Long elastic open neck acoustic resonator in flow

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To cite this version:

Frank Simon. Long elastic open neck acoustic resonator in flow. Internoise 2016, Aug 2016, HAM-

BOURG, Germany. �hal-01399279�

Long Elastic Open Neck Acoustic Resonator in flow

Frank SIMON

Onera Centre de Toulouse, France

ABSTRACT

Passive acoustic liners, used in aeronautic engine nacelles to reduce radiated fan noise, have a Quarter-wavelength behavior, thanks to perforated sheets backed to honeycombs (SDOF, DDOF). So, their acoustic absorption ability is naturally limited to medium and high frequencies because of constraints in thickness.

The ratio "plate thickness / hole diameter" usually centred around 1 generates impedance levels dependent on the incident sound pressure level (SPL) and on the grazing mean flow (by a nonlinear dissipation mechanism of vortex shedding), which penalises the optimal design of liners. Moreover, their physical law is not suited to an absorption to the lowest frequencies, as needed for future Ultra High Bypass Ratio engines with shorter and thinner nacelles (frequencies around 500 Hz). A possible approach could be to link the perforated panel with tubes introduced in the cavity, to shift the resonance frequency to a lower frequency by a prolongation of air column length. Applied for an aeronautical liner, the resonance frequency decreases considerably compared to classical resonator (factor of about 1/5).

The aim of this paper is to describe mathematically this type of concept called LEONAR ("Long Elastic Open Neck Acoustic Resonator) and to validate experimentally its linear behaviour according to SPL and flow.

Keywords: Sound, Insulation, Transmission Resonator, Acoustic Impedance, local reaction, absorbing material I-INCE Classification of Subjects Number(s): 34.3, 35.2.2, 37.1

1. INTRODUCTION

Locally reacting liners, as those used in aeronautical engine nacelles, are generally “sandwich”

resonators with a perforated plate linked to an honeycomb material above a rigid plate. Their absorption behavior can be described approximately with the principle of an Helmholtz resonator. The frequency range of absorption is so essentially controlled by the thickness of the honeycomb cavity ("Quarter-wavelength" behavior). The small size of holes (mostly from 0,4 to 2 mm according to industrial needs), absorbs the energy (thanks to the acoustic boundary layer applied at the internal walls) when a wave is propagated through the resonant cavity (1,2). The impedance can depend non linearly on the incident particle velocity level (or Sound Pressure Level) (1). So, acoustic “vortices” of particle velocity can occur at the resonator surface thus modifying the impedance. Many studies, since Ingard in the 50s (3), have tried to determine the influence of various parameters on the impedance and the absorption of holes. Gaeta & Ahuja (4) show in particular that to increase the perimeter of the hole with the same surface allows to increase the absorption with low magnitudes of particle velocity (< 1 m/s) but has not a significant effect for higher velocities. Above a threshold value of the ratio "vo / v*"

(acoustic or particle velocity / friction velocity of acoustic boundary layer) the hole behavior becomes non linear (5). It appears that the nonlinear dissipation mechanism of vortex shedding is crucial for noise level higher than 120 dB (6), value much lower than in an aircraft engine. Chandrasekharan et al.

(7) lead impedance measurements in a tube and compared results with classical laws of Hersh, Kraft and Candrall & Melling. It is shown that an increase of ratio "lp/d" (plate thickness / hole diameter) increases the frequency band on which there is linear behaviour of the plate with the sound level (between 100 and 150 dB until 6,4 kHz), which allows in particular to choose a liner characteristic more easily.

Finally, in order to enlarge the frequency range of absorption, different types of SDOF liners can be

piled up to constitute 2DOF or 3FOF liners. As for SDOF, the increase of SPL increases their

resistance and decreases their reactance (8). Nevertheless, their physical law is not suited to an

absorption to the lowest frequencies, as needed for future Ultra High Bypass Ratio (UHBR) engines

with shorter and thinner nacelles (frequencies around 500 Hz). A possible approach could be to link the perforated panel with flexible tubes introduced in the cavity, as proposed by Lu et al. (9), thus to shift the resonance frequency to a lower frequency by a prolongation of air column length. The interest of this concept has been proved experimentally by authors but without mathematical model to allow a determination of the absorption frequency range according to dimensional parameters.

The aim of this paper is therefore, firstly, to implement a mathematical model without hypothesis of short tube, in order to describe a concept of perforated plate coupled with tubes of variable lengths that fill a limited volume of a cavity (LEONAR for "Long Elastic Open Neck Acoustic Resonator), then to validate this concept with materials having one or several lengths of tube, flexible or rigid, within different cavities.

