Prediction of tonal ducted fan noise

Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Prediction of tonal ducted fan noise J. de Laborderie 1,n, S. Moreau Department of Mechanical Engineering, University of Sherbrooke, Sherbrooke, Qc, Canada J1K 2R1

a r t i c l e i n f o

abstract

Article history: Received 18 June 2015 Received in revised form 11 December 2015 Accepted 17 February 2016 Handling Editor: P. Joseph

This study introduces an analytical model aiming at predicting the tonal acoustic sources generated and radiated by the rotor–stator interaction in a fan stage. This model is able to cope with complex three-dimensional stator geometries and it fully accounts for cascade effects, characteristic of modern fan stages. It is based on a proper description of the rotor wake coupled with an analytic cascade response function and with an acoustic analogy. The proposed model is validated for the first time against acoustic sources and sound power measurements, on the Advanced Noise Control Fan from NASA. On this configuration representative of an actual fan stage, the model is shown to predict tonal sources and powers accurately, in function of the rotational velocity and of the stator-vane count. Another realistic configuration, namely the low-pressure CME2 research compressor, is then considered in order to demonstrate the suitability of the model to be used as a design tool in an industrial context. A parametric study concerning both the stator vane sweep and lean angles is performed on this rotor–stator stage. The model produces predictions consistent with studies from literature, quantifying the effectiveness of swept and leaned vanes as a tonal noise reduction mechanism. This parametric study allows defining an optimal stator design for minimal noise emission. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Tonal noise Rotor-stator interaction Cascade model Acoustic optimisation

1. Introduction In the future of aircraft propulsion, the Ultra-High Bypass Ratio (UHBR) turbofan architecture is promising as it leads to an overall efficiency improvement and to a decrease of pollutants emission with respect to the current generation of aeroengines. In particular, the gain in propulsive efficiency is partly reached with a larger fan diameter. However the larger nacelle associated with this configuration implies a drag increase that is detrimental to the overall engine efficiency. Thus a UHBR nacelle has to be shortened. This in turn induces shorter rotor–stator distances, more pronounced inlet and outlet distorsions and less efficient passive treatment that reduces fan noise propagating outside of the nacelle. Therefore, in the frame of fan noise source reduction, the current work aims at improving fan noise modelling. As stated by several authors, e.g. Groeneweg et al. [1], Envia et al. [2] and Peake and Parry [3], the interaction of fan wakes with the downstream outlet guide vanes (OGVs) represents the main source of fan noise, a trend growing with the expected decrease of the rotor–stator spacing in UHBR. A blade wake is composed of a mean velocity deficit creating a periodic fluctuation in time and in the azimuthal direction in the stator reference frame. Convected with the mean flow, these wakes interact with the vane leading edges generating pressure fluctuations on the whole vane surfaces. This unsteady loading radiates acoustic waves n

Corresponding author. E-mail addresses: [email protected] (J. de Laborderie), [email protected] (S. Moreau). 1 Present address: CERFACS, CFD team, 31057 Toulouse, France.

http://dx.doi.org/10.1016/j.jsv.2016.02.032 0022-460X/& 2016 Elsevier Ltd. All rights reserved.

Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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propagating within the nacelle, at the blade passing frequency (BPF) and its higher harmonics. This mechanism corresponds to the rotor–stator interaction tonal noise. Moreover turbulent structures present in blade wakes, upon interaction with the downstream vanes, generate random wall pressure fluctuations that radiate to form the rotor–stator interaction broadband noise [4]. The current study focuses on fan tonal noise as it becomes crucial in UHBR because of the less efficient passive treatment as well as the decrease of the rotor–stator gap creating a larger interaction of the rotor wakes with the stator. Several types of Computational Aero-Acoustics (CAA) methods aiming at the prediction of fan tonal noise have been developed [2]. Among them, a first category uses Reynolds-Averaged Navier–Stokes (RANS) equations to compute the viscous blade wakes downstream of the rotor. This vortical excitation is then imposed at the inlet of a stator domain on which the linearised Euler equations are numerically solved. This approach has for instance been implemented within the LINFLUX code, that computes the vane row acoustic response in the frequency domain, by Montgomery and Verdon [5], Verdon [6] and Prasad and Verdon [7]. The method of Atassi et al. [8] also uses linearised Euler equations to predict the acoustic response of the vane cascade. With such an approach, actual vane and duct geometries are accounted for in the source and near-field acoustic prediction. A second category of methods, based on the Unsteady Reynolds-Averaged Navier– Stokes (URANS) equations, has been successfully surveyed by several authors [9–12]. These approaches allow considering a realistic flow, particularly the vane mean loading and the influence of the stator on the rotor wakes, in the acoustic prediction. However the computational cost associated with these numerical methods still prevents them to be routinely used in an industrial context, especially for pre-design and parametric studies. This is the reason why analytical approaches represent an interesting compromise as they provide fast and exact solutions of simplified problems. The response function of Amiet [13], extended by Paterson and Amiet [14], Moreau et al. [15] and Roger et al. [16], deals with the interaction of an aerodynamic gust with a single airfoil in free-field. In this model, suited for low solidity rotors without external casings such as helicopter rotors, propellers, contra-rotating open rotors and ventilators, the airfoil is modelled as an infinitely thin flat plate immersed in a uniform inviscid flow with zero incidence (neither vane camber nor mean loading). The asymptotic analyses of Myers and Kerschen [17] and Evers and Peake [18] allow considering some mean loading effects in the response. For low solidity ducted fans, as proposed by Glegg [19] for instance, the isolated response can be coupled with Green's function tailored to a duct developed by Goldstein [20]. However cascade effect, i.e. the influence of the neighbouring vanes on the source generation and radiation of a given vane, has to be considered in a modern fan stage where vanes overlap. For the interaction of a gust with a rectilinear flat plat cascade, the models are based on the resolution of an integral equation, using the acceleration potential method [21] or the lifting surfaces methods [22], leading to the LINSUB code for instance [23–26]. Namba and Schulten's and Zhang et al.'s model [27–29], also based on a lifting surface method, account for an annular cascade and the casing walls, but are limited to vanes with zero stagger angle and no radial geometrical variations. Another group of cascade models is based on a closed-form analytical solution of the integral equation with the Wiener– Hopf technique, successively extended by Mani and Hovray [30], Koch [31], Peake [32] and Glegg [33]. Posson et al. [34] extended Glegg's work to the calculation of the unsteady vane loading and of the pressure field in the vane passage. In the cascade model of Hanson [4], also based on the Wiener–Hopf technique and Glegg's blade response, the strip theory approach allows accounting for variable stator geometry along the duct height, but the acoustic power is evaluated in free field and decorrelated from one strip to another one. Finally, another approach consists in considering the actual annular distribution of acoustic sources on the stator vanes as well as their radiation within an annular duct, contrary to the free field acoustic radiation from a rectilinear cascade. In the model of Ventres et al. [35], the unsteady vane loading is obtained via a numerical resolution of the integral equation then radiated with the annular duct Green's function of Goldstein [20]. This model, developed for fan tonal and broadband noise applications, has been extended by Meyer and Envia [36], Nallasamy and Envia [37] and Grace et al. [38]. Finally the model of Posson et al. [39], using a purely analytical solution for the acoustic sources and the in-duct radiation with Goldstein's analogy, has been applied and validated for fan broadband noise predictions. Tonal fan noise modelling is challenging mainly because the sound emission is sensitive to the stator geometry and because the excitation and the vane response are correlated along the duct height. This is why the present model, based on the analytical cascade response function of Posson et al. [34], uses a radial strip approach in order to consider a complex stator geometry, with variable stagger, sweep and lean angles for the vanes. Moreover this cascade response is coupled with Green's function developed by Goldstein [20] so as to account for the annular duct geometry and an axial mean flow in the acoustic propagation. The aerodynamic excitation is decomposed into skewed gusts that are coupled with the 3D cascade response to ensure a correct representation of the radially correlated vane response. Although this tonal noise model has already been used in preliminary studies [40,41], the first objective of the paper consists in performing a complete description of this cascade based acoustic model, in focusing particularly on the complex stator geometry and the aerodynamic excitation. Then Section 3 aims at validating the model against experimental data collected on the Advanced Noise Control Fan (ANCF) configuration [42,43]. The latter is representative of an actual fan stage of a modern aeroengine and exhaustive available datasets allow performing trends with the vane count and the rotational velocity. Finally the model is evaluated in Section 4 on the CME2 axial compressor stage [44], at higher reduced frequencies than on the ANCF configuration. Parametric studies on the vane geometry are performed in order to highlight the model capability of handling complex stator configurations and of determining an optimal acoustic design. Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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2. Analytical model for fan tonal noise prediction 2.1. Configuration In this section, the cascade based acoustic model of Posson et al. [34,39], originally developed for fan broadband noise applications, is generalised to fan tonal noise prediction. The configuration treated here, represented in Fig. 1, is a rotor– stator stage mounted in an annular duct. The cylindrical reference frames Rr ðxd ; r d ; θr Þ and Rd ðxd ; r d ; θd Þ are direct and linked to the rotor and the stator respectively, with xd directed downstream. The rotor is composed of B blades and the stator of V vanes. In order to account for complex blade and vane geometries present in modern turbofan engines, i.e. with stagger, sweep and lean angles varying along the duct height, both the rotor and stator rows are split into annular strips along the span. Each strip at radius rd is then unwrapped into a local rectilinear geometry represented in Fig. 2. The direct Cartesian reference frame R0 ðxd ; yd ; zd Þ is attached to the rectilinear cascade of infinite span flat plates. Following a literature review, it appears necessary to propose a unified formulation of the equations in order to account for any rotational direction. Indeed, Fig. 2(a) corresponds to the configuration with the rotational speed Ω o 0 used in the model of Posson et al. [34] for instance, whereas Fig. 2(b) represents the configuration treated in the model of Ventres et al. [35] as well as in its further extensions [36,45] with Ω 4 0. The frames Rr and Rd are linked by the following relation, Ω being a relative speed from now on:

θr ¼ θd  Ωt:

(1)

^ ðr d Þ and lean ψ^ ðr d Þ angles of the vane are defined in Figs. 3 and 4, In the Rd reference frame, the geometrical sweep φ ^ is independent of the rotational looking successively from the side and the front of the cascade. It is worth noting that φ velocity sign, whereas the sign of ψ^ changes with the sign of Ω, as depicted in Fig. 4. Moreover an arbitrary ðyf ; zf Þ surface is introduced, viewed from the side in Fig. 3 (dotted grey line), on which the aerodynamic data (mean flow and rotor wake) are extracted to be input within the model. For an efficient interaction modelling, this surface has to follow the vane leading ^ f ðr d Þ the angle of this surface with rd, the transfer matrix from Rf ðxf ; yf ; zf Þ to R0 is edge radial evolution. With yf ¼ yd and φ

Fig. 1. Fan stage configuration for the acoustic model.

