Sound generation by non-synchronously oscillating rotor blades in turbomachinery

Journal of Sound and Vibration 355 (2015) 150–171

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Sound generation by non-synchronously oscillating rotor blades in turbomachinery Di Zhou a, Xiaoyu Wang a,b,n, Jun Chen a,c, Xiaodong Jing a,b, Xiaofeng Sun a,b a b c

School of Energy and Power Engineering, Beihang University, Beijing, PR China Collaborative Innovation Center for Advanced Aero-Engine, Beijing, PR China IRPHE, CNRS & Ecole Centrale Marseille, 49 rue F. Joliot-Curie, F-13013 Marseille, France

a r t i c l e in f o

abstract

Article history: Received 4 January 2015 Received in revised form 10 May 2015 Accepted 10 June 2015 Handling Editor: P. Joseph Available online 9 July 2015

In this paper, the sound generation by non-synchronously oscillating rotor blades in axial compressor is investigated with emphasis on establishing an analytical model for the corresponding sound field inside an annular duct. In terms of the present model, it is found that the acoustic frequency and propagating modes generated by nonsynchronously oscillating rotor blades are not only associated with the blade vibration frequency and rotational speed, but also depend on the cascade inter-blade phase angle (IBPA) and the interaction between blades, which is clearly distinguished from typical Doppler effect. Moreover, it is also shown that although the IBPA of cascade is nonconstant practically, the characteristics of sound generation are only slightly affected. Besides, the present work has conducted experimental investigations in order to gain insight into the generation mechanism of such complex sound field. Excellent agreement between the model prediction and experimental measurement in the near and far fields is generally observed in the circumstances with different parameter settings. Since the present study links the sound generation with blade oscillation, it would be very helpful to the fault diagnosis of rotor non-synchronous oscillation to some extent. & 2015 Elsevier Ltd. All rights reserved.

1. Introduction In recent years, as the compression capability and efficiency of turbomachinery increasing gradually, the operating circumstance in turbomachinery is also becoming harsher, which can be reflected mainly by the increment of loading on blades. With such complicated and rugged operating conditions, the stress of blades is at comparatively high level, and it can be easy to cause fatigue failure by oscillation. Therefore, how to monitor blade oscillation and give forewarning before the failure happens has become a problem worthy of in-depth study, for the purpose of averting turbomachinery rotor blade fatigue failure. According to some experimental research, while the compressor is operating at certain non-design states, a sort of anomalous sound would be generated inside the compressor. Even though the sound is primarily tonal noise, its frequencies are usually equal to the integer multiples of neither the blade vibrational frequency nor the blade passing frequency. That is, the frequency characteristic of the sound is different with typical Doppler shift phenomenon. In general, since the rotor

n

Corresponding author at: School of Energy and Power Engineering, Beihang University, Beijing, PR China. Tel.: þ86 010 82338262. E-mail address: [email protected] (X. Wang).

http://dx.doi.org/10.1016/j.jsv.2015.06.018 0022-460X/& 2015 Elsevier Ltd. All rights reserved.

D. Zhou et al. / Journal of Sound and Vibration 355 (2015) 150–171

Nomenclature b B c0 fi fφ fz G hγ ^ H i k0 kmn m M n ! n Q α3 α^ Γ mn Δp ρ0 σ τ

blade chord length blade number sound speed unsteady force exerted by the blades on the fluid projection of unsteady force exerted by the blades on the fluid in φ-direction projection of unsteady force exerted by the blades on the fluid in z-direction Green’s function bending vibration pattern displacement amplitudes of bending vibration imaginary unit wavenumber eigenvalues of hard duct eigenfunctions circumferential mode order Mach number in axial direction radial mode order normal vector of solid boundary a summation term of a series of exponential functions wavenumber in z-direction of downstream propagating vortex wave integrated displacement of blade vibration orthogonal value of hard duct eigenfunctions amplitude of blade loading mean density of flow inter-blade phase angle time associated with emission of sound wave

151

Q0

a reduced summation term of a series of exponential functions Q~ width of incident wake ! r ðr; φ; zÞ; ðx1 ; x2 ; x3 Þ the fixed-frame coordinates for an observer !0 0 0 0
y ; y ; y  the fixed-frame coordinates for r ðr ; φ ; z Þ; 1 2 3 a source ! r ″ðr″; φ″; z″Þ a coordinate system that rotates with cascade Rd duct radius Rh hub radius s an integer t time in an observer position T large time interval T ij Lighthill’s stress tensor U mean flow velocity in axial direction V inter-blade phase angle (IBPA) coefficient W flow velocity relative to blade ze reference position of blade torsional vibration α wavenumber in z-direction α1 wavenumber in z-direction of downstream propagating sound wave α2 wavenumber in z-direction of upstream propagating sound wave ψ mn eigenfunctions of a hard duct ω sound frequency Ωs frequency of blade vibration Ω cascade rotational frequency θχ torsional vibration pattern Θ displacement amplitudes of torsional vibration

blades oscillation is largely influenced by vibration of blade disk and non-uniform inlet condition, the inter-blade phase angle (IBPA) between adjacent blades in disk is not zero. Thus, the type of compressor rotor blade oscillation can be called as non-synchronous oscillation phenomenon. In order to explain the frequency characteristic of the sound generated by non-synchronously oscillating rotor blades, Schuster [1] carried out an experimental investigation on the sound generation of a fan at different operating conditions. He observed that the frequency of generated sound is tantamount to non-integral multiples of rotor rotational frequencies, while the fan is operating at the region near the stall margins. Schuster proposed that the sound is generated by the unsteady local pressure field which results from blade oscillation. Although he summarized the frequency and circumferential mode characteristics of the generated sound from experimental results, owing to lack of effective theoretical computing model, his conclusion does not include complete illustration about the phenomenon and interpretations that explain why the energy of specific frequencies as well as circumferential modes is accumulative due to the non-synchronous oscillation of rotor blades. Currently, the methodology of monitoring abnormal acoustic or vibration signal is widely utilized in carrying out fault diagnosis for actual compressor rotor non-synchronous oscillation. Since it is incapable of reliably estimating the frequency property as well as the energy or amplitude of the generated sound by utilizing the existing methodology, the fault monitoring is primarily based on comparison between the benchmark signals with the measured one, and information identification from the comparison. In addition, the fault analysis results depend mainly on the experience of analyzer and can be considerably influenced by the utilized signal processing method. Because the frequency and mode features of the sound field generated by non-synchronously oscillating blades is complicated, the accuracy of currently existing fault monitoring cannot be guaranteed. Hence, it is indispensable for the fault monitoring of rotor non-synchronous oscillation to construct a model which contains complete analysis about the sound generation by oscillating blades, especially, its characteristics associated with frequency and propagating modes. It is noted that a lot of previous work is concerned with the issue about sound generation by a blade row. Some of the investigations [2–5] focus on the solution of the equations of compressible flow through a blade row, in order to obtain the

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z b

M Ω φ

r

R R

Fig. 1. Rotating annular cascade in a duct with infinite axial length.

characteristics of generated sound waves. It can be noticed that all of these previous studies are two-dimensional. Thus, their results are limited on providing sufficient information about the three-dimensional effect of sound propagation. Namba [6] developed a three-dimensional unsteady lifting-surface method to predict the sound generation caused by an annular cascade in distorted inlet flows, which is more applicable and precise. After that, some investigations about sound generation by unsteady loading on rotor blades or stator vanes has been set up by the unsteady lifting surface theory [7–12] and solving three-dimensional linearized Euler equation [13,14]. Although these models can handle more complex situation that more factors such as the effect of interaction between sound source and sound propagation, and the effect of geometry of stator vanes can be contained, they only focus on the sound generation by rotor-stator interaction. However, the acoustic characteristics of frequency and mode are different between the sound field generated by oscillating rotor and that by rotor– stator interaction. In this aspect, Namba [15] utilized his lifting-surface theory to investigate the influence of the interaction between a pair of contra-rotating annular cascades with focus on the aeroelastic problem of oscillating blades, so the study does not include complete analysis about the acoustic features of oscillating blades. Besides, in the work, only relatively ideal situation that IBPA is constant among cascade blades is discussed in order to manage to make the IBPA coefficient become an integer for the convenience of calculations. Anyway, these pioneering investigations done by both Whitehead [3] and Namba [15] have undoubtedly enriched our knowledge on complex vibrational and acoustic phenomena in turbomachinery. However, provided more practical circumstances are considered, it is required to establish a new model with non-constant IBPA. More importantly, emphasis should be placed on how the sound frequency is characterized by a ground-fixed observer, distinguished from typical Doppler effect. In addition, a special focus should be also put on studying how the sound modes generated by cascade blades in different vibrating phase superpose themselves to form new propagating modes. In particular, as far as the latter is concerned, the relevant experimental validation is absolutely necessary. The present work contains a theoretical and an experimental study for the problem of sound generation by a rotating annular cascade with non-synchronously oscillating blades. In the theoretical part, an acoustic model of single oscillating annular cascade in three-dimensional subsonic flow field is built, which is based on the unsteady lifting-surface theory. The frequency, amplitude and circumferential mode characteristics of the sound generated by cascade with non-synchronously oscillating blades are discussed. In the experimental part, an artificial sound generator is utilized to simulate the sound generated by blades oscillation, and finally a comparison between experimental results and the model prediction is made.

