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Non-linear effect of wall linings on sound attenuation in ducts
Journal ofSound
and Vibration (1985) 103(3) 395-415
NON-LINEAR ON
SOUND
EFFECT OF WALL LININGS ATTENUATION
M. NAMBA
AND
IN DUCTS
K. KOBAYASHI~
Department of Aeronautical Engineering, Kyushu University, Hakozaki, Higashi-ku, Fukuoka 812, Japan (Received 18 June 1984, and in revised form 10 December 1984) The equivalent surface source method is extended to the analysis of high intensity sound propagation in a duct whose wall is partially treated with a sound absorbing material. The propagation of sound in the gas is assumed to be linear, but the acoustic resistance of the sound absorbing material is assumed to be a function of the normal acoustic velocity. The problem is reduced to a non-linear integro-differential equation for the fluid particle displacement at the lined wall surface, which can be solved by a successive approximation method. Numerical examples show that the non-linear effect decreases or increases the peak sound attenuation rate of the lowest mode depending upon the linear component of the resistance. The dependence of the attenuation spectrum on modal phase difference of multi-mode incident waves is heavily affected by the non-linear effect. In the case of incident waves of multi-circumferential modes, different circumferential modes are generated by the non-linear effect.
1.
INTRODUCTION
The ducts of current turbofan engines are treated with sound absorbing wall liners which consist of a thin perforated facing sheet over honeycomb cells backed by an impervious wall. Previous experimental studies indicate that the acoustic impedance of such liners is no longer independent of the sound pressure level (ML) when it exceeds about 130 dB.
The threshold value, however, is not definite, and will be lower or higher depending upon facing sheets. The resistance component of the impedance is a function of the normal acoustic velocity at the liner surface, while the reactance component is a weakly decreasing function of it [ 1,2]. The attenuations of high-intensity sound in ducts lined with such a non-linear acoustic material have been calculated by using the one-dimensional transmission line approximation [3], the finite difference approximation [4], the method of multiple scales [5] and the harmonic linearization method [6]. The analytical model treated by these methods has been confined to a two-dimensional duct, with zero mean flow, the lined wall of which is of an infinite length. The attenuation rates per unit duct length predicted by these methods show good agreements with experimental data near the entrance region of lined sections. It seems difficult, however, to apply these methods to a partially lined duct in which the sound wave is partly reflected from the end of the lined section or to the case of multi-mode incident waves. Therefore they are of limited practical use as the analytical method for a non-linear problem in which simple superposition is no longer valid. In this paper the equivalent surface source method developed by Namba and Fukushige [7] is extended to the analysis of high intensity multi-mode sound waves propagating in a duct whose wall is partially treated with a non-linear acoustic material. This method I’Present address Ishikawajima-Harima Heavy Industries Co., Ltd., 229 Tonogaya, Gun, Tokyo
Mizuho-Machi,
Nishitama-
190-12, Japan.
395 0022-460X/85/230395+21
%03.00/O
IQ 1985 Academic
Press Inc. (London)
Limited
396
M. NAMBA
AND
K. KOBAYASHI
allows one to apply the mathematical formulations in the linear analysis to this non-linear analysis with little modification and the problem is ultimately reduced to a non-linear integro-differential equation for the fluid particle displacement at the wall surface, which can be easily solved by a successive approximation method. In the following sections, this analytical procedure is outlined and numerical examples are given to show the non-linear effect of wall linings.
2. PROBLEM
FORMULATION
2.1. ANALYTICAL MODEL AND BASIC EQUATIONS As Figure 1 shows, the analytical model considered in this paper is a channel between two parallel walls of wall to wall height H*, in which a mean subsonic flow is contained. The mean flow state is assumed to be uniform with density p$, speed of sound at and Mach number M( < 1). In the following analysis unstarred symbols denote dimensionless quantities: lengths, velocities, density, pressure, acoustic admittance and time are nondimensionalized with respect to H*, a$, p$, &a$*, (&a$)-’ and H*fa$ respectively.
H=l
Figure 1. The co-ordinate
system of the duct model.
The channel extends infinitely in the circumferential direction z as well as in the mean flow direction x. The acoustic field, however, is assumed to be periodic in the z direction so that an integral multiple of the circumferential wave length of each mode is equal to a circumferential distance 2 W. Hence the model can approximate an annular channel of high hub-to-tip ratio with the circumferential length 2 W. The lower wall (y =0) is rigid everywhere, while the upper wall (y = 1) is rigid except the section from x = 0 to x = L, where it is treated with a locally reacting sound-absorbing liner of non-linear admittance /3. Many investigators have indicated that the acoustic resistance of acoustic duct lining materials is dependent on both the normal acoustic velocity and the tangential mean flow velocity at the material surface. The latter dependence is called the grazing flow effect. Thus the admittance /3 is generally considered to be a function of the normal acoustic velocity and the duct flow velocity. In this paper, the admittance /3 is assumed to be dependent only on the normal acoustic velocity at the lining surface since the paper aims at making clear solely the non-linear effect of the acoustic materials. From the practical point of view, the exclusion of the grazing flow effect will be justified for low mean Mach numbers. One should note, however, that no essential difficulty exists in including the
NON-LINEAR
EFFECT
OF
WALL
391
LININGS
grazing flow effect in the present theory if the dependence of /3 on the mean Mach number can be prescribed. In addition to the above-mentioned non-linearity of the acoustic properties of the lining material, the non-linearity of the gas itself exists in the high-intensity sound propagation, which is significant at SIX’s exceeding 160 dB. In the present analysis, however, the gas non-linearity is not considered. Therefore the validity of the analysis will be limited to the case of SPL’s up to 160dB. Then one obtains a linearized wave equation and a linearized momentum equation, V=F - D=il Dt= = 0,
D;/ Dt = -I’@,
Cl,21
where D/ Dt = d/at + M a/ax,
(3)
and p’(x, y, z, t) and ?(x, y, z, t) denote the acoustic pressure and acoustic fluid velocity, respectively. Since the acoustic admittance of the lined wall depends only on the normal acoustic velocity at the wall surface as mentioned before, the boundary condition at the wall can be described as &=O
aty=O,
(4)
aty=l and0CxCL aty=landx
(5a,b)
where 4, = a;i,,,/at, and i,,, denotes the y component of fluid particle displacement at the wall surface. In addition to the wall boundary condition all the acoustic quantities are required to satisfy the z-wise periodicity condition with a periodic distance 2 W. Since the admittance p in equation (5a) is a function of the absolute value of &,, the boundary condition at the lined surface is non-linear with respect to the acoustic disturbance quantities. Let an incident wave of a single frequency component be considered and higher harmonics generated by the non-linear effect be neglected. Then one can use complex notation as follows: ~~~~~~]
= [ zzii]
e-‘“‘.
(6)
Here o denotes a dimensionless radian frequency. The acoustic resistance of the resistive plate is assumed to be a linear function of l&,1, while the acoustic reactance is assumed to be independent of 1&I. Then $,,/S(l$,,l) includes the product term l&l&,. Expressing &, by 4, = Re (~117~1e-i(w’-cp)), with -ioqW = 01~~l eicp,one can decompose I&J&, into a Fourier series as follows: l&l& = w21n,,,12 Re [(8/3~) e-““‘-“‘+(8/15r) Taking only the fundamental
harmonic component,
-iwdp
e-3i(w’-‘p)+. . .].
one can rewrite equation (5a) as
=P[@/3~r)4~~11.
Equation (7) implies that the non-linear correction for the resistance is proportional 8wln,)/3~, which is similar to that presented in reference [6].
(7)
to
398 2.2.
M. NAMBA INCIDENT
The
ACOUSTIC
AND
K. KOBAYASHI
PRESSURE
present method of analysis is based on regarding the acoustic pressure p as P=Pe+Pd,
(8)
where pe is an incident acoustic pressure component which is defined as the pressure that would be realized if the lined wall were replaced by a solid surface, while pd is a disturbance acoustic pressure component which is created by the presence of the lined wall. The incident acoustic pressure pe must satisfy the rigid wall boundary condition 13p,/dy=O
aty=Oorl.
(9)
Then pe can be generally expressed as
xexp{ -(iwM/a2+~n,=,,~)x+im,~z/
W+icp,,,,},
(10)
where a2=l-M2 (n,_,~)‘={(y2,rr2(n~+m2e/ R me.“, =
(11)
, W’)-w2}/a4,
Pm&l; vLI&)‘> 0
1
(12)
-Wm_nJ;(&_,)‘< 0I ’ &()=1, E, =2(n ZO).
(13)
Equation (10) shows that pe is composed of modal components specified by the notation (m, n,), where m, and n, denote the z-wise wave number and y-wise nodal number respectively. Actually, however, one has only to consider a finite number of cut-on modes which satisfy (fin,_,.)” < 0, since the incident waves are assumed to be generated at a far upstream station (x = -00). Furthermore C,,_ = IC,,,_,,l exp (itP,,& denotes the relative magnitude of the complex amplitude of the (m, n,) mode at the entrance of the lined section (x = 0), and one has that
c * L,,n,12= 1,
me.%
where I* denotes the summation of the cut-on modes only. P, is the pressure amplitude normalized by ~:a$~ and is related to the incident sound pressure level SPL, (dB) by P, = (p;/po*a,*2)10SPLJ20,
(19
where p: = 2.8 x 10e5 Pa is the reference pressure amplitude. One should note that excluding cut-off modes from pe is rather a matter of convenience. In cases where sound sources are located near the lined section, or in the cases of shear flows where the modes are not orthogonal, the cut-off modes also should be included in pe. 2.3. DISTURBANCE ACOUSTIC PRESSURE A lined surface can be represented by a surface of mass source singularity. Let Q(x, y) e-‘“’ be mass source density at the lined surface. Then, as shown in reference [7], the disturbance pressure pd induced by the mass source is given by Pd tx,
Y, z)
=
Q(5,
SVWX-~,Y,
z-5)
dtd5.
