Effect of acoustic treatment on fan flutter stability

Journal of Fluids and Structures 93 (2020) 102877

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Effect of acoustic treatment on fan flutter stability ∗

Yu Sun, Xiaoyu Wang , Lin Du, Xiaofeng Sun School of Energy and Power Engineering, Beihang University, 100191 Beijing, China

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Article history: Received 4 March 2019 Received in revised form 2 November 2019 Accepted 7 January 2020 Available online 31 January 2020 Keywords: Fan flutter stability Acoustically treated wall Aerodynamic interaction Wall impedance condition Energy method

a b s t r a c t This paper presents an investigation of the effect of acoustically treated wall on fan flutter stability. The analysis is based on a three-dimensional analytical model which captures the aerodynamic coupling between oscillating annular rotor and finite-length liner in subsonic flow. With the response function of the entire system constructed as a whole, the perturbed duct field in modal description is determined by a simultaneous solution of the rotor and liner responses. Whether or not the aeroelastic instability occurs under a given wall impedance condition is evaluated through the energy method. Numerical experiments are conducted in the parametric space of inter-blade phase angle, liner distribution and acoustic impedance. The results show that under different conditions the acoustically treated wall can take diverse effects on the fan flutter stability. In the presented cases where the rotor stability is considerably affected, it is found that the aerodynamic interaction between the rotor and the lined section modify the overall distribution of the aerodynamic loadings on blade surface rather than change their magnitudes significantly. Moreover, the aeroelastic response of the oscillating blades turns out to be very susceptive to the reactance variation when the resistance is small, further indicating the important role of the reflections on the liner surface in such aerodynamic interaction. The analysis in general implies that the actual impact of a lined wall on the fan flutter stability depends on the aerodynamic excitation condition mutually determined by duct geometry, rotor state and configurations of liner. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction As a kind of self-excited vibration destructive for aero-engines, fan flutter has been extensively studied since the middle of the last century (Marshall and Imregun, 1996; Srinivasan, 1997) but not been truly solved yet. One of the least understood respects of this problem is the influence of duct boundary condition on aeroelastic stability. Recently, comparing the computational fluid dynamics (CFD) results with those obtained in the rig tests, Lee et al. (2016) and Stapelfeldt and Vahdati (2018) found the reflections from the intake of particular length can result in the stall flutter of a low-speed fan, and further showed that an appropriate intake liner can enhance the aeroelastic stability of the fan blades by attenuating the reflected pressure waves. On the other hand, though uncommon, fan instability brought about by the introduction of acoustic liner to engine nacelle has been observed during the test phase. The knowledge in regard to the effect of acoustic treatment on fan flutter boundary thus is essential for practical application, as it can help not only to avoid the extreme acoustic designs which may trigger aeroelastic instabilities but also to explore the active flutter suppression by means of adjustable wall impedance (Zhao and Sun, 1999). ∗ Corresponding author. E-mail addresses: [email protected] (Y. Sun), [email protected] (X. Wang), [email protected] (L. Du), [email protected] (X. Sun). https://doi.org/10.1016/j.jfluidstructs.2020.102877 0889-9746/© 2020 Elsevier Ltd. All rights reserved.

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Y. Sun, X. Wang, L. Du et al. / Journal of Fluids and Structures 93 (2020) 102877

Nomenclature B c0 Ca i k0 kmn m M n r , ϕ, z Rh Rd t U Vσ x x′ Zs

α β δij ( ) ∆p x ′ ε φm (kmn r) ± γmn , γv H

Θ ρ0 σ τ ω Ω Ωs

Re(), Im()

number of rotor blades mean flow sound speed axial √ length of blade chord, assumed to be constant along blade span = −1 = ω/c0 , wave number eigenvalue for annular hard-walled duct circumferential mode number Mach number of axial mean flow radial mode number cylindrical coordinates, where ϕ increases in the opposite direction of rotor rotation inner duct radius outer duct radius time associated with arrival of disturbances at observation point = M · c0 , axial mean flow velocity inter-blade phase parameter, an integer between −B/2 and B/2 vector coordinate of observation point vector coordinate of source point specific acoustic impedance order of blade bending vibration order of blade torsion vibration Kronecker delta, δij = 1, if i = j; δij = 0, if i ̸ = j amplitude of unsteady blade loading (pressure difference on blade surface) blade displacement function normalized eigenfunction for annular hard-walled duct axial wave numbers of acoustic waves and vortical waves amplitude of circumferential bending displacement of rotor blade amplitude of angular displacement of rotor blade about its torsional axis mean flow density = 2π Vσ /B, inter-blade phase angle time associated with emission of disturbances at source point angular frequency of spinning mode angular speed of rotor rotation angular frequency of blade vibration real and imaginary parts, respectively

Subscripts A D

propagating upstream propagating downstream

For an improved understanding about how acoustically treated wall affects the aeroelastic stability of fan blades, more systematic parametric studies with respect to the relevant factors, including acoustic impedance and inter-blade phase angle (IBPA), are indispensable, which necessitates an effective and meanwhile efficient model. Since in engine nacelle fan and liner interact with each other through acoustic field, a simultaneous solution of their unsteady responses is required for the strict modelling of such aerodynamic interaction. CFD methods undoubtedly play an important role in the refined analyses as they allow the detailed descriptions of real-world fan geometry and flowfield. Nevertheless, the remarkable complexities in the unsteady coupled analysis involving both rotor and liner make the three-dimensional numerical simulations on this issue still challenging and costly. By contrast, in the primary stage a reasonably simplified analytical model is preferred, to provide the indications of the basic effect trends through necessary parametric studies. While in most practical situations fan of a civil engine is enveloped by the side walls acoustically treated for the noise reduction purpose (Envia, 2001), only a few of the flutter analysis models have been developed for the non-rigid wall boundary condition. As early as 1984 Watanabe and Kaji (1984) proved that the aerodynamic damping of oscillating blades is considerably affected by the wall impedance change using a three-dimensional semi-actuator disk model. Then Sun and Kaji (2000, 2002) developed a lifting surface model on the basis of the generalized Green’s function theory (Goldstein,

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3

Fig. 1. Diagram of the lining system for the NASA Lewis 22 inch Advanced Ducted Propeller model scale fan, with the liner segments denoted by A to H (Bielak et al., 1999).

1976) to investigate the similar effect for a linear cascade between two infinite parallel endwalls, one of which is entirely rigid while the other of which is of uniform impedance. Employing the same model, Sun et al. (2001) further verified the possibility of the active flutter control by using a kind of liner with adjustable impedance. However, their solving approach is of great complexity due to the need to derive the Green’s function for the lined duct, since for a lined duct carrying mean flow the eigenvalues are complex and the orthogonality of its eigenfunctions breaks down (Tester, 1973; Zorumski, 1974). Moreover, it is worth noting that there have been no existing theoretical models for cascade flutter which can consider both finite-length liner and annular effect crucial for real turbomachines. The present study is thus motivated to investigate how the flutter stability of a rotor is affected by the finite-length liner in an annular duct. To this end, a three-dimensional analytical model is established based on the transfer element method (TEM) (Sun et al., 2008; Wang and Sun, 2011, 2010), which has been developed by the authors to tackle the problems concerning the aerodynamic couplings among different duct components. In TEM, the duct in question, according to its inner structures, is first divided into several finite-length segments by the cross-section interfaces. The general solution of the perturbed duct field is formulated as the series expansion in terms of the orthogonal modes which satisfy the hard wall boundary condition. Then, by making full use of the mode-matching technique (Zorumski, 1974), the response function of a finite-length segment can be constructed in a unified matrix form, with the modal amplitudes on its interfaces as the independent and dependent variables. Such a matrix is referred to as the transfer element (TE). By applying the interface matching conditions, all the involved TEs are combined together to give the response function of the entire duct system as a whole, thus allowing a simultaneous solution of the disturbances everywhere inside the duct with the aerodynamic couplings therein strictly taken into account. In the present study, two methods of singularity are employed to model rotor and liner in their respective TEs. Firstly, the three-dimensional lifting surface theory (LST) developed by Namba (1977) serves to determine the unsteady aerodynamic loadings on rotor blade in response to the upwash perturbations. Each loaded blade is modelled as a lifting surface of dipole distribution. Secondly, the lined section is treated as the rigid wall distributed with the equivalent surface monopole sources, as suggested by Namba and Fukushige (1980) but with a different singularity treatment (Huang, 1999; Sun et al., 2008). By this means the aforementioned complex eigenvalue problem for soft-walled duct is circumvented and the difficulty arising from the discontinuity of wall impedance is overcome. Finally, the likelihood of flutter, according to the energy method (Carta, 1967), is predicted in terms of the unsteady aerodynamic work exerted on rotor blades. Using this model, the parametric studies with respect to IBPA, liner distribution and acoustic impedance are conducted in an attempt to gain fundamental insights into the acoustic treatment effect on the fan flutter stability. 2. Theoretical model and method of solution 2.1. Model description One of the typical examples of the acoustic treatment design for modern high bypass turbofan engine is the lining system for the NASA Lewis 22 inch Advanced Ducted Propeller model scale fan depicted by Fig. 1 (Bielak et al., 1999). As in this lining system, multiple liner segments at various positions, both upstream and downstream of fan, may be applied to actual fan duct for noise suppression purpose. In this paper, however, to allow a fundamental and meanwhile systematic theoretical investigation we study the liner effect on fan flutter stability with a simplified fan duct model (see Fig. 2). Under the present consideration is a subsonic fan rotor in an annular straight duct whose side walls are entirely hard except for a uniform locally reacting lined section on the casing. The acoustic property of the locally reacting liner in response to the harmonic acoustic perturbation of time dependence eiωt is characterized by its specific acoustic impedance

