PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Effect of acoustic treatment on fan flutter characteristics
Yu SUN; Xiaoyu WANG; Xiaofeng SUN
Beihang University, Fluid and Acoustic Engineering Laboratory, School of Energy and Power Engineering,Beijing, China

ABSTRACTThis paper presents an investigation of the acoustic treatment effect on fan flutter characteristics. A threedimensional model is proposed to predict the unsteady aerodynamic responses of an oscillating rotor in a
finite-length annular duct with the acoustically treated duct walls. In this model, the aerodynamic couplingsof oscillating rotor and acoustic liner are analysed in a strict sense by employing the transfer element method,with the unsteady aerodynamic loading distribution on rotor blades predicted by the three-dimensional lifting
surface theory and the liner modelled as equivalent monopole sources distributed on hard wall. Moreover,combined with a boundary integral approach, the present model implicitly considers the reflections at duct
openings, thus the aerodynamic interactions within such a finite-length duct system can be truly determined.
Within the linear scope, the primary perturbations induced directly by blade oscillation and the scattering
fields resulted from aerodynamic couplings are superposed to give the total blade loadings, such that the rotor
flutter stability can be further evaluated by its unsteady aerodynamic work based on the energy method. A
parametric study for the first-order torsion vibration has been conducted, whose results reveal the noticeable
impact of the acoustic liner on aeroleastic stability under certain circumstances.
Keywords: Fan flutter stability, Acoustic treatment, Transfer element method

1. INTRODUCTION

Extensive research efforts have been paid to fan flutter problems since the middle of last century.
However, a brake is on the development of blade flutter control due to the great complexities arisingfrom the fact that any factors affecting the surrounding flow fields or the aeroelastic characteristics
of cascade structure are likely to further alter the evolution of this kind of self-excited vibration.
Passive suppression techniques of cascade flutter, which were comprehensively assessed by
Bendiksen (1) and have revealed little appreciable breakthroughs in the recent decades, usually comewith a limited range of utility under the extreme and variable operating conditions of aero engines.
By contrast, the concept of suppressing fan flutter with the active control of acoustic treatment onduct wall (2–4) shows a promising prospective. Zhao and Sun (5) have experimentally achieved theactive control of duct wall impedance with a kind of perforated liner with adjustable bias flow backed
up by a hollow cavity of controllable depth, whose impedance characteristics have been investigatedby considerable theoretical and experimental work (6–9). Then we can expect that, once the in-depth
knowledge of the aerodynamic impact of acoustically lined wall on fan flutter stability is gained, thecorresponding control strategy can be devised for flutter suppression purpose.
Watanabe and Kaji (3) have firstly proven that the change of duct wall impedance could
considerably change the aerodynamic damping of oscillating blades. However, the three-dimensionalsemi-actuator disk model they used has been excessively simplified. By determining the Green’s
function for a duct comprised of two parallel infinite flat walls with one of locally reacting impedance,
Sun and Kaji (4) further investigated the soft wall effect on the aeroelastic stability of a linear cascade.
Their model cannot take into account the annular effect critical for real compressors or turbines aswell as the acoustic liners of finite extent. Besides, a troublesome process involved for numericallycomputing the complex eigenvalues for soft walled duct considerably reduces its efficiency. Hence,
more refined and efficient model is still in urgent demand.
In this paper, a three-dimensional semi-analytical model is proposed to predict the unsteady2041

