Analytic solutions for acoustic scattering by blade rows with complex
boundaries: porosity, compliance and impedancePeter J. Baddoo∗ & Lorna J. Ayton
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom

Abstract
The acoustic scattering by an infinite cascade of flat plates is analysed for a range of boundary conditions including
porosity, compliance and impedance. By employing the Wiener-Hopf method, analytic expressions are derived
for the pressure field that are uniformly valid throughout the entire domain. This enables exact solutions for theunsteady lift, sound power output and other acoustic quantities. The Wiener-Hopf analysis shows that complex
boundaries have a significant effect on the modal structure of the cascade’s inter-blade region, whilst the modesin the far-field regions are unchanged. The results demonstrate the considerable impact of small modifications
to the boundary conditions and highlight the importance of the accurate modelling of aeroelastic effects when
predicting sound transmission and generation in turbomachinery.
Keywords: fluid-structure interaction, turbomachinery noise, Wiener-Hopf method

1

INTRODUCTION

Turbomachinery noise remains a significant contributor to overall aero-engine noise [12]. A considerable sourceof broadband noise is the so-called “rotor-stator interaction”, where unsteady wakes shed by rotors interact withdownstream stators. Much progress has been made in understanding rotor-stator interaction noise when theblades are modelled as flat plates [11, 6, 13] or with realistic geometries [2]. However, research into bladeswith complex boundaries – where the blades are not necessarily stationary and rigid – are far less developed,
especially from an analytic standpoint. Consequently, in the present work we derive analytic solutions for the
acoustic scattering by blade rows where the boundary conditions are not limited to a typical rigid no-flux
boundary condition. The method is capable of modelling a range of boundary conditions of practical interest,
some of which are detailed below.
Aspirations for lightweight and efficient engines have driven the design of thinner and lighter blades inturbomachinery [14]. As a result, aeroelastic effects such as flutter and resonance must be considered inmodern turbomachinery design and testing. The rapid and accurate prediction of the aeroacoustic performanceof turbomachinery with consideration of aeroelastic effects is therefore essential in evaluating the appropriatenessof potential blade designs. Analytic solutions are excellent candidates for this task [6, 13, 2], but are presently
limited to rigid blades with no consideration of aeroelastic effects. The present study permits the analysis ofcompliant blades [5], where the plate deforms with a local response to the pressure gradient across the blade.
More general impedance relations with background flow [10] are also amenable to the present approach.
An influential trend in aeroacoustic research is to modify aerofoils with noise reducing technologies. A
popular choice is a poroelastic extension, which was originally inspired by the silent flight of owls [7]. These
extensions have been applied to semi-infinite [8] and finite [1] plates and have demonstrated considerable noise
reductions. The effects of porosity on blades in cascade formations has not received any analytical treatment,
except for recent work by the authors which showed that aerodynamic losses can be mitigated by modifying
the blade spacing appropriately [3]. The approach of the present research permits aeroacoustic analysis ofporous blades under the assumption of a Darcy-type condition where the seepage velocity through the blade is
proportional to the pressure jump across the blade.
In the present research, we extend the work of Glegg [6] and Posson [13] to analyse the acoustic scattering
by a cascade of flat plates with a range of boundary conditions using the Wiener-Hopf method. A significant
advantage of the presented technique is that the method is identical regardless of the boundary condition – the∗ baddoo@damtp.cam.ac.uk

612

only effects of modifying the boundary condition are to modify the kernel in the Wiener-Hopf method. We
consider four possible boundary conditions labelled cases 0–III. Physically, case 0 corresponds to rigid blades;
case I, porous or compliant blades with no background flow; case II, Darcy flow with background flow; andcase III, a general impedance relation. Mathematically, case 0 corresponds to a Neumann boundary; case I, a
Robin boundary; case II, an oblique derivative boundary; and case III, a generalised Cauchy boundary.
We begin by presenting a mathematical model for the blade row in section 2, including the modelling of
the various boundary conditions. We then present some details of the mathematical solution in section 3. In
section 4 we present a range of results including the unsteady lift, sound power output, and scattered pressure
field. Finally, in section 5 we summarise our work and suggest future directions of research.

