not necessarily go through the centre of the circle enclosing the various impedance values (i.e., it is not necessarily the attenuation corresponding to the spatially averaged impedance). This is because the mode shape and attenuation changes are not, in general, balanced with respect to impedance variations from this average impedance. 04

T

l

0-2

0

~ -0-2 6(:18

~. - 0 4

IOdB

6dB

-O 6 -OB -I .O O0

0"2

04

06

08

I0

1"2

14

16

I'B

20

22

Resislonce (R)

Figure 2. Level attenuation isograms on complex outer wall impedance plane for (3,0) mode, 1000 Hz, zero flow. (After reference [351.)

Meanwhile, in work that is just beginning to be reported (see, e.g., references [43]-[45]) Mungur and his co-workers have further developed his computational programme (generally proceeding direct from appropriately linearized versions of the fluid transport equations rather than ria the Pridmore-Brown equation) so that it can be applied to a very wide range of physical situations indeed. The computer programme options available now make it possible to solve problems of propagation of small amplitude, locally adiabatic pressure fluctuations in duct or jet flow conditions, or in the atmosphere, with any or all offactors such as the following being taken into account: different duct cross section geometries, varying cross section geometry, finite duct length, spatial variation of the wall impedance, both transverse and axial variation of both mean velocity and mean temperature.

550

P.E. D O A K

In the course of the development of these computational methods for flow duct acoustics, and of experimental work in the subject, it has become almost painfully obvious that there is an unsolved, and as yet largely unformulated, problem in connection with the duct wall boundary conditions. The computational methods require an actual, or effective, normal acoustic impedance to be prescribed at the flow duct boundaries. The experimental results of Dean [35] and of Plumblee and Dean [36] have shown that, at least for a wide range of acoustical lining materials of practical importance, it is possible to measure such an actual, or effective, acoustic impedance in the flow situation. However, it is not possible as yet to theoretically predict, in anything like a fully satisfactory manner, what this actual, or effective, acoustic impedance should be from knowledge of the acoustic liner's geometry and material properties and of the mean profile and turbulence characteristics of the flow. There is also a problem in predicting the non-linear dependence of the acoustic impedance on the amplitude of the pressure fluctuations, even in the absence of mean flow, but here a recent thorough investigation by Melling [46] has gone a very long way towards providing the information required in practical situations. Not only is there no method of theoretical prediction of the impedance but also experiments have shown (see, e.g., reference [35]) that, for configurations occurring in practical situations, a given acoustic liner that would have only a passive impedance in the absence of flow can have both passive and active components of impedance under flow conditions. In other words, the liner can both absorb acoustic disturbances propagating along the duct and at the same time act as a generator of other acoustic disturbances. In Dean's experiments [35] these two aspects of the liner behaviour appeared to be independent of one another, as the amount of absorption was in reasonable agreement with that calculated from Mungur's theory for the measured, ht situ values of the liner impedance at the frequencies concerned (see Figu re I). The "howl" noise generated by t he liner (which was of perforate type; negligible "howl" was observed with a microporous, felted metal type liner of nominally equivalent actual acoustic impedance in the absence of flow) was at much higher frequencies than that at which acoustic waves were being injected into, and propagated down, the flow duct and these higher "howl" frequencies bore no discernible relation to this "acoustic" frequency. As the "howl" noise can be very intense, it is fortunate indeed that in the great majority of important noise control situations in which perforate and/or microporous types of acoustic duct liners are used in flow ducts (e.g., aircraft engines, ventilating systems, industrial fluid piping systems) the occurrence of howl seems to be very much more the exception than the rule. Nevertheless, it is a phenomenon which needs to be understood, ifonly to make sure that it does not occur, with catastrophic consequences, in some future practical situation. In addition to this requirement, the nature of the possible interactions among the mean flow, the turbulence and the acoustic disturbances (and perhaps thermal disturbances as well) at a porous or flexible boundary of an unsteady flow obviously needs to be understood to provide the basis for proper and complete prediction of the acoustical performance of flow duct liners and for developing practical methods of designing liners for optimum performance. The conceptual formulation of this problem and the development of theoretical and experimental methods for its solution are important and urgent tasks for future research. Equally important and urgent, and of considerably greater immediate pi'actical importance, is the need to rapidly develop and exploit in applications to present engineering problems the in sillt impedance measurement techniques and the computational methods that have been devised by Dean, Plumblee, Mungur, Tester and others. Development of these possibilities can undoubtedly lead to better and more economic methods of acoustic liner design and performance prediction. In this category, too, is the computer programme for predicting performance and optimizing design of internal combustion engine silencers that has been developed by Alfredson and Davies [47].

