An exact analysis of the field radiated by tonal and random non-axisymmetric sources distributed over a disc or cylinder is presented. The analysis is exact, without recourse to near- or far-field approximations, and leads to a direct relationship between source frequency and the nature of the radiated field. The implications of the analysis for a number of applications are discussed, finding in particular that source identification is inherently ill-conditioned as a result of a ‘filtering’ effect that removes information from the radiation field; low-frequency sources generate fields that are indistinguishable from each other; jet noise fields are inherently simpler than the flow that gives rise to them, a finding that has previously been noted for experimental data.
A common problem in the prediction of acoustic, and other, radiation is that of calculating the field produced by a source distributed over a disc or a cylinder, such as loudspeakers, rotors, duct terminations and turbulent jets. There is an extensive literature on these fields, in acoustics, as well as in allied fields such as electromagnetism and optics. This paper presents an exact analysis of such fields, which can be used to draw conclusions about the nature of the field and about a number of problems that arise from it. The analysis was motivated by problems in acoustics, and will be described in these terms, but it should apply to any system governed by a retarded potential.
In the first part of the analysis, a disc source whose strength varies periodically in azimuth is considered, corresponding to the problem of a tonal sound field. This is a generalization of the familiar problem of radiation from an axisymmetric circular baffled piston (Pierce 1989) that has led to the division of the field into a near and far, or radiation, field. The nature of these regions for a non-axisymmetric source has been examined previously (Chapman 1993), with application to high-speed rotor noise. It was found that an exponentially small amount of energy ‘tunnels’ across the boundary between the near and far fields, with the form of the field being set by the relationship of source radius and frequency.
This analysis has been extended to consider the ‘information content’ of the field (Carley 2011), showing that the radiated field has a limited number of degrees of freedom, the number being fixed by the source frequency. This is especially relevant to the inverse problem, where acoustic field measurements are used to attempt to infer the nature of the source, such as for aircraft engines (Holste & Neise 1997; Gérard et al. 2007; Castres & Joseph 2007a,b; Lewy 2008). It has been recognized that the inverse problem is poorly conditioned, requiring special numerical methods for its treatment. Our analysis will demonstrate that this ill conditioning is an unavoidable result of the nature of the source and that there is a limit on the information available for source identification. It is well-known that owing to the arbitrary nature of source terms (Ffowcs Williams 1984), and the filtering effect of propagation, which removes source terms with subsonic phase velocity (Carley 2010), there is no unambiguous relationship that characterizes a source using only field information.
Acoustic field measurements have also been used to try to characterize the noise source in turbulent jets, such as in the near- to far-field correlation technique (Laurendeau et al. 2008), where it is assumed that the near-field data ‘encode’ source information, so that they can be used to draw inferences about the relationship between the source and the radiated noise field. The nature of the jet noise field, and of its relationship to the acoustic source, has long been of interest, and a number of recent experimental and computational studies have produced unexpected results on the relationship between the source and field.
It has long been accepted that turbulent jets are inefficient acoustic radiators, but it is only recently that it has been possible to quantify the simplicity of the noise field, compared with the complexity of the turbulent velocity field of the jet proper. In these studies (Jordan et al. 2007; Schlegel et al. 2012), it has been found that in a Mach 0.9 jet, 90 per cent of the acoustic field energy could be resolved using 24 orthogonal modes, while 350 were required to capture only 50 per cent of the flow energy. A number of explanations have been given for this relative simplicity of the acoustic field such as source cancellation (Michel 2007, 2009) and spatial filtering of the source (Freund 2001).
Our analysis gives an exact formulation for the field radiated by tonal and by random sources, which gives a framework for the understanding of these questions, without requiring assumptions about the nature of the source. In particular, it explains the limits on the source information radiated into the field and allows us to explain many of the features that we have just described.
