Experimental and Computational Study of Near Field/Far Field Correlations in Supersonic Jet Noise

Transcription

1 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition January 2012, Nashville, Tennessee AIAA Experimental and Computational Study of Near Field/Far Field Correlations in Supersonic Jet Noise Ching-Wen Kuo 1, Yongle Du 2, Dennis K. McLaughlin 3 and Philip J. Morris. 4 Penn State University, University Park, PA This paper describes near field/far field correlations for supersonic jet noise with the use of both numerical simulations and experimental measurements. Examples are provided of correlations between the output from an Optical Deflectomter (OD) and far field microphone measurements. Numerical simulations using a modified Detached Eddy Simulation are performed for nozzle geometries typical of high performance military aircraft. Correlations are performed between the simulated flow field fluctuations and the predicted far field acoustic pressure. Comparisons are made between the simulations and experiments. The relationship between the output of the OD and the flow field fluctuations is discussed based on the numerical simulations. The results demonstrate the potential advantages of the complementary use of both unsteady flow measurements and simulations in understanding the noise generation mechanisms in high speed jets. I. Introduction HE flow-acoustic correlation method was originally proposed by Siddon 1. Based on the cause-and-effect T relation, it was argued that the jet noise sources can be identified by measuring the direct correlation between the near-field turbulent fluctuations and the far-field acoustic pressure. Similar attempts were made by Seiner and Reethof 2 but both Sidden and Seiner and Reethof s work raised the important issue of noise generated by the hotwire probe in the flow. Richarz 3 was one of the first to circumvent the probe noise issue by using a non-intrusive laser Doppler velocimeter for the turbulence measurements. At that time, the sampling speed limited the useful data to very low speed jets. Flow-acoustic correlation measurements have been performed with different experimental techniques (see, for example, Schaffar 4 and Richarz 3 ). But, significant advances were made with the introduction of the Molecular Rayleigh Scattering (MRS) technique (see Panda and Seasholtz 5, Panda 6, and Panda et al. 7 ). This technique has been used to explore correlations between flow field turbulence and radiated noise in high speed jets. Recent efforts by Veltin et al. 8 and Kuo et al. 9 have used the Optical Deflectometer (OD) to perform quite extensive correlation measurements of flow field turbulence and far-field noise. The present work involves a close partnership between the experiments and numerical simulations of similar jet flows being conducted at Penn State University. It focuses on space-time correlations obtained both numerically and experimentally between locations in the jet shear layer and slightly outside the jet plumes. Additionally, the numerical analyses and the related experiments were conducted with simultaneous measurements at microphones located in the acoustic far field at the peak noise emission angle. The space-time correlation measurements correlate the noise measurements in the far-field with the jet turbulence or near field fluctuations in the vicinity of the jet plume. The magnitude of the space-time correlation represents the relationship between the near field noise at locations in the vicinity of the jets and the noise radiated at specific polar angles in the acoustic far field. This study seeks to clarify some of the mechanisms of noise generation, in particular how the turbulent noise sources couple to the acoustic field. The next section of the paper describes the experimental facility and the instrumentation and data-processing techniques. It also tabulates the jet flow conditions used in the experiments, which are (mostly) duplicated in the 1 Post-doctoral Scholar, Aerospace Engineering, 29 Hammond Building, Member AIAA. 2 Post-doctoral Scholar, Aerospace Engineering, 229 Hammond Building, Member AIAA. 3 Professor, Aerospace Engineering, 230A Hammond Building, Fellow AIAA. 4 Boeing/A. D. Welliver Professor, Aerospace Engineering, 233C Hammond Building, Fellow AIAA. 1 Copyright 2012 by the, Inc. All rights reserved.

2 numerical simulations. Also summarized are the flow field and acoustic field locations of the microphones and Optical Deflectometer sensing positions. For the most part, the data from the numerical experiments are available anywhere in the computational domain but advanced planning is required to make sure adequate record lengths are saved as the computations near their completion. Following the summary of the experiments is a description of the numerical simulation approach for flow and noise. The results of the experiments and computations are typically presented simultaneously where possible. In this fashion direct comparisons provide a measure of validation of the numerical methods but often an improved physical interpretation is possible because of the more in depth and continuous data available from the numerical simulations. A. Facility and Instrumentation Descriptions II. Description of the Experiments The experiments presented in this paper were conducted in the Pennsylvania State University high-speed jet noise facility shown in Fig. 1 a). In order to produce acoustic measurements that can be compared directly to aircraft engine measurements, the temperature of the jet is an important parameter that needs to be replicated. A hotter jet results in different noise characteristics, due to the increase in jet exit velocity and decrease in jet density. Actual heating of the exhaust air is often performed in facilities such as the ones used at the NASA Glenn Research Center 10. However, it requires an extensive amount of power and infrastructure, raising the overall operating costs of the facility. The present experiments simulate the effects of jet heating using a mixture of helium and air. Kinzie and McLaughlin 11, Doty and McLaughlin 12 and Papamoschou 13 demonstrated that the use of a mixture of helium and air is able to capture the dominant noise characteristics of actual heated jets. Miller and Veltin 14 showed a good agreement of the mean flow properties between experimental data from helium-air mixture jets and a Reynoldsaveraged Navier-Stokes numerical calculation of heated air jets. Recent careful comparisons 15 between the measurements conducted at Penn State with the measurements performed in other facilities have shown very good agreement when matching the acoustic velocity of the helium-air mixture jet to that of a heated jet following the procedure developed by Doty and McLaughlin 12. a) b) Fig. 1. The Pennsylvania State University high speed jet noise facility. a) Schematic. b) Photo. The jet noise anechoic chamber facility is a 5.02 x 6.04 x 2.8 m room covered with fiberglass wedges and with an approximate cut-off frequency of 250 Hz. An exhaust fan, installed in the downstream direction from the jet plenum, collects the jet exhaust and minimizes air recirculation and possible local helium accumulation in the anechoic chamber. Six microphones are positioned at similar radial distances and different polar angles as can be seen in Fig. 1 b). The microphone array is centered at the nozzle exit plane. The microphones are positioned at a grazing incidence to the jet centerline plane. The microphones used are 3.2 mm (1/8 in) pressure-field microphones, type 4138 from Brüel and Kjaer (B&K), and type 40DP from GRAS. 2

