Acoustic scattering in arrays of orifices, slits and tube rows with mean
flow: A comparison
Charles BOAKES; Aswathy SURENDRAN∗ ; Dong YANG; Aimee MORGANSImperial College London, United Kingdom

Abstract
Thermoacoustic oscillations result from positive feedback between acoustic fluctuations and unsteady heat transfer into or out of the system. The thermoacoustic response of combustion system heat exchangers in cross flow
is assumed to be dominated by two effects: acoustic scattering at the heat exchanger tube row and unsteadyheat transfer from the tube row. While the latter causes a significant reduction in the net energy of the oscillations, the influence of both the effects on the acoustic fluctuations needs to be addressed carefully to predict thethermoacoustic response of the heat exchanger. To this end, we adopt a sequential approach where the acoustic
scattering and the heat transfer are assumed to be independent of each other. In the present work, we focus
on the acoustic scattering behaviour of heat exchanger tubes (only) by comparing existing acoustic models fororifices, slits and tube rows and identifying those operating regimes where the different models are valid andoverlap. This will enable us to formulate appropriate acoustic models for the various operating regimes of theheat exchanger tube rows.
Keywords: Acoustic scattering, Heat exchangers, Aeroacoustic models

1.

INTRODUCTION

The present work is motivated by the use of heat exchangers in futuristic aero-propulsion engines (1). Heatexchangers, much like flames and other heat sources, can influence the thermoacoustic behaviour of combustion
systems that include them. Their behaviour is assumed to be dominated by two effects: acoustic scattering at theheat exchanger tube row and the unsteady heat transfer across the tube row. It is therefore crucial to accurately
model these two effects. The present work is dedicated to the acoustic scattering effect of heat exchanger tube
rows.
Heat exchanger tubes are typically circular in cross-section, but can also be approximated by thin plates ofrectangular cross section, separated by a rectangular gap. Studies conducted with the latter configuration haveshown to have a stabilising influence on combustion systems undergoing thermoacoustic instabilities (2). Thereexists vast literature for the acoustic modelling of slits and perforated plates (3), and in the present work, we
focus only on three existing models: a Quasi-steady cylinder model applied to an array of cylindrical tubes,
the Dowling and Hughes slit model applied to rectangular slits and the modified Cummings slit model adapted
for a tube row. The predictions of the acoustic scattering matrices from these models are compared in order toestablish their validity in different operating regimes of the heat exchanger tube rows.
The structure of the paper is as follows: the three existing models are described in Section 2. In Section 3, wecompare the acoustic scattering matrix predictions from the three models and draw our preliminary conclusions
in Section 4.

2.

ACOUSTIC MODELS

The flow past heat exchanger tubes in cross flow is essentially flow through a gap or constriction. This produces
a jet and an associated shear layer downstream of the gap. When a low frequency acoustic wave is introduced∗ Corresponding author: a.surendran@imperial.ac.uk

2126

into the system, it creates high frequency evanescent waves and low frequency planar waves. Typically, theevanescent waves decay faster than the planar waves, and are hence found closer to the gap. Farther fromthe constriction, we have only low frequency planar waves that are one dimensional. Moreover, in most of
the combustion systems undergoing thermoacoustic instabilities, the frequency range of interest is generally∼ O(100)Hz. Therefore, in the present study, we restrict our analysis to low frequency planar 1-D waves. To
enable us to develop a 1-D model for the aeroacoustic behaviour of heat exchangers, we compare the following
existing models which account differently for the gap shape.
• A Quasi-steady model developed for a row of cylinders (Section 2.1)
• The Dowling and Hughes model for thin slits (Section 2.2)
• The modified Cummings model developed for thin slits and applied to tube rows (Section 2.3)
2.1 Quasi-steady cylinder model
The schematic of the tube row geometry considered is illustrated in Figure 1. It consists of two half cylinders
placed inside a duct of height hd and having a bias flow of velocity ug through the gap of height hg between
the cylinders. The bias flow then forms a jet of height h j downstream of the cylinder. It is sufficient to model
this system instead of an array of cylinders as they are equivalent (through the method of image sources).
T1→2 , R1p̂+
1u1

