The Acoustic Analogy and Alternative Theories for Jet Noise Prediction

43 Pages·2002·0.33 MB·English

by  Morris, Philip J.

The Acoustic Analogy and Alternative Theories for Jet
Noise Prediction
(cid:3)
Philip J. Morris
Department of Aerospace Engineering
The Pennsylvania State University
University Park, PA 16802, U.S.A.
and
y
F. Farassat
Aeroacoustics Branch
NASA Langley Research Center
Hampton, VA 23681, U.S.A.
(cid:3)
Boeing/A. D. Welliver Professor, Associate Fellow, AIAA
y
Senior Research Scientist, Associate Fellow, AIAA
Abstract
This paper describes several methods for the prediction of jet noise. All but one of
the noise prediction schemes are based on Lighthill’sor Lilley’s acoustic analogy while
1
the other is the jet noise generation model recently proposed by Tam and Auriault.
In all the approaches some assumptions must be made concerning the statistical prop-
erties of the turbulent sources. In each case the characteristic scales of the turbulence
are obtained from a solution of the Reynolds-averaged Navier Stokes equation using a
k(cid:0)(cid:15)turbulencemodel. Itisshownthat, forthesamelevelofempiricism,TamandAu-
riault’smodel yields better agreement with experimentalnoise measurements than the
acoustic analogy. It is then shown that this result is not because of some fundamental
(cid:13)aw in the acoustic analogy approach: but, is associated with the assumptions made
in the approximation of the turbulent source statistics. If consistent assumptions are
made, both the acoustic analogy and Tam and Auriault’s model yield identical noise
predictions. Thepaperconcludeswitha proposalforanacousticanalogy thatprovides
a clearer identi(cid:12)cation of the equivalent source mechanisms and a discussion of noise
prediction issues that remain to be resolved.
1 Introduction
The prediction of jet noise has been the object of continuous interest and study since the
introduction of the jet engine for commercial use. The earliest theoretical formulation for
2;3
aerodynamic noise was the work of Lighthill . Lighthill’s equation was the (cid:12)rst example
of an \acoustic analogy." The de(cid:12)nition of an acoustic analogy, to be used in this paper,
is any aerodynamic noise theory in which the equations of motion for a compressible (cid:13)uid
are rearranged in a way that seprates linear acoustic propagation e(cid:11)ects. By de(cid:12)nition, this
rearrangement results in a set of equivalent sources that are assumed to be non-negligible in
a limited region of space. In an apparent departure from formulations based on the acoustic
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analogy, Tam and Auriault have recently developed a jet noise prediction method in which
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the sound sources are modeled explicitly and the propagation of sound from these sources
is described by solutions to the linearized Euler equations. These two aerodynamic noise
theories appear to be very di(cid:11)erent. In this paper we reconcile these apparent di(cid:11)erences
and show that if consistent assumptions are made concerning the statistical properties of the
turbulent noise sources, both approaches can yield identical noise predictions. It should be
noted that this reconciliation has only been demonstrated at 90 degrees to the jet axis. The
di(cid:11)erent approaches could lead to very di(cid:11)erent results at other angles.
In ordertomakepredictionsoftheradiatednoiseitisnecessary todescribethe properties
oftheturbulence. Sinceacompletesimulationofthejetturbulenceandthenoiseitgenerates
and radiates are too computationally expensive for high Reynolds number jet (cid:13)ows, noise
predictions are often based on the solution of the Reynolds-averaged Navier Stokes (RANS)
equations using a k (cid:15) turbulence model. Such a solution provides an estimate of the
(cid:0)
amplitudeof the turbulent velocity (cid:13)uctuations as wellas a locallength scale. Since the k (cid:15)
(cid:0)
solution only provides time-averaged properties, it is necessary to make assumptions about
the statistical characteristics of the turbulence in order to predict the noise radiation that is
an inherently unsteady phenomenon. In particular, the axial two-point cross correlation of
the turbulent sources must be approximated. This correlation may be described in either a
(cid:12)xed frame of reference or a reference frame moving with the turbulence.
One of the earliest attempts to couple estimates of the statistical properties of the turbu-
lence froma steady (cid:13)ow prediction with a noise modelbased on the acoustic analogywas the
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MGB approach developed by Mani, Gliebe and Balsa: see Balsa and Gliebe. More recent
extensions have used RANS solutions for the (cid:13)ow (cid:12)eld based on a k (cid:15) turbulence model.
