Interaction of a turbulent-jet noise source with transverse modes in a ...

NASA TP I 248 ,

NASA Technical Paper 1248

\

George P. Succi, Kenneth J. Baumeister, and ,K. Uno Ingard

JUNE 1978

NASA

c.1

-

TECH LIBRARY KAFB, Nhl

NASA Technical Paper 1248

Interaction of a Turbulentget Noise Source With Transverse Modes in a Rectangular Duct

George P. Succi Massachusetts Institute of Technology, Cambridge, Massachusetts Kenneth J. Baumeister Lewis Research Center, Cleveland, Ohio K. Uno Ingard Massachusetts Institute of Technology, Cambridge, Massachusetts

National Aeronautics and Space Administration

Scientific and Technical Information Office 1978

I

INTERACTION OF A TURBULENT-JET NOISE SOURCE WITH TRANSVERSE MODES IN A RECTANGULAR DUCT

by George P. Succi, * Kenneth J. Baumeister, and K. Uno lngardt Lewis Research Center SUMMARY A turbulent jet w a s used to excite transverse acoustic modes in a rectangular duct. The pressure spectrum showed asymmetric singularities (pressure spikes) at the resonant frequencies of the duct modes. This validates previously published theoretical results. These pressure spikes occurred over a range of jet velocities, orientations, and inlet turbulence levels. At the frequency of the spike, the measured transverse pressure shape matched the resonant mode shape. In the experiment, the p r e s s u r e response of a duct to an enclosed air jet w a s measured a t frequencies from 0 to 25 kilohertz and from 0 to 5 kilohertz. The jet noise in these experiments w a s produced by drawing air through an orifice into a 3.811 by 4-in.) rectangular duct. by 10. 16-centimeter (12-

INTRODUCTION In the transmission of sound in a duct with or without flow, large amplitude variations in pressure can be produced by the interaction of the sound with the duct walls. For example, the combustor noise signature in a turbojet engine (such as recently r e ported in refs. 1 and 2) can be modified by the engine ducting. In those studies, the duct flow had some effect; nevertheless, the present study concentrates on the interaction of sound with duct w a l l s in the absence of flow. Therefore, we used a turbulent jet t o excite the transverse acoustic modes in a rectangular duct. Because the jet a r e a w a s much smaller than the duct area, the duct flow w a s nearly zero and is thus neglected. The acoustic response to the jet near an eigenfrequency is identified, and the *Massachusetts Institute of Technology, Cambridge, Massachusetts. Professor of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts.

I.

frequency range is determined for which the acoustic response in a duct differs qualitatively from the response in f r e e space. The qualitative acoustic response to turbulent j e t s in f r e e space has long been known (ref. 3, pp. 123-125). In f r e e space the sound p r e s s u r e spectrum r i s e s smoothly, approximately as the frequency squared, to a peak a t a Strouhal number (St = vD/v. ) of Jet about 0.2 (ref. 4); it then falls off smoothly, approximately as the inverse of the f r e quency squared. In a duct, however, the pressure spectrum of a jet is more complicated than for the same jet in f r e e space. Although the most popular model of the turbulent noise source of a f r e e jet is a quadrupole (refs. 5 and 6), some success has been had in modeling the jet noise source as a distribution of simple (monopole) sources (ref. 7). Also, a turbulent field in the presence of boundaries has been shown to exhibit dipole source t e r m s (ref. 6, ch. 3). If an enclosed jet can be modeled by a monopole source, the pressure spectrum theoretically (refs. 8 and 9) will show asymmetric singularities (pressure spikes) at the duct-mode cutoff frequencies. Theoretically, the p r e s s u r e spectrum of a fan noise source in a semi-infinite duct should show similar asymmetric spikes (ref. 6, p. 335). These spikes have been measured for a simulated fan noise source (ref. 10). Although they do not show the sharp spike at the resonant frequency predicted by theory, the measured spikes a r e asymmetric (ref. 10, fig. 8). In reference 10, however, detailed sound-pressurelevel (SPL) t r a v e r s e s confirm that these peaks correspond to the excitation of higher order modes at the duct-mode resonant frequencies. Since both fan and jet noise sources can be modeled by simple acoustic sources, the simple source model should predict the response of a turbulent jet in a duct. The main purpose of this study w a s twofold: First, to confirm the existence of asymmetric resonance peaks in the acoustic spectrum of an enclosed turbulent jet, to show that the simple source model adequately predicts the resonant frequencies, and to identify the acoustic modes associated with resonance; second, to study the effects of jet velocity, orifice diameter and orientation, Strouhal number, and inlet turbulence. These experiments were performed in a new flow-duct facility; therefore, a secondary purpose is to document the facility and instrumentation. Also, some important aspects of pressure measurements peculiar to duct acoustics a r e discussed as an aid in interpreting flow-duct data. The p r e s s u r e response of a duct to an enclosed turbulent jet w a s measured at f r e quencies of 0 to 25 kilohertz. The jet noise w a s produced by drawing air through orif i c e s of different sizes and orientations at different flow rates.