(16)
NON-LINEAR
EFFECT
OF WALL
Here KP(x - 5, y, z - 5) e-‘“’ denotes the disturbance source e-‘“’ located at (5, 1, 5) and is given by KP(x-[,y,
pressure induced by a unit mass
z-l)=& =f_KP,(x-~,~)~~~~(~-')'~, m
l -se
KP,(x-&y)=
399
LININGS
(17)
m
-iwM(x-c)/n*
m4-l)” n;.
~0s
w
x{io/(a*fk,,)+M sgn(x-01 exp(-%&--~I)-
(18)
2.4. INTEGRO-DIFFERENTIAL EQUATION As shown in reference [7], the y component of acoustic fluid particle displacement nW(x, z) e-‘“* at the wall surface is related to the mass source Q(x, z) e-‘“’ by Q(x, z)= -(-io+Ma/ax)%%(x,z).
(19)
By using equations (8), ( 16) and ( 19), equation (7) can be written as an integro-diff erential equation for 7,(x, z) in the form
-ivdx,
z) = P
[
I[
$477,(x, z)l PAX,1, z)
It should be noted that equation (20) is non-linear with respect to n,(x, z) since /3 is a function of 77,(x, z). In order to solve this non-linear integro-differential equation, let n,,,(x, z) be expanded into a Fourier double series form which satisfies the z-wise periodicity such that %v(x,z)=2 where a,(0)
f A,,@J : In-x v=-,
0) eimaz’W,
(21)
denotes the x-wise mode function and x = (L/2)(1 +cos e).
It is desirable that each (x = 0 and L). Koch and However what is the most paper two types of @JO),
(22)
of 0,(e) shows an appropriate behaviour at the liner edges Mohring [8] investigated various types of edge condition. appropriate edge condition still remains to be settled. In this . i.e.
@,(8)=sin
vB, and
@,(r?)=cos(v-l)e,
(23,24)
are considered. For equation (23) it is assumed the normal displacement of the fluid particle 7, is zero at the liner edges x = 0 and L, while in equation (24) 7, is allowed to show discontinuous finite jumps across the liner edges. Substitution of equation (21) into equation (20) gives f e’ ‘mrz’W f { -2i~~,(S)-B(~~l~u(x,z)l)~~P~,,(x, In=-cc “=,
= P(&
4%v(x, z+.(x,
1, z),
l)]&,,
(25)
400
M. NAMBA
AND
K. KOBAYASHI
where the expression for IKP,,,.(x, y) is given in Appendix I. The liner admittance p (8~ (qJ x, z)l/ (3 7~))also should satisfy the z-wise periodicity and hence can be expanded into Fourier series at each x as follows:
(26) Here PI,&; PAr., ,) implies that &,,,(x; P&,,,,,)
is dependent on all PJ,,,,,, (v’= 1,2,. . . , m’=0,*1,*2 , . . .). Substituting equations (10) and (26) into equation (25), multiplying it by e-ik?n’w and integrating it with respect to z from - W to W, one obtains
‘Pe
c *&
me
p,k-m,,(x;
f’e&,,,) C * n,
e
x exp{ - GUM/a’+ fL,&
c,,e~(-l)“e
+ icp,,,,.),
k = 0, *l, *2,. . . ,
(27)
where denotes the Kronecker delta. Equation (27) is reduced to a set of non-linear algebraic equations for A, by taking a finite number of A, (v = 1,2, . . . , v,; j = 0, *1, *2,. . . , *MM,) and letting equation (27) be satisfied at V, control points x = (L/2) (l+COSBi) (i=l,2,*.*,’ v,,), The non-linear algebraic equations can be solved by using a successive approximation method. Thus the new pth approximation of AvJ, i.e., A$.‘, can be determined as the solution of the linear algebraic equations which are obtained from equation (27) by substituting the known (CL- 1)th approximation of coefficients A(?-‘) for A ,, , in P,,,,l(x; PJvf,,,). In the present analysis, the first approximation A$ is zbtained b; zeglecting the non-linear resistance component. Then the above-mentioned process is repeated until the following convergence criterion is satisfied: 6j,k
-1O-3 s (IAp,$ - IA&‘)l)/IA(I$eI
c 10-3.
(28)
Once the coefficients A, have been determined, every acoustic quantity can be evaluated in terms of the A,‘s as shown in reference [7]. 2.5.
ACOUSTIC
INTENSITY
mean acoustic power through unit cross section of the channel, which is nondimensionalized with respect to &a$‘, is given by w 1 E=1 ;{f(l+M)2(pa+~u)+Mp~+Muti}dydz, (29) 2w I -w I 0 The
where J? and ti denote the complex conjugates of p and u respectively. Here one should note that p = pe +pd and w = u, + ud, where u, and ud denote the axial aCOUStiC dOCitieS of the incident wave and the disturbance wave respectively. (The expressions for u, and ud are shown in reference [7].) Expressing pd and ud in terms of A,, and carrying out the integration in equation (29), one obtains modal expressions for the mean acoustic power as follows: in the lined section, i.e., 0 S x C L, E(x)=
f
f
WI=--m n=ll
J%,“(x),
(30)
NON-LINEAR
EFFECT
OF
WALL
LININGS
401
and in the unlined section, i.e., x < 0 or x > L, E’=
C * E;,,,
(31)
m.n
where superscripts + and - correspond to the values for x > L and x < 0, respectively. Expressions for El,,,,(x) and E& for @,( 0) = sin vf.9are given in reference [7]. Some modifications required for @,( 0) = cos ~0 are described in Appendix I. In terms of the modal transmitted power E “,,, and the incident acoustic power E, (see reference [7]), the modal transmitted power ratio AP,( m, n) and total transmitted power ratio AP, scaled in decibels are given by AP,( M, n) = 10 log,,, (EJ EL,,)
and
AP, = 10 log,,, (EJ E+),
respectively. In the lined section (i.e., 0 ZGx s L), the modal power ratio AP(x; total power ratio AZ’(x) are given by Wx;
m, n) = 10 log,,
[WEL,n(x)l
and
AZ’(x) = 10 log,, [EJE(x)],
(3% 33) m, n) and
(34,35)
respectively.
3. NUMERICAL
RESULTS
In the calculations to be presented, the wall liner is assumed to consist of a thin resistive layer over honeycomb cells backed by an impervious wall. If the dependence of the reactance component of the liner upon the normal acoustic particle velocity at the liner surface is assumed to be negligible, then the non-linear impedance (i.e., the reciprocal of the admittance /3) of such a liner is described by
1/P[@/3~M77,ll=
~~+(~/3~)~~~177~1-~i(~~~I~~-~~t (41,
(36)
where the resistance component is composed of the linear term, RL, and the non-linear term, 8R~+ln,l/(37r). I-Ier e o0 and d are the characteristic radian frequency of the facing sheet and the depth of the honeycomb cavities, respectively. 3.1. SINGLE-MODE INCIDENT WAVE Figures 2 and 3 show spectra of the transmitted power ratio AP, for various mean flow Mach numbers predicted on the basis of the mode function series (23) and (24) respectively. First one can see that the trend of the convection effect of the mean flow is not affected by the presence of the non-linear effect. Thus the sound attenuation is increased for M < 0 (upstream propagation) and is decreased for M > 0 (downstream propagation) for both RN = 0 (linear impedance) and RN = 180 (non-linear impedance). One should note, of course, that no grazing flow effect on the liner impedance has been taken into account in this calculation. Comparison between these figures indicates that prediction by equation (23) gives higher values of AP, than that by equation (24) when the mean flow is present (M # 0). On the other hand there is little difference between both predictions in the case of no mean flow (M = 0). It has been found, however, that the difference between the two predictions becomes distinguishable even for M = 0 when the attenuation is very large (AZ’, > 40 dB), and furthermore the prediction by use of @,( 0) = cos (V - l)e generally gives higher values of AP, which show better agreement with our experimental results. (These results are to be published later.) Therefore numerical examples presented hereafter are limited to those solved by use of @,( 0) = cos (V - I)@, and further only the case of M = 0 is considered.
402
M. NAMBA
AND
K. KOBAYASHI
Figure 2. Sound power attenuation spectra at various mean flow Mach numbers calculated on the basis of R, = 0; - - -, equation (23), i.e., e,(e) = sin I@. L = 5, W = 1, w,, = 5, d = 0.75, S&T.,= 160 dB, R, = 1, -, RN = 180, (0,O) mode incident.
r 2c
9 C Q 1c
Figure 3. Sound power attenuation spectra at various mean flow Mach numbers calculated on the basis of equation (24), i.e., Q”(0) = cos (Y - 1)0. Other conditions are the same as in Figure 2.
NON-LINEAR
EFFECT
OF
WALL
LININGS
403
Figure 4. Comparison of the present analytical results*with the experimental data of Kurze and Allen [4] for the sound R, = 0.95, R,
attenuation rate at the entrance = 150, (0,O) mode incident.
of the lined section.
