Investigating Broadband Shock-Associated Noise of

AIAA 2006-2495

12th AIAA/CEAS Aeroacoustics Conference (27th AIAA Aeroacoustics Conference) 8 - 10 May 2006, Cambridge, Massachusetts

12th AIAA/CEAS Aeroacoustics Conference 08–10 May 2006, Cambridge, MA

Investigating Broadband Shock-Associated Noise of Axisymmetric Jets Using Large-Eddy Simulation Daniel J. Bodony∗ Center for Turbulence Research Stanford University, Stanford, CA 94305-3035 Jaiyoung Ryu† Department of Mechanical Engineering Stanford University, Stanford, CA 94305-4035 Sanjiva K. Lele‡ Departments of Aeronautics and Astronautics and Mechanical Engineering Stanford University, Stanford, CA 94305-4035

A supersonic jet operated at an off-design condition supports a standing wave pattern of alternating compression and expansion waves, often called shock-cells. The interaction of turbulence with the shock-cells leads to an additional source of broadband noise that is preferentially radiated upstream, i.e. towards the engine. In commercial aircraft this condition occurs in the fan stream at the cruise-climb condition and adversely impacts passenger comfort in the forward cabin section. In military aircraft the extra acoustic loading on the aft structures leads to reduced life cycle times. An unheated jet with a design Mach number of 1.95 is studied at onand off-design conditions; for the latter case the underexpanded jet has a fully expanded Mach number of 2.2. The near-fields of the jets are studied with respect to their mean, fluctuation, and spectral characterizations. It is observed that the shear layers of the off-design jet merge one diameter downstream of the on-design jet with a slight reduction in the peak level of velocity fluctuation. The spectral content of the velocity and pressure field, inferred from axial two-point measurements, of both jets is similar. The underexpanded jet differs, however, in that it supports Mach wave radiation stemming from supersonically-convecting instability waves. It is observed that the Mach waves significantly contribute to the near-field pressure.

Nomenclature R, Θ, Φ Re St Dj Ma Mj r, θ, x U, V, W u, v, w

Spherical coordinates Jet Reynolds number = ρU j D j /µ j Strouhal number = f D j /U j Jet diameter = 2ro Jet acoustic Mach number = U j /a∞ Jet Mach number = U j /a j Cylindrical coordinates Mean velocity components Fluctuating velocity components

Subscripts ∞ rms c j

Ambient condition root-mean-square centerline Jet exit condition

Superscripts h·i b (·)

Time and/or azimuthal-average Transformed variable

∗ AIAA

member. [email protected] student member. [email protected] ‡ AIAA member. [email protected] c 2006 by D. J. Bodony, J. Ryu, and S. K. Lele. Published by the American Institute of Aeronautics and Astronautics, Inc. with Copyright permission. † AIAA

1 of 11 American of Aeronautics and Paper 2006-2495Inc., with permission. Copyright © 2006 by D. J. Bodony, J. Ryu, and S. K. Lele. PublishedInstitute by the American Institute of Astronautics Aeronautics and Astronautics,