SYMBOLS Ad

2

cross-sectional a r e a of duct, m 2

a

radius of sphere about a simple source, m

b

duct width, m

cO D

speed of sound, m/sec

d

duct height, m

i

fi

k

wave number (see eq. (6)), m - l

kmn

axial propagation coefficient for mode number (myn), m - l

orifice diameter, m

source wave number of a propagating mode (see fig. 7 and eq. (18))

kg L

length of pipe, m

M

Mach number, vj et/cO Mach number in pipe leading to orifice

MP m

transverse horizontal wave number

N

number of propagating modes

AN

number of propagating modes in Av

n

transverse vertical wave number

nt P

normal to w a l l acoustic pressure, Pa

'ref

reference r m s pressure, Pa

'rms

r m s pressure (see eq.

R

radiative resistance of source, g/sec source strength, m 3/sec

sW

(lo)),

St

Strouhal number, vD/vjet

SPL

sound p r e s s u r e level

T

one period of oscillation, sec

t

time, s e c

V

velocity, m/sec

Vd

duct velocity, m/sec

vj et

velocity of jet, m/sec

X

Pa

transverse horizontal duct coordinate, m

3

Y

transverse vertical duct coordinate, m

Z

axial duct coordinate, m

6.. 1J

Kronecker delta symbol (1 when i = j; 0 when i

Kn,Km,n

eigenvalue (transverse mode in duct), l/m

'mn) n'

mean-square amplitude of cross-duct mode

x

wavelength, c d v , m

V

frequency, Hz

"mn, n '

cutoff frequency, c0/2

i-,

#

j)

Hz

Strouhal frequency, Hz

Av

frequency interval, Hz atmospheric density, g/m 3

PO +mn, +n

wave function

w

angular frequency,

~ T V ,rad/sec

THEORY

A sound wave propagating in a rectangular duct can b e composed of a fundamental plane wave and various transverse waves that depend on the nature of the noise source. The fundamental plane wave, identified as the (0,O) mode, travels in the axial direction and has a uniform pressure distribution across the duct c r o s s section. A transverse wave, identified as the (m, n) mode, has a nonuniform p r e s s u r e distribution across the duct; consequently, it reflects back and forth from the duct walls as it travels along the duct. The maximum number of modes in a sound wave propagating in a hard-wall duct depends on the frequency. For a given mode (i. e., the (m, n) mode) to propagate, its frequency must be greater than the duct "cutoff" frequency. However, a noise source may not excite (at significant magnitude) the maximum number of modes that its f r e quency content would indicate. For example, a loudspeaker that fills the entire duct end might excite a large-amplitude plane wave but greatly reduced-amplitude higher order modes. In any case, as the frequency of the source is increased beyond the mode cutoff frequency, a new mode becomes available for the radiation of sound down the duct. The cutoff frequency for each mode and the p r e s s u r e response of a simple (monopole) source in a duct a r e well known from classical acoustics (ref. 8, pp. 497-502). The classical theory relating to mode cutoff frequency and p r e s s u r e response is briefly 4

.