AERODYNAMIC SOUND AND DUCT ACOUSTICS 5. S O U N D

PROPAGATION

551

IN JET AND INTAKE FLOWS

It has been mentioned in the previous section that, in the development by Mungur and his co-workers of computational methods for predicting sound propagation in flow ducts of varying area of cross section and having mean flow velocity and mean temperature gradients in both transverse and axial directions, the analysis has been based on appropriately linearized versions ofthe full transport equations of the fluid, and not on the Landahl Lilley or PridmoreBrown equations as such. Thus these basic linearized equations are applicable to jet or intake flows as well as to duct flows. Preliminary results relevant to jet flows have been reported [45] and further results are being obtained in a programme of coordinated research at the I.S.V.R., George Washington University (N.A.S.A Langley campus), the Lockheed-Georgia Company and O.N.E.R.A. in France. The programme includes complementary theoretical and experimental investigations and is aimed, in the first instance, at developing valid methods of predicting, for example, both the in-duct and far field "acoustic" pressures caused by a fan operating in an acoustically lined duct of finite length. Figures 3 and 4 (adapted from reference [45]) show the type ofresults that are being obtained. For convenience in producing the results shown in Figures 3 and 4, the jet flow was assumed to issue from a duct fitted with an infinite rigid flange (this is not necessary; some preliminary calculations have also been done for a jet issuing from a hole in a sphere). Also, for convenience, the mean Mach number has been assumed to decay axially as l/(axial distance)'. (For compressible jet flows, decay like the inverse first power of the axial distance is more realistic; calculations done for this decay law show even deeper refraction troughs along the -00

9

^

+ 9 0

~

-I-9

-4 -4

q. o o~ - 8 " O O txl

-14"0 --20"0 dB

-90 ~

Figure 3. Effect of frequency on directivity of(0,0) spherical mode, in sheared flow. Exit Mach number Mo of 0"5. A, k d = 6 n; I~, k d = 8 n; O, k d = 10 n; • k d = 12 n. (After reference [45].)

552

P.

E. DOAK

-1"90"

e

- 90*

Figure4. Effectofsheared flowexit Mach number Moon directivityof(0,2) sphericalmode. Fixed frequency: kd= 6 ~r. O, Mo = 0.1 ; z~, Mo = 0-3; +, Mo = 0'5; D, Mo = 0.7; • Mo = 0"9. (After reference [45].)

axis than are evident in Figures 3 and 4.) The directivities shown in the figures are for certain of the radiating modes (m, n), respectively, in the given mean flow conditions. These modes are in one-to-one correspondence with the familiar, no-flow spherical modes P~(cos 0) etm~h~2~(kr), where P# is the associated Legendre function of the cosine of the polar angle O, ~p is the azimuthal angle, h~2) is the spherical Hankel function of the second kind (for e tot time dependence), and r is radial distance (from the centre ofthe duct exit plane). It can be argued that, like the no-flow radiating modes, these flow modes (re, n) should form a complete set of radiating wave functions which can represent, in linear combination, any pressure distribution over a convenient reference surface (such as a hole, representing the exit plane of a duct, in a sphere). Figure 3 shows the effect of frequency on the directivity pattern ofthe (0, 0) mode, for an exit mean Mach number of 0.5 and an assumed flow profile of the shape indicated. The increasing depth, with increasing frequency, of the refraction trough along the axis is clearly evident. (As mentioned previously, a compressible jet with a slower axial decay than that assumed in preparing Figure 3 would show even deeper troughs.) Figure 4 shows the refractive effects, at a fixed frequency ( k d = 6rr), of various mean flow Mach numbers on the directivity ofthe (0,2) mode. In the no-flow case the directivity ofthis mode would be that of the secondorder Legendre polynomial: namely, (3 cos20 - 1)/2. The increase, with increasing mean flow speed (at a fixed frequency), of the refraction troughs along the axis is evident from the figure. The same computational method can be used for intake flows. Computed results for