2. Theoretical analysis of sound generation by rotating cascade with oscillating blades In the first place, we consider single rotating annular cascade in a duct as depicted in Fig. 1. All blades of the cascade are identical and they are treated as rigid flat plates that their thickness as well as camber is neglected. It is assumed that the duct is a perfect straight annular duct of infinite axial extent, and its wall is completely impenetrable. In addition, it is assumed that the flow in the duct is compressible, inviscid and isentropic. The mean flow inside duct is uniform, and its Mach number is lower than one. The direction of mean flow is constantly axial. Moreover, by comparing with mean flow in the duct, the amplitude of perturbation is small, thus, the analysis theory becomes linear. Ultimately, it is supposed that the blades do not working at stall state.

D. Zhou et al. / Journal of Sound and Vibration 355 (2015) 150–171

153

Fig. 2. The direction of unsteady force.

In consideration of the uniform flow with velocity U, the Lighthill’s equation is 2 0 1 D20 p0 1 ∂ T ij 2 0  ∇ p ¼ ; c20 Dτ2 c20 ∂yi ∂yj

(1)

where ðD0 =DτÞ ¼ ð∂=∂τÞ þ Uð∂=∂yi Þ, and T 0ij is the Lighthill’s stress tensor. According to the generalized Green's function theory, the Green's function equation can be put in the form that   1 D20 G ! !  ∇2 G ¼  δðt  τÞδ x  y ; c20 Dτ2

(2)

the corresponding boundary condition is D0 G ¼0 Dτ

ðt o τÞ: (3)   ! If we consider a moving source with volume of υ, surface area of S y as well as velocity of v, the consequence of solving Eq. (1) is ZT Z ZT Z ZT Z       ∂2 G ! !   f i ∂G dS !   ρ0 v0n D0 G dS ! p0 x ; t ¼ y dτ; T 0ij d y dτ þ y dτ þ ! ! ∂yi ∂yj ∂yi Dτ υ S y S y T

T

(4)

T

where v0n ¼ vn  ni U, and vn is the projection of v in the normal direction of the source surface. Considering the right side of Eq. (4), the first term represents the generation of sound by volume quadrupole source; the second term represents the generation of sound by dipole source which arises from unsteady forces exerted on the fluid by the solid boundaries; finally, the third term represents the sound that generated by monopole source. Whenever rotor blades oscillate with comparatively large amplitude, the flow region between the blades would be influenced sharply and the blade loading would also display extreme unsteadiness. Therefore, the majority of noise is generated by the unsteady force that is imposed by the blades on the fluid. In other words, the dipole source is the major source of sound in this specific issue. We shall neglect the contribution of volume quadrupole source and monopole source, then, we can obtain that ZT Z     !   f i ∂G dS ! p0 x ; t ¼ y dτ; ! ∂y S y i

(5)

    ∂ ∂ ∂ ∂ ∂ ∂ Ωr 0 ∂ ¼ f φ 0 0  f z 0 ¼ f φ 0 0  tan θ  0 ¼ f φ 0 0  ; ∂yi r ∂φ ∂z r ∂φ ∂z r ∂φ U ∂z0

(6)

T

where fi

and the relation of unsteady force direction is illustrated in Fig. 2. It is assumed the unsteady force imposed on the fluid by the blade boundaries are the harmonic time dependence form. The frequency characteristic of the unsteady forces is similar with that of blade oscillation. Furthermore, it is supposed that the oscillation frequency of every blade is equivalent. The positive cascade rotating direction is anticlockwise direction which is described in Fig. 1 and is the same to that of the IBPA. In order to be close to the reality of compressor operation, it is considered that the IBPA is non-constant between rotor blades. And then we can discuss the properties of sound field generated by non-synchronously oscillating rotor. It is assumed that the difference of phase angle between adjacent blades consists of an average value and random variation. Thus, the phase angle of the kth blade is σ k ¼ ðk  1Þ  σ þ

k X

Δσ d ;

d¼1

where σ denotes the average IBPA, and Δσ d is the random variation of phase angle of dth blade.

(7)

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D. Zhou et al. / Journal of Sound and Vibration 355 (2015) 150–171

Next then, the circumferential unsteady force of the kth blade can be written as  k P iΩs τ þ i ðk  1Þσ þ

f φ ¼ Δp  e

Δσ d

; k ¼ 1; 2; …; B; σ A ½ π; þ π Þ:

d ¼ 1

(8)

Then inserting f φ as well as the Green’s function with annular duct eigenfunctions into Eq. (5), the sound generated by the kth blade can be expressed as Z 1 1 X X i ψ mn ðkmn rÞeimφ 2π 0 !  ! Δpψ mn ðkmn r 0 Þe  imðφ  B ðk  1ÞÞ p0 ð x ; tÞ ¼ 2 Γ mn 4π m ¼  1 n ¼ 0 Sð ξ Þ Z 1 Z 1 0 Ωr 0 m eiαðz  z Þ  0Þ  ðα U r β2 α2  2Mk0 α  k20 þ k2mn 1 1  k P Z T i ðk  1Þσ þ Δσ d ! d ¼ 1 dτ dω dα dSð ξ Þ; (9)  eiΩs τ eiωðt  τÞ eimΩτ e T

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! where β ¼ 1  M 2 , φ0 ¼ φ0 þ Ωτ; φ0 as well as z0 , r 0 denote the position of source in the coordinate system ξ that rotates with the blades. Considering all of the blades in the cascade, the total sound pressure is Z 1 1 B X X X i ψ mn ðkmn rÞeimφ 2π 0 ! p0 ð x ; tÞ ¼ 2  ! Δpψ mn ðkmn r 0 Þe  imðφ  B ðk  1ÞÞ Γ mn 4π m ¼  1 n ¼ 0 Sð ξ Þ k ¼ 1 Z 1 Z 1 0 Ωr 0 m eiαðz  z Þ   0Þ ðα 2 U r β α2  2Mk0 α  k20 þ k2mn 1 1  k P Z T i ðk  1Þσ þ Δσ d ! d ¼ 1 dτ dω dα dSð ξ Þ: (10)  eiΩs τ eiωðt  τÞ eimΩτ e T

By utilizing the fact in the generalized function theory that Z T lim e  iðω  Ωs  mΩÞτ dτ ¼ 2πδðω Ωs  mΩÞ; T-1

(11)

T

we can find out only when ω ¼ Ωs þ mΩ, the right side of the preceding equation is not zero. Hence, it is obtained that  k P Z 1 1 B i ðk  1Þσ þ Δσ d imφ X X X i ψ mn ðkmn rÞe 2π 0 ! d ¼ 1  ! Δpψ mn ðkmn r 0 Þe  imðφ  B ðk  1ÞÞ e p0 ð x ; tÞ ¼ eiωt 2π Γ mn Sð ξ Þ k ¼ 1 m ¼ 1 n ¼ 0 Z 1 0 ! Ωr 0 m eiαðz  z Þ  0Þ ðα dα dSð ξ Þ: (12)  U r β2 α2 2Mk0 α k20 þk2mn 1 Utilizing the residue theorem to the second integral in Eq. (12) yields k P Z 1 1 B i Δσ d X X eiωt X ϕmn ðkmn rÞeimφ B 2π !  imφ0 iðm þ σ 2π ðk  1Þ Þ 0 B Δpϕmn ðkmn r Þe  e e d¼1 p ð x ; tÞ ¼  ! 4π m ¼  1 n ¼ 0 κ nm Sð ξ Þ k¼1  0 0 ! m Ωr m Ωr 0Þ 0 iα ðz  z Þe 1 Þeiα2 ðz  z Þ dSð ξ Þ;  Hðz z0 Þð 0 α1 þ Hðz0 zÞð 0 α2 U U r r

0

where

8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > < k20 β2 k2mn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi κnm ¼ > :  i β2 k2mn  k20

ϕmn ðkmn rÞ ¼

1

z 4z0

0

z r z0

2

2

;

(14)

;

(15)

ψ mn ðkmn rÞ pffiffiffiffiffiffiffiffiffi ; Γ mn

(16)

