Enhanced micro-perforated wall-treatments for reducing the bounda- ry layer noise in a low-speed wind-tunnel: modelling, characterization and optimization studies Teresa Bravo 1 Instituto de Tecnologías Físicas y de la Información Consejo Superior de Investigaciones Científicas Serrano 144 28006 Madrid, Spain Cédric Maury 2 Aix Marseille Univ, CNRS, Centrale Marseille Laboratory of Mechanics and Acoustics 4 impasse Nikola Tesla 13013 Marseille, France

ABSTRACT Micro-perforated aero-acoustic liners are robust non-fibrous wall-treatments able to achieve sig- nificant axial attenuation in low-speed ducted flows, if their input impedance is suitably optimized. They are also potential candidates for broadband reduction of the flow-induced noise, useful to increase the signal-to-noise ratio in wind-tunnels test section or for fan noise control when inserted in casing treatments. The current theoretical and experimental study determines under which range of frequencies, holes diameter and free stream flow velocity, micro-perforated partitions are able to efficiently reduce the boundary layer noise, either by absorption or transmission mechanisms, de- pending on the spectral content of the aero-acoustic excitation. The formulated analytical models, either modal- or wavenumber-based, are validated against wind-tunnel measurements of the over- all power injected by a low-speed boundary layer into a micro-perforated partition and its aerody- namic transmission loss. A hole-based Strouhal number is found below which little back-scattering of the wall-pressure components occurs. Optimization studies are performed to determine the range of parameters that enhance the total absorption or transmission of boundary layer noise, with spe- cial emphasis on the low frequency content of the wall-pressures.

1. INTRODUCTION

The reduction of boundary layer noise radiated outward is a major feature encountered in aeronautic transports such as the airframe noise over fuselage surfaces, wings and flaps [1] or in energy pro- duction such as the trailing edge noise generated by windfarm wings [2]. The objective is then to avoid the conversion of turbulent energy into sound by optimizing the geometry. As it will be seen,

1 teresa.bravo@csic.es

2 cedric.maury@centrale-marseille.fr

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such backscattering of the turbulent pressures may not always occur in which case another strategy is to enhance the absorption of aero-acoustic wall-pressures. This has been achieved in aerodynami- cally complex flows through non-fibrous micro-perforated liners (MPLs). For instance, low drag external MPLs have been flush-mounted onto the rear of aircraft fuselage to absorb the wall- pressures induced by contra-rotating open rotors [3]. Circumferential MPL casing treatments have also been studied to develop quieter small-sized contra-rotating fans with minute aerodynamic pen- alties [4].

Back to 1972, it was found from theoretical [5] and experimental [6] studies that a turbulent boundary layer (TBL) developed over a perforated screen may generate additional sound rather effectively due to scattering effects by the individual apertures. This occurs for a hole-based Strou- hal number,   U fd St h h , typically greater than 0.1, with h d the holes diameter, f the frequency

and  U the flow free-stream velocity. For h St below 0.1, conversion of turbulence into sound is rather inefficient and one should then rely on other mechanisms, such as enhanced dissipation and reduced transmission of the aeroacoustic wall-pressures by the wall-treatment. The current study provides physical insights on the potential of a Micro-Perforated Panel-Cavity-Panel (MPPCP) par- tition to maximize the absorption and minimize the transmission of aero-acoustic wall-pressures. Section 2 presents a wavenumber formulation to predict the absorption and transmission of aeroa- coustic wall-pressures through MPPCPs. It is assessed against a modal formulation and wind-tunnel (WT) measurements. Section 3 proceeds with parametric studies. Section 4 compares several opti- misation strategies that maximize either the dissipation or the Transmission Loss (TL) of MPPCPs. 2. BOUNDARY-LAYER NOISE ABSORPTION AND TRANSMSISSION BY A MPPCP: MODELLINGS AND EXPERIMENTS

One considers a MPPCP partition of cavity depth D undergoing an aeroacoustic excitation. The MPP is made up of circular holes of thickness h t and perforation ratio  . Its overall surface imped- ance is given by Maa's model [7] with grazing flow corrections [8] as follows

   



1





             



2 1 i Z M K R F d

1 0 MPP 4 3 8 i

i

k J

t Z e M h

  

  

0 0

h

h

, (1)

i

k



i

k J

h

0

h

with     visc. 2 r d k h h  , the perforate constant, e.g. the ratio of the hole radius to the viscous

boundary layer thickness,      0 visc.  r ,  the dynamic viscosity of the air, 0  the air densi-

ty, 0 c the sound speed,  the angular frequency and 0 0 0 c Z   . The effect of a one-sided grazing flow is to increase the resistance by  0 KMZ , with 15 .0  K , and to decrease the reactance by a

factor      

   M M F with 0 c U M   the Mach number. 2 0    e R accounts for fric- tional loss outer-resistance for holes with rounded edges.