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Fig. 2. The duct model in question: (a) the liner is downstream of the fan (the default case); (b) the liner is upstream of the fan.

Zs , i.e., the impedance made dimensionless by ρ0∗ c0∗ . On the liner surface the impedance boundary condition can then be described by p′∗ /vn∗ = −ρ0∗ c0∗ Zs , where p′∗ and vn∗ = v∗ · n are respectively the acoustic pressure and the normal acoustic particle velocity with the surface unit normal n pointing away from the surface (Myers, 1980; Namba and Fukushige, 1980; Scott, 1946). Unless otherwise stated, the lined section is considered to be downstream of the fan (i.e., the default case depicted by Fig. 2(a)), though the parametric studies regarding the interaction of the fan with the upstream liner (see Fig. 2(b)) will also be discussed to understand the influence of liner position. With our attention in this work restricted to the effect of the side wall impedance condition on the flutter stability, the duct is assumed to be of infinite axial extent such that the contributions from the duct end reflections can be ruled out. The time-mean incoming flow of interest is a uniform axial flow. Hereafter, unstarred symbols denote dimensionless quantities, with length, time and pressure scaled with respect to R∗d , R∗d /U ∗ and ρ0∗ U ∗2 . With the focus on the onset of instabilities, the major effort will be devoted to modelling the aerodynamic interaction between the rotor and the liner, while some basic assumptions are made towards the aerodynamics and blade geometry for simplification of analysis. Firstly, the flow is inviscid and of negligible thermal conductivity. Secondly, assume that the rotor suffering the incipient oscillation is unstalled and the flow is without separation. The B identical, equally-spaced blades of zero thickness and zero camber are vibrating harmonically in the given mode with a single frequency Ωs and a constant IBPA σ , which disturbs the duct field within the linear scope. The unsteady aerodynamic loadings on rotor blades are under fully three-dimensional investigation, while no steady blade loading is considered based on the given assumptions. 2.2. Determination of the perturbed duct field On the basis of linearization, the unsteady perturbed field inside the duct can be split into two parts: the primary field induced directly by the blade oscillation and the secondary field driven by the former and generated due to the aerodynamic interaction within the duct. Correspondingly, we call the unsteady aerodynamic loadings excited directly by the blade vibration the primary blade loadings, while the secondary blade loadings are identified as the difference between the unsteady aerodynamic loadings acting on rotor blades in the lined duct and those generated in the corresponding hard-walled duct. 2.2.1. Excitation disturbances In this work, the blade displacement amplitude normal to blade surface, supposed known as a priori, is defined by Eq. (24) in (Namba et al., 2000):

√ √ ε (r , z) = H · Ca · hα (r)/ 1 + (Ω r /U )2 + Θ · θβ (r) · (z − ze ) · 1 + (Ω r /U )2

(1)

where the first term denotes the blade displacement due to the circumferential bending motion of order α whose normalized radial distribution function is hα (r), and the second term arises from the torsion vibration of order β with the normalized spanwise distribution function θβ (r) (Namba et al., 2000; Nishino and Namba, 2006). The potential phase lag between bending and torsion vibrations is implicitly taken into account by their complex amplitudes H and Θ . Besides, the torsional axis z = ze is set to be the mid-chord line in this work. With the B rotor blades numbered against the direction of rotor rotation, the upwash velocity impinging on the qth blade is formulized by

) D ( ′ ′ ε (r , z ) · eiΩs τ +(q−1)σ (2) Dτ where σ > 0 if the phases of the upwash perturbations acting on the blade denoted by q1 proceed that sensed by the blade denoted by q2 , provided q1 > q2 . According to the flow tangency condition on blade surface, the disturbance velocity uϕˆ =

Y. Sun, X. Wang, L. Du et al. / Journal of Fluids and Structures 93 (2020) 102877

5

Fig. 3. An arbitrary segment of an annular duct, numbered by k.

induced by the primary blade loadings should have the component vϕˆ normal to blade surface that is equal in magnitude but opposite in sign to the upwash velocity uϕˆ , i.e., vϕˆ = −uϕˆ

(3)

Solving the upwash equation (3) by means of the three-dimensional LST (Namba, 1977) (see Appendix B.2), one can determine the primary blade loadings excited directly by the blade oscillation and subsequently the induced disturbances (the primary field) as the excitation sources of the secondary field.

2.2.2. Formulation of the aerodynamic coupling problem To analyze in a strict sense the aerodynamic coupling between the oscillating rotor and the finite-length liner, as previously illustrated, it is preferable to solve for the secondary duct field by using the TEM. With the liner effect modelled by the equivalent surface source method (Namba and Fukushige, 1980), the problem concerning the partially lined duct wall becomes the one of treating the additional surface monopole distribution with the hard wall boundary condition consistently applied. The solution is identical in every respect to that developed in (Wang and Sun, 2010) where the effect of the interaction between fan stator and acoustic treatments on sound attenuation is investigated, except that the stator TE should be replaced with the rotor one and the aeroelastic stability is assessed instead of the acoustic performance. To outline the formulation here, a general framework of TE will be first derived, followed by the specifics for its application to the duct system in question. Consider a finite-length duct segment denoted by index k, the axial coordinates of whose front and back interfaces are zk and zk+1 , as shown in Fig. 3. With the impermeable boundary condition imposed on its hub and casing, the disturbances existing in the kth segment can be expressed in terms of the hard-walled duct modes by p′A,k =

∑∑ m

+

Akmn φm (kmn r)eimϕ eiγmn (z −zk+1 ) eiωt

n

wD′ ,k =

∑∑

m

m

−

Dkmn φm (kmn r)eimϕ eiγmn (z −zk ) eiωt

pD,k =

∑∑

′

n

n

k Vmn

φm (kmn r) r

(4)

eimϕ eiγv (z −zk ) eiωt

where p′A,k and p′D,k are the acoustic waves propagating upstream and downstream respectively and wD′ ,k represents the vortical wave convected downstream without decaying. The index k as subscript and superscript indicates on the interfaces of which segment the disturbances are measured, and so will it be hereafter. Furthermore, the complex modal amplitudes of p′A,k on the back interface of the kth segment are denoted by Akmn , while the modal amplitudes of the k downstream-propagating waves p′D,k and wD′ ,k on the front interface are denoted by Dkmn and Vmn , respectively. Without loss of generality, consider that the upstream-propagating acoustic wave piA,k is generated within this segment due to certain source distribution known as a priori. The transmitted waves p′A,k , p′D,k and wD′ ,k by interacting with its inner structures further arouse the scattering disturbances psA,k , psD,k and wDs ,k . Based on the superposition principle, the sum of all these waves gives the total perturbed field inside the kth segment. Moreover, to simplify the derivation the two adjacent segments, numbered by k − 1 and k + 1, are assumed to be free from any inner sources or scatterings. Therefore, by requiring the continuity of acoustic pressure, axial sound particle velocity as well as circumferential vortex velocity on