Γ mn ¼ 2πΓ mn ; α1 ¼

2

k0 r β2 kmn

( Hðz z0 Þ ¼

2

k0 4 β2 kmn

(13)

Mk0  κnm ; β2

(17) (18)

D. Zhou et al. / Journal of Sound and Vibration 355 (2015) 150–171

α2 ¼

Mk0 þκ nm : β2

155

(19)

Therefore, after the unsteady loading on the blade surface is procured via either numerical computation or experiment, the acoustic pressure can be obtained on the basis of Eq. (13). 3. Some discussions about the characteristics of the sound generated by cascade with oscillating blades 3.1. The frequency and circumferential mode characteristics 3.1.1. The situation that IBPA is constant and IBPA coefficient is an integer It can be noticed that there is a term which includes summation operation in the right side of Eq. (13). The term can be denoted by Q¼

B X

B 2π ðk  1Þ iðm þ σ 2π ÞB

e

i

e

k P d ¼ 1

Δσ d

:

(20)

k¼1

Here, we firstly consider an ideal situation that the IBPA of cascade is constant and the IBPA coefficient V ¼ σB=2π is an integer. Hence, Δσ d is equal to zero and the term can be simplified as ( B B B m þ σ 2π ¼ sB X B 2π ðk  1Þ 0 iðm þ σ 2π Þ B Q ¼ e ¼ s ¼ …;  1; 0; þ1; …: (21) B asB 0 m þ σ 2π k¼1 Only if the relation m ¼ sB  σB=2π is satisfied, the consequences of the operation and sound pressure are not zero. Thus, now the sound pressure can be denoted by Z 1 1 X Beiωt X ϕmn ðkmn rÞeimφ 0 ! Δpϕmn ðkmn r 0 Þe  imφ p0 ð x ; tÞ ¼  ! 4π s ¼  1 n ¼ 0 κnm Sð ξ Þ  ! m Ωr 0 iα1 ðz  z0 Þ m Ωr 0 iα2 ðz  z0 Þ Þe Þe  Hðz  z0 Þð 0  α1 þ Hðz0  zÞð 0  α2 dSð ξ Þ: (22) U U r r We can find that the circumferential mode and frequency of the sound generated by cascade with oscillating blades should satisfy the following relationship: m ¼ sB  V;

(23)

ω ¼ Ωs þ ðsB  V ÞΩ;

(24)

regardless of whether flow exists in duct or not. To be sure, from the mathematical perspective of theoretical analysis, the sound frequency ω can be negative, but in actual measurement situation, the calculated negative frequency does not have physical meaning and it would become positive, besides, only a time phase diversification would be induced. In the following analysis, the sound frequency also satisfies the definition. The sound field contains a number of tones whose frequencies are decided by the frequency of blade oscillation, the rotor rotational speed as well as the circumferential mode. The circumferential mode properties of these tones are determined by the blade number and the IBPA. It can be aware from the model that every tone corresponds to single specific circumferential mode. Moreover, the model also reveals that the frequency detected at a fixed location in the duct and the actual frequency of rotating source is not completely tantamount, and the relationship between them is different with what Doppler effect describes while the blades are oscillating non-synchronously. Since only limited cut-on modes can transmit in duct, the range of s as well as m is finite. Therefore, value of ω only can be a finite series of specific values. According to above analysis, it is thus clear that the frequency and IBPA of blade oscillation can be acquired by measuring and analyzing the generated sound field rather than directly focusing on blade or cascade oscillation. Since the frequency of each blade vibration mode order is known in advance and it would not change a lot in actual sophisticated operation condition, it would be feasible to judge what the vibrational state of blade and which stage of cascade might conk. If the characteristic of the measured sound field is anomalous compared to the normal condition, prior warning and regulation might prevent the compressor from further mechanical failure. When V ¼ 0, the model denoted by Eqs. (23) and (24) can be simplified as m ¼ sB;

(25)

ω ¼ Ωs þ sBΩ;

(26)

which describes the features of sound generated by synchronously oscillating blades. Now, the circumferential modes are equal to the integer multiples of the blade number, and the frequencies of generated sound are tantamount to the sum of the blade oscillation frequency and integer multiples of rotor rotational frequency. That is, the frequency shift phenomenon is satisfied with Doppler effect. Furthermore, when s ¼ 0, there would be a plane wave inside the duct, whose frequency is

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D. Zhou et al. / Journal of Sound and Vibration 355 (2015) 150–171

40

|Q'|

30 20 10 0 40

|Q'|

30 20 10 0 40

|Q'|

30 20 10 0 40

|Q'|

30 20 10 0 -40

-30

-20

-10

0

+10

+20

+30

+40

Circumferential mode order

Fig. 3. Distribution of Q 0 with different circumferential mode orders: (a) σ ¼ 363 ; V ¼ 4, (b) σ ¼ 353 ; V ¼ 3:89, (c) σ ¼ 333 ; V ¼ 3:67, and (d) 3 σ ¼ 31:5 ; V ¼ 3:5.

equal to blade oscillation frequency. The situation is similar with the one that compressor is surging and blade disk is vibrating with circumferentially zero mode order patterns. If there is only one source in the duct, then B ¼ 1 and the model can be simplified as ω ¼ Ωs þ mΩ ¼ Ωs þsΩ:

(27)

This single rotating source mechanism has been investigated by Kameier and Neise [16] before. 3.1.2. The situation that IBPA coefficient is equal to non-integral Non-synchronous oscillation of cascade blades may be caused by the blades-disk coupling vibration of compressor rotor at some non-design states under the condition of uniform or non-uniform inflow. Here, the vibration pattern of blade disk is primarily higher order modes patterns, thus, the blades oscillate with different phase angles. Because of the phenomenon, IBPA coefficient which describes the phase difference of adjacent blades should be an integer. Nonetheless, since the manufacturing as well as assembly of compressor rotor would induce deviation, IBPA coefficient can be non-integral. Hence, the consequences denoted by Eq. (21) would not be strictly correct. According to the conclusion denoted by Eq. (13), it can be aware that the amplitude of generated sound is greatly determined by the module of Q in Eq. (20). Currently, it is also assumed that IBPA is constant and thus Q can be simplified as Q 0 which is stated in Eq. (21). We will provide an example to illuminate the influence of IBPA coefficient on the distribution

of Q 0 (the module of Q 0 ) with different circumferential mode orders. It is assumed that the amount of cascade blades is

B ¼ 40. Fig. 3 illustrates Q 0 with different circumferential mode orders,

when V is either integer or non-integral.

From the preceding analysis, it can be observed that when V ¼ 4, Q 0 is satisfied with Eq. (21). Specifically, Q 0 is equal to 40 when m ¼ 44; 4; þ 36, otherwise it would be zero. However, while IBPA coefficient is gradually deviating from an integer, the distribution of Q 0 is also altering. For instance, as V changing from 4 to 3.5, the modes with high amplitude are gradually changing from m ¼  44, m ¼  4 and m ¼ þ 36 to m ¼  43, m ¼  3 as well as m ¼ þ 37. When IBPA coefficient is non-integral, it can be noted that the orders of high amplitude modes are close to the theoretically predicted values which are obtained by solving Eq. (23). On the other hand, the highest amplitude of all modes would decline, and the amplitudes of

D. Zhou et al. / Journal of Sound and Vibration 355 (2015) 150–171

157

40

|Q|

30 20 10 0 -40

-30

-20

-10

0

+10

+20

+30

+40

Circumferential mode order Fig. 4. Distribution of jQ j with different circumferential mode orders when IBPA is non-constant.

other low amplitude modes would be larger than zero but still comparatively small, which means the sound energy does not disperse a lot. The most extreme situation is depicted in Fig. 3(d) where the number of modes with the highest amplitude are doubled. Therefore, in the situation that IBPA coefficient is non-integral, the tones corresponding to all modes would be generated. The frequencies of these tones are tantamount to the sum of blade oscillation frequency and the product of rotor rotational frequency and the dominant circumferential mode order, and these tones is equally-spaced on both sides of blade oscillation frequency that their intervals is equal to the rotational frequency of rotor. Furthermore, since IBPA coefficient is non-integral, the theoretical circumferential modes of the generated sound can be calculated by applying the developed model and the modes are also equal to an array of non-integrals. This means the actual dominant high amplitude modes would be a series of modes close to those that have been computed and thus the sound energy is majorly centralized in the tones corresponding to those high amplitude modes. Above all, it can be noticed that even though the IBPA coefficient would be non-integral due to the manufacturing and installation of compressor rotor, the characteristics of sound field are also similar with the situation that IBPA coefficient is an integer as discussed previously.

