TY - JOUR
T1 - A Modified Fourier Series Solution for a Thermo-Acoustic Tube with Arbitrary Impedance Boundaries
AU - Xing, Xue
AU - Xu, Qi
AU - Du, Jingtao
AU - Cheng, Li
AU - Liu, Zhigang
N1 - Funding Information:
This work is supported by the Research Grant Council of the Hong Kong SAR [PolyU 152036/18E]; the Fok Ying Tung Education Foundation [Grant No. 161049]; and the National Natural Science Foundation of China [Grant Nos. 11972125 and 11702227].
Publisher Copyright:
© 2020 World Scientific Publishing Europe Ltd.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.
PY - 2020/6/1
Y1 - 2020/6/1
N2 - 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.
AB - 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.
KW - arbitrary impedance boundary
KW - modified Fourier series
KW - Rijke tube
KW - Thermo-Acoustic mode
UR - http://www.scopus.com/inward/record.url?scp=85090201394&partnerID=8YFLogxK
U2 - 10.1142/S1758825120500477
DO - 10.1142/S1758825120500477
M3 - Journal article
AN - SCOPUS:85090201394
SN - 1758-8251
VL - 12
JO - International Journal of Applied Mechanics
JF - International Journal of Applied Mechanics
IS - 5
M1 - 2050047
ER -