.. . .

. . ..

,

,

.. ,

I

reviewed here as a background for the experimental results.

Cutoff Frequency Consider a rectangular duct of width b and height d. The wave equation describing the propagation of an acoustic disturbance in a source-free region with negligible mean flow is

v 2p - - -1= oa2p c i at2 where viscous effects have been neglected. Because of the hard walls the normal particle velocity must be zero, which is equivalent to requiring ap =o a' w a l l s

where nt is the normal to the wall. The axes of the Cartesian coordinate system a r e chosen so that the w a l l s a r e located at x = 0, b and y = 0, d, as shown in figure 1. For a sinusoidal time disturbance Ziwt, the solutions to equation (1) a r e of the form

where

q mn(x,Y ) = c o s ( ~ ) c o s ( ~ )

I

(4)

and Amn is an arbitrary constant. that is,

The functions @mn are orthogonal to one another;

where Ad is L e cross-sectional a r e a of the duct and Amn, the average vaAJeof

Y). The experiments were concerned with the excitation and propagation of the modes represented by the function +mn as they travel down a duct. In particular, p r e s s u r e traverses in the x-direction were made to determine if the Qo0, +lo, and $20 modes were present. The pressure amplitude (spatial) distribution of the first three modes is shown in figure 2. The plus and minus signs on the duct c r o s s sections in figure 2 indicate the spatial domain when the $Jmn t e r m takes on either a positive or a negative sign. (The qmn term is defined in eq. (4). ) Recall from equation (3) that the press u r e P also varies harmonically, as e-iwt. The temporal variation of the $Jlo mode is shown in figure 3. Although a microphone will respond to this instantaneous variation in pressure, a typical measurement system records p r e s s u r e s in t e r m s of the sound p r e s s u r e level (SPL), defined as $J;&

2 'rms SPL = 10 loglo 2

(9)

'ref where Prms is the r m s (root mean square) sound p r e s s u r e defined as

where T is one period of oscillation. The reference r m s pressure Pref is 2X10'5 pascal. Assume, for example, that only one higher order mode is excited. Then substitut-id into equations (9) and (10) and the r e a l component of e ing the product of J/ m, n yields the following SPL dependence on c r o s s section:

6

[

SPL = Constant + 10 loglo cos (m;x)cos(y)] -

The pressure amplitude (spatial) and SPL variations of the first three modes a r e shown in figure 4. The minimum sound pressure levels in figure 4(b) correspond to the dashed partition lines in figure 2. These types of SPL distributions indicate the presence of a particular mode. (The SPL is not, in general, equal to the intensity level in a duct if higher order modes a r e excited (ref. 9). ) We now replace the double index mn by the single index n by arranging the modes in order of increasing vmn. From equation (5), for the nth mode to propagate down the duct, kn must be real. For a given frequency v, kn is r e a l only if (ref. 3, p. 504)

v > v n = -- ‘OKn 2K where vn is the cutoff frequency. That is, for any frequency, only a finite number of modes can propagate. To put it another way, each mode has a cutoff frequency: For frequencies below cutoff, the mode attenuates exponentially; for frequencies above cutoff, the mode propagates along the duct without attenuation. These ideas a r e illustrated in figure 5 for a modal pattern in a rectangular duct. The upper schematic shows decay, v < vn; the lower schematic shows propagation, v L vn. Dissipative mechanisms can cause some attenuation in the propagating wave.

Impedance of a Simple Source The pressure near a simple source and the p r e s s u r e radiated down the duct a r e directly proportional t o the mechanical resistance at the source. However, the duct w a l l s reflect sound back on the source and thus modify that resistance. The mechanical impedance of a point source inside a duct is discussed in the following paragraph. Consider the response of a medium in a duct of cross-sectional a r e a Ad t o a simple (monopole) source of strength Sw and frequency v. The acoustic velocity is uniformly radial at a spherical surface of radius a (a -

10

9

8

7

8

5

6 4

6 4

2 1

2

c

3

(0.1)

~

I ~

1

I

2

1

3

4

Transverse vertical wave number, n (b) Mode diagram for rectangular duct 3.81 cm (1.5 in. ) high by 10.16 cm (4 in. 1 wide. Figure 7.

