Duct modes damping through an adjustable electroacoustic liner under grazing incidence

This paper deals with active sound attenuation in lined ducts with flow and its application to duct modes damping in aircraft engine nacelles. It presents an active lining concept based on an arrangement of electroacoustic absorbers flush mounted in the duct wall. Such feedback-controlled loudspeaker membranes are used to achieve locally reacting impedances with adjustable resistance and reactance. A broadband impedance model is formulated from the loudspeaker parameters and a design procedure is proposed to achieve specified acoustic resistances and reactances. The performance is studied for multimodal excitation by simulation using the finite element method and the results are compared to measurements made in a flow duct facility. This electroacoustic liner has an attenuation potential comparable to that of a conventional passive liner, but also offers greater flexibility to achieve the target acoustic impedance in the low frequencies. In addition, it is adaptive in real time to track variable engine speeds. It is shown with the liner prototype that the duct modes can be attenuated over a bandwidth of two octaves around the resonance frequency of the loudspeakers.

by Guicking and Lorentz (1984) [11], and later by Furstoss et al. (1997) [12], to develop a hybrid passive/active 23 treatment composed of a thin porous layer backed by an air cavity closed by a feedback-controlled loudspeaker. By 24 successfully imposing a pressure release condition at the rear of the porous layer, a purely real surface impedance 25 given by the flow resistance of the porous layer can be achieved and the liner behaves like an active quarter-wave 26 resonator. Direct application of the hybrid passive/active absorber to achieve broadband noise reduction in flow ducts 27 can be found in [13,14,15]. If the desired acoustic resistance for the lined duct is directly related to the porosity and 28 thickness of the resistive layer, the reactance is generally close to zero. As discussed above, however, a purely real 29 surface impedance condition does not lead to optimal attenuation rates of duct modes [5,6,7]. Alternative solutions 30 have been proposed to implement complex acoustic impedances with the help of actively controlled electromechanical 31 actuators. An active SDOF liner for attenuating noise that includes a rigid backplate supporting a piezoelectric patch 32 was proposed and patented by Kraft and Kontos [16], where a microphone is used in combination with a controller to 33 obtain a predetermined acoustic impedance at the panel surface. Zhao and Sun proposed achieving specific impedance 34 condition actively through two controllable variables, the cavity depth of the liner and the bias flow through the 35 orifice [17]. However, the authors concluded that change in reactance was harder to achieve practically using this control is not to absorb or cancel noise but rather to scatter or redistribute acoustic energy among modes to maximize 42 the efficiency of the passive sound absorbing elements [20]. Other promising results can be found in the literature 43 concerning active Helmholtz resonator concepts using a loudspeaker in the resonator cavity to extend the sound 44 absorption capability of the resonator [19]. Collet et al. (2009) investigated the potential of a distributed control scheme 45 to block wave propagation in a given direction in a waveguide. In contrast to the active methods mentioned above, 46 this approach aims at redirecting the sound field without directly interacting with acoustic energy to cancel or absorb it. Recently, the concept of electroacoustic absorber has been introduced as an effective means of damping the duct 48 modes, either using shunt loudspeaker technique [22,23], direct feedback control [24,25], or by self-sensing control of 49 the loudspeaker impedance [26]. Rivet et al. (2017) showed that a loudspeaker and a microphone nearby, both being 50 connected by a model-based transfer function, can be used for matching the impedance of a loudspeaker diaphragm to 51 a target specific acoustic impedance, which has the effect of damping the standing waves in an enclosure [27]. 52 The paper presents an active lining concept based on an arrangement of electroacoustic absorbers and its application 1 to achieve broadband noise reduction in aircraft engine nacelles. Instead of trying to improve a passive DDOF 2 liner, emphasis is placed on achieving adjustable local reaction through active impedance control of the loudspeaker 3 diaphragm. The theoretical analysis is based on a rectangular duct with section lined on one side in the presence of a 4 uniform flow and a multimodal excitation. The remainder of the paper is organized as follow. The design of the baseline 5 electroacoustic absorber from a feedback-controlled loudspeaker is addressed in Section 2 using an impedance-based 6 approach. Section 3 provides an overview of the theoretical framework used in this study to investigate the liner 7 performance in a flow duct. Computer simulations and experimental results are given in Section 4 to show the 8 performance and potential of the proposed active liner. A discussion of the benefits and limitations of this active lining 9 concept to reduce fan noise in aircraft engine nacelles is finally provided to conclude this paper. This section describes the design of an electroacoustic liner concept composed of distributed electroacoustic 12 absorbers. In particular, a locally reacting impedance model is developed on the basis of the electrodynamic loudspeaker 13 in order to achieve a given target specific acoustic impedance. 14 2.1. Locally reacting impedance models 15 The macroscopic properties of acoustic liners are generally characterized by the specific acoustic impedance 16 Z = p/v n , which defines the ratio of the local sound pressure p to the normal component of the particle velocity v n on 17 the lining surface. For convenience, the non-dimensional specific acoustic impedance z in a lined duct is commonly 18 given in terms of resistance θ and reactance χ as [1,2] where ρc is the characteristic impedance of air. Analytical models for predicting impedance of conventional SDOF 20 liners are usually formulated from where D is the lining depth, R is the flow resistance of the face-sheet, km is the facing sheet inertance, cot (kD) is the 22 cavity reactance, and k is the wavenumber. In Eq.
(2), acoustic resistance is mainly related to the open area of the 23 perforated sheet while reactance strongly depends on the depth of the liner cavity. Figure 1 illustrates the schematic 24 representation of a lined duct in the presence of a uniform mean flow. Additionally, for a DDOF liner, the surface impedance can be expressed as [29] where Z 1 is the face-sheet impedance, Z 2 is the septum impedance, d 1 is the face sheet backing space depth, d 2 is the 2 septum backing cavity depth, and d = d 1 + d 2 . Other models based on Helmholtz resonator or mass-spring-system where it is assumed that Z s is independent of the direction of incident sound, i.e. the diaphragm is a locally reacting 20 surface. In order to achieve an actively tunable SDOF system, the target specific acoustic impedance can be formulated 21 from Eq. (5) as where R st , µ 1 and µ 2 are design parameters used to assign a desired resistance, mass and compliance at the loudspeaker 23 diaphragm, respectively.