M = 0, L = 10, W = 1, o0 = 5, d = 0.75,
Figure 4 shows a comparison with the experimental data of reference [4] for acoustic attenuation per unit duct length. The theoretical values of the acoustic attenuation are defined as the x-wise slope of total power ratio Al’(x) at the entrance of the lined section x = 0. Although the theoretical values in Figure 4 have been obtained for L = 10, it is confirmed from Figure 5 that the slope of AZ’(x) at x = 0 is essentially independent of the length L of the lined section unless L is very small. Figure 4 shows that the numerical results agree well with the experimental data. One should note that no distinguishable
Figure 5. Axial variation of the sound power ratios at the peak attenuation frequency and long (L= 10) lined sections. Other conditions are the same as in Figure 5.
w = 1.5 for short (L = 5)
404
M. NAMBA
SPL.
AND
=130
K. KOBAYASHI
dB
20
8 0 c Q IC
C
1
05 w
Figure 6. Sound power attenuation spectra at various incident sound pressure levels. M = 0, L = 5, W = 1, w0 = 5, d = 0.75, RL = 1, R, = 180, (0,O) mode incident.
change occurs in the attenuation rate per unit length at x = 0 by further reducing SPL, below 140 dB, i.e., the non-linear effect is very small for SPL, < 140 dB. Figure 6 shows the effect of the incident sound pressure level SPL, on the attenuation spectrum. The peak sound attenuation at the liner resonance frequency decreases with increasing SPL, while the attenuation at the frequencies apart from the resonance frequency increases slightly with increasing SPL,. The reason why the non-linear effect is distinguishable around the resonance frequency is that an increase in the resistance of the liner with an increase of SPL, appears clearly when the reactance is relatively small. It can be confirmed from numerical results that the spectrum becomes independent of SPL, for SPL, < 130 dB, becoming coincident with the spectrum based on the linear analysis. This result means that the non-linear effect on AP, is of little significance for SPL, < 130 dB. Figures 7 and 8 show the effect of the non-linear resistance coefficient RN. In the case of a relatively large linear resistance RL (Figure 7) increasing RN gives rise to flattening of the attenuation spectrum curve; the non-linearity has an unfavorable effect near the resonance frequency while it has a favorable effect at the frequencies away from the resonance frequency. In the case of a relatively small linear resistance RL (Figure 8), however, increasing RN increases the attenuation for all frequencies. A similar result has been reported in reference [6]. Examples are shown in Figures 9 and 10 for the x-wise variation of the normal particle velocity at the lined surface near the resonance frequency. In the case of a relatively large linear resistance R,_ (Figure 9), the amplitude of the particle velocity almost monotonically decreases as the sound wave goes through the lined section. Such a monotonic decrease can be seen for all SPLe’s but the slope becomes smaller with increasing SPL,. On the other hand the difference between the phases of i,,, and Pe in the lined section is almost constant around 180” and independent of SPL,. In the case of a relatively small linear
NON-LINEAR
EFFECT
OF WALL
405
LININGS
1 RN=0
PI
2(
6 e ci 0 l(
1
0.5
5
w
Figure 7. Variation of sound power attenuation spectrum with increasing non-linear resistivity linear resistivity R, = 1. M = 0, L = 5, W = 1, o0 = 5, d = 0.75, SPL, = 160 dB, (0,O) mode incident.
I-
I
I “II
I
at a high
1
x )-
40
g
a
30
20I_
10
CJL w
Figure 8. Variation of sound power attenuation spectrum with increasing resistivity RL = 0.05. Other conditions are the same as in Figure 7.
non-linear
resistivity
at a low linear
resistance RL (Figure lo), however, the amplitude of the particle velocity shows nonmonotonic variation (especially for SIX, < 140 dB) and its phase difference is no longer constant. This is caused by the fact that higher modes with amplitudes comparable with that of the incident mode can be generated in the lined section.
406
AND
M. NAMBA ,
,
,
,
K. KOBAYASHI 360’
I
,
,
/
I 1
I 2
I 3
I 4
2700 -
900
0
/ 1
, 2
/ 3
/ 4
5
X
O”
-
5
X
Figure 9. Axial variation of normal acoustic particle velocity on the lined surface at the peak attenuation frequency w = 1.3 and a high linear resistivity R, = 1. M = 0, L = 5, W = 1, o0 = 5, d = 0.75, R, = 180, (0,O) mode incident.
f
/
r
fi / 5
5 X
Figure 10. Axial variation of normal acoustic particle velocity on the lined surface at the peak attenuation frequency w = 1.3 and a low linear resistivity RL = 0.05. Other conditions are the same as in Figure 9.
Figure 11 shows the dependence of the peak attenuation at the resonance frequency upon the linear resistance RL and non-linear resistance coefficient RN. It shows that there exists an optimum linear resistance RL which maximizes the peak attenuation for a fixed non-linear resistance coefficient RN, and that it becomes smaller with increasing RN. This suggests that large attenuations can be obtained even at high sound pressure levels if one can control the linear resistance and the non-linear resistance coefficient independently. The ceiling of sound attenuation can be seen between RL = 0.1and 0.3 for RN = 0 where no further increase of the attenuation beyond 70 dB can be obtained. It would be too
NON-LINEAR
EFFECT
OF WALL
407
LININGS
I “77,
1 c 1”)
70 -
60 -
50 z :
40-
% 30 -
20 -
10 -
01
I
0.05
IILl
I
I
,.,I
0.5
0.1
1
RL
Figure 11. Variations of the sound power ratios at the peak attenuation frequency o = 1.4 with increasing linear resistivity for various non-linear resistivities. M = 0, L = 5, W = 1, R,/w, = 0.2, SPL, = 160 dB, (0,O) mode incident.
R,=450
\
0 iii
E
w
: 10 -
0 1
Figure 12. Sound power attenuation d = 0.75, R, = 1, SPL, = 160 dB.
w
spectra
5
for the (0, 1) mode incident
10
wave. A4 = 0, L = 5, W = 1, w0 = 5,
hasty to conclude, however, that the ceiling really exists, .since the reliability of numerical values of the power ratios larger than 60 dB will be rather small. Figure 12 shows the non-linear effect of the liner for the (0,l) mode incident wave. In this case, one can see that increasing the non-linear effect brings about increase of the
408
M. NAMBA AND K. KOBAYASHI
attenuation for all frequencies. It should be noted that the attenuation vanishes at w = 4.2 and 8.4. This is because the lining wall acts as a rigid wall at these frequencies where the reactance component becomes infinite. 3.2. MULTI-MODE
INCIDENT
WAVE
In Figures 13 to 16 the attenuation spectra are given for multi-mode incident waves composed of (0,O) and (0,l) modes. Figures 13 and 15 give the cases of absence of the non-linear effect while Figures 14 and 16 gives the cases of presence of the non-linear effect. In Figures 13 and 14, these two modes are in phase at x = 0 but various ratios of the squared modal amplitudes are considered. Comparison between these figures indicates that the presence of the non-linear effect gives no essential change in the spectrum curve, just increasing the attenuation slightly. In Figures 15 and 16, on the other hand, the squared amplitudes of the (0,O) and (0,l) modes are fixed in the ratio 1: 2 but various phase differences 00,1 between these modes are considered. It can be seen by comparison between these figures that the dependence of the attenuation spectrum on modal phase differences of multi-mode incident waves is heavily affected by the non-linear effect of the liner. For instance the wave of Qi0,1= 180” is most attenuated for RN = 0 (Figure 15), but that of @o,l= 270” is most attenuated for RN = 180 (Figure 16). This seems due to the fact that a change of the modal phase difference gives rise to a substantial variation in the amplitude of the normal acoustic velocity at the lined surface, resulting in a substantial change of the distribution of the acoustic resistance in the lined section.
1 ! \
c* 0,O’ .c*
0.1
=O.l
1.o
I
“--i !:/.\\ !
1 I. 1.
\\\\\ Ii
\ \ :
Figure 13. Sound power attenuation spectra for various modal amplitude ratios of incident waves of two modes (0,O) and (0,l) at zero non-linear resistivity R N=O. M=O, L=5, W=l, w,=5, d=0.75, R,=l, cpo,o= (PO.1 = 0.
NON-LINEAR
EFFECT
OF WALL
409
LININGS
1
c&o:C&l =O.l 1.2 1 :l 1 :o
5 w
Figure 14. Sound power attenuation spectra for various modal amplitude ratios of incident waves of two modes (0,O) and (0,l) at a finite non-linear resistivity RN -- 180 and SPL, = 160 dB. Other conditions are the same as in Figure 13.
I
15
p &,,,= 270’ -
--A\ :
10 1 0 :
5
CI_ 1
5
10
w
Figure 15. Sound power attenuation spectra for various modal phase differences of incident waves of two modes (0,O) and (0, 1) at zero non-linear resistivity R, = 0. M =0, L= 5, W= 1, wo= 5, d =0.75, RL= 1, IC,.,lZ : IC,*,12 = 1 : 2, cpo.0= 0.
410
M. NAMBA
AND
K. KOBAYASHI
1 ! \
! ! I\ \i i’
\ \ 4
10
Figure 16. Sound power attenuation spectra for various modal phase differences of incident waves of two modes (0,O) and (0,l) at a finite non-linear resistivity R N = 180 and SPL,= 160 dB. Other conditions are the same as in Figure 15.
I
G 20
10 -!
