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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2011, Article ID 573209, 9 pages doi:10.1155/2011/573209 Research Article A Theoretical Basis for the Scaling Law of Broadband Shock Noise Intensity in Supersonic Jets Max Kandula ASRC Aerospace, NASA Kennedy Space Center, Merritt Island, FL 32899, USA Correspondence should be addressed to Max Kandula, email@example.com Received 8 September 2010; Accepted 27 January 2011 Academic Editor: Lars Hakansson Copyright © 2011 Max Kandula. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A theoretical basis for the scaling of broadband shock noise intensity in supersonic jets was formulated considering linear shock- shear wave interaction. Modeling of broadband shock noise with the aid of shock-turbulence interaction with special reference to linear theories is brieﬂy reviewed. A hypothesis has been postulated that the peak angle of incidence (closer to the critical angle) for the shear wave primarily governs the generation of sound in the interaction process with the noise generation contribution from oﬀ-peak incident angles being relatively unimportant. The proposed hypothesis satisfactorily explains the well-known scaling law for the broadband shock-associated noise in supersonic jets. 1. Introduction In imperfectly expanded supersonic jets, the rapid varia- tion in the pressure across the nozzle exit is accompanied by a Noise from subsonic jets is mainly due to turbulent mixing, system of steady compression (oblique shock) and expansion according to the theoretical model of Sir James Lighthill waves (Figure 2). The structure of these shock cells was inves- [1, 2]. The turbulent mixing noise is essentially broadband. tigated by Emden , Prandtl , Rayleigh , Pack , In perfectly expanded supersonic jets (nozzle exit plane and others. In general, these shock-expansion units interact pressure equals the ambient pressure), the large-scale mixing with instability waves, vortices, turbulence, and other stream noise manifests itself primarily as Mach wave radiation disturbances in the viscous shear layer that surrounds the [3, 4] caused by the supersonic convection of turbulent inviscid region. The interaction of turbulence with shock eddies with respect to the ambient ﬂuid. In imperfectly waves leads to the generation of the broadband shock expanded supersonic jets (nozzle exit pressure diﬀerent from noise, which is of relatively high intensity and may form a the ambient pressure) typical of jet and rocket exhausts at oﬀ- signiﬁcant component of the overall jet noise, depending on design conditions, additional noise is generated in the form the ﬂow conditions. The peak (characteristic) frequency of of broadband shock-associated noise (BBSN) emanating the broadband shock noise is intimately related to (varies from shock-turbulence interaction  and screech tones  inversely as) the shock-cell spacing which is roughly uniform with the tonal (screech) amplitude shown to be occasioned over several shock cells . A fundamental understanding by shock-acoustic wave interaction . of the mechanism by which turbulence interacts with a shock wave is thus requisite in the analysis of the complex Figure 1 displays a typical narrowband farﬁeld shock noise spectrum, indicating various noise components. Here, phenomena of shock noise generation. the quantity St denotes the Strouhal number (fd /u ), f the j Lighthill and Ribner [14, 15] originally suggested frequency, d thenozzleexit diameter, u the jet exit velocity, that the scattering of eddies by shocks could be a strong j j p the nozzle exit pressure, p the ambient pressure, φ the source of supersonic jet noise. The importance of source e 0 angle from the downstream jet axis, M the nozzle design coherence, however, has not been recognized, so that only Mach number, and M the fully expanded jet Mach number. incoherent and randomly scattered sound waves had been j 2 Advances in Acoustics and Vibration on linear theory for shock-vorticity interaction. It is demon- M = 2, M = 1.5 strated here that the scattering of turbulence by the leading p /p = 0.47, φ = 150 e 0 shock wave is related to the measured shock noise intensity scaling. Flight eﬀects are excluded from consideration here. Also screech eﬀects are not relevant to this investigation. This Screech work is primarily based on . 