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Flat Fresnel-Spiral Acoustic Metamaterials Composed of Several Arms Ventilated Metamaterials for Simultaneous Broadband Sound Absorption and Air Circulation Sanjeet Kumar Singh 1 Department of Design, Indian Institute of Technology, Kalyanpur, Kanpur, Uttar Pradesh 208016, India Shantanu Bhattacharya 2 Microsystems Fabrication Laboratory, Department of Mechanical Engineering, Indian Institute of Technology, Kalyanpur, Kanpur, Uttar Pradesh 208016, India

ABSTRACT With a steep rise in the urban population requiring an increased number of buildings, public and private transport systems, urban noise is posing a serious environmental problem affecting health. To attenuate the effect on the well-being of human health, a variety of conventional sound-absorbing materials suitable at mid and higher-frequency noise absorption are commonly being used but low and mid-frequency noise remains a challenge. These applications are further limited by the ventilation efficiency and acoustic performance in conventional noise barrier limits of their fields. Acoustic metamaterial presents a unique solution as an artificially designed material showing low- frequency noise mitigation. A novel subwavelength device having thickness of 14mm (<2 cm), with potential application in noise mitigation and air ventilation solution is presented herein. In this study, the design and fabrication of a small prototype based on a Fresnel-spiral shape composed of several arms are performed. The acoustic properties of the proposed ventilated metamaterial in terms of sound absorption and air ventilation were determined through experimental and numerical investigations. The experimental results shows excellent sound attenuation with a wide bandwidth (grater than one octave in the range of > 884 Hz), acoustic properties leading to potential applications in urban noise control for low and mid-frequency ranges.

1 sanjeet@iitk.ac.in 2 bhattacs@iitk.ac.in

1. INTRODUCTION

Designing subwavelength ventilation barriers with simultaneous soundproofing is a great challenge and an engineering demand. Due to the increasing number of motorcycles, automobiles, and traffic noise in developing countries with tropical climates, windows are useless. A loud and prolonged noise of drone propellers, airplane turbines, and MRI machines in which sound mitigation and airflow is critical. The researchers used traditional ventilation barriers with absorptive lining or perforated partitions, which cause significant pressure drop but do not allow for free air flow[1],[2]. Recent advancements in acoustic metamaterials have provided a solution for air-permeable barriers that operate at frequencies below 1600 Hz audible range. The advantage of using acoustic metamaterials is that they manipulate and control the sound wave that are often unobtainable through conventional materials, lead to minimizes size and thickness. Researchers developed acoustic ventilated metamaterials based on the locally resonant unit (Helmholtz resonators, quarter-wavelength tubes, membranes, etc.) for noise reduction in low-frequency range with a compact size in such an unrivaled open way[3],[4],[5]. Kumar et al. [6], reported acoustic metamaterials based on a labyrinthine structure composed of six-unit cells with different geometrical configurations to get broadband sound absorption. However, this structure need carefully design and is not easy to manufacture. Researchers used a Fano-like interface to improve ventilation throughout the structure [7],[8],[9] to block low- frequency sound for a narrow working frequency range around every destructive interface frequency, which limit its application in various field of engineering. Broadband absorption with ultra-open acoustic metamaterials at low frequency has been achieved through the coupling of multiple lossy resonators by Xiang et al. [9]. These designs solve broadband issues but having a one-sided opening, and limited bandwidth limits their application in various field. Furthermore, designing thin ventilated metamaterials with excellent sound absorption remains a challenge. To overcome this challenge, the authors proposed a subwavelength thin (1.4 cm to 3.5 cm) metamaterial structure inspired by a flat Fresnel spiral [11]. It is composed of several arms with the ventilation capacity to obtain broadband multiple sound absorption peaks in the range of frequencies 0–1600 Hz. The suggested structure consists of two disks, each disk has four spiral channels and a

2

ventilated opening of 12% (The ventilated area is calculated as, 𝑉 𝑎 = ( 𝑑 𝐷 ⁄ )

× 100% ) at the center

to allow the air to pass through. These spiral channels rotate the acoustic wave around the central orifice and slow down the acoustic wave propagation. To address broadband sound absorption and transmission challenges associated to facades and acoustic barriers, an array of eight identical spiral arm resonators is being coupled. The current study focused on the noise attenuation mediated by the coupling of N spiral resonators, with the goal of achieving a high bandwidth sound absorption and a subwavelength structural thickness. The following is a summary of the content of this article: In Section 2, the author proposes the mathematical model and design strategies of an acoustic meta-structure. Section 3 investigates the effect of geometric configuration and coupling of N spiral resonators on the acoustic characteristics of the proposed metamaterials. In section 4, we performed the FEM CFD simulation to understand its ventilation characteristics.

