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MATRIZANT APPROACH TO ACOUSTIC ANALYSIS OF PERFORATED MULTIPLE PIPE MUFFLERS CARRYING MEAN FLOW
Journal of Sound and Vibration (1996) 191(4), 505–518
MATRIZANT APPROACH TO ACOUSTIC ANALYSIS OF PERFORATED MULTIPLE PIPE MUFFLERS CARRYING MEAN FLOW E. D Department of Mechanical Engineering, Dokuz Eylul University, Bornova, Izmir, Turkey (Received 3 March 1995) A new method based on the matrizant theory is developed for acoustic analysis of perforated pipe muffler components. The analysis is presented in a generality encompassing any number of parallel pipes that communicate along a common length. The utility and potential of the method is demonstrated on a plug muffler and a reverse flow muffler. The analysis shows, for the first time, that the distributed parameter theory of plane wave propagation in a perforated pipe provides a more satisfactory setting than the segmentation method for the analysis of the effects of axially varying quantities such as the mean flow velocity. The effect of the mean flow velocity gradient on the transmission loss of the plug and reversed flow mufflers are computed and are shown to be insignificant for practical purposes. 71996 Academic Press Limited
1. INTRODUCTION
Perforated pipes are extensively used in automobile mufflers to attenuate internal combustion engine exhaust noise. At present, there are two competing methods for practical acoustic modelling of perforated pipe muffler components, namely, Sullivan’s segmentation method [1] and the distributed parameter method. A discussion of these methods has been presented in the book by Munjal [2]. In Sullivan’s segmentation method, the effect of the perforations is lumped into a few discrete planes with solid pipes assumed to be present between the planes. The distributed parameter method, on the other hand, is based on the solution of the one-dimensional equations of conservation of mass and momentum. In practical mufflers employing perforated pipes, the mean flow is either contained in the pipe or forced through the perforations on the pipe. In the latter case, the mean flow velocity will vary along the length of the perforated pipe and Sullivan’s method is well known for its capability to cope with such variations. In contrast, all previous authors who used the distributed parameter method chose to proceed by assuming the mean flow velocity to be uniform. With this simplification made, the governing differential equations emerge with constant coefficients, the solution of which is relatively simple to obtain. Munjal et al. [3] used a numerical decoupling procedure which is based on numerical extraction of the roots of the system characteristic equation. Peat [4] developed an improved numerical decoupling scheme which is based on the determination of the eigenvalues and eigenvectors of the system matrix by Hessenberg decomposition followed by the LR method and claimed that this approach gives better numerical stability. He also attempted to include the mean flow gradient in the analysis by assuming a constant gradient and keeping the gradient terms in the governing equations while using a constant 505 0022–460X/96/140505 + 14 $18.00/0
7 1996 Academic Press Limited
506
.
value for the mean flow velocity to remove the variable terms. This approach is not satisfactory because the variations neglected may be of the similar order as the gradient terms retained. Also, a gradient term appears to be missing in the governing equations used by Peat, who eventually used Sullivan’s segmentation method to compute the effects of constant mean flow velocity gradient on a plug muffler. Similarly, Munjal [2] has presented the distributed parameter method as having the weakness of being limited to the uniform mean flow velocity case and recommended Sullivan’s segmentation method for taking into account the mean flow velocity gradient. The aim of the present paper is to present a new approach for numerical implementation of the distributed parameter method. This approach is called the matrizant method because it is based on the matrizant theory [5]. It is comparable to Peat’s numerical decoupling method in that an eigenvalue problem is required to be solved; however, it has several advantages such as the elegance inherited from the matrizant theory, a more systematic re-arrangement of the governing equations and the capability to deal with the mean flow velocity gradient in a more rigorous theoretical setting and less computational effort than Sullivan’s segmentation method. Thus, contrary to the general belief, ability to deal with a mean flow velocity gradient emerges as the strength, rather than a weakness, of the distributed parameter modelling when the matrizant method is used. The paper will demonstrate application of the matrizant method on a plug muffler and a reversed flow muffler. Other features of the analysis are that the perforated multiple pipe two-port elements are formulated here, for the first time, in scattering matrix forms, and the theoretical formulation is presented in a generality encompassing the perforated pipe arrangements with any number of parallel pipes which communicate along a common length in an enclosing pipe or chamber.
2. ACOUSTIC ANALYSIS OF PERFORATED MULTIPLE PIPES
2.1. With visco-thermal losses neglected, acoustic wave propagation in a uniform pipe carrying mean flow is governed by the linearized forms of the equations of conservation of mass and momentum. Upon assuming uniform mean flow temperature and plane wave propagation with e−ivt time dependence, where i denotes the unit imaginary number, v the radian frequency and t the time, but allowing for axial variations in the mean flow velocity, the equations of conservations momentum and conservation of mass can be expressed, respectively, as (D + MM')p + (MD + M' − ik)V = 0,
(MD + M' − ik)p + DV = mc0 . (1, 2)
Here p denotes the acoustic pressure, V = vr0 c0 where v is the acoustic particle velocity, M is the mean flow Mach number, M = v0 /c0 where v0 denotes the mean flow velocity, m is the mass injected into the pipe per unit volume per unit time, k is the wavenumber, k = v/c0 , c0 is the speed of sound, r0 is the ambient density, D denotes differentiation with respect to x, the distance along the pipe, and M' is the mean flow Mach number gradient, that is, M' = DM. The Mach number is taken positive if the mean flow velocity is in the positive x-direction. Here, it may be of interest to note the M' term in equation (2) is the term which is missing in the equations developed by Peat [4]. Now consider the n-pipe element shown in Figure 1. This consists of n − 1 parallel perforated pipes that communicate with each other along a common length, L, in an enclosing hard walled pipe or chamber. Pipes are numbered from 1 to n, pipe 1 alwas being the enclosing chamber.
507
Figure 1. n-pipe acoustic element: n − 1 perforated pipes enclosed in a hard walled pipe or cavity.
For pipe 1, equation (2) can be expressed as (M1 D + M'1 − ik + b2 + b3 + · · · + bn )p1 + DV1 = b2 p2 + b3 p3 + · · · + bn pn ,
(3)
and, for a perforated pipe, (Mj D + M'j − ik + aj )pj + DVj = aj p1 ,
j = 2, 3, . . . , n,
(4)
where aj = 2sj /rj zj ,
bj = (Sj /S1 )aj ,
j = 2, 3, . . . , n.
(5)
Here S, s, r and z denote the cross-sectional area, the porosity, the hydraulic radius and the hole impedance of a pipe, respectively, and the subscript j refers to pipe j. Similarly, in a n-pipe element, equation (1) is written for pipe j as (D + Mj M'j )pj + (Mj D + M'j − ik)Vj = 0,
j = 1, 2, . . . , n.