3 Besides the microphones, the major instrumentation used was the Optical Deflectometer (OD) which is predominantly sensitive to density gradients in the cross stream direction of the flow field or acoustic near field. The optical deflectometer setup is shown in Fig. 2. The 6 in. parabolic mirror produces a parallel beam in the optics sending side. The parallel beam path is vertical to the jet axis. While the beam traverses across the jet plume, the beam is deflected from its original path due to density gradients inside and around the jet plume. On the optics receiving side, a second parabolic mirror collects the parallel beam and focuses it to a point. The knife edge is positioned to this focus point (48 in.) of the parabolic mirror. By controlling the knife edge position, the light intensity that is detected by Avalanche Photodiodes (APD) can be adjusted. The photodiodes chosen for this application are Hamamatsu APD modules model C The beamsplitter splits the beam to two APDs. Each APD has the capability to traverse to any position of the test section image. In this way, the instrument is able to detect the instantaneous fluctuations in the density gradient. Fig. 2. Schematic of optical deflectometry setup with photo of the breadboard. B. Data Acquisition The analog time histories from the microphones are routed through a Nexus, B&K signal conditioner or a GRAS model 12AN power module and then amplified and filtered for anti-aliasing, thus enabling their accurate digital conversion. The time histories from both APDs and microphones are connected to a high-pass filter. The high-pass filter is set to 500 Hz, removing any undesirable low frequency noise that could contaminate the data. A PCI-6123 National Instruments DAQ board acquires all the time domain data (8 channels simultaneously). The sampling rate is set at 500 khz and 500,000 data points are collected for either pure air jets (unheated jets) or helium-air mixture jets (heat simulated jets). The raw data are then fed into Matlab for data processing. The raw data are split into 4096 point segments and a Hanning window function is applied with 50 percent overlap between each window. A Fast Fourier Transform is calculated in each window and an average value is calculated from all segments. This yields the double sided power spectral densities Sii ( f ), S jj ( f) and the cross spectral density Sij ( f) between any two selected channels. The single-sided power spectral densities Gii ( f ), Gjj ( f) and the cross spectral density Gij ( f) are then computed. The coherence between two selected signals is then given by, 2 G ( ) 2 ij f ij ( f ) (1) Gii ( f ) G jj ( f ) The phase of the cross spectral density Gij ( f) is also computed. If one APD is fixed at one location and the other APD traverses away from it, the relative phase changes between the two signals, related to the different separation distances, provide the phase velocity of the turbulence as a function of frequency. The auto/cross correlation functions between any two selected channels are calculated by taking the inverse Fourier transform of the double sided power spectra densities Sii ( f ), S ( f) and S ( f ) as follows. jj ij 3

4 1 1 1 R ( ) [ S ( f )], R ( ) [ S ( f )], R ( ) [ S ( f )] (2) ii ii jj The cross correlation coefficient is then calculated with the normalization of the maximum auto correlation functions, as defined below. Rij ( ) ij ( ) (3) R (0) R (0) jj 1/2 ii jj ij ij C. Experimental Procedure and Operating Conditions Two sets of experiments have been conducted and are presented in this paper. One involves OD measurements and the other consists of OD-Acoustics measurements. The experiments were conducted using a military-style nozzle, with conical converging and diverging sections and facets to represent the nozzle flaps and seals. The nozzle has a nominal Mach number of 1.5 with equivalent nozzle diameter of 1.72 cm (0.676 in.) at the nozzle exit plane. This military-style nozzle is representative of the exhaust of aircraft engines of the F404 family. The inner contours of the nozzles were provided by General Electric Aviation. These nozzles were designed with a multi-faceted (12 segments) inside conical contour. In general, the expansion portion of the flow contour consists of 12 flat segments that are interleaved to facilitate area adjustment of the operational nozzles. Unlike well designed contoured CD nozzles, shocks and expansions can appear in the nozzle plume and even inside the nozzle, even at perfectly balanced pressure conditions. A schematic of the OD measurements is shown in Fig. 3 a). The corresponding jet operating conditions and the OD sensor locations are tabulated in Table 1. The OD measurements are designed to determine the convection velocity along the jet lip line or corresponding convection velocity between two adjacent positions. Both APDs are initially fixed at the same location (red and blue crosses). One is motorized (blue cross) and programmed to move away downstream to specific distances. At each location of the traversing APD, the signals are acquired simultaneously from both APDs. Figure 3 b) shows a schematic of the OD-Acoustics measurements. The corresponding jet a) b) Fig. 3 Schematic of experimental setup. a) OD measurements. b) OD-Acoustics measurements. 4