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Y. Sun, X. Wang, L. Du et al. / Journal of Fluids and Structures 93 (2020) 102877

its front and back interface one can obtain p′A,k−1 + p′D,k−1 = p′A,k + p′D,k + psA,k + piA,k ⎪ u′A,k−1 + u′D,k−1 = u′A,k + u′D,k + usA,k + uiA,k

⎫ ⎬

for z = zk , r ∈ (Rh , Rd ) ,

⎪ ⎭

wD′ ,k−1 = wD′ ,k

(5)

p′A,k + p′D,k + psD,k = p′A,k+1 + p′D,k+1 ⎪

u′A,k + u′D,k + usD,k = u′A,k+1 + u′D,k+1

⎫ ⎬

for z = zk+1 , r ∈ (Rh , Rd )

⎪ ⎭

wD′ ,k + wDs ,k = wD′ ,k+1

As is done in Eq. (4), the scattering waves are also resolved into the scattering modes psA,k

⎡

s,k

⎡

Am′ µ (z)

⎤

⎤

⎥ ⎢ps ⎥ ∑ ∑ ⎢ ⎢Ds,k′ (z)⎥ φm′ (km′ µ r)eim′ ϕ eiω′ t ⎣ D ,k ⎦ = mµ ⎣ ⎦ µ s,k m′ wDs ,k V (z)

(6)

m′ µ

s,k

k whose modal coefficients Xm′ µ (z) (X = A, D and V ) turn out to be the functions of the primary modal coefficients Cmn (C = A, D and V ): s,k

Xm′ µ (z) =

) ∑∑ ∑ X ′ ∑ ∑ ( Xm′ µ Xm′ µ Xm′ µ m µ k k (z)Cmn = ζCmn (z)Dkmn + ζVmn (z)Vmn ζAmn (z)Akmn + ζDmn m

m

n

n

(7)

C =A,D,V

Xm′ µ

The multipliers ζCmn (z) which relate the scattering modes to the primary modes are referred to as the ‘‘scattering multipliers’’ in this paper. If in the kth segment a scattering mode (e.g. the mode (m′ , µ) of psD,k ) is independent of a Dm′ µ

primary mode (e.g. the mode (m, n) of p′A,k ), the corresponding scattering multiplier (i.e. ζAmn (z)) should then equal zero for zk ≤ z ≤ zk+1 . It is well that, due to the orthogonality of the hard-walled duct eigenfunctions φm (kmn r), taking the double ∫ 2π known ′′ ∫ R integral 0 e−im ϕ R d [·] · r φm′′ (km′′ n′′ r)drdϕ of the both sides of Eq. (5) leads to a series of mode-matching equations, h which can be written in the matrix form

⎡

p,b Ak−1

χ ⎢ mn ⎢χ u,b ⎢ Ak−1 ⎢ mn ⎢ ⎢0 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

χ

p,b Dk−1

p,f

χAk

0

mn

mn

u,b Dk−1

χ

p,f

u,f

0

χ

0

w,b k−1 Vmn

χDk

χV k

u,f

u ,f

mn

mn

χAk

χDk

χV k

0

0

χVw,k f

χApk,b

χVp,kb

χ pk,+f 1

χ pk,f+1

0

mn

χDpk,b

χAuk,b

χDuk,b

χVu,kb

χ uk,+f 1

χ uk,f+1

0

w,b

w,b

w,b

0

0

χ w,k+f 1

mn

mn

p,f

mn

mn

mn

mn

χAk

mn

mn

mn

χDk

mn

mn

mn

χV k

Amn

Amn

Dmn

Dmn

mn

Vmn

⎡ −1 ⎤ ⎤ Akmn ⎢ k−1 ⎥ ⎥ ⎢Dmn ⎥ ⎡ p,f ⎤ I ⎥ ⎢ k−1 ⎥ ⎥ ⎢Vmn ⎥ ⎢ m′′ n′′ ⎥ ⎥ ⎢ I u ,f ⎥ ⎥⎢ ⎥ ⎢Ak ⎥ ⎢ m′′ n′′ ⎥ ⎥ ⎢ mn ⎥ ⎢ ⎥ ⎢ k ⎥ ⎢0 ⎥ ⎥ ⎥ ⎢Dmn ⎥ = ⎢ ⎥ ⎢ k ⎥ ⎢0 ⎥ ⎥ ⎥ ⎢Vmn ⎥ ⎢ ⎥⎢ ⎥ ⎣0 ⎥ ⎦ ⎥ ⎢Ak+1 ⎥ ⎥ ⎢ mn ⎥ 0 ⎦ ⎢ k+1 ⎥ ⎣Dmn ⎦

(8)

k+1 Vmn

where the source terms on the right-hand side are given by

[

p,f

Im′′ n′′ u ,f

Im′′ n′′

] =

1 2π

2π

∫

e 0

−im′′ ϕ

∫

Rd Rh

[

piA,k uiA,k

] · r φm′′ (km′′ n′′ r)drdϕ

(9)

k and the resultant multipliers for the undetermined modal coefficients Cmn are defined as the ‘‘transfer multipliers χC k ’’ mn (C = A, D and V ). The first superscript of χC k , namely, p, u or w , indicates the matching of acoustic pressure, axial sound mn particle velocity or circumferential vortex velocity, respectively. Besides, for a given segment, the transfer multipliers

which involve in the matching equations at its front interface are marked by the second superscript f , while those appearing in the back interface matching equations are denoted by b.

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Table 1 Mode and frequency characteristics. At oscillating rotor

Circumferential mode numbers

Frequencies

Primary modes Scattering modes

m = sB + Vσ m′ = (s + s′ )B + Vσ

ω = Ωs + mΩ ω′ = Ωs + m′ Ω

The transfer element of the kth duct segment is then defined as

⎡ p,f χAk ⎢ umn ⎢χ ,f ⎢ Akmn ⎢ ⎢0 ⎢ TEk = ⎢ ⎢ p,b ⎢χAk ⎢ mn ⎢ u ,b ⎢χ k ⎣ Amn χAw,k b mn

χDpk,f

χVp,kf

χDuk,f

⎥ χVu,kf ⎥ ⎥ mn ⎥ w,f ⎥ χV k ⎥ mn ⎥ p,b ⎥ χV k ⎥ mn ⎥ ⎥ χVu,kb ⎥ mn ⎦ χVw,k b

mn

mn

0

χDpk,b

mn

χDuk,b

mn

χDw,k b mn

⎤

mn

(10)

mn

Xm′ µ

The final expressions of χC k as the functions of the scattering multipliers ζCmn (z) are given in Appendix A. Since mn no special simplifications have been made towards the inner environment of the kth duct segment, the TEk defined by Eq. (10) with the expressions of χC k given by Eq. (A.2) and Eqs. (A.4)–(A.8) is featured with generality. mn

Xm′ µ

Hence, for applying the TEM to the present issue, of fundamental concern are the scattering multipliers ζCmn (z) of the three TEs in question, namely, TE1 for the oscillating rotor, TE2 for the empty segment with the rigid duct walls, and TE3 for the segment with the lined casing (see Fig. 2(a)). Firstly, when it comes to an empty segment with hard duct walls, where no scatterings of disturbances exist, all the scattering multipliers should vanish and thus TE2 takes the simplest form. Besides, the application of the TEM to a finite-length duct segment with one side wall of uniform impedance has been discussed in detail in (Sun et al., 2008), where Eqs. (22) to (25) yield the corresponding scattering multipliers given in Appendix B.1. Lastly, the derivation of the scattering multipliers for the rotor segment based on the three-dimensional LST is briefly outlined in Appendix B.2, and more details concerning LST can be found in the published literatures (Namba, 1977; Wang and Sun, 2010). With the three TEs in question obtained, the simultaneous interface matching equations governing the secondary perturbed field can be written as the matrix equation