(6)
Equations (3), (4) and (6) can be expressed concisely in matrix notation as: A(D)Q(x) = 0
(7)
A A · · · A1nLK Q1 (x)L K GA11 12 13 GG G GA21 A22 O · · · O GG Q2 (x)G GA31 O A33 · · · O GG Q3 (x)G=0, · · · · GG · G G ·· · · · · · · · · · · GG · G G kAn1 O O l k Ann Qn (x)l
(8)
or, in expanded form,
where Qj (x) =
$ %
pj (x) , Vj (x)
0 E x E L,
Aj j =
$
%
D + Mj M'j Mj D + M'j − ik , Mj D + M'j − ik + aj D
j = 1, 2, . . . , n, Aj1 =
$
0 −aj
%
0 , 0
A1j =
$
0 −bj
%
0 , 0
(9, 10) j = 2, 3, . . . , n,
(11)
and a1 = b2 + b3 + · · · + bn by definition and O denotes a 2 × 2 matrix whose elements are all zero. 2.2. Equation (7) constitutes a set of 2n first order ordinary differential equations for a n-pipe element in variables pj and Vj , j = 1, 2, . . . , n. Previous authors [3, 4] using numerical
.
508
decoupling have preferred to proceed with general solution of the governing equations by first collapsing the mass and momentum equations for each pipe into a single differential equation of the second order in D, a process which is easily carried out by eliminating Vj , and then transforming back the resulting set of n second order equations to a system of 2n first order equations. This approach produces the general solutions for the acoustic pressures, pj , and their derivatives with respect to x, p'j . This is rather inconvenient because, for the eventual application of the boundary conditions, the general solutions are required in an impedance matrix or, still better, in a scattering matrix form, and recasting of the native solutions in one of such forms involves cumbersome calculations. A more systematic and elegant approach here, is to proceed directly from equation (7). For this purpose, it is first observed that matrices A1j and Aj1 do not contain the D operator and that matrices Aj j can be written as Aj j =
$
% $
Mj M'j M'j − ik 1 + 0 Mj M'j − ik + aj
%
Mj D, 1
(12)
or, briefly, Aj j = Bj j + Mj D,
j = 1, 2, . . . , n.
(13)
Hence, it follows that, equation (7) can be expressed as DQ(x) = U(x)Q(x),
U(x) = −M−1B(x),
(14)
where
K G M=G G k
M1 O · · · O
O M2 · · · O
···
O
··· · · · ···
O · · · Mn
L G G, G l
(15)
and matrix B(x) has the same form as matrix A, equation (8), but with the diagonal blocks Aj j replaced by matrices Bj j . Note that the matrices Mj are defined in equation (12) and that the elements of matrix B will be functions of x when the mean flow velocity is not uniform along the pipe. Equation (14) is a set of 2n first order ordinary differential equations, in the variables pj and Vj , j = 1, 2, . . . , n, in the standard matrix form and the general solution can be written, by invoking the theory of the matrizant [5], Q(x) = [V]x0 Q(0),
0 E x E L,
(16)
where [V]x0 denotes the matrizant for the interval (0, x). This is a square matrix of order 2n which is a function of U(x) yet to be computed. The advantage of this approach is now clear: putting x = L in the foregoing relationship gives Q(L) = [V]L0 Q(0), which is simply the general solution of the governing equations expressed as a wave transfer relationship between the ends of the n-pipe element in an impedance matrix form, and [V]L0 , that is the matrizant, being the impedance matrix itself. Before embarking on the problem of calculation of the matrizant, the general solution will be derived also in a scattering matrix form. In fact, this is the preferred approach here because the analysis was developed for implementation with the computer code Acoustic Design of Exhaust Mufflers (ADEM) [6], which accepts the scattering matrix representations of acoustic two-port elements directly.
509
To obtain the general solution of the governing equations in a scattering matrix form, − two new variables, namely, p+ j (x) and pj (x), which are subsequently called the wave components, are introduced for each pipe by the transformation
$ % $
pj (x) 1 = Vj (x) 1
%$ %
p+ j (x) , p− j (x)
(17)
j = 1, 2, . . . , n.
(18)
1 −1
or, briefly, Q j (x) = Ej Pj (x),
Obviously, the matrices Ej are the same for all pipes; however, the subscript j will be retained for notational convenience. With this transformation applied, equation (14) becomes, H(x) = −E−1M−1B(x)E,
DP(x) = H(x)P(x),
(19)
where
K G E=G G k
E1 O · · · O
O E2 · · · O
··· ···
O O · · · En
L G G, G l
K P1 (x) L G P2 (x) G P(x) = G · G· G ·· G k Pn (x) l
(20)
By using the matrizant, the general solution of equation (19) can be expressed as P(x) = [T]x0 P(0),
0 E x E L,
(21)
where [T]x0 is the matrizant for the interval (0, x) and is a function of H(x). Putting x = L, one has P(L) = [T]L0 P(0),
(22)
Here, [T]L0 can be regarded as a physical scattering matrix for the n-pipe element. Indeed, with the ends of the n-pipe element connected to ordinary pipes, the wave components − p+ j (x) and pj (x) at x = 0 and x = L will denote the sound waves travelling in +x and −x directions in these pipes, respectively. By carrying out the matrix operations indicated in the second of equations (19), the matrix H(x) can be shown to have the form
K G G H(x) = G G G k
H11
H12
H13
···
H1n
H21 H31 · · · Hn1
H22 O · · · O
O H33 · · · O
··· ···
O O · · · Hnn
···
L G G G, G G l
(23)
L G, G G l
j = 1, 2, . . . , n (24)
where Mj M'j + 2(M'j − ik) + aj Mj M'j + aj K G 1 + Mj 1 + Mj 2Hj j = G Mj M'j + aj Mj M'j + 2(M'j − ik) + aj G k 1 − Mj 1 − Mj
.
510
K G 2H1j = G bj G k 1 − Mj −bj 1 + Mj
L G G, bj G 1 − Mj l −bj 1 + Mj
−aj −aj K G 1 + Mj 1 + Mj 2Hj1 = G aj aj G k 1 − Mj 1 − Mj
L G G, G l
j = 2, 3, . . . , n.
(25, 26)
2.3. To evaluate the matrizant [T]L0 , or, what is the same thing, the scattering matrix, for a n-pipe element, the basic interval 0 E x E L is divided into N parts by introducing intermediate points x1 , x2 , . . . , xN − 1 . For simplicity, the lengths of the parts are assumed to be all equal, i.e., l = xk − xk − 1 , k = 1, 2, . . . , N and xN = L = Nl. Then, by using the well known properties of the matrizant, it can be shown that, [T]L0 = [T]xxNN − 1 . . . [T]xx21 [T]x01 ,
(27)
where [T]xxkk − 1 denotes the matrizant for the interval (xk − 1 , xk ). Now, let jk be a point, always the mid-point in this paper, in the interval (xk − 1 , xk ), k = 1, 2, . . . , N. By regarding l as a sufficiently small quantity, one may assume, for the approximate evaluation of the matrizant, H(x) 3 H(jk ) = Hk , say. Then, Hk is independent of x and the matrizant for the interval (xk − 1 , xk ) can be shown to be [T]xxkk − 1 = eHk l
(28)
and the scattering matrix for a n-pipe element is then given by [T]L0 = eHN l eHN − 1 l . . . eH2 l eH1 l = eHl,
H = HN + HN − 1 + · · · + H 2 + H1 .