AERODYNAMICSOUND AND DUCT ACOUSTICS

553

intake flows show, as is to be expected, refractive peaks (or beams) along the intake axis, in contrast to the troughs along a jet axis. As computational programmes are either available, or under active development, for calculating sound propagation in both flow ducts and in exterior flow situations such as intakes and jets, it is possible to consider assembling such programmes into packages for both prediction and parameter studies of sound radiated from quite realistic theoretical models of, say, a by-pass fan in a fan-jet engine nacelle. From computational programmes such as those described in this and the preceding section, sets of modes for representing, respectively, the in-duct sound field of the fan and the sound field exterior to the nacelle are available. It remains only to suitably connect up these solutions over the intake and jet exit planes. A preliminary study of this connection problem, for a source distribution such as that of a fan in a finite length duct, has been carried out [48], with promising results. This study shed some interesting light, incidentally, on the "mode coupling" concept which has traditionally been a rather awkward matter to deal with when predicting acoustic radiation out of the termination of a duct. As there has been some confusion about mode coupling at duct terminations, it is worth mentioning here that the study [48] has shown that the mode coupling concept does not arise explicitly except when it is assumed that only the modes incident on the termination from within the duct are initially known and hence the problem is to determine not only the transmitted sound field but also the in-duct reflected field. In the study [48] an alternative formulation ofthe problem was adopted. It was shown, explicitly, that in general the radiation from the termination can always be expressed, over the terminating aperture of the duct, in terms of a suitable set of orthogonal termination aperture modes, like those used, for example, by Gutin or Wright [21] for calculations of free-field acoustic radiation from propellers, fans and helicopter rotors. Mutually independent radiation impedance coefficients for each of these modes can then be defined. It is a mathematically interesting point that the problem of defining these coefficients is exactly that of expressing, on the aperture surface, the Green function for the exterior radiation in terms of the aperture modes. Thus, these aperture mode impedance coefficients can be defined in terms of the exterior boundary and propagation conditions only, without any reference to the specific nature of the in-duct sound field. Once they have been obtained in this way, they can be re-expressed as a complete set of similarly independent duct termination impedances. In turn, once these duct termination impedances are known, the complete sound field inside the duct for any prescribed in-duct forcing function (i.e., source distribution) can be calculated, without any further reference to the exterior sound field radiated from the termination. In this formulation the question of mode coupling does not arise at all, explicitly, it is implicitly dealt with in the formal process of coupling, over the terminating aperture, the exterior radiating modes to the in-duct modes via the orthogonal "duct aperture modes". In certain cases of practical interest (for example, when the duct walls in proximity to the aperture are bard) the "aperture modes" can be chosen to be identical to the in-duct modes, with a very appreciable gainin computational simplicity. The general gain in computational simplicity afforded by this formulation of the problem is that what would otherwise be a very lengthy calculation for each in-duct source and/or acoustical lining configuration and each exterior flow and geometric configuration, in every case dependent upon both exterior and in-duct conditions, can be split into two independent calculations. For a given exterior flow :rod geometric configuration, the aperture reflection coefficients can be calculated once and for all, In-duct sound fields can then be calculated for any source, acoustic liner and (compatible) flow conditions in the duct, and the total radiated sound power can be determined in terms of the duct modes, which is obviously very convenient for optimization and parameter studies ofduct silencing configurations. The externally radiated sound field only has to be calculated at all when it is desired to determine the directivity of the radiated sound power.

554

P.E. DOAK

6. "TRUE SOURCE" MODELS: THE MOMENTUM POTENTIAL DESCRIPTION OF UNSTEADY FLUID FLOWS Although the Landahl Lilley equation, iri the special case of a unidirectional, transversely sheared mean flow provides, in the context of a successive approximation scheme, a conceptually and mathematically satisfactory description of the generation and propagation of small amplitude pressure fluctuations'l" in a wide class of fluid mean flows, this description fails conceptually in the general case. In particular, it is not applicable (except in special cases where certain approximations may be made) in jet flows, where mean temperature and mean velocity gradients may have significant components in both axial and transverse directions. Also the Landahl Lilley equation by itself provides no guidance as to what parts of the fluctuating quantities could, or should, be given precise qualitative and quantitative definitions as "turbulent", or "acoustic", or "thermal". The question ofproviding such definitions is not academic. In the first place it is known from the work of Stokes, Kirchoff, Helmholtz and Rayleigh [1] that in a uniformly moving, thermally homogeneous (in the mean) ideal Stokesian fluid small amplitude fluctuations in pressure, particle velocity and temperature can be precisely analysed into acoustic, thermal and vortical parts. This "Rayleigh analysis" of the fluctuations into physically distinct components was indeed the starting point of an interesting successive approximation scheme for describing finite amplitude, unsteady fluid motion [49]. Morfey [50] was the first to realize that if such a successive approximation scheme were to be usefully applied to flows such as those of jets and boundary layers, in which gradients of mean particle velocity and mean temperature can be large; then the possibility of such mean gradients would have to be included in the first-order approximation, instead of confining this to the situation of uniform mean velocity and temperature as in the Chu and Kovasznay scheme [49]. In the second place, intuition based on observation strongly suggests that the basically "acoustic", "turbulent" and "thermal" types of motion clearly evident in the Rayleigh linearized situation also tend to co-exist in many, more general, non-linear fluctuating flows, and that even when non-linear coupling of these separate types of motion exists, it may often be relatively weak and/or localized. In seeking to generalize Rayleigh's "acoustic", "vortical" and "thermal" modes of fluctuating motion it would be natural, in keeping with the usual concepts of fluid dynamics, to regard the particle velocity as the dependent vector field and, say, the pressure and entropy as the two dependent scalar fields. The particle velocity could then be analysed into unique irrotational and solenoidal components, with fluctuations in the solenoidal component hopefully being identified as the "turbulent" or "vortical" part of the particle velocity fluctuations. When such an identification is made, however, the forms of the governing equations of transport of mass, linear momentum and energy in the fluid are not encouraging. Neither the momentum nor energy equation are appreciably simpler than they would be if the velocity field had not been partitioned in this way and the mass transport equation is definitely more complicated. Despite these difficulties, a certain amount of progress can be made by applying the scheme outlined in the previous paragraph to the transport equations. In this connection Crow [51] has obtained a very important result concerning Lighthill's model in a study to ascertain whether or not, for low Mach number situations, it could be regarded as a self-consistent solution of all the relevant transport equations. He concluded that the density fluctuations given by the formal solution could be regarded as valid in this context only if, in the velocity tensor v~vt in the quadrupole source strength density Ltj, the velocity vectors were to be r Strictly speaking, of small amplitude fluctuations in the logarithm of the pressure.

