Near- to far-field characteristics of acoustic radiation through plug flow jets

Transcription

1 Near- to far-field characteristics of acoustic radiation through plug flow jets G. Gabard a Institute of Sound and Vibration Research, University of Southampton, Southampton SO17 1BJ, United Kingdom Received 27 March 2008; revised 22 July 2008; accepted 26 July 2008 This paper reports a theoretical study of the radiation of sound through jet exhausts. It focuses on the transition from near field to far field by considering the features of the near-field solution and how these features translate to the far field. The main focus of this work is the importance in some cases of lateral waves radiating from the jet. While the presence of lateral waves has long been recognized, there has been no systematic investigation of the practical consequences of these waves in the prediction of sound propagation through round jets. The physical mechanisms involved in the generation of these waves are presented as well as the conditions under which they become significant. Another issue is the possibility of channeled waves inside the jet associated with strong sound radiation in the forward arc. This paper also discusses the validity of the far-field approximation when lateral waves are present. It is shown that the standard far-field approximation can be improved by adding correction terms that account for the presence of the lateral waves and channeled waves. The challenge posed to computational aeroacoustics by these near-field effects is also discussed Acoustical Society of America. DOI: / PACS numbers: Py, El JWP Pages: I. INTRODUCTION Predicting the propagation of sound through jets is of particular importance for jet noise modeling and for fan exhaust noise simulations. For both applications it is crucial to account for the refraction of sound through the mixing layer of the jet. For jet noise modeling, a simplified description corresponding to a low frequency approximation of the real problem is that of a point source embedded in a plug flow jet where the mixing layer is described as a cylindrical discontinuity in the mean flow. 1,2 For fan exhaust noise, a classical idealized model is the Munt solution, which involves the radiation of sound from a semi-infinite jet pipe carrying a plug flow jet. 3 These two model problems will be used in the present paper to investigate the transition of the radiated sound from the near field to the far field. In the process the validity of the far-field approximation will be discussed and defined more precisely. While noise radiation in classical acoustics with no flow can be understood relatively easily, for instance, by means of ray acoustics, the jet introduces additional features in the propagation of sound to the far field. The two features to be discussed in detail are the lateral waves radiated downstream of the source and the channeled waves. Although these features are essentially associated with near-field effects due to the vortex sheet, they can influence the radiated sound at large distances from the source, that is, at distances where the far-field approximation would usually apply. The present paper also aims at providing a description of the conditions required for these effects to be important in the context of the plug flow jet so that analytical models such as the Munt solution can be used with more confidence. In fact the present work is a continuation of that reported in Ref. 4 where numerical simulations of the Munt problem were compared to the far-field analytic solution, and unexpected discrepancies were observed. The validity of the far-field approximation is not only a theoretical issue but is also of importance in jet noise or turbomachinery noise measurements where it is important to have microphones located in the geometric far field in order to use the spherical spreading to extrapolate the measurements for larger distances. Measurements can be influenced by near-field effects and nonlinearities if the microphones are too close to the jet. 5,6 The key of the problem is the transmission and reflection of sound at the vortex sheet separating the jet from the ambient flow. Several theoretical analyses were devoted to the interaction of linear acoustic disturbances with a vortex sheet. Earlier work focused on the continuity conditions to be imposed at the vortex sheet, that is, the continuity of pressure and normal displacement. A summary of this work is given by Howe. 7 A central issue is the stability of the vortex sheet, which exhibits a linear Kelvin Helmholtz instability. It induces an acoustic field growing exponentially in the streamwise direction and thus dominates the sound radiated from the acoustic source. The unlimited exponential growth of this linear instability is nonphysical, and its relevance for the sound radiated to the far field is therefore questionable. However, as first noted by Miles, 8 the instability wave should be included in order to obtain a causal solution. The formulation of causal solutions has since been further investigated by Jones, 9,10 Jones and Morgan, 11 and by Crighton and Leppington. 12 Another important aspect, and the topic of this paper, is the presence of lateral waves that can be particularly signifia Electronic mail: gabard@soton.ac.uk J. Acoust. Soc. Am , November /2008/1245/2755/12/$ Acoustical Society of America 2755