1 3 6. 12 1

2.1. Wavenumber model A 2D simplified model is formulated to predict the absorption and transmission properties of an elastic MPPCP of infinite extent undergoing a streamwise aeroacoustic excitation, generalizing the model of Toyoda et al. [9] to a distribution of subsonic and supersonic trace wavenumbers given by Eq. (2) and representative of an aeroacoustic excitation over the MPP surface. It reads:

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       

 

 

k S ˆ



 , (2)

1 ;

k

c y y dd 

A

    



0 2 2 2 0 1

k

A S



 

k k k

y c y c

with dd S ˆ the spatial Fourier transform of the wall-pressures cross-spectral densities, given by the

sum of Corcos-type [10] turbulent components and acoustic diffuse field components. 0 S is the auto-power spectrum, A the magnitude of the acoustic contribution. y  is a frequency-dependent

empirical constant estimated from [11] for the TBL streamwise correlation lengths, y k the stream-

wise wavenumber, c c U k   the convective wavenumber with   U . U c 6 0 the convection velocity

and 0 0 c k   the acoustic wavenumber. Eq. (2) leads to the spectrum of the incident power for an

aeroacoustic excitation over an infinite plane, given by       0; 2 Π 8 inc 0   y dd k S ˆ Z . From the inte- gral representation of the MPP and back panel displacements in terms of the pressure jumps and panels Green’s functions, formulated in the wavenumber domain, one obtains the sound power ra- diated by the back wall of the partition

       

2 2 0 0 trans

k F k Z    , (3)

k

0

y b k k S ˆ k k

d ; 2 Π

y y dd

2 2 0

k

y

0

where  y b k F is the back wall normal displacement per unit incident pressure of trace wavenumber

y k over the MPP. In Eq. (3), only the supersonic wavenumbers contribute to the back wall radia-

tion. The aeroacoustic TL given by   trans inc 10 Π Π log 10 is then readily obtained. The absorption co-

efficient inc flow Π Π   requires calculation of flow Π , the sound power injected by the aeroacoustic excitation into the MPPCP partition. It reads

K

         

1 Π

max

y y dd z y * v y p k k S ˆ k k Q k Q k Z   , (4)

d ; Re 2

0 0 flow

K

where  y p k Q and  y v k Q are respectively the surface pressures and normal particle velocities over

max

the MPP per unit incident pressures of either supersonic ( 2 2 0 0 , y z y k k k k k    ) or subsonic

( 2 0 2 0 j , k k k k k y z y     ) trace wavenumber. The integral in Eq. (4) is evaluated over a finite

interval bounded by   f c k, k K max 4 max  with f k the MPP flexural wavenumber. One considers a MPPCP partition composed of two steel-made simply-supported panels separat- ed by a cavity depth of 0.15 m whose MPP parameters are mm 5 0 . d h  , mm 1  h t , % 52 0 .  

and back panel thickness mm 7 0 . t b  , sketched in Fig. 1(a). The absorption and transmission prop- erties are compared in Fig. 2 against those obtained from a modal formulation [11] assuming a par- tition of size m 35 .0 m 22 .0  and a modal damping ratio of 1% for the panels. It can be seen that the wavenumber model of infinite partition under a TBL well captures the gradual decrease (resp. increase) of the absorption (resp. TL) when increasing the flow velocity, although it overestimates the absorption by 10% and underestimates the TL by up to 25 dB.

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(a) (b) (c)

Multi-channel acquisition system 3D Presure-velocity sensors i _—- Hot-wire Flexible eae Convergent v Wind-tunnel anenometer vith apertures nozzle diffuser I 30 m/s <— ' Test section | Plexiglas panel Laser Vibrometer

Figure 1: (a) Sketch of the WT facility set up to measure boundary layer noise absorption and transmission through a MPPCP; (b) Inner view of the WT test section with the flush-mounted MPPCP (top) and a thick Plexiglas block (bottom) for laser vibrometric measurements of the MPP velocity; (c) Outer view of the WT test section with the transmission enclosure (top) and the laser (bottom).

