A99-27826 AMA-99-l&40

LINEAR CONTRIBUTION OF LARGE-SCALE STRUCTURE TO SUBSONIC JET NOISE

A. A. Ali Department of Mechanical Engineering

Cairo University, Cairo, Egypt

And Reda R. Mankbadi

ICOMP, NASA Lewis Research Center Cleveland, Ohio 44135

EMAIL: fsmankb@lerc.nasa.gov

The linear contribution of the large-scale structure to subsonic jet noise is assessed by using the linearized Euler equations to directly calculate both the sound source and the associated sound field at various Mach numbers. It is shown that the noise source becomes non-compact with increasing the Mach number. The directivity pattern is characterized by a sharp peak at higher Mach numbers or higher Stroubal numbers. By comparing the computed results to measurements, it is concluded that nonlinear effects are more important at lower Mach numbers than at higher Mach numbers.

1. INTRODUCTION There are several mechanisms which control the sound generation in subsonic jets. Tam et al. (1197) studied various data for jet noise and concluded that mixing noise can be split into two elements. One for the noise from large turbulence structures/instability waves of jet flow, the other for the noise from the fine-scale turbulence. In either case, the sound source is the flow fluctuations, but it is not clear how to computational separate the two sources. One might adopt the view that the fine-scale turbulence is random in nature but the “large turbulence” is wavelike and can be described by linear or nonlinear instability theories (Liu, 1989; Morris, 1974; Tam, 1991; Mankbadi, 1992). In principal, all the scales of the flow fluctuations could be numerically resolved via direct numerical

Copyright Q 1999 by the American Instikte of Aeronautics and Astronautics, Inc. All rights reserved.

simulations. But at high Reynolds number this is currently hindered by the computer capabilities. In this paper we adopt the Large-eddy-Simulations (LES) view that the “large turbulence” is the structure resolved numerically, while the fine-grained turbulence is the unresolved, small-scale, high-frequency structure. We focus this study on the contribution of the large structure to subsonic jet noise.

For better understanding of the mechanism involved, one can also split the contribution of the large structure to noise into linear and nonlinear contributions. Examples of nonlinear mechanisms are self- saturation of the sound source due to its large amplitude; the interaction among the various frequency-components of the source which could alter their own development or could

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lead to the generation of new modes. Linear contribution is therefore defined as the contribution to the jet noise resulting from the linear development of the flow fluctuations in complete absence of nonlinear effects.

The question to be addressed herein is whether linear approximation can adequately describe the sound source in subsonic jets and the associated sound field. The idea is not new. By making further approximation by assuming that the instability wave theory describes the linear development of the flow structure in subsonic jets, Crighton & Huerre (1990) have calculated the corresponding generated noise. However, Linearized Euler Equations (LEE) are more general than the linear stability equations. The former fully accounts for nonparallel flow-divergence effects and the flow disturbances are not necessary of the normal-mode decomposition type required for the stability theory. Thus disturbance solution given by LEE can thus be taken as a linear approximation to the flow fluctuations, which are not necessarily instability modes.

For supersonic jet, Mankbadi et al. (1998) have used LEE to calculate both the sound source and the associated acoustic field. It was shown that the predicted field is consistent with the experimental observations of Trout & McLaughlin (1982) indicating that the linear approximation is adequate. For supersonic jets, this could be explained by the dominance of Mach waves emission and by the fact that the fluctuations in supersonic jets are relatively lower than in e subsonic case (Michalke, 1984). For subsonic jets, however, it is not clear what the linear development of the flow fluctuations under diverging mean flow might play as source of jet noise, which is the subject of this study.

The governing equations and discretization scheme is presented in $2. This is foll,owed by discussion of boundary treatments and the computation grid in $3. In $4 results are presented for various Mach numbers and Strouhal numbers. BY comparing the numerical results with the experimental measurements one can conjecture where nonlinear or linear effects dominate. Discussion and conclusion is presented in $5.