2. Disc source
The problem is initially formulated as that of calculating the acoustic field radiated by a monopole source distributed over a circular disc. The system for the analysis is shown in figure 1, with cylindrical coordinates (r,θ,z) for the observer and (a,ψ,0) for the source. All lengths are non-dimensionalized on disc radius. The field from one azimuthal mode of the acoustic source, specified as , has the form , with Pn given by the Rayleigh integral (Goldstein 1974; Carley 2009), 2.1where and k is non-dimensional wavenumber.
The analysis that follows gives an exact formulation for the field radiated by tonal and by random sources, similar to previous work, but with efficient series expansions for radiation calculations (§2a), explicit integral and asymptotic formulae for the source terms (§2b, and extension to arbitrary frequency and azimuthal mode number for random sources (§2d).
(a) Tonal disc source
The analysis of the nature of the sound field from an arbitrary disc source is based on a transformation of the disc to an exactly equivalent line source, shown in figure 2, which shows the new coordinate system (s2,θ2,z) centred on a sideline of constant radius r>1. Under this transformation (Carley 2009), which gathers terms of constant phase to reduce the double integral to a set of single integrals: 2.2and 2.3with . Functions of the form of Kn(r,s2) have been analysed in previous work (Carley 2010) and can be written as 2.4where Uq(s) is a Chebyshev polynomial of the second kind, s=s2−r and the coefficients are functions of r but not of z. Inserting (2.4) into (2.2) gives 2.5and 2.6with .
For z=0, can be evaluated in closed form (Carley 2010): 2.7
For z≠0, there is no closed form solution for , but a series expansion is readily derived, based on a related integral containing the Chebyshev polynomial of the first kind, 2.8Put and ; note that φ is not the usual spherical polar coordinate. Then, using 8.533(1) from G&R (Gradshteyn & Ryzhik 1980), 2.9where is a spherical Hankel function, Pn is a Legendre polynomial, 2.10and jn is a spherical Bessel function. Expanding jn and integrating (G&R, 7.346) termwise gives an expansion for in powers of k.
Given a means of computing Φq, can be evaluated using standard relations for Chebyshev polynomials: 2.11One motivation for working with the integral involving a Chebyshev polynomial of the first, rather than of the second, kind is that the form of the series (2.9) is simpler for Chebyshev polynomials of the first kind. Second, if is to be computed for various values of q, it is computationally simpler to compute Φq for the appropriate values of q and combine them as required in (2.11).
(b) Line source coefficients
Given the formulation of the radiation problem in terms of the equivalent line source, it remains to find the coefficients in the expansion (2.4). These have been found previously (Carley 2011) as the solution of an infinite system of equations, which, although correct, is poorly conditioned and does not have an obvious explicit form. Here, we derive explicit, closed-form expressions for as weighted integrals of the source term sn(a). This is done by equating two exact expressions for the field in the plane z=0, using (2.7), 2.12with . Note that, as r>1, |S−r|<1.
To derive an integral expression for the coefficients , we begin by considering the corresponding expression for a ring source, 2.13so that the line source coefficients for a disc source are given by 2.14
More generally, let us use the orthogonality integral (G&R, 6.538.2) Thus, multiplying (2.15) by J2p+1(k) and integrating over k gives The inner integral is a Weber–Schafheitlin integral (G&R, 6.671.2): as |S−r|<1, put with , and then we obtain 2.19which reduces to (2.17) when q=0.
Expanding and in powers of and thus in powers of S and r (G&R, 1.332) gives expressions for in terms of elliptic-type integrals, 2.21 2.22 and 2.23where the elliptic-type integral Hn,m(λ) can be evaluated using published methods (Björkberg & Kristensson 1987).
(c) Asymptotic behaviour of for large q
To estimate for large q, we use the method of stationary phase (Bender & Orszag 1978). This shows that 2.24where ϕ′(x0)=0, a<x0<b and we take the + if ϕ′′(x0)>0 and the − if ϕ′′(x0)<0. If there are several points of stationary phase, each one contributes. If x0=a or b, the contribution is halved.