(29)
Hence, the problem of evaluation of the system scattering matrix is reduced to the evaluation of an exponential matrix function. The basic expression for the evaluation of an exponential matrix function is eHl = I + Hl +
H2l 2 +· · · , 2!
(30)
where I denotes a unit square matrix of dimension 2n. If l can be regarded as a small quantity of the first order, then the above series can be truncated at the second term, providing extreme simplification in the numerical calculations. The usefulness of such a simplification in practical computation is, however, apt to be limited by slow convergence. An accurate evaluation of the exponential matrix functions is therefore desirable and the following decoupling procedure is proposed here for this purpose. Let L be a diagonal matrix whose elements are the eigenvalues of H, say, lj , j = 1, 2, . . . , 2n, and F the modal matrix whose columns are the corresponding right eigenvectors. It is assumed, for simplicity, that all the eigenvectors are linearly independent. Then, as is well known from the theory of matrices [5], the collineatory transformation H = F−1LF applies, and therefore eHl = F−1 eLlF,
(31)
eLl = 4el1 l el2 l . . . el2n l 7,
(32)
where
in which 47 denotes a diagonal matrix. The eigenvalues and eigenvectors of matrices H can be computed by using a suitable algorithm. ADEM uses a Jacobi type algorithm.
511
Figure 2.(a) Plug muffler; (b) cross-flow expansion element; (c) cross-flow contraction element.
It should be noted that, if the mean flow velocity gradient is zero for all the pipes, then H1 = H2 = . . . = HN = h say, and equation (29) simplifies to [T]L0 = ehL, which can be evaluated directly by using the above described numerical decoupling procedure. 3. APPLICATION TO PRACTICAL MUFFLERS
Two types of mufflers will be considered in this section to demonstrate the utility and potential of the matrizant method of analysis developed in the previous section. First is the so-called plug muffler shown in Figure 2 and the other is the reverse flow muffler of Figure 3(a). There is published data on acoustic performance of both type of mufflers. 3.1. The scattering matrix of this muffler is obtained by combining the cross-flow expansion and contraction elements shown in Figure 2. These are essentially two-pipe elements but they have different boundary conditions. The application of the boundary conditions will be described for this example in some detail because, as far as the author is aware, this is the first paper in which a scattering matrix approach is employed in the acoustic analysis of perforated multiple pipe mufflers. The scattering matrix of a two-pipe element is derived as described in the previous section. For the application of the boundary conditions, it is convenient to express
Figure 3. Reverse flow muffler incorporating a three-pipe element.
.
512
Figure 4. Transmission loss of a plug muffler for M = 0·3 and various porosities, s.
equation (22) in partitioned form as
$ % $
P1 (L) T = 11 P2 (L) T21
T12 T22
%$ %
P1 (0) : P2 (0)
(33)
that is, P1 (L) = T11 P1 (0) + T12 P2 (0),
P2 (L) = T21 P1 (0) + T22 P2 (0).
(34)
With reference to Figure 2(b), the boundary conditions for the plug expansion are given as R1 (L) and R2 (0), where R(x) denotes the reflection coefficient, R(x) = p−(x)/p+(x), and the subscripts refer to the two pipes. The scattering matrix for the plug expansion can now be derived as a relationship between P1 (0) and P2 (L) as follows. Multiplying the first of equations (34) from left by Y1 (L) = [R1 (L) − 1] gives Y1 (L)T11 P1 (0) + Y1 (L)T12 R2 (0)p+ 2 (0) = 0,
(35)
where R2 (0) = {1 R2 (0)} and P(x) denotes the wave components vector P(x) = {p+(x) p−(x)}. Solving for p+ 2 (0) and inserting the result in the second equation of equations (34) gives
$
P2 (L) = T21 −
%
T22 R2 (0)Y1 (L)T11 P1 (0), Y1 (L)T12 R2 (0)
(36)
as required. Clearly, the expression in square brackets is the scattering matrix of the plug expansion element. The boundary conditions for the plug contraction element are given as R1 (0) and R2 (L) and, the scattering matrix is now required as a relationship between P1 (L) and P2 (0). Equations (34) can be manipulated similarly to obtain the following relationship between
513
Figure 5. Effect of the mean flow velocity gradient on transmission loss of a plug muffler; M = 0·3, s = 0·05. ——, Uniform mean flow; - - -, non-uniform mean flow.
these wave components vectors:
$
P1 (L) = T12 −
%
T11 R1 (0)Y2 (L)T22 P2 (0). Y2 (L)T21 R1 (0)
(37)
Again, the scattering matrix of the plug contraction is given by the expression in square brackets. Equations (36) and (37) can now be combined to obtain a scattering matrix relationship for the plug muffler of Figure 2(a). The combination process is as in the transfer matrix method and may require the introduction of a straight annular pipe to join the cavities of the expansion and contraction elements. 3.2. The second application to be considered is the reverse flow muffler in Figure 3. This is essentially a three-pipe element with the boundary conditions R1 (0), R1 (L), R2 (0) and R3 (0). The scattering matrix of the muffler is required as a relationship between P2 (L) and P3 (L). In this case, it is more convenient to shuffle the scattering matrix relationship, equation (22), so that the left and right vectors become {P1 (L) P2 (L) P3 (0)} and {P1 (0) P2 (0) P3 (L)}, respectively. With the elements of the thus transformed scattering matrix denoted by Ti j , i, j = 1, 2, 3, the relationship between P2 (L) and P3 (L) can be shown to be of the form + P2 (L) = T21 R1 (0)p+ 1 (0) + T22 R2 (0)p2 (0) + T23 P3 (L),
(38)
+ where p+ 1 (0) and p2 (0) are determined in terms of P3 (L) by the following equations: + Y1 (L)T11 R1 (0)p+ 1 (0) + Y1 (L)T12 R2 (0)p2 (0) + Y1 (L)T13 P3 (L) = 0,
(39)
+ Y3 (0)T31 R1 (0)p+ 1 (0) + Y3 (0)T32 R2 (0)p2 (0) + Y3 (0)T33 P3 (L) = 0.
(40)
514
.