2. Measurements and Characteristics of Broadband shock noise Broadband Shock Noise Turbulent mixing noise Harper-Bourne and Fisher  were the ﬁrst to identify signiﬁcant features of shock-noise in considerable detail based on their static jet measurements from conical nozzles. The intensity of BBSN is shown to be primarily a function of the nozzle (operating) pressure ratio NPR = p /p ,where p t 0 t is the stagnation (reservoir) pressure. For a given radiation direction, the measured overall sound intensity I has been 0.03 0.1 1 3 observed to scale as St = fd /u j j I ∝ β,(1a) Figure 1: A typical narrowband farﬁeld shock noise spectrum (adapted from Seiner ). where 1/2 β = M − 1,(1b) r j with the isentropic relation between p /p and M expressed t 0 j by Sock cell Shear layer Nozzle 0 γ/(γ−1) p γ − 1 = 1+ M . (1c) p 2 In the preceding relations, the quantity M represents the fully expanded jet Mach number, γ is the isentropic exponent (ratio of speciﬁc heats), and the parameter β characterizes the pressure jump across a normal shock at approach Mach Expansion Compression number M . Figure 2: Shock cell structure in an underexpanded supersonic jet. Figure 3 presents the data for the overall sound power level (OASPL) at 90 deg. to the jet axis, normalized to r/d = 1, are shown for two diﬀerent nozzle diameters (d = 25 mm, and 35 mm) and at a stagnation temperature T = 290 K, predicted without the peak frequency and directivity rela- with r denoting the measurement location. Equation (1a) tionships. It was Harper-Bourne and Fisher who ﬁrst is plotted as a dashed line, and the estimated mixing noise identiﬁed the detailed characteristics of BBSN with the aid based on extrapolation of low-speed data at T = 290 K of measurements from conical (convergent) nozzles and is plotted as a dotted line. Harper-Bourne and Fisher  indicated the importance of source coherence. The charac- found that the parameter β correlated BBSN quite well up teristics of shock noise were also reviewed and discussed in certain values of β (or NPR), say 0.5 <β < 1.2. At large . Howe and Ffowcs Williams  also considered that NPR or β, the data begins to deviate from this law because the primary source of broadband shock-associated noise is a of the presence of a Mach disc, which signiﬁcantly alters consequence of the interaction between large-scale structures the shock-cell structure. As the Mach disc forms, the large (turbulence) and the shock structure. central portion of subsonic ﬂow formed downstream of the Computing shock noise intensity in supersonic jets from Mach disc considerably reduces the noise generation. The ﬁrst principles (on the basis of shock-turbulence interaction) data also reveals that at a high β the turbulent mixing noise is very diﬃcult. The nature of the relevant noise sources is not level is much lower than the underexpanded noise levels. As β well understood . This situation is exempliﬁed by the fact decreases, the mixing noise contribution relative to the total that the theories of both Lighthill and Ribner [19, 20] noise becomes increasingly signiﬁcant. produce shock noise intensity scaling considerably diﬀerent Experiments by Tanna  and of Seiner and Norum from that indicated by the measurements.  provided further insight into the characteristics of shock It is the purpose of this work to investigate the scaling of noise. These data include measurements from convergent- broadband shock noise intensity from considerations based divergent (C-D) nozzles and covered a broad range of jet (dB/Hz) Advances in Acoustics and Vibration 3 Refracted shear-entropy wave Shock 0.8 1 u mixing noise extrapolation (290 K) Incident shear wave Sound θ wave 0.4 0.6 0.8 1 1.2 1.4 1.6 Figure 3: Intensity of broadband noise at 90 deg. to jet axis (from Fisher et al. ). Figure 4: Interaction of a shear wave with shock wave (Ribner conditions (NPR and jet temperature ratio T /T where T ). t 0 t and T are, respectively, the stagnation temperature and the ambient temperature). Both Tanna’s data , covering β ≤ 1(M ≤ 1.41, or p /p ≤ 3.5), and the data of Seiner and j t 0 moving ﬂuid, a general ﬂuctuation can be decomposed into Norum (covering the designMach number M = 1.5 acoustic, entropy, and vorticity