With (2.24) in mind, we write (2.19) as 2.25where . We have so . Thus, the two endpoints, ψ=0 and ψ=π, are points of stationary phase (where β′=0). For the second derivatives, we have Calculating, at ψ=0, S(0)=r−a, , , S′′=ra/S(0) and β′′(0)>0. Similarly, at ψ=π, S(π)=r+a, , , S′′(π)=−ra/S(π) and β′′(π)<0. Hence, as , where so that β(0)=−β1 and β(π)=β1. Thus, I(λ) decays as and so, from (2.25), grows as : 2.26
(d) Random disc source
The analysis given above can be extended to the case of a random source distributed over a disc. The starting point is an expression for the pressure radiated from a source distributed over a unit disc, 2.27where dA=a dψ da, and c is the speed of sound. Then, the correlation between p measured at two points (r1,θ1,z1) and (r2,θ2,z2) is 2.28where the correlation function is defined by 2.29Fourier transforming (2.28) with respect to τ gives the cross spectrum as 2.30and 2.31where Q12 is the correlation between the source at two points (a1,ψ1) and (a2,ψ2), and k=2πf/c.
On the assumption of statistical axial symmetry, the source correlation can depend only on the angular separation between two points ψ2−ψ1, so that Q12 and W12 can be expanded in Fourier series in azimuth, with 2.32where (j=1,2), 2.33and 2.34and an asterisk denotes complex conjugation.
(e) Low-frequency behaviour
While the theory that has been presented so far is general and exact, it is worth examining the properties of the acoustic field at low frequency, , which have been examined before using an earlier version of the method (Carley 2011). From (2.17) and (2.18), we have, exactly, 2.35so that the resulting line source coefficients and are not linearly independent. This has been noted previously (Carley 2011), using a different argument, and has an implication for the complexity of the low-frequency acoustic fields. When , the field is dominated by the contributions of the first two line source modes. The fact that these modes will have the same coefficients, to within a scaling factor, means that the low-frequency acoustic fields of disc sources are identical, whatever might be the details of the source proper. The implications of this finding for jet noise are considered in §4d.
3. Random cylindrical source
As a model problem for jet noise, and in particular of the modal decomposition methods and the near-to-far field correlation method used in studies of jet noise (Laurendeau et al. 2008; Schlegel et al. 2012), we consider a random source distributed over a cylindrical domain. The analysis for a random disc can be extended to the case of a cylindrical source of length 2L with axial coordinate −L≤z≤L, using the same non-dimensionalization as used previously. The source cross-spectrum is Q12(k;a1,ψ1,b1;a2,ψ2,b2), with b1 and b2 source axial coordinates. Using the same symmetry arguments as previously, the cross-spectrum is decomposed, 3.1and the azimuthal components of the cross-spectrum between two field points are given by 3.2and 3.3
(a) Nature of acoustic field
A first set of conclusions can be drawn from the results for the radiation function , in §2a. For large order q, the Bessel function Jq(k) is exponentially small for k<q so that the line source modes with order q>k generate noise fields of exponentially small amplitude. Since has a maximum in the plane z=0, (2.7) says that the whole field is of exponentially small amplitude. This gives an indication of how much of a given source distribution radiates into the acoustic field, as a function of wavenumber k and source order q, without recourse to near-field, far-field or other approximations. Section 4b, giving sample calculations using full numerical evaluation of the appropriate integrals, when compared with the line source method, shows that indeed the radiated field contains only terms owing to line modes with .
To demonstrate the character of the acoustic field of the line source modes, we plot for k=1.5, figure 3, and k=6, figure 4, for various values of q. The plots show logarithmically spaced contours of the real and imaginary parts of the radiated field, similar to those of Chapman (1993), demonstrating the propagating wave character of the acoustic field, and the variation of amplitude with k.
Figure 3 shows the cut-off behaviour as q is increased above k. The q=0 and q=1 modes radiate efficiently into the whole field, as shown by the first two rows of contour plots. At q=2, the field is already reduced in amplitude (note that the contour levels are the same in all plots) and the radiation pattern is becoming more directive, with less radiation into the near-axis region. At q=3, the bottom row of figure 3, the field has almost disappeared, having only a small amplitude and being concentrated in a region near the source plane.