Fig. 2. Local geometry and flow parameters of the fan stage unwrapped at radius rd, for two rotational directions.

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^ d Þ and of the extraction surface ðyf ; zf Þ (dotted grey line, side view). Fig. 3. Definition of the vane geometric sweep angle φðr

Fig. 4. Definition of the vane geometric lean angle ψ^ ðr d Þ for both rotational directions, assuming vanes are inclined towards their suction side (SS).

written as: 2 6 Qf ¼ 4

^f cos φ

0

0

1

^f  sin φ

0

^f 3 sin φ 7 0 5: ^f cos φ

(2)

^ and ψ^ represent the design angles in Rd , the cascade response introduced in Section 2.4 is calculated in the Whereas φ cascade reference frame Rc ðxc ; yc ; zc Þ represented in Fig. 2. Hanson [4] shows that the Rc frame is obtained following three successive rotations from the R0 frame, in this specific order: stagger angle χ, lean angle ψ and sweep angle φ. The transfer matrix from R0 to Rc is, from [4]: 2 6 Q¼4

cos χ cos φ þ sin χ sin ψ sin φ

sin χ cos φ  cos χ sin ψ sin φ

 cos ψ sin φ

cos χ sin φ  sin χ sin ψ cos φ

sin χ sin φ þ cos χ sin ψ cos φ

cos ψ cos φ

It can be shown that

 sin χ cos ψ

cos χ cos ψ

 sin ψ

3 7 5:

(3)

ψ and φ are, from the design angles ψ^ and φ^ : ^; tan ψ ¼ cos χ tan ψ^  sin χ tan φ

(4)

  ^ : tan φ ¼ cos ψ sin χ tan ψ^ þ cos χ tan φ

(5)

The change of coordinates is chosen to only deal with direct reference frames thus implying the direction yd in R0 to be in the opposite direction than θd (in Rd ), for any sign of the rotational velocity (see Fig. 2). In R0 the sign of the vane stagger angle is always opposed to the sign of Ω. The cascade reference frame Rc is fixed such as yc is in the yd direction. This imposes yc to be oriented from the pressure side (PS) to the suction side (SS) of the vane for a negative rotational velocity (Fig. 2(a)), and in the opposite direction (from SS to PS) for a positive rotational velocity (Fig. 2(b)). For both configurations, the vane number ν increases in the yc direction similarly to the convention adopted by Glegg [33]. Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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2.2. Upwash The rotor wakes formed at the blade trailing edges are convected with the subsonic mean flow and interact with the downstream stator vanes, creating unsteady wall pressure fluctuations that correspond to the tonal acoustic sources. More precisely, following the definition of Sutliff et al. [42], the aerodynamic excitation corresponds to the velocity component w normal to the direction given by the mass-averaged absolute flow angle α (see Fig. 2). Assuming that this excitation is identical from one rotor blade to another one, it becomes 2π =B-periodic in the azimuthal direction as well as periodic in time with the blade passing pulsation qBΩ. Using the following convention for the temporal Fourier series: wðxd ; r d ; θd ; tÞ ¼

þ1 X q ¼ 1

~ q ðxd ; r d ; θd Þe  iqBΩt ; W

(6)

with Z 2π =Ω

  ~ q xd ; r d ; θd ¼ Ω W 2π

0

  w xd ; r d ; θd ; t eiqBΩt dt

the upwash can be written as the following Fourier series: wðxd ; r d ; θd ; tÞ ¼

þ1 X

c q ðxd ; r d ÞeiqBθd e  iqBΩt ; W

(7)

  w xd ; r d ; θr e  iqBθr dθr ;

(8)

q ¼ 1

with c q ðxd ; r d Þ ¼ B W 2π

Z 2π =B 0

c q contributes to the cascade response at the qth harmonic and wðxd ; r d ; θr Þ defined in Rr . From Eq. (7) only the coefficient W of the blade passing frequency (BPF). Because of the no-slip condition on both the hub and casing walls, the absolute flow velocity is zero in a ducted fan stage at these locations. Thus the upwash velocity can be Fourier transformed in the radial direction with a period T r ¼ RT  RH to give the double Fourier series: wðxd ; r d ; θd ; tÞ ¼

þ1 X

þ1 X

c q;p ðxd ÞeiqBθd eikr ðpÞrd e  iqBΩt ; W

(9)

q ¼ 1 p ¼ 1

with c q;p ðxd Þ ¼ 1 W Tr

Z

RT RH

c q ðxd ; r d Þe  ikr ðpÞrd dr d ; W

(10)

where kr ðpÞ ¼ 2π p=T r ; p A Z, is the radial wavenumber. This 3D decomposition aims at a better representation of the incident perturbation than the 2D approach described with Eq. (7). Indeed the latter implies that the radial evolution of the wake properties are only treated by variations of amplitudes with the parameter rd. A 3D decomposition allows accounting for the radial gradients of the perturbation [46] as well as for the radial correlation known to be significative for a tonal noise excitation [47]. The principle of the 3D decomposition proposed in Eq. (9) has been validated in [40] but it has never been confronted to an experimental database. 2.3. Aerodynamic wave-vector From aerodynamic measurements or simulations, the mean velocity profile U as well as the upwash w decomposed in Section 2.2 are extracted in the ðyf ; zf Þ surface introduced in Section 2.1. As a general rule for the interaction modelling, it is ^ f ðr d Þ ¼ φ ^ ðr d Þ [48]. A recommended to consider the excitation as close as possible to the vane leading edges, thus implying φ particular feature of the present generalised acoustic model consists in dealing with a non-zero radial velocity component Ur in the duct reference frame Rd in order to improve the realistic flow modelling. From the mean velocity vector UðU xf ; U yf ; U zf Þ in Rf , the velocity components in the cascade reference frame Rc are, using Eqs. (2) and (3): 2 3 2 3 U xf U xc 6 7 6 7 (11) U ¼ 4 U yc 5 ¼ Q  Qf 4 U yf 5: U zf U zc Even if Ur ¼0, the swept vanes imply a non-zero value for Uzc in the cascade reference frame, a case already treated by Glegg [33] and Hanson [4]. From the aerodynamic radial wavenumber kr introduced in Section 2.2, it is assumed that the wavenumber along zf verifies the relation: kzf zf  krd r d :

(12)

From the azimuthal periodicity of the configuration, the aerodynamic wavenumber along θd can only be equal to an infinite amount of countable values. In the context of rotor–stator tonal noise, the azimuthal indices of the excitation are related to Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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the rotational direction such as kθd ¼ signðΩÞqB. From Section 2.1, yd and kθ d θ d ¼

θd are in the opposite directions thus:

kθ d signðΩÞqB r θ ¼ yd ¼ kyd yd ¼ kyf yf ; rd rd d d

(13)

since yf and yd are identical. From the dispersion relation verified by the wave vector K of the aerodynamic excitation:   K  U ¼ qBΩ; (14) where  denotes a scalar product, the wavenumber in the xf direction is:   1 qB 2π p ^ f U zf : qBjΩj þ signðΩÞ U yf  cos φ kxf ¼ U xf rd Tr Fig. 2 illustrates that signðΩÞU yf keeps the same sign for any value of Ω thus ensuring the independence of kxf relatively to the rotational direction. Finally the wave vector of the excitation in the cascade reference frame Rc can be found with: 2 3 2 3 kxf kxc 6 7 6 7 (15) K ¼ 4 kyc 5 ¼ Q  Qf 4 kyf 5: kzc

kzf

A crucial parameter for the cascade response is the inter-vane phase angle linking the responses of two consecutive vanes upon an interaction with a gust. Following [4], it is:

σ ¼ kxc d þkyc h;

(16)

with d and h related to the inter-vane distance g ¼ 2π r d =V, represented in Fig. 2, by: h ¼ g cos χ cos ψ ;

d ¼ gð sin χ cos φ  cos χ sin ψ sin φÞ:

(17)

Note that for Ur ¼ 0 and φ ¼ ψ ¼ 0, it can be shown that the inter-vane phase angle only depends on the harmonic index as σ ¼ signðΩÞ2π qB=V. 2.4. Cascade response function From the global configuration introduced in Section 2.1, the elementary problem, represented in Fig. 5 in Rc corresponds c q;p of wavenumbers ðkxc ; kyc ; kzc Þ convected with an inviscid flow of uniform velocity Uxc impinging to an aerodynamic gust W on the cascade. This rectilinear cascade is composed of infinitely thin flat plates with infinite span in the zc direction. In order to ensure the slip boundary condition on the plate surface, an unsteady pressure loading is created in response to the gust interaction. The cascade response function of Posson et al. [34] based on the previous work by Glegg [33] uses the Wiener–Hopf technique on four sub-problems to resolve the vane loading, with the following boundary conditions: continuity of velocity potential upstream of the leading edge, zero velocity normal to the plate (slip condition) and pressure continuity at the trailing edge (Kutta condition). A closed form expression for the acoustic field and for the vane loading is obtained providing that the non-overlapping distance d (see Eq. (17)) is smaller than the plate chord C, i.e. the vanes b 0 is presented in partially overlap over their whole span. The analytical expression of the elementary vane loading ΔP Appendix A.

Fig. 5. Rectilinear cascade in the Rc reference frame for two rotational directions.