2 cant inside the zone of silence. Lateral waves represent a distinct part of the scattered acoustic field in addition to the direct reflected and transmitted waves. These were first included in the analysis by Gottlieb who derived correction terms for the far-field approximation of the solution. 13 For a planar vortex sheet, the derivation of a complete solution taking both the instability and the lateral waves into account was initiated by Friedland and Pierce 14 and completed by Jones and Morgan. 11 For axisymmetric jets, Morgan derived the exact solution for a fixed monopole embedded in a subsonic plug flow jet. 1 However, lateral waves were not included in the analysis as it was incorrectly assumed that their behavior was similar to that of the two-dimensional vortex sheet. Goldstein also derived solutions in the low frequency limit for convected point sources in an axisymmetric jet, 15,16 but lateral waves were not considered. The present paper aims at providing a detailed account of near-field effects, including lateral waves, in the context of round jets using the plug flow as a simplified model for the shear layer. It should be noted that noise radiation from round jets has been extensively studied in the high frequency limit see, for instance, Ref. 17. However, again lateral waves have seldom been considered. For planar vortex sheets Suzuki and Lele studied both the low and high frequency limits for the lateral waves and compared these results with direct numerical simulations. 18,19 The remainder of this paper is as follows. The plug flow jet and the Munt problem are introduced in the next section together with their analytical solutions. The standard far-field approximation and a uniformly valid approximation are discussed in Sec. III. In Sec. IV, the presence of lateral waves is illustrated and analyzed. The properties of channeled waves are described in Sec. V. II. DESCRIPTION OF THE PROBLEMS The present analysis is concerned with two model problems in aeroacoustics shown in Fig. 1. The first one is the radiation of sound by a fixed point source in a plug flow jet. The second one is the Munt problem, which involves the radiation of sound from a semi-infinite jet pipe carrying a plug flow jet. A. The plug flow jet This is an idealized model for the propagation of sound through a jet mixing layer, and it forms the basis for several jet noise models. 2 The jet is a cylinder with radius R 0 where the flow is purely axial and uniform with velocity v j, density j, and sound speed c j. The jet is embedded in a uniform axial ambient flow with velocity v 0, density 0, and sound speed c 0. The continuity of pressure and normal displacement apply across the vortex sheet separating the jet and the ambient flow. An important aspect of the problem is the refraction of sound by the flow discontinuity across the vortex sheet, which results in the so-called cone of silence. This model represents a low frequency approximation of sound propagation through jets in the sense that the mixing layer is acoustic mode small compared to the acoustic wavelength. The model is presented in nondimensional form with R 0, c 0, and 0 as reference values. For a source Q=r r s ze it im /r s representing a circumferential Fourier component of a time-harmonic mass point source located inside the jet r s 1, the complex amplitude of the pressure field outside the jet can be written as an inverse Fourier transform, pr,z = ambient flow vortex sheet 2 jet nozzle PuH m 2 0 re iuz du, where H m 2 is the Hankel function of the second type and u denotes the reduced axial wavenumber u is the actual axial wavenumber. The superscript 2 for the Hankel function will be omitted in what follows. The solution is given by 1 Pu = i1 um um 1 J m 1 r s, 2 Lu where M 0 =v 0 /c 0, M 1 =v j /c 0, and J m is the Bessel function. The radial wavenumbers in the jet and the ambient flow are 1 u=c um 1 2 u 2 1/2 and 0 u=1 um 0 2 u 2 1/2 with C 1 =c 0 /c j. The branch cuts of 0 and 1 are defined from u 0,1 to u 0,1 i, with u 0 =1/M 0 1 and u 1 =C 1 /C 1 M 1 1. The function Lu is of particular importance as Lu = 0 represents the characteristic equation of the jet, which specifies the modes of oscillation of the jet column. It is defined as Lu = D 1 1 um 1 2 J m 1 0 H m 0 r s point source 1 um 0 2 H m 0 1 J m 1, r r ambient flow vortex sheet z FIG. 1. Top: plug flow jet with a point source. Bottom: the Munt problem. where D 1 = j / 0. One mode of the jet corresponds to the Kelvin Helmholtz instability of the vortex sheet. It is located at u=u 0 in the first quadrant of the complex plane and represents a disturbance growing exponentially in the axial di- jet z 2756 J. Acoust. Soc. Am., Vol. 124, No. 5, November 2008 G. Gabard: Acoustic radiation through plug flow jets

3 branch cut branch cut u 0 u 0 + FIG. 3. Close-up view of the branch points u 0 and u 0 + with the zeros of Lu and the zeros of H m 0. The parameters are =20, m=18, M 1 =0.5, M 0 =0, and C 1 =D 1 =1. problem is slightly modified since the axial phase speed of a downstream mode of the jet can become subsonic with respect to the ambient medium. The consequence of this will be discussed in detail in Sec. V. As noted by Morgan, 1 another consequence is that one has to deform the contour of integration so as to cross the real axis at u c and then to pass below the branch point u 0, as illustrated in Fig. 2. FIG. 2. Locations of the zeros of Lu and the zeros of J m 1 in the complex u plane. Top: with M 0 1 M 1 C 1 2 /1 M 1 2 C 1 2 ; bottom: with M 0 1 M 1 C 1 2 /1 M 1 2 C 1 2. The dashed line is Reu=u c. The thick line is the integration contour. In the bottom figure note the deformation of the integration contour around the branch point u 0. rection. A high frequency approximation yields the following estimate: 20 u 1 0 M 0 + M 1 + i M1 M /2. In addition to the instability mode, there is a series of acoustic modes corresponding to exponentially decaying oscillations of the jet. These modes are located either close to the real axis between u 1 and u + 1 or close to the line Reu=u c =u 1 +u + 1 /2 see Fig. 2. The zeros located in the region /2argu u c 0 represent modes propagating in the positive axial direction, while the zeros in the region /2 argu u c represent modes propagating in the negative direction. The physical relevance of the location of these zeros will be discussed in more detail in Secs. IV and V. Finally, there are two series of zeros clustered around u + 0 and u 0, as shown in Fig. 3 in the lower and upper parts of the complex plane, respectively. The contour of integration for the inverse Fourier transform is also shown in Fig. 2. This contour is defined from i0 to++i0, and it intersects the real axis at u c.in order to obtain a causal solution, it is necessary to deform the contour around the point u 0 corresponding to the instability wave of the vortex sheet. 