5 operating conditions, APD sensor locations and the microphone positions are all tabulated in Table 2. The OD- Acoustics measurements are designed to study the correlation between the near-field flow fluctuations and the farfield noise. The motorized APD is programmed to traverse through certain locations of the near field of the jet plumes. Again at each location of traversing APD, the signal from APD and microphones are recorded simultaneously. Each set of signals recorded from individual microphones positioned at different polar angles is independently correlated to each set of signals acquired from the APD traversing to different locations. Table 1. Table of the jet conditions and the positions of APD sensors for OD measurements Jet Operating Conditions APD Sensor Locations centered at the nozzle exit plane M j TTR M a f c (khz) U j (m/s) x/d y/d to 12, x/d = to 9.2, x/d = to 9.2, x/d = Table 2. Jet operating conditions and the positions of APD sensor and microphones for OD-Acoustics measurements Jet Operating Conditions APD Sensor Locations centered at the nozzle exit plane M j TTR M a x/d y/d , 2, 3, 4 and 5 8, 9, 10, 11 and to 2.5, y/d = Polar Angles & Radius of the Mic Array 20, 25, 30 42, 47, 52 R/D ~ 55 Where possible the numerical simulations were conducted using the same design Mach number nozzle and operating conditions (nozzle pressure ratio NPR, which establishes the jet operating Mach number Mj, and total temperature ratio TTR). A summary of the methods used in the numerical simulations is presented in the next section. A. Simulation Strategy III. Numerical Simulations A hybrid method combining unsteady CFD with an acoustic analogy is used for the jet noise simulations. Details of the numerical methods are given by Du and Morris 16, but some key points are summarized below for convenience. The Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations, supplemented with a variant of the Detached Eddy Simulation (DES), are solved to simulate the development of unsteady turbulent noise sources in the jet flow. A formally 4th order Dispersion-Relation-Preserving (DRP) scheme 17 is used for spatial discretization. The dual time-stepping method is used to advance the development of the unsteady turbulent jet flow, and multigrid and implicit residual smoothing are used to accelerate the convergence of the sub-iterations. A multiblock structured mesh with 6.14M grid points, as shown in Fig. 4, is created for jet flow simulations. The finite nozzle thickness is meshed to trigger the unsteadiness of the jet flow. As shown in Fig. 4 b), the grids near the inside nozzle wall are refined locally to better capture the boundary layer development. Therefore, multiple non-matching block interfaces appear near the nozzle exit. The block interface treatment is described by Du and Morris 16. Around the jet potential core downstream of the nozzle exit, the grids are refined significantly. The average grid sizes are 0.024D from the 5

6 a) Full computational domain and the FWH b) An axial station near the nozzle exit surface (pink lines) c) Grids near the nozzle exit d) Grid details aroud the nozzle lip Fig 4 Computational mesh for the jet flow simulations nozzle exit to x/ D 4, and 0.047D from x/ D 4 to x/ D 10. This enables an estimate of the highest resolvable Strouhal number of approximately 4.0 to be made. In the circumferential direction, 121 grid points are used to represent the non-circular faceted nozzle contour. Once the unsteady turbulent jet flow has reached a statistically stable state, the flow solutions are sampled every two physical time steps on a set of FWH acoustic data surfaces surrounding the shear layers, as shown in Fig. 4 a). Based on the permeable surface Ffowcs Williams & Hawkings solution 18,19, the numerical integration of the unsteady flow solution at the retarded time gives the time-history of the acoustic pressure at a far-field observer. B. Numerical Flow-Acoustic Correlation Experiments This investigation focuses on the cross correlation between a single near-field flow probe and a single far-field acoustic signal. Unsteady jet flow solutions are recorded over a wide region in a symmetry plane. The solutions at individual points are extracted to correlate with the far-field noise prediction at given virtual microphone locations. In the calculation of the time delay, the retarded time at the flow probe is estimated based on a straight path between the flow probe and the far-field microphone. In other words, at zero time delay, the signal of the turbulent fluctuation has a time advance τ= y /a 0 as compared to the noise signal, where y is the distance between the nearfield flow probe and the far-field microphone and a 0 is the ambient speed of sound. IV. Results of Experiments and Computations This section describes the results obtained from both the model scale experiments and the numerical simulations. The results are acquired from both approaches using the same nozzle geometries and identical jet operating conditions. The detailed comparisons provide both an analysis of the jet noise generation mechanisms as well as an 6

7 in-depth assessment of both the experimental technique and the flow and noise simulations. First the mean flow development is considered. Near-field cross-correlations are then examined to study the characteristics of the noise generation mechanisms in the jet shear layer and the near field acoustic region just outside the jet shear layer. Next, near-field to far-field cross-correlations are performed. Comparisons are made in each case between the simulations and experiments. These results demonstrate the potential advantages of the complementary use of both unsteady flow measurements and simulations in understanding the noise generation mechanisms in supersonic jets. A. Mean Flow Calculation and the Experimental Results Figure 5 shows the variation of the time-averaged axial velocity along the centerline predicted for the unheated jet and the heated jet. Flow measurements at similar flow conditions are shown for comparison 20,21,22, since measurements for both jet conditions are not available at present. It is clear that the jet potential core lengths are significantly under-estimated in the jet flow simulations for both the unheated and heated jet. In an accompanying paper 23 it is shown that this may be caused by a nonphysical development of the initial turbulent mixing layer near the nozzle exit, which in turn leads to the mismatch of predicted overall convection speed of the turbulent noise sources and the radiated noise levels in comparison with the experimental measurements. Fig. 5 Normalized jet centerline velocity and velocity fluctuation at various jet operating conditions. (Blue and red curves are the numerical simulations from this study.) Using the FWH theory, far-field noise predictions are made with about 7000 samples of the instantaneous near-field flow solutions. Considering that a 2 db difference of the noise level is observed between the small-scale heatsimulated experimental data of Penn State and the moderate-scale hot jet data of NASA GRC, the agreement is judged to be good if the disparity between the predicted and measured noise level is within 3 db. Figure 6 presents the predicted far-field noise spectra at different observer angles ranging from 30º to 70º relative to the jet downstream axis, to fit the scope of this study. Also shown are the far-field acoustic measurements acquired at NASA GRC with moderate model-scale nozzles and at PSU with small model-scale nozzles. Overall, a good agreement between the simulations and experiments is observed at most resolved frequencies up to St ~3.0. The disparity is well below 4 db. 7

