The Effect of Heat on Turbulent Mixing Noise in Supersonic Jets.

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1 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition 4-7 January 2, Orlando, Florida AIAA 2-29 The Effect of Heat on Turbulent Mixing Noise in Supersonic Jets. Woutijn J. Baars and Charles E. Tinney Aerospace Eng. and Eng. Mechanics, The University of Texas at Austin, Austin, TX 7872, USA Nathan E. Murray and Bernard J. Jansen National Center for Physical Acoustics, University of Mississippi, University, MS 38677, USA Praveen Panickar Combustion Research and Flow Technology Inc., Pipersville, PA 8947, USA The most prominent component of turbulent mixing noise in jets is associated with Mach wave radiation. Large-turbulence structures radiate Mach waves efficiently when they convect supersonic relative to the ambient. An experimental study is conducted on an unheated (T = 286.6K) and heated (T = 2.6K) fully expanded Mach.553 jet to investigate the effect of heat on this radiation process. The acoustic near-field was captured using a line array that comprised ten microphones situated in the hydrodynamic periphery of the dominant sound producing region, which is located downstream of the collapse of the potential core. Space-time correlations revealed a convective speed of the pressure signatures that was slightly larger than sonic, relative to the ambient, for the unheated jet, while being around M =.48 for the heated jet; Mach wave radiation occurred in both cases. A far-field circular arc array with a radius of 58.5 jet exit diameters was centered at the jet exit and consisted of twelve microphones ranging from 2 to 35 relative to the jet axis. A linear coherence and temporal correlation study unveiled mechanisms by which the near- and far-field pressures are coupled. Mach wave radiation is a fairly linear process. From arrival times of the acoustic disturbances traveling from near- to far-field it was found that propagation speeds were uniform in the heated case, while variations in speed, up to 5% above the ambient sound speed, were found for the unheated case. I. Introduction The ability to accurately predict the far-field noise intensity, directivity and spectral distribution from heated supersonic jets continues to be a major challenge in aeroacoustics. In supersonic jets, sound is generated by a combination of three distinct mechanisms,.) turbulent mixing noise, 2.) broadband shock noise, and 3.) screech tones. In the current work, detailed surveys of the near-field pressure and far-field acoustics are conducted on both cold and heated supersonic jets with a matched gas dynamic Mach number of.553. The nozzle is designed to produce perfectly expanded flow at this Mach number and so eliminates the source mechanisms associated with broadband shock noise and screech. In doing so, the effect of heat on turbulent mixing noise can be isolated, which is of practical scientific interest to military aircraft where jet exit velocities and temperatures are high. Similar studies have been conducted in the past. Some include the work by Bridges (26,27),2 of supersonic heated jets (based on the gas dynamic Mach number) and by Seiner et al. (994), 3 Greska et al. (28) 4 and McLaughlin et al. (2) 5 (based on the acoustic Mach number) who collectively investigated PhD Student, AIAA Student Member. Assistant Professor, AIAA Senior Member. Research Scientist & Research Assistant Prof. of Mech. Engineering, AIAA Member. Senior R&D Engineer, AIAA Member. Research Scientist, AIAA Member. of 4 Copyright 2 by the author(s). Published by the American American Institute of Institute Aeronautics of and Aeronautics Astronautics, and Inc., Astronautics with permission.

