The acoustic absorption coeﬃcient of short circular holes sustaininga high Reynolds number bias ﬂow
Renaud GAUDRON(1) , Aimee S. MORGANS(2)(1)

Department of Mechanical Engineering, Imperial College London, UK, r.gaudron@imperial.ac.uk(2)

Department of Mechanical Engineering, Imperial College London, UK

Abstract
The damping of acoustic waves across short circular holes sustaining a high Reynolds number bias ﬂow
is of interest in many engineering applications, such as aircraft and rocket engines liners. This damping
is generated by the interaction between acoustic waves and unsteady vortical structures generated at
the perforation rim. The objective of this work is to determine and compare the acoustic absorption
coeﬃcient of a short circular hole sustaining a high Reynolds number, low Mach number bias ﬂow
for two distinct models. In the ﬁrst model, the acoustic response of the hole is described using an
isentropic area decrease followed by a non-isentropic area increase with large scale ﬂow separation. In
the second model, the acoustic response of the hole is described using Howe’s model assuming thatthe incoming perturbations are hydrodynamically compact. For both models, the acoustic absorption
coeﬃcient is shown to depend on the cross section area ratio θ, Mach number inside the hole M2 andupstream boundary acoustic reﬂection coeﬃcient R1 only. For limited values of θ and M2 , the acoustic
absorption coeﬃcient remains close to zero and does not depend on R1 . On the other hand, for large
values of θ and M2 , the acoustic absorption coeﬃcient is found to be highly dependent on R1 and may
become close to unity, thus indicating that a large fraction of the incoming acoustic energy is damped
across the hole for such parameters.

Keywords: Acoustic absorption, Short circular holes

1

INTRODUCTION

Short circular holes and perforated plates are used for their acoustic damping properties in many engineering applications, ranging from thermoacoustic instability control using perforated liners in aero-engines
or land-based gas turbines [1, 13, 14] to broadband noise absorption in car muﬄers [12]. In most cases,
these short circular holes sustain a high Reynolds number bias ﬂow which signiﬁcantly aﬀects the acoustic
energy balance of the system [6, 11, 15]. The main objective of this work is to predict the acoustic absorption coeﬃcient [9, 10] of short circular holes sustaining a high Reynolds number, low Mach number bias
ﬂow based on two analytical models widely employed in low order network tools used for thermoacoustic
stability prediction [4, 7].
The geometry of interest and the corresponding mean ﬂow considerations are ﬁrst examined in Sec. 2.
Two analytical models predicting the acoustic absorption coeﬃcient generated by a short circular holesustaining a high Reynolds number, low Mach number bias ﬂow are then derived in Sec. 3. The ﬁrst
analytical model, called the 2-step (2S) model and consisting of an isentropic area decrease followed byan area increase with large scale ﬂow separation, is investigated in Sec. 3.1. The second analytical model,
called the quasi-steady Howe’s (QSH) model and accounting for the interaction between vortical structuresgenerated at the burner rim and sound waves, is investigated in Sec. 3.2. The analytical predictions of
the acoustic absorption coeﬃcient according to these two models are then discussed in Sec. 4.

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L

𝑓1

𝑢1

𝑢2

⌀ 2𝑎

𝑔1

|

1

|

𝑢3

𝑓3
𝑔3

𝑥

|

3

2

Figure 1. Sketch representing the hole geometry. The mean ﬂow direction is from left to right.

2

PROBLEM SETTING

The scope of this work is limited to the absorption of linear acoustic waves across single short circularholes sustaining a high Reynolds number, low Mach number bias ﬂow. The ﬂuid is assumed to be aperfect gas. The geometry of interest, represented in Fig. 1, is assumed to be axisymmetric. The single
circular perforation has a length L and a radius a and both the upstream and downstream cross section
areas are equal to A. Three reference locations are subsequently deﬁned. The upstream (respectively
downstream) reference location is denoted by 1 (respectively 3) while the reference location inside theperforation is denoted by 2. Each variable xi deﬁned at reference location i can be decomposed into amean xi and an acoustic xi component such that xi  xi . These variables include the gas temperature
T , density ρ, pressure p, velocity u and sound speed c. Flow variables are assumed to be one-dimensionalin each of these reference sections. The system comprised between section 1 and section 3 is assumed
to be acoustically and hydrodynamically compact. The mean ﬂow variables are assumed to be timeindependent. Body forces, thermal conduction and thermal radiation are neglected. Entropy waves are
not considered in this work. The Riemann invariant associated with sound wave propagation in the ﬂow
direction (respectively opposite to the ﬂow) is denoted by f (respectively g). The harmonic convention
retained in this article is exp(−iωt) following previous works [6, 15]. Nevertheless, the results presented
in Sec. 4 are valid for both harmonic conventions exp(±iωt) since the value of the acoustic absorption
coeﬃcient does not depend on the harmonic convention.
It can be shown that the mean ﬂow is characterized by eight dimensionless numbers:
M1 = u1 /c1
M2 = u2 /c2||