aerodynamic responses of an oscillating rotor in a finite-length annular duct with the acousticallytreated side walls. This model is established based on the transfer element method (10,11), in which
each segment of a duct is modelled as a response function in a unified matrix form, called transfer
element. Combined with a boundary integral approach (12), this method can derive an overall matrix
equation for the entire disturbed field from the continuity conditions imposed on every interface ofduct segments as well as on inlet and outlet duct terminations. By solving this matrix equation one
can simultaneously determine the disturbed fields both inside and outside the duct, with the existing
aerodynamic interactions and the reflections at duct openings thoroughly considered. Moreover, twoanalytical tools have been used to construct the specific transfer elements for rotor and liner,
respectively. First of all, by using Namba’s three-dimensional lifting surface theory (13), the unsteady
aerodynamic loadings and the corresponding induced disturbances on blade surfaces in response to
the oncoming perturbations can be determined from an upwash integral equation. Secondly, by treating
an impedance wall as the corresponding hard wall distributed with the equivalent monopole offluctuating strength, (14), the scattering effect of finite-length lined section can be effectively
modelled. Finally, as indicated by the energy method, the possibility of cascade flutter is evaluated by
the total unsteady aerodynamic work exerted on rotor blades, which is obtained by summing up thecontributions from both the initial blade oscillation and the aerodynamic couplings of oscillating rotor,
acoustic liner and duct openings.
In the following content, a general framework of transfer element will be introduced at first andthen its tailored forms for rotor and liner will be discussed subsequently. With the other relevant
theoretical tools briefly introduced, the complete process for establishing the present model will bepresented. Numerical experiments will be discussed for an improved understanding about how theacoustic treatment on duct wall can influence the rotor flutter stability in a finite-length duct.

2. MODEL AND METHOD DESCRIPTION
2.1

Model Description

Figure 1 –Finite-length annular duct in question
In this paper, we consider a subsonic rotor cascade and a downstream lined section on the casing
in an annular duct of finite axial extent, as shown in Figure 1. The outer and the inner duct radii aredenoted by 𝑅𝑅𝑑𝑑∗ and 𝑅𝑅ℎ∗ , respectively. With the focus in this paper on the basic insights of the
aerodynamic impact, on the onset of instabilities, of the partially lined wall upon the aeroelastic
stability of an oscillating rotor in finite-length duct, the analyses have been developed within thelinear scope with necessary assumptions made towards flow conditions, cascade geometry and bladevibration mode. Suppose the time mean flow is a uniform axial flow without separation, of staticpressure 𝑝𝑝0∗ , density 𝜌𝜌0∗ and velocity 𝑈𝑈 ∗ = 𝑀𝑀𝑐𝑐0∗ (where 𝑀𝑀 < 1 is its Mach number and 𝑐𝑐0∗ is thesound speed). Hereafter, with the asterisk denoting dimensional quantities omitted, all the quantities
will be correspondingly scaled with respect to 𝑅𝑅𝑑𝑑∗ , 𝑅𝑅𝑑𝑑∗ /𝑈𝑈 ∗ and 𝜌𝜌0∗ 𝑈𝑈 ∗2. Further, consider the unstalled
rotor rotating with angular velocity Ω has 𝐵𝐵 identical, equally spaced, zero-thickness blades
operating at zero mean incident. The 𝐵𝐵 blades are vibrating in a given harmonic mode with the sameangular frequency Ω𝑠𝑠 but with a constant inter-blade phase lag σ, which gives birth to the excitation
sources in our problem. All the induced perturbations are small compared to the steady counterparts.
Also note that the unsteady aerodynamic loadings exerted on rotor blades have been investigatedunder fully three-dimensional consideration instead of subject to any simplifications required by strip
theories, while the effect of steady blade loadings has been excluded in the entire discussion.

2042

2.2 Method of Calculation
The difficulties in predicting unsteady rotor response in a partially lined duct of finite axial extent
mainly arise from two respects. Firstly, the disturbances everywhere within a duct are tightly coupled
and simultaneously changing, subject to all the scatterings of inner components as well as thereflections at duct open ends. So one cannot truly determine the disturbed fields unless the response
function of the entire duct system is constructed as a whole, with all the coupling effects taken into
account. Secondly, the capability of treating a wall boundary condition of non-uniform impedance isrequired, in the preference for a time-saving process. To these ends, we suggest a semi-analyticalmodel developed by using the transfer element method combined with a boundary integral equation,in which the three-dimensional lifting surface theory is employed to derive the rotor response function
while the acoustically lined section is equivalently modelled as a rigid wall with the monopole sourcesof fluctuating strength distributed on it.
2.2.1 Transfer Element Method