24
From continuity considerations at the boundary surface of the electroacoustic liner, the normal acoustic velocity v n 25 is related to the diaphragm velocity v over the effective piston area S d by where σ = S d /S is the fractional effective area of a unit cell of surface S , as shown in Fig. 2 (b). Substituting Eq. (7) 1 into Eq. (6), the non-dimensional specific acoustic resistance and reactance in Eq. (1) can be rewritten as Equation (8) gives the relationship between the macroscopic parameters of the liner (θ and χ), the physical 3 parameters of the loudspeaker, and the design parameters used to meet the desired acoustic specifications. For a current-driven loudspeaker, the control law by which a conventional electrodynamic loudspeaker is turned 6 into an electroacoustic absorber can be derived by substituting Eq. (6) into Eq. (4) as and after some manipulations, the corresponding electroacoustic transfer function can be expressed as a second-order where s = jω denotes the Laplace variable. The gain constant, characteristic frequency and quality factor of the 10 numerator of Eq. (10) are given by and for the denominator in Eq. (10), the characteristic frequency and quality factor are Equation (10) corresponds to a tunable equalization filter which features adjustable gain at specified frequencies frequencies are boosted or cut relative to frequencies much above or below the selected center frequency. Table 1 20 examines the characteristics of the transfer function H in Eq. (10) as a function of the design parameters µ 1 , µ 2 and R st .