8 Fig. 6 Comparisons of the acoustic spectra obtained from the simulations and experiments from NASA and Penn State In general the numerical simulations do a good job of predicting the far field noise for both the cold and hot jets. However, an over-prediction of the noise levels of as much as 6 db is observed at mid to high frequencies at some observer angles. Similar over-predictions have also been reported in recent publications 24,25. Attempts have been made by increasing the grid resolution near the nozzle exit but failed to resolve this problem. The results suggest that the nonphysical development of the initial turbulent mixing layer is responsible for the mismatch of the predicted noise spectra with the acoustic measurement, and that further grid refinement alone may not be the solution. Figure 7 shows the auto-correlation of the far-field acoustic pressure from both the measurements and simulations for both jets. The auto-correlation function is interpreted as the spatial coherence at specific noise propagation directions. The premise is that the noise signal caused by the large-scale turbulent structures (Mach wave radiation) is different from the one generated by the fine-scale turbulence. It is reasonable to expect a wider half-width of the auto-correlation function when the measured signal is extracted from the direction where the jet noise is dominated by the large-scale turbulent structures. As shown in Fig. 7 a), while the dominant noise radiation direction is mostly less than 30º for the cold supersonic jet, the half-width of the auto-correlation function crossing zero is observed to be wider in these directions than in the other directions. When the jet is heated the dominant radiation direction is oriented further from the jet centerline to observer angles of 40º ~ 60º. As shown in Fig. 7 b), when the jet is heated, the spatial coherence increases (corresponding to the longer correlation time scale) due to the passage of large-scale structure noise, which is coherent over a longer time scale (see Tam et al. 26 ). The negative peak of the autocorrelation function appearing at 42º represents the maximum product of the positive and negative pressure signal which is close to the dominant noise radiation direction observed in Fig. 6. A similar result is obtained from the simulations as shown in Fig. 7 c). The half-width of the auto-correlation function becomes narrower as the observer angle increases. While the auto-correlation function indicates the spatial coherence at specific noise propagation directions, the cross-correlation function between two polar directions can provide additional information concerning the nature of the noise sources. It is argued that when a set of propagating acoustic wave fronts generated by an extended noise source reaches multiple measured positions simultaneously, higher cross-correlation values will be measured between these noise signals. It is therefore reasonable to expect a higher value of the cross-correlation function 8

9 a) b) c) Fig. 7 Auto-correlation function from measurements and simulations for a Md = Mj = 1.5 jet. a) Experimental data with TTR = 1.0. b) Experimental data with TTR = 3.0. c) Simulations with TTR = 3.0. when both measured signals are extracted within the dominant noise radiation direction. As shown in Fig. 8 a), while the dominant noise radiation direction is generally less than 30º for the cold supersonic jet, the maximum value of the cross-correlation reaches 0.75 with the reference microphone located at 25º. The value of the crosscorrelation rapidly drops below 0.15 with the reference microphone changing to a position at 47º. This characteristic changes when the jet is heated as shown in Fig. 8 b). The maximum value of the cross-correlation is still as high as 0.75 with the reference microphone located at 25º. When the reference microphone changes to a position at 47º, the maximum value of the cross-correlation still maintains a high value. This is consistent with the measurements shown in Tam et al. 26 and indicates that in the heated jet case the range of polar angles for which the large scale structure noise dominates is much higher in the heated jet case. Figure 9 shows the corresponding results from the simulations for the heated jet. Similar trends are observed as in the experimental observations. When the reference microphone changes to the position at 50º, the maximum value of the cross-correlation maintains a relatively high value. Now that nature of the auto/cross-correlations of the far-field pressure signal is established, the far-field pressure signal (effect) is correlated to the near-field fluctuation signal (cause). The flow probe position is located at x/d = 8 and y/d = 1.5. This position is located just outside the jet shear layer. The near-field fluctuation signal is recorded by the APD simultaneously with the far-field pressure signal acquired by microphones. Figure 10 shows the directivity of the peak cross-correlation between the near-field fluctuation signal and far-field pressure signal. In the unheated jet case the maximum cross correlation increases as the angle to the jet downstream axis decreases. It should be remembered that in the unheated case the peak noise direction is at approximately 30 o to the jet downstream axis. In addition, the increase in peak correlation at the small angles also suggests that even though the turbulence is convecting subsonically the coherent large scale structures are contributing to the noise in the peak noise direction. This is discussed further in the appendix. When the jet is heated, the maximum peak appears between 40º~60º as shown in Fig. 10. This is the radiation direction for Mach wave radiation in the heated jet case. 9

10 Ref. Mic at 25 deg Ref. Mic at 47 deg Cross-correlation Cross-correlation a) Mic polar angle (degree) Mic polar angle (degree) Ref. Mic at 25 deg Ref. Mic at 47 deg Cross-correlation Cross-correlation b) Mic polar angle (degree) Mic polar angle (degree) Fig. 8. Maximum cross-correlation function from measurements and simulations for a Md = Mj = 1.5 jet. a) Experimental data with TTR = 1.0. b) Experimental data with TTR = 3.0. Cross-correlation Ref. Mic at 30 deg Mic polar angle (degree) Identical trends are observed from the numerical simulations with relatively higher value of the cross-correlation than the one obtained from the OD experiments. The reason is that the APD sensor receives the signal from the light beam crossing the cylindrical jet plume. The positioning of APD sensor is on a two-dimensional image plane, but the received signal is a three-dimensional integration of the light beam path crossing the jet plume. This probably degrades the quantity of the fluctuation signal and contributes to a lower value of the cross-correlation. On the other hand, the results from the numerical simulations only extract the fluctuation levels from a compact volume (depended on the grid resolution) at x/d = 8 and y/d = 1.5. The higher correlation levels predicted in the simulations is also the result of the lack of decorrelation provided by the smallest turbulent scales, which are not captured in the simulations. Cross-correlation Ref. Mic at 50 deg Mic polar angle (degree) Fig. 9. Maximum cross-correlation function from measurements and simulations for a Md = Mj = 1.5 jet. Simulations with TTR =

