A FINITE ELEMENT METHOD FOR THE MODELING OF THERMO-VISCOUS EFFECTS IN ACOUSTICS

Abstract. A ¯nite element approach to the solution of acoustic ¯eld equations including the e®ects of viscosity and heat conduction is presented. The mathematical model con-sidered is based on the linearized Navier-Stokes, continuity and energy equations. The solution to the ¯eld equations is assumed to be time-harmonic, so that what remains to be solved is the spatial dependence of the amplitudes of primary unknowns. The original set of ¯eld equations may be reduced to a system with the amplitudes of the velocities and temperature as the only unknown ¯elds. This reduced system has often been used as a starting point in the analytical treatment and could also be thought to be a natural starting point for the ¯nite element formulation. However, special care has to be taken in the ¯nite element discretization of the reduced system. Firstly, a stable pair of ¯nite element spaces for the velocities and temperature has to be chosen. Secondly, we show by writing the system in a dimensionless form that in certain cases, commonly met in practice, an additional di±culty known to lead to a ¯nite element locking e®ect is expected. In fact, the same problem is generally met in the ¯nite element analysis of nearly incompress-ible °ows. To avoid unnecessary error ampli¯cation we rewrite the system as a mixed problem with a pressure-like quantity as auxiliary unknown. The resulting formulation is discretized using an enhanced MINI ¯nite element. The reliability and applicability of the resulting ¯nite element method is demonstrated by numerical experiments.