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Moreover the resolution needs the non-overlapped length of the cascade d to be positive, that is the case of the configuration with Ω o0 in Fig. 5(a). In other words, s being the cascade gap, the cascade response function is valid for χ 40, with: d ¼ s sin χ :

(18)

However a simple symmetry consideration between both configurations of Fig. 5 leads to the following relation:

ΔPb 0 ðd; σ Þ ¼  ΔPb 0 ð  d;  σ Þ;

(19)

since the pressure and suction sides of a vane are inverted with respect to the yc direction as in Fig. 5(a) and (b). Eq. (19) allows the cascade response function to be used for any sign of the rotational velocity while jdj oC. It must be mentioned that the semi-numerical method employed for the resolution of the cascade response of Ventres et al. [35] makes possible any value of d. The implementation of this 2D model has allowed confirming the validity of Eq. (19) for 2D gusts in [48]. It has also highlighted the limitation of this semi-numerical cascade response as its accuracy depends of the amount of discretisation points, especially at high frequencies, contrary to the closed-form expression for the cascade response presented in Appendix A. The present analytical cascade response has been validated in a rectilinear configuration in [34] by successful comparisons with alternative models, e.g. the LINSUB model [49], the semi-numerical LINC code [22] and the three-dimensional linearised Euler code of Ataasi et al. [8]. The 3D analytical cascade response is used with a correction proposed in [50] in order to account for annular effects in an otherwise rectilinear problem. Indeed this correction has been shown to strongly reduce unphysical effects related to the strip theory approach applied to a varying geometry along the duct height [48]. Previous studies have also evaluated the vane thickness effect as well as the fluid viscosity effect on fan tonal noise generation and propagation [41]. Vane thickness has been found to modify the distribution of the acoustic energy across the radial modes within each azimuthal mode. Viscous effects are of secondary importance, providing neither vortex shedding nor any secondary flow with their own frequency close to tonal frequencies exists. Besides vane camber has been shown to be relevant for fan tonal noise prediction, and a simple method for considering vane camber effects has been proposed [51]. It is based on the coupling of two cascade response functions with different stagger angles and is not applied for the present study. b 0 ðxc ; zc ; σ ; kzc Þ created by the impingement of a gust on the cascade (Eq. (A.2)), the cascade From the pressure jump ΔP response at the qth harmonic of the BPF is written as:

ΔPb q ðxc ; zc Þ ¼

þ1 X p ¼ 1

c q;p ΔP b 0 ðxc ; zc ; σ ; kzc ðpÞÞeikr ðpÞrd : W

(20)

For the future integration on the vane, the pressure jump needs to be expressed on the chord at constant radius Cd, inclined of φ with the chord C in Rc . Assuming a small value for φ, the pressure jump is [50]:

ΔPb q ðxcd ; zc Þ ¼ ΔPb q ðxc ; zc Þeikzc

sin φxcd

;

(21)

with C d ¼ C= cos φ. In Eq. (20), the summation over the indices p of the radial wavenumber is actually truncated. Indeed for a given pulsation of the incoming gust qBjΩj, only a limited amount of diffracted cascade acoustic modes is cut-on and directly contributes to the cascade response [4]. In addition, because of the rectilinear to annular transformation in the acoustic analogy, some cut-off modes may also have an influence on the response, especially at low frequencies [48]. That is why the criterion on kzc defining the truncation of Eq. (20) corresponds to an extension of the expression given in [4]. It reads: 

qBjΩj qBjΩj  Δkzc okzc o þ Δkzc ; c0 β  U zc c0 β þU zc

(22)

with C Δkzc ¼ 20, and has been shown to be relevant for the present study. 2.5. In-duct acoustic analogy The acoustic analogy is based on the unsteady loading on the whole vane surface, analytically predicted by the cascade response function (Eq. (21)), that is used as dipole sources. Assuming that the stator is placed in an infinite annular duct with a constant section and rigid walls, containing a uniform axial mean flow (U xd ¼ c0 M xd ), Goldstein [20] gives the formulation of the acoustic pressure perceived by an observer located at point x ¼ ðxd ; r d ; θd Þ inside the duct at time t generated by the force f exerted at point x0 ¼ ðx0 ; r 0 ; θ0 Þ and time t0 by the vane surface Sðx0 Þ on the fluid: Z T ZZ ∂Gðx; tjx0 ; t 0 Þ pðx; t Þ ¼ f i ðx0 ; t 0 Þ dSðx0 Þ dt 0 ; (23) ∂x0i Sðx0 Þ T with T large but finite and the monopole and quadrupole terms being neglected. G is Green's function tailored to a rigid annular duct with a uniform axial mean flow thus verifying the boundary conditions on the hub and tip walls. G can be Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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written as: Gðx; tjx0 ; t Þ ¼

Z þ 1  iðγ m;7μ ðx  x0 Þ þ ωðt  t0 ÞÞ þ1 þ1 X Em;μ ðrÞEm;μ ðr 0 Þeimðθ  θ0 Þ i X e  dω; 4π m ¼  1 μ ¼ 0 Γ m;μ κ m;μ 1

(24)

with

Γ m;μ ¼ 2π

Z

RT

RH

E2m;μ ðrÞr dr:

(25)

The radial eigenfunction Em;μ , corresponding to the radial component of the solution of the Helmholtz equation, is a linear combination of Bessel functions [20]. This function Em;μ must satisfy the boundary conditions of the problem, written for a rigid duct as: ∂Em;μ ðrÞ ¼0 ∂r

for r ¼ RH and r ¼ RT :

(26)

The duct eigenvalues χ m;μ are the constants of the Bessel functions allowing the boundary conditions of Eq. (26) to be satisfied. The indices m and μ are associated with the azimuthal and radial mode orders respectively. The axial acoustic 7 wavenumber is γ m; μ , the superscripts þ and  indicating respectively the upstream and downstream propagations:  2   M xd ω=c0 7 κ m;μ ω 7 2 γ m; ¼ with κ ¼  1  M 2xd χ 2m;μ : μ m;μ 2 c0 1 M xd For both configurations in Fig. 5, n represents the unitary vector normal to the vane, directed from the pressure to the suction side, corresponding to n ¼  signðΩÞyc . Using the matrix Q (Eq. (3)), the coordinates of n in the duct reference frame are: 2 3 Q 21 6 7 (27) n ¼ signðΩÞQ  1  yc ¼  signðΩÞ4 Q 22 5: Q 23 The flow being inviscid, the force f in Eq. (23) is only composed of its component normal to the vane, i.e. the pressure distribution, thus: f ¼ f n ¼ ΔPn:

(28)

  ∂G ∂G 1 ∂G ∂G f i ¼ signðΩÞ Q 23 þ Q 22 þ Q 21 ΔP: ∂x0i ∂r 0 r 0 ∂θ 0 ∂x0

(29)

The integrand of Eq. (23) is then:

Using Green's function introduced in Eqs. (24) and (29) becomes:   ∂G ∂ m 7 f i ¼  signðΩÞi  iQ 23  Q 22 þ γ m; Q 21 GΔP: μ ∂x0i ∂r 0 r 0

(30)

Using Eq. (23) applied to tonal noise and dealing only with positive frequencies qBjΩj, the acoustic pressure within the duct for the harmonic q is expressed as: p 7 ðxd ; r d ; θd ; qBjΩjÞ ¼

þ1 X

þ1 X

m ¼ 1 μ ¼ 0

7  iqBjΩjt Em;μ ðr d Þeiðmθd  γ m;μ xd Þ P m; : μ ðqBjΩjÞe 7

(31)

The double infinite sum in Eq. (31) is limited to the acoustic duct modes (m,μ) excited by the rotor–stator interaction that verify the following relation, adapted from Tyler and Sofrin [52], providing a regular equispaced set of rotor blades and stator vanes: m ¼ signðΩÞqB þzV;

z A Z:

(32)

ΩjÞ is the modal pressure coefficient of the duct mode (m; μ) written as:

7 P m; μ ðqBj

7 P m; μ ðqBjΩjÞ ¼

signðΩÞV 2Γ m;μ κ m;μ

Z

RT

RH



 iQ 23

 7 ∂ m 7  Q 22 þ γ m; Q 21 Em;μ ðr 0 Þ  eiðm=r0 yLE ðr0 Þ þ γ m;μ xLE ðr0 ÞÞ IP m;μ dr 0 ; μ ∂r 0 r 0

(33)

^ and yLE ¼ ðr  RH Þ sin ψ^ are the coordinates of the vane leading edge. IP m;μ is the chordwise where xLE ¼ ðr  RH Þ sin φ integral along the xc direction of the pressure distribution in Eq. (21): Z xc;TE 7 dxc ΔPb q ðxc ; zc Þeiðkzc sin φ þ γ m;μ;xcd Þxc = cos φ ; (34) IP m;μ ¼ cos φ xc;LE Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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9

and the acoustic wavenumber in the xcd direction is: 7 7 γ m; μ;xcd ¼ γ m;μ cos χ þ

m sin χ : r0

(35)

It can be verified that the modal pressure coefficient in Eq. (33) keeps the same value for any rotational direction, noting that the sign of the azimuthal index m changes between both configurations of Fig. 2. Finally the acoustic power of the harmonic q reads after integration in a duct section [36]:

Π q7 ¼

þ1 X

þ1 X

m ¼ 1 μ ¼ 0

7 Π q;m; μ¼

2 þ1 X Γ m;μ ð1  Mxd Þ2 qBjΩjκ m;μ  7  2 P m;μ ðqBjΩjÞ : 2 ρ c  1 μ ¼ 0 0 0 qBjΩj=c0 7 M xd κ m;μ

þ1 X m¼

(36)

Among the excited duct modes (Eq. (32)), only the cut-on modes, i.e. verifying the relation κ 2m;μ 40, are considered in the acoustic power calculation of Eq. (36). Our model dedicated to the prediction of tonal fan noise has been fully described in this section. It has been shown that this model is a generalisation of the model of Posson et al. [34,39] for fan broadband noise prediction. As stated in the introduction, this model uses successive developments from Glegg [33] and Hanson [4]. Several original contributions have been introduced within the present model in order to increase the accuracy of the prediction. First an unified formulation of the whole set of equations has been proposed in order to account for both rotational directions of the fan. A radial decomposition for the aerodynamic excitation has also been introduced. It aims at accounting for the radial correlation existing between the excitation and the cascade response along the duct height in the tonal noise context. In order to consider a realistic flow, this model is able to consider the radial component of the mean flow velocity. Moreover the aerodynamic input data can be extracted, from experiments or numerical simulations, on a surface matching the radial shape of the vane leading edge. This ensures an accurate gust–cascade interaction modelling, especially for vanes with complex geometries.

3. Validation of the model on an actual fan configuration 3.1. Description of the Advanced Noise Control Fan The tonal fan noise prediction model detailed in Section 2 is evaluated on the Advanced Noise Control Fan (ANCF) configuration. This test bed has been operated at the NASA Glenn Research Center [53,54] and has supported a large amount of aeroacoustic research directed towards fan noise prediction, control and reduction [55]. The ANCF test case has been chosen as it is representative of an actual fan stage of a modern high bypass ratio turbofan. Indeed the rotor–stator stage represented in Fig. 6 is mounted in a 1.219 m (4 feet) diameter annular duct with a hub to tip ratio of 0.375. At the frequencies of interest, this ensures the propagation of azimuthal and radial acoustic duct modes similarly to the acoustic field present in a nacelle. As shown in Fig. 6, the large scale structures of the inlet flow are reduced by an inflow control device so that no additional tonal noise comes from ingested vortical structures [53]. The particularity of this test rig corresponds to the center body supporting the rotor without any structural element fixed to the outer casing. Thus the rotor alone and rotor–stator configurations can be investigated without any possible strut interactions in the inlet or exhaust part of the duct. This fan rig is placed in an anechoic test facility equipped with aerodynamic and acoustic measurement devices

Fig. 6. Advanced Noise Control Fan test bed, from [55].

Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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[56,57]. In particular two rotating rakes [58] allow capturing the modal content of the acoustic field generated by the fan stage, in a cylindrical section at the inlet and in an annular section at the exhaust, with a hub to tip ratio of 0.5 (see Fig. 6). This system provides the modal decomposition of the acoustic power used in Section 3.4 for the comparisons with the analytical predictions. The rotor is composed of 16 blades with a 0.133 m chord. The stator vane chord is C ¼ 0.114 m with an aspect ratio of C=ðRT  RH Þ ¼ 0.3. It is radially stacked with a 12° twist without any sweep nor lean angle. In this highly configurable fan stage, the stator vane count V can vary between 13 and 30 in keeping the same vane geometry. Thus the solidity changes from CV =2π RT ¼ 0.39–0.84 at the tip. Even with these relatively low solidities, the vane stagger angle close to 0° at the tip ensures that cascade effects are present since the overlapping length is 84 percent of chord with V ¼ 13. The inter-blade row spacing, measured at the hub, can vary between 1/2 and 2 vane chords from the rotor blade trailing edge. One of the stator vanes is instrumented with flush-mounted microphones on each side of the vane [43], as schematically represented in Fig. 7. This allows capturing the unsteady vane loading, partly corresponding to the tonal acoustic sources, that are compared with analytical predictions in Section 3.3. At nominal conditions the rotational velocity is 1800 RPM, giving a blade passing frequency (BPF) of 480 Hz, a peripheral tip Mach number around 0.34 and an axial Mach number around 0.15. The mass flow rate is 56 kg s  1 and the Reynolds number based on the blade chord is around 4  105. 3.2. Reference configurations Among the investigations performed in the ANCF test rig, a particular attention is paid to the work of Sutliff et al. [42,43]. Indeed these references contain the measured vane unsteady loading and modal acoustic powers of the tonal noise generated by the rotor–stator interaction, for several configurations listed in Table 1. These allow performing trend studies with the stator vane count and the rotational speed. The cut-on and excited duct modes at 1 BPF and 2 BPF in these configurations are reported in Table 1. For all these cases, the inter-row spacing is fixed at half a vane chord. Moreover Sutliff et al. [42] compare the modal powers with the predictions of V072 code, corresponding to the cascade based model of Ventres et al. [35] updated by Meyer and Envia [36]. As already mentioned in the introduction and in Section 2.4, this model is close to the present evaluated model, the main differences being the semi-numerical resolution of the 2D cascade response instead of a closed-form expression of the 3D cascade response for the current model. Note that both models use the same acoustic analogy of Goldstein [20] in an annular duct for the acoustic propagation. In Sections 3.4 and 3.4, the acoustic model introduced in Section 2 is evaluated on the ANCF configurations listed in Table 1 against experimental results and V072 analytical predictions. The aerodynamic input data needed for these predictions are also provided in [42] and consist of crossed hot-wire measurements in the plane of the stator vane leading edges, the stator being removed. Thus the convection effects on the rotor wakes are correctly taken into account, but the influence of the upstream potential effect of the stator on these wakes is absent. This potential effect may significantly modify the shape of the rotor wake, as studied by de Laborderie et al. [59] for instance. Therefore, the vortical excitation used in the models does not exactly correspond to the actual excitation impinging on the vanes. The X-shape of the probe,

Fig. 7. Locations of the 39 microphones on one side of the instrumented vane in the stator configuration with 14 vanes, from [43]. The other side of the vane is equipped with the same pattern.

Table 1 Excited and cut-on acoustic duct modes in the ANCF configurations, from [42]. Stator configuration

Frequency

1700 RPM

1750 RPM

1800 RPM

1850 RPM

13 vanes

1 BPF 2 BPF 1 BPF 2 BPF 1 BPF 2 BPF 1 BPF 2 BPF

(3,0) (  7,0) (6,0) (2,0) (4,0) (4,1) – (6,0) – (4,0) (4,1)

(3,0) (  7,0) (6,0) (2,0) (4,0) (4,1) – (6,0) – (4,0) (4,1)

(3,0) (  7,0) (6,0) (2,0) (4,0) (4,1) – (6,0) – (4,0) (4,1)

(3,0) (  7,0) (6,0) (2,0) (4,0) (4,1) – (6,0) – (4,0) (4,1)

14 vanes 26 vanes 28 vanes

Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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11

w [m/s] 3

1

2 0.9 1 0

r/R [−] T

0.8

−1 0.7 −2 −3

0.6

−4 0.5 −5 −6

0.4 −0.1

0 r θ/RT [−]

0.1

Fig. 8. Upwash velocity measured by a cross-hotwire probe in the vane leading edge plane, the stator being removed, over one blade passage, at 1850 RPM, from [42].

placed in a plane perpendicular to the radial direction, provides the axial and tangential velocity components. The experimental uncertainty is around 0.6 m/s on the velocities and 0.2° on the angles [42]. From these two-component hotwire data, the upwash is determined as the velocity component normal to the mass-averaged absolute flow angle [42]. In order to perform relevant comparisons, the upwash over one blade passage and the mean velocity profiles provided in [42] are used as input data for the acoustic predictions with the present cascade model. Although the velocity profiles are available for the four rotational speeds of Table 1, the upwash in the stator leading edge plane is only given at 1850 RPM. It is represented in Fig. 8, where the velocity deficit, indicating the rotor wake, is decreasing with the increasing radius. According to Sutliff et al. [42], the upwash for the other rotational speeds are similar to Fig. 8. However it means that the excitations used for the V072 predictions are not identical to the single one used with the present model. It has to be mentioned that the mean flow properties (temperature and density), not available in [42], are coming from a more recent measurement campaign [56]. The equivalent stagger angle χ of the flat plate cascade used for the V072 predictions, corresponding to a weighted average of the inlet and outlet vane metal angles, is given in the reference paper [42]. 3.3. Unsteady vane loading As described in Section 3.1, one of the ANCF stator vanes is instrumented with embedded microphones, shown in Fig. 7, in order to locally measure the unsteady vane loading. The latter is known to represent the sources of rotor–stator interaction tonal noise, as mathematically expressed in Eqs. (31)–(36). The measurements correspond to the amplitudes (in terms of sound pressure levels (SPL)) and phases of the wall pressure on each side of a vane, for all the cases reported in Table 1. These data are provided in [43]. However, for the sake of brevity, only a single case is analysed in this section, corresponding to the stator with 14 vanes and Ω ¼ 1800 RPM. Indeed for this configuration, the amount of microphones is larger than for the other cases since the sensors line at 85 percent of chord in Fig. 7 is only installed in this case. These experimental data allow evaluating the cascade response function of the acoustic model introduced in Section 2.4, in its 2D and 3D versions. To this end, the measured wall pressures are first transformed into pressure jumps across the vane, knowing that the microphones are mounted by pair at the same chordwise and spanwise locations on each side of the vane. Unlike the acoustic power comparisons performed in Section 3.4, no analytical results from the V072 cascade response function are available preventing any comparisons between both acoustic models. Fig. 9 represents the unsteady loading at 1 BPF along the vane chord at the three spanwise positions shown in Fig. 7, measured by the microphones and predicted by the 2D and 3D versions of the cascade response function. From the vane leading edge to around 60 percent of chord, the SPL shapes are in good agreement all over the span, with a higher loading level in the leading edge region followed by a smooth decrease. This is consistent with the vane interaction noise theory (e.g. [13]) predicting that the main acoustic sources are generated in the leading edge region of the profile where the incoming gust is diffracted. In Fig. 9(a), the 2D response gives a prediction closer to the experimental data than the 3D response, whereas both produce equivalent results in Fig. 9(b) and (c). Nevertheless the measured pressure jumps present a sudden Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

J. de Laborderie, S. Moreau / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

160

160

140

140

140

120 100

Exp. 2D Model 3D Model

80 60

0

0.2

SPL [dB]

160

SPL [dB]

SPL [dB]

12

120 100 80

0.4 0.6 x /C [−]

0.8

60

1

100 80

0

0.2

c

0.4 0.6 x /C [−]

0.8

60

1

0

200

100

100

100

Exp. 2D Model 3D Model

−200

0

0.2

0.4 0.6 x /C [−]

0.8

1

Phase [deg]

200

0

0 −100 −200

0

0.2

0.4 0.6 x /C [−]

c

0.4 0.6 x /C [−]

0.8

1

0.4 0.6 x /C [−]

0.8

1

c

200

−100

0.2

c

Phase [deg]

Phase [deg]

120

0.8

1

0 −100 −200

0

0.2

c

c

Fig. 9. ANCF, stator with 14 vanes, 1800 RPM. Pressure jumps along the vane chord at 1 BPF. Black squares: experimental results from [43]. Grey lines: analytical predictions from the acoustic model presented in Section 2.

0.7

1

1

0.9

0.9

0.9

0.8

0.8

0.8

0.7

r/RT [−]

r/RT [−]

0.8

Exp. 2D Model 3D Model r/RT [−]

0.9

1

r/RT [−]

1

0.7

0.7

0.6

0.6

0.6

0.6

0.5

0.5

0.5

0.5

0.4 80

100 120 SPL [dB]

140

0.4 −200

0 Phase [deg]

200

0.4 80

100 120 SPL [dB]

140

0.4 −200

0 Phase [deg]

200

Fig. 10. ANCF, stator with 14 vanes, 1800 RPM. Pressure jumps along the vane span at 1 BPF. Black squares: experimental results from [43]. Grey lines: analytical predictions from the acoustic model presented in Section 2.

SPL rise starting at around 70 percent of chord in the upper part of the vane (at 0.74RT and 0.91RT). This behaviour is not predicted by the model for which the loading monotonically decreases until the trailing edge. Referring to the original data in [43], it is shown that this local SPL increase only occurs on the vane suction side. Indeed as mentioned in [42], the mean flow incomes on the vane with a relatively high incidence thus creating a flow separation on the downstream part of the vane suction side that is responsible for this high level of pressure fluctuations at these locations. As the model assumes a zero incidence mean flow, this flow separation mechanism cannot be predicted analytically. The flow separation is also responsible for the high level of phase fluctuations observed in Figs. 9(e) and (f). In addition, the discrepancy between the measured and predicted loading phases (Figs. 9(d)–(f)) is partly caused by the stator potential effect not present in the upwash used for the predictions (see Section 3.2). This potential effect modifies the phase of the excitation, as shown in [59], thus directly impacts the phase of the vane response. Figs. 10(a) and (b) compare the loadings in the spanwise direction in the leading edge region (20 percent of chord). The levels match fairly well between the measurements and the predictions in terms of SPL. The phases are also in correct agreement, except for a constant shift existing between 0.5RT and the vane tip, again caused by the stator potential effect on the wake. The global shape of the measured phase, decreasing from hub to tip, indicates that the rotor wake interacts with the vane bottom part before the vane tip. At 85 percent of chord (Figs. 10(c) and (d)), the level of the predicted loading is Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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Fig. 11. ANCF, stator with 14 vanes, 1800 RPM. Pressure jumps along the vane chord at 2 BPF. Black squares: experimental results from [43]. Grey lines: analytical predictions from the acoustic model presented in Section 2.