⎡

χAp0,b

⎢ mn ⎢ χ u ,b ⎢ A0mn ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎤⎡

TE1

TE2

TE3

χDp4,f

mn

χDu4,f

mn

χVw,4 f

mn

A0mn

⎤

⎡

p,f

Im′′ n′′

⎤

⎥ ⎢ 1 ⎥ ⎢ u ,f ⎥ ⎥ ⎢ Amn ⎥ ⎢Im′′ n′′ ⎥ ⎥ ⎥⎢ 1 ⎥ ⎢ ⎥ ⎢Dmn ⎥ ⎢ 0 ⎥ ⎥⎢ ⎥ ⎢ ⎢ p,b ⎥ ⎥ ⎢V 1 ⎥ ⎥ ⎢ mn ⎥ ⎥ ⎢ I ⎢ m′′ n′′ ⎥ ⎥⎢ 2 ⎥ ⎢ ⎥ ⎢ Amn ⎥ u , b ⎥ ⎢I ′′ ′′ ⎥ ⎥ ⎥⎢ ⎢ mw,bn ⎥ ⎥ ⎢D2 ⎥ ⎢ ⎥ ⎥ ⎢ mn ⎥ = ⎢Im′′ n′′ ⎥ ⎥⎢ 2 ⎥ ⎢ ⎥ ⎢ Vmn ⎥ ⎥ ⎢ 0 ⎥ ⎥ ⎥⎢ ⎥ ⎢ ⎥ ⎢ A3 ⎥ ⎢ ⎥ ⎥ ⎢ mn ⎥ ⎥⎢ 3 ⎥ ⎢ 0 ⎥ ⎥ ⎥ ⎢Dmn ⎥ ⎢ 0 ⎥ ⎥⎢ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢V 3 ⎥ ⎢ 0 ⎥ ⎥ ⎢ mn ⎥ ⎥ ⎥ ⎥⎣ 4 ⎦ ⎢ ⎣ 0 ⎦ ⎦ Dmn 4 Vmn

(11)

0

where the first two and the last three equations are resulted from the non-reflecting boundary conditions imposed on the inflow and the outflow cross-sections of the concerned ducted domain. By solving Eq. (11), one can simultaneously determine all the unknown disturbances in the secondary perturbed field, with the scattering effects and the aerodynamic Xm′ µ

interaction considered automatically by the scattering multipliers ζCmn (z) and the coupled TEs. 2.2.3. Modes and frequencies At the rotor in question, the circumferential mode number m′ and the angular frequency ω′ of a scattering mode are related to the incident primary mode by m ′ = s′ B + m ω′ = ω + s′ BΩ

(12)

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Y. Sun, X. Wang, L. Du et al. / Journal of Fluids and Structures 93 (2020) 102877

Fig. 4. (a) The complex work coefficient of the isolated rotor subject to the pure-bending vibration with α = 1, H = 1 and Ωs Ca = 0.5. (b) The unsteady aerodynamic work coefficient of the isolated rotor subject to the bending-torsion vibration with α = β = 1, H = 1, Θ = eiθ and Ωs Ca = 0.1.

It is known that the interaction of acoustic waves with a liner of circumferentially uniform impedance will bring about the energy transfer between different radial modes only but no scatterings among the modes of different circumferential mode numbers. Therefore, for the duct system in question, the primary disturbances resulted directly from the blade vibration as well as the generated scattering waves should satisfy the relations of circumferential mode numbers and frequencies given in Table 1, where the harmonic numbers s (of a primary mode) and s′ (of a scattering mode) can take arbitrary integer value from negative infinity to positive infinity. Then, one can conclude that all the disturbances herein consist of an infinite number of modal components in the uniform mode and frequency characteristics. 2.3. Flutter prediction At last, a criterion is needed for flutter prediction. The energy method claims that flutter likelihood can be predicted by the energy exchange between vibrating blades and surrounding fluids. Hence, the complex work coefficient (Namba et al., 2000; Namba and Nishino, 2006) is introduced as Cw = π

∫

Rd

√

1 + (Ω r /U)2 ·

Rh

Ca

∫

△p(r ′ , z ′ ) · ε(r ′ , z ′ )dz · dr

(13)

0

where the overbar on the blade displacement function ε denotes its complex conjugate. Its imaginary part Im(Cw ) called the unsteady aerodynamic work coefficient, is the dimensionless unsteady aerodynamic work exerted on each blade per cycle of blade oscillation and is proportional to the aerodynamic damping. Assuming zero mechanical damping, the rotor is stable out of aeroelastic concern if Im(Cw ) < 0; otherwise, the small-amplitude oscillation will grow into a flutter. 3. Results and discussion 3.1. Model validation For the purpose of predicting accurately the fan flutter characteristics, before any parameter studies are conducted, the rotor response function used in this work will be examined first. The predicted aerodynamic work coefficients Cw and Im(Cw ) for an isolated oscillating rotor in the straight annular hard-walled duct of infinite axial extent are compared with the results obtained by Namba et al. (2000). Satisfactory agreements of the prediction results for both the pure bending vibration and the coupled bending-torsion vibration are shown in Fig. 4, indicating the aerodynamic blade loadings can be correctly captured by the rotor TE. Since the capability of TEM to predict the acoustic response of finite-length liner has been proven in a previous work (Sun et al., 2008), the present model is then believed to be valid for investigating the aeroelastic responses of the rotor in a partially lined duct of infinite axial extent. 3.2. Parametric studies In this section, some parametric studies will be discussed to gain fundamental insights into the effect of finite-length lined wall on the aeroelastic stability of fan blades. Two illustrative cases are defined in Table 2, whose parameters are assigned with the purpose of illuminating the representative aerodynamic coupling effects and meanwhile guaranteeing fundamental practical interest for part-speed fan flutters. In addition, the axial lengths of the oscillating rotor segment, the gap segment and the lined segment depicted in Fig. 2(a) are given by their default values L1 = 1.0Ca , L2 = 2.0Ca and

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Table 2 Case configurations. Parameter

Rd

Rh

B

Ca

M

Ω

Ωs Ca

H

Θ

Case 1 Case 2

1 1

0.2 0.2

30 30

0.2 0.2

0.35 0.4

2.4744 2.0

0.5 0.5

0 1

1 eiθ (θ = 5π/36)

Fig. 5. The unsteady aerodynamic work coefficient over the whole range of IBPAs, for (a) Case 1 and (b) Case 2.

