Experimental investigation of a phase-cancelling slow-sound metamaterial with mean flow

Richard Martin 1

CAPS Laboratory, Department of Mechanical and Process Engineering, ETH Zürich Sonneggstrasse 3, Zürich 8092, Switzerland

Bruno Schuermans 2

CAPS Laboratory, Department of Mechanical and Process Engineering, ETH Zürich Sonneggstrasse 3, Zürich 8092, Switzerland

Nicolas Noiray 3

CAPS Laboratory, Department of Mechanical and Process Engineering, ETH Zürich Sonneggstrasse 3, Zürich 8092, Switzerland

ABSTRACT Acoustic metamaterials have a wide variety of potential applications in various engineering disciplines. In many cases, the acoustic medium can be assumed to be at rest. However, for a lot of engineering applications a moving fluid is used to transport energy or substances, or to perform mechanical work. For those cases, acoustic waves are advected and acoustic energy can be dissipated or generated. These e ff ects play a significant role for the design of acoustic metamaterials. In this work, an acoustic metamaterial with a low Mach number mean flow is presented. It is an array consisting of an even number of channels. Half of them are equipped with a slow-sound metamaterial, whereas the other half has unaltered sound propagation characteristics. Thus, the resonance frequencies of the channels are shifted. This results in a phase shift in the acoustic velocity response between both types of channels for a given pressure excitation, resulting in phase cancellation. However, this e ff ect is weakened for higher mean flow velocities.

1. INTRODUCTION

Acoustic metamaterials give a new perspective on an old topic: how to manipulate the propagation of sound. In general, sound propagation is governed by the wave equation. This partial di ff erential equation has only one state variable (normally the pressure p ) in at least two dimensions (space and time) and one material parameter (e.g. the speed of sound c 0 ). Naturally, the boundary conditions to the wave equation play a big role for the propagation of sound. Here, acoustic metamaterials are a new and useful paradigm, especially for periodic structures. Instead of solving the wave equation

1 richard.martin@mavt.ethz.ch

2 bschuermans@ethz.ch

3 noirayn@ethz.ch

ante. noize 21-24 AUGUST SCOTTISH EVENT CAMPUS O ? . GLASGOW

for a given structure with its boundary conditions directly, the e ff ect of the macroscopic structure can be modelled by adapting the e ff ective material properties of the acoustic medium accordingly. This results in an artificial medium with properties which are often frequency dependent and which are not bound to the actual microscopic properties of the acoustic medium, hence the name metamaterials. For example, lower or even negative wave propagation speeds can be achieved. Inversely, structures can be designed to achieve certain e ff ective material properties, which cannot be achieved with naturally occurring bulk materials [1]. The previously mentioned reduction of the speed of sound is the main e ff ect utilized for this study. It is analogous to the slow light e ff ect in photonic crystals [2]. Studies dealing with the slow-sound phenomenon were conducted by Aurégan, Groby et al. [3–5], and Jiménez et al. [6–8]. A key principle that was used in the majority of these studies was the shift and accumulation of the resonance frequencies of the main slit or channel due to the slow-sound e ff ect. This accumulation of the main channels resonance frequencies in proximity to the resonance frequencies of the side-wall cavities leads to the band-gap e ff ect, which can be utilized for acoustic liners. However, for this study the focus lies on the shift of resonance frequencies in the low-frequency range significantly below the resonances of the cavities. A phase-cancelling metamaterial utilising this e ff ect was already demonstrated by Kim et al. [9]. They achieved phase cancellation for an upstream disturbance by splitting a waveguide lengthwise into two parts and applying a slow-sound metamaterial to one of the parts. The resulting delay of a disturbance from upstream in the slow-sound channel caused a phase cancellation at the downstream side for a certain frequency, similar to a Herschel-Quincke waveguide. For this work, we regard a metamaterial similar to Kim et al. [9], consisting of channels which are embedded as a rectilinear grid in a wall separating two larger acoustic domains. There is a static pressure di ff erence ∆ ¯ p between the two domains causing a mean flow in each channel. The direction towards the domain with the larger static pressure is henceforth called upstream and the opposite direction is called downstream. On the downstream side of the wall, there is an acoustic disturbance, causing velocity oscillations at the channel outlets. We want to achieve phase cancellation for these velocity oscillations, using a slow-sound metamaterial. This slow-sound metamaterial is reducing the e ff ective speed of sound for half of the channels and is thus shifting its resonance to lower frequencies. The resonance induces a phase-shift in the channel. Thus, above the resonance frequency of the slow- sound channel and below the resonance frequency of the non-slow-sound channel, phase cancellation can be achieved and the mean value of the velocity oscillations will vanish. This behavior is equivalent to a sound-hard wall, while maintaining permeability to a mean flow of air. The remainder of this paper is divided into three sections. Section 2 deals with the theoretical background of sound propagation in tubes, phase cancellation and the slow-sound e ff ect. Section 3 contains a brief description of the metamaterial, its manufacturing, and the experimental setup, followed by the experimental results. Section 4 summarizes the findings and gives an outlook to future work.