I. Introduction en a nozzle is designed to produce a supersonic jet a set of target conditions are given. If the nozzle is operated W at conditions other than those for which it was designed a standing wave will develop within the jet stream due to the pressure mismatch at the nozzle exit. Characteristics of the standing waves, also called shock-cells, depend on the extent of the pressure difference and whether the nozzle exit pressure is below the back pressure (an overexpanded jet) or above the back pressure (an underexpanded jet). As most jets are turbulent the interaction of the turbulence with the standing wave pattern leads to an additional source of noise termed ‘shock-associated noise.’ The additional noise can occur in commercial aircraft in the climb-cruise maneuver when the fan stream becomes supersonic, subsequently annoying passengers sitting in the forward sections. In the case of military aircraft operating at high-thrust conditions shock noise may lead to structural fatigue in the aft structure, shortening the operational life of the vehicle. Shock-associated noise has been characterized experimentally 1–6 and theoretically by Tam and coworkers7–10 and more recently by Ray & Lele.11 A summary of shock-associated noise is given by Tam.12 From the experimental studies it is known that the broadband shock associated noise is preferentially directed upstream, towards the jet nozzle, resulting in a measurable increase in the overall sound pressure levels (OASPLs) for cold jets. For angles closer to the downstream jet axis the shock associated noise contributes less to the OASPL, but is visible in the acoustic spectra. 2 Historically prediction of shock-associated noise has been semi-empirical. For a given operating condition the standing wave pattern can be estimated using the Prandtl-Pack13 solution which also identified the importance of the √ 2 parameter β = M j − 1 where M j is the fully expanded jet Mach number. The multiple-scale interpretation of the shock-cell structure given by Tam et al.8 led to a better description of the standing wave pattern, including the distance between successive cells and the amplitude of the pressure variation. Harper-Bourne & Fisher1 used experimental data in conjunction with modeling to develop a shock-cell noise model based on a phased array. By observing that the shock-cell noise was primarily omni-directional they proposed that it could be modeled as a series of noise sources located along the lip-line, representing the tips of the shocks, radiating with a phase relationship associated with the shock-cell spacing and the convection speed of the turbulence. Their model represents well the directivity and fundamental frequencies of the shock-associated noise. Tam & Tanna7 reinterpreted the problem of shock-associated noise and considered the problem to be one of coherent ‘large-scale’ disturbances interacting with the quasi-periodic standing waves as described by Pack. 13 They assume that instability waves interacting with the shock cells gives rise to the shock-associated noise. Their model leads to a very similar peak frequency prediction to the Harper-Bourne & Fisher model with appropriate choice of axial lengthscale. Tam9, 10 later improved on this model by introducing a stochastic element to represent the shear layer turbulence and by replacing the Prandtl-Pack shock-cell solution with his earlier multiple scales solution. 7 By considering the interaction between the instability waves and the shock cells to first order Tam was able to give an matched asymptotic expansion solution to the shock noise problem but did not pursue a detailed solution in his original paper. Instead he opted for a semi-empirical model to describe the near field pressure and thus the associated far-field radiation. Ray & Lele11 have recently revisited Tam’s formulation in it’s original form. The introduction of large-eddy simulation (LES) as a viable computational technique for jet noise 14 has permitted the reinvestigation of shock-associated noise in a manner not previously available. In particular the fully nonlinear governing equations are solved and the shock-associated noise is captured as an inherent portion of the solution. Availability of the time-dependent, three-dimensional flow field then permits examination of the causal relationship between the instability waves, the turbulence, the shock-cells, and the radiated sound. Such data permits the aforementioned semi-empirical models to be evaluated and additional insight into the sound generation process. Moreover the reduced cost of LES relative to direct numerical simulation (DNS) allows for a parameter study of jet operating conditions and the corresponding changes in the shock-associated noise. In the DNS context Manning & Lele15 and Suzuki & Lele16 investigated shock-associated noise as applicable to jet screech. They found, for their two-dimensional mixing layers, that the generation of sound can be interpreted as a ‘shock leakage’ process. Lui,17 in a DNS of quasi-laminar and turbulent mixing layers (three-dimensional calculations of mixing layers that are two-dimensional in the mean) interacting with a single shock tip, found a linear scaling between the acoustic amplitude and the near-field pressure fluctuation at the shock tip with a roughly omni-directional amplitude. From visualization Lui observed an apparent sound origin slightly downstream of the shock tip due to convection of the sound by the mean flow. Using LES Shur et al.18 simulated two pressure mismatched jets using the MILES approach with numerical shock capturing. Their simulations captured the increased sound due to the presence of the shock cells. At most observer locations the peak frequency of the far-field sound was captured with very good agreement in the upstream direction. For the downstream direction there was a modest underprediction of the OASPL and a 10 degree difference in the peak

2 of 11 American Institute of Aeronautics and Astronautics Paper 2006-2495

directivity for the cold jet. When a heated jet was considered the peak predicted OASPL was better captured but the overall directivity was shifted towards the upstream angles. The remainder of this article discusses the study of shock-associated noise of an underexpanded jet using LES. Details of the simulations are presented prior to near-field comparisons between the pressure matched and mismatched jets. This is followed by conclusions.