- Graphic illustration of mode cutoff. 27

I

-

-

1

I

-

-- -

Noise-generating orifice ,- Microphone location

I

-7

I

-

r I

I

Acoustically,’ lined //

. \

I

, L

i-

Test section

In-line porous

,- Mezzanine floor 15-cm (6-in.)

Figure 8.

- Flow duct system.

-

1.42111 (4ft 8in.)-

1.83 m (6 ft)

m

10 cm (4 in.) I

Test plates \.

Microphone

Muffler \,-

\

I

to vacuum system

additional foam blocks placed in front to reduce orifice flow rate)

Figure 9.

28

(l-

- Flowduct test section.

1

I

Exponential horn (filled with foam) I as uostream poco termination -I

Orifice diameter, cm (in.) 0.64 (114)

1.27 (112)

.2.54(1)

(a) 0.32-Centimeter-thick

plates.

Q)Long tube. Figure 10.

- Orifice plates. 29

"jet

I

Orifice

Downstream

\( ..............................................................................

Figure 11.

- Flow pattern of jet into duct.

Figure 12.

30

- Flow-duct apparatus.

Orifice diameter, D. cm (in.

Orifice orientation

Side

0 0.64 (114) 0 1.27 (1/2) V 2.54(1)

_ . . -,

.

,-

1

V

i- . . I .', .

V

I

-.

I . -

I

1''

00 0

."

1

I

.001

.01

flow rate, kglsec Figure 13. - Mass flow rate as measured by hot-wire anemometer. Mass

I .I

-.

.

-- -

~

Figure 14. -Microphone positioning plate.

-- _.-A

-.

,

.

,

Microphone . power supply

and amplifier

Spectrum

- analyzer and averager

Anemometer

x-Y recorder

Digital

I

Microphone

Anemometer -.

/Test piece Figure 15.

- Instrumentation.

:k %-

e L W L

z o

1

VI I W I n

2 3 Frequency, u, kHz

4

5

(a) Low-frequency range. VI

.-m> Nra,

c

a,

E

Frequency, u, kHz (b) High-frequency range. Figure 16. - Low- and high-frequency spectra i n a rectangular duct. Orifice diameter, 1.27 cm (1/2 in. k orifice orientation, top of duct: Mach number, 0.44.

32

L

n

1

3

2

4

5

0

1

2

3

4

5

Frequency, u. kHz (b) Orifice diameter, D, 1.27 cm (1/2 in. 1,

(a) Orifice diameter, D, 0.64 cm (1/4 in. ). 120-

110 m-

ug

Frequency, u. kHz (c) Orifice diameter, D. 2.54 cm (1 in. 1. Figure 17. - Effect of orifice jet velocity (Mach number) and orifice diameter on acoustic spectra in the low-frequency range (0to 5 kHz).

33

1wI

/ Jet c

Microphone a

>Screech

/ peaks

e

m

U

O'r fin --

0

I

I

5

10

YY-7 15

20

25

0

5

10

15

M

Frequency, v. kHz (b) Orifice diameter, D, 1.27

(a) Orifice diameter, D, 0.64 cm (1/4 in. 1.

-

m O-

cm (1/2 in. 1.

140 -

130 -

V

5

70-

v) 0

1 J I 1 . I. 5 10 15 M 25 .

6o 0

Frequency, u. kHz (c) Orifice diameter, D, 2.54 cm (1in. 1.

Figure 18. - Effect of orifice jet velocity (Mach number) and orifice diameter on acoustic spectra in the high-frequency range 10 to 25 kHz1.

34

25

Orifice diameter.

t

30 dB I

Frequency. u. kHz (a) Strouhal frequency. us. -1.9 kHz.

(b) Strouhal frequency.

ug. -2.5 kHz.