3.1.3. The situation that the IBPA of cascade is non-constant In this section, the acoustic properties of oscillating cascade with non-constant IBPA will be discussed. Although the phase angle deviation of each blade on the cascade might not be equivalent, the variation would generally not be fairly large. In addition, since the deviation Δσ d is usually a small random value which can be positive or negative, and the blade number k P is usually large, the function of the summation term Δσ d can be neglected approximately. Hence, it can be obtained that d¼1

Q  Q0 ¼

B X

eiðm þ σ2πÞ B ðk  1Þ ; B 2π

(28)

k¼1

which means the frequency and circumferential mode features of the generated sound in this practical situation are similar with what have been illustrated previously. In order to verify the above analysis about influence of IBPA deviation on the distribution of jQ j (the module of Q ), a relevant example will be given in the following part. It is assumed that the blade number is B ¼ 40, and the oscillation phase angle difference of each pair of adjacent blades is different that it randomly deviates from 363 . The deviation can be positive or negative but its maximum magnitude is limited to 53 . Fig. 4 shows the distribution of jQ j with different circumferential mode orders. The high amplitude mode orders are m ¼ 44, m ¼ þ 36 and m ¼  4 which is identical with the situation illustrated by Fig. 3(a) that σ ¼ 363 ; V ¼ 4. Furthermore, it can be noticed that the amplitudes of other modes are distributed randomly, and theses amplitudes are larger than zero but still comparatively small. Roughly, the distribution of jQ j in this situation that IBPA is non-constant is similar with that of the former one as depicted in Fig. 3(a). Hence, it can be noted that even though the phase angle difference of each pair of adjacent blades might be slightly dissimilar, the deviation does not bring tremendous influence on the distribution of jQ j. The previous analysis is reasonable.

3.2. The analysis of sound propagation properties It is known that the sound energy of cut-off modes inside duct would decay exponentially with the modes propagating, thus in most cases, only cut-on modes would be considered when we calculate the sound field inside a duct. From Eq. (13), we can find that amplitude of sound pressure is tied up with sound propagation properties. Although the cut-off modes would attenuate, they might also contain considerable energy at the place near sound source due to the effect of κnm . Here, we would introduce an instance about the sound wave propagating downstream. It is assumed that the cut-off ratio, which can be defined as k0 =ðβ  kmn Þ, is 0.95 now, and it will be greater than one if the corresponding mode is cut-on.

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2.5

2

Pcut-off

1.5

1

0.5

0

0

1

2

3

z

4

5

Fig. 5. Relationship between P cutoff of sound modes with different cut-off ratio and the axial distance (dimensions in m) from the cascade to observer.

After separating the term which describes sound propagation from Eq. (13), it can be obtained that  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi iα1 z

i

Mk0 β2

þ

i

2 β2 k2 mn  k0 β2

e e ! qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p0 ð x ; tÞ p κnm 2 2 β2 kmn k0



z

2 β2 k2 mn  k0 β2

z

e p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ P cutoff : 2 2 β2 kmn  k0

(29)

Fig. 5 illustrates the relationship between P cutoff and the axial distance from sound source to observer, when the cut-off ratio is 0.95 and 1.5. It can be observed that the cut-off mode, whose cut-off ratio is 0.95, possesses comparatively greater energy in the near field. After propagating for a comparatively large distance, the cut-off mode attenuates to the level that is similar with the cut-on mode, whose cut-off ratio is 1.5. Considering a more extreme situation that the cut-off ratio of mode is not only less than one but also quite close to one, the sound energy of the mode would be fairly high at location near source, and it would also depreciate very slowly. As far as the preceding results are concerned, the disparity between the acoustic near and far field is very large. Cut-off modes in the near field are so significant that they might contain remarkable sound energy at the location near cascade and might cause fluid–structure–sound interaction in that area. Thus, the measurement results in the far field might leave out some valuable information about the cut-off modes.

3.3. Solution of sound pressure field It can be known from Eq. (13) of Section 2 that when the unsteady aerodynamic force on blade surface is obtained by experiment or numerical computation, the sound generated by oscillating blades can be solved. However, since the source which causes blade oscillation is associated with aeroelasticity and other fields, the present acoustic model does not involve analysis of vibration starting. In order to discuss the amplitude feature of the sound generation, we need to rebuild the unsteady aerodynamic force on blades surface. Hence, in the following part, we will firstly employ typical three-dimensional lifting-surface method to establish a velocity integral equation. Next then, by setting the vibration displacement of blade, the equation would be solved, in which the relevant unsteady aerodynamic force on blade surface can be acquired. After that, the generated sound pressure can be solved. In order to establish a velocity integral equation at blade surface, it is indispensible to build a new coordinate system which is moving with rotating cascade, as shown in Fig. 2, and to procure the normal disturbance velocity that is relative to the coordinate at the blade surface. Then, the pressure disturbance relative to the former rotating coordinate system can be obtained according to previous analysis that Z 1 1 X iBeiΩs t X ψ mn ðkmn rÞeimφ 0 p0q ¼   ! Δpψ mn ðkmn r 0 Þe  imφ 2π s ¼  1 n ¼ 0 Γ mn Sð ξ Þ Z 

1

0

! m Ωr 0 eiαðz  z Þ Þ ð 0 α dα dSð ξ Þ; 2 U β α2  2Mk0 α  k2 þ k2 1 r 0 mn

(30)

! where m ¼ sB σB=2π; φ as well as z, r denote the position of observer in the coordinate system ξ that rotates with the blades. The IBPA is constant and the IBPA coefficient is an integer at present.

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After that, the momentum equation is established in the new rotating coordinate system that ∂v~ φ″ ∂v~ φ″ 1 ∂p0q þW ¼ ; ρ r″∂φ″ ∂t ∂z″

(31)

where the sound disturbance velocity vertical to the blade surface is defined as v0φ″ , and v~ φ″ ¼ v0φ″  eiΩs t . After inserting Eq. (30) into Eq. (31), v0φ″ can be acquired via solving Eq. (31), and v0φ″ can be denoted by Z 1 1 X iBeiΩs t X ψ mn ðkmn rÞeimφ0 0  ! Δpψ mn ðkmn r 0 Þe  imφ 0 v0φ″ ¼ 2πρW s ¼  1 n ¼ 0 Γ mn Sð ξ Þ Z 1 Ω 0 0 ! m Ωr m Ωr 0 eimU ðz  z Þ eiαðz  z Þ Þ dα dSð ξ Þ; (32)  ð α Þð 0  α U r U ðα þ λ þmΩÞðβ2 α2  2Mk0 α  k2 þ k2 Þ 1 r 0 mn U U Ω 0 0 0 where φ0 ¼ φ  Ω U z, φ 0 ¼ φ  U z . According to the former definitions in Eqs. (16) and (17), Eq. (32) can be transformed into Z 1 1 X iBeiΩs t X 0 ϕmn ðkmn rÞeimφ0  ! Δpϕmn ðkmn r 0 Þe  imφ 0 v0φ″ ¼ 2 4π ρW s ¼  1 n ¼ 0 Sð ξ Þ Z 1 Ω 0 0 ! m Ωr m Ωr 0 eimU ðz  z Þ eiαðz  z Þ Þ dα dSð ξ Þ:  ð α Þð 0  α U r U ðα þ λ þmΩÞðβ2 α2  2Mk0 α  k2 þ k2 Þ 1 r 0 mn U U

Utilizing the residue theorem to solve the second integral in Eq. (33) leads to Z 1 1 X  BeiΩs t X 0 ϕmn ðkmn rÞeimφ0  ! Δpϕmn ðkmn r 0 Þe  imφ 0 v0φ″ ¼ 2πρW s ¼  1 n ¼ 0 Sð ξ Þ ( " Ω 0 0 β2 MeimU ðz  z Þ eiα1 ðz  z Þ m Ωr 0 m Ωr ð 0  α1 Þð  α1 Þ  Hðz  z0 Þ U 2κ nm ðMκnm  k0 Þ r U r # 0 Þ iα ðz  z0 Þ 2 imΩ ðz  z 0 M e U e 3 m Ωr m Ωr þ Þð  α3 Þ ð 0  α3 2 2 U U r r k0 þ M 2 kmn ) 0 Þ iα ðz  z0 Þ 2 imΩ ðz  z ! β Me U e 2 m Ωr 0 m Ωr 0  Hðz  zÞ ð α2 Þð  α2 Þ dSð ξ Þ; U 2κnm ðk0 þ Mκ nm Þ r 0 U r

(33)

(34)

where ω α3 ¼  ; U

(35)

the definitions of α1 as well as α2 are identical to that in Eqs. (18) and (19), and they denote that the sound wave transmit downstream and upstream, respectively. While α3 denotes that the unsteady vortex wave transmit downstream. Since the blade is absolutely rigid, the normal disturbance velocity should satisfy the condition of no penetration which can be written as v0φ″ þ w0 ¼ 0;

(36)

0

where w denotes the upwash velocity on the surface of blade. As far as the sound generation by cascade vibration is concerned, the upwash velocity can be acquired through solving the relation between the displacement and velocity of blade vibration that is shown as w0 ðr; zÞ ¼

∂α^ ∂α^ ∂α^ ∂α^ þW ¼ þU ; ∂t ∂z″ ∂t ∂z

(37)

where α^ is the displacement that can include both bending vibration and torsional vibration. α^ can be obtained through the following formula that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 ^ γ ðrÞ Ωr Hbh ^ zÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi αðr; ðrÞðz  z Þ 1 þ þ Θθ : (38) χ e
Ωr2 U 1þ U The first term in the right side of Eq. (38) denotes the displacement of bending vibration, and the second term represents ^ and Θ represent the displacement amplitudes of bending and the displacement of torsional vibration. Furthermore, H torsional vibration, respectively. The bending vibration pattern is denoted by hγ ðrÞ that hγ ðrÞ ¼ where γ is the order of bending vibration pattern.