2. THEORY 2.1. MATHEMATICAL MODEL

Figure 1: Schematic diagram of the (a) Archimedes spiral. (b) Parameters in polar coordinates The spiral curve can be expressed as 𝒓= 𝒂 𝒊 + 𝒃𝜽 (1)

Where 𝑎 𝑖 is the initial radius of the spiral with its final angle θ. Whose 𝑥 and 𝑦 components in polar coordinates are

𝑥= 𝑟cos 𝜃= (𝑎+ 𝑏𝜃) cos 𝜃 (2) and

𝑦= 𝑟sin 𝜃= (𝑎+ 𝑏𝜃) sin 𝜃 (3)

where the spiral growth ratio (parameter b) is determined as

𝑎 𝑓 −𝑎 𝑖

𝑏=

2 𝜋 𝑁 (4)

Where N denotes the number of turns of the spiral The spiral depicted in figure 1 (a) is a one-dimensional curve. This one-dimensional curve is used to create the 3D acoustic wave rectangular spiral channels (RSC). The dimension of the channels having the width w and a height h. The final angle of the spiral can be calculated as

𝑎 𝑓 −𝑎 𝑖

𝜃 𝑓 =

𝑏 (5)

The total length of the 2D RSC is then calcula ted as

𝐿 𝑅𝑆𝐶 = ∫ √𝑟 2 + (𝑑𝑟𝑑𝜃) ⁄ 2 𝑑𝜃 𝜃 𝑓 𝜃 𝑖 (6)

If the RSC is immerse in a sample with diameter “a” , a sound wave entering in the system encounters the following impedance at the entrance of the channel:

Oa

𝑍 𝑅𝑆𝐶 = −𝑗 𝑍 𝑐 cot(𝑘𝐿 𝑅𝑆𝐶 )/∅ (7)

Where 𝑍 𝑐 = 𝜌 0 𝐶 is the characteristics impedance of the channel, the wave number 𝑘= √−𝜔 2 𝜌 0 𝐶 where 𝐶 represent the sound speed in air and 𝜌 0 stand for the air density. The surface porosity 𝜑 is the ratio of the cross sectional area of the sound wave’s entrance to the total area. The entire acoustic impedance of these n spiral resonators can therefore be calculated as follows:

−1

𝑍 𝑎 = [∑ ℎ

1

1

1

(8)

2𝐻 (

𝑍 1 +

𝑍 2 … … …

𝑍 𝑛 )]

n = 1,2 ,3 ,---------- now we can calculate the absorption coefficient of the system

4𝑅 𝑒 (𝑍 𝑎 ) [1+𝑅 𝑒 (𝑍 𝑎 )] 2 +[𝐼 𝑚 (𝑍 𝑎 )] 2 (9)

𝛼=

2.1 METAMATERIAL DESIGN

Figure 2: (a) Conceptual diagram of the ventilation acoustic barrier consisting of unit cells. (b) Schematics of the unit cell composed of eight spiral resonators coupled around the air passage hole. (c) Colored channels show the acoustic wave flow path inside the spiral chambers.

In this study, the author proposed a thin 14 mm ventilated acoustic meta-structure consisting of eight spirals and axially coupled resonators to solve sound mitigation problems. Figure 2(a) shows the conceptual view of proposed ventilated barriers, of which the unit cell is schematically illustrated in figure2 (b). The thin metamaterials proposed by the author are composed of two disks. In each disk, four identical spiral channels are placed at 0 ° ,90 ° ,180 °, and 270 ° with respect to the x-y plane, as shown in figure 2(c). The second disk rotates 45 ° with respect to the first disk and is pasted together. The sound propagation paths of each channel are shown in different colors in figure 2(c). The length of each channel is 0.40896, calculated by using equation no. (6).

The sample was fabricated using a 3D printer, as shown in figure 3(a). Polylactic acid (PLA) with a density of 1.24 kg/cm3 and an elastic modulus of 4.8 GPa, was used in the construction, using a nozzle extruder with a 0.4 mm technique. The samples were tested in an 100 mm diameter impedance tube and made samples accordingly. The acoustic absorption experiment has been performed in laboratory conditions following the two-microphone transfer function following the ASTM E1050-

eo

19 standard[10] and a FEM based simulation, with the help of available commercial software ANSYS 2021 R1 discussed in section 3.