3.3. Figure 4 shows the transmission loss characteristics of the plug muffler considered by Peat [4] for various porosities as computed by using the present matrizant approach. With reference to the notation of Figure 2, the salient dimensions of this muffler are as follows: LA = LB = 0, L = L = 0·3 m, d1 = 0·075 m, d2 = 0·25 m, t = 0·015 m, dh = 0·003 m. Here t and dh denote, respectively, the wall thickness and hole diameter of the perforated piping. The expression used for the hole impedance is the same as that of reference [4]: that is, z = z8mv/r0 c02 (1 + t/dh ) + 0·3M + 1·15Mh − ik(t + 0·25dh ),
(41)
where M denotes the upstream (downstream) mean flow Mach number in the inlet (outlet) pipe and Mh the Mach number of the mean flow through the holes. In reference [4], the fluid viscosity was taken as 1·85 × 10−5 kg/ms, but the mean flow temperature was not specified. The present computation was based on a mean flow temperature of 27°C which, for air, fairly accurately corresponds to the above value of the viscosity. The actual mean flow velocity in each pipe is assumed to vary linearly over the perforates. All end cap reflection coefficients are assumed to be unity because no information to the contrary was given in reference [4]. The transmission loss calculation was based on the anechoic radiation condition. The results of Figure 4 have been computed for M = 0·3, by considering the perforated sections in the expansion and contraction elements as single intervals. For simplicity, the Mach number was interpolated at the mid-point of each interval, although ADEM allows one to select the interpolation point arbitrarily. Thus, the use of a single interval here is tantamount to the assumption that the mean flow in the pipe is uniform with Mach number equal to the axially averaged Mach number of the actual flow, as has been assumed by the previous authors using the distributed parameter method. Hence, Figure 4 is comparable to figures 8(b) and 8(d) of reference [4] for s = 0·15 and 0·05, respectively, where s denotes the porosity of the perforated sections of the expansion and contraction elements. For these cases, and also for the other parameter sets not considered here, the matrizant method gave exactly the same results as the numerical decoupling method. One conclusion that can be drawn from Figure 4 is that, as porosity increases, the muffler tends to behave like a simple expansion chamber. With the matrizant method, the effect of the mean flow velocity gradient over the perforated sections can be taken into account by dividing these sections into a sufficiently large number of parts, as described in section 2. In this example, convergence was very fast and accurate results were obtained by using only three or four divisions. In Figure 5 are compared the transmission loss characteristic computed by using one division and five divisions for each each perforated section for the case of M = 0·3 and s = 0·05. As can be seen, the mean flow velocity gradient has only very little effect on the transmission loss characteristic. This case has been solved also by Peat [4], using Sullivan’s segmentation method. With 12 segments, the effect of the mean flow velocity gradient on transmission loss was found to display the same trend as in Figure 5: that is, no change except a very little increase in the vicinity of the maxima with respect to the uniform mean flow case; however, the results for the uniform mean flow case were a few dB lower than those predicted by the numerical decoupling method. This is considered to be due to the number of segments not being large enough for the convergence to the correct results to have occurred. Indeed, for 20 segments, the segmentation method was found to give consistently closer results to the numerical decoupling results.
515
Figure 6. Transmission loss of a reverse flow muffler for various onset Mach numbers.
Figure 6 presents the transmission loss characteristics of the reverse flow muffler considered in reference [3] as computed by the present matrizant method for M = 0·1, 0·2 and 0·3. With reference to Figure 3, the dimensions of this muffler are as follows: d1 = 0·1481 m, d2 = d3 = 0·0493 m, LA = LB = 0·0064 m, L = 0·1286 m, s2 = s3 = 0·039, t = 0·0008 m, dh = 0·00249 m and the hole impedance is given by z = 2·056(rM/Ls) − i0·95k(t + 0·75dh ),
Figure 7. Transmission loss of the reverse flow muffler, reproduced from reference [3].
(42)
516
.
Figure 8. Transmission loss of the reverse flow muffler for no mean flow, computed by the matrizant method.
where r denotes the pipe radius. End cap reflection coefficients for all pipes are assumed to be unity as no information to the contrary was provided in reference [3]. Similarly, the mean flow temperature is assumed to be 74°C and the mean flow in the chamber is neglected. The mean flow velocity was assumed to vary linearly over the perforated sections of pipes, decreasing from M to 0 in pipe 2 and increasing from 0 to M in pipe 3. The transmission loss calculation was based on anechoic radiation condition. The characteristics shown in Figure 6 were computed by using only one interval for each perforated section, and therefore they correspond to the assumption of uniform mean flow velocity in the perforated section with the axially averaged Mach numbers. The M = 0·1 and 0·2 characteristics for this muffler have been presented also in reference [3], and are reproduced here as Figure 7. The agreement between the present results, Figure 6, and Figure 7 seems to be fairly satisfactory except at certain frequencies where the results of reference [3] appear to be degraded by numerical instability. It may be of interest to note that, although the present results agreed fairly well with those given in reference [3] for M = 0·1 and 0·2, considerable discrepancy was observed for the no mean flow case. The transmission loss characteristic computed for this case by using the matrizant approach is shown in Figure 8 and the corresponding characteristic of reference [3] is reproduced in Figure 7. The discrepancy may be due to the frequency resolution used in plotting the latter not being fine enough. To study the effect of the mean flow gradient in pipes 2 and 3, the perforated sections of these pipes were divided into several intervals. Again, convergence was rapid and occurred for only three or four divisions. Presented in Figure 9 is the transmission loss computed for the case of M = 0·3 by using five divisions. As can be seen, the effect of the mean flow velocity is small and confined to the vicinity of the maxima, as in the case of the plug muffler.
517
Figure 9. Effect of the mean flow velocity gradient on transmission loss of the revese flow muffler; M = 0·3. ——, Uniform mean flow; - - -, non-uniform mean flow.
4. CONCLUSIONS
A new method has been developed for acoustic analysis of perforated pipe mufflers. It has been shown that, contrary to the general belief, the distributed parameter method can be used to obtain accurate predictions of the effect of the mean flow velocity gradient, with versatility comparable to that of the segmentation method and better convergence characteristics. Clearly, this approach is not limited to the variations in the mean flow velocity. Axial variations in the mean flow temperature and hole impedance can similarly be accounted for. The present matrizant formulation provides a systematic and structured setting for both analytical and computational analysis of the problem with or without a mean flow velocity gradient. In this context, the recasting of the governing equations in the form of equation (19) and the methodology introduced for the scattering matrix formulation are features of the present analysis which have been presented here for the first time. For the mufflers considered in this paper, it has been found that the effect of the mean flow velocity gradient can be taken into account fairly accurately by assuming an axially averaged uniform mean flow velocity. The errors caused by this approximation are small and can be ignored in realistic design situations because they are confined to the vicinity of the frequencies where the transmission loss attains maximum values.
ACKNOWLEDGMENT
The author acknowledges with thanks the financial support received in connection with this work from Dokuz Eylul University Research Fund, Turkey.
518
. REFERENCES
1. J. W. S 1979 Journal of the Acoustical Society of America 66, 772–788. A method of modelling perforated tube muffler components: I, Theory; II, Applications. 2. M. L. M 1987 Acoustics of Ducts and Mufflers. New York: John Wiley. 3. M. L. M, K. N. R and A. D. S 1987 Journal of Sound and Vibration 114(2), 173–188. Aeroacoustic analysis of perforated muffler components. 4. K. S. P 1988 Journal of Sound and Vibration 128(2), 199–212. A numerical decoupling analysis of perforated pipe silencer elements. 5. R. A. F, W. J. D and A. R. C 1963 Elementary matrices and Some Applications to Dynamics and Differential Equations. Cambridge: Cambridge University Press. 6. E. D 1992 Report AF-908-91-07-07, Dokuz Eylul University, Turkey. Reactive silencer design for standard automobiles. (In Turkish.)