2. THE GOVERNING EQUATIONS AND THE DISCXETIZATION SCHEME We consider the flow of a subsonic round jet of at Mach number M, issuing from a nozzle of diameter D. The streamwise coordinate is x, the radial direction is r, and the azimuthal direction is 8 The velocities are u, v, w in the x, r, and 0 directions, respectively. Starting from the full compressible, Navier- Stokes equations and linearizing around a given mean flow (I-l, V, W), the Linearized Navier-Stokes equations are obtained. Upon neglecting the viscous effects the Linearized Euler Equations LEE are obtained in the form (see Mankbadi 1994, chapter 8 for more details) :

(1)

where Q = [p, 22, 0, I+, fqT, (2)

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fi=

ri p + 2au-pu= riv+v^u-puv

liw+*u-Ijuw @^+t?)U+(ri-ljU)E

I3 liW+W-(iuw v^w+tiv-flvw fl+2+w-pw=

@^+C)W+(G-(jW)E

g= 1 r

and

0 0

g+2*w--(jw= vw+wv-pvw

0

.

p = (y-l)[e’-(ziU+v^V+~w)

++J=+v=+w=)l ,

J’=(Y -1) P[E +I=+ v=+ W2)] ,

and (a,9,c,e,p,p) =

KPU)’ 1 (PV)’ 9 (PW)’ 9 (Pd’ t P’ ,P ‘1

(3)

(4)

(5)

(6)

(7)

(8)

(9)

Velocities are normalized by the jet exit

centerline velocity, time by D/U,, where sub e denotes the exit conditions. Density is normalized by the exit density, and pressure bY PUe=. The mean flow is given by the analytical fit of Tam & Burton (1984).

The MacCormack 2-4 scheme is used to discretize the above equations. The scheme is fourth-order accurate in space, second-order accurate in time. The operator is split into radial and axial directions as follows

Each operator consists of a predictor and corrector steps using one-sided difference as follows : Predictor

n+; Qi =Qj”-&(7$-8Fie, +F& (11)

Corrector:

Q;+l.$Q;‘t+Ql”

+&(7Fi-8Fi+l+E;3n+~] (12)

and likewise for the radial direction. The sweep directions are reversed between operators to avoid biasing. Flux quantities outside the computational boundaries are obtained via third-order extrapolation of the interior quantities when needed.

3. BOUNDARY TREATMENTS AND COMPUTATIONAL MESH The computational domain is shown in figure 1. The mesh size is 400x300 in he x-r directions. he mesh is uniform in x, but stretches in r with more points concentrated

around the line r =D/2. At x=0, the radial boundary is split into two regimes: an incoming flow regime for small r and a radiation regime at large r. At X=Xmaximum, the radial regime is split into two regimes, at small r an outflow regime and at large r a radiation regime. At r=rmaximumr radiation boundary condition is applied along x. At r = 0, the centerline treatment is applied along x. The radiation, outflow, and centerline treatments follow Mankbadi et al. (1998). Inflow treatment had to be modified, as will be discussed below, since the flow here is subsonic.

At inflow, the disturbances are assumed to be small and time-harmonic, Its radial distribution is of Gaussian type with peak at r=D/2. The effect of inflow disturbances is added to the computed flow various at the inflow at each time step

(13)

In Ali (1997) various inflow treatments were considered. Tam et al. (1996) were chosen as it produced little reflections and will be discussed flow. Asymptotic analysis of LEE produces for outgoing waves

aFouf - I at

-qxd +‘d + I] Fo, R ax Rdr R (14)

where F’“’ = Ffj - F auf (15)

w=qX~+(l-(‘j@)“=] R R (16)

and

R=[x=+r=]“= (17)

F, ,,,(“) is the numerical solution of the discretized, while F is the incoming characteristics. In implementing this method, the derivatives appearing in the right-hand side of equation (14) are split into axial and radial ones. During the axial sweep, equation (14) is applied in the form

a&!? x a (n) 1 = -W [- -] (F,,,-fl at R ax (18)

and the incoming waves are assumed in the form

F = Re{@) &-‘nAf)} (19)

During the radial sweep, equation(14) is applied in the form

After implementing equations (18) and (20) for the axial and radial sweeps, the time derivative of the incoming disturbances is accounted for.