II. Details of the large-eddy simulation A subset of the simulation details most relevant to the prediction of shock-cell noise are given here; a more comprehensive description may be found in Bodony & Lele.19 The simulations solve the filtered compressible equations of motion in cylindrical coordinates with a dynamic subgrid scale model 20 for closure. An optimized compact finite difference scheme is used in the radial and axial directions and Fourier-spectral differentiation is used in the azimuthal direction. The low-dispersion, low-dissipation Runge-Kutta scheme 21 is used for time advancement. No shock capturing schemes are used so that only jets with moderate pressure mismatches are considered. At the lateral and outflow boundaries zonal treatments22 are used which add a forcing term −σ(x)(q − qref ) to the right-hand-side of the equations of motion where σ is a positive semi-definite strength parameter that drives the solution q towards the reference solution qref . Determination of the reference solution utilizes a v2 - f Reynolds averaged Navier-Stokes solution for a jet at the same conditions. At the inflow boundary a hyperbolic tangent velocity profile of the form u/U j = (1 − tanh {(r/r0 − r0 /r)/(4θ0 /r0 )}) /2

(1)

is prescribed with maximum velocity U j and momentum thickness θ0 . From this condition the inflow density is derived from the Crocco-Busemann relation assuming constant static pressure equal to the ambient pressure. To the inflow mean profile unstable solutions to the compressible Rayleigh equation associated with the mean field are added to promote turbulence development. Frequencies in the range of 0.2 ≤ St ≤ 0.6 with azimuthal modes of n = ±1, . . . , ±4 are used with a temporal random walk of their phases to destroy periodicity. For the case of the underexpanded jet the inflow reference solution q ref is described in more detail. Using Eq. (1) a fully expanded mean flow is constructed with the specified nozzle pressure ratio. About the fully expanded mean flow the instability waves are computed as above. To introduce shock-cells the multiple scales solution of the shock structure given by Tam et al.8 is used following the method outlined by Ray & Lele.11 The desired pressure mismatch is used to determine the strength of the shock-cell structure. Only the first three modes of the multiple scales solution are used. All three solutions are combined linearly so that qref = qmean + qinstab + qshock . Their sum is then fed into the LES using a sponge region at the inflow boundary.

III. Presentation of the data Two jets are considered for the remainder of the paper; their properties are given in table 1. Both jets are unheated, i.e. the jet stagnation temperature is the same as the ambient temperature, with a design Mach number of 1.95. The pressure matched jet is the same one described in detail in Ref. 19 and therein labeled M15TR056. The off-design jet has the same design Mach number but is slightly underexpanded with a fully expanded Mach number of 2.2. This latter jet corresponds to an operating point measured by Norum & Seiner 6 (hereafter NS). The jets have the same inlet momentum thickness and forcing algorithm, the same solution algorithm, and have been post-processed in identical manners. Differences between the two solutions are thus due to the presence of the standing wave pattern. A. Near-field Instantaneous snap-shots of the vorticity magnitude and of the dilatation are shown in figure 1 for the pressure matched jet and in figure 2 for the pressure mismatched jet. The presence of the shock cells in the latter image is apparent as is the increased sound radiation in the upstream direction. The measured NS and LES-predicted centerline pressure distribution for the off-design jet is shown in figure 3. The target pressure enforced in the LES is determined by the multiple scales solution as discussed in the previous section and is forced through the inlet sponge region of 0 ≤ x/r0 ≤ 5; beyond x/r0 = 5 there is no sponge forcing. The LES solution exhibits adjustment of the pressure away 3 of 11 American Institute of Aeronautics and Astronautics Paper 2006-2495

