A highly efficient approach to model acoustics with visco-thermal boundary losses

Abstract

For devices such as hearing aids, microphones, micro loudspeakers, and compression drivers, thermal and viscous boundary layer effects are often highly noticeable. These effects can be modeled in the linear regime by the linearized, compressible Navier-Stokes equations. However, the need for resolution of the very thin boundary layers typically makes numerical solutions of these equations computationally very expensive. Based on a boundary-layer analysis, we have derived for the pressure Helmholtz equation what appears to be a new boundary condition that accurately takes visco-thermal boundary losses into account. The model is valid when the wavelength and the minimum radius of curvature of the wall is much larger than the boundary layer thicknesses. In the special case of sound propagation in a cylindrical duct, the model collapses to the classical Kirchhoff solution. We assess the model in the case of sound propagation through a compression driver, a kind of transducer that is commonly used to feed horn loudspeakers. The transmitted power spectrum through the device calculated numerically using our model agrees extremely well with computations using a hybrid model, where the full linearized, compressible Navier-Stokes equations are solved in the narrow regions of the device and the pressure Helmholtz equations elsewhere. However, our model needs two orders of magnitude less memory and computational time than the more complete model.