AERODYNAMIC

SOUND AND DUCT ACOUSTICS

555

replaced, respectively, by their solenoidal parts. The implication of this conclusion is obvious: to avoid obtaining tautological results from such formal solutions some appropriate separate identification of source terms for "acoustic" fluctuations must be made so that these source terms are at least not obviously dependent on the acoustic fluctuations themselves. Essentially the same point can be made by another kind of argument, which has been put forward in detail, independently, in references [25] and [I ] (see pp. 282-289). If one looks at the order of magnitude of the shear refraction term in the Landahl Lilley equation then it becomes at once apparent that inside the turbulent mixing region it is probably small compared with the source terms on the right-hand side of the equation, which are dominated by the "turbulent" velocity fluctuations. However, it is not necessarily small compared with the other propagation terms on the left-hand side of the equation. Thus local measurements of fluctuating velocities and pressures inside the mixing region, unless of presently unattainable accuracy, would not reveal the presence of this shear refraction term at all. Its existence, therefore, depends entirely upon its being theoretically identified. If it is omitted, use of the Landahl Lilley equation to predict the far field radiation of a measured source distribution will give the wrong answer, as all the significant refractive effects will not have been included.'[" It is clear that the general, fundamental issue here is the need for a priori, separate and distinct, identification of cause and effect, which is conceptually applicable both theoretically and experimentally. Without such distinct identifications, the conceptual model is inherently tautological, and thus is capable of undetectably producing nonsensical results--"a rose is a rose is a ...", as Gertrude Stein has put it. If the scheme in which the particle velocity is analysed into irrotational and solenoidal parts is not much help, is there any scheme which is better ? The momentum potential description appears to have promising possibilities [52, 53]. In this description the linear momentum density, pv~, is chosen as the primary dependent vector field, instead of the particle velocity. The linear momentum density is analysed into unique irrotational and solenoidal parts:

O~/Oxl;

Bl -

aA,/Oxl = 0.

Bl = (curl A)l,

(34)

The equation of mass transport is then independent of the solenoidal part of the linear momentum density, and becomes a linear partial differential equation, for arbitrary motion of any non-relativistic continuum:

Op at

a2~ -

-

ax~

--

0.

Moreover, if the motion is time-stationary in respect to the mass density, so that the mass density can be written as the sum of its mean value and a fluctuating part of zero mean,

p(xk, t) = p(xk) + p'(xk, t), then, without any further loss of generality, the mean value of the scalar potential ~,(xk,t), can be taken to be zero [52] so that the equation of mass transport reduces further to Op'

at

0 2 ~," - -

ax~

= o.

(35)

9Given an appropriate thermodynamic equation of state, the mass density can be written as a function of any suitable set of independent thermodynamic variables. For simplicity (and for 1"The reason that "equivalent source" terms for refractive effects, although relatively small in magnitude,

can nevertheless make significant contributions to the far field pressure is, of course, that their correlation volumes are relatively large.