2

MATHEMATICAL FORMULATION

We consider a rectilinear cascade of blades in a uniform, subsonic flow as illustrated in figure 1. As is typicalin these analyses [6, 11], it is useful to rotate the coordinate system so that(x ∗, y ∗, z ∗ ) = x̃ cos( χ∗ ) − ỹ sin( χ∗ ), x̃ sin( χ∗ ) + ỹ cos( χ∗ ), z̃ ,and the x ∗ and y ∗ coordinates are tangent and normal to the blades respectively. The background flow is tangent
unsteady perturbation

∆∗
∗

2b
U∗
ỹ

χ∗

x̃
background flow
Figure 1. A rectilinear cascade of flat plates with complex boundaries.
to the blades, and may have a spanwise component such that U ∗ = (U ∗, 0, W ∗ ). The blades in the cascade havedimensional length 2b∗ and are inclined at stagger angle χ∗ . The distance between adjacent blades is ∆∗ . We
assume that a vortical or acoustic wave is incident on the cascade, resulting in a velocity perturbation u ∗ to themean flow. The Kutta condition is satisfied by ensuring that there is no pressure jump across the wake.
We introduce an acoustic potential function for the scattered field defined by∇φ∗ = u ∗ .
so that conservation of mass yields the convected Helmholtz equation
1 D0 φ ∗− ∇2 φ∗ = 0.
c02 D0 t ∗22(1)

where c0 is the isentropic speed of sound. We suppose that the unsteady perturbation incident on the cascadetakes the formf gφ∗i = A∗ exp i k x∗ x ∗ + k y∗ y ∗ + k z∗ z ∗ − ω∗ t ∗ .
(2)

613

Since the system is infinite in the spanwise direction, the scattered solution u ∗ must also have harmonic
dependence in the z-direction. Accordingly, making the following convective transformation and appropriate
non-dimensionalisationsf gφ∗ (x ∗, y ∗, z ∗, t ∗ ) = U ∗ b∗ φ(x, y) exp iω −M 2 δx + z − t ,ω=b∗ ∗(ω − W ∗ k z∗ ),U∗

w 2 = (Mδ) 2 − (k z / β) 2 − (2 + 2i)k z δM 2 W (1 − W k z ),reduces (1) to
!
∂2
∂2
2 2
+
+
ω
w
φ = 0.
∂ x2 ∂ y2(3)

All undefined quantities used in this non-dimensionalisation are defined in [9]. The blades now have horizontal
spacing d and vertical spacing s.
2.1 Boundary conditions
We now introduce the boundary conditions for the problem. It is sufficient to specify the behaviour along y = ns
for n ∈ Z. In the following analysis, we use ∆n and Σn to denote the difference and sum of a given quantity
either side of the nth blade and wake.
2.1.1 Upstream boundary condition
There may be no discontinuities upstream of the blade row. Consequently, we write∆n φ(x) = 0,

φ < nd.

(4)

2.1.2 Blade boundary condition
Several possible boundary conditions that can be modelled with the present approach. Mathematical analysis reveals that physically relevant boundary conditions can be expressed in terms of φ and its derivatives.
Accordingly, we may characterise all the possible modified boundary conditions in the general form
" #
f
g
∂φ
(x) = −2w0 exp ik x (δ(nd + x) + k y ns)
Σn
∂y
(5)
 
 


+ µ0 ∆n φ (x) + µ1 ∆n φ x (x) + µ2 ∆n φ x, x (x),nd < x < nd + 2,
where the µn are summarised in table 1 for the different boundary conditions. Furthermore, in the present
analysis we do not allow any added mass and therefore enforce that there is no jump in the normal velocity
either side of the plate. Accordingly, we write
" #
∂φ
∆n
(x) = 0,nd < x < nd + 2.
(6)
∂y

2.1.3 Downstream boundary conditions
Downstream we require the pressure jump across the wake to vanish so that ∆n φ (x) = 2πiP exp [iωδx] ,x > nd + 2,
where P is a constant of integration that will be specified by enforcing the Kutta condition.