Figure 2 – An arbitrary segment of an annular duct
As depicted in Figure 2, consider a finite-length segment of an annular duct, denoted by the index
𝑘𝑘. In the cylindrical coordinates (𝑟𝑟, 𝜑𝜑, 𝑧𝑧) fixed to the duct, the axial coordinates of its front and theback interfaces are given by 𝑧𝑧𝑘𝑘 and 𝑧𝑧𝑘𝑘+1 , respectively. As is mentioned earlier, by treating theacoustic liners on the casing wall as the equivalent monopole source distributions, the impermeableboundary condition is imposed on the entire duct wall, and thus in the present approach the acoustic
fields are expressed consistently in terms of rigid duct modes.
𝑖𝑖
Suppose a forward-propagating acoustic wave 𝑝𝑝𝐴𝐴,𝑘𝑘
emanates from certain sources within the 𝑘𝑘 th′duct segment. The primary disturbances transmitting within this segment, i.e., the acoustic wave 𝑝𝑝𝐷𝐷,𝑘𝑘′′
propagating upstream and the vortical wave 𝑤𝑤𝐷𝐷,𝑘𝑘,
propagating downstream, the acoustic wave 𝑝𝑝𝐴𝐴,𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘whose complex modal coefficients are respectively denoted by 𝐷𝐷𝑚𝑚𝑚𝑚 , 𝐴𝐴𝑚𝑚𝑚𝑚 and 𝑉𝑉𝑚𝑚𝑚𝑚 , further arouse the𝑠𝑠𝑠𝑠𝑠𝑠, 𝑝𝑝𝐴𝐴,𝑘𝑘
and 𝑤𝑤𝐷𝐷,𝑘𝑘
by interacting with its inner structures. Here thescattering disturbances 𝑝𝑝𝐷𝐷,𝑘𝑘
subscripts D and A are used to indicate the fluctuations propagating downstream and upstream,
respectively. 𝑚𝑚 denotes the circumferential mode number and 𝑛𝑛 denotes the radial mode number(noting that the lowest radial mode is defined as 𝑛𝑛 = 1 in this paper). The adjacent segments, denotedby 𝑘𝑘 − 1 and 𝑘𝑘 + 1 , are assumed to be free from any inner sources or scattering effects. The
continuity conditions of acoustic pressure, axial sound particle velocity as well as circumferential
vortex velocity on the front and the back interface of the 𝑘𝑘 th segment imply

p′A, k −1 + pD′ , k −1 = p′A, k + pD′ , k + p As , k + p iA, k p′A, k + pD′ , k + pDs , k = p′A, k +1 + pD′ , k +1 
u ′A, k −1 + uD′ , k −1 = u ′A, k + uD′ , k + u As , k + u iA, k  for z = zk ; u ′A, k + u D′ , k + u Ds , k = u ′A, k +1 + u D′ , k +1  for z = zk +1 (1 )wD′ , k −1 wD′ , kwD′ , k + wDs , k wD′ , k +1
=
=


𝑠𝑠,𝑘𝑘
′
It can be proven that the complex modal coefficient 𝑋𝑋𝑚𝑚
′ 𝜇𝜇 ′ of an arbitrary scattering mode (𝑚𝑚 , 𝜇𝜇) is
𝑘𝑘of the primary waves in the form of
the function of the modal coefficients 𝐶𝐶𝑚𝑚𝑚𝑚=
X ms ,′kµ ( z )∑∑ ∑ ℑmn C = D , A, w

X m′µ
Cmn

k
( z )Cmn

(2)

2043

𝑋𝑋 ′ ′

where ℑ𝐶𝐶 𝑚𝑚 𝜇𝜇 is referred to as the “scattering multiplier”. It is well-known that the orthogonality of
𝑚𝑚𝑚𝑚
the eigenfunctions of a hard-walled duct can transform Eqs. (1) into a series of mode-matching𝑘𝑘
equations. Then, we can further define the multiplier of the unknown primary mode coefficient 𝐶𝐶𝑚𝑚𝑚𝑚
in the resulting mode-matching equations as the “transfer multiplier ℘𝐶𝐶𝑚𝑚𝑚𝑚
𝑘𝑘 ”. All the interface matchingconditions lead to an overall matrix equation(3)[℘] ⋅ [Cmn ] =I
The submatrix [℘𝑘𝑘 ] of its coefficient matrix, composed of all the transfer multipliers of the unknown𝑘𝑘(𝐶𝐶 = 𝐴𝐴, 𝐷𝐷 and 𝑉𝑉), is defined as a transfer element. Then we know that once
modal coefficients 𝐶𝐶𝑚𝑚𝑚𝑚
the scattering multiplier ℑ embodied the inner scatterings of a segment is determined, thecorresponding transfer element can be obtained subsequently.
Intuitively, since disturbances propagate in a hollow rigid duct without any scatterings, thescattering multipliers will vanish when it comes to an empty duct segment with hard side walls, thus
leading to the simplest form of the corresponding transfer element. The application of the transfer
element method to a finite-length segment lined with locally reacting liner has been derived in detailin (12), where Eqs. (22-25) can yield the corresponding scattering multipliers.
Table 1 – Mode and frequency characteristics
At the oscillating rotor