M3 = u3 /c3||

α1 = p1 /p2||

Ξ1 = c1 /c2

||

2

||

α2 = p2 /p3||

Ξ2 = c2 /c3

θ = A/πa(1)Where Mi is the mean Mach number at reference location i and αi (respectively Ξi ) quantiﬁes the meanpressure (respectively the mean sound speed) ratio between reference locations i and i + 1. Finally,θ represents the cross section area ratio between reference locations 1 or 3 and reference location 2.
The number of dimensionless parameters can be further reduced by considering the mean conservation
equations. Accordingly, the mean conservation of mass applied between reference locations 1 and 2 andbetween reference locations 2 and 3 leads to [5, 8]:
M2 Ξ1 = α1 M1 θM3 Ξ2 θ = α2 M2

(2)
(3)

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Assuming that viscous eﬀects can be neglected at all three reference locations and neglecting terms oforder M 2 , the mean stagnation enthalpy conservation applied between reference locations 1 and 2 andbetween reference locations 2 and 3 leads to [5]:
Ξ1 = Ξ2 = 1(4)Finally, mean entropy conservation is applied between reference locations 1 and 2 and between reference
locations 2 and 3 for the QSH model introduced in Sec. 3.2. The mean ﬂow is assumed to be isentropic
throughout the system in that case. On the other hand, for the 2S model introduced in Sec. 3.1, the area
increase between reference locations 2 and 3 cannot be assumed to be isentropic because of ﬂow separation[5]. In that case, the mean conservation of momentum is considered between reference locations 2 and 3[2, 5] and it is assumed that the pressure on the lateral surface just after the sudden cross section area
increase is equal to the upstream pressure p1 [2]. By neglecting terms of order M 2 , the mean conservation
equations for both models lead to:
α1 = α2 = 1(5)In the end, the structure of the mean ﬂow is fully determined by two dimensionless numbers: the Mach
number inside the perforation M2 and the cross section area ratio θ. Furthermore, the mean pressure p,mean temperature T , mean sound speed c and mean density ρ are constant throughout the system.

3

ACOUSTIC ABSORPTION COEFFICIENT

The acoustic absorption coeﬃcient Δ of a compact element located between reference locations i and
i + 1 is a function of the associated S-Matrix (sometimes called Scattering Matrix) Si , boundary acoustic
reﬂection coeﬃcient Ri or Ri+1 , Mach numbers Mi and Mi+1 , cross section area ratio θi , mean pressure
ratio αi and mean sound speed ratio Ξi only [5]. For instance, when the upstream acoustic reﬂection
coeﬃcient Ri is known, the acoustic absorption coeﬃcient is given by [5]:
Δ=1−

2

2

(1 + Mi+1 ) |Si (1, 2) + Ri det Si |2 + θi Ξi αi−1 (1 − Mi ) |Si (2, 2)|222

θi Ξi αi−1 (1 + Mi ) |Ri Si (2, 2)|2 + (1 − Mi+1 ) |1 − Ri Si (2, 1)|2

(6)