MATRIZANT APPROACH TO ACOUSTIC ANALYSIS OF PERFORATED MULTIPLE PIPE MUFFLERS CARRYING MEAN FLOW E. D Department of Mechanical Engineering, Dokuz Eylul University, Bornova, Izmir, Turkey (Received 3 March 1995) A new method based on the matrizant theory is developed for acoustic analysis of perforated pipe muffler components. The analysis is presented in a generality encompassing any number of parallel pipes that communicate along a common length. The utility and potential of the method is demonstrated on a plug muffler and a reverse flow muffler. The analysis shows, for the first time, that the distributed parameter theory of plane wave propagation in a perforated pipe provides a more satisfactory setting than the segmentation method for the analysis of the effects of axially varying quantities such as the mean flow velocity. The effect of the mean flow velocity gradient on the transmission loss of the plug and reversed flow mufflers are computed and are shown to be insignificant for practical purposes. 71996 Academic Press Limited
1. INTRODUCTION
Perforated pipes are extensively used in automobile mufflers to attenuate internal combustion engine exhaust noise. At present, there are two competing methods for practical acoustic modelling of perforated pipe muffler components, namely, Sullivan’s segmentation method [1] and the distributed parameter method. A discussion of these methods has been presented in the book by Munjal [2]. In Sullivan’s segmentation method, the effect of the perforations is lumped into a few discrete planes with solid pipes assumed to be present between the planes. The distributed parameter method, on the other hand, is based on the solution of the one-dimensional equations of conservation of mass and momentum. In practical mufflers employing perforated pipes, the mean flow is either contained in the pipe or forced through the perforations on the pipe. In the latter case, the mean flow velocity will vary along the length of the perforated pipe and Sullivan’s method is well known for its capability to cope with such variations. In contrast, all previous authors who used the distributed parameter method chose to proceed by assuming the mean flow velocity to be uniform. With this simplification made, the governing differential equations emerge with constant coefficients, the solution of which is relatively simple to obtain. Munjal et al. [3] used a numerical decoupling procedure which is based on numerical extraction of the roots of the system characteristic equation. Peat [4] developed an improved numerical decoupling scheme which is based on the determination of the eigenvalues and eigenvectors of the system matrix by Hessenberg decomposition followed by the LR method and claimed that this approach gives better numerical stability. He also attempted to include the mean flow gradient in the analysis by assuming a constant gradient and keeping the gradient terms in the governing equations while using a constant 505 0022–460X/96/140505 + 14 $18.00/0
7 1996 Academic Press Limited
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.
value for the mean flow velocity to remove the variable terms. This approach is not satisfactory because the variations neglected may be of the similar order as the gradient terms retained. Also, a gradient term appears to be missing in the governing equations used by Peat, who eventually used Sullivan’s segmentation method to compute the effects of constant mean flow velocity gradient on a plug muffler. Similarly, Munjal [2] has presented the distributed parameter method as having the weakness of being limited to the uniform mean flow velocity case and recommended Sullivan’s segmentation method for taking into account the mean flow velocity gradient. The aim of the present paper is to present a new approach for numerical implementation of the distributed parameter method. This approach is called the matrizant method because it is based on the matrizant theory [5]. It is comparable to Peat’s numerical decoupling method in that an eigenvalue problem is required to be solved; however, it has several advantages such as the elegance inherited from the matrizant theory, a more systematic re-arrangement of the governing equations and the capability to deal with the mean flow velocity gradient in a more rigorous theoretical setting and less computational effort than Sullivan’s segmentation method. Thus, contrary to the general belief, ability to deal with a mean flow velocity gradient emerges as the strength, rather than a weakness, of the distributed parameter modelling when the matrizant method is used. The paper will demonstrate application of the matrizant method on a plug muffler and a reversed flow muffler. Other features of the analysis are that the perforated multiple pipe two-port elements are formulated here, for the first time, in scattering matrix forms, and the theoretical formulation is presented in a generality encompassing the perforated pipe arrangements with any number of parallel pipes which communicate along a common length in an enclosing pipe or chamber.
2. ACOUSTIC ANALYSIS OF PERFORATED MULTIPLE PIPES
2.1. With visco-thermal losses neglected, acoustic wave propagation in a uniform pipe carrying mean flow is governed by the linearized forms of the equations of conservation of mass and momentum. Upon assuming uniform mean flow temperature and plane wave propagation with e−ivt time dependence, where i denotes the unit imaginary number, v the radian frequency and t the time, but allowing for axial variations in the mean flow velocity, the equations of conservations momentum and conservation of mass can be expressed, respectively, as (D + MM')p + (MD + M' − ik)V = 0,
(MD + M' − ik)p + DV = mc0 . (1, 2)
Here p denotes the acoustic pressure, V = vr0 c0 where v is the acoustic particle velocity, M is the mean flow Mach number, M = v0 /c0 where v0 denotes the mean flow velocity, m is the mass injected into the pipe per unit volume per unit time, k is the wavenumber, k = v/c0 , c0 is the speed of sound, r0 is the ambient density, D denotes differentiation with respect to x, the distance along the pipe, and M' is the mean flow Mach number gradient, that is, M' = DM. The Mach number is taken positive if the mean flow velocity is in the positive x-direction. Here, it may be of interest to note the M' term in equation (2) is the term which is missing in the equations developed by Peat [4]. Now consider the n-pipe element shown in Figure 1. This consists of n − 1 parallel perforated pipes that communicate with each other along a common length, L, in an enclosing hard walled pipe or chamber. Pipes are numbered from 1 to n, pipe 1 alwas being the enclosing chamber.
507
Figure 1. n-pipe acoustic element: n − 1 perforated pipes enclosed in a hard walled pipe or cavity.
For pipe 1, equation (2) can be expressed as (M1 D + M'1 − ik + b2 + b3 + · · · + bn )p1 + DV1 = b2 p2 + b3 p3 + · · · + bn pn ,
(3)
and, for a perforated pipe, (Mj D + M'j − ik + aj )pj + DVj = aj p1 ,
j = 2, 3, . . . , n,
(4)
where aj = 2sj /rj zj ,
bj = (Sj /S1 )aj ,
j = 2, 3, . . . , n.
(5)
Here S, s, r and z denote the cross-sectional area, the porosity, the hydraulic radius and the hole impedance of a pipe, respectively, and the subscript j refers to pipe j. Similarly, in a n-pipe element, equation (1) is written for pipe j as (D + Mj M'j )pj + (Mj D + M'j − ik)Vj = 0,
j = 1, 2, . . . , n.