2 the effects of temperature ratio on the jet potential core length, near-field pressure intensity, far-field spectra and space-time correlations. Where the development of more robust and accurate acoustic analogies are concerned, one may wish to review the work of Tam et al. (28). 6,7 While a great number of these surveys have focused on jet noise from supersonic jets in general, very few have attempted to isolate the effects of heat on turbulent mixing noise alone. Turbulent mixing noise, which occurs in both unheated and heated jets, consists of two components as described by Tam & Chen (994) 8 and recently by Tam et al. (28). 6 The first component is generated by large-turbulence structures or instability waves in the hydrodynamic region that causes Mach waves to radiate to the far-field. This process becomes efficient when the large-scale turbulence convects at supersonic speeds relative to the ambient. The uniform convection of these structures reflects a harmonic supersonic wave in a panel that radiates the waves at an angle φ = π/2 µ = π/2 sin (a /U c ) as schematically indicated in figure. The intensity of the wave radiation decays rapidly beyond the radiation angle φ, and therefore, this component is only noticeably present at downstream angles (roughly θ 6 ). a acoustic wave fronts traveling to far-field µ φ U c nozzle θ large-turbulence structures moving at speed U c x, jet axis Figure. Simplified schematic of the Mach wave radiation mechanism in the near-field of the jet (ambient speed of sound a and uniform convective speed U c). The Mach wave radiation process is highly complex in reality due to the growth, saturation and decay of the instability waves, non-uniform convective speeds along the jet axis, and other factors such as refraction in the case of non-isothermal jets. A more detailed description of the Mach wave radiation mechanism can be found in the literature. 3,6 9 The other component of turbulent mixing noise is associated with the fine-scale turbulence in the jet. It causes an omni directional propagation of sound and is the dominant source of sound at angles exceeding the Mach wave radiation angle. Details on fine-scale turbulence noise can be found in the work by Tam et al. (25). It has to be noted that the exact sources of far-field noise is still an intense topic of debate in the jet noise community. Various acoustic analogies, originally proposed by Lighthill, have been presented 8,,2 and experimental investigations have been performed. 6 However, a uniform model of the sources of sound in the broad spectrum of jet flows, especially unheated and heated supersonic jets, has not been established yet and is therefore an extensive topic of research at the moment and foreseeable future. In the current work, an experimental study is conducted on an unheated (T = 286.6K) and heated (T = 2.6K) fully expanded Mach.553 jet to investigate the effect of heat on the Mach wave radiation process. Near-field pressure signatures were acquired using a line array situated in the hydrodynamic periphery of the dominant sound producing region. Far-field pressures were captured by a far-field circular arc array with a radius of 58.5 jet exit diameters and was centered at the jet exit. Synchronized pressure measurements in the near- and far-field region were obtained and will contribute to the database of heated jets for verification purposes. Moreover, a linear coherence and temporal correlation study will be applied to unveil mechanisms by which the near- and far-field pressures are linearly coupled. II.A. Experimental Facility II. Experimental Arrangement The jet acoustic measurements were performed in the anechoic chamber of the National Center for Physical Acoustics (NCPA) - University of Mississippi in March 2. The fully anechoic chamber has dimensions of 5.8m 6.m 2.4m, measured from the wedge tips, and has a 2Hz low-frequency cut-off. A nozzle rig is positioned inside the chamber as presented in figure 2. The chamber allows flow to pass by the wedges on 2 of 4

3 the entrance and eductor walls as it is important during jet runs to maintain a low room temperature for having accurate refraction effects and acoustic far-field measurements. The supersonic nozzle used in this study represents a shock free, fully expanded, nozzle of the F44-4 engine (without center body) that can be installed on the F/A-8. The sub-scale nozzle has a design exit Mach number of M e =.553 at a total temperature of T = 2.6K (38 F) (ratio of specific heats γ =.366 at the jet exit) and has an exit diameter of D j = 5mm. The Method of Characteristics was used in the design of the nozzle with parameters corresponding to conditions. This will cause a slight deviation from fully expanded conditions when the nozzle is operated at the same Mach number under cold conditions, however, this effect can be neglected. The nozzle rig comprises a swirl can propane burner system upstream of the nozzle contraction for obtaining jet conditions. The burning process is lean, so the combustion output is assumed to be equivalent to air with a specific gas constant of R = 287.5J/kg/K. Figure 2. Nozzle hardware, near-field line array of microphones (right side), and far-field arc array (left, in background) in the NCPA anechoic chamber. The near-field pressure data was acquired using a line array of ten /4 inch PCB model 2A22 microphones positioned in the vertical plane at a % slope as indicated in figure 3. The microphones had an equidistant separation of = 6/5 located in the range = [.4,2.]