waves (perturbations). The and 2, and M = 1to2.37 or β = 0 to 2.15) suggest trends acoustic waves (isentropic pressure ﬂuctuations) propagate similar to those indicated by the data of Harper-Bourne and with the acoustic speed c relative to the moving ﬂuid, while Fisher  to the extent that the overall intensity of shock- the vorticity and the entropy waves are convected with associated noise is principally a function of jet pressure ratio, the ﬂuid. Linear analyses of a single wave (shear/vorticity, scales as I ∝ β , and is independent of jet temperature acoustic, or entropy) interaction with a shock wave were ratio (eﬄux temperature) and emission angle. The data by carried out by Blokhintzev , Burgers , Ribner [14, Krothapalli et al.  for broadband shock noise for M in 15, 19, 20], Moore , and McKenzie and Westphal . the range of 1.24 to 1.66 suggest that the shock noise intensity With regard to broadband shock noise, we are primarily follows the β dependence for both the stationary ambient concerned here with the generation of acoustic waves by the and in forward ﬂight. interaction of a shock wave with an incident vorticity wave Directivity and spectral characteristics of BBSN were in- (Figure 4) in our endeavor to investigate shock-turbulence vestigated experimentally by Tanna , Norum and Seiner interaction. , Pao and Seiner , Krothapalli et al. , and Jothi According to the linear theory, for suﬃciently high and Srinivasan . Detailed measurements by Norum and angles of incidencefor thewaveahead of theshock,the Seiner  suggest that the shock noise is fairly directional incident wave vector k has a nonzero imaginary part. Under at lower values of β and approaches omnidirectionality. such circumstances, the refracted (or generated) acoustic Test data by Tanna  reveal the peak frequency (which wave is not an inﬁnite plane wave; instead, it exhibits an represents an important characteristic) with the angle of exponential decay as it propagates downstream behind the observation. Pao and Seiner  indicate that the power shock. The incidence angle that separates the plane wave spectral density (dB/Hz) increases as ω below the peak acoustic response from the decaying ones is termed the −2 frequency and decays as ω beyond the peak frequency. critical angle. The critical angle is close to 90 deg. for incident Measurements by Jothi and Srinivasan  suggest that at acoustic waves, and roughly 60 deg. for incident vorticity higher pressure ratio exceeding about two, noncircular jets and entropy waves . Lineartheory predictsthat most are quieter relative to circular jets by as much as 10 dB. transmission and generation coeﬃcients are peaked near the critical angle. From a theoretical point of view, the actual transmission/generation coeﬃcients are independent of the 3. Studies on Shock-Turbulence Interaction incident wavelength in the linear limit . 3.1. Linear Theories. Broadly speaking, the decomposition of A turbulent velocity ﬁeld can be represented as a super- a general ﬂuctuation into acoustic, vorticity, and entropy position or spectrum of elementary waves distributed among waves is well known (Kovasznay ). In general, any plane all orientations and wavelengths in accordance with Fourier’s wave (acoustic, vorticity/shear, or entropy) interacting with integral theorem. The waves are transverse for weak turbu- a shock undergoes transformation and at the same time gen- lence because of the constraint of incompressibility (even erates the other two waves (Zang et al. ). In a uniformly though convected at high speed). Thus, a single wave can be Intensity (dB), corrected to r/d = 1 j 4 Advances in Acoustics and Vibration 115 3 −3 −6 −9 0 10 20 30 40 50 60 70 80 90 0.1 1 10 Incident angle, θ (deg) Figure 6: Dependence of acoustic response to vorticity waves Theory  incident on a Mach 8 shock (solid lines: linear theory, circles from Theory  nonlinear Euler simulations; from Zang et al. ). Data  Figure 5: Intensity of broadband noise according to the theories of Ribner and Lighthill . where P(θ) is the transfer function for sound wave genera- tion, and the special symbol [uu] stands for the longitudinal spectral density of u in wave-number space k,where k is a three-dimensional vector . The wave number is deﬁned interpreted physically as a plane sinusoidal