Fig. 12. ANCF, stator with 14 vanes, 1800 RPM. Pressure jumps along the vane span at 2 BPF. Black squares: experimental results from [43]. Grey lines: analytical predictions from the acoustic model presented in Section 2.

well below the experimental data in the upper part of the vane, confirming the previous observation related to the flow separation on the vane suction side in the trailing edge region. The vane loadings at 2 BPF are compared in Figs. 11 and 12. Similarly to the case at 1 BPF, the SPL values are higher in the leading edge region than in the rest of the chord, a behaviour correctly predicted by the model. The agreement between the experimental data and the 3D cascade response are very satisfactory at 0.74 and 0.91RT. The flow separation is again visible in the SPL data in Figs. 11(b) and (c) from 70 percent of chord, but the level rises are less pronounced that at 1 BPF. The measured and analytical loading phases (Figs. 11(d)–(f)) are significantly different, especially in the upper part of the vane where the large variation of the phases indicates a complex flow behaviour not predicted by the model. Yet some of the phase shifts in the measurements are also questionable and suggest larger experimental uncertainties on the phase at this frequency. At 0.49RT (Fig. 11(a)), the 3D model under-predicts the loading by about 10 dB and the 2D model by a larger value. This is also visible in Fig. 12(a) representing the loading evolution in the spanwise direction. Sharp deficit peaks appear in the analytical responses around 0.49RT and 0.83RT, whereas the measured and predicted vane loading compare fairly well on the rest of the vane height. Sudden phase changes are present at the same radial locations in Figs. 12(b) and (d). These analytical discontinuities originate from the coupling of the strip theory with a cascade response function, as already discussed in [33,60] for instance. Indeed the variation of the rectilinear cascade geometry at each strip makes possible diffracted cascade acoustic modes and inter-vane channels modes to become cut-on at specific spanwise locations. Following a Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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120

120

115

115

110 Exp. V072 2D Model 3D Model

105

100 1600

1700

1800 Ω [RPM]

110

100 1600

2000

95

85

1900

2000

2000

1700

1800 Ω [RPM]

1900

2000

95

85

1800 Ω [RPM]

1900

100

90

1700

1800 Ω [RPM]

105

90

80 1600

1700

110

SWL [dB]

100

110

105

Exp. V072 2D Model 3D Model

105 SWL [dB]

1900

SWL [dB]

SWL [dB]

14

80 1600

Fig. 13. ANCF, stator with 13 vanes. Black lines: experimental and analytical results from [42]. Grey lines: analytical predictions from the acoustic model presented in Section 2.

procedure similar to [48], it can be shown that the discontinuity at 0.49RT is caused by a cascade acoustic mode, generated by the interaction of the gust (q ¼ 2, kzc ¼ 0) with the cascade, becoming cut-on at this location. At 0.83RT, the inter-vane channel mode, created by a zero radial wavenumber gust, becomes cut-on at this position for 2 BPF and is responsible for the loading discontinuity. These two discontinuities, present over the whole chord, are also visible at 85 percent of chord in Fig. 12(c). Note that a cascade acoustic mode is also becoming cut-on near the vane tip in Fig. 10 explaining the differences between the 2D and 3D responses at this location. On the rest of the vane span, the predicted levels are lower than the experimental data, a feature again caused by the flow separation taking place on a large part of the vane suction side. As previously noted, the experimental phase data present large uncertainties at this frequency limiting the comparisons in Figs. 12(b) and (d). The comparisons performed in this section show that the differences between the 2D and the 3D cascade responses increase with the frequency of the excitation. Indeed, a single gust of zero radial wavenumber contributes to the 2D response, whereas the amount of skewed gusts contributing to the 3D response rises with the frequency. This 3D behaviour improves the modelling accuracy relatively to the actual interaction [48]. Moreover, the local discontinuities created by the coupling of the strip theory with a cascade response function are smaller with the 3D model than with the 2D model, leading to a more accurate prediction with the 3D model, e.g. in Figs. 11(a) and (c). Thus the analysis of the predicted unsteady vane loading with respect to wall pressure measurements allows reasonably validating the 3D cascade response function in an actual fan configuration. The main discrepancies encountered in the comparisons have been explained, as they result from the measured upwash not accounting for the stator potential effect as well as a flow separation region on the upper part of the vane suction side. 3.4. Modal acoustic power Using the measured aerodynamic data of Sutliff et al. [42], among which the upwash shown in Fig. 8, the cascade model detailed in Section 2 is applied to the whole set of configurations listed in Table 1. For a complete evaluation, both the 3D and the 2D version of the model, i.e. with the decomposition of Eq. (7), are considered. In this section, the sound power level (SWL) of Eq. (36) is presented for each harmonic of the BPF and for the upstream and downstream propagations. In Figs. 13–16, these Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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120

115 SWL [dB]

SWL [dB]

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Exp. V072 2D Model 3D Model

115

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100 1600

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105

1700

1800

1900

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100 Exp. V072 2D Model 3D Model

95

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Ω [RPM]

1900

SWL [dB]

SWL [dB]

Ω [RPM]

90 1600

15

100

95

90 1600

2000

1700

Ω [RPM]

1800 Ω [RPM]

Fig. 14. ANCF, stator with 14 vanes. Black lines: experimental and analytical results from [42]. Grey lines: analytical predictions from the acoustic model presented in Section 2.

120 115 110

115 SWL [dB]

SWL [dB]

120

Exp. V072 2D Model 3D Model

105

110 105

100

100

95

95

90 1600

1700

1800 Ω [RPM]

1900

2000

90 1600

1700

1800 Ω [RPM]

1900

2000

Fig. 15. ANCF, stator with 26 vanes. Black lines: experimental and analytical results from [42]. Grey lines: analytical predictions from the acoustic model presented in Section 2.

analytical predictions are compared with experimental power values obtained from the rotating rakes installed in the inlet and exhaust part of the duct (see Fig. 6) as well as with the predictions of the 2D cascade model V072 reported in [42]. In the stator configuration with 13 vanes, only the duct mode (3,0) is excited and cut-on at 1 BPF (Table 1). For this case, Figs. 13(a) and (b) show an excellent agreement between the analytical predictions and the experimental values, for the upstream and downstream propagations respectively. Indeed the results of the model are within 2.4 dB from the measured values upstream. In Fig. 13(a) the 3D model predicts the power with a gap of 0.4 dB at 1800 RPM, the V072 value being at Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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16

120

115 SWL [dB]

SWL [dB]

115

110

105

100 1600

120

Exp. V072 2D Model 3D Model

110

105

1700

1800 Ω [RPM]

1900

2000

100 1600

1700

1800 Ω [RPM]

1900

2000

Fig. 16. ANCF, stator with 28 vanes. Black lines: experimental and analytical results from [42]. Grey lines: analytical predictions from the acoustic model presented in Section 2.

0.7 dB. Downstream (Fig. 13(b)), the 3D model results vary between 1 and 1.8 dB with respect to the V072 predictions. For both upstream and downstream propagations, the experimental trend shows an increase of sound power with the rotational velocity, whereas all analytical predictions appear relatively flat. The difference between upstream and downstream powers is more pronounced experimentally than with the predictions. At 2 BPF, the modes ( 7,0) and (6,0) propagate within the duct. The sum of their SWL is plotted upstream and downstream in Figs. 13(c) and (d) respectively. The present cascade model provides a relatively poor agreement with the V072 prediction, between 7 and 11 dB upstream and around 7 dB downstream. If the 3D model and V072 results present a similar trend with Ω, they both are far from the experimental trends. The latter show large and non-monotonic SWL variations with the rotor speed, that are partly explained in [42] as an acoustic reflection in the exhaust section where the rotating rake is mounted. It must also be mentioned that for all the comparisons performed in this section, the analytical results are obtained from an acoustic analogy in a constant section annular duct with RH =RT ¼ 0:375, whereas the measurements at the inlet and exhaust rotating rakes take place in sections with RH =RT ¼ 0 and 0.5 respectively. Thus reflections and standing waves may appear in the actual duct geometry thus explaining some differences with the analytical results. In Figs. 14(a) and (b), corresponding to the stator with 14 vanes at 1 BPF, only the duct mode (2,0) is present. For both the upstream and downstream propagations, the 2D and 3D versions of the model are similar and very close to the V072 predictions (around 0.5 dB). The analytical results are in fair agreement with the experimental data upstream since the gap is about 2 dB, whereas this difference is larger downstream (between 7 and 8 dB). It is interesting to note that the predicted trend with Ω is now similar to the measured one. At 2 BPF, a single azimuthal mode is created, with two radial components (4,0) and (4,1). The upstream and downstream predictions given by the 3D model (triangles in Figs. 14(c) and (d)) are in correct agreement with the measured data since the differences are within 2 dB at 1700 and 1750 RPM and 3 dB at 1800 and 1850 RPM (except at 1850 RPM downstream where the experimental value shows an odd sharp drop). Therefore the 3D model provides better results upstream, and worse downstream, than the V072 code. With the 26 vane stator, all the duct modes are cut-off at 1 BPF. Figs. 15(a) and (b) show the acoustic power at 2 BPF, corresponding to the single (6,0) mode. Again the evolution of the experimental data with the rotational speed is significative, with an increase followed by a strong decrease of the levels upstream and by a slight decrease downstream. No acoustic model is able to predict this measured shape, partly influenced by the actual duct acoustic propagation as stated above. Moreover, according to [42], the main reason for this particular experimental trend was not definitely found, but could result from mode trapping and scattering. Only for this configuration, it is remarkable to note that the 2D and 3D models clearly present opposite trends. Both provide results below the V072 predictions, between 7.5 and 0.5 dB upstream and around 5 dB downstream for the 3D model. Again no duct mode is cut-on at 1 BPF in the 28 vane stator configuration. At 2BPF, a single azimuthal mode with two radial components (4,0) and (4,1) propagates. For the upstream propagation (Fig. 16(a)), it is shown that the analytical predictions are similar since they are all comprised within 1 dB. They are very close to the measured power, at least for the available data. Downstream (Fig. 16(b)), the evaluated 3D model provides results closer to the experimental data than the other 2D models and under-predicts the measurements by 0.4–1.6 dB. In summary the present comparisons of tonal acoustic powers have allowed the evaluation of the cascade model introduced in Section 2 relatively to measured data and analytical predictions of an existing code. As explained in Section 3.2, the V072 code and the 2D version of the model should be similar since they both use a 2D cascade response function coupled with an acoustic analogy within an annular duct. Figs. 14(a), (b) and Figs. 16(a), (b) indeed show very close results between these two codes. However in some cases, e.g. in Figs. 13(a), (b) and Figs. 14(c), (d), the 2D results present a similar trend but are separated by a shift. For the other cases, the 2D predictions are relatively different. From this analysis, it may be inferred that the input data used for V072 and the 2D models are not exactly equivalent, especially concerning the mean flow values that Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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are not provided in [42] and come from a more recent study [56]. Therefore, this shows that these acoustic models are particularly sensitive to the input data. As described in Section 2, the 3D cascade model has been designed to improve the fan noise prediction. In half of the studied cases, the 3D model provides better or equivalent (less than 1 dB difference) results than the V072 code relatively to the experimental data. Moreover, the cases where the 3D model gives worse than expected results correspond to the configurations for which the measured powers are highly influenced by a complex acoustic duct propagation not accounted for in the model (Figs. 13(c), (d) and Figs. 15(a), (b)). This phenomenon is related to the variations of the duct section [42]. Therefore, following the assessment of the 3D cascade response in Section 3.3, a reasonable validation of the 3D model is provided. It is shown to accurately predict tonal noise created by the rotor–stator interaction, in the limits of the assumptions that mainly consist of an acoustic propagation within a constant section annular duct and of the absence of the acoustic transmission and reflection through the fan. The influence of the stator potential effect on the rotor wake, seen in Section 3.3, must also be added to the sources of discrepancies with the measured acoustic powers. Moreover, except for one stator configuration, the trend with the vane count and the rotational speed is predicted by the 3D model. It should also be emphasised that this is the first detailed comparison of this analytical model for high-speed rotor–stator interaction noise with detailed source and noise measurements.