614

(7)

Case

Model

µ0

µ1

µ2

Case 0 [6, 13]

rigid, impermeable

0

0

0

Case I [5, 8, 9]

porous, compliant(no background flow)

CI

0

0

Case II

porous, compliant(background flow)iωδCI I−CI I

0Case III [10]impedance−2ω2 (1 + W k z )CI I I−2iM 2 ωCI I I

CI I I

Table 1. Summary of possible boundary conditions and corresponding µ0 , µ1 and µ2 values for equation (5).
The CI , CI I and CI I I each represent constants that control the porosity, compliance or impedance.
Additionally, the normal velocity across the wake must vanish, i.e.
" #∂φ(x) = 0,x > nd + 2.
∆n
∂y

(8)

This completes the description of the mathematical model.

3

SOLUTION

We now present the mathematical solution to the Helmholtz equation (3) subject to the boundary conditions (4),(5), (6), (7) and (8).
As is typical in cascade acoustics problems, we employ integral transforms to obtain a solution that is
uniformly valid [11, 6, 13]. However, φ is discontinuous across each blade and wake in the y-direction.
Therefore, ∂φ/∂ y possesses non-integrable singularities thus preventing the application of a Fourier transform.
Consequently, we must regularise the derivatives of φ to remove these non-integrable singularities. To this end,
we introduce introduce generalised derivatives and write
" #
∞
∞
X
X
 
∂ 2 φ ∂˜2 φ∂φ
0=−∆n φ (x)δ (y − ns) −(x)δ(y − ns),
∆n
2
˜
∂y
∂ y2
∂y
n=−∞
n=−∞

(9)

where ∂˜ represents the partial derivative with discontinuities removed. The second term in (9) vanishes because
we do not allow added mass so there is zero jump in normal velocity across the blade (6) and wake (8).
The scattered solution must obey the same quasi-periodicity relation as the incident field (2). Consequently,
the scattered acoustic potential function in the entire plane may be reduced to a single channel in the domain
by writing
0

φ(x + nd, y + ns) = φ(x, y)einσ ,(10)

where the inter-blade phase angle for φ is σ 0 = k x (δd + k y s). Substituting (9) into the Helmholtz equation (3)and applying the phase angle relation (10) yields
∞
X
 
0
∂2 φ ∂2 φ
2 2
+
+
ω
w
φ
=
∆0 φ (x − nd)δ 0 (y − ns)einσ .
∂ x2 ∂ y2
n=−∞

(11)

615

Reference

gap-to-chord
ratio
∆/2

stagger angleχMach numberM

reduced frequency
ωinter-blade
phase angleσGlegg [6]

0.6

40

0.3

0–30

3π/4

Posson [13]sec (20◦ )

20◦

0.2

5

3π/4

Table 2. Summary of parameters used in results section. These cases both considered a convected gust.
Following Glegg [6], we Fourier transform in the x- and y-directions, and invert the Fourier transform in they-direction to obtain
" #
Z ∞
∂φ
(x) = −4π
D(γ) j (γ)e−iγx dγ,
(12)
Σ0
∂y
−∞
where j is defined in [6]. The problem is now to find D(γ) which represents the Fourier transform of the jumpin acoustic potential either side of the blade and wake. We now solve equation (12) subject to the remaining
boundary conditions applied on y = 0 (4,5,7). Consequently, we have an integral equation (12) subject to mixed
value boundary conditions (4,5,7). We solve this system with the Wiener-Hopf method using a similar approach
to Glegg [6], and the solution for D is given byw0
w0 δ(ω − k x )e2i(γ+δk x )+(2π) 2 i(γ + δk x )K− (−δk x )K+ (γ) (2π) 2 i(γ + δk x )(γ + δω)K+ (−δk x )K− (γ)−∞∞
X
K+ (θ n+ )(A n + Cn ) e2i(γ−θn ) K− (θ n− ) X Bn
−
−
·
,
·
i(γ + δω)(γ − θ n− ) K− (γ)γ − θ n+ K+ (γ)n=0
n=0