11 Fig. 10. Directivity of the peak cross-correlation from the OD-mic measurements and numerical simulations for a Md = Mj = 1.5, TTR = 1.0 for left figure and TTR = 3.0 for right figure. (Probe located at x/d = 8, y/d = 1.5.) B. Near-field Cross-Correlations The first comparisons from both experimental data and numerical simulations shown here are the characteristics of the flow fluctuations along the jet lip line. Figure 11 a) shows autospectra of the Optical Deflectometer signal measured along the lip line of the jet and at different downstream locations x/ D 4 to 9. While these measurements are not enough to accurately measure the centroid of the noise production region for each frequency component, they do highlight the fact that the highest frequency components are strongest at small downstream distances and the lower frequencies dominate further downstream. The peaks of the autospectra shift from a) b) Fig. 11 Turbulent fluctuation spectra from the OD measurements and numerical simulations conducted at M j = 1.5, TTR = 1 along nozzle lip line. a) OD measurements. b) u predicted by numerical simulations. 11

12 approximately St 0.6 at x/ D 5to St 0.1or lower at x/ D 9. The amplitude also increases monotonically, as is expected in the annular mixing region of the jet. These measurements therefore conform to a description for the noise source distribution: the high frequency noise is mostly generated close to the nozzle exit and the low frequency noise is produced predominantly farther downstream. To further examine the details of the turbulent mixing layer development, Fig. 11b) shows the predicted power spectral density of u' (normalized by the fully expanded jet velocity U j ) at several locations along the lip line of the M j 1.5 unheated jet. A direct comparison with the OD is not appropriate because the OD is predominantly sensitive to cross stream density gradient (not the u' of the computations). Nevertheless comparisons of the spectral shapes are helpful to the analysis. All positions downstream of the axial location x/d 2.0 for the jet show similar energy spectra as measured in a typical turbulent mixing layer. This suggests that the mixing layer becomes fully turbulent downstream of x/d 2.0 and the self-similarity is correctly captured in both data sets. At upstream locations, a gradual increase of the intensity over the entire frequency range is seen and a strong peak appears at the frequency corresponding to St 0.3. The peak is identified as the Strouhal instability, corresponding to a particular frequency of the disturbance that is selectively amplified in the shear layer. Experimental measurements exhibit the similar behavior around the Strouhal number St In the high-speed jet noise laboratory at Penn State the OD measurements produce correlation or cross spectral analysis data with a second OD sensor or with a microphone signal. An example of such correlation data is shown in Fig. 12 for correlation plots shown for eight separation distances together with the auto correlation produced by two sensors at the same measurement position. These data were measured and computed for positions in the shear layer of an unheated Mj = 1.5 jet. The values of the time delays for each separation distance are used to produce an ensemble average estimate of the convection velocity of the jet turbulence for the center of the jet shear layer in the x/ D 4 to 5 and from x / D = 8 to 9 ranges of downstream distance. It can be seen that the convection velocity decreases with downstream distance. The numerical computations more closely match the measurements at the downstream distance except that a higher correlation is predicted. The measurements shown in Fig. 12 and similar experimental and computational data (at various x/d positions) have been assembled to show the bulk convection velocities for the Md = Mj = 1.5 jet (shown in Fig. 13). In the case of the Optical Deflectometer, only cold jet data are judged reliable for reasons explained by Doty 12 and by Veltin et al. 8. Assuming no such problem with the hot jet numerical simulations, the predictions from the hot jet computations are included on the graph. The agreement between the cold jet measurements and computations is quite reasonable and similar estimates from the computations of velocity fluctuations and density fluctuations produce very similar results. As noted earlier there are no OD measurements along the lip line with which to compare the computation data with for hot jets. The predictions show that the heated jet has a lower Uc/Uj than the unheated jet, as observed in some previous experimental measurements 29. However, other measurements suggest that the ratio of convection velocity to jet exit velocity is relatively independent of jet temperature (see Bridges 30 ). It has been shown 30,31 that in the jet downstream the convection velocity gradually increases to a maximum at the end of the potential core and decays monotonically after that. These trends are both observed from the experimental data and numerical simulations. However, the bulk convection velocity is slightly lower in the numerical simulations, especially in regions close to the nozzle exit. The predictions of the density and velocity fluctuation levels show similar trends. This supports the concept of OD technique that the measured density fluctuation level can reasonably correspond to the velocity fluctuation level in the jet shear layer. Since the normalized bulk convection velocity (normalized to the jet velocity at the nozzle exit) is given, this information helps to describe in which regions of the jet shear layer there could be supersonically convecting flow structures in the jet downstream direction that would cause Mach wave radiation. It is also enlightening to examine the estimates of phase velocity derived from cross spectral estimates of the respective time signals. Shown in Fig. 14 are the phase velocity estimates as a function of frequency (nondimensionalized as Strouhal number). These data are calculated from the phase of the cross spectral estimates for numerous probe separation distances. This information helps to describe that at specific measured position of the jet shear layer there could be supersonically convecting flow structure at certain frequencies along the jet downstream and cause the Mach wave radiation. It is observed that the low frequency components convect more slowly than the high frequency components. When the Strouhal number reaches around unity, the normalized 12

13 x/d = 4 x/d = 8 Fig. 12 Cross-correlation coefficient function from numerical calculations (based on u ) and OD measurements between one APD sensor at x/d = 4 to 8 and another APD sensor traversing downstream along the lip line for a Md = Mj = 1.5 cold jet. (Top row: experimental results, bottom row: calculations) convection velocity remains barely changed. There is a reasonable agreement between the experimental data and numerical simulations for the frequency-dependent phase velocity. Fig. 13 Comparison of the bulk convection velocity obtained from the OD measurements and numerical calculations for a Md = Mj = 1.5 jet. (Experimental data were conducted with cold jet, computations based on different flow quantities include hot and cold jets.) 13