L3 = 2.0Ca , respectively. In view of the fact that fan flutters encountered in most practical cases are characterized by the low-order modes, we limit the present investigation to the first-order vibration modes, i.e. α = β = 1 in Eq. (1). The calculations indicate that among all the admissible IBPAs, for both Case 1 and Case 2, only the mode (0, 1) for Vσ = 0 and the mode (1, 1) for Vσ = 1 are cut-on in the infinitely long annular duct with entirely rigid side walls (noting that in this paper the mode (m, n) represents the mode of the circumferential mode number m and radial mode order n, with the first radial mode denoted by n = 1). As shown in Table 1, all the disturbances generated in the present duct system are composed of an infinite number of discrete frequency components. Nevertheless, the computation results show that, for the present two cases, due to the cut-off phenomena retaining only the lowest three radial modes (i.e. n = 1, 2, 3) of the lowest circumferential mode set (i.e. m = Vσ ) in the numerical computation for Eq. (11) already ensures adequate accuracy. This implies that only the low-order modes of the lowest frequency ω = Ωs + Vσ Ω make the non-negligible contributions to the rotor-liner interaction sound field, which allows us, without loss of generality, to represent the acoustic property of the uniform lined section by single specific impedance Zs (the actual acoustic impedance of a given liner is generally frequency-dependent and if the incident acoustic waves of multiple frequencies are of interest, certain impedance model should be selected and applied to model the liner effect) . 3.2.1. Flutter predictions for the whole range of IBPAs Fig. 5 shows the predictions of the unsteady aerodynamic work coefficient Im(Cw ) for the entire IBPA range. The comparisons are conducted between the results for the duct with entirely rigid walls and those of the cases including the lined section on the casing. It is seen that IBPA has a significant influence on the aeroelastic impact of a lined section of given impedance. In other words, the actual impedance wall effect on flutter stability is strongly affected by the mutual interference condition among blades, disk and surrounding flows. Hence the evident effect of a given liner may only arise for particular IBPAs. For the rotor subject to the first-order torsion vibration in Case 1, it is found from Fig. 5(a) that with the liner of specific impedance Zs = (0.01, −2.4) added to the casing, Im(Cw ) is reduced by around 80% at Vσ = 4 compared to its corresponding rigid wall value. More pronounced stabilizing effect of the lined wall is observed under the condition Zs = (0.1, −3.0) and Vσ = 2, where Im(Cw ) becomes negative from the positive value predicted under the hard wall condition. This implies such a liner can stabilize the oscillating rotor which would flutter in the hard-walled duct. The conspicuous modifications of the flutter stability due to the liner effect are also observed for the rotor undergoing the coupled bending-torsion vibration in Case 2. As shown in Fig. 5(b), at Vσ = 2 where the rotor is proven to be stable in the hard-walled duct, with the liner of Zs = (0.05, −3.9) added Im(Cw ) is further reduced to a negative value twice larger in magnitude, which in the linearized scope indicates more energies will be absorbed by the surrounding fluids from the vibrating blades; however, if Zs = (0.05, −3.2) instead, the introduction of such a liner will in turn deteriorate the aeroelastic response of the rotor and further trigger a flutter. It should then be noted that the modifications of rotor aerodynamic damping due to acoustic treatment on duct wall become rather critical for the operation conditions near the flutter boundary, under which a slight change might result in or prevent a flutter.

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Fig. 6. The UBLD plotted on the projection of blade surface in the r–z plane, for Case 1 with Vσ = 0 and Zs = (0.01, −0.6): (a) the absolute value; (b) the phase angle.

Fig. 7. The UBLD plotted on the projection of blade surface in the r–z plane, for Case 1 with Vσ = 2 and Zs = (0.1, −3.0): (a) the absolute value; (b) the phase angle.

3.2.2. Unsteady blade loading distribution (UBLD) In order to obtain more in-depth knowledge of the aerodynamic interaction between the lined wall and the oscillating rotor, in Figs. 6 to 9 we plot the primary UBLD directly resulted from the blade vibration, the secondary UBLD due to the aerodynamic coupling between the rotor and the liner, and the total UBLD as the sum of them under three different conditions. First of all, the results for Case 1 are presented. Fig. 6 shows the predicted UBLDs under the condition Vσ = 0 and Zs = (0.01, −0.6). It can be seen that compared with the primary unsteady blade loadings, the total aerodynamic loadings acting on rotor blade reveal no significant variation in the overall magnitude, or more specifically, their overall magnitude slightly decreases. Meanwhile, some visible modifications in their distribution on blade surface are observed in terms of both loading magnitude and phase. As is shown in Fig. 5(a), in the presence of the liner the unsteady aerodynamic work Im(Cw ) exerted on rotor blade increases by around 34% compared to its rigid wall value. Thus one can conclude that under this condition, the rotor-liner interaction results in the minor reduction in the aerodynamic damping by modifying the distribution of the unsteady aerodynamic loads on blade surface, even though the overall blade loading magnitude turns out to slightly decrease. Furthermore, it is seen from Fig. 7 that, under the circumstance Vσ = 2 and Zs = (0.1, −3.0) where a considerable and critical reduction in Im(Cw ) is predicted in Fig. 5(a), with the secondary field added the magnitudes of the unsteady blade loadings are not decreased in general. It is the phase distribution of the unsteady blade forces which turns into an approximately symmetric distribution about the torsional axis on the mid-chord line that makes the dominant contribution to balance the aerodynamic blade loadings. Nagai et al. (1996) have explored the flutter control for a linear cascade using an actuator surface on the duct wall which radiates sound waves as surface monopole sources, i.e., in the same nature as liner. Though no evidence was given, they expected the flutter suppression might be achieved by the wellknown ‘‘anti-sound’’ (Williams, 1984), the strategy to cancel the primary perturbed field by introducing the secondary field

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11

Fig. 8. The UBLD plotted on the projection of blade surface in the r–z plane, for Case 2 with Vσ = 2 and Zs = (0.05, −3.2): (a) the absolute value; (b) the phase angle.

that is exactly out of phase. However, the influence mechanism identified from Fig. 7 differs fundamentally from the ‘‘antisound’’. Understandably, among all the possible cases where the rotor-liner interaction make a stabilizing contribution, it is a very low probability event that the secondary field is generated in antiphase with the primary field and thus the direct reduction in unsteady blade loading amplitudes is achieved by the cancellation; by contrast, it is more likely that the UBLD is balanced to some degree by the secondary field in various forms. It should also be pointed out that in this case the liner is not sound-absorbing for the instability waves generated by the oscillating blades, as the secondary UBLD is not negligible but comparable to the primary UBLD. However, if it is the acoustic reflections from other components in engine nacelle, such as intake (Stapelfeldt and Vahdati, 2018) or adjacent blade rows, that drive the rotor oscillation, a liner in between with sound-absorbing impedance for the reflected waves is believed to have stabilizing effect as well. In contrast to the stabilizing example shown in Fig. 7, a typical destabilizing example is given by Fig. 8 for the rotor undergoing the bending-torsion vibration in Case 2 with Vσ = 2 and Zs = (0.05, −3.2). As shown in Fig. 5(b), with the liner added the unsteady aerodynamic work under this condition becomes positive from the negative rigid wall value, indicative of a flutter triggered by the rotor-liner interaction. Similar to Fig. 6(a), Fig. 8(a) shows a small amount of reduction in the overall magnitude of the total unsteady blade loadings when compared to the primary loadings. On the other hand, the pronounced change in the phase distribution of the unsteady aerodynamic loadings on blade surface is observed in Fig. 8(b). The blade loading phase distribution which becomes somewhat antisymmetric about the mid-chord torsional axis for the total unsteady blade loadings is largely responsible for the increase of the unsteady aerodynamic work. In other words, under this circumstance the secondary perturbed field generated due to the rotor-liner interaction intensifies the unbalance of the UBLD, thus making the oscillating rotor considerably destabilized. Since there is no fundamental distinction in the rotor-liner interaction mechanism among different blade vibration modes, for brevity the following analyses will be presented for Case 1 only. 3.2.3. Influence of liner distribution Further, the dependence of the rotor flutter stability on the liner distribution has been investigated by numerically changing the axial gap L2 between the rotor and the liner as well as the axial length L3 of the lined section. Both the default case where the liner is downstream of the rotor (see Fig. 2(a)) and the modified case where the liner is upstream of the rotor (see Fig. 2(b)) have been investigated. Firstly, for Case 1 with Vσ = 0 where only the plane wave is cut-on among all the hard-walled duct modes, as the gap between the rotor and the liner gradually increases, the phases of the cut-on disturbances arriving at the lined section should change sinusoidally, which justifies the periodicity observed in Fig. 9(a). Besides, the change of Im(Cw ) against the axial length increase of the lined section with Zs = (0.01, −0.6) also reveals a remarkable quasi-sinusoidal pattern as shown in Fig. 9(b). The interaction with the liner of such a small resistance will result in non-negligible phase shift but limited energy absorption of the acoustic waves, which accounts for the periodicity of the Im(Cw ) change with the slowly decreasing amplitude in Fig. 9(b). Secondly, for Case 1 with Vσ = 2 (where all the hard-walled duct modes are evanescent with the mode (2, 1) being weakly cut-off) and Zs = (0.1, −3.0), the analysis for the results presented in Fig. 10(a) is developed as follow. When the gap between the rotor and the liner is not too large, the cut-off primary disturbances emanated from the rotor can still arrive at the lined section and further arouse the scattering waves on the liner surface. If the secondary field can significantly alter the primary UBLD, a stabilizing or destabilizing effect will be observed. On the other hand, since the amplitudes of all the primary disturbances of cut-off modes will attenuate exponentially on their way to the lined wall, if the gap between the rotor and the liner is large enough, the existence of the liner cannot affect the rotor behavior anymore. Under the same condition, a quasi-periodic change with the decreasing amplitude of Im(Cw ) against the liner

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Fig. 9. The dependence of the unsteady aerodynamic work coefficient on: (a) the gap between the rotor and the liner; (b) the axial length of the acoustically lined wall. Case 1 with Vσ = 0 and Zs = (0.01, −0.6).