. The acoustic signature in the far-field was acquired using a circular arc array in the horizontal plane. The array was centered at the nozzle exit plane and had a radius of 58.5D j. Twelve microphones were positioned in the range θ = [2,35 ], as indicated in figure 4. Synchronized data of all near- and far-field microphones were acquired at a sampling frequency of 2kHz for a total of samples. nozzle x = side view, (x, z)-plane () (.) x, jet axis nozzle lip.4 D j line array, = 6/5 D j z 2. D j Figure 3. Location of the near-field line array of ten PCB model 2A22 microphones. II.B. Experimental Conditions Experiments were performed at two conditions, henceforth referred to as the cold and case. The nozzle was operated at fully expanded conditions during both cases, thus the jet exit Mach number, M j, was controlled to be equal to the design exit Mach number. The total jet temperature was T = 286.6K for the cold case and T = 2.6K for the case. The jet exit conditions are calculated from the isentropic relations and the dynamic viscosity is based on the Sutherland law 3 of 4

4 58.5 D j θ = 2 θ = 5 θ = 9 θ = 75 θ = 2 θ = 6 θ = 27.5 θ = 35 top view, (x, y)-plane θ = 52.5 θ = 45 θ = 37.5 θ = 3 nozzle θ y x, jet axis Figure 4. Location of the far-field circular arc array of twelve microphones centered on the jet exit. ( ) 3/2 µ j Tj T ref + s = µ ref T ref T j + s, () where µ ref =.76 5 Ns/m 2, T ref = 273K and s = K for air. The ratio of specific heats is solely dependent on static temperature according to γ (T) =.428 ( T ), (2) where T is in units of Kelvin. The experimental conditions and flow quantities at the jet exit (subscript j ), ambient (subscript ) and stagnation (subscript ) are summarized in table. Furthermore, nondimensional groups are presented. The acoustic Mach number, M a, and the convective Mach numbers associated with the sources of Mach wave radiation observed by Oertel were computed as follows and are used throughout the literature on supersonic jets. 4 M a = U j a, M c = U j + a j U j, M c =, a j + a a j + a M co = U j + a j a j + a. (3) III.A. III. Flow Characteristics in the Near-Field of the Jet Convection along the Near-Field Array The pressure signatures captured by the near-field line array represent the convective nature of the most energetic instability modes. The raw fluctuating pressures are shown in space-time format in figure 5. The fluctuations are roughly a factor 5 larger during the case, which will result in a higher overall sound pressure level (OASPL) at the location of the array as will be discussed in section IV. The convective speed of the prominent pressure signatures is quantified by space-time correlations that are shown in figure 6 (serrated pattern due to the limited spatial resolution). For the cold case, the convective speed is constant over the length of the array and is estimated as U c = 2U j. This shows good agreement with the measurements of the phase velocity of the axial instability waves by Troutt & McLaughlin (982) 3 in a cold M j = 2. jet, who concluded that the convective speed is U j for a broad range of wavenumbers. Considering the case, the convective speed is U c =.62U j at the beginning of the array and is fairly constant up to = 3.5 whereafter it decreases to U c =.37U j at the end of the array. The decrease in 4 of 4

5 Table. Experimental conditions (Mj was controlled to be the fully expanded Mach number Me =.553). cold measured quantities Mj p [kp a] p [kp a] 4.9 T [K] T [K] cold calculated quantities Tj [K] γj µj [N s/m2 ] aj [m/s] a [m/s] Uj [m/s] fc = Uj /Dj [khz] cold non-dimensional quantities Ma Mc Mc Mco Tj /T ρ = ρ /ρj Rej (b), case (a), cold case t [ms] 5 2 t [ms] Figure 5. Space-time contours of the fluctuating pressure along the near-field array (in 3 P a). convective speed is the result from the deceleration of the large-turbulence structures traveling downstream in the post-potential core region. No deceleration is observed in the cold case due to the positional difference of the line array relative to the length of the potential core (section III.B). The Mach numbers, based on the ambient