Some configurations are so simulated and tested to evaluate the relevance of theoretical approach and the linearity of behavior vs. incident sound pressure level. The impedance of such a material is also determined in a duct with a high grazing flow (Mach number 0.3) for comparison with its impedance without flow.

2. DESCRIPTION OF LEONAR

The resonator is composed of a perforated plate whose holes are connected to hollow flexible tubes, inserted in a cavity and opened at the end (Figure 1).

Figure 1 – Illustration of resonator with hollow tubes in cavity and physical principle

The parameters describing the resonator are, respectively: the thickness l

and the porosity σ

of plate, the inner radius r

and outer radius r

of tubes, the length of tubes l

, the thickness of cavity h.

Examples of materials in different configurations are shown below.

Figure 2 – Type 1 samples with tubes of variable lengths to place above a cavity

Figure 3 – Type 2 samples with tubes (g) to place above a cavity (d)

Figure 4 – Type 2 samples with tubes of variable lengths, partitioned (g) or not (d)

The propagation of waves along hollow tubes (direction x) can be shown as a linear combination of

"propagational", "thermal" and "viscous" modes (10). We assume that the wavelength of waves is much larger than the inner diameter of a tube.

To simplify the mathematical description, the theory of sound propagation is shown in narrow channels with plane parallel plates (distance: 2 r

).

The pressure field in presence of visco-thermal effects is solution of the classical equation:

0

2

2

¸ =

¹

¨ ·

© + §

∇ p

p ω c ₍₁₎

and can be expressed as following:

( ) ^{x , r A cos( q r ) ( e e )}

p =

+

⁽²⁾

with q

the propagation constant in the transverse direction of the channel.

r

x 0

r i

boundary layers

Figure 5 – Illustration of a channel with boundary layers

As the associated transverse velocity must vanish at the inner boundaries, the transverse propagation constant (with the assumption of q

r

<< 1 ) is given by:

( ) ( ) ( ) ( )

i v i

h

r

F k r

r k F r k F q c

− +

− −

=

¸ ¹

¨ ·

©

= §

1

2

1

2

ω γ

where ( ) ( )

X X X tan

F =

(3)

with

-

h h

k i δ

= 1 +

where

ω δ ρ

p

h

C

K

= 2 (thermal boundary layer thickness)

-

v v

k i δ

= 1 +

where

ρω μ

δ

= ² ( viscous boundary layer thickness).

The average axial velocity has the following form:

( )

(

) (

)

x

Aq F k r e e

u = 1 − −

ωρ ⁽⁴⁾

As

2 2

q

q c ¸ −

¹

¨ ·

©

= § ω , the complex propagation constant in the axial direction of the channel q is determined simply by:

( ) ( ) ( )

i h

r k F

r k F q c

−

−

¸ +

¹

¨ ·

©

= §

1 1 1 γ

ω (5)

For circular tubes, r

should be replaced by r

/ 2 in equations (3), (4) and (5).

One can notice that an assumption of low q

l

leads, for instance, to the Melling formulation (11).

The theory based on reference (12) provides nevertheless a better estimate of the impedance than reference (11) for high values of l

.

Then, one assumes that:

• the continuity of pressure and mass flow between tubes and the surrounding cavity is verified at the end of tubes

• the transmitted waves propagate in the rigid cavity, without loss, mainly in the direction of thickness, as for a classical resonator.

•

The specific impedance of the structure is given thanks to the plate porosity:

p t s s s

c ir Z c x

Z

σ ρ

ρ ^{= + = (6)}

with Z

the impedance at the opening of tubes

, then the absorption coefficient, in normal incidence can be calculated.

3. VALIDATION

The validation is led on different types of resonator (with flexible tubes or not) and variable lengths

(with 1 or 2 tube lengths in parallel).

3.1 Type 1 Resonator

The impedance is obtained in an impedance tube equipped with three microphones for pressure measurements. On the opposite side of the tube, the loudspeaker generates a broadband random noise propagated in plane waves from 100 to 5,000 Hz from 100 to 145 dB.

The standard measurement method for two microphones is used in accordance with references (13-15). The three microphones taken by pair allow to satisfy the total frequency range.

First tests have been led for type 1 samples whose characteristics are specified in Table 1.

Table 1 – Characteristics of type 1 samples l

(mm) s

(%) r

(mm) r

(mm) l

(mm) h (mm)

1 1.92 0.35 0.55 10,20,

30,60, 90

35.9

It appears that these materials have a linear behavior according to the incident acoustic pressure level, representative of constant impedance and absorption coefficient (ex. for sample with l

=20 mm in Figure 5), while a sample with only the perforated plate (without tubes) generates a large variation of absorption (Figure 6). Non-linearity are due to acoustic vortices around the holes for a high ratio of

" r

/ l

" (hole radius / plate thickness) (6,7). So, to increase artificially the plate thickness, by the extension of tubes, prevent the presence of vortices.