21
As shown in Tab. 1, H is a second-order band-reject transfer function when the design parameters are such that H degenerates into an all-pass filter for µ 1 = µ 2 = S d R st /R ms . For µ 1 > µ 2 , H is a low-boost filter (ω p < ω z ) that can 28 strengthen the energy of a specific frequency band below the natural frequency of the loudspeaker without filtering 29 out the high frequencies, as a low-pass filter does. For µ 1 < µ 2 , on the other hand, H is a high-boost filter that can 30 strengthen the energy of a specific frequency band above the natural frequency of the loudspeaker. In both cases of low-31 or high-boost, the design parameters allows adjusting Q p and Q z . Note that for µ 1 = 1, K = 0 and the liner is passive.

32
When some of the open loop zeros, i.e. the roots of the numerator of Eq. (10), lie on the right-hand side of the complex 33 s-plane, the system is non-minimum phase and is conditionally stable. Note also that Q z may be negative if µ 1 > 1 and 34 µ 2 > 1, in which case H is a non-minimum phase filter, while Q p must always be positive.
35 Table 1: Main features of the electroacoustic transfer function (10) as a function of the design parameters µ 1 , µ 2 and R st .
Design settings Characteristic frequency Q factors Filter function   Figure 3 illustrates the waveguide and coordinate system considered in this study 1 . The acoustic medium is a 6 frictionless, homogenous (ideal) fluid of mass density ρ subject to a steady axial flow with mean velocity u 0 that is 7 assumed to be uniform over the cross section. The processes associated with wave motion are isentropic and fluctuating where M = u 0 /c is the flow Mach number, k = ω/c is the wavenumber, ω is the angular frequency (in rad/s), and 13 c is the speed of sound in air (in m/s). 14 • For rigid-walled parts, the boundary conditions are given by where L y and L z are the cross-sectional dimensions in the direction y and z, respectively. • The boundary condition in the source plane located at x = 0 (see Fig. 3) is a second-order radiation boundary 1 condition defined in the frequency domain as [37] where the incident pressure field is a superposition of plane waves where p is the complex sound pressure and w = ρcv x is an auxiliary variable that is related to the complex acoustic 16 velocity v x along the duct axis. More details on the derivation of Eq. (18) are given in Appendix A. 17 The acoustic attenuation due to the electroacoustic lining is evaluated from the average intensity over a duct 18 cross-section located downstream of the treated section. The IL is obtained according to 19 IL = 10 log 10 I x1 I x2 where the subscript 1 refers to the rigid-walled duct including the DDOF liner and subscript 2 refers to the rigid-walled 20 duct including the DDOF liner and the electroacoustic liner, as shown in Fig. 3. In this section, the decomposition of the acoustic pressure field is presented to provide the expression of sound 2 excitation. In the duct section upstream of the treatment, the general solution of the boundary value problem given by 3 Eqs. (13) and (14) can be obtained by separation of variables, and the acoustic pressure can be expressed in terms of 4 rigid duct modes as [38] where p mn are modal coefficients and, for a duct with rectangular cross-section, the normalized eigenfunctions are [39] where δ 0m is the Kronecker delta function. where The condition for propagation of a given mode (m, n) is that k xmn is real, i.e. γ mn 0, and the corresponding cut-on where p and v are vectors of modal coefficients, respectively, and Ψ is a vector of mode shape functions.

16
Without prior knowledge on source distribution or modal content in the experimental wind tunnel, we assumed that In the case of the plane wave (m = n = 0), we have δ 0m = δ 0n = 1 and γ mn = 1, and it follows that the total sound 1 power transported by the duct is Assuming now that the total sound power is equally distributed over N m propagating modes, i.e. such that 3 Π mn = Π/N m , the corresponding modal amplitudes can be expressed in terms of the plane wave amplitude A 00 as Equation (30) is used to define the sound excitation in Section 4.1.