from the NS data beyond the first two diameters and evolves with a slightly different shock-cell wavelength, with the two being out of phase by x/r0 = 20. The LES pressure amplitude is also reduced relative to the NS data. Along the jet centerline the addition of the shock-cells is clearly seen in the centerline mean density as shown in figure 4. The underexpanded jet has a 1D j longer potential core (see figure 5). It is observed that the presence of the shock cells on the mean and fluctuating fields is apparent approximately 6D j downstream of the potential core collapse. In the axial velocity root-mean-square of the underexpanded jet in figure 6 the peak level drops by 6% with the addition of the shock-cells with a broadening of the axial extent over which the fluctuations are appreciable. a Spectra corresponding to two-point correlations for the axial velocity are presented in figure 7 along the jet centerline and in figure 8 along the nominal lip line of r/r0 = 1. Lip-line pressure spectra are shown in figure 9. From the velocity spectra in figures 7 and 8 it is observed that apart from a drop in the overall axial velocity fluctuation amplitude in the underexpanded jet the spectral distributions are similar. From the pressure spectra in figure 9 one finds that the pressure matched and mismatched jets are qualitatively the same though with a change in the peak response at kr0 = 1.3 for the on-design case and kr0 = 1.0 in the off-design. The change in the wavenumber of the peak is consistent with the change in the wavelength of the standing wave patterns between the two jets. (Note that the on-design case exhibits a weak shock-cell pattern most easily visible in the centerline mean density of figure 4.) Observe that standing wave spacing in the on-design jet is approximately 2D j while that of the underexpanded jet is 2.5D j . It is also seen that the broadness of the spectrum for the underexpanded jet is narrower around the peak than what is found for the pressure-matched jet. The spectra for the underexpanded jet also exhibit a notable peak wavenumber for all axial locations considered while in the pressure matched jet the peak disappears sufficiently far downstream. B. Mach wave radiation and the near-field pressure Visible in the off-design jet shown in figure 2 in the dilatation field contours are a train of roughly parallel lines at an angle of 110 degrees from the downstream axis. Such structure is not visible in the on-design jet of figure 1. The cross-correlation of the pressure field, taken at the point (r, x) = (23, 17)r 0, and the axial velocity taken at all x along the jet lip-line, i.e. the correlation hp0 (X, t)u0 (Y, t + τ)i with X = (23, 17)r0 and Y = (x, 1)r0 , is shown in figure 10. The spectra of the temporal cross correlation for the fixed points x ∈ {15, 20, 25, 30}r0 are also shown. Clearly a regular pattern is observed with a well defined set of frequencies. The convection velocity implied in the two-point space-time correlation is U c /a∞ = 1.07 ± 0.02 which is consistent with the expected phase speeds of the St = 0.2, 0.27 and 0.31 instability waves (see figure 11) as observed in the temporal spectrum. These frequencies correspond directly to the center frequencies forced into the calculations. Around x/r0 = 30 it appears that the u-p correlation is reduced. It also appears that the Mach wave radiation is antisymmetric (±1 mode) circumferentially (recall that the axisymmetric mode is not forced in these calculations). Measurement of the Mach angle of (113 ± 2)◦ implies a phase velocity of 1.09a∞ which is supersonic relative to the ambient speed of sound and in agreement with the estimate from the cross-correlation of figure 10. Although the pressure-matched jet supports an axisymmetric mode capable of supersonic phase velocity it is not visible in the data. In figure 12 the pressure distribution on a cylindrical surface parallel to the jet axis and placed 5D j from the centerline is given. Apparent in both is the presence of upstream and downstream propagating sound. (The slopes of the isocontours are < a∞ in the downstream direction and < −a∞ in the upstream direction.) The downstreampropagating radiation is evidently stronger than the upstream-propagating radiation as would be expected from farfield measurements. It is also noted that the upstream radiation is stronger for the off-design jet than for the on-design jet, consistent with the presence of shock-associated noise. Around x/r 0 = 27 for both jets one finds a combination of upstream and downstream traveling sound. That the isobars exhibit a slope less than a∞ in magnitude implies that the sound is traveling obliquely to the surface. At locations near x/r0 = 27 there are visible instances of zero-slope isobars indicating sound radiating normal to the jet axis. At the extreme axial ends of the pressure surfaces one finds an increased slope of value nearing a ∞ , implying the sound is predominately traveling parallel to the measurement surface. In figure 13 the azimuthal correlations of the pressure fluctuations on the same cylindrical measurement surface located at r/D j = 5 at various axial locations are given. Nearer to the inlet the on-design jet exhibits a decorrelation angle roughly 60◦ with little antisymmetry. The underexpanded jet shows a similar decorrelation angle but with a significant antisymmetric component, indicating the pressure field is dominated by the antisymmetric Mach waves discussed previously. Farther downstream the influence of the Mach waves reduces and the azimuthal correlations a In the review by Bodony & Lele14 it was noted that the imposition of instability waves in calculations of turbulent jets led to a reduced axial extent of urms . Similar conclusions are likely to hold for simulations of underexpanded jets using instability wave forcing.


between the two jets become similar. That the correlation does not cross zero at the two locations farthest downstream indicates an axisymmetric pressure field has been created.