Separation

T2→1 , R2

1

2

hd

hg

ug

p̂−
1

j
hj

T1→2 , R1
p̂+
2

p̂+
1

u2

u1

p̂−
2

p̂−
1

1

hd

hg

ug

Figure 1. Schematic for the flow domain in the Quasi-steady cylinder model
The quasi-steady model developed for tube rows can be found in Surendran et al. (4). This approach, first proposed by Ronneberger (5), assumes that the time-dependent acoustic perturbations of the flow are sufficiently
slow such that the unsteady terms in the relevant 1-D conservation equations can be neglected. This is valid
for small Strouhal numbers (St = ωd/u) and small Helmholtz numbers (He = ωd/c), where ω is the angularfrequency, d is the diameter of the tube, u is the velocity and c is the speed of sound. The latter requiresthat the acoustic wavelength is much larger than the tube diameter, leading to a compact tube row assumption. It is also assumed that the acoustic wavelength is much larger than the mixing zone downstream of the
tube row. Consequently, no phase changes in the acoustic scattering properties are expected across the tube row.
The duct is divided into three regions: Region 1, region j and region 2. Region 1 and 2 have uniform flows
and are upstream and downstream of the cylinders, respectively. Region j lies between regions 1 and 2 andcontains the cylinders, the jet and the mixing region. Assuming isentropic and irrotational flow between theregions 1 and j, the conservation of mass and energy under the assumptions of quasi-steadiness and perfect gas
behaviour can be applied. This yields Eqs. (1)-(3), where the variables ρ, p and γ denote the density, pressure
and ratio of specific heats, respectively, while the subscripts indicate the corresponding regions.
hd ρ1 u1 = h j ρ j u j
γ p1
1
γ pj
1 2
u +
= u2j +
2 1 γ − 1 ρ1
2
γ −1 ρj
 γ
p1
ρ1
=
pj
ρj

(continuity)(1)(energy)(2)(isentropic)(3)

Downstream of the mixing region, the flow is uniform and non-isentropic. Hence, the conservation of mass andmomentum are used to link regions j and 2 as shown in Eqs. (4)-(5). Neglecting heat transfer, viscous andfrictional losses at the wall, conservation of energy can be applied across regions 1 and 2 as given in Eq. (6).
h j ρ j u j = hd ρ2 u2(continuity)(4)

hd p j + h j ρ j u2j = hd p2 + hd ρ2 u22(momentum)

(5)

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1
γ p1
γ p2
1 2
= u22 +u +2 1 γ − 1 ρ1
2
γ − 1 ρ2(energy)(6)

These equations (Eqs. (1)-(6)) are then linearised about a mean condition by decomposing u, p and ρ into
the sum of a mean component (¯) and a small acoustic perturbation component (0 ), while neglecting higher
order terms of the primed quantities (6). These decompositions are shown in Eq. (7) and the derivation of thelinearised equations can be found in Appendix D of (7). The acoustic perturbations are related to the complex−
0
amplitudes of the forward and backward travelling pressure waves p̂+
1,2 and p̂1,2 as shown in Eq. (8), where sis the entropy perturbation, c p is the specific heat at constant pressure and the subscript i denote the region.
Substituting these expressions for the velocity, pressure and density perturbations yields a set of equations interms of p̂±
1,2 that can be manipulated to form a scattering matrix. In the absence of an incoming entropy wave
(s01 = 0), the scattering matrix will be of the form shown in Eq. (9). Here, T1→2 and R1 are the transmission andreflection coefficients for a wave incident from upstream and T2→1 and R2 are the transmission and reflection
coefficients for a wave incident from downstream.
p = p̄ + p0

u = ū + u0−
p0i = p̂+i + p̂i

u0i =

−
p̂+
i − p̂iρ̄i ci + 
p̂2
T
= 1→2
p̂−
R1
1

R2
T2→1

ρ = ρ̄ + ρ 0(7)p̂+ + p̂− ρ̄iρi0 = i 2 i − s0i
cp
ci

(8)

  +
p̂1
p̂−
2

(9)

2.2 Dowling and Hughes slit model
The Dowling and Hughes model describes the acoustic scattering behaviour of thin rectangular slits with bias
flow by extending the analysis of Howe (8). The flow domain, as shown in Figure 2, consists of two infinite
half planes connected by the slit. The infinitely long slit plate can be approximated to the geometry shown,
again through the method of image sources. Each slit has a width of hg and is spaced hd distance apart, having
a bias flow velocity of ug though the slits. The flow is assumed to be incompressible and hence ρ̄1 = ρ̄2 = ρ0 .
Separation