conditions, and associated with the convective speeds along the array are given as cold Mcon = 2Uj =.4, a Mcon =.62Uj =.48. a (4) The convective Mach number should be corrected for large inclinations of the array to reflect the convection of the structures parallel to the jet axis. Here, the inclination is small and can be neglected. The convective Mach number is close to sonic for the cold case. Furthermore, a necessary condition for fully developed Mach wave radiation is Mc >.25.4 During the cold condition the radiation process may therefore not be as efficient as during the condition. Nonetheless, Mach wave radiation is expected in both cases with a radiation angle φ given by (illustrated in figure ) φ= π π µ = sin 2 2 Mcon. (5) For the cold case, φ = 5.9, and is expected to be constant (uncurved Mach waves are eminated) along the axial range spanned by the array. During the condition, φ = 47.5, and will decrease along the array range since a decrease in convective speed will result in steeper Mach waves. 5 of 4

6 (a), cold case (b), case U c = 2U j 5 4 U c =.62U j 3 U c =.27a j. 3 U c =.97a j τ [ms] τ [ms] Figure 6. Space-time correlations along the near-field array. The gray contour map illustrates the space-time correlation with the first microphone as a reference. The white line indicates the estimated convective speed, U c, along the array (for the case based on the first three microphones). III.B. Axial Distributions of Flow Quantities Characteristic lengths of the turbulent mixing region of the jet are necessary for describing the location of potential sound sources. Axial distributions of the flow quantities along the jet axis are presented in this section along with the comparison of the experimental data with empirically determined formulations. Pitot pressures, static pressures and total temperatures were measured along the centerline for the jet a and were used to compute the axial distributions of the Mach number, velocity and thermodynamic quantities. This is an iterative process, although local thermodynamic equilibrium is assumed, due to the varying ratio of specific heats γ (T (x)), Eq. 2 and the presence of a normal shock in front of the pitot tube in the supersonic core region. The axial distributions are presented in figure 7.. M > M <.6.4 M/M j U/U j T/T j p/p j T /T j U c =.62U j near-field array p /p j Figure 7. Axial distribution of flow quantities along the jet centerline. The location of the near-field array and sonic point (dashed line) are indicated. The convective speed along the first microphones as obtained from space-time correlations is about equal to the core velocity at that location. ( jet only) Numerous studies were concerned with the decay of the jet centerline velocity and how to collapse the axial distributions. Witze (974) 4 presented empirical relationships to determine the length of the potential core and the deceleration trend along the centerline for subsonic and supersonic jets. The location where a conditions were slightly different than listed in table. M j =.57, T j /T = 2.5 and ρ = of 4

7 the potential core collapses, denoted as x c, was proposed to be x c /D j = and the centerline velocity decay is given as U (x) U j 2κρ ( = exp, where ρ = ρ /ρ j, (6) 2κ ( ) ρ X c ). (7) X c is the non-dimensional correlation parameter core length and has a universal value of.7. The proportionality constant κ was empirically found to be κ =.63 ( M 2 j ).5. (8) This expression is applicable from the jet exit to the sonic point. Downstream of the sonic point a modification has to be made, the details of which are described in the literature. 4 These empirical formulations were successfully verified with data from subsonic cooled, heated and ambient temperature jets and supersonic cooled jets. The application of this formulation results in figure 8a where the axial distribution of the centerline velocity is presented as function of the axial correlation parameter, κx(ρ ) = 2κ ( ) (ρ ). The theory under predicts the collapse of the potential core which is estimated from the data as x h c/d j = (a) (b). M > M <. M > M < x w c D j = 5.23 x w c D j = 6.95 U/Uj.6 x h c D j = 6.79 U c =.62U j U/Uj.6 x h c D j = 6.79 U c =.62U j.4 near-field array Witze (974) κx(ρ e ).5.4 near-field array Witze* (974) κx(ρ e ) Figure 8. Centerline velocity decay of the jet. (a) Compared to theoretical formulation proposed by Witze (974), 4 (b) Compared to theoretical formulation by Witze (974) 4 modified by the suggestion of Lau (98). 