4. Application of the model to an axial compressor stage 4.1. Description of the CME2 compressor stage Following the validation of the tonal noise prediction model with an actual fan configuration, an axial compressor stage setup is now chosen in order to illustrate the range of applications of this model. The CME2 low pressure research compressor (Laboratoire de Mecanique de Lille – Snecma Moteurs, France) is composed of a 30-blade rotor upstream of a 40vane stator [44]. It is mounted in a convergent annular duct with a hub-to-tip ratio of RH =RT ¼ 0:77 at the leading edge of the stator vanes and an outer casing of 0.55 m constant diameter (Fig. 17). The vanes have a uniform aspect ratio of C=ðRT  RH Þ ¼ 1:25 and are stacked without any sweep nor lean angles. Their stagger angle varies from 17.5° at the hub to 12.5° at the tip. The solidity at midspan of the stator being CV =2π RT ¼ 2:01, cascade effects on the noise sources are expected to be significant, thus justifying the use of the present acoustic model. The gap between the rotor and the stator varies from 17 percent at the hub to 29 percent of the vane chord at the tip. The rotational speed is 6300 RPM at nominal conditions, implying a BPF of 3150 Hz. At this operating point, the mass flow rate is 11 kg s  1 and the total pressure ratio of the stage is 1.14. The compressor operates in a fully subsonic regime with a 0.33 axial Mach number at the inlet. A detailed unsteady compressible flow simulation of this configuration has recently been performed and validated in [59] with available experimental data from [44]. The Navier–Stokes solver Turb'Flow, developed at Ecole Centrale de Lyon in France, was chosen for this simulation as it is specifically designed for turbomachinery applications. This solver is based on a finite volume formulation and solves the conservative equations on a multiblock structured grid [61]. This URANS wallresolved simulation has been performed on one tenth of the compressor, the computational domain being composed of three rotor blades and four stator vanes. The flowfield is transferred from the rotating grid to the fixed grid through an

Fig. 17. CME2 research compressor (unshrouded).

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Fig. 18. Instantaneous axial Mach number field at midspan of the CME2 compressor. w [m/s]

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interface using a high-order Fourier decomposition and recomposition with adequate angular phase in the azimuthal direction. The conservative flow variables are spatially discretised with the second-order Jameson centered scheme. Kok's k–ω turbulence model [62] is used along with a limiter for the production of turbulent kinetic energy. The time marching is performed with an explicit single-time-stepping method based on a second order Runge–Kutta scheme using up to five steps. As an illustration of the flow topology, Fig. 18 presents an instantaneous axial Mach number field at midpsan of the compressor. The boundary layers on both sides of the rotor blades merge at the trailing edges to form the wake. The latter are convected with the mean flow and interact with the stator vanes. This unsteady flow simulation has two main purposes in this aeroacoustic context. On the one hand, it provides the aerodynamic excitation w for the model. As shown in [59], the upwash from a rotor–stator simulation has to be extracted as close as possible to the vane leading edge (around 1 percent of chord upstream of the vane) during a complete cycle, and averaged in the rotor reference frame, in order to account for the convection by the flow and the stator blockage effects on the wake. This excitation is represented in Fig. 19 over a rotor blade passage. On the other hand, the unsteady wall pressure fluctuations, partly corresponding to the tonal acoustic sources, can be recorded on every gridpoint of the vane surfaces. These data have been used in a CAA method developed in [48] that has allowed to perform preliminary assessments of the present acoustic model. 4.2. Parametric studies As stated in the introduction, the acoustic model is particularly suited to perform parametric studies on complex geometries, contrary to a CAA approach. Thus it perfectly meets the requirements of a pre-design tool in an industrial context. In order to demonstrate these features, a parametric study is performed on the CME2 compressor. From the original configuration introduced in Section 4.1, the sweep and lean angles of the stator vanes are progressively modified. The influence of these geometrical variations can then be directly surveyed on the aerodynamic excitation, the acoustic sources and the acoustic power prediction. The effect of swept stator vanes in rotor–stator tonal noise generation has been known from early studies, e.g. by Adamczyk [63] and Hayden et al. [64], since this geometrical modification may lead to noise reduction with respect to radially stacked Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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Fig. 20. Upwash averaged in the rotor reference frame and extrapolated from the URANS flow field until the stator leading edge plane with φ^ ¼  201.

vanes. More recent works [33,65–67] give some physical insights into the mechanism related to this noise reduction. The rotor–stator interaction is highly dependent of the orientation of the rotor wakes with respect to the vanes leading edges. An increase of the rotor–stator spacing enhances the bending of the wakes by the mean flow that is swirling within this gap. In order to obtain a similar wake–vane interactions for UHBR engines with short nacelles, that present a reduced fan-OGV spacing, swept and leaned stator vanes have to be introduced [66]. Sweeping the vanes in the backward direction, i.e. with φ^ o 0, reduces the mean flow velocity component along the chord Uxc and extends the distance between the upstream rotor and the stator. Consequently the aerodynamic excitation is damped on the additional convection distance up to the vane leading edge, thus creating a lower vane response. Furthermore the main effect of sweeping stator vanes consists of large phase variations of the excitation along the vane leading edge direction. Therefore the vane submitted to this excitation also presents a significant phase variation along the direction zc. This phase evolution from strips to strips leads to a lower integrated response on the vane compared to a radial vane because of possible cancellation of the source terms, thus to a reduction of the emitted tonal noise. Coupling sweep and lean angles of a vane can enhance the noise reduction efficiency. Envia and Nallasamy [68] propose some guidelines for designing quieter stators: imposing backward sweep angles and lean angles in the rotational direction. These design rules were shown to be efficient on an actual fan stage [69]. 4.2.1. Sweep effects ^ has The first surveyed parameter corresponds to the sweep angle of the CME2 stator vane. According to [68] and Fig. 3, φ to be negative to obtain a tonal noise reduction with respect to the original stator geometry. Although the model is able to handle variable sweep angle along the duct height, it is chosen to keep this angle uniform in the present study for each case. The sweep angle varies between 0° and  30° with a 5° step, i.e. a typical range for modern compressors and fans [70]. The aerodynamic excitation, extracted in the original configuration and represented in Fig. 19, has to be extrapolated until the leading edge plane of the swept stator for each case. This is performed using a linear extrapolation from the simulated ^ . For instance, Fig. 20 represents the upwash velocity extrapolated until the flowfield in the inter-row gap, for each value of φ ^ ¼  201. By comparisons with the upwash extracted on a radial plane in leading edge plane of the stator swept with φ Fig. 19, it can be seen that the main sweep effects on the excitation consist of a decrease of the upwash amplitude, a more pronounced bending of the wake and a larger wake width. These effects are more significant going from hub to tip since the ^ j in the axial direction from the radial plane. This additional distance is leading edge plane is located at Δx ¼ ðr d  RH Þ tan jφ ^ ¼ 201. It is particularly interesting to note the variation of the significant since it corresponds to Δx ¼ 0:3 C at the tip for φ rotor wake shape with the orientation of the stator leading edge plane since this directly influences the wake–stator interaction as shown later. The bending of the wake seen by the swept vane tends to increase the amount of wake intersections per vane. This design objective has to be followed to ensure noise reduction according to [68]. For a quantitative analysis of the upwash evolution with the sweep angle, Fig. 21 presents the Fourier coefficients (Eq. (8)) of the excitation for 1 and 2 BPF along the dimensionless coordinate zc ¼ ðzc zcH Þ=ðzcT  zcH Þ. Concerning the amplitudes of these coefficients, Figs. 21(a) and (c) clearly show the expected decrease of the excitation with the increase of the sweep angle (in absolute value). In this configuration, it is observed that the decrease of the amplitudes is quasi-linear with the ^ . This decrease is significant since the amplitudes are globally divided by a factor of two for a 15° variation of variation of φ the sweep angle. Figs. 21(b) and (d) show the phase of these coefficients along zc . As expected, the phase variations are ^ ¼ 301. Again this feature is more significant in the upper part of the duct than in ^ ¼ 01 to φ increasing when going from φ the lower part. Following the sweep effects on the aerodynamic excitation, the cascade response of the model (Section 2.4) is surveyed ^ . Fig. 22 presents the cascade response along zc , at 15 percent of chord from the leading edge, in relatively to variations of φ Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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terms of amplitudes. For 1 BPF, the behaviour of the response is similar to the one of the upwash, i.e. a quasi-linear decrease ^ j. The model is thus able to provide quantitative trends of the cascade response with the in amplitude with the increase of jφ ^ j also leads to a decrease of the cascade response amplitude, parametric variation of the sweep angle. For 2 BPF, increasing jφ ^ ¼ 01), the cascade response exhibits a sharp nevertheless following a different pattern. Indeed, for the original vane (φ pressure peak at zc ¼ 0:87. As already encountered in Section 3.3, this feature is caused by an inter-vane channel mode becoming cut-on at this location, upon the impingement of the gust with jpj ¼ 1 [48]. This phenomenon is directly linked to ^ , both the the local geometry of the equivalent rectilinear cascade and to the value of the incident velocity Uxc. By varying φ geometry and the mean flow are changed. Therefore the cut-off frequencies of inter-vane channel modes are different for ^ ¼ 01 to each sweep angle value. In Fig. 22(b), the pressure peak is indeed shifted towards the duct mid-height from φ φ^ ¼  201, indicating that the corresponding inter-vane channel mode becomes cut-on at lower radii when increasing jφ^ j. This feature is coupled to a global decrease of the cascade response with the variation of the sweep angle, thus leading to the oscillatory shapes of the model response observed in Fig. 22(b). Fig. 23 represents the phase of the cascade response over the vane surface for specific values of the sweep angle at 1 and 2 BPF. For the original vane geometry, the phase of the loading is quite radially uniform at 1 BPF, and already presents some Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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radial variations at 2 BPF. For these two harmonics, the phase variations significantly increase in the zc direction for larger ^ . This feature is caused by the phase variations of the excitation highlighted in Figs. 21(b) and (d). As a result, the values of φ cascade response is progressively decorrelated in the radial direction. This is known to be the main effect expected with swept stator vanes that leads to possible cancellations when the response is integrated over the vane [68]. Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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Fig. 24. Total acoustic power for each harmonic of the BPF.