D(γ) =(13)

where K represents the new Wiener-Hopf kernel defined by
K (γ) = j (γ) +1 µ0 − iµ1 γ − µ2 γ 2 .
4πAll new variables in (13) are defined in an analogous way to [6]. We note that the solution is similar to Glegg’s
solution for the rigid cascade [6], except the original Wiener-Hopf kernel j is now replaced with the modified
kernel K. We note that the poles of K are the same as j, whereas the zeros of K are generally distinct fromthose of j. The original kernel j is recovered when the boundaries are rigid and µ1 = µ2 = µ3 = 0.
We now invert the Fourier transform in the x-direction to obtain an expression for the acoustic field that is
uniformly valid throughout the entire domain. The inversion is achieved by splitting the physical domain in asimilar manner to [13]. Contour integration yields the final expression for the acoustic potential function, whichtakes a similar form to that of [13], and yields analytic expressions for the velocity and pressure fields.

4

RESULTS

In order to enable comparison against previous research, we consider the test cases analysed by Glegg [6] andPosson et al. [13] as outlined in table 2.
During the solution to the Wiener-Hopf problem associated with the scattering by the blade row withcomplex boundaries, we observed that the major difference with the rigid case is that the duct modes are
modified. Consequently, we expect that modifications to the boundary conditions will have a strong impact on
the pressure gradient across each blade, leading to significant effects on the unsteady loading. This hypothesis
is tested in figure 2, where the unsteady loading for a rigid cascade is compared against the loading for a range616

of µ0 values. The effects of complex boundaries on the unsteady loading are most significant at low frequencies(ω < 8) and are far less significant at higher frequencies (ω > 8) after the second acoustic mode is cut-on.
Moreover, we observe a shift in the cut-on frequency of the passage modes, as reflected by the shifts in thepeaks near ω ≈ 12.
Analytic expressions for the sound power output are available by a similar method to Glegg [6], andthe downstream power output is plotted in figure 3. As anticipated by the Wiener-Hopf analysis, the cut-on
frequencies of the acoustic modes are unchanged by the effects of complex boundaries. We note that there is alarge jump in the sound power output for the case µ0 = −5 just above the cut-on frequency of the first mode.
However, beyond this frequency, the sound power output monotonically decreases as µ0 decreases. Similarly
for the second and third modes, the sound power generated by blades with a small amount of compliance isgenerally much smaller than that for rigid blades µ0 = 0.
Varying µ1 has the effect of increasing or decreasing the upstream and downstream pressure fields, as
illustrated in figure 4. For example in figures 4c and 4d there is an increase in the downstream and upstreampressure fields respectively, whereas in figure 4b both the upstream and downstream pressure fields are reduced.

1.5

Cp

1

0.5

0

Downstream sound power output

10
12
14
18
20
16
ω
Figure 2. Unsteady lift for a range of frequencies. The aerodynamic and aeroacoustic parameters are definedin table 2 and correspond to those in figure 3 of [6]. In all plots µ1 = µ2 = 0 and the colours correspond toµ0 = 0, µ0 = −1 and µ0 = −5.
2

0

2

4

8

6

·10−2

1.5
1
0.5
0

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

ω
Figure 3. Downstream sound power output for a range of frequencies. The aerodynamic and aeroacoustic
parameters are defined in table 2 and correspond to those in figure 9 of [6]. In all plots µ1 = µ2 = 0 and thecolours correspond to µ0 = 0, µ0 = −1 and µ0 = −5. The line styles correspond to the first (), second(), and third () modes respectively.

617

These effects can be attributed to the absorption and production of energy via the impedance condition applied
to the blades.
(a)

(b)

4
y/2

y/2

4

2

0
−4

0

−2

2

4

2

0
−4

6

0

−2

x/2
(c)

4

6

2

4

6

(d)

4
y/2

4
y/2

2
x/2

2

0
−4

0

−2

2

4

2

0
−4

6

0

−2

x/2
−0.10

−0.08

−0.06

x/2
−0.04

−0.02

0.00

0.02

0.04

0.06

0.08

0.10

normalised pressure, p
Figure 4. Scattered pressure for the parameters defined in Posson et al. [13] and reiterated in table 2. The
boundary data is defined with µ0 = µ2 = 0 and (a) µ1 = 0 (i.e. rigid), (b) µ1 = −2/3i, (c) µ1 − 4/3i, (d)µ1 = 4/3i.