(6)
Equations (3), (4) and (6) can be expressed concisely in matrix notation as: A(D)Q(x) = 0
(7)
A A · · · A1nLK Q1 (x)L K GA11 12 13 GG G GA21 A22 O · · · O GG Q2 (x)G GA31 O A33 · · · O GG Q3 (x)G=0, · · · · GG · G G ·· · · · · · · · · · · GG · G G kAn1 O O l k Ann Qn (x)l
(8)
or, in expanded form,
where Qj (x) =
$ %
pj (x) , Vj (x)
0 E x E L,
Aj j =
$
%
D + Mj M'j Mj D + M'j − ik , Mj D + M'j − ik + aj D
j = 1, 2, . . . , n, Aj1 =
$
0 −aj
%
0 , 0
A1j =
$
0 −bj
%
0 , 0
(9, 10) j = 2, 3, . . . , n,
(11)
and a1 = b2 + b3 + · · · + bn by definition and O denotes a 2 × 2 matrix whose elements are all zero. 2.2. Equation (7) constitutes a set of 2n first order ordinary differential equations for a n-pipe element in variables pj and Vj , j = 1, 2, . . . , n. Previous authors [3, 4] using numerical
.
508
decoupling have preferred to proceed with general solution of the governing equations by first collapsing the mass and momentum equations for each pipe into a single differential equation of the second order in D, a process which is easily carried out by eliminating Vj , and then transforming back the resulting set of n second order equations to a system of 2n first order equations. This approach produces the general solutions for the acoustic pressures, pj , and their derivatives with respect to x, p'j . This is rather inconvenient because, for the eventual application of the boundary conditions, the general solutions are required in an impedance matrix or, still better, in a scattering matrix form, and recasting of the native solutions in one of such forms involves cumbersome calculations. A more systematic and elegant approach here, is to proceed directly from equation (7). For this purpose, it is first observed that matrices A1j and Aj1 do not contain the D operator and that matrices Aj j can be written as Aj j =
$
% $
Mj M'j M'j − ik 1 + 0 Mj M'j − ik + aj
%
Mj D, 1
(12)
or, briefly, Aj j = Bj j + Mj D,
j = 1, 2, . . . , n.
(13)
Hence, it follows that, equation (7) can be expressed as DQ(x) = U(x)Q(x),
U(x) = −M−1B(x),
(14)
where
K G M=G G k
M1 O · · · O
O M2 · · · O
···
O
··· · · · ···
O · · · Mn
L G G, G l
(15)
and matrix B(x) has the same form as matrix A, equation (8), but with the diagonal blocks Aj j replaced by matrices Bj j . Note that the matrices Mj are defined in equation (12) and that the elements of matrix B will be functions of x when the mean flow velocity is not uniform along the pipe. Equation (14) is a set of 2n first order ordinary differential equations, in the variables pj and Vj , j = 1, 2, . . . , n, in the standard matrix form and the general solution can be written, by invoking the theory of the matrizant [5], Q(x) = [V]x0 Q(0),
0 E x E L,
(16)
where [V]x0 denotes the matrizant for the interval (0, x). This is a square matrix of order 2n which is a function of U(x) yet to be computed. The advantage of this approach is now clear: putting x = L in the foregoing relationship gives Q(L) = [V]L0 Q(0), which is simply the general solution of the governing equations expressed as a wave transfer relationship between the ends of the n-pipe element in an impedance matrix form, and [V]L0 , that is the matrizant, being the impedance matrix itself. Before embarking on the problem of calculation of the matrizant, the general solution will be derived also in a scattering matrix form. In fact, this is the preferred approach here because the analysis was developed for implementation with the computer code Acoustic Design of Exhaust Mufflers (ADEM) [6], which accepts the scattering matrix representations of acoustic two-port elements directly.
509
To obtain the general solution of the governing equations in a scattering matrix form, − two new variables, namely, p+ j (x) and pj (x), which are subsequently called the wave components, are introduced for each pipe by the transformation
$ % $
pj (x) 1 = Vj (x) 1
%$ %
p+ j (x) , p− j (x)
(17)
j = 1, 2, . . . , n.
(18)
1 −1
or, briefly, Q j (x) = Ej Pj (x),
Obviously, the matrices Ej are the same for all pipes; however, the subscript j will be retained for notational convenience. With this transformation applied, equation (14) becomes, H(x) = −E−1M−1B(x)E,
DP(x) = H(x)P(x),
(19)
where
K G E=G G k
E1 O · · · O
O E2 · · · O
··· ···
O O · · · En
L G G, G l
K P1 (x) L G P2 (x) G P(x) = G · G· G ·· G k Pn (x) l
(20)
By using the matrizant, the general solution of equation (19) can be expressed as P(x) = [T]x0 P(0),
0 E x E L,
(21)
where [T]x0 is the matrizant for the interval (0, x) and is a function of H(x). Putting x = L, one has P(L) = [T]L0 P(0),
(22)
Here, [T]L0 can be regarded as a physical scattering matrix for the n-pipe element. Indeed, with the ends of the n-pipe element connected to ordinary pipes, the wave components − p+ j (x) and pj (x) at x = 0 and x = L will denote the sound waves travelling in +x and −x directions in these pipes, respectively. By carrying out the matrix operations indicated in the second of equations (19), the matrix H(x) can be shown to have the form
K G G H(x) = G G G k
H11
H12
H13
···
H1n
H21 H31 · · · Hn1
H22 O · · · O
O H33 · · · O
··· ···
O O · · · Hnn
···
L G G G, G G l
(23)
L G, G G l
j = 1, 2, . . . , n (24)
where Mj M'j + 2(M'j − ik) + aj Mj M'j + aj K G 1 + Mj 1 + Mj 2Hj j = G Mj M'j + aj Mj M'j + 2(M'j − ik) + aj G k 1 − Mj 1 − Mj
.
510
K G 2H1j = G bj G k 1 − Mj −bj 1 + Mj
L G G, bj G 1 − Mj l −bj 1 + Mj
−aj −aj K G 1 + Mj 1 + Mj 2Hj1 = G aj aj G k 1 − Mj 1 − Mj
L G G, G l
j = 2, 3, . . . , n.
(25, 26)
2.3. To evaluate the matrizant [T]L0 , or, what is the same thing, the scattering matrix, for a n-pipe element, the basic interval 0 E x E L is divided into N parts by introducing intermediate points x1 , x2 , . . . , xN − 1 . For simplicity, the lengths of the parts are assumed to be all equal, i.e., l = xk − xk − 1 , k = 1, 2, . . . , N and xN = L = Nl. Then, by using the well known properties of the matrizant, it can be shown that, [T]L0 = [T]xxNN − 1 . . . [T]xx21 [T]x01 ,
(27)
where [T]xxkk − 1 denotes the matrizant for the interval (xk − 1 , xk ). Now, let jk be a point, always the mid-point in this paper, in the interval (xk − 1 , xk ), k = 1, 2, . . . , N. By regarding l as a sufficiently small quantity, one may assume, for the approximate evaluation of the matrizant, H(x) 3 H(jk ) = Hk , say. Then, Hk is independent of x and the matrizant for the interval (xk − 1 , xk ) can be shown to be [T]xxkk − 1 = eHk l
(28)
and the scattering matrix for a n-pipe element is then given by [T]L0 = eHN l eHN − 1 l . . . eH2 l eH1 l = eHl,
H = HN + HN − 1 + · · · + H 2 + H1 .