IV. Conclusions Using large-eddy simulation two supersonic jets with a design Mach number of 1.95 were investigated when operated at different pressure ratios. One jet was perfectly matched while the other was underexpanded with a fully expanded Mach number of 2.2. These conditions correspond to jets for which measurements were taken by Norum & Seiner.6 It was observed that the off-design jet had a longer potential core with modestly (6%) reduced levels of axial velocity fluctuations. One-dimensional spectra of the axial velocity and of the pressure were qualitatively similar, aside from a change in the overall levels, with the exception of the underexpanded jet exhibiting a narrower spectral peak at a smaller wavenumber than was found for the pressure-matched jet. The change in the peak wavenumber was consistent with the change in the shock-cell spacing between the two jets. The near pressure field for the off-design jet was dominated by Mach wave radiation in the upstream portions of the flow field but eventually relaxed to a state more similar to the on-design jet. Similarly the azimuthal correlation of the near pressure field was predominately antisymmetric for the underexpanded jet in the upstream region, relaxing towards the correlation of the pressurematched jet downstream.

V. Acknowledgments Support from the Aeroacoustics Research Consortium is gratefully acknowledged. Computational support for the simulations was provided by the U.S. Department of Defense.

References 1 Harper-bourne, M. and Fisher, M. J., “The Noise from Shock Waves in Supersonic Jets,” Proceedings No. 131 of the AGARD Conference on Noise Mechanisms, Brussels, Belgium, AGARD, 1973, pp. 11.1–11.13. 2 Tanna, H. K., “An Experimental Study of Jet Noise Part II: Shock Associated Noise,” J. Sound Vib., Vol. 50, No. 3, 1977, pp. 429–444. 3 Seiner, J. M. and Norum, T. D., “Experiments of shock associated noise on supersonic jets,” AIAA Paper 79-1526, 1979. 4 Seiner, J. M. and Norum, T. D., “Aerodynamic aspects of shock containing jet plumes,” AIAA Paper 80-0965, 1980. 5 Seiner, J. M. and Yu, J. E., “Acoustic near field and local flow properties associated with broadband shock noise,” AIAA Paper 81-1975, 1981. 6 Norum, T. D. and Seiner, J. M., “Measurements of Mean Static Pressure and Far-Field Acoustics of Shock-Containing Supersonic Jets,” Tech. Rep. NASA TM-84521, NASA, 1982. 7 Tam, C. K. W. and Tanna, H. K., “Shock Associated Noise of Supersonic Jets from Convergent-Divergent Nozzles,” J. Sound Vib., Vol. 81, No. 3, 1982, pp. 337–358. 8 Tam, C. K. W., Jackson, J. A., and Seiner, J. M., “A Multiple-Scales Model of the Shock-Cell Structure of Imperfectly Expanded Supersonic Jets,” J. Fluid Mech., Vol. 153, 1985, pp. 123–149. 9 Tam, C. K. W., “Stochastic Model Theory of Broadband Shock Associated Noise from Supersonic Jets,” J. Sound Vib., Vol. 116, No. 2, 1987, pp. 265–302. 10 Tam, C. K. W., “Broadband Shock-Associated Noise of Moderately Imperfectly Expanded Supersonic Jets,” J. Sound Vib., Vol. 140, No. 1, 1990, pp. 55–71. 11 Ray, P. K. and Lele, S. K., “On sound generated by instability wave/shock cell interaction in supersonic jets,” AIAA Paper 2006-0620, Presented at the 44th Aerospace Sciences Meeting and Exhibit, Reno, NV, 9–12 Jan, 2006. 12 Tam, C. K. W., “Jet Noise Generated by Large-Scale Coherent Motion,” Aeroacoustics of Flight Vehicles: Theory and Practice. Volume I: Noise Sources, NASA Langley Research Center, Hampton, Virginia, 1995. 13 Pack, D. C., “A Note on Prandtl’s Formula for the Wave-Length of a Supersonic Gas Jet,” Quart. Journ. Mech. and Applied Math., Vol. 3, No. 2, 1950, pp. 173–181. 14 Bodony, D. J. and Lele, S. K., “Review of the current status of jet noise predictions using large-eddy simulation (invited).” AIAA Paper 2006-0468, Presented at the 44th Aerospace Sciences Meeting and Exhibit, Reno, NV, 2006. 15 Manning, T. A. and Lele, S. K., “A numerical investigation of sound generation in supersonic jet screech,” AIAA/CEAS Paper 2000-2081, 2000. 16 Suzuki, T. and Lele, S. K., “Shock leakage through an unsteady vortex-laden mixing layer: application to jet screech,” J. Fluid Mech., Vol. 490, 2003, pp. 139–167. 17 Lui, C. C. M., A Numerical Investigation of Shock-Associated Noise, Ph.D. thesis, Stanford University, Palo Alto, California 94305, Sept. 2003. 18 Shur, M. L., Spalart, P. R., Strelets, M. K., and Garbaruk, A. V., “Further Steps in LES-Based Noise Prediction for Complex Jets,” AIAA Paper 2006-0485, Presented at the 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, 9–12 Jan, 2006. 19 Bodony, D. J. and Lele, S. K., “On Using Large-Eddy Simulation for the Prediction of Noise from Cold and Heated Turbulent Jets,” Phys. Fluids, Vol. 17, No. 085103, 2005.