2. THEORY

In this section the theoretical basics for phase cancellation between channels with di ff erent resonance frequencies, and the slow-sound e ff ect will be discussed. Figure 1a shows the schematic of the metamaterial, which was already briefly described in Section 1. The channel in which the sound propagates with the speed c 0 , is called channel A. In channel B the e ff ective speed of sound in lengthwise direction is reduced by a slow-sound metamaterial, which is implemented as an array of side-wall Helmholtz resonators, or cavities. On the upstream side, there is a static pressure ¯ p u > ¯ p d causing a mean flow in both types of channels ˙ m A , B . The pressure oscillations at the upstream side are assumed to be p u = 0. On the downstream side there is atmospheric pressure ¯ p d and a disturbance in form of p d . For the frequency of interest, the outlet admittances Y A , B are of opposite phase due to the shift of the resonance frequency of the slow-sound channel B.

10 4 a b

downstream upstream

Y A Y B

10 2

Y

˙ m A

c A = c 0

Y A

| Y | [–]

¯ p u > 0 ¯ p d = 0 p u = 0 p d , 0

10 0

˙ m B

c B < c 0

Y B

10 -2

cavity

0 200 400 600 800 10 -4

f [Hz]

Figure 1: (a) Basic concept for a phase-cancelling metamaterial with a slow-sound channel A and a non-slow-sound channel B (b) Admittances of the individual channel openings at downstream side and mean admittance

2.1. Phase cancellation In pipes, only plane waves propagate in axial direction below the cut-on frequency of higher order modes. The wave equation reduces then to the one-dimensional case [10] " ∂ 2

∂ t 2 − c 2 z ∂ 2

# p = 0 , (1)

∂ z 2

where solutions are of the form p ( z , t ) = ( C 1 e − i k z z + C 2 e i k z z ) e i ω t . These solutions can be used to derive the velocity oscillations v = − i / ( Z 0 k 0 ) · ∂ p /∂ z at the outlets of the channels A and B, using the previously discussed boundary conditions. The normalized admittance Y = Z 0 v / p at the downstream side of the channels is then Y d = − i cot ( k z L eq ) k z

k 0 , (2)

where L eq is the e ff ective length of the channels, due to the inertia of the air in front of the inlet and outlet. The e ff ective axial wavenumber is k z = k 0 for the non-slow-sound channel A and k z > k 0 for the slow-sound channel B. To achieve phase cancellation, Y d , A = − Y d , B has to be achieved for a certain frequency. Figure 1b shows the curves of the channel admittances calculated with Equation 2. For the slow-sound channel, a constant factor for the wave numbers is assumed ( k B / k 0 = const > 1.). At the point above f = 200 Hz, where | Y A | = | Y B | , the admittances of the channels are of opposite phase. Thus, the average admittance of the metamaterial vanishes Y → 0, which corresponds to the behavior of a sound hard wall for this frequency.

2.2. Slow-sound e ff ect As it was already established in the previous section, waves propagate in hard-walled tubes with the speed of sound c 0 or the wave number k 0 = ω/ c 0 . This changes, however, when the walls are compliant. The wave propagtion is then split into an axial part with the wave number k z and a radial part with the wave number k r , such that k 2 0 = k 2 z + k 2 r . Because k r is complex, the axial wave number k z can become larger than k 0 , which corresponds to a slowing down of the sound. From the momentum equation and the boundary conditions of a duct with circular cross-section, the following equation can be derived [10] J m ( k r r i ) ( k r r i ) J ′ m ( k r r i ) = j Z w

1 k 0 r i . (3)

ρ 0 c 0

which is called the dispersion relation, where J m is the Bessel function of the first kind, r i is the inner radius of the tube and Z w is the side-wall impedance. This equation can be solved numerically for k z .

Because Z w is frequency dependent, the e ff ective axial speed of sound is frequency dependent as well. In the low frequency limit however, it is mainly depending on the wall compliance. A compliant wall can be realized in various ways. Here, the side-wall has openings to cavities. Where the size of the cavity defines the e ff ective compliance of the wall, and by this the slow-down factor of the speed of sound k 0 / k B , or c B / c 0 .

3. EXPERIMENTS

Two types of impedance tube experiments are conducted to verify the working principle of the metamaterial. First, the downstream admittance is determined with an open-ended tube. This setup approximates the idealized p u = 0 boundary conditions. For the second experiment, the tube is closed, and the transfer matrix is determined for three flow rates. The implementation of the acoustic metamaterial with its cavities is shown in Figure 2a. In this cutaway view, the 16 channels, 8 of each type, can be seen. The channels of type B are connected via 4 holes each to the cavities with rectangular cross section. In total, the metamaterial is composed of 10 layers with 8 cavities each, resulting in a total length of L = 135 mm. The metamaterial has a quadratic cross-section of approximately 62 × 62 mm 2 , which ensures a transition fit into the impedance tube. Figure 2b shows the manufacturing process using a Prusa MK3S printer with PETG filament. Note, that the structure is self-supporting, so no additional support structures are needed for the printing process.

a b

c

AE AE LS LS AMM

˙ m

mics

Figure 2: (a) Screenshot of the Prusa slicer (b) 3D printing of the model (c) Impedance tube setup for experiment (not to scale) with open end (solid lines only) and transfer matrix measurements with flow (solid and dotted lines). Both sides of the tube are equipped with anechoic ends (AE) and loudspeakers (LS). The acoustic metamaterial (AMM) is placed flush to the flange of the open end. Microphones (mics) are used to reconstruct the Riemann invariants in the tube.