η ηðrÞ ; ¼ ηmax ηðRd Þ

(39)

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The torsional vibration pattern is denoted by θχ ðrÞ that θχ ðrÞ ¼ sin ð

χπ r Rh  Þ; 2 Rd Rh

(40)

where χ is the order of torsional vibration pattern. For the purpose of simplifying the calculation, we only consider the first-order mode of bending and torsional vibration. Hence, γ as well as χ are equal to one, and the vibration pattern can be written as hγ ðrÞ ¼

ηðrÞ ; ηðRd Þ

    ηðrÞ ¼ 0:707 coshðyÞ  cos ðyÞ  0:518 sinhðyÞ  sin ðyÞ ; yðrÞ ¼ 0:597π

r Rh ; Rd Rh

π r  Rh Þ: θχ ðrÞ ¼ sin ð  2 Rd  Rh

(41) (42) (43)

(44)

After procuring the upwash velocity, Eq. (36) can be solved directly, since there is only one unknown variable. By inserting the obtained disturbance velocity into Eq. (34), the unsteady force on the blade surface can be acquired. Then, the generated sound is completely known. 3.4. Verification of the model Now it is indispensible to verify the validity of the previously developed model. We have compared the calculation results of an example, which predicts the sound field generated by compressor wake–rotor interaction, with the results of Namba [6] to verify the model. It is assumed that the rotor blade number is B ¼ 40 and the chord length of blade in axial direction is a constant b ¼ 0:054978. Moreover, the rotational frequency of rotor is Ω ¼ 294:4536 Hz, and the hub-tip ratio is h ¼ Rh =Rd ¼ 0:5. The Mach number of mean flow inside the duct is M a ¼ 0:35. The influence of wake in front of rotor blades is regarded as unsteady flow disturbance, and the rotational frequency of the disturbance is Ωw ¼ 210:347Hz whose direction is the same as the rotating direction of compressor rotor. The disturbance in rotating coordinate system is denoted as w0i;φ″ ¼ 

Aq ðrÞ  ðΩw þ ΩÞ  r  eiNw ½ðΩw þ ΩÞt  φ þ ΔφðrÞ  zΩw =U ; W

(45)

where Ωw is the rotational frequency of unsteady flow disturbance, Aq ðrÞ ¼ εU and ε is a small value. Furthermore, ΔφðrÞ ¼ ΔθT ðr  Rh Þ=ðRd Rh Þ and it means the disturbance is skewed with respect to the radial direction. ΔθT denotes the level of disturbance oblique at the position near the duct wall. The spanwise direction lifting coefficient is defined as R b=2 0  b=2 Δp dz C L ðrÞ ¼ (46) ~ πρbW  Q ðr Þ where Q~ ðrÞ ¼ Aq ðrÞ  ðΩw þΩÞ  r=W, Q~ ðr Þ is the width of incident wake. The lifting coefficient is a non-dimensional parameter. Fig. 6(a) illustrates the computation results of the lifting coefficient with different radial distribution of incoming flow. Here, it is supposed that the amount of periodical disturbance of incoming flow is N w ¼ 35. Fig. 6(b) shows the sound energy of downstream and upstream propagating sound wave with different amount of disturbance. It can be observed that both the results show excellent agreement with that of Namba [6]. Thus, the present model is valid. 3.5. Prediction of sound generated by synchronously oscillating blades After verifying the model, next we will firstly solve the sound filed generated by cascade with synchronously oscillating blades. It is assumed that the cascade is installed in an annular duct of infinite axial extent with the duct radius is Rd ¼ 0:4m and the hub-tip ratio is h ¼ Rh =Rd ¼ 0:5 as depicted in Fig. 1. There is uniform flow with Mach number 0.5 in the duct, and the rotational frequency of cascade is Ω ¼ 110 Hz. Besides, the cascade consists of 24 blades, the chord length of the blade in axial direction is b ¼ 0:1m and IBPA is zero degree. Then, only the first-order blade bending vibration will be considered, the ^ ¼ 0:01m. On the basis of the previous analysis, the vibration frequency is Ωs ¼ 940 Hz, and the vibration amplitude is H frequency, circumferential modes of the generated sound field can be acquired, as shown in Table 1 (only considering s ¼ 3  þ 3). It can be observed that the circumferential modes distribution is symmetric about zero mode order. Fig. 7 illustrates the calculation results about the sound pressure modal amplitude of upstream propagating sound wave at z ¼  0:1b which is an axial location close to the blade leading edge (z ¼ 0 is the axial location of blade leading ledge). From this figure, it can be found that the modal amplitude of m ¼ 0; n ¼ 1 mode order is comparatively higher than that of other modes. As the radial mode order increasing, the modal amplitudes of the modes with same circumferential mode

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Fig. 6. The computation results of sound generation by compressor stator-rotor interaction: (a) lifting coefficient with different radial distribution of incoming flow, (b) sound energy of downstream (Ed) and upstream (Eu) propagating sound wave with different amount of disturbance. Table 1 The prediction of sound generated by the sample synchronously oscillating annular cascade. s

ω(Hz)

m













3 2  1 0 þ1 þ2 þ3

8860 6220  3580 þ 940 þ 3580 þ 6220 þ 8860



72 48 24

0 þ 24 þ 48 þ 72

|Amn(r, z)|/dB

160 130 100 70 40 -72

-48

-24

0

+24

+48

+72

Circumferential mode order Fig. 7. The sound pressure modal amplitude of upstream propagating sound wave at z ¼  0:1b.

order decreases. When the circumferential mode order increases, the modal amplitudes would decreases, except those whose radial mode order is equal to n ¼ 3. 3.6. Prediction of sound generated by non-synchronously oscillating blades Currently, we will solve the sound field generated by non-synchronously oscillating blades. Similarly, only the first-order bending vibration of rotating cascade blade is considered. The parameters about the rotating cascade, the duct and mean flow are the same to the previous sample in Section 3.5. Since the blades now are non-synchronously oscillating, the IBPA is assumed to be 601. On the basis of the developed model, the frequency, circumferential mode and cut-off ratio of the sound generated by above cascade can be acquired, as shown in Table 2 (Only considering s ¼  3 to þ3 and n ¼ 1 3). Here, it is evident that

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Table 2 The prediction of sound generated by the sample non-synchronously oscillating annular cascade. s