5
A parametric study was carried out to evaluate the influence of the design parameters R st , µ 1 and µ 2 on the liner 6 specific acoustic impedance. The results presented below correspond to a set of specific values and are intended to show 7 the capability of the electroacoustic liner to achieve broadband performance. Table 3 summarizes the liner acoustic 8 performance for the studied configurations. The comparison of the IL calculated and measured from the values of Tab. 9 3 is presented in Section 4.4. The determination of the optimal attenuation at specified frequencies with respect to the 10 specific acoustic impedance of the liner is discussed in Section 4.5.  Tab. 3. As discussed above, these transfer functions correspond to the control law by which the loudspeaker is driven 13 from the sound pressure at its diaphragm. As shown in Fig. 4, different frequency response functions can be obtained 14 depending on the design parameters. When µ 1 = µ 2 < S d R st /R ms as in cases B, F and G, the electroacoustic transfer 15 function is a band-reject filter whose center frequency is the natural frequency of the loudspeaker, as indicated in Tab. 1.

16
As can be seen in Fig. 4  21 Figure 5 shows the non-dimensional specific acoustic impedance calculated from Eq. (6) for the studied configu-

22
rations. This calculation result illustrates the specific acoustic impedance that is assigned to each diaphragm for the 23 control settings listed in Table 3. As shown in Fig. 5, the design parameters used in the control law (9) make it possible 24 to adjust the specific acoustic impedance of the loudspeakers diaphragm. By varying the ratio µ 1 /µ 2 for a given value 25 of R st = ρc/2 < R ms /S d (see cases B, C, D, E, H, and I), it can be observed that the reactance of the loudspeaker 26 diaphragm is changed: the resonance frequency (where the non-dimensional specific acoustic reactance curve crosses 27 zero) is shifted and the slope is decreased compared with the uncontrolled loudspeaker (case A). Conversely, increasing 28 the value for R st for a constant ratio µ 1 /µ 2 (see cases E, F, and G in Fig. 5 (b)) results in an increase of the diaphragm 29 specific acoustic resistance without modifying its reactance. Not shown in this paper, decreasing the value for R st 30 causes a decreases of the diaphragm resistance while leaving the reactance unchanged accordingly. into the wind tunnel that connects two reverberation chambers in which the sound pressure level (SPL) is measured 5 to determine the acoustic attenuation provided by the treatment under grazing flow conditions [43]. The first duct 6 cut-on frequency for M = 0 is 567 Hz for the no-flow case, which implies a multimodal propagation for f > 567 Hz.

7
The IL was measured by comparing the change in SPL due to the insertion of the electroacoustic liner into the duct 8 connecting the two reverberation chambers, with air flowing through the treated section to simulate engine conditions. 9 Sound excitation is generated in the sending (source) reverberation chamber using acoustic drivers. Over the frequency  The electroacoustic liner prototype shown in Fig. 7 (a) consists in an arrangement of 3 rows of 10 unit cells, each    Numerically computed and experimentally obtained acoustic performance comparisons are presented in Fig. 8. 9 Numerical predictions of the sound attenuation are in good agreement with the measured data, both in the presence of 10 flow and in the absence of flow. The differences observed between measurements and simulations are partly due to some 11 variation in the loudspeakers electromechanical parameters, which can be up to 10% between any two loudspeakers.

12
This dispersion causes the diaphragms to present different resonance frequencies and internal mechanical losses. In the 13 experimental results presented above, these differences are not compensated in the control law. Moreover, the gain all the loudspeakers. This explains why the performance is overestimated in the simulations. Note also that the load 17 impedance of the reverberation chamber is not included in the simulation, which may also explain some of the observed 18 discrepancies between calculated and measured data. As expected, the active electroacoustic liner prototype provides a 19 very high level of IL in the low frequency range around the loudspeaker resonance frequency (see cases B and E). As 20 shown in Fig. 8, the peak of maximum attenuation obtained experimentally for the configurations studied is about 15 21 dB at 0.94 · f n , i.e. around the natural frequency of the loudspeakers. By adjusting the design parameters, furthermore, 22 a good attenuation can be achieved in a wide frequency range, between f n /2 and almost 2 · f n Hz for the configurations 23 studied in this paper. As expected by varying the ratio µ 1 /µ 2 , the peak of attenuation is shifted in a frequency range 24 that is close to the center frequency of the corresponding electroacoustic filter, which is given by µ 2 /µ 1 · f n (see ratio µ 1 /µ 2 = 4 and 5.8 dB at 1.72 · f n for case I with a ratio µ 1 /µ 2 = 0.3, is consistent with the theory. parameters can be derived from Eq. (8). Figure 9 shows the constant attenuation contours in dB with and without 5 airflow computed as a function of the liner complex impedance at some specified frequencies (see Tab. 3). As shown in 6 Fig. 9, the maximum IL level for the flow duct model and a multimodal excitation is frequency dependent. Overall, it is 7 found that the optimal acoustic resistance and reactance both increase with frequency, with or without flow. As can 8 be seen in Fig. 9 (a), the maximum attenuation below the natural frequency of the loudspeaker is expected for a low Hz. As can be seen in Fig. 9, the lower the frequency, the smaller the optimal attenuation area in the complex s-plane.