(cid:0)
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This has been referred to as the MGBK method: see Khavaran et al. and Khavaran. In
developing a solution to the acoustic analogy equations these methods assume a form for the
two-point cross correlation function for the turbulent sources in a moving frame of reference.
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Conversely, Tam and Auriault, who also use RANS solutions for the (cid:13)ow (cid:12)eld based on a
k (cid:15) turbulence model, describe the two-point cross correlation in a (cid:12)xed reference frame.
(cid:0)
2
Itisshowninthispaperthat,forthesameRANSsolution,jetnoisepredictionsmadewith
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the Tam and Auriault model give much better agreement with experimental measurements
at 90 degreees to the jet axis than methods based on the acosutic analogy. However, it will
also be shown that this is not due to any inherent (cid:13)aw in methods based on the acoustic
analogy: but, is associated with the assumptions made concerning the statistical properties
of the turbulent sources. Both approaches yield identical noise predictions at 90 degrees to
the jet axis if consistent descriptions of the turbulent sources are chosen.
This paper is organized as follows. First a noise prediction formula based on Lighthill’s
acoustic analogy is developed. It is denoted here as Model I based on the acoustic analogy.
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The features of Tam and Auriault’s noise prediction model are then described. Another
model based on the solution to Lighthill’s equation is then formulated using the solution
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procedure of Tam and Auriault. This is designated as Model II based on the acoustic
analogy. The reasons for the di(cid:11)erences between the noise predictions obtained with the
di(cid:11)erentapproaches arediscussed andthese di(cid:11)erences arereconciled. Finally,analternative
form of acoustic analogy is proposed. It is argued that this form allows for the easier
identi(cid:12)cation of the equivalent noise source mechanisms.
2 Lighthill’s Acoustic Analogy: Model I
In this section we develop a prediction scheme based on the solution to Lighthill’s equation.
The details of the methodology do not follow those of the MGBK approach exactly. This is
becausethefocusofthepresentpaperisonnoiseradiationat90degreestothejetaxis,where
mean (cid:13)ow/acoustic interaction e(cid:11)ects are negligible. It is at this angle that the speci(cid:12)cation
of the source and its assumed relationship to the k (cid:15) solutions is best assessed. Also, it
(cid:0)
is intended to keep the assumptions made and the level of empiricism used as consistent as
possiblebetween thedi(cid:11)erentschemes described inthispaper. So, thepossibleimportanceof
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the e(cid:11)ects of anisotropy of the turbulence on the radiated noise, as proposed by Khavaran,
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is not included. The analysis in this section follows that given by Goldstein and Lilley in
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Chapter 4 of Hubbard. It is repeated in suÆcient detail here to emphasize any assumptions
made and the di(cid:11)erences with the alternative approaches presented in later sections of the
paper.
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Lighthill’s equation may be written in Cartesian tensor form as,
2 0 2 0 2
@ (cid:26) 2 @ (cid:26) @ Tij
2 co = (1)
@t (cid:0) @xi@xi @xi@xj
where Tij is the Lighthill stress tensor given by
2
Tij = (cid:26)uiuj +Æij (p po) co((cid:26) (cid:26)o) (2)
(cid:0) (cid:0) (cid:0)
(cid:2) (cid:3)
and primes denote perturbations about the basic state denoted by a subscript o. co is a
constant speed of sound that is sensibly taken to be the speed of sound in the uniform
medium surrounding the source region. ui is the instantaneous velocity vector. Viscous
terms have been neglected in the Lighthill stress tensor. In the subsequent analysis it is
assumed that the departures from isentropic behavior are everywhere small and that the
o
(cid:13)ow is at relatively low Mach number. Since we are concentrating on noise radiation at 90
to the jet axis it is also assumed that the primary contributions to the Lighthillstress tensor
involve products of velocity (cid:13)uctuations. The terms that are linear in the (cid:13)uctuations on
the right hand side of Eqn. (1) should be regarded as terms associated with the propagation
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of the sound and be placed on the left hand side of the equation: see Lilley. These e(cid:11)ects
o
are negligibleat 90 to the jet axis so this assumption is reasonable here. So we approximate
the Lighthill stress tensor by
0 0
Tij = (cid:26)suiuj (3)
where (cid:26)s is the mean density in the source region. In the far (cid:12)eld the density (cid:13)uctuation is
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readily shown to be given by
2
0 1 1 @ x y
(cid:26) (x;t) = 4 2Txx y;t j (cid:0) j dy (4)
4(cid:25)cox @t (cid:0) co
ZVZ(y)Z (cid:18) (cid:19)
where Txx is the component of the Lighthill stress tensor in the direction of the far (cid:12)eld
observer and x = x y x .
j (cid:0) j (cid:25) j j
The far (cid:12)eld spectral density for the intensity is related to the Fourier transform of the
autocorrelation function of the far (cid:12)eld pressure.