ω (Hz)

m













3 2  1 0 þ1 þ2 þ3

7420 4780  2140 þ 500 þ 3140 þ 5780 þ 8420

76 52  28  4 þ 20 þ 44 þ 68

Cut-off ratio n¼ 1

n¼ 2

n¼ 3

0.797 0.741 0.599 0.825 1.206 1.052 1.008

0.725 0.656 0.501 0.483 0.967 0.919 0.910

0.679 0.605 0.445 0.307 0.839 0.840 0.849

the frequency and circumferential mode features of the sound generated by non-synchronously oscillating blades are quite different with the one of synchronously oscillating blades. The modes distribution is not symmetric about the zero mode order now. 2 2 According to Eq. (29), it is clear that β2 kmn would tend to be equal to k0 and thus the denominator of each term would tend to be zero when the cut-off ratio approaches to one. Even though the numerator would cause the sound pressure qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 amplitude to decay exponentially, the attenuation would be at slow rate since β2 kmn  k0 =β2 is extremely small, as described in Fig. 5. Notwithstanding, if cut-off ratio is much less than one and thus the denominator would be comparatively large, the sound pressure of that mode would show an exponent decay. Fig. 8 illustrates the calculation results about the sound pressure modal amplitude of upstream propagating sound wave at different axial locations. By comparing the results at z ¼  0:1b, it can be found that the results are entirely different with the previous one in Section 3.5. The modal amplitude of m ¼  4; n ¼ 1 mode order is comparatively higher than that of other modes. As the radial mode order increasing, the amplitudes of the modes with same circumferential mode order are generally decreasing. Similarly, with the increase of circumferential mode order, the modal amplitudes also decline. In addition, the cut-on modes are denoted by blue arrows. It can be noticed that the modes with the cut-off ratio which is far less than one is sharply dissipating as the distance between the observer and blades increasing. However, the (m ¼ 4, n ¼ 1) mode, which is cut-off, is dominant in amplitudes at the location near the blades as shown in Fig. 8(a) and (b). Though the mode is cut-off, its amplitude level is comparatively high since its cut-off ratio is close to one, and the mode is probable to cause complex sound phenomenon related to the acoustic resonance in turbomachinery [17]. Therefore, if the results measured in the far field are barely adopted without considering this factor, certain sound modes that generated by oscillating blades might not be detected and the relative analysis might be unreliable. Even though the present model does not pay attention to the source that cause blade oscillation, the frequency of blade oscillation can be inferred from the relation between sound frequency and circumferential mode that is described by the developed model. Then, the vibrational modes and corresponding type of blade vibration can be also deduced. In addition, in terms of the theoretical analysis on the frequency and propagation features of the generated sound, the particular position of blade failure or rupture can be preliminarily determined.

4. Experiment In order to verify the theoretical model about the frequency as well as circumferential mode properties and then clearly exhibits the discrepancy of sound generated by synchronous and non-synchronous blade oscillation, a simplified validation experiment has been carried out. Since it is harmful and hard to induce oscillation on a real compressor rotor, a simulation facility is established, which includes a rotating flat disk and loudspeakers mounted on it, to replace oscillating compressor rotor blades. In fact, the test duct is a straight circular duct and there is roughly no flow, whereas it does not affect the applicability of the proposed model.

4.1. Experimental set-up The experimental set-up consists of drive motor, sound source and test duct; those three parts are illustrated in Fig. 9. The diameter of circular test duct is D ¼ 600:3 mm. The sound source consists of eight equally-spaced loudspeakers mounted on a flat disk, which is presented in Fig. 10. Every loudspeaker represents single oscillating blade, and its frequency range is 700  6000 Hz. The diameter of the disk is Dd ¼ 584:6 mm and the highest rotational speed of the disk is 1200 rev= min. Control signal transmits from power amplifier to the loudspeakers via a high-speed hollow slip ring. The drive motor of this system is a servomotor whose speed can be controlled accurately, and its speed range is 200  3000 rev= min. Moreover, the frequency of the sound generated by motor and slip ring majorly ranges from 100 Hz to 500 Hz. The test duct contains a section for sound fields monitoring at the downstream of the sound source. The section is equipped with ten wall-flush mounted 1/4-in. microphones which are circumferentially equally-spaced at far-field

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|Amn(r, z)|/dB

160 130 100 70 40 -76

-52

-28

-4

+20

+44

+68

+44

+68

+44

+68

Circumferential mode order

|Amn(r, z)|/dB

160 130 100 70 40 -76

-52

-28

-4

+20

Circumferential mode order

|Amn(r, z)|/dB

160 130 100 70 40 -76

-52

-28

-4

+20

Circumferential mode order Fig. 8. The prediction of modal amplitude of upstream propagating sound wave at different axial locations: (a) z ¼  0:1b, (b) z ¼  2:1b, and (c) z ¼  10:1b. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.) 1716 1218

A

440

R 25

Test duct

ϕ600.3

Drive motor

1303

Sound source

Fig. 9. Schematic of experimental set-up (dimensions in mm).

axial position “A”. In addition, there is a microphone at axial position “R” which is close to the sound source and can be treated as a reference position.

4.2. Measurement of frequency and circumferential mode amplitude Since the flow velocity in the test duct is quite small, the influence of flow on the propagation of sound wave in the duct is negligible. Furthermore, there is no anechoic treatment at the end of duct and the space of test is semi-reverberate. Consequently the data acquired by the stationary microphones on the wall can only be used to evaluate the frequency and circumferential mode characteristics of sound field, but not to quantify the absolute sound levels. On the basis of the

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ϕ 46.2

ϕ 584.6

Flat disk

Loudspeaker

ϕ 186

Slip ring

R232

Fig. 10. Schematic of rotating sound source on a flat disk (dimensions in mm). Table 3 Experimental parameters. Parameters

Case 1

Case 2

Case 3

Case 4

Case 5

B Ω ðHzÞ Ωs ðHzÞ σ ðdegÞ

4

4

4

8

8

10

10

0

10

10

900

900

900

900

900

0

90

90

33.5

Non-constant

experimental goal, it is indispensible to procure the information of sound pressure spectrum as well as that of circumferential mode amplitude. Only considering circumferential modes, the sound pressure inside the duct can be written as pðr; φ; z; ωÞ ¼

þ1 X

Am ðr; z; ωÞe  imφ ;

(47)

m ¼ 1

where Am denotes the amplitude of a circumferential mode with order m. The amplitude depends on the radial position, the axial position as well as the frequency. In order to detect modal amplitude, it is required to obtain the sound pressure at several different circumferential positions with a given radial position in a constant axial plane of test duct. In addition, to suppress the interference of background noise that primarily comes from the servomotor and slip ring, the pressure signals at the reference position should be imported. The mode detection methodology used here is adopted from the one described by Liu et al. [18], which can be called as method of cross-correlation with reference signal. The method mainly includes two steps. Firstly, the complex form sound pressure of every circumferential location should be obtained by calculating the cross spectrum of the signals, which are measured at that particular position and the reference position. Secondly, the mode spectrum which describes the modal amplitude of specific frequency can be procured via computing Space domain discrete Fourier Transform (SFT) of the known complex form sound pressure. Nonetheless, during SFT, the phenomenon of mode aliasing cannot be averted. It is clear about that, with limited measurement positions around the circumference, only limited modes can be detected. In the current experiment, ten equally spaced microphones can only discern ten circumferential mode orders ranging from  4 to þ5; other higher mode orders are indistinguishly aliased to those ten mode orders. The information of frequency spectrum can be obtained from the sound pressure signal by time domain Fast Fourier Transform (FFT). Moreover, in order to obtain high resolution of frequency, the data sampling rate and sampling time should be set at comparatively high level.

4.3. Experiment results In this section, the measurement results as well as the comparison between model prediction and measured data will be illustrated. The measurement is carried out on a number of different conditions, and here only a small sample of the measured data which are comparatively representative are reported. Currently, the physical parameters of the preceding model are replaced by several analogous factors which are controllable in experiment. In detail, the blade number is substituted by the amount of loudspeakers; the frequency of blade vibration is replaced by the frequency of sinusoidal loudspeaker signal; the rotor speed is simulated by the rotational speed of disk which loudspeakers are mounted on; and the IBPA is realized by the initial phase difference of signal between adjacent loudspeakers. In Table 3, the critical experimental parameters of chosen cases are presented, and here the rotational speed is denoted by the form of rotational frequency. Besides, other parameters are approximately identical and it should be mentioned that the power of each loudspeaker used is same.

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Table 4 Model prediction for case 1. Predicted data

Near field

Far field

s m ¼ sB ω ¼ Ωs þ mΩ ðHzÞ

–

1

0

þ1

–

0

–

4

0

þ4

–

0

–

860

900

940

–

900

120 90

Sound pressure level/dB

60 30 120 90 60 30 820

860

900 Frequency/Hz

940

980

Fig. 11. Far field (solid line) and near field (dashed line) sound pressure spectra for case 1. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

110

930

100

920 Frequency/Hz

90 910 80 900 70 890

60

880

50

870

40 -4

-3

-2

-1

0

+1

+2

+3

+4

+5

Circumferential mode order Fig. 12. 2-D circumferential mode spectra for case 1.

In the following part, the frequency spectra of generated sound measured by far-field microphone as well as near field microphone, and the two-dimensional circumferential mode spectra at the dominant tones' frequencies are displayed. On the ground of these data, the comparison between predicted and measured data can be direct and explicit.

4.3.1. Simulation of sound generation by synchronously oscillating rotor blades Since the condition of synchronous blade oscillation is considered, the phase difference of adjacent loudspeakers in case 1 is zero. At present, the frequency of loudspeaker signal is 900 Hz that just below the cut-on frequency of the fourth higher order mode in the test duct which is f 40  956 Hz. According to the model for synchronously oscillating blades that are illustrated by Eqs. (25) and (26), there would be single dominant tone in the far sound field of the test duct, whose dominant circumferential mode is m ¼ 0. The model prediction for case 1 is depicted in Table 4.