4
Maximum IL is much more sensitive to small variations in impedance in lower than in high frequencies. in real-world applications is therefore the capability of the active electroacoustic absorbers to interact with the fan  This active liner concept is based on a multi-channel decentralised control system. Each processor (or node) receives a single sensor signal (here the average pressure of the four microphone around the diaphragm) from which a single 23 control signal is delivered to the loudspeaker driver. The decentralised control scheme allows for parallel computation 24 of control variables but absence of communication between controllers may limit the achievable performance. The 1 advantage of the decentralised architecture is, however, to be robust even if some of the control units are disabled.

2
During the experiment in the wind tunnel, some control units were disabled without disturbing the other electroacoustic 3 absorbers.

4
In contrast to ducted ventilation systems and mufflers where the small cross-section predominantly allows the 5 propagation of plane waves, the transverse dimensions of high-bypass turbofan engine nacelle permit wave propagation 6 of many higher-order modes. To approximate such a multimodal excitation, the simulations were carried out in the 7 frequency domain by generating a constant modal sound intensity distribution [42] in the source plane at the duct 8 inlet. In a flow duct facility such as that shown in Fig. 6, on the other hand, the phase angles of the propagation modes 9 depend, among other things, on the space-time pattern of the fan noise source distribution. If source distributions with 10 temporal and spatial incoherence are assumed, the phase angles vary randomly with space and time, as discussed by 11 Doak [39]. In this study, modal contributions were applied in the source plane without taking into account temporal and 12 spatial incoherence between them. To provide a more realistic excitation, it would be interesting to consider the effects feedback-controlled loudspeaker membranes in a decentralised scheme, and its application to duct modes damping.

20
This electroacoustic liner was theoretically studied using a lumped element model and its ability to achieve a desired 21 specific acoustic impedance in a flow duct was evaluated. The IL of the liner was calculated numerically for various 22 design parameters and compared to measurements performed in a flow duct facility. The parametric study carried out by 23 simulation showed the influence of the design parameters on the overall performance of the liner. This electroacoustic 24 liner has an attenuation potential comparable to that of a conventional passive liner, but offers greater flexibility to 25 achieve a given complex acoustic impedance in the low frequencies, which is moreover adaptive in real time. It is 26 shown that the acoustic resistance and reactance of the liner can be tuned independently, which allows the duct modes 27 to be attenuated over a bandwidth of two octaves around the resonance frequency of the loudspeakers. The level of 28 peak attenuation and target frequency are, for a given duct geometry and flow condition, controlled by the diaphragm 29 acoustic resistance R st and reactance through the ratio µ 2 /µ 1 , respectively, and related to the percentage of effective 30 area σ of the liner. When the target frequency is below the natural frequency of the loudspeaker, the target resistance 31 R st decreases rapidly and may be much less than ρc. In addition to offering an adaptive locally reacting impedance, one 32 advantage of the multi-channel decentralised control system used to implement the liner is that it remains robust even if 33 some of the control units are disabled. In future work, it is planned to combine the decentralised control scheme with a 34 distributed control law [21] to improve performance.