4. RESULTS In what follows results are presented for the sound source and the associated acoustic field for a round jet at various subsonic speeds and various Strouhal numbers. We will denote the time-dependent flow fluctuations in the jet flow as the sound source. The Strouhal number St is St=fD/U,. The emission angle 8 is measured from the jet centerline in the streamwise direction.

Figure 2 shows the instantaneous pressure disturbance along the r/D =0.5 line at St=O. 1 for various Mach numbers. The

253

figure shows that the code can predict the wavelike nature of the sound source with minimum dispersive errors. The source amplifies according to the linear stability theory and later decays due to the flow divergence effects. We note that with increasing Mach number, the source extends over a longer streamwise distance, i.e., becomes non-compact.

Figure 3 shows the sound source at a fixed Mach number of 0.9 and various Strouhal numbers. We note that as the Strouhal number increases, the source becomes more compact, amplifies and decays closer to the jet exist. This is consistent with the experimental observations of Moore (1977), Chan (1974), and the nonlinear computation of Mankbadi (1985).

Figures 4-6 show the pressure disturbances field over the full computational domain containing both the flow fluctuations and the radiated field. We first note that the acoustic field is free from reflections indicating that the numerical scheme is accurate and that the boundary treatments are reflections-free. Figure 4 shows snap shots of the pressure disturbances at Mach number = 0.8 at various Strouhal numbers. As the Strouhal number increases the directivity becomes characterized with a sharp peak- emission angle that increases with increasing St. In figure 5 the Strouhal number is fixed at St=0.3 and the Mach number is varied. The figure indicates that with higher Mach numbers, the peak-emission angle becomes sharper and is lower with increasing Mach number. The root-mean-square (RMS) of the pressure disturbance corresponding to figure 4 is shown in figure 6. The looped shaped contours are apparent in the figure, with the emission peak angle sharper and higher with increasing St.

Figures 7-9 show the computed

directivity of the sound intensity in comparison wit the experimental observations of Lush (1971) for several Mach numbers. Results for St=O. 1, 0.3 and 1.0 are shown in figures 7,8, and 9, respectively. The agreement is better at higher Mach numbers indicating that the linear approximation adequately represents the sound source and the radiated field for higher Mach numbers.

Figures 10, 11, and 12 show the dependence of the sound intensity on the Mach number at various Strouhal numbers for emission angles 15, 45 and 60 degrees, respectively. The total acoustic power is compared with the experiment in figure 13. For all emission angles the figures show that the agreement with experiment is satisfactory at low Strouhal numbers. At high Strouhal numbers where the amplitude is large (see figures 2 & 3), nonlinear effects become important and the linear approximation is not adequate.

5. DISCUSSION AND CONCLUSION Linear computations were presented in which the sound source and the associated acoustic fields are directly calculated for subsonic jet at various Mach numbers. The flow fluctuations, representing the sound source, are given by the solution of the linearized Euler equations. The linear Euler equations encompass the linear stability solution, but is more general in that the flow divergence effects are fully accounted for and the disturbances are not necessary stability modes. The captured fluctuations at a given frequency are thus the Kelvin-Hehnholtz modes plus other modes at the same frequency.

In general the linear approximation to the sound source and the noise field seems to indicate that the linear mechanism is the

254

dominant mechanism even at Mach numbers as low as 0.5. At lower Mach numbers the linear vs. the linear contribution depend on the Strouhal number. At the low Strouhal number range, where the amplitude is not too high, the linear mechanism dominates over the nonlinear one.

For a given Strouhal number, the captured structure is f wavelike nature, but not exactly harmonic in x. It can be viewed as composed of various streamwise Fourier components. Thus, for a given frequency, there are several streamwise-Fourier components with various streamwise wavelengths. There are components with supersonic speeds, even if the flow is subsonic. Thus, Mach waves emission could be responsible for subsonic jet noise provided the flow divergence is accounted for.