1
0 0
1 p (x;t)p (x;t+(cid:28)) i!(cid:28)
S(x;!) = h ie d(cid:28) (5)
2(cid:25) (cid:26)oco
(cid:0)Z1
0 2 0
where denotes an ensemble average. Since, in the far (cid:12)eld, p = co(cid:26) we obtain
h i
1
2 2
1 @ Txx @ Txx i!(cid:28)
S(x;!) = 3 5 2 2 (y1;t1) 2 (y2;t2) e dy1dy2d(cid:28) (6)
32(cid:25) (cid:26)ocox @t @t
(cid:0)Z1V(Zy1)V(Zy2) (cid:28) (cid:29)
where
x y1
t1 = t j (cid:0) j (7)
(cid:0) co
x y2
t2 = t+(cid:28) j (cid:0) j (8)
(cid:0) co
If the turbulent statistics are assumed to be stationary and the usual far (cid:12)eld approximation
is made we obtain,
1
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1 @ i!(cid:28)
S(x;!) = 3 5 2 4 Txx(y1;t)Txx(y2;(cid:28)o) e dy1dy2d(cid:28) (9)
32(cid:25) (cid:26)ocox @(cid:28) h i
(cid:0)Z1V(Zy1)V(Zy2)
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where
x (y2 y1)
(cid:28)o = t+(cid:28) + (cid:0) (10)
x (cid:1) co
The two{point cross correlation function of the Lighthill stress tensor in a (cid:12)xed reference
frame may be denoted by,
Rf(y1;(cid:17);(cid:28)) = Txx(y1;t)Txx(y2;t+(cid:28)) (11)
h i
where (cid:17) = y2 y1, then,
(cid:0)
1
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! x (cid:17)
S(x;!) = 3 5 2 Rf(y1;(cid:17);(cid:28))exp i! (cid:28) dy1d(cid:17)d(cid:28) (12)
32(cid:25) (cid:26)ocox (cid:0) x (cid:1) co
(cid:0)Z1V(Zy1)VZ((cid:17)) (cid:18) (cid:20) (cid:21)(cid:19)
2;3
As noted by Lighthill and others, it is best to include as many properties of the source
as possible prior to any modeling of the turbulent sources. To include the e(cid:11)ects of source
convection the statistical properties of the sources may be described in a moving frame of
reference. This also has the advantage that it is the temporal variation in this frame that
controls the noise radiation. In a (cid:12)xed reference frame, the temporal variation is dominated
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by convection e(cid:11)ects. For example, as noted by Goldstein, a frozen pattern of turbulence
convecting subsonically would radiate no noise. However, its local time variation would
depend on the convection velocity and the turbulent length scales and would not be zero.
Let
(cid:24) = (cid:17) icoMc(cid:28) (13)
(cid:0)
where, i is a unit vector in the direction of the mean (cid:13)ow and Mc is the convection Mach
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number of the turbulent eddies. This gives
1
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! x (cid:24)
S(x;!) = 3 5 2 exp i! (1 Mccos(cid:18))(cid:28)
32(cid:25) (cid:26)ocox (cid:0) (cid:0) x (cid:1) co (cid:2)
(cid:0)Z1V(Zy1)VZ((cid:24)) (cid:26) (cid:20) (cid:21)(cid:27)
Rm(y1;(cid:24);(cid:28))d(cid:24)dy1d(cid:28) (14)
where Rm denotes the two-point cross correlation function of the Lighthill stress tensor in
the moving reference frame. Also,
cos(cid:18) = x1=x (15)
The wavenumber/frequency spectrum of the turbulent sources is given by
1
1 i(!(cid:28)(cid:0)(cid:11)(cid:1)(cid:24))
H(y1;(cid:11);!) = 4 e Rm(y1;(cid:24);(cid:28))d(cid:24)d(cid:28) (16)
(2(cid:25))
VZ((cid:24))(cid:0)Z1
where (cid:11) is a wavenumber vector. This describes the spatial and temporal periodicity of the
source. Then,
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(cid:25)! 1 !x
S(x;!) = 5 2 H y1; ;!(1 Mccos(cid:18)) dy1 (17)
2(cid:26)ocox xco (cid:0)
V(Zy1) (cid:20) (cid:21)
This shows that the far (cid:12)eld noise depends on the components of the source wavenum-
ber/frequency spectrum with a wavenumber that gives a sonic velocity in the direction of a
far (cid:12)eld observer and at a Doppler shifted frequency.