The experimental setup is depicted in Figure 2c. For the open-end configuration, three microphones are mounted between the loudspeaker and the metamaterial. The metamaterial is mounted flush to the flange of the open end of the tube. The behavior of the channels of type A

Feature ype i esiveter Hil cert peineter Cverhang perimeter nea nil So int

is determined by plugging the channels of type B at both ends with Allen bolts, leaving only the channels of type A open. The opposite is done to determine the behavior of the channels of type B. For the closed configuration, three additional microphones are mounted on the upstream side of the metamaterial. The tubes are terminated with near-anechoic ends, consisting of horns and boxes with absorbing foam. The upstream side of the impedance tube can be pressurized through a port in the anechoic end, resulting in the mean flow ˙ m , which is measured using a mass flow meter. The loudspeakers force the system in stepped frequency sweeps. The Fourier coe ffi cients of the microphone measurements are used to calculate the Riemann invariants with the multiple microphone method [11], from which the downstream admittance and transfer matrix can be determined. Figure 3 shows the results for both types of experiments. For the experiment with the open end, the admittances determined while blocking the channels are scaled by a factor of two to obtain the admittances Y A and Y B for an equivalent metamaterial consisting only of channels of type A, or type B respectively. The phase-cancelling e ff ect predicted by the simplified model of the admittance, shown in Figure 1b, can be seen at f = 216 Hz. At this point, Y A and Y B are of equal magnitude. However, the phase di ff erence between both channel types is only 0 . 61 π instead of 1 π , so the phase cancellation is not perfect and thus, the averaged admittance Y does not vanish. The reason for this is that the channels are not purely reactive, as assumed for Equation 2.

open end

closed tube

u = 0 . 0 m / s u = 1 . 8 m / s u = 9 . 6 m / s

Y A Y B Y

| Y | [m 3 / (Pa s)] ∠ Y [rad]

10 -5

10 -5

10 -6

10 -6

1.5

1.5

1

1

0.5

0.5

0

0

-0.5

-0.5

-1

-1

0 250 500 750 1000 -1.5

0 250 500 750 1000 -1.5

f [Hz] f [Hz]

Figure 3: Admittances measured for open-end and closed-tube experiments. For the open-end experiments, the channels A are closed to measure the admittance Y B and vice versa. In the closed- tube configuration, the transfer matrix of the full system is obtained, from which the averaged admittance Y with idealized upstream boundary conditions can be derived. The experiment is repeated for three mass flow rates.

For the closed configuration with mean flow, the averaged admittance of the metamaterial Y is calculated via the transfer matrix. With this approach, the boundary conditions can be chosen freely. Here, ideal p u = 0 boundary conditions are assumed at the upstream side. The frequency of the phase-cancelling e ff ect is shifted to slightly higher frequencies ( f = 230 Hz), because closing the tube changes the end-correction term. Nevertheless, the averaged admittances Y are qualitatively equal for the open and closed configurations. If the static pressure at the upstream side ¯ p u is increased, the mass flow and thus the velocity in the channels increases as well. The mean flow velocity u is calculated for

the area of the impedance tube with u = ˙ m / ( ρ A ), assuming an incompressible flow. Due to the area contraction of the metamaterial, the flow velocity in the channels is approximately 8.5 times higher than in the impedance tube. For increasing mass flow rates, the phase-cancelling e ff ect gets weaker. One likely reason for this is the increased acoustic resistance of the system, which is caused primarily by vortex shedding at the outlet of the metamaterial [12].

4. CONCLUSIONS AND OUTLOOK

In this study an acoustic metamaterial (AMM) is presented, which utilizes the slow-sound e ff ect to shift the resonance frequency of channels in a flow duct to achieve phase-cancellation of the admittances. Thus, the AMM behaves like a hard wall for the frequency of interest, while allowing a mean flow to pass through. One drawback of the AMM is the creation of constructive interferences due to the shift of the slow-sound channels’ resonances to lower frequencies. For high mean flow rates, the AMM loses its phase-cancelling capability. Thus, special care has to be taken, when designing such an AMM for applications with considerable mean flow. In future studies, more sophisticated modelling approaches can be employed to investigate the metamaterial and to develop design strategies. One possibility for this are acoustic network models, which are based on analytically derived transfer matrices [10]. Furthermore, if there are clearly defined objectives for an application of this AMM, optimization techniques can be used to find optimal solutions, for example with respect to the static pressure drop and acoustical performance.

ACKNOWLEDGEMENTS

This research is part of the TORCH project and financed by the European Research Council (ERC) [grant number 820091].

REFERENCES

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