(29)
Hence, the problem of evaluation of the system scattering matrix is reduced to the evaluation of an exponential matrix function. The basic expression for the evaluation of an exponential matrix function is eHl = I + Hl +
H2l 2 +· · · , 2!
(30)
where I denotes a unit square matrix of dimension 2n. If l can be regarded as a small quantity of the first order, then the above series can be truncated at the second term, providing extreme simplification in the numerical calculations. The usefulness of such a simplification in practical computation is, however, apt to be limited by slow convergence. An accurate evaluation of the exponential matrix functions is therefore desirable and the following decoupling procedure is proposed here for this purpose. Let L be a diagonal matrix whose elements are the eigenvalues of H, say, lj , j = 1, 2, . . . , 2n, and F the modal matrix whose columns are the corresponding right eigenvectors. It is assumed, for simplicity, that all the eigenvectors are linearly independent. Then, as is well known from the theory of matrices [5], the collineatory transformation H = F−1LF applies, and therefore eHl = F−1 eLlF,
(31)
eLl = 4el1 l el2 l . . . el2n l 7,
(32)
where
in which 47 denotes a diagonal matrix. The eigenvalues and eigenvectors of matrices H can be computed by using a suitable algorithm. ADEM uses a Jacobi type algorithm.
511
Figure 2.(a) Plug muffler; (b) cross-flow expansion element; (c) cross-flow contraction element.
It should be noted that, if the mean flow velocity gradient is zero for all the pipes, then H1 = H2 = . . . = HN = h say, and equation (29) simplifies to [T]L0 = ehL, which can be evaluated directly by using the above described numerical decoupling procedure. 3. APPLICATION TO PRACTICAL MUFFLERS
Two types of mufflers will be considered in this section to demonstrate the utility and potential of the matrizant method of analysis developed in the previous section. First is the so-called plug muffler shown in Figure 2 and the other is the reverse flow muffler of Figure 3(a). There is published data on acoustic performance of both type of mufflers. 3.1. The scattering matrix of this muffler is obtained by combining the cross-flow expansion and contraction elements shown in Figure 2. These are essentially two-pipe elements but they have different boundary conditions. The application of the boundary conditions will be described for this example in some detail because, as far as the author is aware, this is the first paper in which a scattering matrix approach is employed in the acoustic analysis of perforated multiple pipe mufflers. The scattering matrix of a two-pipe element is derived as described in the previous section. For the application of the boundary conditions, it is convenient to express
Figure 3. Reverse flow muffler incorporating a three-pipe element.
.
512
Figure 4. Transmission loss of a plug muffler for M = 0·3 and various porosities, s.
equation (22) in partitioned form as
$ % $
P1 (L) T = 11 P2 (L) T21
T12 T22
%$ %
P1 (0) : P2 (0)
(33)
that is, P1 (L) = T11 P1 (0) + T12 P2 (0),
P2 (L) = T21 P1 (0) + T22 P2 (0).
(34)
With reference to Figure 2(b), the boundary conditions for the plug expansion are given as R1 (L) and R2 (0), where R(x) denotes the reflection coefficient, R(x) = p−(x)/p+(x), and the subscripts refer to the two pipes. The scattering matrix for the plug expansion can now be derived as a relationship between P1 (0) and P2 (L) as follows. Multiplying the first of equations (34) from left by Y1 (L) = [R1 (L) − 1] gives Y1 (L)T11 P1 (0) + Y1 (L)T12 R2 (0)p+ 2 (0) = 0,
(35)
where R2 (0) = {1 R2 (0)} and P(x) denotes the wave components vector P(x) = {p+(x) p−(x)}. Solving for p+ 2 (0) and inserting the result in the second equation of equations (34) gives
$
P2 (L) = T21 −
%
T22 R2 (0)Y1 (L)T11 P1 (0), Y1 (L)T12 R2 (0)
(36)
as required. Clearly, the expression in square brackets is the scattering matrix of the plug expansion element. The boundary conditions for the plug contraction element are given as R1 (0) and R2 (L) and, the scattering matrix is now required as a relationship between P1 (L) and P2 (0). Equations (34) can be manipulated similarly to obtain the following relationship between
513
Figure 5. Effect of the mean flow velocity gradient on transmission loss of a plug muffler; M = 0·3, s = 0·05. ——, Uniform mean flow; - - -, non-uniform mean flow.
these wave components vectors:
$
P1 (L) = T12 −
%
T11 R1 (0)Y2 (L)T22 P2 (0). Y2 (L)T21 R1 (0)
(37)
Again, the scattering matrix of the plug contraction is given by the expression in square brackets. Equations (36) and (37) can now be combined to obtain a scattering matrix relationship for the plug muffler of Figure 2(a). The combination process is as in the transfer matrix method and may require the introduction of a straight annular pipe to join the cavities of the expansion and contraction elements. 3.2. The second application to be considered is the reverse flow muffler in Figure 3. This is essentially a three-pipe element with the boundary conditions R1 (0), R1 (L), R2 (0) and R3 (0). The scattering matrix of the muffler is required as a relationship between P2 (L) and P3 (L). In this case, it is more convenient to shuffle the scattering matrix relationship, equation (22), so that the left and right vectors become {P1 (L) P2 (L) P3 (0)} and {P1 (0) P2 (0) P3 (L)}, respectively. With the elements of the thus transformed scattering matrix denoted by Ti j , i, j = 1, 2, 3, the relationship between P2 (L) and P3 (L) can be shown to be of the form + P2 (L) = T21 R1 (0)p+ 1 (0) + T22 R2 (0)p2 (0) + T23 P3 (L),
(38)
+ where p+ 1 (0) and p2 (0) are determined in terms of P3 (L) by the following equations: + Y1 (L)T11 R1 (0)p+ 1 (0) + Y1 (L)T12 R2 (0)p2 (0) + Y1 (L)T13 P3 (L) = 0,
(39)
+ Y3 (0)T31 R1 (0)p+ 1 (0) + Y3 (0)T32 R2 (0)p2 (0) + Y3 (0)T33 P3 (L) = 0.
(40)
514
.
