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Stability analysis for the onset of thermoacoustic oscillations in a gas-filled looped tube

Published online by Cambridge University Press:  13 February 2014

H. Hyodo  and
N. Sugimoto
H. Hyodo
Affiliation:
Department of Mechanical Science, Graduate School of Engineering Science, University of Osaka, Toyonaka, Osaka 560-8531, Japan
N. Sugimoto*
Affiliation:
Department of Mechanical Science, Graduate School of Engineering Science, University of Osaka, Toyonaka, Osaka 560-8531, Japan
*
†Email address for correspondence: sugimoto@me.es.osaka-u.ac.jp
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Abstract

This paper develops a stability analysis for the onset of thermoacoustic oscillations in a gas-filled looped tube with a stack inserted, subject to a temperature gradient. Analysis is carried out based on approximate theories for a thermoviscous diffusion layer derived from the thermoacoustic-wave equation taking account of the temperature dependence of the viscosity and the heat conductivity. Assuming that the stack consists of many pores axially and that the thickness of the diffusion layer is much thicker than the pore radius, the diffusion wave equation with higher-order terms included is applied for the gas in the pores of the stack. For the gas outside of the pores, the theory of a thin diffusion layer is applied. In a section called the buffer tube over which the temperature relaxes from that at the hot end of the stack to room temperature, the effects of the temperature gradient are taken into account. With plausible temperature distributions specified on the walls of the stack and the buffer tube, the solutions to the equations in both theories are obtained and a frequency equation is finally derived analytically by matching the conditions at the junctions between the various sections. Seeking a real solution to the frequency equation, marginal conditions of instability are obtained numerically not only for the one-wave mode but also for the two-wave mode, where the tube length corresponds to one wavelength and two wavelengths, respectively. It is revealed that the marginal conditions depend not only on the thickness of the diffusion layer but also on the porosity of the stack. Although the toroidal geometry allows waves to be propagated in both senses along the tube, it is found that the wave propagating in the sense from the cold to the hot end through the stack is always greater, so that a travelling wave in this sense emerges as a whole. The spatial and temporal variations of excess pressure and mean axial velocity averaged over the cross-section of a flow passage are displayed for the two modes of oscillations at the marginal state. The spatial distribution of mean acoustic energy flux (acoustic intensity) over one period is also shown. It is unveiled that the energy flux is generated only in the stack, and it decays slowly in the other sections by lossy effects due to a boundary layer. Mechanisms for the generation of the acoustic energy flux are also discussed.

JFM classification

Acoustics: Acoustics Instability: Instability Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 741 , 25 February 2014 , pp. 585 - 618
Copyright
© 2014 Cambridge University Press 

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References

Backhaus, S. & Swift, G. W. 1999 A thermoacoustic Stirling heat engine. Nature 399, 335–338.Google Scholar
Backhaus, S. & Swift, G. W. 2000 A thermoacoustic Stirling heat engine: detailed study. J. Acoust. Soc. Am. 107, 3148–3166.Google Scholar
Berson, A. & Blanc-Benon, Ph. 2007 Nonperiodicity of the flow within the gap of a thermoacoustic couple at high amplitudes. JASA Express Lett. 122, EL122–EL127.Google Scholar
Berson, A., Michard, M. & Blanc-Benon, Ph. 2008 Measurement of acoustic velocity in the stack of a thermoacoustic refrigerator using particle image velocimetry. Heat Mass Transfer 44, 1015–1023.Google Scholar
Garrett, S. L. 2004 Thermoacoustic engines and refrigerators. Am. J. Phys. 72, 11–17.Google Scholar
Guedra, M. & Penelet, G. 2012 On the use of a complex frequency for the description of thermoacoustic engines. Acta Acust. Acustica 98, 232–241.CrossRefGoogle Scholar
Guedra, M., Penelet, G., Lotton, P. & Dalmont, J.-P. 2011 Theoretical prediction of the onset of thermoacoustic instability from the experimental transfer matrix of a thermoacoustic core. J. Acoust. Soc. Am. 130, 145–152.Google Scholar
Lieuwen, T. C. 2012 Unsteady Combustor Physics, chap. 6. Cambridge University Press.Google Scholar
Morfey, C. L. 2001 Dictionary of Acoustics, p. 112. Academic Press.Google Scholar
Penelet, G., Job, S., Gusev, V., Lotton, P. & Bruneau, M. 2005 Dependence of sound amplification on temperature distribution in annular thermoacoustic engines. Acta Acust. Acustica 91, 567–577.Google Scholar
Rott, N. 1969 Damped and thermally driven acoustic oscillations in wide and narrow tubes. Z. Angew. Math. Phys. 20, 230–243.Google Scholar
Shimizu, D., Nishikawa, K. & Sugimoto, N. 2012 Numerical simulations of thermoacoustic oscillations in a looped tube. In Proc. 19th Intl Symp. on Nonlinear Acoustics, State-of-the-Art and Perspectives (ed. Kamakura, T. & Sugimoto, N.), AIP Conf. Proc. no. 1474, pp. 299–302. American Institute of Physics.Google Scholar
Shimizu, D. & Sugimoto, N. 2010 Numerical study of thermoacoustic Taconis oscillations. J. Appl. Phys. 107, 0349101–11.Google Scholar
Sugimoto, N. 2010 Thermoacoustic-wave equations for gas in a channel and a tube subject to temperature gradient. J. Fluid Mech. 658, 89–116.Google Scholar
Sugimoto, N. & Takeuchi, R. 2009 Marginal conditions for thermoacoustic oscillations in resonators. Proc. R. Soc. A 465, 3531–3552.Google Scholar
Sugimoto, N. & Yoshida, M. 2007 Marginal condition for the onset of thermoacoustic oscillations of a gas in a tube. Phys. Fluids 19, 074101.Google Scholar
Swift, G. W. 2002 Thermoacoustics: A Unifying Perspective for Some Engines and Refrigerators. Acoustical Society of America.Google Scholar
Ueda, Y. & Kato, C. 2008 Stability analysis of thermally induced spontaneous gas oscillations in a straight and looped tube. J. Acoust. Soc. Am. 124, 851–858.CrossRefGoogle Scholar
Yazaki, T., Biwa, T. & Tominaga, A. 2002 A pistonless Stirling cooler. Appl. Phys. Lett. 80, 157–159.Google Scholar
Yazaki, T., Iwata, A., Maekawa, T. & Tominaga, A. 1998 Traveling wave thermoacoustic engine in a looped tube. Phys. Rev. Lett. 81, 3128–3131.Google Scholar