Numerical Simulation of Elastic Micro-Perforated Plates Including Viscous and Thermal Effects

Abstract eng:
The aim of this work is the mathematical modelling and the numerical simulation of the timeharmonic motion of elastic micro-perforated plates (MPP), surrounded by a viscous compressible fluid. Micro-perforated plates are widely used as passive sound absorbers. For that reason, a variety of works have been devoted to model the acoustic impedance of such absorbing systems. Most of those models presented in the literature assume rigid plates and perforations with a very simple geometry. Consequently, the impedance of the MPP plates (which can be understood as a first-order approximation of the DtN operator mapping the surface jump pressure into the normal velocity on the MPP) have been estimated by using semi-empirical derivations (see [1] or, more recently, the revision of MPP models in [2]). The proposed mathematical modelling of MPPs is based on a coupled linear system of Partial Differential Equations, which takes into account: the linear elastic behaviour of the MPP structure and the compressible and viscous properties of a surrounding Newtonian fluid. In addition, the motion of the acoustic fluid is not assumed isothermal and hence the temperature has to be also included as an unknown field in the fluid domain. Obviously, the numerical simulation of this coupled system in a physical domain, whose size is several orders of magnitude larger than the typical size of the perforations, will become unaffordable its computational cost, not only for the required very detailed meshes used in the spatial discretization but also for the number of unknown fields (solid displacement, pressure, velocity and temperature in the fluid domain) to be discretized. However, other numerical methodology can be utilized is these large-scale problems at the low frequency regime and involving MPPs with arbitrary geometry perforations: if the entire coupled system is solved numerically in a reduced periodic domain (involving only a single perforation neighbourhood of the elastic structure) and the surface jump pressure and the normal velocity associated to the MPP are computed, then its complex-valued impedance can be obtained. Once this first-order approximation of the DtN operator is computed, its numerical frequency response could be used in real-life problems involving large computational domains without requiring high resolution meshes and discarding viscous and thermal effects (negligible at large scales) To illustrate and validate this numerical methodology, the computed acoustic impedance is compared with the results provided by rigid plates with other classical semi-empirical models existing in the literature.

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