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Recent Advancements in Helmholtz Resonator Based Low-Frequency Acoustic Absorbers: A Critical Review

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Abstract

Helmholtz resonator (HR) is an elementary resonating structure predominantly used for acoustic wave manipulation. The sound absorption capabilities of HR are well examined and widely accepted, and it has extensive applications in engineering acoustics. Perhaps, low-frequency sound mitigation is a major technological challenge wherein, HR based absorbers play a pivotal role. In this review, the recent advancements in various HR based sound absorbers are considered in general and low-frequency absorbers in particular for a detailed comparison and critical evaluation. Since the majority of the reported investigations have numerical predictions to corroborate the experimental findings, a detailed review of analytical and computational methods is necessary. Initially, finite element computations of a conventional HR are performed to assess the efficacy of trusted simulation techniques such as thermo-viscous, narrow-region and poro-acoustics models. Then, the structural aspects and noise absorption characteristics of various alterations of conventional HR configurations are critically examined using an analytical approach. Thereafter, a detailed appraisal of the low frequency sound attenuation properties of different HR combinations such as arrays of resonators, hybrid models, and acoustic metamaterials is performed. Moreover, a non-dimensional performance parameter is introduced for uniform comparison among available absorbers and to identify suitable candidates for efficient low-frequency acoustic attenuation. Finally, different optimization approaches including forward and inverse design strategies for selecting appropriate sub-wavelength HR designs for targeted low-frequency noise mitigation are also provided. The development of effective strategies for the creation of HR structures amenable to the real-life industrial environment that provide low-frequency acoustic attenuation is discussed as a future direction.

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Abbreviations

\(l_{N}\) :

Neck length of Helmholtz resonator

\(d_{N}\) :

Neck diameter of Helmholtz resonator

\(l_{C}\) :

Cavity length of Helmholtz resonator

\(d_{C}\) :

Cavity diameter of Helmholtz resonator

\(S_{N}\) :

Cross sectional area of neck

A :

Cross sectional area of HR

V :

Volume of HR cavity

L :

The difference between the peak and valley values of the sound pressure level

\(H_{12}\) :

Transfer function

\(x_1\) :

The distance from the first microphone to the absorber

s :

Distance between microphones

R :

Reflection coefficient

\(l_{e}\) :

Effective neck length

c :

Velocity of sound in air

k :

Wavenumber

p :

Acoustic pressure

\(\textbf{v}\) :

Fluid velocity

T :

Temperature

K :

Thermal conductivity of air

\(C_{p}\) :

Specific heat at constant pressure

\(z_{0}\) :

Impedance of air

\(P_0\) :

The equilibrium pressure

\(T_0\) :

The equilibrium temperature

\(Z_{HR}\) :

The acoustic impedance of Helmholtz resonator

\(Z_{N}\) :

The acoustic impedance of neck of Helmholtz resonator

\(Z_{C}\) :

The acoustic impedance of cavity of Helmholtz resonator

t :

Thickness of wall

\(c^c\) :

Complex sound speed

\(k^c\) :

Complex wavenumber

\(k_v\) :

Viscous wavenumber

\(k_h\) :

Thermal wavenumber

\(N_{Pr}\) :

Prandtl number

\(K_{eq}\) :

Equivalent bulk modulus

\(F(\omega )\) :

Correction function for complex effective fluid density

\(G(\omega )\) :

Correction function for complex effective bulk modulus

\(\alpha\) :

Absorption coefficient

\(\eta\) :

Dynamic viscosity of air

\(\rho\) :

Density of air

\(\rho ^c\) :

Complex density

\(\rho _{eq}\) :

Equivalent complex density by JCA model

\(\gamma\) :

Ratio of specific heats

\(\omega\) :

Angular frequency

\(\delta\) :

The end correction length

\(\Lambda\) :

Viscous characteristics length

\(\Lambda ^{\prime }\) :

Thermal characteristics length

\(\tau\) :

Tortuosity

\(\phi\) :

Porosity

\(\psi ^v\) :

Function of viscous field

\(\psi ^h\) :

Function of thermal field

\(J_n\) :

Bessel function of first kind with order n

\(l_p\) :

Length of untapered section of neck in Helmholtz resonator with tapered neck

\(r_{o}\) :

The outlet radius of tapered neck

\(r_{i}\) :

The inlet radius of tapered neck

\(z_i\) :

The acoustic impedance at the inlet of tapered neck

\(Z_{THR}\) :

The total acoustic impedance of Helmholtz resonator with tapered neck

\(V_e\) :

Effective volume of cavity

\(q_0\) :

The viscous permeability

\(\chi\) :

Tortuous ratio of rough neck

\(\sigma _s\) :

The static flow resistivity of smooth neck

\(\sigma _r\) :

The static flow resistivity of rough neck

\(\tau _r\) :

The tortuosity of rough neck

\(\epsilon _a\) :

The relative axial roughness of neck

\(\epsilon _c\) :

The relative circumferential roughness of neck

\(\beta _a\) :

The wavenumber of surface roughness of neck

\(\beta _c\) :

The wavenumber of circumferential surface roughness

b :

The wavelength of roughness of neck

\(\delta _r\) :

The radial fluctuation of axial roughness of neck

\(e_c\) :

The radial fluctuation of circumferential roughness of neck

y :

The ratio of end radius of roughened neck to viscous boundary layer thickness

\(\psi\) :

The perforation ratio

\(\delta _1\) :

The depth correction factor of cavity of Helmholtz resonator with roughened-embedded neck

\(\delta _2\) :

The correction factor of cavity of Helmholtz resonator with roughened-embedded neck

\(Z_{RHR}\) :

The total acoustic impedance of Helmholtz resonator with roughened-embedded neck

\(S_{aa}\) :

The effective area of the inserted tapered neck

\(d_{N1}\) :

The outlet diameter of inserted tapered neck

\(d_{N2}\) :

The inlet diameter of inserted tapered neck

\(k_x\) :

Perforation constant

\(R_{ext}\) :

The resistive term of end correction of the inserted tapered neck

\(X_{ext}\) :

The reactive term of end correction of the inserted tapered neck

\(R_s\) :

The surface resistance of inserted tapered neck

CMA :

Coefficient of maximum absorption

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  1. Department of Mechanical Engineering, College of Engineering Trivandrum (Government of Kerala), Thiruvananthapuram, Kerala, 695016, India

    K. Mahesh & R. S. Mini

  2. Micro/Nanofluidics Research Laboratory, Department of Mechanical Engineering, College of Engineering Trivandrum (Government of Kerala), Thiruvananthapuram, Kerala, 695016, India

    S. Kumar Ranjith

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Mahesh, K., Ranjith, S.K. & Mini, R.S. Recent Advancements in Helmholtz Resonator Based Low-Frequency Acoustic Absorbers: A Critical Review. Arch Computat Methods Eng 31, 2079–2107 (2024). https://doi.org/10.1007/s11831-023-10038-7

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