= ic SERRE y

Figure 2: Simulated absorption spectra (a) and TL (b) of a MPPCP partition assuming a modal (sol- id) and a wavenumber (dashed) formulation undergoing a TBL with free-stream velocity of

1 s m 10  (green) and 1 s m 50  (blue), or an acoustic diffuse field (red).

2.2. Comparison against wind-tunnel measurements Experiments have been set up in a low-speed wind-tunnel, shown in Fig. 1, to estimate the absorp- tion and transmission performance of a MPPCP partition flush-mounted on the top wall of the test section and excited by a fully-developed TBL of free-stream velocity of 1 s m 30  together with an acoustic field whose contribution A to the aeroacoustic wall-pressures was estimated to be 8 % between 100 Hz and 3200 Hz [13]. Both aluminium clamped panels have dimensions of

m 47 0 m 38 0 . .  , are 1 mm thick and the MPP has holes diameter 0.5 mm and a perforation ratio of 0.6 %. Part of the input power structurally dissipated by the front panel was measured by laser vibrometry through the optically transparent bottom wall of the test section whereas part of the power injected through the holes and either dissipated through or radiated by the apertures was es- timated from flush microprobe measurements of the spatially-averaged wall-pressures and from

 MPP Z in Eq. (1). The MPPCP radiates into a wall-treated enclosure and a pair of pressure- velocity probes displaced over the radiating surface estimates the radiated sound power.

Figure 3 shows a reasonable correlation between the measured absorption and transmission pro- perties of the MPPCP and those predicted by the wavenumber model. The correlation significantly improves when considering the modal simulations. Figure 1(a) shows that, below 150 Hz, a large part of the input power is injected into the front panel structural resonances (as verified by the PCP simulations) whereas, between the resonances, it is essentially transmitted by the partition, due to inefficient back-scattering of the turbulent wall-pressured by the apertures. As frequency increases, both models and the measurement show that the absorption steeply decays due to efficient back- scattering of the small-scale turbulence into sound.

Figure 3: Absorption spectra (a) and transmission loss (b) of a MPPCP excited by a TBL with free- stream velocity of 1 s m 30  : comparisons between simulations from modal (MPPCP, solid blue; PCP, solid grey) and wavenumber (MPPCP, dashed green) formulations, and measurements (MPPCP, dashed red with squares) while neglecting the cavity pressures in the injected power. 3. PARAMETRIC STUDIES

Wavenumber- and modal-based simulations have been carried out assuming a MPPCP with alumin- ium front and back panels of thickness 1 mm, separated by an air gap of 0.03 m and excited by aer- oacoustic wall-pressures. Figure 4 shows the influence of the flow velocity and the acoustic com- ponent on the absorption and transmission properties of the MPPCP partition as a function of the hole-based Strouhal number.

It can be seen from Figs. 4(e) to (h) that the maximum absorption values significantly decrease when  U increases from 1 s m 30  to 1 s m 80  over the range of Strouhal numbers   Aero H Max max , h , h , h h St , St St St   where H , h St and Aero , h St are respectively associated to the Helm- holtz frequency and to the MPP aerodynamic coincidence frequency. One notes the invariance of

Max , h St that stays below 0.02 despite the crossing between H , h St and Aero , h St at 1 s m 70  and above which there is significant back-scattering of the input power. Figs. 4(a) to (c) show a uniform reduc- tion by 10 dB of the TL over the whole range of h St when  U increases from 1 s m 30 

to 1 s m 80  .

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(a) 0.02 St, 0.002 (d) J =30ms! (b) 00 40 60 80 100 A (%)

0.02 St, Aero 0.002 20 40 60 A(%) 80 100 0.02 fa = i) Aero 0.002 20 40 60 A(%) 80 100

Figure 4: Wavenumber-based predictions of the transmission loss (a-c) and absorption spectra (d-f) of a MPPCP undergoing an aeroacoustic excitation with free-stream velocities 1 s m 30  (a,d),

1 s m 50  (b,e) and 1 s m 70  (c,f) as a function of the acoustic contribution  % A and of the hole- based Strouhal number h St ; also shown are the Helmholtz (red dashed) and aerodynamic coinci- dence (black dashed) frequencies. 4. OPTIMISATION STUDIES

Since MPPCPs are mostly efficient for absorbing the low frequency components of the aero- acoustic wall-pressures, a question is to determine which cost function (absorption, dissipation or transmission) and which frequency-weighting should be considered for the optimization of the liner constitutive parameters, namely h d ,  , h t and D . Global particle swarm algorithm has been used to find out the set of optimal parameters that maximize the overall frequency-weighted coefficients,

inte, r. no oise — 21-24 AUGUST SCOTTISH EVENT CAMPUS ? O ? ? GLASGOW Uy, =50ms! (c) Uy, =80ms! a

  

f

n f

n n f f f f f c c with    , c or      related to absorption, transmission or

  

max

max

min d d

f

f

min

dissipation, Hz 20 min  f and Hz 10 4 max  f .