5 An experimental study by Lau et al. (979) 6 focused on isothermal jets at three different Mach numbers and observed that, in the supersonic range, Witze s formulation under predicts x c /D j. They stated that the density variation seems to be taken into account inappropriately. Therefore, the range of applicability of Witze s formulation seems to exclude supersonic jets. Lau et al. (979) 6 therefore proposed another relationship for determining x c /D j, being x c /D j = M 2 j. (9) This empirical relation is based on their isothermal jet experiments, which holds to a jet Mach number of about.5. After that, the potential core length is under predicted for isothermal jets. But, when heating the jet to T j /T >.5 the potential core shrinks and their relationship of isothermal jets seems to be applicable as was observed by Lau (98). 5 For the condition in the present study, Eq. 9 yields x c /D j = 6.86, which agrees fairly well with the experiments. Furthermore, it was proposed that for heated jets (subsonic and supersonic) x c /D j varies as (T j /T ) while for unheated jets it varies as (T j /T ). Witze (974) 4 on the other hand distinguished subsonic jets, x c /Dj (T j /T ) 8, (since κ is now 7 of 4

8 dependent on the density/temperature gradient as well) from supersonic jets, x c /Dj (T j /T ). When using the suggestion of Lau (98) 5 and adapting Witze s formulation (Eq. 6-8,, denoted Witze*), the experimental velocity profile shows a reasonable agreement with theory as presented in figure 8b. Another empirical curve to determine x c /D j was introduced by Greska et al. (28) 4 as x c /D j = 3.34exp(.43M j M c ). () This relation was based on supersonic cold and jets. The predicted core lengths and comparison with the other theories and data is presented in table 2. Assuming that Witze s formulation predicts the core length accurately for the cold case, the position of the array relative to the core length is x/x c = [.,2.], and x/x c = [.7,3.] for the case. Table 2. Summary of x/x c (collapse of the potential core) based on data and theory. data Witze (974) 4 Witze* (974) 4 Greska (28) 4 cold IV. Acoustic Near-Field and Far-Field of the Cold and Hot Jet Overall sound pressure levels (OASPL) at the location of the near- and far-field arrays are shown in figure 9. The OASPL in the near-field is considerably higher during the condition due to an increase in turbulence fluctuations that are mainly caused by the higher jet exit velocity. The OASPL is roughly constant over the first 3% of the array for the cold case. It is believed that the most intense noise region is just after the collapse of the potential core, and thus causes the broadband peak in OASPL to be in that axial range (array starts at x/x c =.). For the case, the array starts at x/x c =.7 and thus shows a decay in OASPL from the first sensor on. The reader is referred to Greska et al. (28) 4 for more near-field OASPL data considering supersonic jets. OASPL [db], pref = 2µPa cold 5 o 4 OASPL [db], p ref = 2µPa: Figure 9. Overall sound pressure level (OASPL) in the near-field and far-field of the cold and jet. cold 2 o 9 o 6 o 3 o o Concerning the far-field, it was argued by Tam et al. (992) 7 that the direction of peak noise radiation is close to the direction of Mach wave radiation when it is the dominant sound producing mechanism. Thus for the cold case the OASPL is expected to peak around φ = 5.9 which is outside of the range of the array. However, it is observed that the OASPL still has an increasing trend when moving to the most shallow angles. Besides the Mach wave radiation phenomenon, refraction can cause a higher noise intensity close to the jet axis due to the fact that there is a positive temperature gradient in the normal direction to the jet axis; the region surrounding the jet axis will act as an acoustic waveguide. 8 The peak noise radiation direction for the case is found as 45 which corresponds well with the Mach wave radiation angle φ = For both the cold and case, the OASPL drops at angles beyond the Mach wave angle and seems to stabilize at 8 of 4

9 high angles (θ > 5 ). This is in agreement with theory stating that the sound generated by the fine-scale turbulence is the prominent component at high angles and has an omni directional radiation pattern. Far-field power spectral densities are presented in figure. The spectra are relative peaky at shallow angles (cold θ 37.5 ; θ 52.5 ) and are an indication that the Mach wave radiation is the prominent source. Broadband, lower amplitude, spectra are observed at sideline and upstream angles (cold θ 52.5 ; θ 9 ) and represent the broadband nature of the fine-scale turbulence mixing noise. At intermediate angles the spectra are governed by both components from the turbulent mixing noise. These far-field observations are well-known and can be fitted to two universal spectra as documented in the literature. 