R1

T2→1 , R2
2

hd

hg

ug

j
hj

T1→2 , R1
p̂+
2

p̂+
1

u2

u1

p̂−
2

p̂−
1

T2→1 , R2

1

2

hd

hg

ug

j

p̂+
2
u2

p̂−
2

Figure 2. Schematic for the flow domain in the Dowling and Hughes slit model.
Howe (8) described the attenuation of acoustic waves incident on screens with circular perforations by assumingthat the acoustic losses are due to the unsteady convection of vorticity from the hole rims. Using stagnation
enthalpy B = p0 /ρ0 + 12 u2 as the acoustic variable, the acoustic field is described using Howe’s analogy (9),reducing the problem to that of solving an inhomogeneous Helmholtz equation in stagnation enthalpy (Eq. (10)).
(∇2 + k2 )B = −∇ · (Ω0 × ūg ),(10)

where k is the wave number, Ω0 is the amplitude of linear vorticity perturbations generated by the incident
acoustic wave, and ūg is the mean convection velocity of the shed vorticity. It is assumed that vorticity isconvected without any dissipation by the mean flow, and its strength is determined by the Kutta condition.
Dowling and Hughes solved Eq. (10) in the limit of small open area ratio (η = hg /hd ) and small Helmholtz
number. They showed that the transmission and reflection coefficients can be determined by Eqs. (11) and (12),

with

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T1→2 = ρ0 ω V̇ / (khd ) ,(11)

R1 = 1 − T1→2 ,(12)iπη/ (2StMg )ρ0 ω V̇=,khdiπη/ (2StMg ) − ln (πη) + ln 2/Φ(St)

(13)







1
πI0 (St) e−St + 2i sinh (St) K0 (St)hihi
Φ(St) = 1 −,St ln 2  πe−St I (St) + I0 (St) + 2i sinh (St) K0 (St) − K (St) 
1
1
St ln 2St ln 2(14)

where V̇ is the perturbation volume flux through the slit, St = ωhg /(2ug ), Mg = ūg /c is the mean Mach numberof the bias flow and Im and Km are the modified Bessel functions of order m. Again, the scattering matrix
would be of the form shown in Eq. (9).
2.3 Modified Cummings slit modelCummings (10, 11) developed a model to describe the acoustic transmissions through duct terminations using
unsteady Bernoulli’s equation. The configuration studied in (10) is shown in Figure 3a, where an incompressible
uniform mean flow with superimposed plane acoustic waves travels through a converging nozzle of circular crosssection. The flow then separates at the nozzle exit to form a jet with height hvc (vena contracta). The velocities
at 1 and 2 can be related to the gap velocity ug through the open area ratio η (= hg /hd ) and the contraction
coefficient σ = hvc /hg as u1 = ηug and u2 = ug /σ . In the present study, we adapt a similar flow structure forthe slit (see Figure 3b), where the uniform flow upstream of the slit separates at the slit edge forming a jet.
1
1

hg
hg

2

u1

hd
hd

p̂+
1

2

T1→2 , R1
T1→2 , R1
p̂+
1

hvc

u1
u2

u2

hd
p̂−
1

1

u1

hd
hvc

1

u1

hg

T2→1 , R2
T2→1 , R2
hvc
p̂+
2
2
hvc
p̂+
2
2
u2
u2
h
g

p̂−
2

p̂−
1

(a) Converging nozzle flow (Cummings andEversman (10))p̂−2(b) Flow though a slit (Present study)

Figure 3. Geometry used in the modified Cummings model.
Applying unsteady Bernoulli’s equation from region 1 to 2 yields
Z 2
1

ρ0


dug
ρ0 2
dx +
u2 − u21 = p1 − p2 .
dt
2Defining an effective length L for the slug of fluid undergoing unsteady motion at the slit as= ρ0 L (dug /dt), the Cummings equation (Eq. (15)) can be written as
ρ0 L

2 1 − η 2 σ 2du0g+ ρ0 ūg + u0g= p1 − p2 .
dt
2σ 2

(15)
R2

1 ρ0 (dug /dt) dx(16)

Cummings (11) argued that for high amplitude pressure fluctuations, the jet formed at the orifice can grow anddiminish periodically due to flow reversal, causing L to vary. However, for low amplitude pressure perturbations
that do not lead to flow reversal, Luong et al. (12) suggested that L = 2l0 + lw , where l0 ≈ (π/4) hg is the end
correction contributed by the duct termination with irrotational flow on one side and lw is the thickness of theorifice. Subtracting the steady contribution from Eq. (16), and for small η, one can obtain the equation for
perturbations as (12)
du0g u0g
u0g
pI+ 2 ūg += ,(17)(2l0 + lw )
dt
σ
2
ρ0
where pI = p01 − p02 . Albaharna (13) linearised Eq. (17) by neglecting the higher order terms i.e., products ofperturbation terms, and evaluated the transmission and reflection coefficients. This linearised equation (Eq. (18))
is referred to as the modified Cummings model in the present paper.
(2l0 + lw )du0g u0g ūg
pI
+ 2 =
dt
σ
ρ0

(18)

2129

Manipulating the results in (13) for an acoustic wave incident from the upstream side with temporal variationeiωt (see Figure 3b), we obtain the transmission and reflection coefficients as!
4iβ StMg2 + (Mg /σ )2 + 1
2,(19)
T1→2 =
1 + ηMg 4iβ StMg /η + Mg / (ησ 2 ) + 2/ (1 + ηMg )!