12 For sufficiently high jet Mach numbers, the branch point u 0 can move to the right of u c this is the case when M 0 1 M 1 C 2 1 /1 M 2 1 C 2 1. In that situation the physics of the B. The Munt problem The Munt problem is an extension of the problem presented above where the jet exhaust is represented by a semiinfinite duct. The source of sound is an incident mode propagating from inside the duct toward the duct exhaust. This acoustic mode radiates out of the duct through the mixing layer and to the far field. This is a simplified description of the sound radiation from turbofan exhausts. An analytical solution for this problem was first derived by Munt using the Wiener Hopf technique. 3 Several extensions of the original Munt problem have been recently developed In Ref. 20, it was shown that it is possible to evaluate exactly the analytical solution in the near field that is, without the far-field approximation. The pressure field of the incident acoustic mode is pr,z,= mn re i + mn z im, where mn r is the + mode shape, mn is the axial wavenumber, and m and n are the azimuthal and radial mode orders. It is possible to obtain either the noncausal solution, which ignores the instability wave of the vortex sheet, or the causal solution, which includes the instability. 12,23 In the latter case it is also possible to separate explicitly the instability from the rest of the solution, and thus the causal acoustic field alone can be obtained. When the full Kutta condition is applied at the duct lip, the solution for the Munt problem is given by 20 Pu = i mn mn 1 + mn K + mn 1 um u u + mn 0 uk + uh m 0, 3 where 0 =1 M 0 1u 1/2 and 0 =C 1 M 1 C 1 1u 1/2 the branch cuts are the same as for 0 and 1. The addition of the semi-infinite duct introduces a major difficulty since one has to find two split functions, K + u and K u, regular and nonzero in the upper and lower halves of the complex u J. Acoust. Soc. Am., Vol. 124, No. 5, November 2008 G. Gabard: Acoustic radiation through plug flow jets 2757

4 plane, respectively, such that Ku=K + u/k u. This factorization of the kernel is not trivial but will not be described here see Ref. 24. The Wiener Hopf kernel is defined by Ku=Lu/J m 1 /H m 0, and it has the same zeros as the function Lu for the jet column presented above. In addition, the poles of Ku corresponds either to zeros of J m 1 or to zeros of H m 0. The first case represents acoustic modes in an infinite straight circular duct. As illustrated in Fig. 2, these poles are located either on the real axis between u 1 and u 1 + for cut-on modes or on the line Reu =u c for cut-off modes. The zeros of H m 0 represent solutions to the exterior problem, that is, the diffraction of sound by an infinite rigid cylinder embedded in the ambient flow. These poles are clustered near u 0 and u 0 +, as shown in Fig. 3 see p. 373 of Ref. 25 for a description of the zeros of H m. III. FAR-FIELD ASYMPTOTIC SOLUTION Although it is possible to evaluate the exact solutions of the plug flow jet and of the Munt problem in the near field using Eq. 1, it is common practice to use the far-field approximation of these solutions. It is justified by the fact that the observer is generally many wavelengths away from the source embedded in the jet or from the duct exhaust. The advantage of the far-field approximation is that one does not need to evaluate the complex integral for the inverse Fourier transform. Also the far-field approximation provides a clear distinction between the geometrical spreading and the directivity of the acoustic field. In preparation for the analysis of the transition from near to far field, the derivation of the standard far-field approximation is reviewed in this section, and a uniformly valid approximation is also derived. A. The standard far-field approximation For the present analysis it is necessary to make explicit the behavior at infinity of the Hankel function in the integrand of the inverse Fourier transform Eq. 1. To that end we consider pr,z = im+1/2 2 3 PuQ 0 r e iuz i 0 r du, 0 r which is an exact reformulation of Eq. 1. The additional function Q in the integrand is the ratio between the Hankel function and its asymptotic expression for a large argument see item in Ref. 25, Qz = z 1/2 H m ze iz im/2 i/4. 2 When zm 2 1/4, we have Qz1. This approximation is almost systematically used in previous work but is not mandatory. We now introduce the following change of variables corresponding to a Lorentz transform that converts the ambient flow into a steady medium: r = Rˆ sin ˆ, z = 1 M0 2 Rˆ cos ˆ, 4 = 2 1 M0 ˆ, u = M M. 0 The solution written with this Lorentz transform is the acoustic field perceived by an observer moving with the ambient flow. Equation 4 now reads ˆ i prˆ m+1/2 e iˆ Rˆ M 0 cos ˆ,ˆ = M 2 0 ˆ Rˆ sin ˆ 1/2 PuQˆ Rˆ 1 2 sin ˆ 1 2 1/4 e ˆ R ˆ q d, with q= i 1 r 2 sin ˆ i cos ˆ. When ˆ Rˆ, the complex integral in Eq. 5 can be approximated using the method of steepest descent see Chap. 4 of Ref. 26. The function q is analytic except for the branch cuts of the square root, which correspond to the branch cuts of 0 and go from 1 to1i in the complex plane. It has a single first-order saddle point at s =cos ˆ, where we have q s = i and q s =i/sinˆ 2. From that saddle point, a path of steepest descent can be defined by q=q s s 2, where s is a real parameter. This yields the following parametrization of the path of steepest descent: = cosˆ cos 1 1 is 2 with s 0. The contour can be deformed onto the path of steepest descent provided that any poles or branch cuts crossed in the process are properly accounted for. We will deal in detail with this aspect in the next section. On the path of steepest descent, the integrand decays exponentially away from the saddle point, and only the contribution from that point is significant. The integral in Eq. 5 can then be approximated by 2 ˆ Rˆ q s Pu s Qˆ Rˆ sin 2 ˆ 1 s 2 1/4 e ˆ R ˆ qs, where u s =u s and the argument of the square root is taken to be argd s =/4, with d denoting an element of the steepest descent path at s see p. 382 of Ref. 26. The asymptotic far-field solution is given by prˆ,ˆ im+1 Pu s Qˆ Rˆ sin 2 ˆ Rˆ e iˆ Rˆ 1 M 0 cos ˆ M 0 We now apply the inverse Lorentz transform by introducing the spherical coordinates R- in the physical frame of reference defined by z=r cos and r=r sin. This yields pr, im+1 Pu s e irs R 1 M0 2 sin 2 Q R sin2 1 M 0 2 sin 2 1/2, where S=1 M 0 2 sin 2 1/2 M 0 cos /1 M 0 2 accounts for the Doppler effect of the ambient flow. The standard spherical spreading of the acoustic waves in the far field is not yet apparent in Eq. 8 due to the dependence of the J. Acoust. Soc. Am., Vol. 124, No. 5, November 2008 G. Gabard: Acoustic radiation through plug flow jets