7 (a), cold case (b), case Spp(f) [db/hz], pref = 2µP a θ = 2 θ = 3 θ = 37.5 θ = 45 θ = 52.5 θ = 6 θ = 75 θ = 9 θ = 5 θ = 2 θ = 27.5 θ = 35 Spp(f) [db/hz], pref = 2µP a θ = 2 θ = 3 θ = 37.5 θ = 45 θ = 52.5 θ = 6 θ = 75 θ = 9 θ = 5 θ = 2 θ = 27.5 θ = 35 4 f [Hz] 5 4 f [Hz] 5 Figure. Far-field power spectral densities (5% bandwidth moving filter applied). Near-field power spectral densities are presented in figure. General trends are similar for the cold and case. Firstly, the spectra shift to lower frequencies when moving downstream, reflecting the characteristic trend of turbulence evolution in jet flows. Secondly, the peak at the mid-range frequencies represents the most energetic instability waves. Since Mach wave radiation is caused by these waves, the peak frequencies of the far-field spectra at the angles lower than the Mach wave radiation angle must be fairly equal to the instability wave frequency range. 7 The range is about Hz 2Hz for the cold jet and Hz 4Hz for the jet. Thirdly, an interesting observation is the second hump at low-frequencies, 2Hz 3Hz, that arises near the end of the array, which is not observed in subsonic jets. 9 The physical relevance must be questioned since this frequency range is close to the chamber cut-off frequency of 2Hz. V.A. V. Linear Coherent Mechanisms in the Jet Linear Coherence Signatures in the Near-Field The degree of linear coherence of the near-field instability waves, and their induced pressure signatures, is presented in figures 2 & 3 for the cold and case respectively. Linear coherence spectra are shown along the array for all microphones with respect to a reference microphone at location x = x, according to 2 γ 2 xx (f) = p(x;f) p (x ;f) 2 p(x;f) p (x;f) p(x ;f) p (x ;f), () where p(x;f) is the temporal Fourier transform of the near-field pressure p(x;f) = F [p(x,t)]. There is strong spatial coherence in all cases, for example, when considering a threshold the structures are coherent up to 3 at the chamber cut-off frequency of 2Hz. There is no observable coherence for frequencies exceeding khz. The most energetic instability waves are very coherent when convecting along the array as can be seen from the widening of the coherence band around Hz. From a comparison of the cold 9 of 4

10 (a), cold case (b), case Spp(f) [db/hz], pref = 2µP a x/d =.4 j x/d = 2.5 j x/d = 3.5 j 9 x/d = 4.6 j x/d = 5.7 j = x/d = 7.8 j x/d = 8.9 j x/d = 2. j x/d = 2. j 7 f [Hz] 5 Spp(f) [db/hz], pref = 2µP a x/d =.4 j x/d = 2.5 j x/d = 3.5 j 9 x/d = 4.6 j x/d = 5.7 j = x/d = 7.8 j x/d = 8.9 j x/d = 2. j x/d = 2. j 7 f [Hz] 5 Figure. Near-field power spectral densities (5% bandwidth moving filter applied). and jet it can be concluded that a faster decay in coherence occurs for the heated jet. This indicates a more rapid decay in turbulent eddies. A plausible explanation is the further downstream location of the array with respect to the core length in the heated case, however, Doty and McLaughlin (22) 2 already suggested that eddies in jets decay faster based on two-point space-time correlation measurements inside the jet flow using optical deflectometry. f [Hz] f [Hz] x /D j =.4 x /D j = 2.5 x /D j = 3.5 x /D j = 4.6 x /D j = 5.7 x /D j = 6.7 x /D j = 7.8 x /D j = 8.9 x /D j = 9.9 x /D j = 2. Figure 2. Linear coherence spectra, γ 2 xx (f), for all microphones along the near-field array with respect to a reference microphone at location x = x, cold case. of 4

11 f [Hz] f [Hz] x /D j =.4 x /D j = 2.5 x /D j = 3.5 x /D j = 4.6 x /D j = 5.7 x /D j = 6.7 x /D j = 7.8 x /D j = 8.9 x /D j = 9.9 x /D j = 2. Figure 3. Linear coherence spectra, γ 2 xx (f), for all microphones along the near-field array with respect to a reference microphone at location x = x, case. V.B. Linear Coupling between the Near-Field and Far-Field This section is concerned with an attempt to unveil the linear mechanisms by which the near- and far-field are coupled. First, linear coherence spectra between all near- and far-field microphones are computed according to γ 2 xθ(f) = p(x;f) p (θ;f) 2 p(x;f) p (x;f) p(θ;f) p (θ;f), (2) where p(θ;f) is the temporal Fourier transform of the far-field pressure p(θ;f) = F [p(θ,t)]. Coherence spectra are presented in figure 4 for all near-field microphones and far-field microphones up to an angle of θ = 6. An absence of coherence was found for far-field microphones at higher angles; this indicates that the fine-scale turbulent mixing noise is uncorrelated, as would be expected. More importantly, it suggests that the linear coherence between the near-field and downstream angles of the far-field is caused by the Mach wave radiation mechanism, which is thus fairly linear in nature. A simplified illustration of the Mach wave radiation as it occurs in cold versus, perfectly expanded, supersonic jets is presented in figure 5. In order to interpret the linear coupling two observations are made: ) Mach waves that are generated more downstream have steeper Mach angles due to lower convective Mach numbers, propagate at shallower angles to the jet axis (φ > φ 2 > φ 3, φ 4 > φ 5 > φ 6 ), and their