Circumferential mode numbers

Primary waves′

𝑚𝑚 = 𝑠𝑠𝑠𝑠 + 𝑉𝑉𝜎𝜎

′

′ )𝐵𝐵

Scattering waves
𝑚𝑚 = 𝑠𝑠 𝐵𝐵 + 𝑚𝑚 = (𝑠𝑠 + 𝑠𝑠
Note: the inter-blade phase parameter 𝑉𝑉𝜎𝜎 = 𝜎𝜎𝜎𝜎/2𝜋𝜋.

+ 𝑉𝑉𝜎𝜎Frequencies′𝜔𝜔 = Ω𝑠𝑠 + 𝑚𝑚Ω

′𝜔𝜔 = 𝑠𝑠 𝐵𝐵Ω + 𝜔𝜔 = Ω𝑠𝑠 + 𝑚𝑚′ Ω

Based on the three-dimensional lifting surface theory, the scattering multipliers in the transfer
element for a rotor of 𝐵𝐵 identical, equally spaced blades in an annular hard-walled duct can be
written as follow:ℑCmnm′µ ( z ) = ∫
X

Rd

Rh

Ca

∫ ∆p
0

Cmn

(r ′, z ′, ω ) ⋅ K X (r ′, r , z − z ′ B, Ca , ω ′, m′, µ ) ⋅ dz ′dr ′(4)where 𝐶𝐶/𝑋𝑋 = 𝐷𝐷, 𝐴𝐴 and 𝑉𝑉 and 𝐶𝐶𝑎𝑎 denotes the axial length of blade chord, which is assumed to beconstant along the blade span. The coordinates of source point and observation point in the cylindricalframe of reference fixed to the duct are respectively given by (𝑟𝑟 ′ , 𝜑𝜑′ , 𝑧𝑧 ′ ) and (𝑟𝑟, 𝜑𝜑, 𝑧𝑧). △ 𝑝𝑝 𝐶𝐶𝑚𝑚𝑚𝑚 is
the unsteady blade loading on rotor blade surface resulted from the primary mode (𝑚𝑚, 𝑛𝑛) of the′′′, 𝑝𝑝𝐴𝐴,𝑘𝑘or 𝑤𝑤𝐷𝐷,𝑘𝑘) with the incident frequency ω and the modal coefficient
upwash disturbance (𝑝𝑝𝐷𝐷,𝑘𝑘
equal to unit. Moreover, the kernel function 𝐾𝐾𝑋𝑋 indeed denotes the mode coefficient 𝑋𝑋𝑚𝑚′ 𝜇𝜇′ of thescattering disturbances induced by a row of pressure dipoles of unit fluctuating amplitude.
It is known that the circumferential mode numbers and the angular frequencies of the scatteringwaves generated at such a rotor are related to those of the primary disturbances by=′ s ′B + m
m(5)
ω ′ =ω + s′BΩ
Therefore, it can be proven that both the primary disturbances and the scattering disturbances
generated at the oscillating rotor in question are comprised of an infinity number of the modal
components in the uniform mode and frequency characteristics shown in Table 1.
So far, all the scattering multipliers for the duct segments involved in the present model have beenbriefly discussed, and the relevant transfer elements can be built up correspondingly.
2.2.2 Boundary Integral Equation
In order to match the disturbed fields inside of the duct with the radiation fields outside of it, thematching conditions on the duct openings based on the classic Helmholtz integral (14)
∂G 0 ( P, Q) ∂Ψ (Q) 0
C ( P) p( P) =(6)
∫S Ψ (Q) ∂n − ∂n G ( P, Q) ⋅ dS (Q)2where the velocity potential Ψ = 𝑝𝑝 ′ 𝑒𝑒 −𝑖𝑖𝑘𝑘0𝑀𝑀𝑀𝑀/(1−𝑀𝑀 ) (𝑘𝑘0 = 𝜔𝜔⁄𝑐𝑐0 ) and 𝐺𝐺 0 is the free-space Green’s
function, is applied in combination with the transfer element matrix in Eq. (3) to construct an overall
response function of the duct system in a matrix form (10).