14 a) 33b) Fig. 14 Frequency-dependent convection velocity from a) the OD measurements between one APD sensor at x/d = 4 and another APD sensor traversing downstream along the lip line. b) prediction from computation of density fluctuations (both data sets are for a Md = Mj = 1.5 cold jet). C. Near-field to Far-field Cross-correlation Now flow-field/near-field correlations with the acoustic far-field experiments are conducted with simultaneous OD and far-field microphone measurements. The strongest correlations were established with the OD sensor just outside the jet flow where the uncorrelated hydrodynamic density fluctuations do not decorrelate the signals from the OD-microphone measurements. Figure 15 shows an example of the cross correlations determined from the ODmicrophone signals. Fig. 15 a) shows results for the unheated, subsonically convecting jet (M a = 1.28) and 15 b) shows results for the heated, supersonically convecting jet (M a = 1.85). M a is the ratio of the jet exit velocity to the ambient speed of sound. a) TTR = 1, f c = 33.7 khz, mic at 27. b) TTR = 2.2, f c = 50 khz, mic at 45. Fig. 15 Cross correlation between the optical sensor (starting from x/d = 5 along y/d = 1.5 moving downstream) and the microphones (located at 27 with R/D = 50.5 and 45 with R/D = 45.5) from the measurements with the M d 1.5 CD nozzle (D = 0.5 in.) operating at M j

15 The fact that such a good correlation was obtained between the OD sensors and the far acoustic field led to an examination in more detail of the near field region just outside the turbulent flow field, in the vicinity of y / D = 1.5. A schlieren image of this region is shown in Fig. 16. Fig. 16 Schematic of the projected wavelength in the acoustic field and the flow-field wavelength in jets. Despite the significant difference in acoustic and convective Mach numbers in the two jets whose data are shown in Fig. 15, the character and magnitude of the correlations are quite similar. At the OD positions of y/d = 1.5 and x/d ranging from 5 to 7 it is expected that the density and pressure fluctuations are propagating at the ambient acoustic velocity predominantly in the maximum noise emission direction. It might be expected that a projected wavelength p (not a wavelength component) could be extracted from the differences in the delay times between the correlation peaks (of Fig. 15) with each separation of the OD sensors. In the Mach wave radiation model, this p would be approximately equal to the dominant flow fluctuation wavelength f. (All of these wavelengths are schematically depicted on the schlieren image recorded for the M a = 1.85 jet in Fig. 16.) An effort has initiated to make OD correlation measurements in this region just outside the flow field as visualized in Fig. 16. Figure 17 shows the traverse positions of the two OD sensors for the following series of experiments. Measurements in the cold jets proved to have too low signal to noise levels to provide the fidelity necessary, so shown here are data for a simulated hot jet (TTR = 3.0). Figure 18 shows the cross correlation plots for two of the 4 traverses depicted in Fig. 17. The OD traverses were oriented at 0 o and 10 o for these data. Figure 19 shows the data from the remaining two OD traverses depicted in Fig. 17, corresponding to traverses at 20 o and 50 o. The correlation plots are analyzed in the same fashion as are those that are measured along the lip line of the jet resulting in convection velocity estimates in the direction of probe separation. Fig. 17. Schematic of APD sensor positions for each configuration. 15

16 a) y/d = 2 along 0º line. b) y/d = 1.75 along 10º line. Fig. 18. Cross-correlation coefficient function from the OD measurements between one APD sensor at x/d = 8, y/d = 1.75 to 2.0 and another APD sensor traversing away along 0º and 10º lines for a Md = Mj = 1.5, TTR = 3.0 jet. The approach taken is to estimate the frequency content of the most energetic fluctuations. It is difficult to establish this with any accuracy so estimates are made from the far field acoustic spectra, since the ultimate interest is the generated noise. It is noted that for the TTR = 3.0 jet the far field noise peaks between Strouhal numbers of 0.2 and 0.3. Using these two non-dimensional frequencies, the pseudo convection velocities are converted to wavelengths using the standard formula: = Uc / f, or in non-dimensional form D = (Uc / Uj) / St. The four convection velocity / traverse direction combinations convert to wavelengths that can then be plotted as shown in Fig. 20. Note that the origin of these wavelength components is at x / D = 0 and y / D = 1.5, where for convenience the real origin of the traverses is at x / D = 8. The final action with this processing is to plot a line indication the Mach angle of the wave field that will produce a peak in the far field noise at a polar angle of 50 degrees from the axis. This is the maximum noise radiation direction for Mj = 1.5 jets with temperature ratios of TTR = 3.0. The excellent agreement of this idealized Mach wave suggests that the interpreted turbulence convection velocity, together with the wave fronts interpreted from the near field OD measurements reinforce the current understanding of the process of Mach a) y/d = 1.5 along a 20º line. b) y/d = 1.5 along a 50º line. Fig. 19. Cross-correlation coefficient function from the OD measurements between one APD sensor at x/d = 8, y/d = 1.5 and another APD sensor traversing away along 20º and 50º lines for a Md = Mj = 1.5, TTR = 3.0 jet. 16