3.3. Figure 4 shows the transmission loss characteristics of the plug muffler considered by Peat [4] for various porosities as computed by using the present matrizant approach. With reference to the notation of Figure 2, the salient dimensions of this muffler are as follows: LA = LB = 0, L = L = 0·3 m, d1 = 0·075 m, d2 = 0·25 m, t = 0·015 m, dh = 0·003 m. Here t and dh denote, respectively, the wall thickness and hole diameter of the perforated piping. The expression used for the hole impedance is the same as that of reference [4]: that is, z = z8mv/r0 c02 (1 + t/dh ) + 0·3M + 1·15Mh − ik(t + 0·25dh ),
(41)
where M denotes the upstream (downstream) mean flow Mach number in the inlet (outlet) pipe and Mh the Mach number of the mean flow through the holes. In reference [4], the fluid viscosity was taken as 1·85 × 10−5 kg/ms, but the mean flow temperature was not specified. The present computation was based on a mean flow temperature of 27°C which, for air, fairly accurately corresponds to the above value of the viscosity. The actual mean flow velocity in each pipe is assumed to vary linearly over the perforates. All end cap reflection coefficients are assumed to be unity because no information to the contrary was given in reference [4]. The transmission loss calculation was based on the anechoic radiation condition. The results of Figure 4 have been computed for M = 0·3, by considering the perforated sections in the expansion and contraction elements as single intervals. For simplicity, the Mach number was interpolated at the mid-point of each interval, although ADEM allows one to select the interpolation point arbitrarily. Thus, the use of a single interval here is tantamount to the assumption that the mean flow in the pipe is uniform with Mach number equal to the axially averaged Mach number of the actual flow, as has been assumed by the previous authors using the distributed parameter method. Hence, Figure 4 is comparable to figures 8(b) and 8(d) of reference [4] for s = 0·15 and 0·05, respectively, where s denotes the porosity of the perforated sections of the expansion and contraction elements. For these cases, and also for the other parameter sets not considered here, the matrizant method gave exactly the same results as the numerical decoupling method. One conclusion that can be drawn from Figure 4 is that, as porosity increases, the muffler tends to behave like a simple expansion chamber. With the matrizant method, the effect of the mean flow velocity gradient over the perforated sections can be taken into account by dividing these sections into a sufficiently large number of parts, as described in section 2. In this example, convergence was very fast and accurate results were obtained by using only three or four divisions. In Figure 5 are compared the transmission loss characteristic computed by using one division and five divisions for each each perforated section for the case of M = 0·3 and s = 0·05. As can be seen, the mean flow velocity gradient has only very little effect on the transmission loss characteristic. This case has been solved also by Peat [4], using Sullivan’s segmentation method. With 12 segments, the effect of the mean flow velocity gradient on transmission loss was found to display the same trend as in Figure 5: that is, no change except a very little increase in the vicinity of the maxima with respect to the uniform mean flow case; however, the results for the uniform mean flow case were a few dB lower than those predicted by the numerical decoupling method. This is considered to be due to the number of segments not being large enough for the convergence to the correct results to have occurred. Indeed, for 20 segments, the segmentation method was found to give consistently closer results to the numerical decoupling results.
515
Figure 6. Transmission loss of a reverse flow muffler for various onset Mach numbers.
Figure 6 presents the transmission loss characteristics of the reverse flow muffler considered in reference [3] as computed by the present matrizant method for M = 0·1, 0·2 and 0·3. With reference to Figure 3, the dimensions of this muffler are as follows: d1 = 0·1481 m, d2 = d3 = 0·0493 m, LA = LB = 0·0064 m, L = 0·1286 m, s2 = s3 = 0·039, t = 0·0008 m, dh = 0·00249 m and the hole impedance is given by z = 2·056(rM/Ls) − i0·95k(t + 0·75dh ),
Figure 7. Transmission loss of the reverse flow muffler, reproduced from reference [3].
(42)
516
.
Figure 8. Transmission loss of the reverse flow muffler for no mean flow, computed by the matrizant method.
where r denotes the pipe radius. End cap reflection coefficients for all pipes are assumed to be unity as no information to the contrary was provided in reference [3]. Similarly, the mean flow temperature is assumed to be 74°C and the mean flow in the chamber is neglected. The mean flow velocity was assumed to vary linearly over the perforated sections of pipes, decreasing from M to 0 in pipe 2 and increasing from 0 to M in pipe 3. The transmission loss calculation was based on anechoic radiation condition. The characteristics shown in Figure 6 were computed by using only one interval for each perforated section, and therefore they correspond to the assumption of uniform mean flow velocity in the perforated section with the axially averaged Mach numbers. The M = 0·1 and 0·2 characteristics for this muffler have been presented also in reference [3], and are reproduced here as Figure 7. The agreement between the present results, Figure 6, and Figure 7 seems to be fairly satisfactory except at certain frequencies where the results of reference [3] appear to be degraded by numerical instability. It may be of interest to note that, although the present results agreed fairly well with those given in reference [3] for M = 0·1 and 0·2, considerable discrepancy was observed for the no mean flow case. The transmission loss characteristic computed for this case by using the matrizant approach is shown in Figure 8 and the corresponding characteristic of reference [3] is reproduced in Figure 7. The discrepancy may be due to the frequency resolution used in plotting the latter not being fine enough. To study the effect of the mean flow gradient in pipes 2 and 3, the perforated sections of these pipes were divided into several intervals. Again, convergence was rapid and occurred for only three or four divisions. Presented in Figure 9 is the transmission loss computed for the case of M = 0·3 by using five divisions. As can be seen, the effect of the mean flow velocity is small and confined to the vicinity of the maxima, as in the case of the plug muffler.
517
Figure 9. Effect of the mean flow velocity gradient on transmission loss of the revese flow muffler; M = 0·3. ——, Uniform mean flow; - - -, non-uniform mean flow.
4. CONCLUSIONS
A new method has been developed for acoustic analysis of perforated pipe mufflers. It has been shown that, contrary to the general belief, the distributed parameter method can be used to obtain accurate predictions of the effect of the mean flow velocity gradient, with versatility comparable to that of the segmentation method and better convergence characteristics. Clearly, this approach is not limited to the variations in the mean flow velocity. Axial variations in the mean flow temperature and hole impedance can similarly be accounted for. The present matrizant formulation provides a systematic and structured setting for both analytical and computational analysis of the problem with or without a mean flow velocity gradient. In this context, the recasting of the governing equations in the form of equation (19) and the methodology introduced for the scattering matrix formulation are features of the present analysis which have been presented here for the first time. For the mufflers considered in this paper, it has been found that the effect of the mean flow velocity gradient can be taken into account fairly accurately by assuming an axially averaged uniform mean flow velocity. The errors caused by this approximation are small and can be ignored in realistic design situations because they are confined to the vicinity of the frequencies where the transmission loss attains maximum values.
ACKNOWLEDGMENT
The author acknowledges with thanks the financial support received in connection with this work from Dokuz Eylul University Research Fund, Turkey.
518
. REFERENCES
1. J. W. S 1979 Journal of the Acoustical Society of America 66, 772–788. A method of modelling perforated tube muffler components: I, Theory; II, Applications. 2. M. L. M 1987 Acoustics of Ducts and Mufflers. New York: John Wiley. 3. M. L. M, K. N. R and A. D. S 1987 Journal of Sound and Vibration 114(2), 173–188. Aeroacoustic analysis of perforated muffler components. 4. K. S. P 1988 Journal of Sound and Vibration 128(2), 199–212. A numerical decoupling analysis of perforated pipe silencer elements. 5. R. A. F, W. J. D and A. R. C 1963 Elementary matrices and Some Applications to Dynamics and Differential Equations. Cambridge: Cambridge University Press. 6. E. D 1992 Report AF-908-91-07-07, Dokuz Eylul University, Turkey. Reactive silencer design for standard automobiles. (In Turkish.)