Fig. 10. The dependence of the unsteady aerodynamic work coefficient on: (a) the gap between the rotor and the liner; (b) the axial length of the acoustically lined wall. Case 1 with Vσ = 2 and Zs = (0.1, −3.0).

length increase is depicted in Fig. 10(b). It is noteworthy that the flutter boundary Im(Cw ) = 0 is crossed in both Fig. 10(a) and (b), indicating that the distribution of liner, which in essence affects the mode propagation characteristics in a softwalled duct and the phase shift of acoustic waves due to the scatterings on liner surface, takes an important role in the aerodynamic interaction with an oscillating rotor and further in the impact of acoustically treated wall on flutter stability. In Figs. 9 and 10 the changing curves of Im(Cw ) predicted for the upstream liner and those for the downstream liner reveal similar effect trend. This observation confirms that there is no fundamental difference in the influence mechanism on fan flutter stability for liners at different locations. Furthermore, the quantitatively different effects of the upstream and the downstream liners on the aerodynamic damping can be explained to some degree by the consideration of the following two aspects. Firstly, the liner location have a direct influence on the acoustic field generated in the fan duct. Secondly, as the initial aerodynamic state of the oscillating rotor determines the primary UBLD, the interaction sound field (i.e., the secondary perturbed field) with different energy and phase distribution would change the UBLD and consequently the unsteady aerodynamic work to different extents. 3.2.4. Influence of acoustic impedance Fig. 11 shows the contour maps of the unsteady aerodynamic work Im(Cw ) as a function of the specific acoustic impedance Zs for Case 1 with Vσ = 0 and Vσ = 2, respectively. The given impedance range with specific reactance from −5 to 5 and specific resistance from 0 to 3 largely covers the values considered in acoustic treatment design. Again, the results for the upstream liner and those for the downstream liner are presented for comparison. It can be found that though the specific changing patterns of Im(Cw ) are different for the upstream and the downstream liners as well as for Vσ = 0 and Vσ = 2, there are some common features regarding the basic effect trend. For each contour map in Fig. 11 there is a localized small-resistance region where Im(Cw ) shows the greatest deviation from its rigid wall value, e.g. for the reactance around −3 in Fig. 11(b). This region points to the most effective impedance with which the liner can bring about the maximum change in the aerodynamic damping of fan blades, and its small resistance is to some degree indicative of

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Fig. 11. The unsteady aerodynamic work coefficient versus the specific acoustic impedance, for Case 1 with (a) Vσ = 0 (dash dot line: the major contour of the rigid wall value Im (Cw ) = −0.377) and (b) Vσ = 2 (dash dot line: the major contour of the rigid wall value Im (Cw ) = 0.224; dash line: the flutter boundary Im (Cw ) = 0).

the limited sound absorption at the concerned frequency by such a liner. Apart from this region, some drastic changes in Im(Cw ), e.g. for the reactance between −2 and 0 in Fig. 11(b), are observed as the resistance approaches to zero. For a closer observation, with the specific resistance held constant, the changing curves of Im(Cw ) with the specific reactance are given in Fig. 12 for the downstream liner case, where the reactance range has been further extended out of aeroelastic stability concern. When the resistance is very small, the aerodynamic damping of the oscillating blades is found to be so sensitive to the reactance variation that the oscillating curves of Im(Cw ) are observed. As the resistance increases a bit, the drastic oscillation of Im(Cw ) against the change of the reactance is gradually damped while the reactance values corresponding to those peaks in the oscillating curves almost remain unchanged. Then, with the specific resistance increased up to around 0.5, the oscillation vanishes. It can be concluded that, in this case, as the resistance increases the liner impact upon the aeroelastic responses of the oscillating blades is generally weakened, which may be due to that the rise in the acoustic energy absorbed on the liner surface makes the secondary field carrying insufficient energy less influential. Such tendency is reconfirmed in Fig. 13. For Case 1 and Vσ = 2, with the specific reactance of the liner held constant to be −3.0, a sharp increase of Im(Cw ) is observed with the specific resistance varied from zero to unity. Then with the resistance further increasing, the growth of Im(Cw ) is gradually slowing and Im(Cw ) eventually turns into the positive value corresponding to the rigid wall condition. So far, one can conclude that an evident influence on the aerodynamic damping of an oscillating rotor is very likely to be brought about by the lined wall of small resistance and particular reactance, in other words, the low energy absorption and appropriate phase shift of disturbances on the liner surface. It is known that the propagating (cut-on or nearly cut-on) acoustic waves excited by fan flutter are generally generated at the frequencies ω = Ωs + Vσ Ω , where Vσ in most cases is very small. Acoustic liner designed for noise reduction purpose, however, are optimized for attenuating the acoustic waves of much higher frequencies and much shorter wave lengths, such as the fan noise generated at the first two blade passing frequencies (ω = BΩ or 2BΩ ). Thus within the impedance range of acoustic interest, due to the frequency mismatch it is the sound reflection effect of a liner that is considered to be of further significance for flutter instability waves. Furthermore, to seek an explanation of the drastic oscillation of Im(Cw ) against the reactance variation when the resistance is small, in the following contents we will explore the mutual aerodynamic excitation within the duct by studying the decoupled responses of the oscillating rotor and the acoustically lined segment. 3.2.5. Decoupled reflection coefficient and mutual aerodynamic excitation Indeed, the mutual aerodynamic excitation between the oscillating rotor and the lined section is only contributed by the acoustic waves denoted by the red and blue arrows in Fig. 14. Now we decouple the rotor and the liner to investigate their reflection coefficients for the effective incident disturbances respectively. Firstly, assuming that a + modal wave piA,2 = Aimn φm (kmn r)eimϕ eiγmn (z −z2 ) eiωt with Aimn = eiIphase propagating upstream is incident through the back interface of the oscillating rotor segment, the reflected wave propagating downstream is of the form p′D,2 =

∑∑ m′

µ

Dbm′ µ φm′ (km′ µ r)eim ϕ e ′

iγ −′ (z −z2 ) iω′ t m µ e

(see Fig. 14(a)). Then for the oscillating rotor segment in Fig. 14(a), the reflection

coefficient of the mode (m′ , µ) with respect to the incident mode (m, n) is defined by the ratio of their modal amplitudes measured on the back interface z = z2 , i.e., RCrotor = Dbm′ µ /Aimn

(14)

Secondly, as shown in Fig. 14(b), for the lined segment with its casing wall of uniform impedance, assume that the − acoustic mode piD,2 = Dimn φm (kmn r)eimϕ eiγmn (z −z3 ) eiωt propagating downstream is incident through its front interface, thus

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Fig. 12. The variation of the unsteady aerodynamic work coefficient against the change of the specific reactance with the specific resistance held constant, for Case 1 with (a) Vσ = 0 and (b) Vσ = 2.

Fig. 13. The variation of the unsteady aerodynamic work coefficient against the change of the specific resistance with the specific reactance fixed as −3.0, for Case 1 with Vσ = 2.

Fig. 14. The reflection of disturbances: (a) at the rotor cascade; (b) at the acoustically lined segment.

giving rise to the reflected wave p′A,2 =

∑ µ

+

Afmµ φm (kmµ r)eimϕ eiγmµ (z −z3 ) eiωt propagating upstream. In a similar fashion, the

reflection coefficient, for the lined segment, of the mode (m, µ) with respect to the incident mode (m, n) is then defined

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Fig. 15. The reflection coefficients of the modes (0, 1), (0, 2) and (0, 3) with respect to the incident acoustic mode (0, 1) for the lined segment with Re (Zs ) = 0.01, Case 1: (a) the real part; (b) the imaginary part.

