Viscothermal accoustics using finite elements

Abstract

Hearing aids contain miniature loudspeakers and microphones. Following the trend of miniaturization, hearing aids and the acoustic transducers inside become smaller and smaller. This presents new challenges for engineers who develop such transducers. Because of the small geometries, the viscothermal boundary layer effects cannot be neglected in their acoustic models. This thesis presents four numerical viscothermal acoustic models that can aid the engineer in the development of, for example, these miniature loudspeakers and microphones. The ideal viscothermal acoustic model for a design environment would be computationally efficient, applicable for arbitrary geometries and usable in fluid structure interaction problems. These three aspects are satisfied to a different degree for each of the four presented models. The aspects of applicability for arbitrary geometries led to choosing the finite element method (FEM) as the numerical solution framework for the models. The software Comsol is used to implement the presented viscothermal acoustic models. This has the added advantage that structural finite elements supplied by Comsol can be used in fluid structure interaction problems. The first of the presented models is a finite element implementation of the fully coupled viscothermal acoustic equations: the linear time harmonic Navier-Stokes equations. This general model requires a minimum of four field values in 3-D: three velocity components and the temperature. However, a mixed FEM formulation with the pressure as an additional field is used to ensure a good convergence rate. The drawback of this model is that it requires large computational resources, especially in 3-D. Some authors label viscothermal acoustics as a `three wave theory' with coupled viscous, thermal and acoustic waves. The viscous and thermal waves damp the acoustic waves. It is possible to make accurate models using the approximation that the acoustic wave does not influence the viscous and thermal waves. The other three of the four viscothermal acoustic models use this approximation. Two of these models are known in the literature. These models are computationally efficient, but have the disadvantage that they are not applicable for arbitrary geometries: one model is for waveguides below the cut-off frequency and the other model is for geometries of which all characteristic lengths are much larger than the viscous and thermal boundary layer thicknesses. The third model is new and does not have this disadvantage. It can be used for arbitrary geometries and is computationally much more efficient than the fully coupled model. The viscothermal acoustic models are validated by means of measurements, an analytic model and mutual comparisons. Besides the relatively simple examples that are used for the validation, a model of a hearing aid loudspeaker is presented. The model has a good correspondence to the measurements. Several parameter studies for this loudspeaker are presented. Each of the four presented viscothermal acoustic models has its own advantages, disadvantages and limitations. Together, they form a set of analysis tools for the engineer that can be used to develop small acoustic transducers, or efficiently solve many other viscothermal acoustic problems.