5 20 log 10 (Q) function Q on R. To recover the standard 1/ R spherical spreading, one must assume that Q1, which is valid when Rm 2 1/41 M 0 2 sin 2 1/2 / sin 2. Note that the factor Q does not depend on the details of the jet and the source. Figure 4 shows the factor Q as a function of for various azimuthal orders m. It is clear that the approximation Q1 is not valid close to the jet axis, especially for high azimuthal orders, since Q increases very rapidly as decreases. However, it should be mentioned that for large azimuthal orders, sound is primarily radiated at large angles so the approximation Q1 should not impair the prediction of the main peak of directivity. Although not shown in Fig. 4, it was found that the influence of M 0 on Q is negligible for small angle. B. A uniformly valid approximation The standard asymptotic far field Eq. 8 is only valid if there is no pole in the integrand close to the path of steepest descent and if no pole has been crossed when the contour is deformed onto the path of steepest descent. This requirement is generally not satisfied, and the standard approximation Eq. 8 is not uniformly valid over the range of direction. Consider that Pu has a pole close to the path of steepest descent, say, at = p. The contribution of the integrand in the vicinity of the pole is not negligible compared to the contribution of the saddle point. Therefore, the following term should be added to approximation 6 of the integral see Sec. 4.4a of Ref. 26: with φ FIG. 4. Amplitude of the factor Q in db with R=80 and M 0 =0. Thick solid line: m=1; thin solid line: m=5; dashed line: m=10; dot-dashed line: m=15. Âp Qˆ Rˆ 1 p 2 sin ˆ s p ˆ Rˆ 1 p 2 1/4 e ˆ R ˆ q s Cs p ˆ Rˆ, Cz =1 i wzz for Imz 0, where w is the error function defined by item of Ref. 25. Â p is the residue of Pu at = p. The position of the pole in the complex s plane is s p =q s q p, where the branch of the square root is chosen so that s p p s / 2/q s as p s the argument of the square root in this last expression is argd s =/4. This yields a first correction term f p that needs to be added to the standard approximation Eq. 7, i f p Rˆ m+1/2 Â p e iˆ Rˆ 1 M 0 cos ˆ,ˆ = s p Rˆ 1 2 p 1/4 21 M 2 0 sin ˆ 1/2 Cs p ˆ Rˆ Qˆ Rˆ 2 1 p sin ˆ. This expression can also be written with the physical spherical coordinates R and, i m+1/2 2 1 M0 A p e irs f p R, = s p R 20 u 0 sin 1 M 2 0 sin 2 1/4 Cs p ˆ Rˆ Q0 u p r, 9 where A p is the residue of Pu at u=u p. This term becomes significant when q s q p 1 Oˆ Rˆ, and it plays a similar role to the transition functions near shadow region boundaries in ray acoustics see p. 405 of Ref. 26. The correction term described above accounts for the presence of a pole in the vicinity of the path of steepest descent. Another term should be added in the asymptotic expression if a pole is crossed when the contour of integration is deformed onto the path of steepest descent. This term corresponds to the residue of the integrand in Eq. 5, g p Rˆ,ˆ = im+3/2 2ˆ Âp e ˆ R ˆ im0 cos ˆ +q p 1 M 2 0 Rˆ sin ˆ 1/2 1 2 p 1/4 Qˆ Rˆ 1 p 2 sin ˆ HIms p, where H is the Heaviside function. The sign in this expression depends on the position of the pole relative to the path of steepest descent. With the physical spherical coordinates this expression reads 2e irs ˆ R ˆ 2 sp g p R, = i m+3/2 A p r0 u p Q 0 u p rhims p. 10 For a series of poles u p p=1,...,n, the uniformly valid asymptotic solution in the far field is given by 1/2 1 M 2 0 sin 2 pr, im+1 Pu s e irs R 1 M0 2 sin 2 Q R sin2 N + f p R, + g p R,. p=1 11 Although the two terms f p and g p are discontinuous when the sign of Ims p changes, the sum f p +g p is continuous. Jones and Morgan derived similar correction terms for a twodimensional problem with a planar vortex sheet. 11 C. Instability Note that in Eq. 8 the Kelvin Helmholtz instability of the vortex sheet is not accounted for, so Eq. 8 represents J. Acoust. Soc. Am., Vol. 124, No. 5, November 2008 G. Gabard: Acoustic radiation through plug flow jets 2759

6 the far-field approximation of the acoustic solution alone. The instability wave is associated with the pole of the integrand located in the first quadrant of the complex u plane, as shown in Fig. 2. The corresponding pressure field is readily obtained by taking its residue in Eq. 1, pr,z = ia 0 H m 0 u 0 re iu 0 z Hz, where A 0 is the residue of Pu at u 0. This pressure field is exponentially growing in a conical region defined by with tan *= Imu 0 /Im 0 u 0. Outside this region it is exponentially decaying and negligible. 20 IV. LATERAL WAVES The transition of the acoustic solutions from the near field to the far field is now discussed. The first feature is the presence of lateral waves. A. Observations and physical principles Results are presented here for the Munt problem with a circular duct carrying a jet with Mach number v j /c j =0.447, sound speed c j /c 0 =1.02, and density j / 0 =0.96. The ambient medium is steady. The Helmholtz number based on the duct radius is This configuration is representative of an aeroengine bypass duct at approach conditions. To study the transition from the near-field solution to the standard far-field approximation, the pressure is plotted along a circle centered on the exhaust plane. The radius of the circle is then varied from R=3.3 to 38 the latter corresponding approximately to the distance used for engine certification measurements. Figure 5 shows the results obtained with the analytical model for an incident plane wave. With the far-field solution, the cone of silence is approximately 47 wide. The directivity then reaches a sharp peak followed by a series of lobes of directivity. For the near-field solutions outside the cone of silence, the pattern of the far-field approximation quickly emerges as the distance of the observer is