0.02 H = Ss B _ - 80 100 0.02 Se Hs S — 0.002 ie) 20 40 60 80 100 0 20 40 60 80 100 A(%) A(%) H | 20 0.8 TL (dB) rption

Figure 5: Wavenumber-based simulations of the acoustical properties of a MPPCP whose parame- ters are optimized to maximize the total (green), inverse (blue) and inverse squared (red) absorption (a) and TL (b), and those of a MPPCP with nominal parameters (thin black), all undergoing an aeroacoustic excitation with acoustic magnitude % 8  A and free-stream velocity 1 s m 30  .

80 100 0.002 ero 20 40 60 A(%) 80 100 0.002 20 40 60 A(%) 80 100 Abso:

As shown in Fig. 5(a) and Table 1, 2  n gives more weight to the absorption of the low- frequency components by decreasing   Aero H Max max f, f f  , e.g. the bandwidth of inefficient

backscattering. This is achieved at the expense of a drop in the absorption above Max f . 1  n achieves a good compromise to enhance absorption of the low frequency components without pe- nalizing too much the high frequencies.

Since high TL values are only achieved at high frequencies, Fig. 5(b) shows that only minimiza- tion of the total transmission coefficient is efficient to enhance the TL over the full bandwidth. Fig- ure 6 shows that maximization of the dissipated power leads to similar results than absorption opti- mization with comparable variations of the optimal parameters in Table 1. Inverse frequency- weighting is also the retained criterion that enhances the dissipation of the low-frequency compo- nents over a broad bandwidth. However, the dissipation criterion, that mostly occurs below Max f ,

fails to enhance the TL that reaches its highest values above Max f . Table 1 also shows that Max , h St consistently stays below 0.02 for all the optimization criteria.

Figure 6: Wavenumber-based simulations of the dissipation (a), absorption (b) and TL (c) of a MPPCP whose parameters are optimized to maximize the total (green), inverse (blue) and inverse squared (red) dissipation, and those of a MPPCP with nominal parameters (thin black), all undergo- ing an aeroacoustic excitation with acoustic magnitude % 8  A and free-stream velocity 1 s m 30  . Table 1: MPPCP optimal parameters and the Helmholtz and aerodynamic coincidence frequencies.

MPPCP optimized parameters

Optimised frequency-averaged

Optimised frequency-averaged

Optimised frequency-averaged

dissipation

absorption

t ransmission

0 f 1  f 2  f 0 f 1  f 2  f 0 f 1  f 2  f d (mm) 0.1 0.2 0.4 0.1 0.2 0.6 0.04 0.1 0.05  (%) 2 2 2 2 2 2 2 2 2 t (mm) 0.1 0.2 2 0.1 0.1 1.1 2 2 1.1 D (cm) 3.1 5 5 3.1 5 5 0.5 0.5 5

H f (Hz) 2965 1776 520 2965 1953 701 1646 1646 702

Aero f (Hz) 628 366 31 628 442 57 31 31 57

Max , h St 0.010 0.011 0.007 0.010 0.012 0.014 0.002 0.005 0.001

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5. CONCLUSIONS

A wavenumber-based formulation is used to predict the absorption and transmission properties of a MPPCP excited by aeroacoustic wall-pressure fields. When compared against a full modal formula- tion and aeroacoustic measurements in a low-speed wind-tunnel, the simplified model well captures the large absorption of the low-frequency components up to a Strouhal number that stays lower than 0.02. It also predicts the uniform decrease of TL when increasing the flow speed or the magnitude of the acoustic component. This model enabled to perform cost-efficient optimization studies. Max- imization of the inverse-frequency averaged absorption appears to be the most appropriate cost function to achieve broadband absorption of the excitation low-frequency components whereas total minimization of the transmission ensures a wideband increase of the TL. 6. ACKNOWLEDGEMENTS

This study was funded in Spain by the Ministerio de Economía y Competitividad project TRA2017- 87978-R, AEI/FEDER, UE, and the mobility program ILINK+2018. It was supported in France by the ANR VIRTECH (ANR-17-CE10-0012-01). 7. REFERENCES

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