wave of shearing motion (Batchelor ). According to linear interaction by analysis (LIA), the vorticity waves incident at angles beyond 2π ω acriticalangle k =|k|= =,(3b) λ c θ = θ (M ) (2) c c 1 where λ is the acoustic wavelength and ω the circular fre- generate acoustic waves which decay as they propagate quency. Considering that the initial turbulence is isotropic, downstream. In (2), M refers to Mach number upstream of Ribner [14, 15] tabulated the transfer function P(θ)and the critical angle. Calculations of the linear theory were the shock. Lighthill and Ribner [14, 15, 19] conducted the- performed for an upstream Mach number range of 1 <M < oretical analysis on acoustic noise generation by shock 10. For a one percent turbulence, the postshock noise level is predicted to exceed 140 dB for all preshock Mach numbers wave/turbulence interaction. In both Ribner’s and Lighthill’s theories, the turbulence is treated in eﬀect as a frozen spatial above 1.05. pattern with neglect of temporal ﬂuctuations. 3.1.2. Lighthills’s Theory. Lighthill consideredthe gen- 3.1.1. Ribner’s Analysis. Ribner  studied in detail the eration of sound due to the interaction of turbulence with interaction between a vorticity wave and a shock wave. very weak shock waves (acoustic-like waves), by aid of his Ribner [14, 19] extended this analysis to consider a spec- general theory of sound generated aerodynamically [1, 2]. trum of incident vorticity waves (in three dimensions) The weak shock is represented by an acoustic step function. and computed, for an isotropic incident spectrum, detailed In Lighthill’s theory, the assumptions are more restrictive statistics of the downstream ﬂowﬁeld with emphasis on than in Ribner’s analysis in the sense that both the shock the generated noise. The basic building blocks of Ribner’s and the turbulence are weak. As a result, the rippling motion linear theory are oblique plane sinusoidal waves of vorticity of the shock as well as the diﬀerences in the turbulence (shear waves), see Figure 6. These represent single spectral intensity across the shock are suppressed. The ratio of freely (monochromatic) waves composed of (in 3D) an instanta- scattered acoustic energy to the kinetic energy of turbulence neous snapshot of arbitrary ﬂow. The waves are considered traversed by the shock wave is expressed relative to a frame to interact independently with the shock, and then the moving with the ﬂuid, whereas Ribner’s analysis deals with a waves are superposed to represent turbulence upstream and frame attached to the shock. For a direct comparison, Ribner downstream of the shock. The detailed statistical formalism  converted the results of Lighthill tothe shock-ﬁxed was worked out in Ribner  and partly summarized by reference frame. Ribner . Figure 5 shows a comparison of the scattered sound The mean spectral sound pressure is expressed by  intensity (in SPL) between Ribner’s result and that of Lighthill , as presented by Ribner . Signiﬁcant 3 discrepancy is noted between the two results. A critical ( ) [ ] p = |P θ | uu d k,(3a) 0 discussion of this comparison is provided by Ribner . OASPL (dB) δp /(p δu /c ) 2 1 1 1 Advances in Acoustics and Vibration 5 Note that the results of Ribner and Lighthill shown here are seem to be impractical for conditions involving strong shock not to scale. wavesand high Reynoldsnumberturbulence onaccountof resolution requirements of shock waves and turbulence. 3.2. Nonlinear Euler Simulations. Since the shock weakens as the Mach number tends to unity, the shock front will 3.5. Experimental Data. With regard to experimental data, undergo greater distortions from an incident wave of ﬁxed it is found that in general compression enhances turbulence amplitude. Thus, nonlinear eﬀects ought to be increasingly and expansion suppresses it. Measurements by Barre et al. important for lower Mach numbers (Zang et al. ). Zang at M = 3 suggest that the shock wave increases the et al.  validated the linear analysis of McKenzie and West- longitudinal ﬂuctuating velocity in agreement with Ribner’s phal  by comparisons with their numerical solution of theory . As indicated by Ribner , the measured nonlinear 2D Euler equations. Although restricted in terms ampliﬁcation ratio of mean square longitudinal component 2 2 of the incident angle of the disturbance, it was shown that of turbulence velocity (u /u ) is close to the theoretical value 2 1 the linear analysis was valid over a surprisingly large range of about 1.5 as predicted by Ribner’s theory [14, 15]at M = of shock strengths and disturbance amplitudes (Figure 6). 