peak frequency reduces due to the decay in peak frequency of the most energetic instability wave, 2) the imaginary cone that bounds the region influenced by Mach wave radiation is smaller for the cold case due to lower convective Mach numbers when perfectly expanded cold versus jets are considered. It has to be noted that a discrete set of three Mach wave patterns is sketched in figure 5 for simplification. In reality, a continuous Mach wave pattern (curved) occurs where the effects of various directivities, propagation frequencies, and refraction due to temperature gradients are embedded. High peak values of coherence (.45.75) are observed for the cold jet at θ = 2 (figure 4), where the highest value corresponds to the coherence with the last microphone on the line array. The decay in coherence cannot be the result from Mach waves radiating to high far-field angles (observed by the upstream part of the near-field array), since Mach wave radiation is mostly confined to φ 5.9 for the cold case (based on Eq. 5). A plausible explanation for the decay in coherence can be the increased geometrical distance between the microphones and the absorption and nonlinear phenomena associated with that, however, this is not expected to result in such a dramatic drop. When θ = 3 is considered, the coherence with downstream near-field of 4

12 γ 2 xθ(f) γ 2 xθ(f) γ 2 xθ(f) =.4 = 2.5 = 3.5 = 4.6 = 5.7 = 6.8 = 7.8 = 8.9 = 2. = f [Hz] θ = =.4 = 2.5 = 3.5 = 4.6 = 5.7 = 6.8 = 7.8 = 8.9 = 2. = 2. f [Hz] θ = 37.5 θ = 52.5 γ 2 xθ(f) γ 2 xθ(f) γ 2 xθ(f) =.4 = 2.5 = 3.5 = 4.6 = 5.7 = 6.8 = 7.8 = 8.9 = 2. = =.4 = 2.5 = 3.5 = 4.6 = 5.7 = 6.8 = 7.8 = 8.9 = 2. = 2. f [Hz] θ = f [Hz] θ = 45 θ = 6 f [Hz] f [Hz] Figure 4. Linear coherence spectra, γ 2 xθ (f), between the individual microphones in the near- and far-field of the cold (gray) and (black) jet. sensors decreases as would be expected. The peak coherence with the upstream sensors remains roughly equal, indicating some evidence of the effects illustrated in figure 5. Considering the case, the effect of Mach wave radiation is fully developed and the characteristics of figure 5 must therefore be more observable. This is indeed the case, since coherence of upstream near-field microphones with far-field sensors is actually increasing up to θ = 45 ; the coherence with downstream sensors is decreasing. When approaching the Mach wave radiation angle based on Mconv, φ = 47.5, the coherence starts to drop significantly. While the general trends are in agreement with expectations, the cause of the large amplitude difference between downstream and upstream sensors on the near-field array needs to be a topic of further investigation. For the cold case, the peak frequency of coherence decreases when considering near-field microphones further downstream on the array, which again is evidence that the coherence is mainly caused by Mach wave radiation, since the frequencies of the instability waves decrease along the axial range of the array (figure ). Interestingly, less drift in coherence peak frequency occurs for the case. A preliminary analysis concerned with acoustical arrival times is a second attempt to study near- far-field coupling. When temporal cross-correlations between individual near- and far-field sensors are computed, one distinct time-delay peak was observed (for θ 6 ). This time delay is the arrival time of the acoustical disturbances traveling from near- to far-field and is presented in figures 6a & 6b. The trend is as expected 2 of 4

13 deflection due to refraction x c x/x c = [., 2.] φ c max φ φ 2 φ 3 cold, dt dn > potential core x, jet axis deflection due to refraction near-field array x/x c = [.7, 3.] φ h max φ 4 φ 6 φ 5, dt dn < Figure 5. Simplified illustration of the Mach wave radiation mechanism in a supersonic, perfectly expanded, cold versus jet (not to scale). since the geometrical distance b decreases when moving downstream on the near-field array and moving to shallower angles in the far-field. A proper scaling of the near- to far-field trends is taken into account by computing average propagation speeds (U p (x,θ) = ) which are shown in figures 6c & 6d distance arrival time (a), cold case (b), case (c), cold case (d), case θ 2 θ 3 θ 37.5 θ 45 θ 52.5 θ τ [ms] θ 2 θ 3 θ 37.5 θ 45 θ 52.5 θ τ [ms] θ [degr] θ [degr] Figure 6. (a) & (b) Acoustic arrival times of disturbances traveling from near- to far-field microphones. (c) & (d) Average propagation speed, U p (x, θ), between near- and far-field microphones. For the case, the average propagation speed is fairly constant (U p (x,θ) = 355m/s) between all microphones as would be expected. U p is slightly higher than the ambient sound speed, a, which is attributed to faster propagation in the high temperature shear layer of the jet. An interesting observation is made for the cold case when there is undeveloped Mach wave radiation. Average propagation speeds vary from U p 345m/s up to U p 39m/s, the latter being 5% higher than the ambient speed of sound. Current research focuses on the physcial interpretation of these results. b It can be argued that the jet is axisymmetric in a statistical averaged sense, therefore, the geometrical distances between the near- and far-field microphones are taken as if the line array was located in the plane of the arc array ((x, y)-plane, figure 4). 3 of 4