An equivalent expression can be derived when the downstream acoustic reﬂection coeﬃcient Ri+1 is known[5]. Without loss of generality, results will be solely presented in terms of the upstream acoustic reﬂection
coeﬃcient Ri in this work.
The acoustic response of a short circular hole sustaining a high Reynolds number, low Mach number bias
ﬂow is described here using two distinct models. The 2-step (2S) model, described in Sec. 3.1, corresponds
to an isentropic area decrease followed by a non-isentropic area increase with large scale ﬂow separation[5]. On the other hand, the quasi-steady Howe’s (QSH) model, described in Sec. 3.2, corresponds toan hydrodynamically compact isentropic element where acoustic damping arises because of the coupling
between unsteady vortical structures generated at the perforation rim and acoustic waves [6].
3.1 2-STEP (2S) MODEL
The 2-step (2S) model is a sequence of two more fundamental models. Viscous eﬀects are assumed tobe negligible across the cross section area contraction, between reference locations 1 and 2 in Fig. 1.
As a consequence, the acoustic response of this subsystem is described through a transfer matrix M2S
12corresponding to a low Mach isentropic area decrease [5]. On the other hand, large scale ﬂow separation
occurs at the sudden area expansion, between reference locations 2 and 3 in Fig. 1. Hence, this subsystemcannot be considered to be isentropic [2] and its acoustic response is described through a transfer matrix

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Table 1. Expressions for the acoustic absorption coeﬃcient Δ for various limiting cases.
Limiting case
Anechoic boundary(R1 =0)2-step (2S) model
1−

Rigid boundary(R1 =1)
Small hole(θ  1)

Quasi-steady Howe’s (QSH) model
2
1 + 4M2 θ−1 M22 2 − θ−1 − θ + 4
2(2 − M2 (2 − θ−1 − θ))

1−



2
1 + 1 + 4M2 θ−1 (M2 θ)
2(1 + M2 θ)

0

1−

0
2

|2R1 + M2 (2 − θ) (R1 − 1)| + 4
2|2 + M2 (2 − θ) (R1 − 1)| + 4|R1 |2

1−

2|R1 (1 − M2 θ) + M2 θ| + 1|R1 |2 + |1 + M2 θ (1 − R1 ) |2

Large hole(θ ∼ 1)|1 − R1 |2
M2 θ−1 + θ − 2
2
1 + |R1 ||1 − R1 |2
2M2 θ
1 + |R1 |2

No mean ﬂow(M2 =0)00

M2S
23 corresponding to a low Mach non-isentropic area increase with ﬂow separation [5]. Both transfer
2S
matrices M2S
12 and M23 depend on the Mach number inside the hole M2 and the cross section area ratio
θ only:








1 M2 θ−1 − θ
1 −2M2 θ−1 θ−1 − 1

2S
2S
M23 =
(7)
M12 =


0
θ
0
θ−1
The acoustic response of the system between reference locations 1 and 3 in Fig. 1 is then described through2S
2S
the transfer matrix M2S
13 = M23 M12 . The acoustic response of short circular holes is represented using
this approach in many low order network tools used for thermoacoustic stability prediction [2, 7]. The
2S
describing the acoustic response of the whole system is then given by:
S-matrix S13
2S
=
S13

1
2 − M2 (2 − θ−1 − θ)



2



−M2 2 − θ−1 − θ
−M2 2 − θ−1 − θ
2(8)All terms of order M 2 and higher are neglected in Eq. (8). This expression is then inserted in Eq. (6),thus leading to:


Δ

2S






1 + 2M2 θ−1 |2R1 + M2 2 − θ−1 − θ (R1 − 1)|2 + 4 1 − 2M2 θ−1=1−
4 (1 + 2M2 θ−1 ) |R1 |2 + (1 − 2M2 θ−1 ) |2 + M2 (2 − θ−1 − θ) (R1 − 1)|2

(9)Equation (9) is the expression of the acoustic absorption coeﬃcient of a short circular hole sustaininga high Reynolds number, low Mach number bias ﬂow according to the 2-step (2S) model for a given
upstream acoustic boundary condition. Several limiting cases based on Eq. (9) and corresponding to
upstream anechoic (R1 =0) or rigid (R1 =1) boundary conditions, small (θ  1) or large (θ ∼ 1) circularholes and the absence of a mean ﬂow (M2 =0) are summarized in Table 1. The acoustic absorption
coeﬃcient Δ cancels out whenever M2 =0 or R1 =1 for distinct reasons. For the ﬁrst limiting case, there2115