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120

Sound pressure level/dB

90 60 30 120 40Hz

90 60 30 820

860

900

940

980

Frequency/Hz Fig. 13. Far field (solid line) and near field (dashed line) sound pressure spectra for case 2. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Fig. 11 shows the sound pressure spectrum measured by far-field microphone as well as near field microphone in case 1. The near field spectrum contains the loudspeaker signal frequency plus multiple tones at 40 Hz intervals, these tones are denoted by red arrows in the figure. The interval between adjacent tones is 40 Hz that equals the product of loudspeaker number and its rotational frequency. It can be noticed that the spectrum measured in the far field is dominated by the loudspeaker signal above the background noise level more than 50 dB, as predicted by Eqs. (25) and (26) for case 1. The dominant tone is denoted by a blue arrow. Since the higher-order modes would decay sharply in the far field, the additional tones in the near field vanish in the far-field spectrum. However, there are some unexpected tones of lower level situate near the dominant tone in the far-field spectrum. They separated regularly at 10 Hz intervals which is equal to the rotational frequency of the sound source disk. Fig. 12 depicts the circumferential mode amplitudes at dominant tone’s frequency and unexpected tones' frequencies of the far field. Here, it is necessary to introduce a parameter which called “Target versus Actual” (TVA)-level to assess the amplitude difference between the level of target mode and the highest one of the remaining modes [19]. In general, if its value is greater than 10 dB, the target mode is dominant at the specific frequency. As expected, the dominant circumferential mode of dominant tone in the far field is m ¼ 0, which is completely accord with the prediction. Additionally, it can be noticed that each unexpected tone also corresponds to one dominant mode. For example, the dominant circumferential mode of the tone at 870 Hz is m ¼  3. The frequencies of these unexpected tones are tantamount to the sum of loudspeaker signal frequency and the product of rotational frequency and the dominant circumferential mode order. Such phenomenon can be also observed in the succeeding results. Since the loudspeakers are not perfectly equallyspaced on the rotating disk and the loudspeaker signal might be distorted slightly, the situation is similar with the one described in Section 3.1.3 that IBPA coefficient is non-constant. According to the previous conclusions, the tones corresponding to all modes would be generated. Because of the cut-off effect, there would be finite tones that can propagate to the far field. Furthermore, although the phase deviation which is induced by installation as well as signal distortion of each loudspeaker might not be equivalent, the deviation would generally not be fairly large and it would be usually a small random value which can be positive or negative. Hence, even though the phase angle difference of each pair of adjacent loudspeakers might be slightly dissimilar, the additional phase deviation would not lead tremendous influence on the mode determination state which is mainly determined by the mean phase difference of loudspeakers. In the experimental results, it can be noticed that both the frequency spectra and circumferential mode spectra results show good agreement with the previous conclusions. In addition, the detailed experimental results of the situation that IBPA coefficient is non-constant are illustrated in the subsequent case 5. 4.3.2. Simulation of sound generation by non-synchronously oscillating rotor blades In this section, the experimental results including case 2 to case 5 are displayed. Since the condition of non-synchronous blades oscillation is considered, the phase difference of loudspeakers in all these cases is not zero, which can be noticed in Table 3. 4.3.2.1. The situation that IBPA coefficient is an integer. Firstly, the sound pressure spectrum of case 2 is described in Fig. 13. At present, the initial phase difference of adjacent loudspeaker signal is the unique factor which differs from case 1.

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Table 5 Model prediction for case 2. Predicted data

Near field

Far field

s m ¼ sB  σB=2π ω ¼ Ωs þ mΩ ðHzÞ

–

1

0

þ1

–

0

–

5

1

þ3

–

1

þ3

–

850

890

930

–

890

930

þ1

110

930

100

920 Frequency/Hz

90 910 80 900 70 890

60

880

50

870

40 -4

-3

-2

-1

0

+1

+2

+3

+4

+5

Circumferential mode order Fig. 14. 2-D circumferential mode spectra for case 2. Table 6 Model prediction for case 3. Predicted data

Near field

Far field

s m ¼ sB  σB=2π ω ¼ Ωs þ mΩ ðHzÞ

–

1

0

þ1

–

0

–

5

1

þ3

–

1

þ3

–

900

900

900

–

900

900

þ1

120 90

Sound pressure level/dB

60 30 120 90 60 30 820

860

900 Frequency/Hz

940

980

Fig. 15. Far field (solid line) and near field (dashed line) sound pressure spectra for case 3.

Consequently the initial phases of each loudspeaker signal in case 2 are different. On the basis of the previous model which is illustrated by Eq. (23) as well as Eq. (24), we can anticipate that there would be two dominant tones in the far sound field, and they corresponds to the circumferential mode m ¼  1 and m ¼ þ 3, respectively, as exhibited in Table 5.

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110

120

100 100

90 80

80 70 60

60

50 40

40 -4

-3

-2

-1

0

+1

+2

+3

+4

+5

Circumferential mode order Fig. 16. Circumferential mode spectrum at 900 Hz for case 3.

120 90

Sound pressure level/dB

60 30 120 90 60 30 820

860

900 Frequency/Hz

940

980

Fig. 17. Far field (solid line) and near field (dashed line) sound pressure spectra for case 4.

In Fig. 13, the dominant tones in the near sound field are also denoted by red arrows, and that in the far sound field are denoted by blue arrows. It can be aware that the frequencies of dominant tones do not equal the frequency of loudspeaker signal either in the far field or near field. Since the loudspeaker number and rotational frequency are not changed, the interval between adjacent dominant tones is 40 Hz which is equal to that of case 1, and the spacing is denoted by horizontal arrow in the figure. The frequencies of the dominant tones in the far field are 890 Hz and 930 Hz, which are conformed to the prediction. Furthermore, the dominant tones in the far field still show typically more than 50 dB emergence above the background noise as before. Fig. 14 shows the acoustic far-field mode spectrum at dominant tones' frequencies of case 2. At present, the dominant circumferential modes of tones at 890 Hz and 930 Hz are still in accordance with the prediction. At 890 Hz, although it can be observed that the amplitude of m ¼  1 is higher than that of other mode orders, the TVA-levels is less than 10 dB slightly. Moreover, the unexpected tones also appear in case 2. The regular pattern of their frequency and circumferential mode distribution is identical to that illustrated before. 4.3.2.2. The situation that rotating speed is zero. In order to verify that the model is feasible when the rotational speed is zero, case 3 was carried out. In the light of Eqs. (23) and (24), we can acquire the predicted data for case 3 which are displayed in Table 6. It can be noticed that there would be single tone in the far sound field and its frequency is equivalent to the actual frequency of loudspeaker signal. However, the circumferential mode characteristic of the tone is comparatively complex, which corresponds to double modes including m ¼  1 and m ¼ þ 3. Fig. 15 exhibits the sound pressure spectrum. Obviously, there is only one dominant tone corresponding to 900 Hz which is the actual frequency of loudspeakers signal. Furthermore, Fig. 16 presents the circumferential mode spectrum at 900 Hz. In fact, It can be noticed that the amplitudes of m ¼ 1 and m ¼ þ 3 are greater than that of other modes over 10 dB, which

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110

930

100

920 Frequency/Hz

90 910 80 900 70 890

60

880

50

870

40 -4

-3

-2

-1

0

+1

+2

+3

+4

+5

Circumferential mode order Fig. 18. 2-D circumferential mode spectra for case 4. Table 7 Model prediction for case 5. Predicted data

Near field

Far field

s m ¼ sB  σB=2π ω ¼ Ωs þ mΩ ðHzÞ

–

1

0

þ1

–

–

 10

2

þ6

–

2

–

800

880

960

–

880

0

120 80Hz 90

Sound pressure level/dB

60 30 120 90 60 30 820

860

900

940

980

Frequency/Hz Fig. 19. Far field (solid line) and near field (dashed line) sound pressure spectra for case 5.

shows good consistency between prediction and measurement. Besides, these results also indicate that the preceding model is applicable to the circumstance of quiescence. 4.3.2.3. The situation that IBPA coefficient is non-integral. From the previous Section 3.1.2, it can be known that when the IBPA coefficient is non-integral, the frequency and circumferential mode features would be so complicated that the tones corresponding to all modes would be generated. In case 4, the phase difference of loudspeakers is 33:53 and thus the IBPA coefficient is tantamount to 67=90. Fig. 17 shows sound pressure spectrum for case 4. It is obvious that the acoustic near field spectrum contains the loudspeaker signal 900 Hz plus multiple tones at 10 Hz intervals, as predicted by the previous model. Because only the circumferential modes m ¼  3;  2;  1; 0; þ 1; þ 2; þ 3 are cut-on, the additional tones disappear in the acoustic far-field spectrum. Moreover, since now the IBPA is equal to 33:53 , the theoretical circumferential mode of the acoustic far field can be calculated with value at m   0:74, which means the dominant modes would be both m ¼ 0 and m ¼  1 that are the two integers close to 0.74. Hence, the frequencies of dominant tones should be 900 Hz and 890 Hz.