W
Figure 17. Modal and total sound power ratios for a multi-circumferential mode incident wave at zero non-linear resistivity RN = 0.M = 0,L = 3. W = 1.5, o0 = 2.5, d = 033, RL = 0.5, (1,O) and (-1.0) mode incident wave with C,., = C_,,.
NON-LINEAR
EFFECT
OF WALL
LININGS
411
60-
50iis 0
40-
+=. E z
Figure 18. Modal non-linear resistivity
3o
and total sound power ratios for a multi-circumferential mode incident wave at a finite R, = 316 and SPL, = 160 dB. Other conditions are the same as in Figure 17.
As examples of incident waves of multi-circumferential modes, incident waves composed of (m, n,) and (-M,, n,) modes have been considered. In this case the resistance of the lined wall is no longer circumferentially uniform due to the presence of the non-linear term 8RN 1i/,1/(3 7~)which is a function of z. As a result, modes of circumferential wave numbers not equal to m, or -m, can be generated. (The mathematical proof of this mechanism is given in Appendix II.) Examples are given in Figures 17, 18 and 19 for the case where the incident waves are composed of (1,0) and (-1,O) modes with the ratio of the pressure amplitude 1: 1, i.e., C,,0 = C_,,,. Since the acoustic energy of an (m, n) mode is equal to that of a (-m, n) mode in this case, the energy of the (m,n) mode in these figures implies the sum of energies of (m, n) and (-m, n) modes. Figures 17 and 18 show modal transmitted sound power spectra for the liner with RN = 0 and 180, respectively. One should note that existence of modes other than those of the incident wave implies the modal scattering of energy. As Figure 17 shows, modal transfer of sound energy occurs only between the modes of different nodal number n when the non-linear effect of the liner is absent (RN = 0).As shown in Figure 18, however, the non-linear effect brings about transfer of sound energy between the modes of different circumferential wave number m as well as between the modes of different nodal number n. However the energies of the higher circumferential modes are much smaller than that of the incident wave, and hence these modes give little contribution to the total sound power. Comparison between these figures suggests that there seems to exist a threshold frequency (o + 3.5 in this case) below which sound power attenuation is significantly suppressed by the ngn-linear effect. Figure 19 gives an example of the axial variation of the modal sound power in the lined section. It shows again that modes of circumferential wave number not equal to Im,l(= 1) are generated by the non-linear effect. However these modes are cut-off and
412
M. NAMBA AND K. KOBAYASHI
50
m z! -z i. x P Q
.._..I” (1,2)
30 )
/
\.
20
/---’
\
-
1’
.-7Tyi1
10
0 X/L
Figure 19. Axial variations of modal and total sound power ratios in the lined section for a multi-circumferential mode incident wave at o = 2.5. Other conditions are the same as in Figure 18.
energies carried by these modes tend to zero near the entrance and exit of the liner section (x = 0, L). A similar phenomenon can be seen also for the (1,1) and (1,2) modes which are cut-off modes too.
4. CONCLUSION
A singularity method in which a lined wall surface is regarded as a surface with equivalent surface mass sources has been applied to evaluate the attenuation of sound waves in a channel partially treated with a non-linear acoustic material. The method can be easily applied to the case of multi-mode incident waves. The non-linearity of wall linings does not always have an unfavorable effect on sound attenuation of the lowest mode near the resonance frequency. There exists an optimum linear resistance which maximizes the peak attenuation for a fixed non-linear resistance coefficient. The optimum linear resistance decreases as the non-linear resistance coefficient increases. In the case of multi-mode incident waves, the dependence of the attenuation on modal phase difference is heavily affected by the non-linear effect. On the other hand, waves of different circumferential modes are generated by the non-linear effect when incident waves are composed of multi-circumferential modes. Comparison with experimental data indicates that the edge condition which allows finite jumps of particle displacement across the liner edges is suitable in the absence of
NON-LINEAR
EFFECT OF WALL
413
LININGS
mean flow. However further investigation is needed of the edge condition as well as of the grazing flow effect when mean flow is present.
REFERENCES 1. U. INGARD and H. ISING 1967 Journal of the Acousticul Society of America 42, 6-17. Acoustic nonlinearity of an oriface. 2. W. E. ZORUMSKI and T. L. PARROT 1971 NASA TN 6196. Nonlinear acoustic theory for rigid porous materials. 3. U. INGARD 1968 Journal ofthe Acoustical Society of America 43, 167-168. Nonlinear attenuation of sound in a duct. 4. U. J. KURZE and C. H. ALLEN 1971 Journal of the Acoustic& Society of America 49, 1643-1653. Influence of Bow and high sound level on the attenuation in a lined duct. 5. A. H. NAYFEH and M.-S. TSAI 1974 Journal of the Acoustical Society of America 55,1166-l 172. Nonlinear acoustic propagation in two-dimensional ducts. 6. M.-S. TSAI 1980 American Institute of Aeronautics and Astronautics Journal 18, 1180-1185. Harmonic linearization method for high-intensity sound. I. M. NAMBA and K. FUKUSHIGE 1980 Journal of Sound and Vibration 73, 125-146. Application of the equivalent surface source method to the acoustics of duct systems with non-uniform wall impedance. 8. W. KOCH and W. MOHRING 1983 American Institute of Aeronautic und Astronautics Journal 21, 200-213. Eigensolutions for liners in uniform mean flbw ducts.
APPENDIX
I: EXPRESSIONS
FOR
In a manner similar to the formulation cos (v - 1) 0 can be expressed by
ZKP,,,,,(x, y), R,,,(B),
L,,,(O)
AND
S,,,(h)
for @,( 0) = sin ~0, ZKP,,,.(x, 1) for @, (@)=
m
ZKPrn,“(X, 1) =
H
iwLcos(v-l)cpsin~-2M(v-1)sin(v-l)(o}{KP’S’(x-~,
1)
0
(Al) where the expressions for KP’S’(x - 5, 1) and KP’,R’(x - 5, 1) have been given in reference [7]. Furthermore, ZKPS,(x) is expressed as follows: ZKPS,(x)
=
io ff31T[ 2
~{sL,(e)-sL,-,(e)}-2(V-i)MsL,-,(e)
-5 a,(e)
=
1
I
i0{SM,(e)-SM,_,(e)}-
l-(-l)” V
log 4 6v.o- rr log
+
szkf,(e) =
4(v-l)M L
SM,-0)
I
,
(A21
2 sin ve
sm ve) log 1~0s 8 --OS cp[ dp,
&‘.
(A3) (A4)
In evaluation of SL,( 8) and Sk&,( Z3),the integrations in equations (A3) and (A4) should be performed numerically.
414
M. NAMBA
AND
K. KOBAYASHI
Expressions for I?,,,,( 19), L,,,( 0) and S,,,,(A) as given for @,( f3)= sin v6 in reference [7] should be replaced for @,( 0) = cos (v - l)e by .9
1 1
Rn.“(e) = “z,(-l)YAv,, L,,,(e)
IOne I
! I -%t,n~
X
XeiWM5'a2
e”“.“~
0
S,,,(h)
(A9
x{ioLcos(v-l)cpsinp-2M(v-l)sin(v-l)rp}dq, ?I = eiA F (-l)“A,, eihcos~ v=, I0 x{ioLcos(v-l)cpsincp-2M(v-l)sin(v-l)cp}dp.
(A61
APPENDIX II: THE MECHANISM OF GENERATION OF MODES OF DIFFERENT CIRCUMFERENTIAL WAVE NUMBERS It can be proved by using equation (27) that the non-linear effect brings about generation of modes of different circumferential wave numbers in the case of incident waves composed of multi-circumferential modes. Here two cases of incident waves are considered, i.e., (i) a single (m, n,) mode incident wave and (ii) an incident wave composed of (-m, n,) and (m, n,) modes. One should note that for k # 0 Plil(T
647)
0) = 0,
since the wall lining is assumed to be z-wise uniform. Case (i). Since the incident wave is composed of the (m, n,) mode only, Cln,, = 0,
(A8)
in equation ( 10) for rn # M,. From equations (A7), (A8) and (27), one can obtain, forj # A’?
“J
=
()
m,,
(A9)
7
and from equation (21) 1%(X, r)l=2
00 c A1:!nc@“(@ .
(AlO)
v=,
Hence the non-linear resistance term proportional can obtain for k # 0 P,k,(-Y R-4$4
to In,,,] is independent
A’?’ YJ
=
(All)
= 0.
Repeating this process, one obtains, for all p’s and for j # ()
*
of z. Thus one
m,
(A121
Hence equation (A12) and modal expressions for the mean acoustic power show that modes except those of m = m, have no energy. In other words, no modes of different circumferential wave numbers can be generated in Case (i). Case (ii). Similarly to Case (i), one can have, for m # km, c In,”= 0.
(A13)
C In,“, = C-m&.
(Al4)
Here it is assumed, for simplicity, that
NON-LINEAR
EFFECT
OF
One should note that IKP,,,,(x, y) = IKP_,,.(x, and (27), one can obtain for j # m, A”! = 0 YJ
and
WALL
415
LININGS
y). Then from equations (A13), (A14)
A(” = A(‘ ) “,--me # 0. u.4
(A15, A16)
Then IG(x,
r)l = 4 ,.!, A$<@,(@) lcos (m,rz/ W.
(A17)
Since the non-linear resistance term is a function of z, one can have, even for k # 0, P,LI(x; R/4$)
# 0.
(A181
Repeating this process gives A’?’ VJ # 0 7
for j f f m, and for p f 1. This result means that modes of circumferential not equal to m, can be generated in Case (ii).