REFERENCES Ali, A., 1997 “On the Linear

Contribution to Subsonic jet noise, ” M. S. Thesis

Chan, Y. Y., 1974, “Spatial Waves in Turbulent Jets, ” Phys. Fluids, vol. 17, pp. 46-53.

Crighton, D. G. and Huerre, P., 1990, “Shear Layer Pressure Fluctuations and Superdirective Acoustic Sources, ” J. Fluid Mechanics, Vol. 220, 1990, pp. 355- 368.

Liu, J. T. C., 1989, “Coherent Structure in Transitional and Turbulent Shear Flows, ” Ann. Rev. of Fluid Mech., vol. 21, pp. 285-3 15

Lush, P. A., 1971, “Measurements of Subsonic Jet noise and Comparison with Theory, ” J. Fluid Mechanics, Vol. 46, pt. 3, 1971, pp. 477-500.

Mankbadi, R.R., 1992, “Dynamics and Control of Coherent Structure in

Turbulent Jets, ” Appl. Mech. Reviews, vol. 45, pp. 219-248.

Mankbadi, R. R., 1994, “Transition, Turbulence, and Noise, ” Kluwer Academic, Boston. .

Mankbadi, R. R., 1985, “On the Interaction Between Fundamental and Subharmonic Instability Waves in a Turbulent Jet, ” J. Fluid Mechanics, vol. 160, pp. 385-419.

Mankbadi, R. R., Hixon, D. R., Shih, S-H, and Povinelli, L. A., 1998, “Linearized Euler Equations as a Tool in Jet Noise Prediction,” AZAA Journal, Vol. 36, No.2, February 1998, pp. 140-147.

Michalke, A., 1984, “Survey of Jet Instability Theory, ” Progress in Aerospace Sciences, Vol. 21, pp. 159-199.

Moore, C. J., 1977, “The Role of Shear-Layer Instability Waves in Jet Exhaust Noise, ” J. Fluid Mech., vol. 80, pp. 321- 367.

Morris, P, 1974, “A model for the Structure of Jet Turbulence as a Source of Noise, ” AIAA Paper 74-l

Seiner, J. M., McLaughlin, D. K., Liu, C. H., 1982, “Supersonic Jet Noise Generated by Large-Scale Instabilities, ” NASA TP-2072

Tam, C. K. W., 1991, “Jet Noise Generated by Large-Scale Coherent Motion,” Acoustics of Flight Vehicles: Theory and Practice, Vol. 1: Noise Source, edited by H. H. Hubbard, NASA RP-1256, Chapt. 6

Tam, C. K. W. and Burton, D. E., 1984, “Sound Generated by Instability Waves in Supersonic Flows. Part 2. Axisymmetric Jets,” J. FZuid Mechanics, vol. 138, 1984, pp. 273-295.

Tam, C. K. W., Golebiowski, M., and Seiner, J. M. 1996, “On the Two Components of Turbulent Mixing Noise from Supersonic Jets,” AIAA Paper 96-1716.

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Tam, C. K. W., Fang, J. and Kurbatskill,K. A., 1996, “Inhomogeneous Radiation Conditions Simulating Incoming Acoustics Waves for Computational Aero- Acoustics, ASME/Cairo University International Congress on Fluid Dynamics and Propulsion, December 1996.

Trout, T. R. and McLaughlin, D. K., 1982, “Experiments on the Flow and Acoustic Properties of a Moderate Reynolds number Supersonic Jets, ” J. of Fhid Mech., vol. 116, pp. 123-156.