3. Density ﬂuctuations in high-speed jets were investigated In this plot, δp represents the amplitude of the acoustic by Panda and Seasholtz . pressure generated downstream of the shock, δu is the There are also important studies dealing with the funda- amplitude of the incident vorticity wave, p is the mean mental interaction between vorticity and an isolated shock. pressure upstream of the shock, and c is the sound speed The interaction of a shock with a longitudinal vortex was upstream of the shock. The comparisons suggest that the treated by Erlebacher et al. on the basisof analytical linear theory is fairly accurate for a wide range of incident (linear and nonlinear) theories and numerical simulations. angles up to the critical angle. Although comparisons were Grasso and Pirozzoli  solved 2D Euler equations with the made for both the incident and the vorticity waves, only the aid of a higher-order ﬁnite volume-weighted ENO scheme comparisons for the incident vorticity waves are indicated in in their study of sound generation in the interaction of a Figure 6. shock wave with a cylindrical vortex. In this connection, they also derived a Green’s function for the acoustic analogy for a general vortex structure to analytically characterize 3.3. Instability Wave Theories. Tam  formulated a the shock-vortex interaction. Direct noise computation in stochastic model theory of the BBSN of axisymmetric subsonic and supersonic jets was reviewed by Bailly et al. supersonic jets by considering the dynamics of weakly . Avital et al.  investigated Mach wave radiation by nonlinear interaction between the downstream propagating mixing layers. linear instability waves in the mixing layer and shock- cell structures. On account of the solution complexity, a semiempirical (less general) shock-noise model was arrived 4. Proposed Model at, valid for slightly imperfectly expanded supersonic jets.An increase in the spectral peak associated with the BBSN is The discrepancy between the theories of Lighthill and attributed to the convective ampliﬁcation of the sources. of Ribner  in comparison with the experimental data for The theory was extended  to moderately imperfectly the scaling of shock noise intensity (as evident from Figure 5) expanded jets with the aid of empirical modiﬁcations to requires further investigation. This discrepancy is attributed the amplitude of the waveguide modes of the shock cell. to the fact that in their theories the turbulence is treated The speciﬁc role of instability wave-shock cell interaction is eﬀectively as a frozen spatial pattern without regard to the discussed in the reviews [36–38] on supersonic jet noise. temporal ﬂuctuations. There is thus a deﬁciency in applying linear theory to real turbulence, which consists of transient 3.4. DNS Simulations. The simplest circumstance in which phenomena and not steady plane waves . Also, three- turbulence interacts with a shock wave is the case of isotropic dimensional simulation is needed to accommodate vortex turbulence interacting with a normal shock (transverse stretching . vorticity ampliﬁcation). Lee et al. [39, 40]and Mahesh et The irregularity and disorderliness characterizing turbu- al.  performed DNS simulation of the interaction of 3D lence involve the impermanence of the various frequencies isotropic turbulence up to M = 3. Detailed comparisons and of the various periodicities and scale (Hinze ). of DNS results to Ribner’s linear analysis [15, 20]were Strictly speaking, the instantaneous physical interaction made. DNS calculations [39, 40] and numerical simulations process (shock/vorticity) cannot be represented by time- by Rotman  show that the vorticity ampliﬁcation averaging. In view of these circumstances, it is plausible predictions are in good agreement with the linear theory. that the peak angle of incidence is representative of the Satisfactory agreement between the DNS simulations  shock-shear wave interaction insofar as the scaling of and the linear theory is noticed