To this point, other than the far (cid:12)eld assumption, no approximations have been made.
However, to proceed further, it is necessary to introduce a model for the two point cross
correlation. It is usually assumed that, in the moving frame of reference, the correlation
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takes on a Gaussian form (see Ffowcs Williams and Lilley ),
2
2 2 4 (cid:24) 2 2
Rm(y1;(cid:24);(cid:28)) = A (cid:26)susexp j 2j !s(cid:28) (18)
"(cid:0) ‘s (cid:0) #
where, ‘s isa characteristiclength scale, !s is acharacteristic frequency inthe movingframe,
us is a velocity scale that characterizes the turbulent velocity (cid:13)uctuations, and A determines
the magnitude of the correlation. Here, it assumed that the characteristic length scale is the
same in all directions. This restriction could easily be relaxed. It should be noted that this
is simply a model for the turbulent statistics in a moving reference frame. It is not an exact
relationship. Then,
!x
H y1; ;!(1 Mccos(cid:18))
xco (cid:0)
(cid:26) (cid:27)
2 2 3 2 2 2
A 2 4(cid:25) ‘s ! (1 Mccos(cid:18)) ((cid:11)‘s)
= 4(cid:26)sus exp (cid:0) 2 exp (19)
(2(cid:25)) !s ((cid:0) 4!s ) ((cid:0) 4 )
where, (cid:11) is the magnitude of the wavenumber vector. Now,
!s‘s us
(cid:11)‘s = m (20)
co (cid:24) co (cid:24)
where m is a characteristic Mach number for the turbulence and provides a measure of the
compactness of the source region. For compact sources, m << 1, so that at 90 degrees to
the jet axis we obtain,
2 4 2
A 2 4 3 3 ! !
S(x;!) = 5 2 (cid:26)sus‘s!s exp 2 dy1 (21)
32(cid:25)(cid:26)ocox !s (cid:0)4!s
VZ (cid:18) (cid:19) (cid:26) (cid:27)
If a RANS k " solution is available it is possible to determine the contribution of each
(cid:0)
elemental volume in the numerical grid to the radiated noise spectrum. Here, k and (cid:15) are
the turbulent kinetic energy and visous dissipation rate per unit mass respectively. From
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equation (21) this contribution is given by
2 4 2
A 2 4 3 3 ! !
dS(x;!) = 5 2 (cid:26)sus‘s!s exp 2 dV (22)
32(cid:25)(cid:26)ocox ( !s (cid:0)4!s )
(cid:18) (cid:19) (cid:18) (cid:19)
The length and time scales may be obtained from the k " solution. We assume that
(cid:0)
3=2
!s = 2(cid:25)=(cid:28)s; (cid:28)s = c(cid:28)(k=(cid:15)); and ‘s = c‘(k =(cid:15)): (23)
The k (cid:15) solutions indicate that, along the location of maximum shear, both (cid:28)s and ‘s vary
(cid:0)
nearly linearly with axial distance. Then, with us = 2k=3,
p
2 3 2 4 2
A c‘ (cid:25) 2 7=2 ! !
dS(x;!) = 3 5 2 (cid:26)sk exp 2 dV (24)
c(cid:28) 9(cid:26)ocox ( !s (cid:0)4!s )
(cid:18) (cid:19) (cid:18) (cid:19)
In this form there are only two combinations of constants that may be determined by com-
parison with experiments. The factor
4 2
! !
exp (25)
2
!s (cid:0)4!s
(cid:18) (cid:19) (cid:18) (cid:19)
describes how each volume element contributes to a range of frequencies about the local
characteristic frequency !s.
Figure (1) shows a comparison of the predicted radiated noise with experimental data
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by Tanna et al. The one-third octave experimental data have been converted to spectral
density assuming a smooth spectrum. The jet isoperating at Mj = 0:911and Tj=To = 0:975.
o
The jet diameter is 0:0508 m and the observer location is at 90 to the jet axis at a distance
of 72 jet diameters. The k (cid:15) solution has been obtained using the code developed by Thies
(cid:0)
12 12
and Tam. The grid is described by Thies and Tam. It grows in physical size in a stepwise
manner as the solution is marched in the axial direction. For the present calculations the
solutions are saved at every quarter of a jet diameter downstream. The volume of the
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