3. RESULTS AND DISCUSSION 3.1. ACOUSTIC PERFORMANCE MEASUREMENT The overall acoustic performance of the proposed meta cell depends upon the cross-section of the input, output interface and the eight-side branch spiral resonators. The ratio of the open area in the ventilated metamaterial is ∅= 𝑑 2 𝐷 2 ⁄ and 𝑆 𝑜 = 𝑤× ℎ is the cross-sectional area of the spiral channels. The eight spiral resonators are of equal length. The spiral resonators can be considered as quarter-wave resonators and their resonance frequency can be calculated by 𝑓 𝑟 = 𝑛𝐶4𝐿 𝑅𝑆𝐶 ⁄ , where n=1,3,5…., c is the sound speed in air =343m/s and 𝐿 𝑅𝑆𝐶 is the length of the spiral channel that is calculated by equation (6). The first four resonance frequencies of each resonator are approximately 𝑓 1 = 199.35Hz, 𝑓 2 = 598.05Hz, 𝑓 3 = 996.75 Hz and 𝑓 4 = 1395.45 Hz. The geometric parameter cross-ponding of each spiral resonator is 𝑤= 3𝑚𝑚 𝑎𝑛𝑑 ℎ= 3𝑚𝑚 , the number of turns of the spiral is two, and the length of the spiral channel is approximately 𝐿 𝑅𝑆𝐶 = 350.5 𝑚𝑚 . The central circular hole radius 𝑟= 17𝑚𝑚 . Figure 3 (b) shows the numerical and experimental results of the sound absorption coefficient (abs) for the proposed air-ventilated metamaterial. All the simulation are performed using the commercial software ANSYS 2021 R1 acoustic module. The acoustically hard boundary conditions were applied at the interface of air and rigid body because, the young modulus and mass density of the 3D printed materials (PLA) are higher than the air. As a result, sound waves are barely able to get through the structure. At the source end, the radiation boundary condition was applied for non-reflecting field. A plane wave boundary condition was used to simulate the incident sound wave. In our study, the

𝑡ℎ of the shortest incident simulated sound wavelength. The numerical and experimental measurements were performed, and the maximum sound absorption coefficient reached 0.91 at 1344 Hz and 0.6 at 992Hz. The relative broadband sound absorption (>50%) was obtained over a frequency range 884Hz to >1600Hz of more than one octave. The solid blue line represents the experimental results obtained by the acoustic impedance tube test, while the solid orange line represents the numerical simulation (FEM) without taking into account viscothermal losses. Near the resonance frequencies, The FEM simulation and experimental results are quite different, which is consistent with the influence of viscothermal losses on the sound absorption performance [11],[12]. The experimental result shows that the absorption coefficient after the first resonance frequency gradually decreases and then gradually increases when viscothermal effect are included, but in FEM simulation the absorption valley disappears under ideal conditions[13] and shows narrow sudden abs valley. Other factors that affect the resonance are sometimes known as non-ideal effects, like form factors, effects on edges, manufacturing imperfections, etc.

element size of the mesh was taken less than the 1 6 ⁄

Figure 3: (a)Air ventilated 3D printed meta unit cell. (b) Sound absorption spectrum of unit cell with all the spiral resonators with same channel length. Given the axial coupling capabilities of eight spiral resonators in various axial orientations, the following two designs will be carried out to better understand the low frequency broadband sound absorption scenarios: (1) increasing the number of spiral resonators and (2) the effect of geometry configuration without changing the thickness of the structure. 3.2. THE EFFECT OF INCREASING THE NUMBER OF SPIRAL COUPLED RESONATORS The thin, broadband, multiple-frequency air-ventilated acoustic absorber is highly desired in noise control engineering. In this paper, the author achieved narrow and broadband multiple-frequency sound absorption by increasing the number of spiral resonators. In one disk, there are four spiral resonators of the same length. Initially, the author reported the absorption spectra of two disks (a total of eight resonators), as shown in figure 3 (a). Now authors investigated the effect of coupling of more than eight spiral resonators on the sound absorption spectrum, as shown in figure 4(b).

Figure 4: (a) Sound pressure level (SPL) distribution of ventilated acoustic meta-structure. (b) Variation of sound absorption spectrum concerning the increasing number of disks or resonators.

