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Thermoacoustic-wave equations for gas in a channel and a tube subject to temperature gradient

Published online by Cambridge University Press:  19 August 2010

N. SUGIMOTO
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 general theory for linear propagation of acoustic waves in a gas enclosed in a two-dimensional channel and in a circular tube subject to temperature gradient axially and extending infinitely. A ‘narrow-tube approximation’ is employed by assuming that a typical axial length is much longer than a span length, but no restriction on a thickness of thermoviscous diffusion layer is made. For each case, basic equations in this approximation are reduced to a spatially one-dimensional equation in terms of an excess pressure by making use of a method of Fourier transform. This equation, called a thermoacoustic-wave equation, is given in the form of an integro-differential equation due to memory by thermoviscous effects. Approximations of the equations for a short-time and a long-time behaviour from an initial state are discussed based on the Deborah number and the Reynolds number. It is shown that the short-time behaviour is well approximated by the equation derived previously by the boundary-layer theory, while the long-time behaviour is described by new diffusion equations. It is revealed that if the diffusion layer is thicker than the span length, the thermoviscous effects give rise to not only diffusion but also wave propagation by combined action with temperature gradient, and that negative diffusion may occur if the gradient is steep.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 658 , 10 September 2010 , pp. 89 - 116
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions. Dover.Google Scholar
Blackstock, D. T. 2000 Fundamentals of Physical Acoustics. John Wiley & Sons.Google Scholar
Chester, W. 1964 Resonant oscillations in closed tubes. J. Fluid Mech. 18, 44–64.Google Scholar
Christensen, R. M. 1982 Theory of Viscoelasticity: An Introduction, 2nd edn. Academic Press.Google Scholar
Gel'fand, I. M. & Shilov, G. E. 1964 Generalized Functions, vol. 1. pp. 115–122. Academic Press.Google Scholar
Howe, M. S. 1998 Acoustics of Fluid–Structure Interactions. Cambridge University Press.CrossRefGoogle Scholar
Kramers, H. A. 1949 Vibrations of a gas column. Physica. 15, 971–984.Google Scholar
Kirchhoff, G. 1868 Ueber den Einfluß der Wärmeleitung in einem Gase auf die Schallbewegung. Ann. Phys. Chem. 134, 177–193.Google Scholar
Oberhettinger, F. 1957 Tabellen zur Fourier Transformation. Springer.Google Scholar
Ooura, T. & Mori, M. 1991 The double exponential formula for oscillatory functions over the half infinite interval. J. Comput. Appl. Math. 38, 353–360.Google Scholar
Rayleigh, Lord 1945 The Theory of Sound vol. 2, pp. 319–328. Dover.Google Scholar
Reiner, M. 1964 The Deborah number. Phys. Today 17, 62.CrossRefGoogle Scholar
Rott, N. 1969 Damped and thermally driven acoustic oscillations in wide and narrow tubes. Z. Angew. Math. Phys. 20, 230–243.CrossRefGoogle Scholar
Rott, N. 1973 Thermally driven acoustic oscillations. Part II. Stability limit for helium. Z. Angew. Math. Phys. 24, 54–72.Google Scholar
Shimizu, D. & Sugimoto, N. 2009 Physical mechanisms of thermoacoustic Taconis oscillations. J. Phys. Soc. Japan. 78, 094401 (1–6).Google Scholar
Shimizu, D. & Sugimoto, N. 2010 Numerical study of thermoacoustic Taconis oscillations. J. Appl. Phys. 107, 034910 (1–11).Google Scholar
Sone, Y. 2002 Kinetic Theory and Fluid Mechanics. Birkhäuser.Google Scholar
Sugimoto, N. 1989 ‘Generalized’ Burgers equations and fractional calculus. In Nonlinear Wave Motion (ed. Jeffrey, A.), pp. 162–179. Longman Scientific & Technical.Google Scholar
Sugimoto, N. & Horioka, T. 1995 Dispersion characteristics of sound waves in a tunnel with an array of Helmholtz resonators. J. Acoust. Soc. Am. 97, 1446–1459.CrossRefGoogle Scholar
Sugimoto, N. & Shimizu, D. 2008 Boundary-layer theory for Taconis oscillations in a helium-filled tube. Phys. Fluids 20, 104102 (1–11).Google Scholar
Sugimoto, N. & Takeuchi, R. 2009 Marginal conditions for thermoacoustic oscillations in resonators. Proc. R. Soc. A 465, 3531–3552.CrossRefGoogle Scholar
Sugimoto, N. & Tsujimoto, K. 2002 Amplification of energy flux of nonlinear acoustic waves in a gas-filled tube under an axial temperature gradient. J. Fluid Mech. 456, 377–409.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 (1–13).Google Scholar
Swift, G. W. 1988 Thermoacoustic engines. J. Acoust. Soc. Am. 84, 1145–1180.Google Scholar
Swift, G. W. 2002 Thermoacoustics: A Unifying Perspective for Some Engines and Refrigerators. Acoustical Society of America.Google Scholar
Taconis, K. W., Beenakker, J. J. M., Nier, A. O. C. & Aldrich, L. T. 1949 Measurements concerning the vapour–liquid equilibrium of solutions of He3 in He4 below 2.19°K. Physica. 15, 733–739.CrossRefGoogle Scholar
Tanner, R. I. 1985 Engineering Rheology. Oxford University Press.Google Scholar
Tijdeman, H. 1975 On the propagation of sound waves in cylindrical tubes. J. Sound Vib. 39, 1–33.CrossRefGoogle Scholar
Yazaki, T., Tashiro, Y. & Biwa, T. 2007 Measurements of sound propagation in narrow tubes. Proc. R. Soc. A 463, 2855–2862.Google Scholar
Yazaki, T., Tominaga, A. & Narahara, Y. 1980 Experiments on thermally driven acoustic oscillations of gaseous helium. J. Low Temp. Phys. 41, 45–60.Google Scholar
Weston, D. E. 1953 The theory of the propagation of plane sound waves in tubes. Proc. Phys. Soc. Lond. B 66, 695–709.CrossRefGoogle Scholar
Zwikker, C. & Kosten, C. W. 1949 Sound Absorbing Materials, pp. 34–40. Elsevier.Google Scholar