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

0 500 1000 1500 2000 2500

Fréquence (Hz) 104 dB

133 dB

Frequency (Hz)

Figure 5 – Effect of SPL (dB) for a type 1 sample with lt =20 mm.

Coef. d'absorption (paroi perforée + cavité)

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

0 500 1000 1500 2000 2500

133.0 dB (100-2500 Hz) 123.0 dB (100-2500 Hz) 113.0 dB (100-2500 Hz) 104.5 dB (100-2500 Hz) 128.0 dB (1328 Hz)

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Fréquence (Hz) 104 dB

133 dB

Frequency (Hz)

Figure 6 – Effect of SPL (dB) for a classical resonator

(perforated plate σ

=4.7 %, r

=0.55 mm, l

=1 mm and h = 35.9 mm).

One can notice, also, that the frequency range of absorption is very different: around 260 Hz, for the resonator with tubes, vs 1300 Hz for the classical resonator.

The comparison of absorption coefficient for all samples (Figure 7) confirms that the length of tube allows to shift the frequency range of absorption (resonator thickness lower than λ /30). Nevertheless, an increase of length can be associated with a reduction of absorption coefficient, i.e. if l

> h, essentially because of a significant reduction of cavity volume.

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

0 500 1000 1500 2000 2500

Perforated layer Exp. 10 mm Exp. 20 mm Exp. 30 mm Exp. 60 mm Exp. 90 mm length of tubes

Figure 7 – Effect of tube length for type 1 samples , vs. classical resonator (cf. figure 6).

Simulations of reactance r (cf. equation (6)) are led with previous model for several samples (Figure 8) . The satisfying comparison with experimental results allows to be confident in the determination of the frequency range of absorption, relative to "0" reactance.

The simulation of reactance for tubes satisfying the characteristics of Table 1 but placed in front of a rigid background (Figure 9), i.e without cavity, shows that the maximum absorption is obtained at high frequencies (from 2760 Hz). So, it appears that the coupling with the surrounding cavity is predominant to generate an absorption in low frequency range.

-20 -10 0 10 20 30 40

100 200 300 400 500 600 700 800 900 1000

Simu. 10 mm Simu. 20 mm Simu. 30 mm Exp. 10 mm Exp. 20 mm Exp. 30 mm

Figure 8 – Comparison of simulated /experimental reactance for type 1 samples

-1000 -900 -800 -700 -600 -500 -400 -300 -200 -100 100 200 300 400 0

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Simu, 10 mm Simu, 20 mm Simu, 30 mm

Figure 9 – Simulated reactance for type 1 samples without cavity

3.2 Type 2 Resonators 3.2.1 Tests in impedance tube

Figures 11 and 12 show a comparison on absorption coefficients simulated and measured in impedance tube for 2 configurations of resonator defined by the following characteristics:

- Configuration "mono tubes" : so-called long tubes (tube "1") or so-called short tubes (tube "2" cf.

figure 3) with a length ratio of 3.

The other characteristics are similar for these 2 resonators: i.e. the thickness l

, the porosity σ

of plate, the inner radius r

and outer radius r

of tubes, the thickness of cavity h.

The so-called long tubes have a length lower than the thickness of cavity in order to use fully the effect of cavity volume.

- Configuration "double tubes" : so-called length and short tubes introduced simultaneously in the perforated plate tubes (Figure 4).

The total porosity is therefore doubled.

Two samples ate tested, partitoned (tubes "1" and "2" in different cavities) or not (tubes "1" and "2" in the same cavity).

With several tube lengths, the configurations of tube, introduced by transfer matrices, are simply parallelized.

One can notice, for "mono tubes" (Figure 10), an high absorption with an equivalent level whatever the medium frequency of interest. On the other hand, the frequency range decreases vs. the medium frequency.

The simultaneous using of two tube lengths (configuration "double tubes") in a same cavity increases the medium frequency of absorption (Figure 11), without extending the frequency range, which is not interesting for target applications. But if the two lengths of tube are located in two cells, there are 2 frequency area of high absorption around the frequency range without partition.

So, a distribution with several lengths of tube in independent cells can allow to extend the frequency range, as with 2DOF or 3DOF liners, but with a low thickness of cavity.

Moreover, it can be observed that simulations are representative of physical phenomena, whatever the

tested configurations.

