 Open Access
 Total Downloads : 116
 Authors : Myunggon Yoon
 Paper ID : IJERTV6IS120030
 Volume & Issue : Volume 06, Issue 12 (December 2017)
 Published (First Online): 04122017
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Acoustic Model of a Duct with a Varying Cross – Sectional Area and a Non – zero Mean Flow
MyungGon Yoon Department of Precision Mechanical Eng. Gangneung – Wonju National University
Wonju 26403, Republic of Korea
AbstractIn this paper we propose a semianalytic onedimensional acoustic model of a duct which has a continuously varying crosssectional area under the non zero mean flow condition. Our idea is to approximate the region of varying area as a sequence of abrupt area change. It is shown that the commonly employed approximation method in which the area varying zone is replaced with one abrupt area change can result in significant errors in terms of acoustic transfer matrix.
KeywordsOnedimensional acoustic model, Acoustic Transfer Matrix

INTRODUCTION
A precise acoustic model of a duct with a continuously varying crosssectional area, such as the interval [x1, x2] in an acoustic model of a gas turbine engines in Fig. 1, is a difficult task. The acoustic model of the area con tracting part [x1, x2] should represent physical behaviors but at the same time it need be mathematically simple and explicit to be integrated for an acoustic analysis of the whole system.
[x1, x2] is not significant [1]. In addition, some recent results can handle more general cases but the model is too complicated and implicit, e.g. see [2], [3].An intuitively appealing and commonly used approach in practice is to approximate the area varying region [x1, x2] as having an abrupt area change at xe [x1, x2] as depicted in Fig. 2. This approach can be naturally generalised to introduce a sequence of tiny area jumps as shown in Fig. 3 instead of one.
A fundamental question is whether or not the acous tic model corresponding to njump approximation con verges to a certain model in an appropriate sense as the number of jumps n increases. In addition, assuming that the manyjump model converges, our next question is whether or not it is possible to cleverly choose the fictitious area jump position xe in Fig. 2 of the single jump model such a way that the associated model can be reasonably close to multistep model. These two questions are investigated in this paper.
0
R u0 u2
u3 R4 u
0
r1
re
r2
x0 x1 x2 x3 x4
Fig. 1: A simple acoustic model of gas turbine
Some results exist in literature for the cases when there is no mean flow and the area contraction over the interval
x1 xe x2
Fig. 2: Single step approximation
Icu
A combination of the results (1), (3) and (4) gives
u0 = u10
u11
u12
pI cuI
lx2
= S(, s) pI
lx1
(6)
r1 = y0
y1 y2 y
S(, s) := S11(, s) S12(, s)l (7)
3
x = x x x x
u1(n1)
x
yn = r2
= x
S21(, s) S22(, s)
where 0 1 denotes the relative location of area
1 10 11
12 13
1n 2
jump defined
Fig. 3: Mutiple jump approximation
:= xe x1 . (8)
x2 x1

ACOUSTIC MODEL

Single step model
In onedimensional acoustic model of a duct in Fig. 2, we need to characterize the relations between pressure and volume velocity perturbations between x1 and x2.
For the intervals [x1, xe) and (xe, x2], acoustic waves

Multiple step model
Â· Â· Â·
We divide the interval [x1, x2] into n subintervals [x1(k1), x1k)] for k = 1, , n where x1 = x10 and x1n = x2 as shown in Fig. 3. Each subinterval is composed of a wave propagation for a distance := (x2 x1)/n and then a sudden area contraction. This can be written as
simply propagate over a duct of a constant area and thus it follows that
pI
l
I x1k
= T k
k1
pI
l
I x1(k1)
(k = 1, Â· Â· Â· , n) (9)
pI l
cu
I
1 e +s + e s e +s e s1 pI l
e
2
e
e
where the transfer matrix T k
1
given
between subelements is
+
+
x
=
1
1 s
1
e1s
1
1 s
1
+ e1s
cu
I
x
k1

l
k 1 1 (1k)u1(k1)/c1
where the superscript I and overline denote perturbation
and mean value, respectively. Moreover and c denote density and sound speed and
Tk1 := 2
0 1/1k
1k
1k
1k
1k
e + + e
e + e 1
e + e
e+ + e
(10)
Â± = xe x1

1k 1k
1k 1k
1
c Â± u0
y2 ( k(r /r 1) + n \2
Similarly we have
y2
k
1k :=
k
=
2
1
(11)
k1
(k 1)(r2/r1 1) + n
pI
1
e +s + e s e +s e s
=
I
x
2
2
e
l
cu
2
1
2
2
2
2
cu
I
(12)
1
+
2
i=1
2
2
x
u
1k
:= u
1(k1)
/
1k
= u /
i
cu
2
l
pI
s
e s e +s + e s
e
(3)
Â± := (x2 x1)/n
(13)
1k
, 2Â± = (x2 xe)/(c Â± u1), u1 = u0/ and = (r2/r1)2 c1 Â± u1(k1)
denotes the area ratio.
1 2
At the point of abrupt area change xe, it holds that
Form a sequential application of the above transfer matrix for subintervals, the transfer matrix of the area varying interval [x , x ] can be explicitly written as
pI l
cuI
1 ()M1l pI l l
xe
xe+
l
22
=
0 1/
cuI
(4)
pI
I
= M (n, s)
pI
I
() = 2+323
(5)
x2
n
M (n, s) = T k
x1
:= M11
(n, s) M12
(n, s)l .
which follows from the continuity condition and the stagnation pressure loss across the area jump [4].
k=1
k1
M21(n, s) M22(n, s)
(14)