556

P.E. DOAK

convenience of application to problems of aerodynamic sound generation and propagation), suppose that these are the logarithmic pressure variable, r - Inp, and the logarithmic entropy variable, a - In(p/p~). Then Op'/Ot can be written as

ap'

(3p '~ Or'

( ap I aa'

o-7- =

where r' and a' are the fluctuating parts of r and a, respectively, and, of course, (ap/ar) and (Op]aa) are known as functions o f r and a from the given equation of state. One is then at liberty to define the "acoustic" part of the scalar potential as satisfying

02~b" [Op~ Or" ax--~ " k-~r ] ~ -

(36)

axl -

(37)

and the "thermal" part as satisfying

The "vortical" part of the linear momentum density is defined as the fluctuating part of the solenoidal component: i.e., B;. Then the total fluctuating linear momentum density is, automatically, as it were, a linear superposition of the vortical, acoustic and thermal components, thus defined:

(pvt)' =--~; " Bi

axi

axl "

(38)

Further, through the definitions (36) and (37), the acoustic and thermal potentials are strongly associated, respectively, with pressure fluctuations and entropy fluctuations. Obviously, when, as is so often the case in situations of interest in practice, the fluctuations in the thermodynamic quantities are small relative to their respective mean values (in this formulation there is no a priori reason why either the mean linear momentum density or its vortical fluctuations need also be small in such situations), then (Op/Or) and (Op/Oa) in equations (36) and (37) can be replaced by their respective mean values, so that, given these mean values, the relationships between ~,,~and Or'lOt, and between ~p" and Oa'/Ot, become linear, in these fluctuating quantities. The equation of mass transport, in the momentum potential description, is thus satisfied by the definitions themselves of the mean, acoustic, vortical and thermal parts of the linear momentum density. Also, the defining condition OB;/Ox~=O (or B ' - c u r l A', divA'=0) determines one of the three components of B; in terms of the other two. Given the mean values, then, the three scalar components of the fluctuating part of the equation of transport of linear momentum and the fluctuating part of the equation of transport of energy are sufficient, with boundary conditions, to determine the remaining four dependent, fluctuating field variables: two components of the vortical linear momentum density fluctuations, B~, and the two components, acoustic (~,~) and thermal (~,~,) of the scalar potential of the linear momentum density. (Alternatively, of course, the pressure and entropy variables, r' and a', could be used in place of the acoustic and thermal momentum potentials, ~,~ and ~k~,.)The forms of these equations, and some discussion of them, are available in reference [53]. To date, explicit results of any detail that have been worked out in terms of the momentum potential description are restricted to the case where the fluctuating thermodynamic quantities

lar

t ~ t are relatively small (p/t ,P/P and/or appreciably less than unity) but where the fluctuations in the vortical part of the linear momentum density are not necessarily

A E R O D Y N A M I C S O U N D A N D D U C T ACOUSTICS

557

relatively small (i.e., IB'~I/~x/'~ may not be small compared with unity). As has been mentioned, this is a special case of wide practical applicability. There is no apriorilimitation on the magnitudes of either the mean linear momentum (i.e., mean Mach number, in effect) or the mean temperature, or on the magnitudes of their respective gradients. For this situation, it is permissible to linearize the governing equations in the acoustic and thermal scalar potentials, but not in the fluctuating vortical momentum or the mean quantities. A generalized, inhomogeneous, convected "acoustic wave equation" for the acoustic scalar potential can then be derived [53, see equation (40)]. As the terms in this equation which explicitly involve the coefficients of viscosity and thermal conductivity can be shown to be usually of secondary importance, the main features of this equation can be discerned in its form for an inviscid, non-heat conducting fluid:

(

'

0x, 0xj tSu ~ ax 2

D t ' ] T,

ax, 0x, tW k - - - F ] - ~

J"

(39)

Here D~j[Dt 2 is a tensor material second derivative, D~I

0z

_B,

0z

BIBj 0 z

Ot2=6u-~+ Zp atTxj ) ~2 0x]'

(40)

and the notation ( )' signifies the fluctuating part of the quantity in the brackets. The most important feature of equation (39), in the context of the philosophy behind the momentum potential approach, is that its right-hand side is explicitly dependent upon only mean, vortical and thermal quantities, and not at all on acoustic quantities. Thus, in terms of the unique resolution of the linear momentum density fluctuations into vortical, acoustic and thermal fluctuating components (see equation (3)), equation (39) has an unambiguous, non-tautological cause-effect interpretation. It has been shown explicitly that equation (39) includes the Landahl Lilley equation as a special case, for a transversely sheared, unidirectional mean flow [53]. Inspection of equation (39) shows that its terms representing shear refraction arise from 0 2 D 8 ~,)~. axt axj Dt 2 Since both the mean and fluctuating parts of B~ appear on the left-hand side of equation (39), it includes, in its homogeneous form, a self-consistent representation of convection and refraction of the acoustic disturbances by both the mean flow and the "turbulent" fluctuations,

