Modeling viscous and thermal effects in acoustic actuators

Abstract

Viscous and thermal effects are usually neglected when investigating acoustic phenomena. This is admissible if the corresponding boundary layer thickness remains small compared to the relevant problem dimensions. For air at standard atmospheric conditions the viscous boundary layer thickness is about 70 µm at a frequency of 1 kHz, which makes viscous effects relevant in important novel applications like micro perforated panels (MPPs) and acoustic actuators based on MEMS technology. A similar argument holds for the inclusion of thermal effects. Several approaches exist to model viscous and thermal effects: Of course the flow field can be described by the full Navier Stokes equations, which need to be coupled to the flexible solid at the common interface. The resulting set of equations may be solved by, several numerical schemes, all of which share a large computational effort often rendering these methods unfeasible, e.g. for industrial applications. Another common approach is to linearise the compressible flow equations for small (acoustic) disturbances. Especially for cases without stationary background flow, this is often admissible, and the resulting equations can be solved employing the finite element method (FEM) [2]. Another approach is to employ an acoustic formulation in therms of the acoustic wave equation to model the flow field, and incorporate viscous and thermal effects in the boundary layers via an impedance type boundary condition [1]. This approach is computationally cheap, but is only applicable when the modeling assumptions of fully developed boundary layers are fulfilled. Based on the example of a piezoelectric MEMS actuator, we demonstrate the importance of modeling viscous and thermal effects, with respect to the accuracy of the resulting acoustic field. We use a FEM formulation of the linearised compressible flow equations implemented in the open source finite element code openCFS [3]. We show how non-conforming interfaces can be used to couple solid and fluid domains without requiring conforming mesh at the interface. Finally, we investigate what assumptions in terms of mesh size, approximation order, and fluid domain size are suitable to minimize the numerical effort while maintaining acceptable accuracy.

REFERENCES [1] Berggren, M.; Bernland, A.; Noreland, D.: Acoustic boundary layers as boundary conditions, Journal of Computational Physics, vol. 371, pp. 633–650, 2018. [2] Toth, F.; Hassanpour Guilvaiee, H.; Jank, G.: Acoustics on small scales, e & i Elektrotechnik und Informationstechnik, 2021. [3] Toth, F.; Kaltenbacher, M.; Wein, F.: Https://opencfs.org.