A NUMERICAL CASE STUDY
The following parameters were chosen for a numerical case study ;
L = x2 x1 = 60, r1 = 32.5, r2 = 10.6 (15)
in [millimeter] unit and c = 345 [meter/sec].
A. Model Convergence
3
2
1
8
6
4
2
0
400
200
0
200
400
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
0
0 0.5 1 1.5 2 2.5 3
200
100
0
Fig. 6: Bode plot of M21(n, s) for n [2, 10] and u1 = 0
10
5
100
0 0.5 1 1.5 2 2.5 3
0
0 0.5 1 1.5 2 2.5 3
Fig. 4: Bode plot of M11(n, s) with n [2, 10] and
u1 = 0
3
2
1
100
0
100
200
0 0.5 1 1.5 2 2.5 3
0
0 0.5 1 1.5 2 2.5 3
300
200
100
0
0 0.5 1 1.5 2 2.5 3
Fig. 7: Bode plot of M22(n, s) for n [2, 10] and u1 = 0
We have computed the Bode plots of four component Mij(n, s) of the acoustic transfer matrix in (14). Thehorizontal axis in Fig. 47 denotes the normalized fre quency w/Lc (w = 2f ) , L = x2 x1 and the arrows
indicates the direction of increasing n [2, 10].
The overall results strongly suggests that the multiple
step model quickly converges to a limit model as the
Fig. 5: Bode plot of M12(n, s) for n [2, 10] and u1 = 0
number of steps increases.
B. Resonance frequency shift
The antiresonance frequency around w/Lc 1 in Fig. 4 was zoomed with a larger n [5, 30] in Fig. 8. An interesting observation here is that, as n increases, the antiresonance frequency slightly decreases toward w/Lc = 0.968 roughly. Another way of looking at this
result is that the area variation gave rise to an increased duct length L L/0.968 = 1.03L (3%) effectively.
4
3
2
1
0
0 0.5 1 1.5 2 2.5 3
3
0.3
0.25
0.2
2
1
0
0 0.5 1 1.5 2 2.5 3
0.15
0.1
0.05
0
0.85 0.9 0.95 1 1.05 1.1 1.15
Fig. 9: Bode magnitude plots of M11(n, jw) and
M12(n, jw) with mean flow M1 [0, 0.02]
the onejump model (7) such that the resulting model becomes similar to the multijump model (14) ?
Seeking an answer to this question, we have compared the Bode plot of S(, s) with M (10, s) in Fig, 1013 with a zeromean flow. The thick cyan lines in those
Fig. 8: First antiresonance frequency of M11(n, s) on
n [5, 30] and u1 = 0
C. Mean flow effect
One of the most surprising result in this paper is that the acoustic transfer matrix significantly depends on mean flow.
One can see this clearly in the Bode magnitude plot of M11(n, s) (upper) and M12(n, s) (lower) in Fig. 9 where n = 10 and the Mach number M1 := u0/c [0, 0.02].
Note that the tiny mean flow u0 = 6.9 (meter/sec) of the case M1 = 0.02 can virtually eliminate the anti resonance peak around w/Lc = 1. Our computation revealed, but not included here, that the mean flow dependency however is less significant in the cases of another two components M21(n, s) and M22(n, s).
D. Dependency on jump position
Let us consider the next question ; Is it possible to cleverly choose a fictitious area jump position xe of
figures denote the Bode plot of M (10, s).
{ }
An interesting fact is that, in the case of S11, S22 shown in Fig. 10 and Fig. 13, both and 1 give the same results.
Note from Fig. 10 that either a small = 0.1 or a large = 0.9 results in S11(, s) close to M11(10, s). In contrast, in other three cases in Fig. 1113, the single step model becomes similar to multiple step model with a moderate choice 0.5 in overall.
These observation suggest that the single step model is an intrinsically different from the multiple step model and thus it is an inaccurate acoustic model, irrespectively of how to choose a step location.

CONCLUSION
We developed a semianalytic onedimensional acous tic model of a duct with a continuously varying cross sectional area under the nonzero mean flow condition. Our numerical case study suggests that the acoustic transfer matrix of a multiple step approximation for
10
0.5
5
10
0.3,0.7
0.2,0.8 5
0.9
0
200
100
0
100
0.1,0.9
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
0
0
100
200
300
0.1
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
Fig. 10: Bode plot of {M11(10, s), S11(, s)} Fig. 12: Bode plot of {M21(10, s), S11(, s)}
10
0.1
5
0
10
5
0.9
0
0.1,0.9
0.2,0.8
0.3,0.7
0.5
400
300
200
100
0
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
100
0
100
200
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
Fig. 11: Bode plot of {M12(10, s), S11(, s)}
the varying crosssectional region converges to a limit matrix as the number of steps increases. In addition, it was found that a single step model is an erroneous approximation irrespectively of the choice of a step location.
REFERENCES

R. D. Ayers, L. J. Eliason, and D. Mahgerefteh, The conical bore in musical acoustics, American Journal of Physics, vol. 53,
Fig. 13: Bode plot of {M22(10, s), S11(, s)}
no. 6, pp. 528537, 1985.

Q. Min, W. He, J. T. Q. Wang, and Q. Zhang, A study of stepped acoustic resonator with transfer matrix method, Acous tical Physics, vol. 60, no. 4, pp. 492498, 2014.

A. Gentemann, A. Fischer, S. Evesque, and W. Polifke, Acoustic transfer matrix reconstruction and analysis for ducts with sudden change of area, in 9th AIAA/CEAS Aeroacoustics Conference and Exhibit, 2003.

M. L. Munjal, Acoustics of Ducts and Mufflers, 2nd ed. Wiley, 2014.