2044

2.2.3 Upwash Velocity
Assuming all the blades are vibrating harmonically with the same angular frequency Ω𝑠𝑠 , the blade
displacement function ξ(𝑟𝑟 ′ , 𝑧𝑧 ′ ) in this paper is chosen to be exactly the same with Eq. (24) in Ref.(15):ξ (r , z ) = H ⋅ Ca ⋅ hα (r ) / 1 + ( Ωr / U ) + Θ ⋅ θ β (r ) ⋅ ( z − ze ) ⋅ 1 + ( Ωr / U )
2

2

(7)

The first term on the right-hand side of Eq. (7) represents the blade displacement due to the bending
vibration of order 𝛼𝛼 while the second term is contributed by the torsion vibration of order 𝛽𝛽. The
normalized radial distribution functions of the bending vibration and the torsion vibration are ℎ𝛼𝛼 (𝑟𝑟)and 𝜃𝜃𝛽𝛽 (𝑟𝑟), and the complex vibration amplitudes are 𝐻𝐻 and Θ, respectively. In this paper, thetorsional axis 𝑧𝑧 = 𝑧𝑧𝑒𝑒 is assumed to be located on the mid-chord position. Taking the blade oscillationin the given manner ξ(𝑟𝑟, z)𝑒𝑒 𝑖𝑖Ω𝑠𝑠𝑡𝑡+(𝑗𝑗−1)𝜎𝜎 (𝑗𝑗 = 1, ⋯ , 𝐵𝐵) as the excitation source, the upwash velocity ofthe disturbances acting on rotor blade surfaces can then be determined by=u ϕˆ D ξ (r ′, z ′) ⋅ eiΩsτ + ( j −1)σ  / Dτ

(8)

2.2.4 Unsteady aerodynamic work
With Namba’s definition (15) of the complex work coefficient=
Cw π ∫RdRhCa

1 + Ω 2 r 2 / U 2 ∫  p(r , z ) ⋅ ξ (r , z ) ⋅ dzdr
0(9)

introduced to the present model, the likelihood of cascade flutter can be predicted by its imaginary
part Im [𝐶𝐶𝑤𝑤 ] (here the overline represent complex conjugate while R e [ ∙ ] and Im [ ∙ ] in this paper
denote the real and the imaginary parts of a complex value, respectively). Indeed, Im [𝐶𝐶𝑤𝑤 ] is theunsteady aerodynamic work exerted on rotor blade during one period of blade oscillation, thus referred
to as the unsteady aerodynamic work coefficient. According to the energy method, the cascade canmaintain stable if Im [𝐶𝐶𝑤𝑤 ] < 0; otherwise, a flutter is expected.

