A weakly nonlinear theory of the thermoacoustic instability in gas-filled tubes is developed in the time domain by exploiting the difference between the instability time scale and the period of standing waves. By carrying the expansion to fourth order in the perturbation parameter, explicit results for the initial growth, nonlinear evolution, and final saturation are obtained. The dependence of the saturation amplitude upon the temperature difference in the stack, the tube geometry, stack plate spacing, Prandtl number, and other parameters is illustrated.
Topics
Thermoacoustics
REFERENCES
1.
Akhavan
, R.
, Kamm
, R. D.
, and Shapiro
, A. H.
(1991
). “An investigation of transition to turbulence in bounded oscillatory Stokes flows. I. Experiment
,” J. Fluid Mech.
225
, 395
–422
. 2.
Atchley, A. A., Bass, H. E., and Hofler, T. J. (1990). “Development of nonlinear waves in a thermoacoustic prime mover,” in Frontiers in Nonlinear Acoustics, edited by M. F. Hamilton and D. T. Blackstock (Elsevier, New York), pp. 603–608.
3.
Brewster
, J. R.
, Raspet
, R.
, and Bass
, H. E.
(1997
). “Temperature discontinuities between elements of thermoacoustic devices
,” J. Acoust. Soc. Am.
102
, 3355
–3360
. 4.
Cao
, N.
, Olson
, J. R.
, Swift
, G. W.
, and Chen
, S.
(1996
). “Energy flux density in a thermoacoustic couple
,” J. Acoust. Soc. Am.
99
, 3456
–3464
. 5.
Cooper
, W. L.
, Yang
, K. T.
, and Nee
, V. W.
(1993
). “Fluid mechanics of oscillatory modulated flows and associated applications in heat and mass transfer—A review
,” J. Energy Heat Mass Transfer
15
, 1
–19
. 6.
Gopinath
, A.
, Tait
, N. L.
, and Garrett
, S. L.
(1998
). “Thermoacoustic streaming in a resonant channel: The time-averaged temperature distribution
,” J. Acoust. Soc. Am.
103
, 1388
–1405
. 7.
Hinch, E. J. (1991). Perturbation Methods (Cambridge University Press, Cambridge).
8.
Karpov, S. (2000). “Nonlinear Phenomena in Thermoacoustics and Bubbly Liquids,” Ph.D. thesis, Johns Hopkins University.
9.
Karpov
, S.
, and Prosperetti
, A.
(1998
). “Linear thermoacoustic instability in the time domain
,” J. Acoust. Soc. Am.
103
, 3309
–3317
. 10.
Kevorkian, J., and Cole, J. D. (1996). Perturbation Methods in Applied Mathematics, 2nd ed. (Springer, New York).
11.
Morse, P. M., and Feshbach, H. (1953). Methods of Theoretical Physics (McGraw-Hill, New York).
12.
Murdock, J. A. (1991). Perturbations (Wiley, New York).
13.
Naylor, A. W., and Sell, G. R. (1982). Linear Operator Theory in Engineering and Science (Springer, New York), p. 502.
14.
Rott
, N.
(1969
). “Damped and thermally driven acoustic oscillations in wide and narrow tubes
,” Z. Angew. Math. Phys.
20
, 230
–243
. 15.
Rott
, N.
(1976
). “Thermally driven acoustic oscillations. IV. Tubes with variable cross section
,” Z. Angew. Math. Phys.
27
, 197
–224
. 16.
17.
Rott
, N.
(1983
). “Thermally driven acoustic oscillations, VI. Excitation and power
,” Z. Angew. Math. Phys.
34
, 609
–626
. 18.
Swift
, G. W.
(1988
). “Thermoacoustic engines
,” J. Acoust. Soc. Am.
84
, 1145
–1180
. 19.
Watanabe
, M.
, Prosperetti
, A.
, and Yuan
, H.
(1997
). “A simplified model for linear and nonlinear processes in thermoacoustic prime movers. I. Model and linear theory
,” J. Acoust. Soc. Am.
102
, 3484
–3496
. 20.
Wheatley, J. (1986). “Intrinsically irreversible or natural heat engines,” in Frontiers in Physical Acoustics, edited by D. Sette (North-Holland, Amsterdam), pp. 35–475.
21.
Worlikar
, A. S.
, and Knio
, O. M.
(1996
). “Numerical simulation of a thermoacoustic refrigerator. I. Unsteady adiabatic flow around the stack
,” J. Comput. Phys.
127
, 424
–451
. 22.
Worlikar
, A. S.
, and Knio
, O. M.
(1999
). “Numerical study of oscillatory flow and heat transfer in a loaded thermoacoustic stack
,” Numer. Heat Transfer
A35
,49
-65
. 23.
Worlikar
, A. S.
, Knio
, O. M.
, and Klein
, R.
(1998
). “Numerical simulation of a thermoacoustic refrigerator. II. Stratified flow around the stack
,” J. Comput. Phys.
144
, 299
–324
. 24.
Yuan
, H.
, Karpov
, S.
, and Prosperetti
, A.
(1997
). “A simplified model for linear and nonlinear processes in thermoacoustic prime movers. II. Nonlinear oscillations
,” J. Acoust. Soc. Am.
102
, 3497
–3506
.
This content is only available via PDF.
© 2000 Acoustical Society of America.
2000
Acoustical Society of America
You do not currently have access to this content.