4iβ StMg /η + Mg / ησ 2
R1 =
,
(20)
4iβ StMg /η + Mg / (ησ 2 ) + 2/ (1 + ηMg )where St = ωhg /(2ug ) and β is the end correction coefficient. Albaharna used σ = 0.75 as suggested andexperimentally validated by Cummings (11), and used the end correction model for rectangular perforations
suggested in (14), to define β . However, this model does not hold for situations where the slit length  slit
width as β becomes infinite. Therefore, in the present study, we use the end correction for a rectangular orifice
in a baffle wall as given in (15) i.e., πη  1 πη 
1
1
l0
+ cot
(21)
= ln tan
β=
hg
π
2
4
2
4

3.

RESULTS FOR MODEL COMPARISONS

The transmission, reflection and absorption coefficients for the different models (and their corresponding geometries) are compared in this section. The absorption coefficient (α) is defined as the ratio of the acoustic
energy absorbed to the acoustic energy incident (16). Across any two regions denoted by 1 and 2, α1→2 canbe expressed as#"2
2(1 + M2 )2 (ρ̄1 c1 /ρ̄2 c2 ) A2 /A1p̂−(1 − M1 )2 + p̂+
2
1
,
(22)
α1→2 = 1 −22p̂+(1 + M1 )2 + p̂−(1 − M2 )2 (ρ̄1 c1 /ρ̄2 c2 ) A2 /A1
1
2
where A is the area of cross section. For the analysis provided in this paper, we have used the systemproperties shown in Table 1. Here, d is the diameter of the cylinder and µ is the dynamic viscosity.
Table 1. System properties
Property
d
ρ
c

Unit
[m]
[kg/m3 ]
[m/s]

Value
0.02
1.2
700

Property
µ
η
γ

Unit
[Pa s]-

Value
1.8×10−5
0.1
1.4

Figure 4 shows the coefficients predicted by the three models for various excitation frequencies (in terms of theSt), Mg = 0.1 and η = 0.1. The Dowling and Hughes model (D&H) and the modified Cummings model (MC)are for rectangular slits and the Quasi-steady model (QS) is for tube rows. It can be observed that the D&H andMC models are consistent and exhibit similar behaviour in frequency, whereas the QS model, experimentally
validated in (4), does not exhibit any dependence on frequency. It can also be noted that at very low Strouhal
numbers (St ≤ 0.4), the D&H and MC models also predict quasi-steady behaviour, reinforcing the low Strouhal
1 numbers assumptions used in the QS model for similar1flow and geometry considerations.
and low Helmholtz
However, the QS 1model predicts a higher transmission and lower reflection and1 absorption coefficients comparedto the other twomodels.
0.8
0.8
1
1
0.8
0.8
0.6
0.6
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.4
0.4
0.2
0.2
0
0
0
0
1
2
3
4
0
1
2
3
4
0
1
2
3
4
Strouhal number (St)
0.2
0.2
0
0
4 4. Transmission,0
1
1 for Mg 2= 0.1 and 3
4
Figurereflection
and 2
absorption 3
coefficients4for an upstream0perturbation
0
0
η
=
0.1.
0 Strouhal
123(St) 40
1
2
3
4
numberStrouhal number (St)

3

2130

|Rα1 |1→2

α1→2

|R1 |

|R|1 ||T1→2|T1→2 |

1

Modified Cummings (Slit)
Dowling & Hughes (Slit)
Quasi-steady (Cylinder)

The predictions of the different models for a given frequency such that St → 0 (= 10−6 ), η = 0.1 and for various
gap Mach number Mg are shown in Figure 5. Though the overall trend exhibited by the different coefficientsare similar, the QS model is seen to again predict higher transmission and lower reflection coefficients forlower Mg values. For the transmission coefficient, as Mg increases, the MC model is seen to diverge from the
D&H model, though the reflection and absorption coefficients are consistent. Since the QS model has been
experimentally verified for tube rows (4) and the D&H model has been experimentally verified for plates withslits (6), it can be concluded that tube rows cannot be approximated as rectangular slits as was proposed in (13).
1
0.8