3. RESULTS AND DISCUSSION

3.1 Case Configuration
For illustrative purpose, a subsonic rotor subject to the first-order torsion vibration (i.e., β = 1
and 𝐻𝐻 = 0 in Eq. (7)) is considered in this paper. The case configurations are given as follow: 𝑅𝑅ℎ =0.2 and 𝑅𝑅𝑑𝑑 = 1 ; 𝑀𝑀 = 0.35 ; Ω = 2.4744 , 𝐵𝐵 = 30 , 𝐶𝐶𝑎𝑎 = 0.2 and Ω𝑠𝑠 𝐶𝐶𝑎𝑎 = 0.5 ; the complex
amplitude of the torsion vibration Θ = 𝑒𝑒 𝑖𝑖Φ where Φ = −10° . Four transfer elements should beconstructed respectively for the four duct segemnts depicted in Figure 1, i.e., the inlet segment empty
with hard walls, the second segment containing the oscillating rotor, the third segment containing a
locally reacting liner and the outlet segment empty with the hard walls. Their default axial lengths are
𝐿𝐿𝑖𝑖𝑖𝑖 = 3𝐶𝐶𝑎𝑎 , 𝐿𝐿𝑅𝑅 = 𝐶𝐶𝑎𝑎 , 𝐿𝐿𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 = 2𝐶𝐶𝑎𝑎 and 𝐿𝐿𝑜𝑜𝑜𝑜𝑜𝑜 = 𝐶𝐶𝑎𝑎 , respectively. Two inter-blade phase parameters areinvestigated in this work: first, 𝑉𝑉𝜎𝜎 = 1 where only the lowest radial, first circumferential mode (i.e.,
𝑚𝑚 = 1, 𝑛𝑛 = 1) is cut-on in the hard-walled duct of infinite axial extent, and second, 𝑉𝑉𝜎𝜎 = 2 where all
the hard-walled duct modes are cut-off.
3.2 Numerical Experiments
Figure 3 shows the unsteady aerodynamic work coefficient Im [𝐶𝐶𝑤𝑤 ] plotted against the variationof the liner reactance Im [𝑍𝑍𝑠𝑠 ] with its resistance R e [𝑍𝑍𝑠𝑠 ] held constant as 0.01, 0.10, 0.50, 1.00 and
3.00, respectively. The impedance ranges in these two plots are given to be very broad, out ofaeroelastic stability concern rather than any practical acoustic consideration. For the oscillating rotorof concern in the finite-length hard-walled duct, the unsteady aerodynamic work coefficient Im [𝐶𝐶𝑤𝑤 ] =0.02071 with 𝑉𝑉𝜎𝜎 = 1. With the liner added on the casing, a critical value of the specific reactancearound -2.4 is found in Figure 3(a). Within a proper range, where the reactance is smaller than thecritical value, the rotor which would flutter in the rigid duct can remain stable with the stabilizing
effect of the liner; otherwise, the rotor will flutter and the existence of the liner may even considerably
exaggerate the unbalance of the unsteady aerodynamic blade loading distribution due to blade
oscillation. For 𝑉𝑉𝜎𝜎 = 2, the rotor is deeply unstable in the finite-length rigid wall, where Im [𝐶𝐶𝑤𝑤 ] =0.2579. As shown in Figure 3(b), with the liner placed on the prescribed section of a small resistance2045

and a favourable specific reactance around -3, the rotor is predicted to be effectively stabilized for𝑉𝑉𝜎𝜎 = 2.