increased, with the lobes and zeros of the solutions rapidly approaching that of the far-field solution. It can therefore be concluded that the far-field asymptotic solution is a good approximation of the exact solution in this range of directions. However, two important differences can be observed between the far-field solution and the series of near-field solutions. Firstly inside the cone of silence, as the distance is increased, the pressure amplitude does not converge to the farfield asymptotic solution at least for distances up to R=38, which corresponds in this case to more than 117 wavelengths and remains comparable in amplitude to the main lobe of directivity at around 90 db. This observation suggests that there is a feature of the sound radiated in the cone of silence in the near field that is not captured by the far-field approximation. Also the fact that the far-field asymptotic solution is not recovered even at 117 wavelengths from the duct exhaust indicates that the common definition of the far field does not apply in that case. Secondly, there are significant differences for the main lobe of directivity both in terms of direction and amplitude. While a sharp peak at 48 with an amplitude of 105 db is observed in the far-field solution, the main lobe in the near FIG. 5. Transition from near to far field for the Munt problem with an incident plane wave top with mode 2,0 center and mode 5,2 bottom. All the results have been scaled for R=38. field is smooth, located at approximately 46 and with an amplitude of 93 db. In fact the sound amplitude observed in the near field of the cone of silence is almost as high as the main lobe of directivity. These differences are particularly significant since the direction and amplitude of the main lobe are often the main information to be obtained from fan noise simulations, and they also have a significant impact on the predictions of perceived noise levels J. Acoust. Soc. Am., Vol. 124, No. 5, November 2008 G. Gabard: Acoustic radiation through plug flow jets

7 FIG. 6. Exact near-field solutions at R=38. Left: first radial mode with varying azimuthal order. Right: axisymmetric modes m=0 with varying radial order. The agreement between the near-field and far-field solutions tends, however, to improve for higher-order modes. Two examples are given in Fig. 5 with the modes 2,0 and 5,2. While there are still significant differences at short distances from the exhaust, the near-field solution eventually collapses onto the far-field approximation. This is also illustrated in a different way in Fig. 6 where the directivity at a fixed distance R=38 is plotted for various azimuthal and radial orders. While the lateral waves are more important for the plane wave, their amplitudes remain significant for higher azimuthal and radial orders. The sound field observed in the near field inside the cone of silence is due to the presence of lateral waves. They are found in problems of wave propagation across interfaces between media with different propagation speeds, such as underwater acoustics, linear elasticity, and electromagnetism see Sec. 30 of Ref. 27 and see also Ref. 28. In the context of aeroacoustics, the presence of lateral waves has long been recognized in the theoretical analysis of an acoustic wave interacting with a planar vortex sheet More recently Suzuki and Lele investigated numerically and analytically the properties of refracted arrival waves in planar mixing layers. 18,19 However, the practical consequences of these waves for the prediction of noise radiation from round jets have not yet been studied. In the context of the Munt problem, the physical mechanism involved in the presence of lateral waves is the following. When well cut-on acoustic modes radiate from the duct exhaust that is, low-order modes at relatively high Helmholtz numbers, the angle of incidence of the waves on the vortex sheet is small. This results in a weak transmission of acoustic energy through the vortex sheet into the ambient flow diffraction theory shows that transmission decreases and then vanishes for shallow angles of incidence. Hence, the sound waves are partly confined inside the jet due to significant reflections at the vortex sheet. In that case, acoustic waves are allowed to propagate far away from the duct exhaust within the jet and do not decay as fast as the standard decay rate. This is clearly visible in Fig. 7 where the pressure amplitude in the jet is still significant 16 wavelengths away from the exhaust. This is also shown in Fig. 5 where the sound pressure level inside the jet is visible close to the axis and is actually higher than that in the ambient flow by more than 10 db. A consequence is that the lateral waves in the ambient flow are generated continuously along a large portion of the vortex sheet, which can extend far away from the duct exhaust. In contrast, the sound corresponding to the typical lobes of directivity observed in the far field radiates from a small region near the exhaust. The generation of lateral waves for the point source in a plug flow jet can be understood in a similar way. FIG. 7. Real part of pressure in the near field for the Munt problem with the plane wave. J. Acoust. Soc. Am., Vol. 124, No. 5, November 2008 G. Gabard: Acoustic radiation through plug flow jets 2761