20 Germano, M., Piomelli, U., Moin, P., and Cabot, W. H., “A dynamic subgrid-scale eddy viscosity model,” Phys. Fluids A, Vol. 3, No. 7, 1991, pp. 1760–1765. 21 Hu, F. Q., Hussaini, M. Y., and Manthey, J. L., “Low-dissipation and low-dispersion Runge-Kutta schemes for computational acoustics,” J. Comp. Phys., Vol. 124, 1996, pp. 177–191. 22 Bodony, D. J., “Analysis of Sponge Zones for Computational Fluid Mechanics,” J. Comp. Phys., Vol. 212, 2006, pp. 681–702.

Md Mj Re θ0 0 u /U j at inlet Nr × N θ × N x

on-design 1.95 1.95 336,000 0.045D j 0.02 128 × 32 × 256

off-design 1.95 2.20 394,000 0.045D j 0.02 128 × 32 × 256

Table 1. Physical and numerical parameters used in the simulations.

20

r/r0

10

0

-10

PSfrag replacements

-20 0

10

20

30

40

50

x/r0 Figure 1. Contours of the norm of the vorticity (color) overlaid upon contours of the dilatation field for the pressure matched jet.


20

r/r0

10

0

-10

PSfrag replacements

-20 0

10

20

30

40

50

x/r0 Figure 2. Contours of the norm of the vorticity (color) overlaid upon contours of the dilatation field for the pressure mismatched jet. The contour levels of the vorticity and of the dilatation are the same as in figure 1.

1.5

1

p/p∞ − 1

0.5

0

−0.5

PSfrag replacements

−1 0

5

10

15

20

25 x/r0

30

35

40

45

Figure 3. Comparison of LES-predicted centerline pressure (—) against Norum & Seiner 6 measurement (– –).


2 1.8

(ρ − ρ∞ )/(ρ j − ρ∞ )

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

PSfrag replacements

0 0

10

20

30 x/r0

40

50

Figure 4. Centerline mean density for the on-design (– –) and off-design (—) jets.

1.1 1 0.9

Uc /U j

0.8 0.7 0.6 0.5 0.4

PSfrag replacements

0.3 0

10

20

30 x/r0

40

50

Figure 5. Centerline mean axial velocity for the on-design (– –) and off-design (—) jets.


urms /U j

0.1

0.05

PSfrag replacements

0 0

10

20

30 x/r0

40

50

Figure 6. Centerline mean axial velocity root-mean-square for the on-design (– –) and off-design (—) jets.