(A19) waves numbers
and Vibration (1985) 103(3) 395-415
NON-LINEAR ON
SOUND
EFFECT OF WALL LININGS ATTENUATION
M. NAMBA
AND
IN DUCTS
K. KOBAYASHI~
Department of Aeronautical Engineering, Kyushu University, Hakozaki, Higashi-ku, Fukuoka 812, Japan (Received 18 June 1984, and in revised form 10 December 1984) The equivalent surface source method is extended to the analysis of high intensity sound propagation in a duct whose wall is partially treated with a sound absorbing material. The propagation of sound in the gas is assumed to be linear, but the acoustic resistance of the sound absorbing material is assumed to be a function of the normal acoustic velocity. The problem is reduced to a non-linear integro-differential equation for the fluid particle displacement at the lined wall surface, which can be solved by a successive approximation method. Numerical examples show that the non-linear effect decreases or increases the peak sound attenuation rate of the lowest mode depending upon the linear component of the resistance. The dependence of the attenuation spectrum on modal phase difference of multi-mode incident waves is heavily affected by the non-linear effect. In the case of incident waves of multi-circumferential modes, different circumferential modes are generated by the non-linear effect.
1.
INTRODUCTION
The ducts of current turbofan engines are treated with sound absorbing wall liners which consist of a thin perforated facing sheet over honeycomb cells backed by an impervious wall. Previous experimental studies indicate that the acoustic impedance of such liners is no longer independent of the sound pressure level (ML) when it exceeds about 130 dB.
The threshold value, however, is not definite, and will be lower or higher depending upon facing sheets. The resistance component of the impedance is a function of the normal acoustic velocity at the liner surface, while the reactance component is a weakly decreasing function of it [ 1,2]. The attenuations of high-intensity sound in ducts lined with such a non-linear acoustic material have been calculated by using the one-dimensional transmission line approximation [3], the finite difference approximation [4], the method of multiple scales [5] and the harmonic linearization method [6]. The analytical model treated by these methods has been confined to a two-dimensional duct, with zero mean flow, the lined wall of which is of an infinite length. The attenuation rates per unit duct length predicted by these methods show good agreements with experimental data near the entrance region of lined sections. It seems difficult, however, to apply these methods to a partially lined duct in which the sound wave is partly reflected from the end of the lined section or to the case of multi-mode incident waves. Therefore they are of limited practical use as the analytical method for a non-linear problem in which simple superposition is no longer valid. In this paper the equivalent surface source method developed by Namba and Fukushige [7] is extended to the analysis of high intensity multi-mode sound waves propagating in a duct whose wall is partially treated with a non-linear acoustic material. This method I’Present address Ishikawajima-Harima Heavy Industries Co., Ltd., 229 Tonogaya, Gun, Tokyo
Mizuho-Machi,
Nishitama-
190-12, Japan.
395 0022-460X/85/230395+21
%03.00/O
IQ 1985 Academic
Press Inc. (London)
Limited
396
M. NAMBA
AND
K. KOBAYASHI
allows one to apply the mathematical formulations in the linear analysis to this non-linear analysis with little modification and the problem is ultimately reduced to a non-linear integro-differential equation for the fluid particle displacement at the wall surface, which can be easily solved by a successive approximation method. In the following sections, this analytical procedure is outlined and numerical examples are given to show the non-linear effect of wall linings.
2. PROBLEM
FORMULATION
2.1. ANALYTICAL MODEL AND BASIC EQUATIONS As Figure 1 shows, the analytical model considered in this paper is a channel between two parallel walls of wall to wall height H*, in which a mean subsonic flow is contained. The mean flow state is assumed to be uniform with density p$, speed of sound at and Mach number M( < 1). In the following analysis unstarred symbols denote dimensionless quantities: lengths, velocities, density, pressure, acoustic admittance and time are nondimensionalized with respect to H*, a$, p$, &a$*, (&a$)-’ and H*fa$ respectively.
H=l
Figure 1. The co-ordinate
system of the duct model.
The channel extends infinitely in the circumferential direction z as well as in the mean flow direction x. The acoustic field, however, is assumed to be periodic in the z direction so that an integral multiple of the circumferential wave length of each mode is equal to a circumferential distance 2 W. Hence the model can approximate an annular channel of high hub-to-tip ratio with the circumferential length 2 W. The lower wall (y =0) is rigid everywhere, while the upper wall (y = 1) is rigid except the section from x = 0 to x = L, where it is treated with a locally reacting sound-absorbing liner of non-linear admittance /3. Many investigators have indicated that the acoustic resistance of acoustic duct lining materials is dependent on both the normal acoustic velocity and the tangential mean flow velocity at the material surface. The latter dependence is called the grazing flow effect. Thus the admittance /3 is generally considered to be a function of the normal acoustic velocity and the duct flow velocity. In this paper, the admittance /3 is assumed to be dependent only on the normal acoustic velocity at the lining surface since the paper aims at making clear solely the non-linear effect of the acoustic materials. From the practical point of view, the exclusion of the grazing flow effect will be justified for low mean Mach numbers. One should note, however, that no essential difficulty exists in including the
NON-LINEAR
EFFECT
OF
WALL
391
LININGS
grazing flow effect in the present theory if the dependence of /3 on the mean Mach number can be prescribed. In addition to the above-mentioned non-linearity of the acoustic properties of the lining material, the non-linearity of the gas itself exists in the high-intensity sound propagation, which is significant at SIX’s exceeding 160 dB. In the present analysis, however, the gas non-linearity is not considered. Therefore the validity of the analysis will be limited to the case of SPL’s up to 160dB. Then one obtains a linearized wave equation and a linearized momentum equation, V=F - D=il Dt= = 0,
D;/ Dt = -I’@,
Cl,21
where D/ Dt = d/at + M a/ax,
(3)
and p’(x, y, z, t) and ?(x, y, z, t) denote the acoustic pressure and acoustic fluid velocity, respectively. Since the acoustic admittance of the lined wall depends only on the normal acoustic velocity at the wall surface as mentioned before, the boundary condition at the wall can be described as &=O
aty=O,
(4)
aty=l and0CxCL aty=landx
(5a,b)
where 4, = a;i,,,/at, and i,,, denotes the y component of fluid particle displacement at the wall surface. In addition to the wall boundary condition all the acoustic quantities are required to satisfy the z-wise periodicity condition with a periodic distance 2 W. Since the admittance p in equation (5a) is a function of the absolute value of &,, the boundary condition at the lined surface is non-linear with respect to the acoustic disturbance quantities. Let an incident wave of a single frequency component be considered and higher harmonics generated by the non-linear effect be neglected. Then one can use complex notation as follows: ~~~~~~]
= [ zzii]
e-‘“‘.
(6)
Here o denotes a dimensionless radian frequency. The acoustic resistance of the resistive plate is assumed to be a linear function of l&,1, while the acoustic reactance is assumed to be independent of 1&I. Then $,,/S(l$,,l) includes the product term l&l&,. Expressing &, by 4, = Re (~117~1e-i(w’-cp)), with -ioqW = 01~~l eicp,one can decompose I&J&, into a Fourier series as follows: l&l& = w21n,,,12 Re [(8/3~) e-““‘-“‘+(8/15r) Taking only the fundamental
harmonic component,
-iwdp
e-3i(w’-‘p)+. . .].
one can rewrite equation (5a) as
=P[@/3~r)4~~11.
Equation (7) implies that the non-linear correction for the resistance is proportional 8wln,)/3~, which is similar to that presented in reference [6].
(7)
to
398 2.2.
M. NAMBA INCIDENT
The
ACOUSTIC
AND
K. KOBAYASHI
PRESSURE
present method of analysis is based on regarding the acoustic pressure p as P=Pe+Pd,
(8)
where pe is an incident acoustic pressure component which is defined as the pressure that would be realized if the lined wall were replaced by a solid surface, while pd is a disturbance acoustic pressure component which is created by the presence of the lined wall. The incident acoustic pressure pe must satisfy the rigid wall boundary condition 13p,/dy=O
aty=Oorl.
(9)
Then pe can be generally expressed as
xexp{ -(iwM/a2+~n,=,,~)x+im,~z/
W+icp,,,,},
(10)
where a2=l-M2 (n,_,~)‘={(y2,rr2(n~+m2e/ R me.“, =
(11)
, W’)-w2}/a4,
Pm&l; vLI&)‘> 0
1
(12)
-Wm_nJ;(&_,)‘< 0I ’ &()=1, E, =2(n ZO).
(13)
Equation (10) shows that pe is composed of modal components specified by the notation (m, n,), where m, and n, denote the z-wise wave number and y-wise nodal number respectively. Actually, however, one has only to consider a finite number of cut-on modes which satisfy (fin,_,.)” < 0, since the incident waves are assumed to be generated at a far upstream station (x = -00). Furthermore C,,_ = IC,,,_,,l exp (itP,,& denotes the relative magnitude of the complex amplitude of the (m, n,) mode at the entrance of the lined section (x = 0), and one has that
c * L,,n,12= 1,
me.%
where I* denotes the summation of the cut-on modes only. P, is the pressure amplitude normalized by ~:a$~ and is related to the incident sound pressure level SPL, (dB) by P, = (p;/po*a,*2)10SPLJ20,
(19
where p: = 2.8 x 10e5 Pa is the reference pressure amplitude. One should note that excluding cut-off modes from pe is rather a matter of convenience. In cases where sound sources are located near the lined section, or in the cases of shear flows where the modes are not orthogonal, the cut-off modes also should be included in pe. 2.3. DISTURBANCE ACOUSTIC PRESSURE A lined surface can be represented by a surface of mass source singularity. Let Q(x, y) e-‘“’ be mass source density at the lined surface. Then, as shown in reference [7], the disturbance pressure pd induced by the mass source is given by Pd tx,
Y, z)
=
Q(5,
SVWX-~,Y,
z-5)
dtd5.