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

200 400 600 800 1000 1200 1400 1600 1800 2000 Fréquence (Hz)

Simu. 14 mm tube 1 Exp. 15 mm tube 1 Simu. 5 mm tube 2 Exp. 5 mm tube 2

Figure 10 – Comparison of simulated /experimental absorption coefficient for type 2 samples - configuration "mono tubes"

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

200 400 600 800 1000 1200 1400 1600 1800 2000

Fréquence (Hz)

Simu. Double - sans partition Exp. Double - sans partition Simu. Double - avec partition Exp. Double - avec partition

Figure 11 – Comparison of simulated /experimental absorption coefficient for type 2 samples - configuration "double tubes"

3.2.2 Tests in aeroacoustic bench

Finally, a sample of "mono tubes" with so-called long tubes has been manufactured to be tested with grazing flow in the aeroacoustic test bench B2A (Figure 12).

The Aero-Thermo-Acoustic test bench is specifically used to perform in-duct flow Laser Doppler Velocimetry (LDV) or pressure measurements along an acoustic liner in presence of a grazing flow (16).

Two loudspeakers can generate plane waves, up to 3000 Hz, in the wind tunnel (cross-section of 50×50 mm² ) with a turbulent flow (maximum bulk Mach number = 0.5). The testing cell can contain, in its lower part, a sample of material to be studied (30 x 150 mm²).

In the present case, the acoustic pressure field is acquired upstream, in front of and downstream of the sample, on the opposite wall (Figure 13).

Computations rely on the resolution of the 2D linearized Euler equations in the harmonic domain, spatially

discretized by a discontinuous Galerkin scheme (17), which presents advantageous properties. First, it is

weakly dispersive and dissipative. In addition, boundary conditions are imposed through fluxes, which is

particularly robust and straightforward.

Figure 12 – Scheme of aero-acoustic bench B2A with testing cell

UPSTREAM DOWNSTREAM

Sample length

Figure 13 – Location of microphones on the upper wall of testing cell (green circles)

A measurement of acoustic transmission loss is firstly led without grazing flow, by separation of upstream and downstream waves with 2 pairs of microphones located on both sides of the test cell (Figure 14).

upstream

p

p

downstream

p

downstream

p

Figure 14 – Location of upstream and downstream microphones for separation of upstream and

downstream waves

It appears on figure 15 that the maximum of transmission loss is reached around 1000 Hz, frequency of maximum absorption observed in the impedance tube (cf. figure 11).

0 2 4 6 8 10 12

400 600 800 1000 1200 1400 1600

Fréquence (Hz)

dB

Frequency (Hz)

Figure 15 – Transmission Loss (dB) for a type 2 sample - configuration "mono tubes" so-called long tubes (tube "1"), without grazing flow

The figure 16 shows then simulated and measured pressure fields at 1000 Hz, with a grazing flow at bulk Mach number 0.3.

Figure 16 – Comparison of simulated /experimental pressure fields (dB) along the testing cell (respectively in red and green symbols) for a type 2 sample - configuration "mono tubes" so-called long tubes (tube "1"),

at 1000 Hz and bulk Mach number 0.3

The impedance deduced by direct simulation and comparison with measured data in B2A is close to values

measured in impedance tube (so without flow) (Figure 17).

-6 -4 -2 0 2 4 6

400 600 800 1000 1200 1400 1600

Fréquences (Hz)

Im péda nce ré dui te

Résistance - tube à impédance Réactance - tube à impédance Résistance - simu. Directe - B2A Réactance - simu. Directe - B2A

Resistance - impedance tube Reactance - impedance tube Resistance - direct simu. - B2A Reactance - direct simu. - B2A

S p e ci fic im p e d anc e

Frequency (Hz)

Figure 17 – Impedance deduced from tests in B2A at bulk Mach number 0.3 and in impedance tube, for a type 2 sample - configuration "mono tubes" so-called long tubes (tube "1")

These results confirm the linear behaviour of LEONAR resonator vs. grazing flow, contrary to a classical resonator with the same porosity.

4. CONCLUSIONS

This experimental and theoretical study has shown that the introduction of tubes in a cavity of a conventional resonator generates a significant shift of the frequency range of absorption towards lower frequencies, thus by a prolongation of air column length. So, the thickness can reach a value lower than λ /30.

Despite the simplicity of theoretical approach, the impedance can be determined precisely with dimensional parameters.

It appears that the reaction of LEONAR is independent of sound pressure level (observed in impedance tube) and of Mach number up to Mach 0.3 (observed in aeroacoustic test bench), which is an asset for an aeronautic application.

These first results allow to consider that these resonators with linear behavior could be an alternative to classical resonator with requirements of robustness and cleaning theoretically satisfied as for classical resonator.

ACKNOWLEDGEMENTS

The research presented in this paper was led as part of French projects, ANR MANSART (Matériaux sANdwicheS ARchiTecturés).

Many thanks for the manufacturing achieved by V. Fascio (ATECA).

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