10

0

0

10

S uu /(U 2j r0 )

S uu /(U 2j r0 )

10

−2

−2

10

−4

10

PSfrag replacements

−4

10

PSfrag replacements −1

10

10

0

S uu /(U 2j r0 ) 1

10

−1

10

0

10

1

10

kr0

kr0

Figure 7. Axial spectra corresponding to the two-point correlation hu0 (x)u0 (ξ)i measured along the jet centerline. Left figure: pressure matched jet. Right figure: underexpanded jet. Legend: —, x/r0 = 5; − −, x/r0 = 9; − · −, x/r0 = 15; · · · , x/r0 = 20; −−, x/r0 = 28; − ◦ −, x/r0 = 36. The straight line has a slope of −5/3.


10

0

0

10

−1

10

S uu /(U 2j r0 )

S uu /(U 2j r0 )

10 −2

−2

10

−3

10

10

−4

PSfrag replacements

10

S uu /(U 2j r0 )

10

−4


10

10

0

−5

1

−1

0

10

10

1

10

10

kr0

kr0

Figure 8. Axial spectra corresponding to the two-point correlation hu0 (x)u0 (ξ)i measured along the jet lip-line (r/r0 = 1). Left figure: pressure matched jet. Right figure: underexpanded jet. Legend: —, x/r0 = 5; − −, x/r0 = 9; − · −, x/r0 = 15; · · · , x/r0 = 20; −−, x/r0 = 28; − ◦ −, x/r0 = 36. The straight line has a slope of −5/3.

−2

S pp /(ρ2j U 4j r0 )

10

10

S pp /(ρ2j U 4j r0 )

10

−2

−3

10

−4

10

−4 −5

10


PSfrag replacements

10

10

−6 −1

10

10

0

S pp /(ρ2j U 4j r0 ) 1

−1

0

10

10

1

10

10

kr0

kr0

Figure 9. Axial spectra corresponding to the two-point correlation h p0 (x) p0 (ξ)i measured along the jet lip-line (r/r0 = 1). Left figure: pressure matched jet. Right figure: underexpanded jet. Legend: —, x/r0 = 5; − −, x/r0 = 9; − · −, x/r0 = 15; · · · , x/r0 = 20; −−, x/r0 = 28; − ◦ −, x/r0 = 36. The straight, solid line has a slope of −11/3 and the straight, dashed line has a slope of −7/3.

−6

4

x 10

3.5

Sˆ up (St; x)

3 2.5 2 1.5 1

PSfrag replacements

PSfrag replacements

0.5 0 0.1

St Sˆ up (St; x)

0.15

0.2

0.25

0.3

0.35

0.4

St

Figure 10. Spectra of the temporal cross-correlation of pressure-velocity for the underexpanded jet. Legend: —, x/r 0 = 15; – –, x/r0 = 20; – · –, x/r0 = 25; · · · , x/r0 = 30.


0.8

0.8 n=0 n=1 n=2 n=3

0.7

cph /U j

cph /U j

n=0 n=1 n=2 n=3

0.6

0.7

0.6

PSfrag replacements

PSfrag replacements

0.2

0.3

0.4 St

0.5

0.6

0.2

0.3

0.4 St

0.5

0.6

Figure 11. Phase speeds of the linear instability waves forced at the jet inlet. Left: pressure-matched jet. Right: pressure mismatched jet. Horizontal line corresponds to a∞ /U j , the ambient speed of sound. In the legend of each figure the n refers to the azimuthal mode number.

1

1

0.8

0.8

0.6

0.6

0.4

0.4 pp

R (θ)/R (0)

0.2 0

pp

Rpp(θ)/Rpp(0)

Figure 12. Two-point space-time correlation of pressure taken on a surface at r/ D j = 5 from the centerline. Left figure: pressure-matched jet. Right figure: underexpanded jet.

0.2 0

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

−0.8 0

20

40

60

80

θ

100

120

140

160

−0.8 0

180

20

40

60

80

θ

100

120

140

160

180

Figure 13. Azimuthal correlation of pressure field taken on a cylindrical surface at r/ Dj = 5. Left figure: pressure matched jet. Right figure: underexpanded jet. Legend: —, x/r0 = 5; – –, x/r0 = 15; – · –, x/r0 = 25; · · · , x/r0 = 35; ––, x/r0 = 45.