(16)
NON-LINEAR
EFFECT
OF WALL
Here KP(x - 5, y, z - 5) e-‘“’ denotes the disturbance source e-‘“’ located at (5, 1, 5) and is given by KP(x-[,y,
pressure induced by a unit mass
z-l)=& =f_KP,(x-~,~)~~~~(~-')'~, m
l -se
KP,(x-&y)=
399
LININGS
(17)
m
-iwM(x-c)/n*
m4-l)” n;.
~0s
w
x{io/(a*fk,,)+M sgn(x-01 exp(-%&--~I)-
(18)
2.4. INTEGRO-DIFFERENTIAL EQUATION As shown in reference [7], the y component of acoustic fluid particle displacement nW(x, z) e-‘“* at the wall surface is related to the mass source Q(x, z) e-‘“’ by Q(x, z)= -(-io+Ma/ax)%%(x,z).
(19)
By using equations (8), ( 16) and ( 19), equation (7) can be written as an integro-diff erential equation for 7,(x, z) in the form
-ivdx,
z) = P
[
I[
$477,(x, z)l PAX,1, z)
It should be noted that equation (20) is non-linear with respect to n,(x, z) since /3 is a function of 77,(x, z). In order to solve this non-linear integro-differential equation, let n,,,(x, z) be expanded into a Fourier double series form which satisfies the z-wise periodicity such that %v(x,z)=2 where a,(0)
f A,,@J : In-x v=-,
0) eimaz’W,
(21)
denotes the x-wise mode function and x = (L/2)(1 +cos e).
It is desirable that each (x = 0 and L). Koch and However what is the most paper two types of @JO),
(22)
of 0,(e) shows an appropriate behaviour at the liner edges Mohring [8] investigated various types of edge condition. appropriate edge condition still remains to be settled. In this . i.e.
@,(8)=sin
vB, and
@,(r?)=cos(v-l)e,
(23,24)
are considered. For equation (23) it is assumed the normal displacement of the fluid particle 7, is zero at the liner edges x = 0 and L, while in equation (24) 7, is allowed to show discontinuous finite jumps across the liner edges. Substitution of equation (21) into equation (20) gives f e’ ‘mrz’W f { -2i~~,(S)-B(~~l~u(x,z)l)~~P~,,(x, In=-cc “=,
= P(&
4%v(x, z+.(x,
1, z),
l)]&,,
(25)
400
M. NAMBA
AND
K. KOBAYASHI
where the expression for IKP,,,.(x, y) is given in Appendix I. The liner admittance p (8~ (qJ x, z)l/ (3 7~))also should satisfy the z-wise periodicity and hence can be expanded into Fourier series at each x as follows:
(26) Here PI,&; PAr., ,) implies that &,,,(x; P&,,,,,)
is dependent on all PJ,,,,,, (v’= 1,2,. . . , m’=0,*1,*2 , . . .). Substituting equations (10) and (26) into equation (25), multiplying it by e-ik?n’w and integrating it with respect to z from - W to W, one obtains
‘Pe
c *&
me
p,k-m,,(x;
f’e&,,,) C * n,
e
x exp{ - GUM/a’+ fL,&
c,,e~(-l)“e
+ icp,,,,.),
k = 0, *l, *2,. . . ,
(27)
where denotes the Kronecker delta. Equation (27) is reduced to a set of non-linear algebraic equations for A, by taking a finite number of A, (v = 1,2, . . . , v,; j = 0, *1, *2,. . . , *MM,) and letting equation (27) be satisfied at V, control points x = (L/2) (l+COSBi) (i=l,2,*.*,’ v,,), The non-linear algebraic equations can be solved by using a successive approximation method. Thus the new pth approximation of AvJ, i.e., A$.‘, can be determined as the solution of the linear algebraic equations which are obtained from equation (27) by substituting the known (CL- 1)th approximation of coefficients A(?-‘) for A ,, , in P,,,,l(x; PJvf,,,). In the present analysis, the first approximation A$ is zbtained b; zeglecting the non-linear resistance component. Then the above-mentioned process is repeated until the following convergence criterion is satisfied: 6j,k
-1O-3 s (IAp,$ - IA&‘)l)/IA(I$eI
c 10-3.
(28)
Once the coefficients A, have been determined, every acoustic quantity can be evaluated in terms of the A,‘s as shown in reference [7]. 2.5.
ACOUSTIC
INTENSITY
mean acoustic power through unit cross section of the channel, which is nondimensionalized with respect to &a$‘, is given by w 1 E=1 ;{f(l+M)2(pa+~u)+Mp~+Muti}dydz, (29) 2w I -w I 0 The
where J? and ti denote the complex conjugates of p and u respectively. Here one should note that p = pe +pd and w = u, + ud, where u, and ud denote the axial aCOUStiC dOCitieS of the incident wave and the disturbance wave respectively. (The expressions for u, and ud are shown in reference [7].) Expressing pd and ud in terms of A,, and carrying out the integration in equation (29), one obtains modal expressions for the mean acoustic power as follows: in the lined section, i.e., 0 S x C L, E(x)=
f
f
WI=--m n=ll
J%,“(x),
(30)
NON-LINEAR
EFFECT
OF
WALL
LININGS
401
and in the unlined section, i.e., x < 0 or x > L, E’=
C * E;,,,
(31)
m.n
where superscripts + and - correspond to the values for x > L and x < 0, respectively. Expressions for El,,,,(x) and E& for @,( 0) = sin vf.9are given in reference [7]. Some modifications required for @,( 0) = cos ~0 are described in Appendix I. In terms of the modal transmitted power E “,,, and the incident acoustic power E, (see reference [7]), the modal transmitted power ratio AP,( m, n) and total transmitted power ratio AP, scaled in decibels are given by AP,( M, n) = 10 log,,, (EJ EL,,)
and
AP, = 10 log,,, (EJ E+),
respectively. In the lined section (i.e., 0 ZGx s L), the modal power ratio AP(x; total power ratio AZ’(x) are given by Wx;
m, n) = 10 log,,
[WEL,n(x)l
and
AZ’(x) = 10 log,, [EJE(x)],
(3% 33) m, n) and
(34,35)
respectively.
3. NUMERICAL
RESULTS
In the calculations to be presented, the wall liner is assumed to consist of a thin resistive layer over honeycomb cells backed by an impervious wall. If the dependence of the reactance component of the liner upon the normal acoustic particle velocity at the liner surface is assumed to be negligible, then the non-linear impedance (i.e., the reciprocal of the admittance /3) of such a liner is described by
1/P[@/3~M77,ll=
~~+(~/3~)~~~177~1-~i(~~~I~~-~~t (41,
(36)
where the resistance component is composed of the linear term, RL, and the non-linear term, 8R~+ln,l/(37r). I-Ier e o0 and d are the characteristic radian frequency of the facing sheet and the depth of the honeycomb cavities, respectively. 3.1. SINGLE-MODE INCIDENT WAVE Figures 2 and 3 show spectra of the transmitted power ratio AP, for various mean flow Mach numbers predicted on the basis of the mode function series (23) and (24) respectively. First one can see that the trend of the convection effect of the mean flow is not affected by the presence of the non-linear effect. Thus the sound attenuation is increased for M < 0 (upstream propagation) and is decreased for M > 0 (downstream propagation) for both RN = 0 (linear impedance) and RN = 180 (non-linear impedance). One should note, of course, that no grazing flow effect on the liner impedance has been taken into account in this calculation. Comparison between these figures indicates that prediction by equation (23) gives higher values of AP, than that by equation (24) when the mean flow is present (M # 0). On the other hand there is little difference between both predictions in the case of no mean flow (M = 0). It has been found, however, that the difference between the two predictions becomes distinguishable even for M = 0 when the attenuation is very large (AZ’, > 40 dB), and furthermore the prediction by use of @,( 0) = cos (V - l)e generally gives higher values of AP, which show better agreement with our experimental results. (These results are to be published later.) Therefore numerical examples presented hereafter are limited to those solved by use of @,( 0) = cos (V - I)@, and further only the case of M = 0 is considered.
402
M. NAMBA
AND
K. KOBAYASHI
Figure 2. Sound power attenuation spectra at various mean flow Mach numbers calculated on the basis of R, = 0; - - -, equation (23), i.e., e,(e) = sin I@. L = 5, W = 1, w,, = 5, d = 0.75, S&T.,= 160 dB, R, = 1, -, RN = 180, (0,O) mode incident.
r 2c
9 C Q 1c
Figure 3. Sound power attenuation spectra at various mean flow Mach numbers calculated on the basis of equation (24), i.e., Q”(0) = cos (Y - 1)0. Other conditions are the same as in Figure 2.
NON-LINEAR
EFFECT
OF
WALL
LININGS
403
Figure 4. Comparison of the present analytical results*with the experimental data of Kurze and Allen [4] for the sound R, = 0.95, R,
attenuation rate at the entrance = 150, (0,O) mode incident.
of the lined section.