Fig. 16. The reflection coefficients of the modes (2, 1), (2, 2) and (2, 3) with respect to the incident acoustic mode (2, 1) for the lined segment with Re (Zs ) = 0.01, Case 1: (a) the real part; (b) the imaginary part.

by the ratio of their modal amplitudes measured on the front interface z = z3 , i.e., RCliner = Afmµ /Dimn

(15)

Note that all the waves concerned here have the same mode and frequency characteristics (given by Table 1) with those generated by the oscillating rotor and the liner together in the original system. Fig. 15 shows that when the resistance is small, the reflection coefficient of the liner itself to the incident wave of the mode (0, 1) with a frequency as high as that generated by the rotor in Case 1 is also drastically oscillating against the reactance increase. On the other hand, when the incident wave of the lined segment is of the mode (2, 1), which is cut-off in the infinitely long hard-walled duct, RCliner turns out to vary smoothly with the change of the reactance, as is shown in Fig. 16. However, for the original system, as shown in Fig. 12(b) the oscillation of Im(Cw ) against the reactance variation also exists for Case 1 with Vσ = 2 and Re(Zs ) = 0.01. Therefore, the acoustic liner should not be the only one to take the blame. In Fig. 17, for both Vσ = 0 and Vσ = 2 in Case 1, the reflection coefficient of the oscillating rotor RCrotor varies along a wave-like pattern, with the phase angle Iphase of the incident wave piA,2 increasing from 0 to 2π . However, in the absence of blade vibration, the reflection coefficient of a rigid rotor with respect to the incident waves of fixed mode and frequency is proven to be a constant, which indicates the dependence of RCrotor on Iphase here is directly resulted from the blade oscillation. From the obtained results we can infer that, when the resistance is very small, the response of a finite-length liner itself to flutter instability waves can be very sensitive to the reactance variation. Apart from that, if particular reflection condition is composed between an oscillating rotor and a finite-length lined wall of a small resistance, the mutual aerodynamic excitation can result in the susceptibility of the rotor response to the reactance variation as well.

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Fig. 17. The reflection coefficients of the lowest three modes for the oscillating rotor segment, Case 1: (a) with respect to the incident acoustic mode (0, 1) and Vσ = 0; (b) with respect to the incident acoustic mode (2, 1) and Vσ = 2.

4. Conclusions In this paper, the effect of the finite-length lined wall on the flutter stability of an annular rotor has been investigated. Using a three-dimensional analytical model developed on the basis of the TEM, the aerodynamic coupling between rotor and liner have been captured by the simultaneous solution of their unsteady responses. Parametric studies with respect to IBPA, liner distribution and acoustic impedance have been performed, which provide some insights worthy of concern in turbomachinery engineering into the role of side wall impedance condition in fan flutters. It is found that, under some circumstances the lined wall does take appreciable impact on the aeroelastic stability of the fan blades, which becomes rather critical under the operating conditions near the flutter boundary. The stabilizing effect of the lined wall should be attributed to the favorable secondary field provided with respect to the initial aerodynamic state of the oscillating blades; such secondary field generated due to the rotor-liner interaction is more likely to balance the overall distribution of the unsteady aerodynamic loadings on blade surface rather than directly bring about the notable reduction in their magnitudes. On the contrary, if the secondary field intensifies the unbalance of blade loading distribution, the fan flutter stability will be deteriorated. When the resistance is very small, the aerodynamic damping of the oscillating rotor blades turns out to be very sensitive to the reactance change, which can be interpreted as a result of the strong acoustic reflections existing between the oscillating rotor and the liner when little acoustic energy is absorbed on the liner surface. In practice, compared with the sound absorption effect, the sound reflection effect of a liner designed for noise reduction purpose is considered to be of further significance for flutter instability waves in view of the frequency mismatch. In general, it is the combined condition of the duct geometry, the structural and aerodynamic state of rotor as well as the configurations of liner that determines the aerodynamic excitation within a duct, which further plays the decisive role in the actual influence of acoustically treated wall on fan flutter stability. In all this study we have focused exclusively on the qualitative analysis of the fundamental mechanism as well as the basic trends based on the present model, which is simplistic in its description of flowfield, duct geometry and rotor dynamics. To capture fan flutter characteristics in realistic engine nacelle, however, the mutual effect of acoustic treatment and other important factors must be taken into account. These include the aerodynamic influence factors such as the acoustic reflections at duct openings, the aerodynamic interaction with adjacent blade row and swirling flow. Further consideration of the aeroelastic factors including accurate location of elastic axis, bending-torsion coupling and mistuning requires a detailed structural analysis. And if fan blades of low mass ratio and stiffness are of concern, the fluid-structure coupling and the rotor-liner interaction should be modelled simultaneously, which remains an immense challenge even to the state-of-the-art numerical methods. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by National Natural Science Foundation of China under Grant No. 51676008 and Grant No. 51790514.

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Fig. 18. The blade force triangle in the rotor-fixed coordinates.

Appendix A. Transfer multipliers For the kth duct segment depicted in Fig. 2, it can be deduced from the continuity of acoustic pressure enforced on its front interface that +

iγ −′′ ′′ (zk −zk−1 ) m n

− Akm′′ n′′ eiγm′′ n′′ (zk −zk+1 ) − Dkm′′ n′′ ∑∑ ∑∑ ∑∑ A ′′ ′′ Am′′ n′′ Am′′ n′′ k (zk ) − Vmn ζwmmn n (zk ) = Imp,′′f n′′ (zk ) − Dkmn ζDmn − Akmn ζAmn

Akm−′′1n′′ + Dkm−′′1n′′ e m

m

n

n

m

(A.1)

n

On the right-hand side of Eq. (A.1) is the known source term, while on the left-hand side, the first three terms correspond to the transmitted waves while the last three terms are resulted from the scattering effect. Further, the transfer multipliers defined by

] [ + Am′′ n′′ (zk ) χApk,f = − δm′′ m δn′′ n eiγmn (zk −zk+1 ) + ζAmn mn ] [ Am′′ n′′ p,f (zk ) χDk = − δm′′ m δn′′ n + ζDmn

(A.2)

mn

p,f

χV k = −ζ mn

Am′′ n′′ (zk ) Vmn

can transform Eq. (A.1) into

∑∑{ m

−

p,f

p,f

p,f

k −1 −1 (δm′′ m δn′′ n )Akmn + (δm′′ m δn′′ n eiγmn (zk −zk−1 ) )Dkmn + χAmn Akmn + χDmn Dkmn + χVmn Vmn

}

= Imp,′′f n′′

(A.3)

n

which is ready to be cast into a matrix form. In a similar fashion, it can be deduced from the other five matching equations of Eq. (5) that:

• for the continuity condition of axial sound particle velocity on the front interface [ ] + γm+′′ n′′ + γmn Am′′ n′′ iγmn (zk −zk+1 ) χAuk,f = − δm′′ m δn′′ n e + ζ (z ) k + mn ω + U γmn ω′′ + U γm+′′ n′′ Amn [ ] + − γm′′ n′′ γmn Am′′ n′′ χDuk,f = − δm′′ m δn′′ n + ζ (z ) k − mn ω + U γmn ω′′ + U γm+′′ n′′ Dmn χVu,kf = − mn

γm+′′ n′′ ω′′ + U γm+′′ n′′

(A.4)

A ′′ ′′

m n ζVmn (zk )

• for the continuity condition of circumferential vortex velocity on the front interface χAw,k f = χDw,k f = 0 mn

mn

χVw,k f = δm′′ m δn′′ n

(A.5)

mn

• for the continuity condition of acoustic pressure on the back interface [ ] Dm′′ n′′ χApk,b = δm′′ m δn′′ n + ζAmn (zk+1 ) mn [ ] − Dm′′ n′′ p,b χDk = δm′′ m δn′′ n eiγmn (zk+1 −zk ) + ζDmn (zk+1 ) mn

χVp,kb = ζ mn

Dm′′ n′′ (zk+1 ) Vmn

(A.6)