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Acoustic Radiation I-

ll-l-l--l-lll--l

Jcoustic R 1

i gifferent inflow I Centreline conditicn Dounaary co,pi:cnb++ ---- ----

1 I L IIIIII11101IIIIIII J Acoustic Radiation

Figure (I) Computational Domain

257

ew.E4 1 200E-4

1

1 Ma05 1 M = 0.4

-2WE4 I I I I 20.00 asa w.ao @mu

M = 0.63 St = 0.1

!a)

2CCE-r

1

-2cQEJ ’ I , I I I

20.00 40.00 60.00 80.00

xir

Cc)

-1.WE4 -

-200E4 -1,

2o.w 40.00 sass ml00

8.OOE-J -

M = 0.8 St = 0.1

4.00EJ - (\

O.WE+U -

rt

4.00&l- u

4.caE-r I I I I 20.W 40.00 00.00 &3.W

dr

(4

r/r

Figure (2) Instantaneous Pressure Disturbance along the r/Dj = 0.5 Line at St=O.I ((a)M=O. 4, (b)M=O. 5, (c)M =O. 65, (d)M=O. 8)

258

2WE-3 1 M = 0.9 !a=03

l .ME.3 l.WE.3

r

I O.WEd 5

2

ia O.OOE+O

L

-l.WE-3 -l.coE-3

-2.WE-3 I g 1 I I x1.00 46.00 60.00 80.00

x/r

(aj (b)

r .OOE-3 2WE-3

3’ l.WE-3

M=O.Y M=OS St = 1.0

LWE-3 St=05

u O.WE+O-

-2WE-3 3,

2O.W 40.00 !SO.W 80.90

r/r

O.WE+o

t e -1.WE-3 -

-UOE-3

4.0053 I I I 1 20.00 d0.W 8400 80.00

z/r

Cc) (4

i d.WE-3 ’ I

2OfW I I 1

40.W 60.00 80.00

dr

Figure (3) Instantaneous Pressure Dirtwbance along the r/D = 0.5 Line at Ll60.9 ((a)St==. I, (b)St==.3, (c)St=O.S, (d)St=I.O)

259

Cc) (4

Figure (4) Snapshot of the Pressure Disturbance Field at 11kO.8, ((a)St=O.l, (b)St=O.3, (c)St=O.S, (d)St=l.O)

260

(4

“t “k

Figure (5) Snapshot of the Pressure Disturbance Field at SMl.3, ((a)M=O.4, (b)M=0.5, (c)M=O.65, (d)M=O.8, (e)M=O.9)

261

(a)

(4 (4

Figure (6) Root-Mean-Square Pressure contours at AkO.8, ((a)St=O.I, (b)Sktl.3, (c)St=O.S, (d)St=I.O)

262

-J , , , I '1

Boo ULOD mm - loprn Emission Angle (degrees)

(4

Figure (7) DirectiviQ of Sound Inten& at St&I, ((a)M=O. 4, (b)M4.65, (c)&1=0.9)

263

- , I i . I a I I mm am mm 80.03 IO&D2

Emission Angle (degrees)

(4

n f----.

- I I I ’ 1 ZUO ram - DllllD lm.00

Emission AJI@ (degrees)

(c)

Figure (8) Directivity of Sound Intensity at St=O.3, ((a)M=OA, (b)M=O.65, (c)M=O.9)

264

(4

M=0.65 I- CampuWiwai I

M = 0.9 St = 1.0

- Computationnl

I Experhnrnl

Figure (9) Directivity of Sound Xntensily at St=I.O, ((a)M=O.4, (b)M=O.65, (c)M=O.9)

265

St = 0.1

@=I50

[l]

Icnrn -

mm-

St=03 - Computatloml

8=W q ExpdmmW

(4

Figure (IO) Velocity Dependence of Sound Intensity at 15 “, ((a)St==J, (b)St=O.3, (c)St=I.O)

266

l .

. l

(4

. .

Figure (II) Velocity Dependence of Sound Intensity at 4.5: ((a)St==J, (b)St=O.3, (c)St=I.O)

267

St=03 - Computwiond

Figure (12) Velocity Dependence of Sound Inter&y at JO “, ((a)St==.I, (b)St=O.3, (c)SC=I.O)

266

Irnrn 7 I

- compemiood St = 0.1 q ExpdmmM

Figure (13) Velocity Dependence of The Acoustic Power, ((a)St=O.I, (b)St=O.3, (c)9=1.0)

269