Thermoacoustic effects on the propagation of non-planar sound in a circular duct

Abstract

This paper examines thermoacoustic effects on the propagation of non-planar sound in a circular duct subjected to an axial temperature gradient. Of particular concern are thermoviscous diffusive effects, which are taken into account by the boundary-layer approximation in a framework of the linear theory. For disturbances expanded into Fourier and Fourier–Bessel series in the azimuthal and radial directions, respectively, the pressure in each mode is described by a one-dimensional, dispersive wave equation, if non-diffusive propagation is assumed. When the diffusive effects are included, each radial mode is coupled to the other radial modes through the boundary layer. Focusing on a single azimuthal and radial mode only, the dispersion relation for the propagation along an infinite duct of a uniform gas is first derived. Effects of the temperature gradient are then examined by solving boundary-value problems for a duct of finite length in four typical cases. Assuming that the wall temperature increases exponentially along the duct, eigenfrequencies and decay rates in the lowest axial mode are obtained as well as axial distributions of the sound pressure and the axial velocity in the duct. The frequency and the decay rate increase as the temperature ratio at both ends becomes higher. It is found from the acoustic energy equation that the dispersion combined with the diffusion acts to reduce the damping and that the temperature gradient makes little contribution to the production of the energy. However, it is unveiled that the non-uniformity in temperature yields thermoacoustic sound confinement in the vicinity of the cold end.