The parametric study on the vane sweep angle is finally performed in terms of radiated sound power level (SWL) within the duct. Fig. 24 shows the values predicted by the model for each harmonic of the BPF previously investigated in terms of excitation and cascade response. The excited and radiated acoustic duct modes have been summed for each harmonic. As ^ j. Indeed, the expected from the previous analysis, the predicted acoustic power globally decreases with an increase of jφ combined effects of reducing the amplitude of the upwash and decorrelating the excitation lead to lower values of the emitted sound power. This study emphasises the efficiency of swept vanes to provide a tonal noise reduction, since a value of 5 dB reduction can be expected on the acoustic power at 1 BPF for a 10° increase of the sweep angle. As shown in Fig. 24 ^ j. The efficiency of (b), the slope is higher at 2 BPF for which a reduction of 10 dB can be found with a 10° increase of jφ acoustic power reduction thus increases with the tone harmonics. Overall the predicted evolution of acoustic power with the vane sweep angle, as well as the amount of sound reduction obtained, are in good agreement with reported results from tests or models [65,68,70]. The sweep angle variation similarly affects the upstream and the downstream radiations, as shown by the slopes in Figs. 24(a) and (b). However a non-monotonic behaviour of the acoustic power is found for high ^ j, as already encountered in [68] for instance. For 1 BPF, the acoustic power radiated downstream monotonically values of jφ varies with the sweep angle, whereas this evolution is non-monotonic upstream. Thus the upstream propagation is more sensitive to the axial location of the sources than the downstream propagation, a feature already mentioned in [65]. Note that this parametric study has been performed in using an extrapolation to define a realistic rotor wake in function of the sweep angle. Within the limits of the error associated with this extrapolation, imposing the constraints of minimising the sound emission for each tone in each direction would lead to the optimal design of a stator composed of vanes with a 20° sweep angle. 4.2.2. Sweep and lean effects Following the above parametric study on vane sweep angle only, the objective now consists of coupling lean and sweep angles. Indeed a proper combination of these design parameters may lead to an even more acoustically efficient rotor–stator stage. According to Envia and Nallasamy [68], the lean angle ψ^ must be oriented in the rotational direction to provide a noise reduction, i.e. with a negative value here (see Fig. 4). Indeed with this configuration, the amount of wake intersections per vane is increased. Due to this negative lean, the vane leading edge will see the rotor wake with a time delay. In other words, a phase shift is introduced in the upwash excitation to account for this delay, such as the upwash w extracted over an inclined plane becomes: ~ ¼ we  ikxc rd ψ^ sin χ ; w

(37)

where r d ψ^ represents the tangential distance at radius rd caused by the lean angle with respect to a straight vane [68]. Therefore the phase variation of the excitation is enhanced, and so should be the phase of the vane response. It can be thus expected to obtain lower values for the integrated vane loadings due to cancellations of the response caused by this highly varying phase response. In order to evaluate this effect on the CME2 compressor configuration, the lean angle of the vanes varies between 0° and  20°, for each of the swept stator configuration already surveyed. For the sake of brevity, the upstream and downstream radiated acoustic powers are directly presented in Fig. 25. Fig. 25 shows that, for each value of the lean angle, the evolution of the sound power with the sweep angle corresponds to a global decrease. For each case, the slopes of the curves are similar between each other and do not seem to depend on a specific lean value. At 2 BPF, Figs. 25(c) and (d) clearly show the expected influence of the lean angle. Indeed, for most of the sweep angle values, an increase of jψ^ j leads to a tonal noise reduction. This reduction is not linear, contrary to the sweep effect, and is of the order of 7 dB at 2BPF, for a 0–20° lean variation. Leaned vane alone is thus not a noise reduction mechanism as efficient as swept vane, but the combination of both can lead to drastic reductions, e.g. 25–30 dB at 2 BPF, Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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confirming the results of previous studies [68]. A different trend is observed for the power predictions at 1 BPF in Figs. 25 (a) and (b). At this frequency, increasing the value of the lean angle does not necessarily lead to a power reduction, and may even increase the level of the acoustic emission. It means that the phase shifts brought by the swept and leaned geometry do not lead to the expected cancellations in terms of integrated vane loading, but rather to an increase of the sources. Indeed, similarly to sweep effect, the lean effect efficiency increases with the tone harmonics [68]. Therefore Figs. 25(a) and (b) show that the blade passing frequency of 3150 Hz seems too low for a positive power reduction in this configuration. The ^ with ^ ¼  201, again appears to represent the optimal value for φ efficient stator design found in the sweep study, i.e. with φ ^ ¼ 201 and ψ^ ¼  201 corresponds to the most efficient design among variable lean angles. Indeed the configuration with φ the tested cases. At 1 BPF, this results in an almost equivalent design with respect to the swept stator only, with a slight increase of 2 dB upstream and an interesting reduction of 4 dB downstream. It is thus shown that, on this compressor, a ^ of a combined swept and leaned parametric study focused on vane sweep only is able to provide the optimal value for φ vane design. Moreover the surveyed noise reduction mechanism highly depends on the harmonic since its efficiency increases with the frequency. Finally a clear trend in terms of noise reduction is found with the sweep angle whereas this trend appears more complex with the lean angle.

5. Conclusion In this paper, a detailed description of an analytical model for tonal fan noise prediction has been performed. The physical mechanism of interest consists of the tonal noise generated by the rotor–stator interaction. Indeed it corresponds to one of the main noise contributions in current turbofan engines and its influence is expected to grow in future Ultra-High Bypass Ratio turbofan engines. The presented model has been designed to precisely account for complex stator vanes geometries, with variable stagger, sweep and lean angles along the duct height, as tonal noise is sensitive to these parameters. A procedure has been proposed to use the rotor wake as the input excitation for the model. Namely, it is first recommended to extract the upwash, from measurements or flow simulations, in a surface matching the radial evolution of Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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the vane leading edges, and second to decompose this excitation into three-dimensional skewed gusts. The acoustic sources are provided by a cascade response function used within a radial strip theory, so that the radial correlation characteristics of both the excitation and the response, typical for tonal noise, are considered. Moreover cascade effects occurring on noise generation and propagation for blade rows with overlapping elements are fully accounted for. Finally the sources are used as acoustic dipoles in an acoustic analogy tailored to an annular duct containing a uniform mean flow. This ensures the radiated acoustic energy to be distributed on the actual cut-on and excited duct modes. In view of existing cascade models, the present model has been generalised to account for both rotational directions. The proposed model has then been validated on an actual fan stage configuration. It should be mentioned that this is the first detailed comparison of this model with noise sources and far-field measurements. The Advanced Noise Control Fan (ANCF), extensively tested at the NASA Glenn research center, is particularly suited for these comparisons. Indeed it is a low hub-to-tip ratio fan-OGV stage, with instrumented vanes (microphones) and duct (rotating rakes), installed in an anechoic environment. The data are available for several stator vane counts and several rotational velocities. Using an experimental dataset for the rotor wake, the cascade response of the model has first been compared with unsteady pressure measurements on the instrumented vane surface. It has been found that the 3D version of the model provides a more accurate cascade response than the 2D model. These comparisons have led to a reasonable validation of the 3D response function, noting that two sources of discrepancies have been explained. On the one hand, a flow separation in the upper part of the vane suction side enhances wall pressure fluctuations, that cannot be predicted by the inviscid flow model. On the other hand, the measured rotor wake used by the model does not account for the stator potential effect, whereas it influences the experimental vane response. The measured acoustic duct modes in the inlet and exhaust sections of the duct have then been compared with the results of the model in terms of acoustic power. Parametric studies with rotational velocities and vane counts have been performed, as well as comparisons with results from the V072 cascade model. In half of the studied cases, the proposed model has been shown to predict the radiated acoustic power with equivalent or more accuracy than the V072 model. For some other cases, the measurements have been highly influenced by a complex acoustic propagation within the duct, not accounted for in any of the models. The capability of the model to predict trends with vane counts and rotational velocities has been demonstrated on this configuration. This fan noise prediction model has been developed for research and industrial purposes. The model is thus expected to be included in an optimisation procedure aiming at designing quieter fan stages or more efficient acoustic treatments. Thus it has to cope with parametric studies focused on the stator geometry. This feature has been demonstrated on the CME2 compressor stage. The excitation and mean flow properties have been extracted from an existing detailed unsteady simulation of this rotor–stator configuration to be input within the model. The parametric study has first focused on the sweep angle of the vanes as results from the literature show this is an efficient tonal noise reduction mean. Indeed swept vanes induce radial phase shifts in the excitation thus in the vane response, leading to lower integrated vane loadings. The model has correctly reproduced this behaviour, showing the expected trend in terms of vane loading evolution with the sweep angle (amplitude and phase). The model has also allowed predicting the acoustic power emitted by the stage in function of the vane sweep angle. An optimal design has been found with a  20° sweep angle, yielding to a 10 dB reduction at 1 BPF and more than 15 dB reduction at 2 BPF with respect to a radially stacked stator. These values are within the range of what can be expected with this design. A two-parameter study has finally been performed concerning the vane sweep and lean angles. A proper orientation of the latter is indeed expected to enhance the noise reduction provided by the swept stator alone. This behaviour has been correctly reproduced by the model, showing that this noise reduction mechanism is highly dependent on the tone harmonics. A stator composed of vanes with a 20° sweep angle and a  20° lean angle has been determined as the optimal acoustic design, bringing approximately an additional reduction of 7 dB with respect to the optimal swept design at 2 BPF. Finally the proposed fan tonal noise model has been validated against experimental data and its pre-design tool characteristics have been demonstrated. Future research aiming at improving the accuracy of the prediction will mainly focus on the acoustic analogy used to radiate the acoustic sources.