B;, as well, of course, as of refractive effects due to the variation of the sound speed, Vr-~, with mean temperature. When the mean linear momentum and the mean temperature are both slowly varying (i.e., do not change much over a distance of an acoustic wavelength), and when scattering of the acoustic disturbances by the vortical fluctuations is neglected, equation (39) reduces to an equation for the fluctuating pressure variable:

fi2 ] OX!OXj

Ot 2

2v

OXjOt

~. ~ - - - - 7 ~ . ~ j

L~. \ v

]

fi Dt 2

" (41)

The left-hand side of this equation is clearly very similar to that of Phillips (equation (31)), for the same conditions of approximation. However, in equation (41) the right-hand side is clearly a source distribution of quadrupole order (being the double divergence of a tensor).

558

P . E . DOAK

In fact, what equation (41) represents is an important generalization of Crow's result [51 ] to situations of arbitrary mean Mach number and temperature distribution (as long as these are slowly varying). In short, equation (41) shows that a valid generalization of Lighthill's acoustic analogy model to provide a physically realistic model, locally, for acoustic generation and propagation in turbulent flows of slowly varying mean Mach number and temperature (but otherwise arbitrary), is a coJwectedinhomogeneous acoustic wave equation in which the quadrupole moment density of the source distribution depends only upon the vortical and thermal components of the fluctuations in linear momentum density, and on the mean linear momentum density, mass density and temperature. Equation (39) itselfis, in turn, the unambiguous and internally self-consistent generalization (in the context of the full momentum potential description) of equation (41) to situations of arbitrary variation of mean momentum, mass density and temperature. Finally, for situations where vortical and thermal fluctuations may be negligible, equation (39) reduces to a similarly unambiguous and internally self-consistent homogeneous "acoustic wave equation", both defining and describing small amplitude, homentropic pressure fluctuations in an inviscid, non-heat-conducting fluid in (a priori) arbitrary non-uniform mean motion and having an arbitrary non-uniform mean temperature [53]:

Ox, aXs 5u 2 0 x ~ ~'~i - 2~ a-~sat- O, OS-~x~ ~,~.=0,

(42)

the pressure fluctuations, p', being related to the scalar acoustic momentum potential by

ap.--= at

~ a2r

(43)

ax] "

In equation (42), ~ appears as the convection velocity because, under the iinearizing assumptions for which the equation is valid,/~//5 can be replaced by ~j//5 = 5~. In view of the internally self-consistent progression of expressions such as equations (41), (42) and (39) for defining and describing the generation and propagation of acoustic disturbances in progressively more and more complicated fluid flows, it would appear that, for the purposes of the theory of sound generated aerodynamically, at least, a "classical acoustics theory" in terms of the momentum potential description might well be a conceptually more advantageous limiting ease than either of the more traditional types in terms of pressure or velocity potential. Some reference is necessary to the rather vexing problem of "acoustic" energy in unsteady fluid flows. The most recent comments on the difficulties of making practical use o f a n y of the presently available definitions of"acoustic energy flow" appear to be those of Tester [41 ], in connection with problems of acoustic propagation in flow ducts. Tester's paper contains references to other recent and relevant work on the subject, notably that of Cantrell and Hart, Morley and Mtihring. In the momentum potential description, the a priori identification of what is vortical, acoustic and thermal, respectively, ensures that no difficulties can arise in defining and classifying vortical, acoustic and thermal components, and the consequent vortical-acoustic, acoustic-thermal and thermal-vortical interaction components, of the potential energy density, the kinetic energy density and the flux ofenergy. On the other hand, no work has yet been done on the forms of the energy expressions when the momentum potential description is used to see whether or not any of these forms have interesting possibilities in respect, say, to the problem of deciding just where the radiated acoustic energy comes from, or to the even more interesting problem of establishing what, if any, components of the energy flux vector may be conserved (i.e., be solenoidal). These interesting questions remain for the future.

AERODYNAMICSOUND AND DUCT ACOUSTICS

559

ACKNOWLEDGMENTS Much o f the work described and discussed in this review is that of the author's colleagues at the University o f S o u t h a m p t o n - - p a r t i c u l a r l y Professor G. M. Lilley, Drs P. D. D e a n , t A. Kapur, C. L. Morley, P. Mungur,~ and B. J. Tester,'[" and H. E. Plumblee (also o f the Lockheed-Georgia Company). The a u t h o r is especially grateful to these and his other colleagues for producing so much excellent work requiring citation and comment. All opinions expressed in this review, as distinct from facts, are o f course the sole responsibility o f the author.