M = 0, L = 10, W = 1, o0 = 5, d = 0.75,
Figure 4 shows a comparison with the experimental data of reference [4] for acoustic attenuation per unit duct length. The theoretical values of the acoustic attenuation are defined as the x-wise slope of total power ratio Al’(x) at the entrance of the lined section x = 0. Although the theoretical values in Figure 4 have been obtained for L = 10, it is confirmed from Figure 5 that the slope of AZ’(x) at x = 0 is essentially independent of the length L of the lined section unless L is very small. Figure 4 shows that the numerical results agree well with the experimental data. One should note that no distinguishable
Figure 5. Axial variation of the sound power ratios at the peak attenuation frequency and long (L= 10) lined sections. Other conditions are the same as in Figure 5.
w = 1.5 for short (L = 5)
404
M. NAMBA
SPL.
AND
=130
K. KOBAYASHI
dB
20
8 0 c Q IC
C
1
05 w
Figure 6. Sound power attenuation spectra at various incident sound pressure levels. M = 0, L = 5, W = 1, w0 = 5, d = 0.75, RL = 1, R, = 180, (0,O) mode incident.
change occurs in the attenuation rate per unit length at x = 0 by further reducing SPL, below 140 dB, i.e., the non-linear effect is very small for SPL, < 140 dB. Figure 6 shows the effect of the incident sound pressure level SPL, on the attenuation spectrum. The peak sound attenuation at the liner resonance frequency decreases with increasing SPL, while the attenuation at the frequencies apart from the resonance frequency increases slightly with increasing SPL,. The reason why the non-linear effect is distinguishable around the resonance frequency is that an increase in the resistance of the liner with an increase of SPL, appears clearly when the reactance is relatively small. It can be confirmed from numerical results that the spectrum becomes independent of SPL, for SPL, < 130 dB, becoming coincident with the spectrum based on the linear analysis. This result means that the non-linear effect on AP, is of little significance for SPL, < 130 dB. Figures 7 and 8 show the effect of the non-linear resistance coefficient RN. In the case of a relatively large linear resistance RL (Figure 7) increasing RN gives rise to flattening of the attenuation spectrum curve; the non-linearity has an unfavorable effect near the resonance frequency while it has a favorable effect at the frequencies away from the resonance frequency. In the case of a relatively small linear resistance RL (Figure 8), however, increasing RN increases the attenuation for all frequencies. A similar result has been reported in reference [6]. Examples are shown in Figures 9 and 10 for the x-wise variation of the normal particle velocity at the lined surface near the resonance frequency. In the case of a relatively large linear resistance R,_ (Figure 9), the amplitude of the particle velocity almost monotonically decreases as the sound wave goes through the lined section. Such a monotonic decrease can be seen for all SPLe’s but the slope becomes smaller with increasing SPL,. On the other hand the difference between the phases of i,,, and Pe in the lined section is almost constant around 180” and independent of SPL,. In the case of a relatively small linear
NON-LINEAR
EFFECT
OF WALL
405
LININGS
1 RN=0
PI
2(
6 e ci 0 l(
1
0.5
5
w
Figure 7. Variation of sound power attenuation spectrum with increasing non-linear resistivity linear resistivity R, = 1. M = 0, L = 5, W = 1, o0 = 5, d = 0.75, SPL, = 160 dB, (0,O) mode incident.
I-
I
I “II
I
at a high
1
x )-
40
g
a
30
20I_
10
CJL w
Figure 8. Variation of sound power attenuation spectrum with increasing resistivity RL = 0.05. Other conditions are the same as in Figure 7.
non-linear
resistivity
at a low linear
resistance RL (Figure lo), however, the amplitude of the particle velocity shows nonmonotonic variation (especially for SIX, < 140 dB) and its phase difference is no longer constant. This is caused by the fact that higher modes with amplitudes comparable with that of the incident mode can be generated in the lined section.
406
AND
M. NAMBA ,
,
,
,
K. KOBAYASHI 360’
I
,
,
/
I 1
I 2
I 3
I 4
2700 -
900
0
/ 1
, 2
/ 3
/ 4
5
X
O”
-
5
X
Figure 9. Axial variation of normal acoustic particle velocity on the lined surface at the peak attenuation frequency w = 1.3 and a high linear resistivity R, = 1. M = 0, L = 5, W = 1, o0 = 5, d = 0.75, R, = 180, (0,O) mode incident.
f
/
r
fi / 5
5 X
Figure 10. Axial variation of normal acoustic particle velocity on the lined surface at the peak attenuation frequency w = 1.3 and a low linear resistivity RL = 0.05. Other conditions are the same as in Figure 9.
Figure 11 shows the dependence of the peak attenuation at the resonance frequency upon the linear resistance RL and non-linear resistance coefficient RN. It shows that there exists an optimum linear resistance RL which maximizes the peak attenuation for a fixed non-linear resistance coefficient RN, and that it becomes smaller with increasing RN. This suggests that large attenuations can be obtained even at high sound pressure levels if one can control the linear resistance and the non-linear resistance coefficient independently. The ceiling of sound attenuation can be seen between RL = 0.1and 0.3 for RN = 0 where no further increase of the attenuation beyond 70 dB can be obtained. It would be too
NON-LINEAR
EFFECT
OF WALL
407
LININGS
I “77,
1 c 1”)
70 -
60 -
50 z :
40-
% 30 -
20 -
10 -
01
I
0.05
IILl
I
I
,.,I
0.5
0.1
1
RL
Figure 11. Variations of the sound power ratios at the peak attenuation frequency o = 1.4 with increasing linear resistivity for various non-linear resistivities. M = 0, L = 5, W = 1, R,/w, = 0.2, SPL, = 160 dB, (0,O) mode incident.
R,=450
\
0 iii
E
w
: 10 -
0 1
Figure 12. Sound power attenuation d = 0.75, R, = 1, SPL, = 160 dB.
w
spectra
5
for the (0, 1) mode incident
10
wave. A4 = 0, L = 5, W = 1, w0 = 5,
hasty to conclude, however, that the ceiling really exists, .since the reliability of numerical values of the power ratios larger than 60 dB will be rather small. Figure 12 shows the non-linear effect of the liner for the (0,l) mode incident wave. In this case, one can see that increasing the non-linear effect brings about increase of the
408
M. NAMBA AND K. KOBAYASHI
attenuation for all frequencies. It should be noted that the attenuation vanishes at w = 4.2 and 8.4. This is because the lining wall acts as a rigid wall at these frequencies where the reactance component becomes infinite. 3.2. MULTI-MODE
INCIDENT
WAVE
In Figures 13 to 16 the attenuation spectra are given for multi-mode incident waves composed of (0,O) and (0,l) modes. Figures 13 and 15 give the cases of absence of the non-linear effect while Figures 14 and 16 gives the cases of presence of the non-linear effect. In Figures 13 and 14, these two modes are in phase at x = 0 but various ratios of the squared modal amplitudes are considered. Comparison between these figures indicates that the presence of the non-linear effect gives no essential change in the spectrum curve, just increasing the attenuation slightly. In Figures 15 and 16, on the other hand, the squared amplitudes of the (0,O) and (0,l) modes are fixed in the ratio 1: 2 but various phase differences 00,1 between these modes are considered. It can be seen by comparison between these figures that the dependence of the attenuation spectrum on modal phase differences of multi-mode incident waves is heavily affected by the non-linear effect of the liner. For instance the wave of Qi0,1= 180” is most attenuated for RN = 0 (Figure 15), but that of @o,l= 270” is most attenuated for RN = 180 (Figure 16). This seems due to the fact that a change of the modal phase difference gives rise to a substantial variation in the amplitude of the normal acoustic velocity at the lined surface, resulting in a substantial change of the distribution of the acoustic resistance in the lined section.
1 ! \
c* 0,O’ .c*
0.1
=O.l
1.o
I
“--i !:/.\\ !
1 I. 1.
\\\\\ Ii
\ \ :
Figure 13. Sound power attenuation spectra for various modal amplitude ratios of incident waves of two modes (0,O) and (0,l) at zero non-linear resistivity R N=O. M=O, L=5, W=l, w,=5, d=0.75, R,=l, cpo,o= (PO.1 = 0.
NON-LINEAR
EFFECT
OF WALL
409
LININGS
1
c&o:C&l =O.l 1.2 1 :l 1 :o
5 w
Figure 14. Sound power attenuation spectra for various modal amplitude ratios of incident waves of two modes (0,O) and (0,l) at a finite non-linear resistivity RN -- 180 and SPL, = 160 dB. Other conditions are the same as in Figure 13.
I
15
p &,,,= 270’ -
--A\ :
10 1 0 :
5
CI_ 1
5
10
w
Figure 15. Sound power attenuation spectra for various modal phase differences of incident waves of two modes (0,O) and (0, 1) at zero non-linear resistivity R, = 0. M =0, L= 5, W= 1, wo= 5, d =0.75, RL= 1, IC,.,lZ : IC,*,12 = 1 : 2, cpo.0= 0.
410
M. NAMBA
AND
K. KOBAYASHI
1 ! \
! ! I\ \i i’
\ \ 4
10
Figure 16. Sound power attenuation spectra for various modal phase differences of incident waves of two modes (0,O) and (0,l) at a finite non-linear resistivity R N = 180 and SPL,= 160 dB. Other conditions are the same as in Figure 15.
I
G 20
10 -!
W
Figure 17. Modal and total sound power ratios for a multi-circumferential mode incident wave at zero non-linear resistivity RN = 0.M = 0,L = 3. W = 1.5, o0 = 2.5, d = 033, RL = 0.5, (1,O) and (-1.0) mode incident wave with C,., = C_,,.