Figure 3 – Variation of Im[𝐶𝐶𝑤𝑤 ] against the change of the specific impedance 𝑍𝑍𝑠𝑠 : (a) 𝑉𝑉𝜎𝜎 = 1; (b) 𝑉𝑉𝜎𝜎 = 2.
Moreover, the unsteady aerodynamic responses of the oscillating rotor is found to be very sensitive
to the reactance change when the specific resistance is small. When 𝑍𝑍𝑠𝑠 = 0.01, as the reactance varies
from -5 to 5, an oscillating curve of Im [𝐶𝐶𝑤𝑤 ] is observed both in Figure 3(a) for 𝑉𝑉𝜎𝜎 = 1 and in Figure3(b) for 𝑉𝑉𝜎𝜎 = 2. With the resistance increasing to 0.10, the drastic oscillation of unsteady aerodynamic
work against the increase of the reactance is damped a little, while the reactance values corresponding
to those peaks appearing in the curves almost remain unchanged. With the resistance further increased
up to 0.50, 1.00 and 3.00, the curves of Im [𝐶𝐶𝑤𝑤 ] become smooth. Moreover, larger the resistance is,weaker the liner impact upon the rotor responses becomes. Therefore, we guest the phase change ofdisturbances, which is usually more sensitive to reactance than resistance and turns out to be evident
only if less disturbances are absorbed on the liner surface, may take a dominant role in the actual
effect of the aerodynamic interactions between oscillating rotor and acoustic liner on flutter stability.
To further investigate the influence of phase shift as well as mode propagation characteristics on
this issue, the predictions have also been made by numerically changing the rotor-liner gap as well as
the axial length of the liner. For the calculations based on the varying gap, an additional hard-walled
hollow segment of the axial length 𝐿𝐿𝑔𝑔𝑔𝑔𝑔𝑔 is introduced between the rotor and the liner, with 𝐿𝐿𝑖𝑖𝑖𝑖 =3𝐶𝐶𝑎𝑎 , 𝐿𝐿𝑅𝑅 = 𝐶𝐶𝑎𝑎 , 𝐿𝐿𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 = 2𝐶𝐶𝑎𝑎 and 𝐿𝐿𝑜𝑜𝑜𝑜𝑜𝑜 = 𝐶𝐶𝑎𝑎 kept constant. And when the axial length of the liner
𝐿𝐿𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 is changed, we still take 𝐿𝐿𝑖𝑖𝑖𝑖 = 3𝐶𝐶𝑎𝑎 , 𝐿𝐿𝑅𝑅 = 𝐶𝐶𝑎𝑎 and 𝐿𝐿𝑜𝑜𝑜𝑜𝑜𝑜 = 𝐶𝐶𝑎𝑎 . Firstly, for 𝑉𝑉𝜎𝜎 = 1 with the linerof the specific impedance 𝑍𝑍𝑠𝑠 = (0.01, −0.36), the dependencies of Im [𝐶𝐶𝑤𝑤 ] upon the rotor-liner gap
and the liner length are shown in Figure 4, where and the results obtained for the infinitely long duct
(denoted by ILD) and those for the finite-length duct (denoted by FLD) are presented in comparison.
In a similar fashion, Figure 5 shows the corresponding results for 𝑉𝑉𝜎𝜎 = 2 with 𝑍𝑍𝑠𝑠 = (0.1, −3.0).
For 𝑉𝑉𝜎𝜎 = 1, the mode of 𝑚𝑚 = 1 and 𝑛𝑛 = 1 is cut-on in the duct with entirely rigid walls. By
gradually increasing the gap, the phase of the disturbances incident onto the lined section will change
sinusoidally, which justifies the periodicity observed in Figure 4(a). Besides, the change of the rotor’sunsteady aerodynamic work against the increase of the liner length also reveals a quasi-sinusoidal
pattern, whose periodicity is likely due to the relatively evident effect of the reactance embodied by
the phase shift while the small but finite resistance may contribute to its slowly decreasing amplitude.
For 𝑉𝑉𝜎𝜎 = 2, where all the modes generated in the hard-walled duct are cut-off, when the gap
between the rotor and the liner is not too large, the decaying primary disturbances emitted at the rotor
cascade can still arrive at the liner segment and further arouse the scattering waves by interacting withthe lined wall. If the scattering fields can significantly alter the primary blade loading distribution, astabilizing or destabilizing effect will be observed. On the other hand, since all the primary
disturbances of cut-off modes will attenuate exponentially on their way to the acoustically lined wall,
if the rotor-liner gap is large enough, the liner cannot affect the rotor responses anymore, which isexactly consistent with the results given in Figure 5(a). Besides, remarkable quasi-periodic change ofthe unsteady aerodynamic work against the increase of 𝐿𝐿𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 is observed in Figure 5(b) with 𝑉𝑉𝜎𝜎 = 2
and 𝑍𝑍𝑠𝑠 = (0.1, −3.0). One possible reason behind it can be the modification of mode propagation
characteristics in a duct with the side walls of considerable lined area, compared with the duct of thesame geometry but with entirely hard walls.

2046

Figure 4 – Dependency of Im[𝐶𝐶𝑤𝑤 ] on (a) the rotor-liner gap and (b) the axial length of the liner, with 𝑍𝑍𝑠𝑠 =(0.01, −0.36), 𝑉𝑉𝜎𝜎 = 1.

Figure 5 –Dependency of Im[𝐶𝐶𝑤𝑤 ] on (a) the rotor-liner gap and (b) the axial length of the liner, with 𝑍𝑍𝑠𝑠 =(0.1, −3.0), 𝑉𝑉𝜎𝜎 = 2.
Furthermore, the remarkable derivations of the results obtained for the finite-length duct fromthose obtained for the duct of infinite axial extent have been observed for all the predictions of Im[𝐶𝐶𝑤𝑤 ]in Figures 4 and 5. This implies that the effect of acoustic treatment on fan flutter stability is strongly
affected by the reflections at duct openings. Besides, it is found that the difference in Im[𝐶𝐶𝑤𝑤 ] are nota constant as the liner distribution changes, which indicates there is a coupling effect of theacoustically treated wall and the duct terminations. Since the unsteady aerodynamic responses of fan
blades in a real engine nacelle are subject to the mutual acoustic couplings with both the acousticallytreated side walls and the duct terminations, implementing the boundary conditions reflecting thephysical essence is then of vital importance for the accurate fan flutter predictions.