We designed three samples: (1) Three disks with 12 resonators and a thickness of 21mm (2) four disks with 16 resonators and a thickness of 28mm (3) five disks with 20 resonators and a thickness of 35 mm. Figure 4(a) shows the sound pressure level (SPL) in dB of the third sample The sound absorption spectrum of all proposed three meta-structure is shown in figure 4 (b). when we increased the number of disks, we got one narrow sound absorption peak at 224 Hz and two broadband sound absorption around 850 Hz and around 1300 Hz. As we increase the no. of the disk, the structure shows broadband sound absorption, and the maximum absorption value is 0.65, 0.6, and 0.8 at 224 Hz,900 Hz, and 1300 Hz, respectively. Over a band of frequencies in the range of 600 Hz to 1600 Hz, the absorption drop to zero around the 1000 Hz because of not considering the visco- thermal losses in the FEM simulation as we explained in section 3.1. 3.3. THE EFFECT OF INCREASING THE NUMBER OF SPIRAL TURNS This section will consider a set of three different meta-structure, whose thickness is constant (14 mm), formed by two disks. The geometric parameters cross-ponding to the first AVM (acoustic ventilated metamaterial) are w=2mm, h=6mm, and the number of revolutions of spiral N=3 and second AVM is w=1mm, h=6mm, and N=3. Finally, the third AVM is set as w=1mm, h=6mm, and N=4. This meta-unit shows a wider sound absorption bandwidth with multiple absorption peaks between 950 Hz to 1600 Hz, with an absorption amplitude of 0.6 to 0.85, as shown in figure 4(b). As we increase the number of turn (third sample), one new narrow absorption peaks obtained along with broadband sound absorption. The results reveal that the spiral channels' geometric configuration does effect on adjusting the multiple sound absorption peak or broadband sound absorption.

Figure 5: (a) Designed meta-structure of N=4 turn spiral ventilated sound absorber. (b) Variation of sound absorption spectrum with respect to changing the geometric configuration of spiral channel. As the result, it is obvious that the geometry of the meta structure can be optimized further to extend the frequency bandwidth and high absorption towards the lower frequency region below 900 Hz. 4. THE VENTILATION PERFORMANCE The proposed meta structure's ventilation effectiveness is numerically examined. A FEM based computational fluid dynamics (CFD) module available in ANSYS 2021 R1 software is utilized for

investigating the airflow performance of the proposed design. The developed numerical model has unit cell at the center and two air domains, 1 and 2 as shown in figure 6(a).

Figure 6: (a) Schematic diagram of the simulation model for measuring the ventilated performance of a unit cell (No. of disk is 2) developed in ANSYS CFD module. (b) Airflow performance of the proposed meta structure. No-slip boundary condition (velocity components are zero) is applied at the fluid-wall interface. Fluid flow rate was applied at the front face of air domain 1 and the end face of the air domain 2 set to be 1bar (atmospheric pressure). In this simulation, air at 25 ℃ was was allowed to pass through this sample at various rates, and the pressure drop responses were draw out for various air entry velocities. It has been observed that when the pressure drops across the proposed design increases, the airflow rate increases as shown in figure 6(b). As increasing air flow rate, resulting in a high-pressure drop ( ∆𝑝= −𝜎 𝑓 𝑉 𝑎 , where 𝑉 𝑎 is the airflow velocity) across the acoustic metamaterial by wind and buoyancy force [14],[9]. Therefore, ventilation capacity can be tuned as per requirement. 4. CONCLUSION

To conclude, we proposed a planner-profile meta-unit and thin subwavelength acoustic air-ventilated barrier, that enable airflow to pass while blocking sound across a wide range of frequency spectrum. The reported acoustic metamaterials here consist of N spiral resonators' axial coupling to obtain greater bandwidth and multiple sound absorption peaks, which can be possibly used where air- permeable is required. Such design effectively blocks the 90% of the incident energy within in the

884 -1600 Hz frequency range, while its structure thickness is only 14mm (approximately 𝜆 26 ⁄ ).

The optimization of the structure for low frequencies below 800 Hz will be investigated in the future to get broadband sound absorption and minimize the thickness through the coupling of spiral resonator units or increasing the sound propagation path. Further, more research is required in this area of ventilation performance with the changing of the cross-section of the air passage hole and the geometric configuration, that will be investigated in future. 6. REFERENCES

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