NON-LINEAR
EFFECT
OF WALL
LININGS
411
60-
50iis 0
40-
+=. E z
Figure 18. Modal non-linear resistivity
3o
and total sound power ratios for a multi-circumferential mode incident wave at a finite R, = 316 and SPL, = 160 dB. Other conditions are the same as in Figure 17.
As examples of incident waves of multi-circumferential modes, incident waves composed of (m, n,) and (-M,, n,) modes have been considered. In this case the resistance of the lined wall is no longer circumferentially uniform due to the presence of the non-linear term 8RN 1i/,1/(3 7~)which is a function of z. As a result, modes of circumferential wave numbers not equal to m, or -m, can be generated. (The mathematical proof of this mechanism is given in Appendix II.) Examples are given in Figures 17, 18 and 19 for the case where the incident waves are composed of (1,0) and (-1,O) modes with the ratio of the pressure amplitude 1: 1, i.e., C,,0 = C_,,,. Since the acoustic energy of an (m, n) mode is equal to that of a (-m, n) mode in this case, the energy of the (m,n) mode in these figures implies the sum of energies of (m, n) and (-m, n) modes. Figures 17 and 18 show modal transmitted sound power spectra for the liner with RN = 0 and 180, respectively. One should note that existence of modes other than those of the incident wave implies the modal scattering of energy. As Figure 17 shows, modal transfer of sound energy occurs only between the modes of different nodal number n when the non-linear effect of the liner is absent (RN = 0).As shown in Figure 18, however, the non-linear effect brings about transfer of sound energy between the modes of different circumferential wave number m as well as between the modes of different nodal number n. However the energies of the higher circumferential modes are much smaller than that of the incident wave, and hence these modes give little contribution to the total sound power. Comparison between these figures suggests that there seems to exist a threshold frequency (o + 3.5 in this case) below which sound power attenuation is significantly suppressed by the ngn-linear effect. Figure 19 gives an example of the axial variation of the modal sound power in the lined section. It shows again that modes of circumferential wave number not equal to Im,l(= 1) are generated by the non-linear effect. However these modes are cut-off and
412
M. NAMBA AND K. KOBAYASHI
50
m z! -z i. x P Q
.._..I” (1,2)
30 )
/
\.
20
/---’
\
-
1’
.-7Tyi1
10
0 X/L
Figure 19. Axial variations of modal and total sound power ratios in the lined section for a multi-circumferential mode incident wave at o = 2.5. Other conditions are the same as in Figure 18.
energies carried by these modes tend to zero near the entrance and exit of the liner section (x = 0, L). A similar phenomenon can be seen also for the (1,1) and (1,2) modes which are cut-off modes too.
4. CONCLUSION
A singularity method in which a lined wall surface is regarded as a surface with equivalent surface mass sources has been applied to evaluate the attenuation of sound waves in a channel partially treated with a non-linear acoustic material. The method can be easily applied to the case of multi-mode incident waves. The non-linearity of wall linings does not always have an unfavorable effect on sound attenuation of the lowest mode near the resonance frequency. There exists an optimum linear resistance which maximizes the peak attenuation for a fixed non-linear resistance coefficient. The optimum linear resistance decreases as the non-linear resistance coefficient increases. In the case of multi-mode incident waves, the dependence of the attenuation on modal phase difference is heavily affected by the non-linear effect. On the other hand, waves of different circumferential modes are generated by the non-linear effect when incident waves are composed of multi-circumferential modes. Comparison with experimental data indicates that the edge condition which allows finite jumps of particle displacement across the liner edges is suitable in the absence of
NON-LINEAR
EFFECT OF WALL
413
LININGS
mean flow. However further investigation is needed of the edge condition as well as of the grazing flow effect when mean flow is present.
REFERENCES 1. U. INGARD and H. ISING 1967 Journal of the Acousticul Society of America 42, 6-17. Acoustic nonlinearity of an oriface. 2. W. E. ZORUMSKI and T. L. PARROT 1971 NASA TN 6196. Nonlinear acoustic theory for rigid porous materials. 3. U. INGARD 1968 Journal ofthe Acoustical Society of America 43, 167-168. Nonlinear attenuation of sound in a duct. 4. U. J. KURZE and C. H. ALLEN 1971 Journal of the Acoustic& Society of America 49, 1643-1653. Influence of Bow and high sound level on the attenuation in a lined duct. 5. A. H. NAYFEH and M.-S. TSAI 1974 Journal of the Acoustical Society of America 55,1166-l 172. Nonlinear acoustic propagation in two-dimensional ducts. 6. M.-S. TSAI 1980 American Institute of Aeronautics and Astronautics Journal 18, 1180-1185. Harmonic linearization method for high-intensity sound. I. M. NAMBA and K. FUKUSHIGE 1980 Journal of Sound and Vibration 73, 125-146. Application of the equivalent surface source method to the acoustics of duct systems with non-uniform wall impedance. 8. W. KOCH and W. MOHRING 1983 American Institute of Aeronautic und Astronautics Journal 21, 200-213. Eigensolutions for liners in uniform mean flbw ducts.
APPENDIX
I: EXPRESSIONS
FOR
In a manner similar to the formulation cos (v - 1) 0 can be expressed by
ZKP,,,,,(x, y), R,,,(B),
L,,,(O)
AND
S,,,(h)
for @,( 0) = sin ~0, ZKP,,,.(x, 1) for @, (@)=
m
ZKPrn,“(X, 1) =
H
iwLcos(v-l)cpsin~-2M(v-1)sin(v-l)(o}{KP’S’(x-~,
1)
0
(Al) where the expressions for KP’S’(x - 5, 1) and KP’,R’(x - 5, 1) have been given in reference [7]. Furthermore, ZKPS,(x) is expressed as follows: ZKPS,(x)
=
io ff31T[ 2
~{sL,(e)-sL,-,(e)}-2(V-i)MsL,-,(e)
-5 a,(e)
=
1
I
i0{SM,(e)-SM,_,(e)}-
l-(-l)” V
log 4 6v.o- rr log
+
szkf,(e) =
4(v-l)M L
SM,-0)
I
,
(A21
2 sin ve
sm ve) log 1~0s 8 --OS cp[ dp,
&‘.
(A3) (A4)
In evaluation of SL,( 8) and Sk&,( Z3),the integrations in equations (A3) and (A4) should be performed numerically.
414
M. NAMBA
AND
K. KOBAYASHI
Expressions for I?,,,,( 19), L,,,( 0) and S,,,,(A) as given for @,( f3)= sin v6 in reference [7] should be replaced for @,( 0) = cos (v - l)e by .9
1 1
Rn.“(e) = “z,(-l)YAv,, L,,,(e)
IOne I
! I -%t,n~
X
XeiWM5'a2
e”“.“~
0
S,,,(h)
(A9
x{ioLcos(v-l)cpsinp-2M(v-l)sin(v-l)rp}dq, ?I = eiA F (-l)“A,, eihcos~ v=, I0 x{ioLcos(v-l)cpsincp-2M(v-l)sin(v-l)cp}dp.
(A61
APPENDIX II: THE MECHANISM OF GENERATION OF MODES OF DIFFERENT CIRCUMFERENTIAL WAVE NUMBERS It can be proved by using equation (27) that the non-linear effect brings about generation of modes of different circumferential wave numbers in the case of incident waves composed of multi-circumferential modes. Here two cases of incident waves are considered, i.e., (i) a single (m, n,) mode incident wave and (ii) an incident wave composed of (-m, n,) and (m, n,) modes. One should note that for k # 0 Plil(T
647)
0) = 0,
since the wall lining is assumed to be z-wise uniform. Case (i). Since the incident wave is composed of the (m, n,) mode only, Cln,, = 0,
(A8)
in equation ( 10) for rn # M,. From equations (A7), (A8) and (27), one can obtain, forj # A’?
“J
=
()
m,,
(A9)
7
and from equation (21) 1%(X, r)l=2
00 c A1:!nc@“(@ .
(AlO)
v=,
Hence the non-linear resistance term proportional can obtain for k # 0 P,k,(-Y R-4$4
to In,,,] is independent
A’?’ YJ
=
(All)
= 0.
Repeating this process, one obtains, for all p’s and for j # ()
*
of z. Thus one
m,
(A121
Hence equation (A12) and modal expressions for the mean acoustic power show that modes except those of m = m, have no energy. In other words, no modes of different circumferential wave numbers can be generated in Case (i). Case (ii). Similarly to Case (i), one can have, for m # km, c In,”= 0.
(A13)
C In,“, = C-m&.
(Al4)
Here it is assumed, for simplicity, that
NON-LINEAR
EFFECT
OF
One should note that IKP,,,,(x, y) = IKP_,,.(x, and (27), one can obtain for j # m, A”! = 0 YJ
and
WALL
415
LININGS
y). Then from equations (A13), (A14)
A(” = A(‘ ) “,--me # 0. u.4
(A15, A16)
Then IG(x,
r)l = 4 ,.!, A$<@,(@) lcos (m,rz/ W.
(A17)
Since the non-linear resistance term is a function of z, one can have, even for k # 0, P,LI(x; R/4$)
# 0.
(A181
Repeating this process gives A’?’ VJ # 0 7
for j f f m, and for p f 1. This result means that modes of circumferential not equal to m, can be generated in Case (ii).
(A19) waves numbers