30 dB

0

1

2 3 Frequency, u. kHz

4

5

(c) Strouhal frequency, us.-5.7 kHz. Figure 19.

- Sound pressure levels for orifices with approximately equal peak Strouhal frequencies (us= 0.2 v/D).

35

3

VI

,-Low-velocity jet noise

E n

Frequency Figure 20.

-

- Schematic representation of noise sources i n flow duct.

Mach number

m

U

Frequency, (a) Low-frequency spectra.

u, kHz 0)High-frequency spectra.

Figure 21. - Effect of velocity of flow entering orifice from long tube o n acoustic spectra. Tube length, 1.52 m (60in. t. tube inside diameter, 1.27 cm (112 in. ).

36

I

m

a In

2 .-.

Microphone

l'K

Mach

1 2 O k

0

70k 6o0

Mode

I

I

1

(LO)

(2.0)

I

I

2

3

I

(0.1)

(1.1)

5

4

Frequency, u, kHz Figure 22 - Low-frquency spectra observed when jet and duct axis are parallel. Orifice diameter, D, 1.27 cm (1/2 in. 1.

Microphone location Upstream -_---Downstr ey;, '\\

/ , /

w-

0

e

Mach number

l l O r

Frequency, Figure 23. spectra.

u. kHz

- Upstream- and downstream-radiated

acoustic

37

Stationary microphone

P ,-

- -- ..- Moving microphone

1.5r

El a l

g

c

0

-E

1.0

.-Ee

U

m

c .-0

I= .-

c m c v)

L

0 2

0 5

E

.5

0

-I

e

a

2

2

0

L

Y

Axial position. z

-

Figure 24. - Minimum-to-peak sound pressure level variations as function of microphone location.

38

Microphone location on duct cross section

__--r-l

I

0

1 Figure 26.

I

2 Frequency.

4

I

~~

4

5

u. kHz

- Effect of microphone location on detected spectra.

39

.

-

.

3. Recipient's Catalog No

2. Government Accession No.

1. Report No.

NASA TP-1248 5. Report Date

4. Title and Subtitle

June 1978

INTERACTION O F A TURBULENT-JET NOISE SOURCE WITH TRANSVERSE MODES IN A RECTANGULAR DUCT

6. Performing Organization Code

8. Performing Organization Report No

P. Succi, Massachusetts Institute of Technology; Kenneth J. Baumeister, Lewis Research Center; and K. Uno Ingard, Massachusetts _ . Institute of Technology

7. Author(s) George

E-9462 --

IO. Work Unit No.

9. Performing Organization Name and Address

505-03

National Aeronautics and Space Administration Lewis Research Center Cleveland, Ohio 44135 .

11. Contract or Grant No

.

.

13. Type of Report and Period Covered

.

12. Sponsoring Agency Name and Address

Technical Paper

National Aeronautics and Space Administration Washington, D. C. 20546 , . .

.

~

_

4. Sponsoring Agency Code

_

15. Supplementary Notes

~

-

~.

16. Abstract

A turbulent jet was used to excite transverse acoustic modes in a rectangular duct. The pressure spectrum showed asymmetric singularities (pressure spikes) at the resonant f r e quencies of the duct modes. This validates previously published theoretical results. These pressure spikes occurred over a range of jet velocities, orientations, and inlet turbulence levels. At the frequency of the spike, the measured transverse pressure shape matched the resonant mode shape.

.

.~

17. Key Words (Suggested by Author(s) )

18. Distribution Statement

Noise; Turbulent; Jet; Duct; Modes; Singularities

19. Security Classif. (of this report)

Unclassified

Unclassified - unlimited STAR Category 01

1

20. Security Classif. (of this page)

Unclassified

-

-

..

21.

NO.

offpager

..

.

22. Price'

A0 3 __

NASA-Langley, 1978

National Aeronautics and Space Administration

THIRD-CLASS B U L K R A T E

Washington, D.C. 20546

Postage and Fees Paid National Aeronautics and Space Administration NASA451

Official Business Penalty for Private Use, $300

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