with regard to ampliﬁcation the BBSN is concerned. Accordingly, it is postulated here (of turbulent kinetic energy) and anisotropy downstream of that the shock-vorticity interaction at the peak incidence the shock (representing the ratio of longitudinal to transverse governs the generation of sound. Also, it is assumed that velocity ﬂuctuation). the interaction of turbulence with the leading shock cell Although DNS solutions provide the most accurate forms the maximum contribution to the intensity of sound, representation of the shock/turbulence interaction, they and that the sound contribution due to the interactions at 6 Advances in Acoustics and Vibration the subsequent shock cells is of secondary nature (subsidiary importance). With the above postulate, the linear acoustic response (acoustic pressure rise) for shock-vortex interaction (vor- −2 ticity waves incident on a shock) is computed for various upstream Mach numbers. In this context, as pointed out by Zhang et al.  that among the linear analyses, the work by −4 McKenzie and Westphal is moreaccessibleand tractable than the earlier pioneering studies of Ribner [14, 15]and others, while yielding equivalent results. In consequence, −6 our calculations will be based on the work of . For normal shocks and an ideal gas, the relation for the acoustic −8 response in dimensionless form becomes relatively simple 0 10 20 30 40 50 60 70 and is expressed by (see [32, Equation (42)]) θ (deg) M = 1.2 M = 3 1 1 2 2 δp −4γM M − 1 sin θ 1 − (u /u )tan θ 2 1 1 2 1 i 1 M = 1.4 M = 4 1 1 = , M = 1.6 M = 5 p /c δu γ +1 D 1 1 1 1 1 M = 1.8 M = 6 1 1 (4a) M = 2 M = 7 1 1 M = 2.5 M = 8 1 1 where Figure 7: Pressure rise due to shock/vorticity ampliﬁcation accord- ing to linear theory. 1/2 u tan θ 2 i 2 2 2 2 D = 1+ M + tan θ 1 − M +2M M 1 − , i 2 1 1 1 u tan θ 1 c Figure 7, is plotted as function of β in Figure 8.The sound (4b) pressure level (OASPL) is given by 1/2 ⎛ ⎞ γ − 1 M +2 2 1 2 = ,(4c) δp ⎝ ⎠ M 2γM − γ − 1 1 OASPL = 20 log dB, (3) ref ρ γ − 1 M +2 2 1 1 = = ,(4d) where δp is obtained from (4), and p is the reference 2 2 ref u ρ γ +1 M 1 2 −5 2 sound pressure (2 × 10 N/m ). It is revealed that the scaling of intensity very nearly varies as β for a wide range of tan θ = ,(4e) 2 β between 0.2 and 2.0. In this range, the present theory yields R 1 − M 4.2 I ∝ β . (4) 2 2 c 1+ γ − 1 /2 M 2 1 R = = . (4f ) 2 2 c 1+ γ − 1 /2 M Beyond this range, there is seen a change in slope in the 1 2 intensity variation. A direct comparison of the scaling based on the proposed In the preceding equations, ρ denotes the density, and the model and the experimental data of Tanna is presented subscripts 1 and 2 denote the upstream and downstream of in Figure 9. The OASPL data are obtained from an under- thenormalshock,respectively. expanded nozzle with T /T = 1 (cold jet) at φ = 135 deg. t 0 On the basis of (4a)–(4f), the acoustic response has been For scaling purposes, the model results presented in Figure 9 computed as a function of the incident angle for several are adjusted such that at β = 0.7, the prediction matches upstream Mach numbers from 1.2 to 8 (Figure 7). The results the OASPL data of 117 dB (taken for reference purposes). point out that the peak angle of incidence and the associated The predictions from the proposed model substantially agree acoustic response varies with the Mach number. The acoustic with the data in the range of 0.3 <β< 1.0. Recall that for low pressure increases with an increase in Mach number. Notice values of β less than about 0.3, the turbulent mixing noise that the computed acoustic pressure at M = 8isidentical becomes signiﬁcant. It is known that beyond about β = 1, a to that shown in Figure 6, as computed by Zang et al. , Mach disc is formed, which alters the shock-cell structure. thus verifying the present calculations. It should be pointed The large central portion of subsonic ﬂow that develops out that the results for various upstream Mach numbers as downstream of the Mach disc considerably diminishes the shown in Figure 7 are originally obtained by the author. noise generation. The satisfactory explanation of the β scaling law by the proposed theory suggests that the hypothesis of peak in- 5. Results cidence angle for the generation of sound by shock-vorticity Based on the foregoing