0.6

0.6

0.6

|R1 |

0.4

0.4

0.2
0

α1→2

1
0.8

|T1→2 |

1
0.8

0.1

0.2

0.3

0

0.4

0.4
0.2

0.2
0

Modified Cummings (Slit)
Dowling & Hughes (Slit)
Quasi-Steady (Cylinder)

0

0

0.1
0.2
0.3
0.4
Gap Mach number (Mg )

0

0.1

0.2

0.3

0.4

Figure 5. Transmission, reflection and absorption coefficients for an upstream perturbation for St → 0 andη = 0.1.
The discrepancy between the predictions of the QS and MC models can be explained by the different incompressible steady flow loss coefficients implied by Eq. (5) and Eq. (16). The loss coefficient defined as
K=

∆p0
,
0.5ρ0 u2g

(23)

and applied to the QS and MC models give
KQS = (1 − η j )2 ,


1
2
−
η
KMC =
.
σ2

(24)
(25)

|T1→2 |

0.4

3

4

0

|T1→2 |

0

1

0.8

0.6

0.6

0.4

0.2
0.6

0.2

0.8
|R1 |

0.4
0.8

0.6

1

0.2

02 3
0.4
0

0

4

0.4

0

1
2
3
4
Strouhal number (St)MC Adjusted K (Slit)
Dowling & Hughes (Slit)
Quasi-steady (Cylinder)

0.4
0.8
0.2
0.6

0.2
0

0.61

0

|R1 |

0.8

1

α1→2

|R1 |

0.61

1

α1→2

where η j = h j /hd . Since η, η j and σ are less than 1, KMC > KQS , demonstrating that the MC model implies
a greater loss or dissipation than the QS model. If we treat σ or the contraction coefficient as an arbitrary
quantity, KMC can
1 be matched with KQS or any other given loss coefficient.1 The effect of this matching is
depicted in Figure 6 and 7, where the low Strouhal limit of the MC model now coincides with the QS model.
In Figure 7, the small deviation between the two models at higher Mach numbers may be due to the different
linearised formsof Eqs. (5) and (16). Such an adjustment might allow the0.8
use of the MC model to tube rows,
0.8
provided an appropriate end correction is applied.

1

02 3
0.4
0

4

1
2
3
4
1
2
3
Figure 6. Transmission,reflection andnumber
absorption coefficients(St) for an upstream
0.2 Strouhal
0.2 perturbation for Mg = 0.1 and4

η = 0.1. The contraction coefficient in the modified Cummings model is adjusted to match KMC with KQS .

0

CONCLUSIONS

1

2

3

4

0

1
2
3
4
StrouhalnumberIn order to describe the acoustic scattering behaviour of heat exchanger tube rows, weconsidered three
existing (St)
4.

0

0

models: the Quasi-steady cylinder model applied to tube rows, the Dowling and Hughes slit model applied

2131

1

0.8

0.8

0.8

0.6

0.6

0.6

|R1 |

0.4

0.4

0.2
0

α1→2

1

|T1→2 |

1

0.2
0

0.1

0.2

0.3

0.4

0

MC Adjusted K (Slit)
Dowling & Hughes (Slit)
Quasi-steady (Cylinder)

0.4
0.2

0

0.1
0.2
0.3
0.4
Gap Mach number (Mg )

0

0

0.1

0.2

0.3

0.4

Figure 7. Transmission, reflection and absorption coefficients for an upstream perturbation for St → 0 andη = 0.1. The contraction coefficient in the modified Cummings model is adjusted to match KMC with KQS .
to rectangular slits and the modified Cummings slit model. The predictions for the transmission, reflection
and absorption coefficients for the three models show that the slit models are more dissipative than the tube
row model, indicating that heat exchanger tube rows cannot be approximated as rectangular slits. However,predictions from the modified Cummings model, corrected for the incompressible steady flow loss coefficient,gave better agreement with the results of the Quasi-steady model, for low Strouhal and Helmholtz numbers.
One of the advantages of using such a corrected loss coefficient approach is that this information is readily
available through empirical correlations or from steady CFD simulations, for most geometries widely used inengineering applications, and hence can be easily incorporated into the MC model. Presently, studies are being
undertaken to extend the approach given in this paper to accommodate heat transfer in the flow domain.

ACKNOWLEDGEMENTS
We gratefully acknowledge the financial support from the European Research Council Grant AFIRMATIVE(2018-23). We also acknowledge the technical inputs and discussions undertaken with Dr. Ignacio Duran, Reaction Engines Limited, U. K.

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