8 B. Analysis In theoretical studies of sound interacting with planar vortex sheets, the lateral waves are associated with a branch cut in the integrand of the inverse Fourier transform Eq. 1. When the contour is deformed onto the steepest descent path, it is indented around the branch cut, and it is this contribution of the contour that accounts for the lateral waves. This branch cut is associated with the transverse wavenumber 1 in the region with a faster propagation speed. As indicated in Sec. I, previous work on lateral waves focused on planar vortex sheets, and it turns out that the situation is different for cylindrical vortex sheets. As shown in Appendix A, the branch cuts of 1 are not present in the integrands. So apart from the instability wave, the path of steepest descent does not need to be indented. The lateral waves are described by the analytical solutions in a different way. In fact their presence can be traced back to the contribution of the fastest downstream modes of the jet. These modes correspond to the zeros of Lu located close to u 1 +. As can be seen in Fig. 2, these zeros can be close to the real axis. When this is the case, these modes only slowly decay in the axial direction and, if excited, will generate significant oscillations of the vortex sheet far away from the duct exhaust. It is possible to obtain an approximation for the location of these zeros in the complex u plane, + D 11 + mn M mn H m 0 + mn 1 + mn M 0 2 H m 0 + mn J mj mn J m j mn M 1C M 2 1 C 2 1 u 1, mn 12 where j mn denotes the zeros of J m z, so we have 1 + mn = j mn. Details are given in Appendix B. Although this expression provides only a qualitative estimate of the location of the zeros, it is interesting to note that the zeros move closer to the real axis as the frequency increases or as the relative density of the jet decreases. The presence of these zeros close to the real axis renders the standard far-field approximation 8 inaccurate. As explained in Sec. III B, Eq. 8 is valid as long as no pole is located in the vicinity of the steepest descent path. As the far-field angle is reduced from /2 to 0, the saddle point moves closer to the pole, and then for some critical angle c the pole is located right on the steepest descent path. For close to or smaller than c, it is necessary to use the uniformly valid approximation 11, which accounts for the presence of this pole. An example of results obtained with the modified asymptotic solution is given in Fig. 8, where the contributions of the first and second fastest modes of the jet are taken into account. The correction terms have two effects. First, the sharp peak observed in the standard far-field solution is significantly reduced. Second, the lateral waves present in the cone of silence are now included in the farfield approximation. This demonstrates that their presence is controlled by the amplitude of these modes of the jet. The time required to compute the correction terms for the farfield solution is negligible, but the accuracy of the asymptotic solution is significantly improved. SPL far field direction FIG. 8. Comparison of the standard asymptotic far field thin solid line, with the modified asymptotic solutions including the contribution from the fastest jet mode dashed line and the contributions from the fastest two jet modes dot-dashed line. The near-field solution is also plotted thick solid line. All solutions are for R=38. In order to provide a more practical account of the influence of lateral waves, Fig. 9 shows the directivity obtained when all the source is composed of all the cut-on modes of the duct with uncorrelated phases and an equal energy distribution. This source is often used as a first approximation for broadband fan noise. When compared with the exact nearfield solution, the far-field approximation appears to overpredict the amplitude of the peak by more than 1 db and its location by more than 6. Also the pressure amplitude inside the cone of silence is significantly underestimated. The use of correction terms to account for the fastest mode of the jet improves the results especially inside the cone of silence. C. Directivity and rate of decay Now that the lateral waves have been identified in the analytical solution, it is interesting to investigate their properties by inspecting Eqs. 9 and 10. Firstly, these equations can be simplified by assuming Q1. For Eq. 9 we can also scaled SPL far field direction FIG. 9. Sound pressure level at R=38 for broadband noise: Far-field approximation thick solid line; near-field solution thin solid line; corrected far-field solution dashed line J. Acoust. Soc. Am., Vol. 124, No. 5, November 2008 G. Gabard: Acoustic radiation through plug flow jets

9 use Cz z 2 /2asz this result can be obtained by using items and of Ref. 25. Therefore, the correction term f p behaves asymptotically as follows: f p R, = i m+1/2 1 M0 2 A p e irs 2s p 3 R 2 20 u p sin 1 M 0 2 sin 2 3/4, when R. It appears then that this correction term decays like R 2 since A p and s p are independent of R. This is similar to what Suzuki and Lele obtained for the three-dimensional planar shear layer. 18,19 If one notes that s 2 p = i i cosˆ ˆ p, where ˆ p=cos 1 p, then it follows that the magnitude of s p is minimum when ˆ =ˆ m=reˆ p. Due to the s 3 p factor, this correction term has a strong directivity around ˆ m and becomes weaker as the far-field direction moves away from ˆ m. The other correction term g p is zero for any angle greater than ˆ c since the path of steepest descent does not cross the pole for these directions. The critical angle ˆ c is the direction for which s p is real, and the pole u p lies on the path of steepest descent. The decay of g p is given by the factor e ˆ Rˆ 2 s p/ r in Eq. 10. The far-field direction ˆ m is again of interest since it also corresponds to s 2 p being purely imaginary. So for ˆ =ˆ m the correction term would decay like r 1/2. However, since ˆ m is greater than ˆ c, this algebraic decay of g p is not observed, and in fact g p exponentially decays for any direction below ˆ c. Note, however, that for relatively small R the actual rate of the exponential decay can be very small, and thus the amplitude of the lateral waves can remain significant everywhere inside the cone of silence see, for instance, the results in Fig. 5. However, as R increases, the rate of exponential decay increases and g p remains significant only for directions close to ˆ c. From a physical point of view, the lateral can be understood as conical waves i.e., plane waves in the r-z plane generated at the vortex sheet. The angle c at which these plane waves are radiated is similar to the Mach angle of the disturbances inside the jet. Although these waves are present in the cone of silence in the near field, they will eventually leave the cone of silence when propagating to the far field and will generate a narrow beam of sound at the critical angle c this is the sharp peak observed in Fig. 5. Finally, it should be noted that both f p and g p will eventually become negligible compared to the standard