is no source of acoustic damping because large scale ﬂow separation does not occur in a quiescent ﬂuid.
On the other hand, when the upstream boundary reﬂection coeﬃcient is equal to unity, the downstream
boundary reﬂection coeﬃcient is automatically equal to unity as well for the 2S model, this being due tothe compactness assumption. As a consequence, all of the incoming acoustic energy is reﬂected at thesystem boundaries thus leading to the absence of acoustic damping.
3.2 QUASI-STEADY HOWE’S (QSH) MODEL
The quasi-steady Howe’s (QSH) model is obtained from Howe’s model, which quantiﬁes the acoustic
damping due to the conversion of acoustic energy into unsteady vortical energy across a short circularhole sustaining a high Reynolds number bias ﬂow [6]. It is further assumed here that the incoming
perturbations are hydrodynamically compact, e.g. that the Strouhal number St  1. The expression for
the transfer matrix representing the acoustic response of the system between reference locations 1 and 3
in Fig. 1 is simpliﬁed accordingly using a Puiseux series expansion for St → 0. This simpliﬁed transfer
QSH
and the associated S-matrix S13
are then given by:
matrix MQSH
13






1
M2 θ
1 −2M2 θ
1

QSH
QSH
S13 =
(10)
M13 =


1 + M 2 θ M2 θ
0
1
1
Once again, all terms of order M 2 and higher are neglected in Eq. (10). The expression for the S-matrix
is then inserted in Eq. (6), thus leading to:


Δ

QSH




1 + 2M2 θ−1 |R1 + M2 θ (1 − R1 ) |2 + 1 − 2M2 θ−1=1−(1 + 2M2 θ−1 ) |R1 |2 + (1 − 2M2 θ−1 ) |1 + M2 θ (1 − R1 ) |2

(11)

Equation (11) is the expression of the acoustic absorption coeﬃcient of a short circular hole sustaining ahigh Reynolds number, low Mach number bias ﬂow according to the quasi-steady Howe’s (QSH) model fora given upstream acoustic boundary condition. The same limiting cases as for the 2S model are derived
based on Eq. (11) and summarized in Table 1. For the same reason as before, the acoustic absorption
coeﬃcient according to the QSH model cancels out when the upstream boundary reﬂection coeﬃcient isequal to unity. Furthermore, the acoustic damping mechanism considered in the QSH model vanishes in
the absence of a mean ﬂow, thus leading to a zero acoustic absorption coeﬃcient in that case.

4

RESULTS

The predictions for the acoustic absorption coeﬃcient Δ of a short circular hole sustaining a high Reynolds
number, low Mach number bias ﬂow according to the 2S and QSH models are now discussed. Results for
upstream anechoic boundary conditions R1 = 0 are represented as functions of the Mach number M2 forvarious cross section area ratios θ in Fig. 2-(Left) and vice versa in Fig. 2-(Right). The solid and dottedlines represent the predictions according to the 2S and QSH models respectively.
For all θ and M2 investigated, the predicted acoustic absorption coeﬃcients lie between 0, corresponding to an unaﬀected acoustic energy balance, and 0.5, corresponding to a 50% reduction of the acoustic
energy across the system. For limited cross section area ratios, corresponding to θ = 2 and θ = 10 in
Fig. 2-(Left), Δ is larger for the QSH model compared to the 2S model for all Mach numbers M2 investigated. Likewise, for limited Mach numbers, corresponding to M2 = 0.01 in Fig. 2-(Right), Δ is larger
for the QSH model compared to the 2S model for all cross section area ratios θ investigated. At larger
Mach numbers M2 (respectively cross section area ratios θ), there is a critical value of θ (respectively
M2 ) where the predictions according to the 2S model surpass those of the QSH model.

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1

1

0.5

0.5

0

0

0.02

0.04

0.06

0.08

0.1

0

1

10

20

30

40

50

Figure 2. Acoustic absorption coeﬃcient of a short circular hole sustaining a high Reynolds number,low Mach number bias ﬂow for upstream anechoic boundary conditions as a function of M2 for various θ(Left) and vice versa (Right). (Solid line): 2S model. (Dotted line): QSH model.