Figure 4 shows similar behaviour, but with the higher wavenumber k=6, the cut-off happens at a higher mode order, q=8, and the lower order modes, especially q=0 and q=1, have a more complicated field pattern, with a pronounced null on a line through the origin. The q=4 mode has a simpler directivity, but a greater amplitude, shown by the larger number of contours present: again, the same contour levels have been used in all plots.
(b) Numerical performance
As a check on the calculation method and on the convergence to the correct value of (2.1), figure 5 shows a sample evaluation of Pn(k,r,z) using full numerical evaluation of the integral and the line source method with an increasing number of line source modes included, for k=6 and n=4. The radial source term sn(a)≡1. As predicted, the results for two and four modes do not capture the behaviour correctly, while six modes are sufficient to come very close to the correct result. When eight modes are included, the result is indistinguishable from full numerical evaluation.
For a random source, we assume a valid functional form for the source correlation, based on that given by Michalke (1983) for ring sources, with the addition of a radial correlation term, 4.1with α being an azimuthal length scale and β controlling the correlation in radius. Equation (4.1) can be interpreted as the product of the local source strengths q(a1) and q(a2) with a coherence function, given by the exponentials, which is symmetric in source position and has unit value when the source points coincide. For the calculations that follow, q(a)≡1. The azimuthal components of Q12 can be evaluated (Michalke 1983) as 4.2where In is a modified Bessel function.
Figure 6 shows similar results for a random disc source with the same parameters as in figure 5 and the correlation function parameters set to α=β=1/2. Again the effect of the number of line source modes included is demonstrated by the convergence to the full numerical evaluation, with the computed correlation given by a truncated series, where Q=2,4,6,8. As before, when Q→k, the line source results converge to the full numerical evaluation, with the higher order modes being ‘cut-off’ by the exponentially small amplitude of their radiated field.
Finally, we present a check on the evaluation of for a random cylindrical source. Here the correlation function has a Gaussian axial correlation term added, 4.3with Ω=[(b1−b2)/γ]2+[(a1−a2)/β]2 4.4Again, figure 7, the line source approach gives results indistinguishable from full numerical evaluation for . A last point to note is that the decomposition technique is very much faster than full numerical evaluation of the radiation integrals, especially if the field is to be evaluated at a large number of points, making it a practical noise prediction technique, as well as an analytical tool.
The analysis presented so far has a number of implications for noise control and for source identification methods; jet noise is considered in §4d. The main point of the work presented is that it disconnects source from radiation effects, by condensing all of the information about the source into the coefficients and all information about radiation into the function . It is then clear that the radiated acoustic field depends only on the non-dimensional wavenumber k, which fixes the number of line source modes that radiate, in effect fixing the number of source coefficients that contribute to the final field.
The implication of this finding is that acoustic sources whose line source coefficients differ only for generate acoustic fields that cannot be distinguished from each other. In source identification problems, this means that the acoustic field, near or far, has only a finite number of degrees of freedom. Thus, any attempt to infer details of the source beyond this number of degrees of freedom is probably futile: the information about the source is not available in the acoustic field, even without considering the effects of background noise and other measurement errors. This may well explain the poor conditioning of many source identification methods, where the addition of large numbers of sensors has failed to improve the inversion technique.
The obverse of this observation is that if the acoustic field is to be controlled, only a finite number of degrees of freedom are required of an active controller, because only those parts of the source that actually contribute to the acoustic field need be controlled for. Similarly, if noise is to be reduced at source, the reduction effort should concentrate on reducing the contribution of those, lower order, modes that generate the perceptible acoustic field.