premise, the intensity of BBSN taken interaction is plausible. This forms an important contribu- at the peak incidence angle, as obtained from the results of tion of the present work. δp /(p δu /c ) 2 1 1 1 Advances in Acoustics and Vibration 7 an incident acoustic wave and a shock wave, and obtained a remarkable agreement with data for the screech amplitude for fully expanded Mach number M up to 2. The predicted directivity pattern is also satisfactory when compared with measurements. Since the screech amplitudes are considerably larger as relative to broadband shock noise levels, it is −10 hardly surprising that the proposed model based on linear −20 formulation is able to describe the intensity scaling law for broadband shock noise. As indicated in Zang et al. , the −30 linear theory is found to be valid for extraordinarily large amplitudes, suggesting that the region of validity of linear −40 theory is indeed much broader than one would generally −50 expect. Quoting Zang et al. , in some of the examples 0.1 1 10 of their numerical Euler simulations, the postshock velocity ﬂuctuationswere ofnearly the same orderasthe mean stream velocity! Present theory Referring to the notable diﬀerence, for strong shocks, Figure 8: Intensity of broadband noise according to the theory of between the linear approach of Ribner and the weakly maximum incidence angle. nonlinear approach of Lighthill , it may be ascribed to diﬀerences in their assumptions and simpliﬁcations in treating the statistics of the interaction process. 7. Conclusion A physical basis is proposed for the scaling of the broadband shock-associated noise in supersonic jets considering linear interaction between the shock wave and the vorticity wave 110 considering the peak incidence angle for the turbulence. The hypothesis that the generation of sound at peak incidence angle is important is shown to satisfactorily describe the experimental scaling law for the broadband shock-associated noise intensity in imperfectly expanded supersonic jets. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 Nomenclature Present theory c: Speed of sound Data  d : Nozzle diameter f:Frequency Figure 9: Comparison of the present theory and the data of Tanna  for the intensity of the broadband shock noise. I : Overall sound intensity, δp /(ρ c ) 2 0 k: Incident wave vector k:Wavenumber M:Machnumber The determination of the directionality eﬀects and M :Design Mach number spectral distribution of the BBSN are outside the scope of M : Fully expanded jet Mach number the present investigation, which is mainly concerned with p: Pressure the scaling law for broadband shock noise intensity. The fact r: Distance to measurement location that only a single shock-cell/vortex interaction is considered R: c /c 2 1 here indicates that the shock noise intensity obtained by St: Strouhal number, fd /u j j the present formulation is essentially omnidirectional. It is T:Temperature believed that the present investigation would be helpful in u: Velocity our understanding of supersonic jet noise [51, 52]and its u :Jetexitvelocity suppression by active control such as water injection [53–55]. γ:Isentropicexponent δp : Amplitude of the acoustic pressure rise δu : Amplitude of the incident vorticity 6. Discussion wave With regard to the validity of the linear theory, Kandula φ: Angle from downstream jet axis  recently applied the linear theory to the production of λ: Acoustic wavelength screech noise, regarded as a consequence of interaction of ρ:Density OASPL (dB) OASPL (dB) 8 Advances in Acoustics and Vibration θ :Critical angle  M. J. Fisher, P. A. Lush, and M. Harper Bourne, “Jet noise,” ω: Circular frequency, 2πf . Journal of Sound and Vibration, vol. 28, no. 3, pp. 563–585,  M. S. Howe and J. E. Ffowcs Williams, “On the noise generated Subscripts by an imperfectly expanded supersonic jet,” Philosophical Transactions of the Royal Society of London, vol. 289, no. 1358, e: Nozzle exit pp. 71–314, 1978. t:Stagnation (reservoir)  A. Krothapalli, P. T. Soderman, C. S. Allen, J. A. Hayes, and 0: Ambient S. M. Jaeger, “Flight eﬀects on the far-ﬁeld noise of a heated 1: Upstream of shock supersonic jet,” AIAA Journal, vol. 35, no. 6, pp. 952–957, 2: Downstream of shock.  H. S. 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