far-field approximation for any direction. Thus Eq. 7 represents the leading-order term in the far-field approximation. D. Amplitude The condition for the presence of lateral waves can also be inferred from Eqs. 9 and 10. The correction terms are proportional to the amplitude of the mode of the jet represented by the residue A p of Pu. Therefore, the fastest modes of the jet must be strongly excited by the source of sound for the lateral waves to be efficiently generated. In the case of the Munt problem, Eq. 3 shows that A p is proportional to 1/u p + mn. This confirms the observation made in Sec. IV A that lateral waves are efficiently generated by well cut-on modes since their axial wavenumbers are close to those of the fastest jet modes. FIG. 10. Magnitude of J m 1 u p r s as a function of the point source position for different modes of the jet. Solid line: m=0; dashed line: m=3; dot-dashed line: m=6; dotted line: m=9. Top: fastest modes; bottom: second fastest modes. For the plug flow jet, the location of the point source influences the magnitude of A p through the factor J m 1 u p r s. Its magnitude is plotted in Fig. 10. For a point source close to the axis, only the axisymmetric mode m=0 is excited, while higher-order modes are more efficiently generated when the source moves closer to the vortex sheet. E. Discussion It is generally assumed that the far-field asymptotic solution can be applied at several wavelengths from the duct exhaust. However the results in Fig. 5 show that inside the cone of silence the exact solution is still very different from the asymptotic solution at large distances from the jet exhaust. This is because a large portion of the vortex sheet acts as a source of sound for the ambient medium. Therefore, to define the far-field region where this approximation is valid, it is not appropriate to consider only the distance from the duct exhaust. It is preferable instead to think of the vortex sheet as a distributed source of sound that radiates in the ambient medium. The presence of lateral waves can also have some consequences for computational aeroacoustics. Firstly, the sharp J. Acoust. Soc. Am., Vol. 124, No. 5, November 2008 G. Gabard: Acoustic radiation through plug flow jets 2763

10 SPL far field direction FIG. 11. Standard far-field approximation for a point source in a jet dashed line. Modified approximation including the contribution from the channeled mode solid line. directivity peaks observed in Figs. 5 and 6 are the results of a precise interference between sound waves generated over a large portion of the vortex sheet. It is likely that even a small amount of error present in a numerical simulation will dissolve this interference and result in a smoother directivity pattern. Secondly, far-field solutions are generally obtained from numerical simulations by means of Kirchhoff or Ffowcs-Williams Hawkings surfaces. For problems similar to the Munt problem, an open cylindrical surface is generally used. The difficulty introduced by the lateral waves is that this cylinder has to extend far away from the duct exhaust to encompass most of the sound radiated from the jet. This implies that very large computational domains should be used. V. CHANNELED JET MODES Another feature that deserves a closer inspection is the presence of sharp peaks in the forward arc of the far-field approximation. An example is shown in Fig. 11 with the problem of a point source in a plug flow jet. The jet Mach number is 0.8 with unit density and sound speed ratios. The ambient medium is at rest and the Helmholtz number is 20. The source position is r s =0.9 with an azimuthal order m =19. A peak significantly larger than the main lobe is observed at 136 in the analytical far-field solution. Note that similar peaks can also be observed in the Munt solution. This feature originates from the presence of a zero of Lu close to the real axis between u 0 and u 0 +. This zero represents a mode of the jet propagating in the positive axial direction with a slowly decaying amplitude since its axial wavenumber is almost real the statement in Ref. 4 that such zeros lie exactly on the real axis was incorrect. For this mode, there is a very slow transfer of acoustic energy to the ambient flow, and the mode is channeled inside the jet. It is useful to have a closer look at the condition under which a mode of the jet is able to radiate sound in the ambient flow. To that end, consider relatively high frequencies and azimuthal orders so that the curvature of the jet is negligible on the acoustic length scale. The vortex sheet can then be thought of as a planar surface with some prescribed oscillations radiating in the semi-infinite ambient flow this is the so-called wavy wall problem. The dispersion relation in the ambient flow can be written as k z M 0 2 = k z 2 + k r 2 + k 2, where k r is the wavenumber in the direction normal to the wavy wall and k z and k are wavenumbers in the directions parallel to the wall. For the jet mode we have k z =u and k =m/r=m since the jet radius is 1. For the mode to radiate sound k r should be real, which yields the condition û 0 u û 0 + on the reduced axial wavenumber u with û 0 = 1 1 M0 2 m 2 / 2 M 0 1 M When this condition is satisfied, the mode radiates sound in the ambient flow, which is to say that the total phase speed is supersonic. Otherwise the mode induces only an evanescent disturbance in the vicinity of the vortex sheet but does not transfer energy to the ambient medium. In that case the jet effectively acts as a waveguide and the disturbances are channeled inside the jet. With the far-field approximation, the saddle point moves from u 0 + to u 0 when varies from 0 to. This range of values of u corresponds to disturbances with supersonic axial phase speed. The peak observed in Fig. 11 corresponds to a mode of the jet with a subsonic total phase speed i.e., acoustic waves are channeled inside the jet and does not radiate sound but with a supersonic axial phase speed so it lies in the region spanned by the saddle point. Channeled modes can appear in the standard far-field solution when their axial wavenumbers are in the range u 0 uû 0. From Eq. 13 it appears that this is more likely