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Figure 2. The transfer impedance as a function of the Strouhal number for a plate thickness tp = 1 mm and a perforation center radius ac = 0.4 mm. The blue lines represent the linear simulation results (three tapering configurations) and the red line represents the nontapered analytical model. The black symbols represent the nonlinear simulation results (three tapering configurations) and the red dots represent the nontapered semianalytical model.
Sr (a nondimensional measure of the degree of nonlinearity, where a smaller Sr represents a more nonlinear system). The number also depends on the excitation frequency and the dimension of the perforate. Nonlinear effects will be more important at lower frequencies and for smaller perforations.
For some simple perforation geometries, such as cyl- inders in a regular pattern, there are analytical and semianalytical models for computing the transfer imped- ance. The models include descriptions of the viscous (and thermal) losses, end correction approximations, hole– hole interaction models, approximate nonlinear effects, and the presence of a bias or grazing flow.
Using a detailed thermoviscous acoustics simulation model, the analytical approximations and the restric- tions on the geometry can be surpassed. Detailed transfer impedance data can be simulated within minutes, for example, if the perforations are tapered, as seen in Figure
1 (COMSOL, 2021). There is generally no restriction on the geometric configuration. The system of equations is solved with the COMSOL Multiphysics® software using the Thermoviscous Acoustics physics interface for the linear analysis (COMSOL, 2020). Nonlinear effects can be included using the Nonlinear Thermoviscous Acoustic Contributions feature. The model is based on the finite element formulation using a Galerkin least square stabili- zation technique. The governing equations are the full set of acoustic nonlinear first-order perturbation equations.
The transfer impedance for a specific setup (0.4 mm perfo- ration center radius ac and 1 mm plate thickness tp) for the tapered perforation geometry is depicted in Figure 2. The impedance is depicted as a function of the Strouhal number. The results are given for three tapering angles defined by the ratio of a/b (see inset in Figure 2). The COMSOL Multiphys- ics® simulation results comprise linear and nonlinear model results (blue lines and black symbols). For comparison, linear and nonlinear (Temiz et al., 2016) analytical models are also depicted (red line and red dots). The analytical models are only valid for a cylindrical perforate (when a/b = 1).
For many acoustics applications, it is important to know the transfer impedance of a perforate in order to control the damping properties. Using simulation, it is possible to get highly accurate transfer impedance data for all perforate configurations and
hole geometries in a short amount of time. Acquiring the same experimental data is time consuming, and variations in the geometry require many test specimens. Analytical models are fast but restricted to a few simple geometries. The computed transfer impedance values can then, for example, be tabulated and used in a larger system simulation as a boundary condition or in a lumped model. The flexibility of simulations gives engineers the possibility to test and opti- mize many designs in a time-efficient manner.
References
COMSOL (2020). Acoustics Module, COMSOL Multiphysics, Version 5.6, 2020. https://www.comsol.com/acoustics-module. Accessed April 29th 2021.
COMSOL (2021). "Nonlinear Transfer Impedance of a Perforate", COMSOL, 2021. https://www.comsol.com/model/99471. Accessed April 29th 2021. Temiz, A., Tournadre, J. , Arteaga, I.L., and Hirschberg, A. (2016). “Non-
linear acoustic transfer impedance of micro-perforated plates with
circular orifices,” The Journal of Sound and Vibration, 366, 418-428. Vorländer,M.(2020)."AreVirtualSoundsReal,"AcousticsToday,16(1),46-54.
About the Author
Mads Herring Jensen
mads@comsol.dk
COMSOL A/S
Diplomvej 373, 2000 Kgs. Lyngby, Denmark
Mads Herring Jensen joined COMSOL
in 2011 and is the technology manager for the acoustics products. Mads has a PhD in computational fluid dynamics from the Technical University of Denmark. Before joining COMSOL, Mads worked in the hearing aid industry for
five years as an acoustic finite element expert.
Summer 2021 • Acoustics Today 41