14 Concluding Remarks An experimental study was conducted on an unheated (T = 286.6K) and heated (T = 2.6K) fully expanded Mach.553 jet to investigate the effect of heat on turbulent mixing noise and the associated Mach wave radiation mechanism. The radiation process is influenced by many factors, such as refraction, growth and decay of the Mach angles, dispersion and nonlinear propagation. The current work presented a survey of both near- and far-field pressure signatures. Additionally, a linear coherence study between the near-field and far-field unveiled quantitative information of how the two pressure fields are linearly coupled. Unresolved issues, such as the physical interpretation of the acoustical arrival times that show different trends for the cold versus case are addressed with ongoing research. Acknowledgments The authors wish to offer this posthumous acknowledgment to Dr. John M. Seiner whose mentoring, leadership, and vision have touched not only the authors but many others in the jet noise research community. The research presented herein was made possible by the efforts Dr. Seiner made to establish world-class facilities at the National Center for Physical Acoustics during his tenure at the University of Mississippi. References Bridges, J., Effect of Heat on Space-Time Correlations in Jets, 2th AIAA/CEAS Aeroacoustics Conference, AIAA, Cambridge, Massachusetts, may Bridges, J. and Wernet, M. P., Effect of Temperature on Jet Velocity Spectra, 3th AIAA/CEAS Aeroacoustics Conference, AIAA, Rome, Italy, may Seiner, J. M., Bhat, T. R. S., and Ponton, M. K., Mach Wave Emission from a High-Temperature Supersonic Jet, AIAA Journal, Vol. 32, No. 2, 994, pp Greska, B., Krothapalli, A., Horne, W. C., and Burnside, N., A Near-Field Study of High Temperature Supersonic Jets, 4th AIAA/CEAS Aeroacoustics Conference, AIAA, Vancouver, British Columbia, Canada, may McLaughlin, D. K., Bridges, J. E., and Kuo, C.-W., On the scaling of small, heat simulated jet noise measurements to moderate size exhaust jets, 6th AIAA/CEAS Aeroacoustics Conference, AIAA, Stockholm, Sweden, jun 2. 6 Tam, C. K. W., Viswanathan, K., Ahuja, K. K., and Panda, J., The sources of jet noise: experimental evidence, Journal of Fluid Mechanics, Vol. 65, 28, pp Tam, C. K. W., Mach Wave Radiation from High-Speed Jets, AIAA Journal, Vol. 47, No., oct 29, pp Tam, C. K. W. and Chen, P., Turbulent Mixing Noise from Supersonic Jets, AIAA Journal, Vol. 32, No. 9, sep 994, pp McLaughlin, D. K., Morrison, G. L., and Troutt, T. R., Experiments on the instability waves in a supersonic jet and their acoustic radiation, Journal of Fluid Mechanics, Vol. 69, 975, pp Tam, C. K. W., Pastouchenko, N. N., and Viswanathan, K., Fine-Scale Turbulence Noise from Hot Jets, AIAA Journal, Vol. 43, No. 8, aug 25, pp Tam, C. K. W., Supersonic Jet Noise, Annu. Rev. Fluid Mech., Vol. 27, 995, pp Morris, P. J. and Farassat, F., Acoustic Analogy and Alternative Theories for Jet Noise Prediction, AIAA Journal, Vol. 4, No. 4, apr 22, pp Troutt, T. R. and McLaughlin, D. K., Experiments on the flow and acoustic properties of a moderate-reynolds-number supersonic jet, Journal of Fluid Mechanics, Vol. 6, 982, pp Witze, P. O., Centerline Velocity Decay of Compressible Free Jets, AIAA Journal, Vol. 2, No. 4, apr 974, pp Lau, J. C., Effects of exit Mach number and temperature on mean-flow and turbulence characteristics in round jets, Journal of Fluid Mechanics, Vol. 5, 98, pp Lau, J. C., Morris, P. J., and Fisher, M. J., Measurements in subsonic and supersonic free jets using a laser velocimeter, Journal of Fluid Mechanics, Vol. 93, 979, pp Tam, C. K. W., Chen, P., and Seiner, J. M., Relationship Between Instability Waves and Noise of High-Speed Jets, AIAA Journal, Vol. 3, No. 7, jul 992, pp Blackstock, D. T., Fundamentals of Physical Acoustics, John Wiley & Sons, Inc., New York, 2. 9 Tinney, C. E. and Jordan, P., The near pressure field of co-axial subsonic jets, Journal of Fluid Mechanics, Vol. 6, 28, pp Bendat, J. S. and Piersol, A. G., Engineering applications of correlation and spectral analysis, Wiley, New York, Doty, M. J. and McLaughlin, D. K., Two-Point Correlations of Density Gradient Fluctuations in High Speed Jets Using Optical Deflectometry, 4th Aerospace Sciences Meeting and Exhibit, AIAA, Reno, Nevada, jan of 4