17 St = 0.2 St = deg Mach Wave 2.50 y / D Fig. 20 Plot of components of the wavelengths measured for the two Strouhal number spectral components in their measurement directions. wave radiation from convectively supersonic, supersonic jets. A discussion of the differences between subsonic and supersonic convection and radiating and evanescent waves is given in the appendix. Efforts to construct similar wave fronts from similar measurements made in the cold convectively subsonic, supersonic jets did not produce data whose projected acoustic field wave fronts matched the wavelengths of the flow field turbulence. The noise generation process in these jets is considerably more complex. In such jets, because of the growth and decay of the turbulent structures, the actual wavenumber content does not reside solely at the wavenumber corresponding to the convection velocity but is quite broad. This enables disturbances that are convecting subsonically to contain radiating components. This is discussed further in the appendix. In further processing of near field to far field data (such as that shown in Fig. 15), the peak values of each crosscorrelation are extracted and peak cross-correlation contours are plotted in Fig. 21. This shows the peak crosscorrelation contours for the cold supersonic jet, M j = 1.47 (M a = 1.28). As observed in the noise directivity, the dominant noise emission angle is typically less than 30 for these cold supersonic jets. The intensity of the peak cross-correlation contour is therefore strongest for a microphone polar angle of 20 to 30. The peak crosscorrelation values for polar angles above 30 are negligibly small. This meets the expectation that the peak crosscorrelation levels will also correspond to a microphone position of between 20 and 30 degrees to the jet axis (in the far-field). The region of highest correlation occurs just outside the jet lip-line, presumably outside the jet flow itself and closely aligned with the axial location of the maximum noise generation region. As shown in Fig. 21 x / D Fig. 21 Peak cross-correlation contour from the OD measurements and numerical calculations (based on density) between the probe traversing at different x and y locations around a Mj = 1.47 cold jet and far-field microphones (R/D ~ 55) at 30º. 17

18 from both the experimental data and numerical simulations, the dominant noise radiation regions move outward away from the jet centerline. The movement of the highly correlated region to a slightly larger radius is likely associated with the spreading of the jet shear layer. It is noted that the numerical simulations produce data, based on the density property, that are in good agreement with the experiment. D. Statistics of the Near-field Flows The optical deflectometer senses fluctuations in the density gradients. The OD also includes the effects of integration along the senor beam as it traverses the unsteady jet flow. It has been conjectured that the density gradient fluctuations would be correlated with the radial velocity fluctuations in an axisymmetric jet. To provide some evidence to support this conjecture, the numerical simulations were used to find correlations between the radial density gradients and the turbulent velocity fluctuations. Figures 22 and 23 show predicted correlation coefficients between the radial velocity fluctuations and the radial density gradients. Figure 22 shows contours of cross correlation coefficient and Fig. 23 shows radial profiles at selected axial locations. Results for both the unheated and heated jets are shown. No clear trend is visible in the radial traverses though the contours do show limited regions of moderate correlation. TTR=1.0 TTR=3.0 Fig. 22. Contours of cross correlation coefficient between radial velocity and radial density gradients. TTR=1.0 TTR=3.0 Fig. 23. Radial traverses showing cross correlation coefficient between radial velocity and radial density gradients 18

19 From the linearized momentum equation it is expected that the radial pressure derivative would be related to the time derivative of the radial velocity fluctuation. In turn, in an acoustic field the pressure and density derivatives are proportional. Thus it would be expected that a strong correlation would exist between the radial density gradient and the time derivative of radial velocity fluctuation. Figures 24 and 25 show predicted correlation coefficients between the time derivative of the radial velocity fluctuations and the radial density gradients. Figure 24 shows contours of cross correlation coefficient and Fig. 25 shows radial profiles at selected axial locations. In this case there is a strong correlation of -1.0 at all locations outside the jet shear layer. This is consistent with the density gradient being out of phase with the time derivative of the radial velocity (as indicated in the linearized momentum equation). However, there is little correlation in the turbulent mixing region. These results indicate that the relationship between the flow fluctuation measured by the OD and turbulent velocity fluctuations in the jet shear layer is not as previously conjectured and such a relationship remains to be established. TTR=1.0 TTR=3.0 Fig. 24. Contours of cross correlation coefficient between time derivative of the radial velocity and radial density gradients. TTR=1.0 TTR=3.0 Fig. 25. Radial traverses showing cross correlation coefficient between time derivative of radial velocity and radial density gradients 19

20 V. Discussion and Conclusions The present study describes near/far field correlations for supersonic jets with the use of both numerical simulations and experimental measurements. The close partnership between the model scale experiments and numerical simulations for jets operating at identical jet conditions and geometries provides a strong assessment of the numerical simulations as well as offering improved opportunities for understanding the flow physics. The experiments provide physical measurements and the numerical simulations provide more in-depth data not available to measurements. Both approaches have their strengths and in combination they provide a powerful tool. There are many features of the experimental measurements that are duplicated well by the numerical simulations. The auto spectra in the far field are predicted quite well, especially in the region of peak noise radiation. There is some over-prediction of the noise levels at some frequencies and the potential reasons for these differences are discussed. It is noted that there is good agreement at all frequencies and in all directions between the small scale experiments conducted at Penn State and the moderate scale model experiments conducted at NASA Glenn Research Center. In the unheated jet case the predicted spectra of velocity fluctuations on the jet lip line agree well with the experimental data produced by the Optical Deflectometer. There are some differences in the initial stages of the jet shear layer s development. This is believed to be the cause of some of the differences between the predicted and measured far field noise spectra. The numerical predictions have been used to help understand the relationship between the OD measurements and the turbulent velocity fluctuations. No simple relationship was identified, though a clear correlation was established between the density gradient fluctuations and the time derivative of the radial velocity fluctuations in the acoustic near field. Polar cross correlations of the far field noise are consistent between the measurements and predictions and are also in agreement with published observations. Near field cross correlations in the unheated jet show agreement between experiments and simulations for the bulk convection velocities as well as the frequency dependent phase velocity. The predictions suggest that the convection speed decreases with jet heating. This trend has been observed in some previous experiments, but remains an open question. An analysis of two point cross correlations in the jet near field shows that the measurements are compatible with Mach wave radiation in the heated jet case. However, such a simple analysis is not successful in the unheated jet case. In the appendix, an analysis of the noise field generated by a simple wavepacket is used to explain the differences. The goal of the present paper has been to demonstrate how the coupling of experiments and numerical simulations provides a powerful tool for the analysis of noise generation in high speed jets. It is planned to continue this approach with improvements to the quality of the numerical simulations and as additional measurement techniques are brought to bear. Appendix: Wavepackets with Subsonic and Supersonic Phase Velocities In the main body of the paper the wavelengths of the near field fluctuations were estimated based on the convection velocities measured at different angles. Differences were observed between the convectively subsonic and supersonic cases. In this appendix a simple example is given that helps to explain these observations. It is shown how the wave fields differ in the subsonic and supersonic cases by separation of the radiating and non-radiating components of the wavepacket. For simplicity a two-dimensional example has been considered, but the same approach would apply and similar conclusions would be drawn from a three-dimensional analysis. Consider a pressure fluctuation on y 0 given by, p x,0, t A ( x )exp i x t, (1) where the variables have been non-dimensionalized with respect to a reference density and length scale and the constant speed of sound for y 0. The pressured is assumed to satisfy the two-dimensional wave equation and a p xyt,, pxy ˆ, exp i t pˆ xy, satisfies the equation, periodic solution of the form is sought. Then 20