REFERENCES 1. P. E. DOAK 1972 Journal of Sound and Vibration 25, 263-335. Analysis of internally generated sound in continuous materials: 2. A critical review of the conceptual adequacy and physical scope of existing theories of aerodynamic noise, with special reference to supersonic jet noise. 2. M. J. LIGHrmLL 1952 Proceedings of the Royal Society A211, 564-578. On sound generated aerodynamically: 1. General theory. 3. LORD RAYLEIGH1877 Theory of Sound (two volumes). New York: Dover Publications, second edition, 1945 re-issue. 4. P. E. DOAK 1965 Journal of Sound and Vibration 2, 53-73. Analysis of internally generated sound in continuous materials: (I) Inhomogeneous acoustic wave equations. 5. P. M. MORSEand H. FESHBACH1953 Methods of TheoreticalPhysics (two volumes). New York: McGraw-Hill Book Company. 6. P. E. DOAK 1960 Proceedings of the Royal Society A254, 129-145. Acoustic radiation from a turbulent fluid containing foreign bodies. 7. N. CURLE 1955 Proceedings of the RoyalSociety A231, 505-514. The influence of solid boundaries upon aerodynamic sound. 8. J. E. Ft:owcs WILLIAMSand D. L. HAWKINGS1969 Journal of Sound attd Vibration 10, 10-21. Theory relating to the noise of rotating machinery. 9. D. G. CRIGHrON 1972 Journal of FluM Mechanics 51,357-362. Radiation from vortex filament motion near a half-plane. I 0. D.S. JONES1972JoarnalofSolmdand Vibration 20, 71-78. Diffraction theory: a briefintroductory review. I i. P. E. DOAK 1965 Proceedings of the Fifth International Congress on Acoustics, LiOge, Paper K56. Multipole analysis of acoustic radiation. 12. P. E. DOAK 1968 in Noise and Acoustic Fatigue (edited by D. J. Mead and E. J. Richards). Chapter 1 : An introduction to sound radiation and its sources. London: John Wiley and Sons, Ltd. 13. P. A. LUSH 1971 Journal of Fhdd Mechanics 46, 477-500. Measurements of subsonic jet noise and comparison with theory. 14. P. A. Lush and R. H. BURRIN 1972 United States Air Force Technical Report AFAPL-TR-72-53 Vohmte V. An experimental investigation of jet noise variation with velocity and temperature. 15. M. J. FISHER,P. A. LUSH and M. HARPER BOURNE 1973 Journal of Sound and Vibration 28, 563-585. Jet noise. 16. G. R. MACGREGOR,H. S. RIaNER and H. LAM 1973 Journal of Sotmdand Vibration 27, 437--454. "Basic" jet noise patterns after deletion of convection and refraction effects: experiments vs theory. 17. R. MANI 1972JoarnalofSoundand Vibration 25,337-347. A moving source problem relevant to jet noise. 18. R.G. HOCH,J. P. DUPONCnEL,B. J. COCKINGand W. D. BRYCE1973 Journal of Sotmd and Vibration 28, 649-668. Studies of the influence of density on jet noise. 19. C.L. MORFEY1973 Journal of Sound and Vibration 28,587-618. Rotating blades and aerodynamic sound. 20. M.V. Lowso~ and J. B. OLLERHEAD1969Journal of Sound and Vibration 9, 197-222. A theoretical study of helicopter rotor noise. 1"Now at the Lockheed-Georgia Company. .1:N.ow at George Washington University, N.A.S.A. Langley Research Center Graduate Program.

