A Modified Fourier Series Solution for a Thermo-Acoustic Tube with Arbitrary Impedance Boundaries

Rijke tube is a benchmark model widely used in thermo-acoustic community. As an alternative to existing modeling methods, this work proposes a modified Fourier series solution for modal characteristic analyses of a one dimensional (1D) thermo-acoustic system. The proposed modeling framework allows the consideration of arbitrary impedance boundaries owing to the special feature of the Fourier expansion series enriched by boundary smoothing polynomial terms. Thermo-acoustic Helmholtz governing equation coupled with a first-order heat release model is discretized through Galerkin procedure. Thermo-acoustic modal parameters are obtained by solving a standard quartic matrix characteristic equation, different from conventionally used root searching based on a transcendental equation. Numerical examples are presented to validate the proposed model through comparisons with results reported in the literature. Influences of boundary impedance are analyzed. Results reveal a quantitative relationship between the thermo-acoustic instability and heat source position with respect to the acoustic mode shapes. Results also show the existence of a sensitive zone, in which the thermo-acoustic modal behavior of the impedance-ended (IE) tube shows drastic changes with the boundary impedance. Meanwhile, a stable zone can be achieved upon a proper setting of the boundary impedance through suitable combination of the real and imaginary parts of the impedance.

The influence of impedance mismatch of the medium with gradient change of impedance on acoustic characteristics

Medium with gradient change of acoustic impedance is a new type acoustic structure. It is composed of several materials and has a continuous gradient composition and structure whose specific acoustic impedance varies smoothly across the layer. It has been used in various practical situations such as acoustic rectifiers, medical devices, weakening vibration and reducing noise. The purpose of this paper is to investigate the effect of boundary impedance mismatch on acoustic characteristics of gradient impedance media. The propagation mechanism of acoustic waves in medium with gradient change of specific acoustic impedance was calculated by transfer matrix method. By setting the different ratios of the acoustic impedance at interfaces of matching layer, the relationship between specific acoustic impedance mismatch and acoustic characteristics are derived and the corresponding conclusions are drawn. These conclusions provide an idea for the preparation of relevant medium.