5

CONCLUSIONS

We have derived an analytic solution for the scattering of an unsteady perturbation incident on a rectilinear
cascade of flat plates with complex boundaries. The analytic nature of the solution means that it is extremely
rapid to compute, and offers physical insight into the role played by different boundary conditions. In contrast
with previous studies that focussed on the effects of rigid plates [6, 13], the formulation of the present researchallows a range of boundary conditions to be studied with minimal effort, such as porosity, compliance andflow impedance. In terms of the spectral plane, the effect of modifying these boundary conditions is tochange the locations of the zeros of the Wiener-Hopf kernel, whilst the poles are unchanged. Accordingly, themodal structure of the far-field scattered pressure is invariant under modifications to the flat plates’ boundary
conditions. Conversely, the modal structure of the near-field region undergoes large deformations since the zerosof the kernel correspond to the duct modes of the inter-blade region. This has a strong effect on the surfacepressure fluctuations and unsteady loading, which has implications for the far-field sound. Future work will618

focus on more detailed analysis of elastic plates [1, 8] in order to explore the important effects of flutter andresonance.
Finally, this study shows that modified boundary conditions have a large impact on gust-cascade interaction
noise and offers a useful design tool that can model aeroacoustic and aeroelastic effects.

ACKNOWLEDGEMENTS
P. J. B. acknowledges support from EPSRC grant 1625902 and L. J. A. acknowledges support from EPSRC
grant EP/P015980/1.

REFERENCES[1] L. J. Ayton. Acoustic scattering by a finite rigid plate with a poroelastic extension. Journal of Fluid
Mechanics, 791:414–438, mar 2016.
[2] P. J. Baddoo and L. J. Ayton. An analytic solution for gust-cascade interaction noise including effects ofrealistic aerofoil geometry [SUBMITTED]. Journal of Fluid Mechanics.
[3] P. J. Baddoo and L. J. Ayton. On periodic singular integral equations with application to porous cascadesof aerofoils [SUBMITTED]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering
Science.
[4] A. V. Cavalieri, W. R. Wolf, and J. W. Jaworski. Numerical solution of acoustic scattering by finite
perforated elastic plates. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences,472:20150767, 2016.
[5] D. G. Crighton and F. G. Leppington. Scattering of aerodynamic noise by a semi-infinite compliant plate.
J. Fluid Mech., 43(4):721–736, oct 1970.
[6] S. A. L. Glegg. The response of a swept blade row to a three-dimensional gust. Journal of Sound andVibration, 227(1):29–64, 1999.
[7] R. R. Graham. The Silent Flight of Owls. Journal of the Royal Aeronautical Society, 38(286):837–843, 1934.
[8] J. W. Jaworski and N. Peake. Aerodynamic noise from a poroelastic edge with implications for the silent
flight of owls. Journal of Fluid Mechanics, 723:456–479, 2013.
[9] A. Kisil and L. J. Ayton. Aerodynamic noise from rigid trailing edges with finite porous extensions.
Journal of Fluid Mechanics, 836:117–144, 2018.
[10] M. Myers. On the acoustic boundary condition in the presence of flow. Journal of Sound and Vibration,
71(3):429–434, 1980.
[11] N. Peake. The interaction between a high-frequency gust and a blade row. J. Fluid Mech, 241:261–289,
1992.
[12] N. Peake and A. B. Parry. Modern Challenges Facing Turbomachinery Aeroacoustics. Annual Review of
Fluid Mechanics, 44(1):227–248, 2012.
[13] H. Posson, M. Roger, and S. Moreau. On a uniformly valid analytical rectilinear cascade response function.
Journal of Fluid Mechanics, 663:22–52, 2010.
[14] G. Saiz. Turbomachinery Aeroelasticity Using a Time-Linearised Multi Blade-row Approach. PhD thesis,
Imperial College London, 2008.

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