(d) Jet noise
An issue that has been of some interest for many years in studies of jet noise is the disparity between the complexity of the turbulent flow that gives rise to the noise field, and the relative simplicity of the field itself, exemplified by the finding that for a Mach 0.9 jet, 24 orthogonal modes were sufficient to capture 90 per cent of the energy of the acoustic field, but 350 were required to resolve 50 per cent of the flow energy (Jordan et al. 2007). Various explanations have been put forward for this disparity, concentrating on the whole on mechanisms that affect radiation efficiency (Michalke 1983; Freund 2001; Michel 2007, 2009), or discussing the distinction between a ‘source’ term, which radiates, and a ‘flow’ term, which does not (Sinayoko & Agarwal 2012). The analysis presented in this paper gives an interpretation of the sources in terms of radiating and non-radiating terms, purely on the basis of the properties of the acoustic source term, and the wavenumber of the radiation.
A first point to note is that the maximum noise level for a subsonic jet lies in a frequency range , with k based on the jet exhaust diameter. This means, as has been noted in experimental work, that the noise field is composed of azimuthal modes of order , as would be expected from the analysis this far. Thus, the observations made in §2e can be used to help explain certain features that have been noted in experimental studies of jet noise fields.
A recent paper (Schlegel et al. 2012) generalizes the standard technique of proper orthogonal decomposition (POD) to identify orthogonal modes of an aerodynamic field that contribute most to the observed, acoustic field. In a study of a Mach 0.9 jet, it was found (Schlegel et al. 2012, fig. 9) that 24 modes contribute 90 per cent of the correlated noise. We tentatively propose, on the basis of the analysis given here, that such a small number of modes is a result of the radiation properties of the source, and that certain other features of the acoustic field can be explained using arguments based on the radiation properties outlined so far.
The standard theory of radiation from turbulent flows remains that of Lighthill (1952, 1954), although a number of theories modified for particular purposes or insights have been developed. The essential principle remains the same, however, in that the radiated noise, which here we view in terms of the cross-spectrum of the acoustic signal measured at two points in the field, is given, exactly, by an integral over the source region, where the source includes all mean and fluctuating flow effects (Lighthill 1952, eqn 15), subject to some differential operator, 4.5where is the cross-spectrum of the source at two points in the source volume, and index i runs over the different combinations of source term available, with being the spatial differential operator corresponding to a given source combination. This approach, which lumps source and propagation effects together, is supported by computational results (Bogey et al. 2001). The source terms are given by the two-point, second- and fourth-order velocity correlations in the flow, and are not mutually uncorrelated. The analysis presented so far demonstrates that only a finite number of modes can radiate to the acoustic field, imposing a limit on the complexity of that field, without recourse to assumptions about the nature of the acoustic source or of the flow that gives rise to it.
Secondly, in another experimental study, ‘the first POD mode is shown to give a near-perfect representation of the fluctuation energy radiation at low angles’, i.e. near the jet axis, ‘larger numbers of modes being necessary to completely reproduce the radiation characteristics at higher angles’, i.e. nearer the jet exhaust plane (Koenig et al. 2010). These findings are readily explained by examination of figure 3, bearing in mind that the noise field is dominated by two line source modes, whose coefficients are not linearly independent, so that the field radiated by the source effectively has only one orthogonal mode per source term. In figure 3, the radiation pattern of the first four line source modes is shown, and it is clear that the near-axis field contains contributions only from the first two modes, while and only appear nearer the source plane, and then only very weakly. Again, we note that this is purely a result of the radiation properties of the source decomposition employed, and does not require that any assumptions be made about the nature of the source, of the aerodynamic processes which give rise to it, or whether the observer is in the near or far field, as in the earlier analysis of Michalke & Fuchs (1975).
We have presented an analysis which gives the radiated field of tonal and random disc sources without requiring near- or far-field approximations, splitting the problem into purely source-related terms and functions that affect only radiation. The nature and complexity of the radiated field are then fixed by the non-dimensional wavenumber of the source, showing, among other things, that low-frequency sources generate indistinguishable fields. We conclude that many effects reported in the literature, such as the poor conditioning of source identification methods and the relative simplicity of jet noise fields, can be explained via our analysis, on the basis of the radiation characteristics of circular and cylindrical sources, without recourse to other arguments or approximations.
- Received June 16, 2012.
- Accepted August 15, 2012.
- This journal is © 2012 The Royal Society