21 pˆ x pˆ y p 2 2 ˆ 0. (2) Now, introduce the Fourier transform with respect to x and its inverse, ˆ P s, y p x, y exp isx dx, (3) 1 pˆ x, y P s, y exp isx ds. 2 (4) It is then readily shown that p 1 ˆ x, y,0 exp, 2 P s i y sx ds (5) where, 2 2 s with Arg 0 /2, and P s,0 A( x)exp i s x dx. (6) To determine the far field radiation polar coordinates x Rcos, y Rsin can be introduced and the method of stationary phase used to evaluate the inverse Fourier transform. This gives, lim pˆ x, y P cos,0 sin exp i R / 4. R 2 R (7) A simple wavepacket model that has been used before has 2 2 A( x) exp x / b (8) Then, 2 2 cos b pˆ x, y b sin exp exp i R /4. 2R 4 (9) Note that the directivity is controlled by two factors - sin and the exponential term. The former always weights the solution. The maximum of the latter term depends on whether / is greater than or less than unity. That is, whether the wavepacket phase velocity is supersonic or subsonic, respectively. If it is assumed that the convection velocity of the turbulence in the jet shear layer is given byuc 0.70U j, then the corresponding wavepacket phase velocities are given by / 0.87and / 1.5for the unheated and heated jets respectively. The values of and b are set to 2 and 1.0 respectively. The wavenumber spectra for the subsonic and supersonic cases are shown in Fig. A1. The radiating components of the wavenumber spectrum are given by s. 1 o In the supersonic case the peak in the far field will occur close to cos / 48. In the subsonic case, the directivity is affected more strongly by the sin weighting. In the near field the wavepacket consists of radiating and evanescent waves. Figure A2 shows the instantaneous total, radiating and evanescent wave fields for the supersonic case. The radiating and non-radiating components were obtained by zeroing the integrand in Eqn. (5) for s and s, respectively. In this case the sound is beamed to approximately 48 o as expected. The radiating and total fields are almost identical, even for small values of y. The non-radiating field is an order of magnitude smaller than the radiating field (note the change of contour levels in the non-radiating case). If the wavelengths are estimated at different angles to the downstream axis they all reflect the trace of an acoustic wave beaming to the preferred radiation direction. This result is consistent with the schlieren images in excited high speed jets by Kearney-Fisher et al

22 Fig. A1. Wavenumber transform of wavepacket pressure variations at y=0. a) b) c) Fig. A2. Instantaneous pressure for, a) total, b) radiating, and c) non-radiating, fields for wavepacket with supersonic phase velocity Figure A3 shows the same fields in the subsonic phase velocity case. In this case the different components of the total field have very different characteristics. In the very near field the radiating and non-radiating components are of the same order of magnitude. This gives the effect of curvature to the wave fronts as the subsonically convecting non-radiating field transitions into the radiating field. Again, this is consistent with the schlieren images of excited jets by Kearney-Fischer et al. 32 In this case, if the wavelength is estimated in the xdirection for small values of y it is consistent with the wavelength of the fluctuations in the subsonically convecting wavepacket. However, if the wavelength is estimated in the direction of peak sound radiation the wavelength is longer and consistent with an acoustic wave. This result helps to explain some of the cross correlation measurements described in the main body of the paper. 22

23 a) b) c) Fig. A3. Instantaneous pressure for, a) total, b) radiating, and c) non-radiating, fields for wavepacket with subsonic phase velocity. Acknowledgments This research was supported in part by the Strategic Environmental Research and Development Program under task WP-153. Additional funding has been provided under a NAVAIR Propulsion & Power SBIR/STTR with Innovative Technologies and Applications supported by the US Navy. References 1 Siddon, T. E., Noise source diagnostics using causality correlations, AGARD CP 131, Noise Mechanisms, 71-1:7-13, Seiner, J. M. and Reethof, H., On the distribution of source coherency in subsonic jets, AIAA Paper No. 74-4, Richarz, W. G., Direct correlation of noise and flow of a jet using laser Doppler, AIAA Journal, Vol. 18, No. 7, 1980, pp Schaffar, M., Direct measurements of the correlation between axial in-jet velocity fluctuations and far field noise near the axis of a cold jet, Journal of Sound and Vibration, Vol. 64, No. 1, 1979, pp Panda, J. and Seasholtz, R. G., Experimental investigation of density fluctuations in high-speed jets and correlation with generated noise, Journal of Fluid Mechanics, Vol. 450, 2002, pp Panda, J., Identification of noise sources in high speed jets via correlation measurements-a review, AIAA Paper ,