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• for the continuity condition of axial sound particle velocity on the back interface [ ] + γm−′′ n′′ γmn Dm′′ n′′ u,b ′′ ′′ χAk = δm m δn n (zk+1 ) ζ + + ′′ mn ω + U γmn ω + U γm−′′ n′′ Amn [ ] − γm−′′ n′′ − γmn Dm′′ n′′ u,b iγmn (zk+1 −zk ) χDk = δm′′ m δn′′ n (zk+1 ) + ′′ ζ − e mn ω + U γmn ω + U γm−′′ n′′ Dmn χVu,kb = mn

γm−′′ n′′ ω′′ + U γm−′′ n′′

(A.7)

D ′′ ′′

m n (zk+1 ) ζVmn

• for the continuity condition of circumferential vortex velocity on the back interface m n χAw,k b = ℑAmn (zk+1 )

V ′′ ′′

mn

χDw,k b = ℑDmmnn (zk+1 ) V ′′ ′′

(A.8)

mn

χVw,k b = δm′′ m δn′′ n eiγv (zk+1 −zk ) + ℑ mn

Vm′′ n′′ Vmn (zk+1 )

Appendix B. Scattering multipliers

B.1. Scattering multipliers of the liner TE Based on the equivalent surface source method extended to the modelling of finite-length liner in annular ducts (Sun et al., 2008), it can be proven that the scattering multipliers involved in the transfer element of a finite-length segment with the acoustically treated casing wall can be written in the form + ω + U γmn ′′ ′′ QCnmn+ 2 κn′′ ,m − ω + U γmn ρ0 R d ′′ iγ − (zk+1 −zk ) n′′ − Dm′′ n′′ (zk+1 ) = δm′′ m ζCmn φm (kmn′′ Rd ) e mn′′ QCmn 2 κn′′ ,m

ρ0 Rd

A ′′ ′′

m n ζCmn (zk ) = δm′′ m

φm (kmn′′ Rd )

(B.1)

where

κn,m ≡

√

k20 − (1 − M 2 )k2mn

⎧√ ⎨ k2 − (1 − M 2 )k2 mn 0 √ = ⎩−i (1 − M 2 )k2 − k2 mn 0

for k20 ≥ (1 − M 2 )k2mn (cut-on) for k20 < (1 − M 2 )k2mn (cut-off)

(B.2)

with k0 = ω/c0 . As in most existing theories, the interaction of liner with vortical waves has been ignored. Thus we have Vm′′ n′′ ≡ 0. ζCmn For the locally reacting liner of uniform specific impedance Zs whose axial extent is l = zk+1 − zk , ( )∑ ± −iγ ± l γmn qπ l(e mn′′ cos qπ − 1) Cmn ′′ U n′′ ± QCmn = 1 + Vq , (C = A, D) (B.3) ± 2 2 2 2 ω γmn ′′ l − q π q C

where Vq mn (q = 1, 2, 3, . . .) are determined from ∞ ∑

C

(zjq + δjq · ρ0 c0 Zs )VqCmn = Ij mn

(B.4)

q=1

with +

A Ij mn

=−

Dmn

=−

2jπ (cos jπ − e−iγmn l ) 2

+ 2 γmn l − j2 π 2

φm (kmn Rd )

(B.5)

−

2jπ (eiγmn l cos jπ − 1)

φm (kmn Rd ) −2 2 γmn l − j2 π 2 [ ]2 ∫ l ( ) ∫ l ( ) iγ ± (z −z ′ ) ρ0 Rd ∑ φm (kmµ Rd ) jπ z U ∂ kπ z ′ ′ ± m µ zjk = sin ω + U γmµ e 1+ sin dz dz l κµ,m l iω ∂ z ′ l 0 0 µ Ij

(B.6)

(B.7)

Y. Sun, X. Wang, L. Du et al. / Journal of Fluids and Structures 93 (2020) 102877

19

B.2. Scattering multipliers of the rotor TE For the annular rotor of B identical, equally-spaced blades encircled by the hard duct walls, based on the general integral acoustic analogy (Goldstein, 1976), the pressure fluctuations induced by the undetermined UBLD can be formulated by p′ (x, t) =

T

∫

−T

B ∑

∫

S(τ ) q=1

q ∆˜ pj (x′ , τ ) ·

∂G · dSq (x′ )dτ ∂ x′j

(B.8)

where G x′ , τ ⏐ x, t is the Green’s function for the annular hard-walled duct containing uniform mean flow and Sq (x′ ) ⃗ϕˆ (see Fig. 18). In Eq. (B.8) the dipole nature of denotes the qth rotor blade surface whose unit normal vector is e ⃗ϕˆ is explicitly exhibited, upon which the theory of the unsteady aerodynamic blade loading ∆˜ pq (x′ , τ ) = ∆˜ pq (x′ , τ )e representing blade by lifting surface of pressure dipole distribution has been well established. Assuming the harmonic upwash disturbances impinging on each blade is of the same angular frequency λ but with a constant IBPA σ , the unsteady blade loading exerted on the qth blade has the form

⏐

(

)

∆˜ pq (x′ , τ ) = ∆p(x′ ) · ei(q−1)σ eiλτ

(B.9)

Resorting to the momentum equation, one can derive the induced velocity as a function of ∆p(x′ ) from Eqs. (B.8) and (B.9). Expanding ∆p(x′ ) in terms of the Glauert series and further employing the method of finite radial mode expansion proposed by Namba (1977) leads to the final expansion of ∆p(x′ ) as a finite series of the eigenfunctions of circumferential mode number m = 0

∆p =

J ∑

[ φ0 (koj r ) A1j cot ′

ϑ′

j=1

2

+

I ∑

] Aij sin(i − 1)ϑ

′

(B.10)

i=2

with the coordinate transformation Ca (

Ca sin ϑ ′ dϑ ′ (z ′ ∈ [0, Ca ] , ϑ ′ ∈ [0, π ]) (B.11) 2 2 Such treatment ensures the Kutta condition to be satisfied at the trailing edge and makes the infinite ∆p at the leading edge multiplied by zero during the integration over ϑ ′ in Eq. (B.8). By requiring the upwash equation (3) satisfied at I × J collocation points (rj , ϑi ) on blade surface, the unknown blade force coefficients Aij (i = 1, . . . , I ; j = 1, . . . , J) can be determined from the I × J simultaneous algebraic equations, where the orthogonality of φ0 (kon r) contributes to accelerate the convergence considerably. Here, we directly present the final forms of the scattering multipliers required in the TE of a rotor in the annular hard-walled duct: z′ =

1 − cos ϑ ′ , dz ′ =

A ′′ ′′

m n ℑCmn (z) =

)

−BCa 8π κn′′ ,m′′

π

∫ 0

∫

Rd Rh

∆pCmn (r ′ , ϑ ′ ) · φm′′ (km′′ n′′ r ′ ) ·

(

m′′ r′

− γm+′′ n′′

Ωr′ U

) e

iγ +′′ ′′ (z −z ′ ) −im′′ Ω z ′ U m n e dr ′

sin ϑ ′ dϑ ′

( ′′ ) ∫ π ∫ Rd −BCa m Ωr′ iγ − (z −z ′ ) −im′′ Ω z ′ − U ∆pCmn (r ′ , ϑ ′ ) · φm′′ (km′′ n′′ r ′ ) · − γ e m′′ n′′ e dr ′ sin ϑ ′ dϑ ′ ′′ ′′ m n ′ 8π κn′′ ,m′′ 0 r U Rh ) ′′ 2 iγ ′′ (z −z ′ ) −im′′ Ω z ′ ( ′′ ∫ π ∫ Rd ′ U V ′′ ′′ m −BCa m M e v e Cmn ′ ′ ′ ′′ Ω r m n ′′ ′′ ′′ ∆p (r , ϑ ) · φm (km n r ) · − γ dr ′ sin ϑ ′ dϑ ′ ℑCmn (z) = v 2 ′ 2 k2 4π ρ0 U 0 r U k + M Rh 0 m′′ n′′ D ′′ ′′

m n (z) = ℑCmn

(B.12) where ∆p (r , ϑ ) represents the amplitude of the unsteady pressure difference acting on blade surface due to the impingement from the mode (m, n) of the upwash disturbance (p′A,k , p′D,k or wD′ ,k ) with its incident frequency ω = λ + mΩ and its modal coefficient Cmn equal to unit. Cmn

′

′

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