to happen for large azimuthal orders m. The physical principles associated with the channeled waves are the following. The mode of the jet involved propagates downstream but only very weakly radiates sound in the ambient medium due to its subsonic total phase speed. The key is that the axial wavenumber of this mode corresponds to disturbances propagating in the upstream direction in the ambient medium. So although the waves transmitted in the ambient medium are originally located downstream of the source, they will propagate in the upstream direction and they will eventually evolve into a narrow beam of sound observed in the forward arc of the standard far-field approximation. It should be noted that this situation is quite different from the lateral waves described in the previous section. Both lateral waves and channeled waves are due to disturbances propagating on long distances inside the jet with a limited but nonzero transmission of sound through the vortex sheet. However, the rate of decay of channeled waves was found to be at least an order of magnitude smaller than that of lateral waves. As explained in Sec. III B, the standard far-field approximation 8 is inaccurate when poles are in the vicinity of the path of steepest descent and when the contributions of poles crossed when deforming the contour are important. This is the case with channeled modes, and one has to in J. Acoust. Soc. Am., Vol. 124, No. 5, November 2008 G. Gabard: Acoustic radiation through plug flow jets

11 clude correction terms 9 and 10. As shown in Fig. 11, these correction terms remove the peak observed in the standard approximation. In addition a strong pressure field is added inside the cone of silence owing to the fact that in the near field these waves are generated downstream of the source. The behavior of the correction terms has already been discussed above, and the conclusions from Secs. IV B IV D also apply in the case of channeled waves and are not repeated here. VI. CONCLUDING REMARKS The present analysis shows that the standard definition of the far field does not necessarily apply to problems involving sound propagation through a plug flow jet. The distance at which the standard far-field asymptotic solution becomes valid depends not only on the frequency and on the ambient flow properties but also on the properties of the jet and on the nature of the source of sound. For instance, lateral waves are generated by acoustic disturbances propagating over many wavelengths inside the jet with only limited decay in amplitude. As a result, some care should be exercised when far-field solutions are used for the plug flow jet and the Munt problem. If these analytical solutions are used as simple engineering models to assess trends between different configurations, the far-field approximation should provide reliable information except maybe when the source of sound excite only well cut-on modes. If, however, these models are used to benchmark computational methods, it is recommended to use the exact near-field solutions as a comparison with the far-field asymptotic solutions might be inaccurate. For an observer in the ambient flow, it can be misleading to understand the radiated field only in terms of the source of sound inside the jet either a duct modes or a point source. Rather it is more appropriate to consider the corresponding distributed source at the vortex sheet. The extent of this distributed source is directly related to the amount of sound reflection inside the jet and to the nature of the source. In practice the mixing layer of a jet has a finite thickness and grows axially. The extent to which the features observed with the plug flow jets translate to more realistic flow profiles remains to be investigated. It will be worth investigating the properties of lateral waves and of the far-field approximation in the context of round jets with finite mixing layers. ACKNOWLEDGMENTS This work was supported by the European Community through the TURNEX project Technical officer: Daniel Chiron. The author is grateful to Dr. Brian Tester, Dr. Sjoerd Rienstra, and Dr. Chris Powles for several helpful discussions during the course of this work. APPENDIX A: BRANCH CUTS OF THE INTEGRANDS For the plug flow jet the integrand is given by Eq. 2. One can write Lu J m 1 r s = J m 1 J m 1 r s D 1 1 um H m 0 1 um 0 2 H m 0 1J m 1. J m 1 The Bessel function and its derivative can be written as infinite products see item of Ref. 25, J m z = zm 2 m 1 z2 2 for m 0, m! n=1 z m 1 j mn J m z = 2 2 m 1 z2 for m 0, m 1! n=1 j mn where j mn and j mn are the zeros of J m z and J m z, respectively. It follows from these expressions that for any m0, both J m 1 /J m 1 r s and 1 J m 1 /J m 1 are regular functions of 2 1 alone and hence do not include the branch cuts of 1. It also follows that these two functions are nonsingular at u 1 where 1 =0. Therefore, the branch cuts of 1 are not present in the integrand for the plug flow jet, which is also analytic at u 1. For the Munt problem, the integrand is given by Eq. 3. The Wiener Hopf kernel can be written as Ku = 0 u 1 ukˆ u with Kˆ u = D 1 1 um 1 2 1J m 1 J m 1 1 um 0 2 H m 0 0 H m 0. From the analysis above, it appears that the branch cuts of 1 are not present in Kˆ. The split functions can be written as K u= 0 u 1 ukˆ u, so the integrand now reads Pu = i mn1 0 + mn 1 + mn K + mn 1 um 0 2. u + mn 0 ukˆ +uh m 0 It follows that the branch cuts of 1 do not appear in the integrand for the Munt problem. APPENDIX B: APPROXIMATION OF THE ZEROS We seek an approximation for the zeros of Lu that correspond to the fastest modes of the jet column. As shown in Fig. 2, it appears that these zeros are close to the axial wavenumbers + mn of the duct modes given by J 1 =0. It is therefore convenient to rewrite Lu=0 as fuj 1 =1, where fu = 1 um 0 2 H m 0 1 D 1 1 um H m 0 J m 1. An approximation of a zero of Lu can be obtained by substituting u by + mn +u and by using a first-order Taylor expansion with respect to u. This yields u =f + mn 1 + mn J m j mn 1. Equation 12 is obtained by using 1 1 = M 1 C M 2 1 C 2 1 u. J. Acoust. Soc. Am., Vol. 124, No. 5, November 2008 G. Gabard: Acoustic radiation through plug flow jets 2765