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.

中文翻译：

---

具有任意阻抗边界的热声管的修正傅里叶级数解

Rijke管是热声界广泛使用的基准模型。作为现有建模方法的替代方法，这项工作提出了一种改进的傅立叶级数解决方案，用于一维 (1D) 热声系统的模态特性分析。由于边界平滑多项式项丰富的傅里叶展开级数的特殊特征，所提出的建模框架允许考虑任意阻抗边界。热声亥姆霍兹控制方程与一阶热释放模型耦合通过 Galerkin 程序离散化。热声模态参数是通过求解标准的四次矩阵特征方程获得的，不同于传统使用的基于超越方程的根搜索。通过与文献中报告的结果进行比较，给出了数值示例来验证所提出的模型。分析了边界阻抗的影响。结果揭示了热声不稳定性和热源位置之间关于声模式形状的定量关系。结果还表明存在一个敏感区域，其中阻抗端（IE）管的热声模态行为随边界阻抗发生剧烈变化。同时，通过适当组合阻抗的实部和虚部，可以在适当设置边界阻抗的情况下实现稳定区域。结果揭示了热声不稳定性和热源位置之间关于声模式形状的定量关系。结果还表明存在一个敏感区域，其中阻抗端（IE）管的热声模态行为随边界阻抗发生剧烈变化。同时，通过适当组合阻抗的实部和虚部，可以在适当设置边界阻抗的情况下实现稳定区域。结果揭示了热声不稳定性和热源位置之间关于声模式形状的定量关系。结果还表明存在一个敏感区域，其中阻抗端（IE）管的热声模态行为随边界阻抗发生剧烈变化。同时，通过适当组合阻抗的实部和虚部，可以在适当设置边界阻抗的情况下实现稳定区域。