Figure 3. Acoustic absorption coeﬃcient of a short circular hole sustaining a high Reynolds number,low Mach number bias ﬂow according to the 2S model as a function of the phase (x-axis) and modulus(y-axis) of the upstream acoustic reﬂection coeﬃcient R1 . (Left) M1 = 0.01 - (Center) M1 = 0.05 (Right) M1 = 0.1. (Top) θ = 2 - (Middle) θ = 10 - (Bottom) θ = 50.

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Figure 4. Acoustic absorption coeﬃcient of a short circular hole sustaining a high Reynolds number,low Mach number bias ﬂow according to the QSH model as a function of the phase (x-axis) andmodulus (y-axis) of the upstream acoustic reﬂection coeﬃcient R1 . (Left) M1 = 0.01 - (Center)
M1 = 0.05 - (Right) M1 = 0.1. (Top) θ = 2 - (Middle) θ = 10 - (Bottom) θ = 50.
The impact of the upstream acoustic reﬂection coeﬃcient R1 on the acoustic absorption coeﬃcient Δ is
now investigated for the 2S model, as shown in Fig. 3, and for the QSH model, as shown in Fig. 4. For
both models, three diﬀerent Mach numbers M2 and cross section area ratios θ are considered.
The evolution of the acoustic absorption coeﬃcient as a function of the modulus and phase of the upstream
acoustic reﬂection coeﬃcient for given values of M2 and θ are quite similar for the 2S and QSH models.
For limited values of M2 and θ, corresponding to Fig. 3-(Top Left) and Fig. 4-(Top Left), the acoustic
absorption coeﬃcient is close to zero whatever the acoustic reﬂection coeﬃcient R1 . For sizeable valuesof M2 and θ, the acoustic absorption coeﬃcient highly depends on both the modulus and phase of R1 .
The largest variations for the acoustic absorption coeﬃcient according to the QSH model are observed for
M2 = 0.1 and θ = 10 in Fig. 4-(Middle Right), where the minimum and maximum values of the acoustic
absorption coeﬃcient are respectively ΔQSH ∼ 0 and ΔQSH = 0.8988. On the other hand, the largest
variations for the acoustic absorption coeﬃcient according to the 2S model are observed for M2 = 0.05 andθ = 50 in Fig. 3-(Bottom Center) with corresponding minimum and maximum values given by Δ2S ∼ 0
and Δ2S = 0.8581 respectively. For both models, most of the acoustic energy entering the system isdamped across the short circular hole for certain speciﬁc parameters. On the other hand, the acoustic
energy balance is barely aﬀected for other sets of parameters.

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5

CONCLUSIONS

Analytical expressions for the acoustic absorption coeﬃcient of a short circular hole sustaining a highReynolds number, low Mach number bias ﬂow were derived for two distinct models. In the ﬁrst model,
called the 2-step (2S) model, the acoustic response of the hole was represented using an isentropic area
decrease followed by a non-isentropic area increase with large scale ﬂow separation. In the second model,
called the quasi-steady Howe’s (QSH) model, the acoustic damping generated across the hole was described using Howe’s model in the limit of small Strouhal numbers, corresponding to hydrodynamicallycompact perturbations. For both models, it was shown that the acoustic absorption coeﬃcient was afunction of the cross section area ratio θ, Mach number inside the hole M2 and modulus and phase of the
upstream acoustic reﬂection coeﬃcient R1 . Simpliﬁed expressions for the acoustic absorption coeﬃcientwere derived for both models for upstream anechoic boundary conditions R1 = 0 and for small (θ  1)and large (θ ∼ 1) holes. It was ﬁnally shown that the acoustic absorption coeﬃcient was equal to zerofor an upstream rigid boundary condition (R1 = 1) or in the absence of a mean ﬂow (M2 = 0) for boththe 2S and QSH models.
The predictions of the acoustic absorption coeﬃcient according to the 2S and QSH models were subsequently investigated. It was demonstrated that the impact of the upstream acoustic reﬂection coeﬃcient
R1 was limited for small values of the cross section area ratio θ and Mach number M2 . On the other
hand, as M2 or θ became sizable, the predicted acoustic absorption coeﬃcient became highly dependent
on the upstream acoustic reﬂection coeﬃcient R1 for both models. For such conditions, the acoustic
energy balance was shown to be unaﬀected for certain values of R1 while most of the acoustic energy was
found to be damped across the system for other values of R1 .

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