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21. S. E. WRIGHr 1971 Journal of Soandand Vibration 17, 437--498. Discrete radiation from rotating periodic sources. 22. C.L. MOREEVand H. K. TANNA1971 JournalofSolmdand Vibration 15, 325-351. Sound radiation from a point force in circular motion. 23. B. D. MUGRIDGEand C. L. MOREEY 1972 Journal of the Acoustical Society of America 51, 14111426. Sources of noise in axial flow fans. 24. O. M. PHILLIPS 1960 Journal of Fhdd Mechanics 9, 1-28. On the generation of sound by supersonic turbulent shear layers. 25. G. M. LILLE',' 1972 in United States Air Force Technical Report AFAPL-TR-72-53-Vohtme IV. Theory of turbulence generated jet noise: generation of sound in a mixing region. 26. D. C. PRID,',IORE-BROWN1958 Journal of Fhtid Mechanics 4, 393--406. Sound propagation in a fluid flowing through an attenuating duct. 27. D. H. TACK and R. F. LAt~mERr 1965 Journal of the Acoustical Society of America 38, 655-666. Influence of shear flow on sound attenuation in a lined duct. 28. P. MUNGURand G. M. L. GLADWELL1969 Journal of Sound and Vibration 9, 28-48. Acoustic wave propagation in a sheared fluid contained in a duct. 29. P. MONGUR and H. E. PLUMBLEE 1969 NASA SP-207: Basic Aerod)'tlamic Noise Research, 305-327. Propagation and attenuation of sound in a soft-walled annular duct containing a sheared flow. 30. S. P. PAO 1971 Jottrnal ofSottnd attd Vibration 19, 401-410. A generalized theory on the noise generation from supersonic shear layers. 31. A. MtCHALKE1970 DeutscheLttft.tmdRalanfahrtFB70-57. A wave model for sound generation in circular jets. 32. P. J. MORRm 1972 Ph.D. Thesis, University of Southampton. The structure of turbulent shear flows. 33. H. E. PLUMBLEEand R. H. BORRIN 1972 United States Air Force Technical Report AFAPL-TR72-53- Vohtme II. Future studies for definition of supersoinc jet noise generation and reduction mechanisms. 34. B. J. TESTER 1973 Journal of Soundand Vibration 28, 151-203. The propagation and attenuation of sound in lined ducts containing uniform or "plug" flow. 35. P. D. DEAN 1972 Ph.D. Thesis, University of Southampton. On the measurement of the local impedance of the walls of flow ducts and its use in predicting sound attenuation. 36. H. E. PLUMBI.EE and P. D. DEAN 1973 Journal of Sound and Vibration 28 (in press). Sound measurements within and in the radiated field of an annular duct with flow. 37. S. D. SAVKAR 1971 Joltrnal of Sottnd and Vibration 19, 355-372. Propagation of sound in ducts with shear flow. 38. J. F. UNRUH and W. EVERSMAN1972 Journal of Solmd and Vibration 23, 187-197. The utility of the Galerkin method for the acoustic transmission in an attenuating duct. 39. B.J. TESTER1973 Journal of Sound and Vibration 27, 477-513. The optimization of modal sound attenuation in ducts, in the absence of mean flow. 40. B.J. TESTER1973 Jottrnal of Sound attd Vibration 27, 515-531. Ray models for sound propagation and attenuation in ducts, in the absence of mean flow. 41. B. J. TESTER 1973 Journal of Sound and Vibration 28,205-215. Acoustic energy flow in lined ducts containing uniform or "plug" flow. 42. B.J. TESTER 1973 Journal of Sound and Vibration 28, 217-245. Some aspects o f " s o u n d " attenuation in lined ducts containing inviscid mean flows with boundary layers. 43. A. KAPURand P. MtJNGUR 1972 Journal of Sound and Vibration 23,401-404. On the propagation of sound in a rectangular duct with gradients of mean flow and temperature in both transverse directions. 44. A. KAPUR, A. CUMMINGSand P. MUNGUR 1972 Journal of Sound and Vibration 25, 129-138. Sound propagation in a combustion can with axial temperature and density gradients. 45. P. MUNGUR, H. E. PLUMBLEEand P. E. DOAK 1972 Proceedings llleme Colloque d'Acottstiqlte Aeronautique, Totdottsc. On the influence of jet flow on sound radiation. 46. T. H. MELLIN~ 1973 Journal of Sound and Vibration 29 (in press). The acoustic impedance of perforates at medium and high sound pressure levels. 47. R. J. ALFREDSON and P. O. A. L. DAVIES 1971 Journal of Sound attd Vibration 15, 175-196. Performance of exhaust silencer components. 48. P. E. DOAK 1970 Lockheed-Georgia Company Research)~lemorandum ER-10462. Excitation, transmission and radiation of sound from sources in ducts. Part III: Sound field of a source distribution in a hard-walled duct with reflecting terminations and no mean flow.

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49. B. T. CHO and L. S. G. KOVASZNAY 1958 Journal of Fhdd Mechanics 3, 494-514. Non-linear interactions in a viscous heat-conducting compressible gas. 50. C. L. MOP,FEY 1970 Ph.D. Thesis, University of Southampton. Theory of sound generation in ducted compressible flows, with applications to turbomachinery. 51. S. C. CROW 1970 Studies in Applied AIathematics XLIX, 21-44. Aerodynamic sound emission as a singular perturbation problem. 52. P. E. DOAK 1971 Journal of Sound and Vibration 19, 211-225. On the interdependence between acoustic and turbulent fluctuating motion in a moving fluid. 53. P. E. DOAK 1973 Journal of Sound and Vibration 26, 91-120. Analysis of internally generated sound in continuous materials: 3. The momentum potential field description of fluctuating fluid motion as a basis for a unified theory o f internally generated sound.