Acknowledgements The authors wish to acknowledge Dr. Daniel L. Sutliff, from NASA Glenn Research Center, for having provided us with the geometry and the tests data of the Advanced Noise Control Fan (ANCF), and for his fruitful discussions, that have allowed to evaluate the fan tonal noise prediction model on a representative fan test case.

Appendix A. Analytical expression of the cascade response b 0 ðxc Þ, corresponding to the source term used in the tonal The result of the analytical derivation of the vane loading ΔP noise model presented in this paper, is recalled here. From the velocity potential scattered outside of the vane cascade, provided by Glegg [33], in response to the impinging gust: w0 eið  ωexc t þ kxc xc þ kyc yc þ kzc zc Þ ;

(A.1)

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Fig. A1. Definition of the segments of the rectilinear cascade necessary for the pressure jump calculation across a vane.

Posson et al. [34] extended this work to the calculation of the potential inside the inter-vane channels as well as to the pressure jump across the vanes. For a proper use of the Wiener–Hopf resolution method [34], the cascade must be split into the three segments I, II (or II0 ) and III, represented in Fig. A1. Segment II is considered when d o C=2 (Fig. A1(a)) and segment II0 when d 4 C=2 (Fig. A1(b)). Segments I and III represent diffraction problems at the leading edge and at the trailing edge respectively, linking the inter-blade channels mode existing in II (or II0 ) with the radiated cascade modes (these modes are b 0 ðxc Þ reads, according to the considered segment: discussed in Section 3.3). The vane loading ΔP

ΔPb 0 ðxc Þ ¼ ðDP2 þDP3 þ DP4 þ DP5 þ DP6Þðxc Þ on segment I; ΔPb 0 ðxc Þ ¼ ðDP6 þDP8 þDP9Þðxc Þ on segment II; ΔPb 0 ðxc Þ ¼ ðDP2 þDP3 þDP4 þDP5 þ DP10 þ DP11 þDP12Þðxc Þ on segment II'; ΔPb 0 ðxc Þ ¼ ðDP8 þ DP9 þ DP10 þDP11 þ DP12Þðxc Þ on segment III:

(A.2)

The expressions of the DP terms are: X ρ iw0 J ðδn Þðωg þ δn U xc Þð2  e  iðσ þ δn d þ ðn  1Þπ Þ Þ  0

e  iδn xc 2 J  ð  kxc Þðδn þ kxc Þβ h dn ϑn  1  X Xρ ð2π Þ2 Bl J ðεl ÞJ ðδn Þðωg þ δn U xc Þ þ  0 DP3ðxc Þ ¼  2  e  iðσ þ δn d þ ðn  1Þπ Þ e  iδn xc 2 ðδn  εl Þβ h dn ϑn  1 n Z 1l Z 1 DP2ðxc Þ ¼ 

nZ1

þ

DP4ðxc Þ ¼

X ρ0 π ζ qþ Dðλqþ Þeiðσ þ λq þ q dþ

sin ðλ

qAZ

DP5ðxc Þ ¼ 

ω λ β η

Xρ π Bl ðωg þ εl U xc Þeiðεl ðd  xc Þ þ σ Þ 0 ð  1Þl  1  cos ðεl d þ σ Þ

lZ1

DP6ðxc Þ ¼ 

σ Þðζ

þ ðd  xc ÞÞ ð g þ q U xc Þ þ 2 þ þ q Þ q d þ Sq h

X X 2 ρ ið2π Þ2 ðAn þ C n ÞJ ðεl ÞJ ðδn Þeiðεl ðC  xc Þ  iδn CÞ 1  cos ðσ þ εl d þðl  1Þπ Þ þ  0  ðεl  δn Þ β 2 h dl θ l  1 l Z 1n Z 0

DP8ðxc Þ ¼ 

X

2 ρ0 iw0 ðωg þ δn U xc ÞJ  ðδn Þ 

δ

β h dn ϑn  1

n Z 1ð n þ kxc ÞJ  ð kxc Þ

2

  1  cos σ þ δn d þ ðn  1Þπ e  iδn xc

X X2 ρ ð2π Þ2 Bl J ðεl ÞJ ðδn Þðωg þ δn U xc Þ 1  cos ðσ þ δn d þ ðn 1Þπ Þ þ  0 e  iδn xc  DP9ðxc Þ ¼  ðδn  εl Þ β 2 h d n ϑn  1 n Z 1l Z 1 DP10ðxc Þ ¼

X X ρ ið2π Þ2 ðAn þ C n ÞJ ðεl ÞJ ðδn Þeiðεl ðC  xc Þ  iδn CÞ eiðσ þ εl d þ ðl  1Þπ Þ 2 þ  0 ðεl  δn Þ β 2 h dl θ l  1 l Z 1n Z 0 

DP11ðxc Þ ¼ 

X ρ0 π ζ q Dðλq Þe  iðσ þ λq qAZ

DP12ðxc Þ ¼ 





ðd þ xc ÞÞ



ðωg þ λq U xc Þ

sin ðλq d þ σ Þðζ q d þ S q hβ ηq Þ 2

X ρ i π ðAn þ C n Þe  iðσ þ δn ðd þ xc ÞÞ 0 nZ1

ð  1Þn  1  cos ðδn d þ σ Þ

:

All the new terms appearing in the expressions of the vane pressure jumps are defined below. ρ0 is the mean flow density. The variables introduced by Glegg [33] are: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M ¼ U xc =c0 ; β ¼ 1  M 2 ; ωg ¼ ωexc  kzc U zc ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 s ¼ d þh ; se ¼ d þ β h ; tan χ e ¼ d=βh;     κ ¼ ωg = c0 β2 ; κ 2e ¼ κ 2  kzc =β 2 ; ξ ¼ k0xc  κ M; sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 lπ δl ¼ κ M  ϑl  1 ; εl ¼ κ M þ ϑl  1 ; ϑl ¼ κ 2e  ; βh Please cite this article as: J. de Laborderie, & S. Moreau, Prediction of tonal ducted fan noise, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.02.032i

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qffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ þ κ Md 2π q 2 f q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ηq7 ¼  f q sin χ e 7 cos χ e κ 2e  f q ; 2 2 2 d þβ h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   02 λq7 ¼ κ M þ ηq7 ; ζq7 ¼ ζ λq7 ; ζ ðkxc Þ ¼ ðωg þ k0xc U xc Þ2 =c20  k02 xc  kzc ;   ρ ¼ σ þ κ Md; d0 ¼ 2; dnðn a 0Þ ¼ 1; Z ¼  iξ=π βh logð2 cos χ e Þ þ χ e d :

σ ¼ kxc d þkyc h;

It must be mentioned that the wavenumbers of the excitation and of the cascade response in the zc direction are the same 0 (kzc ¼ kzc ) since the rectilinear flat plate cascade has an infinite span (Section 2.4). The function J þ and J  are defined as:    0  β κ e sin β κ e h eZ ∏1 ð1  ξ=θl Þ     1 l ¼ 0 ; J þ kxc ¼

 ∏ 4π cos β κ e h  cos ρ q ¼  1 ð1  ξ=ηq Þ  0  e  Z ∏1 ð1  ξ=ϑl Þ : J  kxc ¼ 1 l ¼ 0 ∏q ¼  1 ð1  ξ=ηqþ Þ

The Fourier transform of the velocity potential across a vane is decomposed into the four terms:  0  Dð1Þ kxc ¼

iw0 0 0 ; ð2π Þ2 ðkxc þ kxc ÞJ  ð  kxc ÞJ þ ðkxc Þ 0 X  0  An eiðkxc  δn ÞC J  ðδn Þ Dð2Þ kxc ¼  0 0 0 ; ω þ kxc U xc Þðkxc  δn ÞJ  ðkxc Þ ið g nZ1 1 X  0  Dð3Þ kxc ¼ 

Bl J þ ðεl Þ 0 0 ;  εl ÞJ þ ðkxc Þ ðk xc l¼1

1 X  0  Dð4Þ kxc ¼ 

0

C n eiðkxc  δn ÞC J  ðδn Þ 0 0 0 : ω þk ið g xc U xc Þðkxc  δn ÞJ  ðkxc Þ n¼1

The methodology proposed by Glegg [33] is used to determine the coefficients An, Bl and Cn. Namely the latter are supposed to be finite so that they can be manipulated under matrix form. Defining A ¼ ½An n A N , B ¼ ½Bl l A N , C ¼ ½C n n A N , F ¼ ½F l;n ðl;nÞ A N2 and L ¼ ½Ll;n ðl;nÞ A N2 , the coefficients are: B ¼ F  ðA þ CÞ; C ¼ L  B; with An ¼

w0 ðωg þ δn U xc Þ

; ð2π Þ ðδn þ kxc ÞJ 0þ ðδn ÞJ  ð  kxc Þ 2

J  ðδn Þeiðεl  δn ÞC ; iðωg þ εl U xc Þðεl  δn ÞJ 0 ðεl Þ iðωg þ δl U xc ÞJ þ ðεn Þ : Ll;n ¼ ðεn  δl ÞJ 0þ ðδl Þ

F l;n ¼ 

The variable S q7 is 1 or þ1, such as, for all integer q, the following expression is always valid:

ζq7 h ¼ S q7 ðλq7 d þ σ  2π qÞ: þ



Finally Dðλq Þ and Dðλq Þ used within the expressions of the DP terms are: 1     X  iw0 Bl J þ ðεl Þ þ ð1;3Þ D λq ¼ λqþ ; q A Z;  þ þ þ þ ¼D 2 ð2π Þ ðλq þ kxc ÞJ þ ðλq ÞJ  ð  kxc Þ l ¼ 1 ðλq  εl ÞJ þ ðλq Þ  1     X ðAn þ C n Þeiðλq  δn ÞC J  ðδn Þ  ð2;4Þ D λq ¼  λq ; q A Z:    ¼D ið ω þ λ U Þð λ  δ ÞJ ð λ Þ g xc n  q q q n¼1

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