4. CONCLUSIONS

Primarily on the basis of the transfer element method, a semi-analytical model is developed in thispaper to predict the unsteady aerodynamic responses of an oscillating rotor in a partially lined annular
duct of finite axial extent. A parametric study exploring the space of acoustic impedance and liner
distribution has been performed. Under certain circumstance, both the stabilizing and the destabilizing
effects of the acoustically lined section on the rotor flutter stability have been observed, where theduct end reflections are found to play a crucial role. In most cases, the impact of a liner turns out tobe evident only when its resistance is small, and under such condition the unsteady aerodynamic
responses of the oscillating rotor is found to be very susceptive to the reactance variation. It is the
combined condition of duct geometry, the structural and aerodynamic state of rotor as well as the2047configurations of acoustic treatment that mutually determines the actual impact of acoustic treatment
on the flutter stability of a ducted fan.

ACKNOWLEDGEMENTS
The financial supports of the NSFC (Grant No. 51676008 and 51790514) are gratefully
acknowledged.

REFERENCES

1. Bendiksen OO. Recent developments in flutter suppression techniques for turbomachinery rotors. J
Propuls Power. 1988 Mar;4(2):164–71.
2. Sun X, Jing X, Zhao H. Control of blade flutter by smart-casing treatment. J Propuls Power. 2001
Mar;17(2):248–55.
3. Watanabe T, Kaji S. Possibility of cascade flutter suppression by use of non-rigd duct wall. In: Whitehead
DS, editor. Proceeding of the Third International Symposium on Aeroelasticity in Turbomachinery.
Cambridge; 1984. p. 261–76.
4. Sun X, Kaji S. Control of blade flutter using casing with acoustic treatment. J Fluids Struct. 2002
Jul;16(5):627–48.
5. Zhao H, Sun X. Active control of wall acoustic impedance. AIAA J. 1999 Jul;37(7):825–31.
6. Howe MS. On the theory of unsteady high Reynolds number flow through a circular aperture. Vol. 366,
Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 1979.
p. 205–23.
7. Hughes IJ, Dowling AP. The absorption of sound by perforated linings. J Fluid Mech. 1990 Sep
26;218(1):299.
8. Jing X, Sun X. Experimental investigations of perforated liners with bias flow. J Acoust Soc Am. 1999
Nov;106(5):2436–41.
9. Eldredgel JD. On the interaction of higher duct modes with a perforated liner system with bias flow. J
Fluid Mech. 2004 Jul 10;510(510):303–31.
10.Sun X, Wang X, Du L, Jing X. A new model for the prediction of turbofan noise with the effect of locallyand non-locally reacting liners. J Sound Vib. 2008 Sep;316(1–5):50–68.
11.Wang X, Sun X. On the interaction of a fan stator and acoustic treatments using the transfer element
method. Fluid Dyn Res. 2010 Feb;42(1):015005.
12. Wang W, Atalla N, Nicolas J. A boundary integral approach for acoustic radiation of axisymmetric bodies
with arbitrary boundary conditions valid for all wave numbers. J Acoust Soc Am. 1997 Mar;101(3):1468–
78.
13.Namba M. Three-dimensional analysis of blade force and sound generation for an annular cascade indistorted flows. J Sound Vib. 1977 Feb;50(4):479–508.
14.Namba M, Fukushige K. Application of the equivalent surface source method to the acoustics of ductsystems with non-uniform wall impedance. J Sound Vib. 1980 Nov;73(1):125–46.
15.Namba M, Yamasaki N, Nishimura S. Unsteady aerodynamic force on oscillating blades of contra-rotating
annular cascades. In: Proceedings of the 9th International Symposium on Unsteady Aerodynamics,
Aeroacoustics and Aeroelasticity of Turbomachines. Lyon; 2000. p. 375–86.

2048