170

D. Zhou et al. / Journal of Sound and Vibration 355 (2015) 150–171

110

930

100

920 Frequency/Hz

90 910 80 900 70 890

60

880

50

870

40 -4

-3

-2

-1

0

+1

+2

+3

+4

+5

Circumferential mode order Fig. 20. 2-D circumferential mode spectra for case 5.

According to Fig. 17, it can be noticed that the actual dominant tones' frequencies in near and far fields conform to the prediction, but their amplitudes are just greater than other tones slightly. Actually, according to the analysis in Section 3.1.2, it can be known that as the loudspeakers number increasing, the effect of non-integer IBPA coefficient would be less and thus the dominant tones would be more distinct. In consideration of the circumferential mode amplitudes at the frequencies of dominant tones which are displayed in Fig. 18, it can be also noted that the dominant modes at these tones are in complete accord with the prediction. The TVAlevels of these tones are also greater than 10 dB. 4.3.2.4. The situation that IBPA is non-constant. Next then, in case 5, we will consider a more practical situation that the initial phase difference of each pair of adjacent loudspeakers signal is different, which is similar with what is illustrated in Section 3.1.3. In case 5, the average phase angle difference between adjacent loudspeakers signal is about 903 . However, the actual phase angle difference slightly varies from the average value. The deviation is randomly positive or negative, and its maximum magnitude is limited to 153 . According to the previous analysis, we can predict the dominant tone in the far field has a frequency value at 880 Hz by utilizing the model illustrated by Eq. (23) and Eq. (24), and its dominant circumferential mode order is equal to 2 (m ¼ 2). Specifically, only the average phase angle difference is considered in prediction, and the model prediction for case 5 is described in Table 7. Fig. 19 shows sound pressure spectrum for case 5. It can be noticed that the frequencies of dominant tones in the acoustic near field are 880 Hz and 960 Hz. Since now eight loudspeakers are operating, the frequency spacing of adjacent dominant tones is 80 Hz. In the far field, only the tone at 880 Hz is cut-on, and its level is higher than the background noise level over 50 dB. Excellent agreement between measurement and prediction is again exhibited. Fig. 20 displays the acoustic far-field modal spectrum at dominant tones' frequencies of case 5. At present, the dominant circumferential mode of tone at 880 Hz is still in accordance with the prediction. Similarly, the additional tones which are so-called unexpected tones in case 1 and case 2 also appear. The regular pattern of their frequency and circumferential mode distribution is identical to that illustrated before. According to the previous analysis in Section 3.1.3, it is indicated that when the number of operating loudspeakers increases, the influence of phase difference deviation on the frequency and circumferential mode characteristics of the generated sound would accordingly decrease. 5. Conclusions In this article, the sound generation by non-synchronously oscillating rotor blades in axial compressor is investigated. An analytical model to predict the corresponding acoustic field has been developed based on unsteady lifting-surface theory. Furthermore, a simulation experiment has been carried out to verify the model. As far as the frequencies and propagating modes are concerned, the agreement between the experiment results with the model prediction is quite remarkable. On the basis of this investigation, some conclusions about the sound generation by non-synchronously oscillating rotor blades in axial compressor are made as follows:

 The developed model clearly manifests the relationship between cascade blade oscillation and the generated sound field.



It can be noticed that the frequency and circumferential mode of generated sound are directly connected with the frequency of blade oscillation, the rotating speed, and the blade number as well as the IBPA. Hence, it would be practicable to monitor compressor rotor blade oscillation by measuring and analyzing the generated sound field with noncontact transducers mounted statically, and especially to detect those oscillation parameters like IBPA, which are quite difficult to procure via common methods. The sound generation by non-synchronously oscillating rotor blades is quite different with that of synchronously oscillating rotor blades. Specifically, the frequency is tantamount to non-integral multiples of the sum of rotor rotational

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frequencies and blade oscillation frequency, while the situation of synchronously oscillating rotor blades is completely disparate. When the IBPA coefficient is not an integer, the tones corresponding to all circumferential modes would be generated. The frequencies of these tones are equal to the sum of blade oscillation frequency and the product of rotor rotational frequency and the dominant circumferential mode order, but the sound energy primarily concentrates on several particular modes among them. Consequently the characteristics of sound field in the situation that IBPA coefficient is not integer are not much different with the one that IBPA coefficient is an integer. Practically, it can be conscious of that although the phase angle difference of each pair of adjacent blades on the cascade might be slightly different, the deviation does not bring tremendous influence on the distribution of sound energy. The developed model is also applicable to analyze the sound generation by non-synchronously oscillating rotor blades in this realistic circumstance.

Acknowledgments The financial supports from the National Natural Science Foundation for Youth in China (Grant no. 51106005) as well as the 973 Program (Grant no. 2012CB720201) are gratefully acknowledged. References [1] S. Bill, Axial fan tone noise induced by separated tip flow, flutter, and forced response, Proceedings of 11th AIAA/CEAS Aeroacoustics Conference, Monterey, California, AIAA-2005-2876, 2005. [2] S. Kaji, T. Okazaki, Propagation of sound waves through a blade row: I. Analysis based on the semi-actuator disk theory, Journal of Sound and Vibration 11 (3) (1970) 339–353. [3] S. Kaji, T. Okazaki, Propagation of sound waves through a blade row: II. Analysis based on the acceleration potential method, Journal of Sound and Vibration 11 (3) (1970) 355–375. [4] D.S. Whitehead, Vibration and sound generation in a cascade of flat plates in subsonic flow, Aeronautical Research Council Reports and Memoranda No. 3865, 1970. [5] S.N. Smith, Discrete frequency sound generation in axial flow turbomachines, Aeronautical Research Council Reports and Memoranda No. 3709, 1973. [6] M. Namba, Three-dimensional analysis of blade force and sound generation for an annular cascade in distorted flows, Journal of Sound and Vibration 50 (4) (1977) 479–508. [7] J.B.H.M. Schulten, Sound generated by rotor wakes interacting with a leaned vane stator, AIAA Journal 20 (10) (1982) 1352–1358. [8] J.B.H.M. Schulten, Vane stagger angle and camber effects in fan noise generation, AIAA Journal 22 (8) (1984) 1071–1079. [9] J.B.H.M. Schulten, Vane sweep effects on rotor/stator interaction noise, AIAA Journal 35 (6) (1997) 945–951. [10] X. Wang, X. Sun, On the interaction of a fan stator and acoustic treatments using the transfer element method, Fluid Dynamics Research 42 (1) (2010) 1–17. [11] X. Sun, X. Wang, L. Du, X. Jing, A new model for the prediction of turbofan noise with the effect of locally and non-locally reacting liners, Journal of Sound and Vibration 316 (1) (2008) 50–68. [12] X. Wang, X. Sun, A new segmentation approach for sound propagation in non-uniform lined ducts with mean flow, Journal of Sound and Vibration 330 (10) (2011) 2369–2387. [13] J.M. Verdon, Linerized unsteady aerodynamic analysis of the acoustic response to wake/blade-row interaction, Contractor Report CR-2001-210713, NASA, 2001. [14] D. Prasad, J.M. Verdon, Validation of a three-dimensional linearized Euler analysis for classical wake/stator interactions, Proceedings of ASME TURBO EXPO 2002, Amsterdam, Netherlands, 2002. [15] M. Namba, R. Nishino, Unsteady aerodynamic response of vibrating contra-rotating annular cascades part I: description of model and mathematical formulations, Transactions of the Japan Society for Aeronautical and Space Sciences 49 (165) (2006) 175–180. [16] F. Kameier, W. Neise, Rotating blade flow instability as a source of noise in axial turbomachines, Journal of Sound and Vibration 203 (5) (1997) 833–853. [17] B. Hellmich, J.R. Seume, Causes of acoustic resonance in a high-speed axial compressor, Journal of Turbomachinery 130 (3) (2008) 031003. [18] J.M. Liu, F. Holste, W. Neise. On the azimuthal mode structure of rotating blade flow instabilities in axial turbomachines, Proceedings of the Second AIAA/ CEAS Aeroacoustics Conference, State College, PA, AIAA paper-96-1741, 1996. [19] S. Pieter, J. Zillmann. In-duct and far-field mode detection techniques, Proceedings of the 13th AIAA/CEAS Aeroacoustics Conference, Roma, Italy, AIAA Paper-2007-3439, 2007.