Hostname: page-component-77c89778f8-rkxrd Total loading time: 0 Render date: 2024-07-16T17:44:39.692Z Has data issue: false hasContentIssue false
  • English
  • Français

Evolution of weakly nonlinear waves in a cylinder with a movable piston

Published online by Cambridge University Press:  26 April 2006

Meng Wang  and
D. R. Kassoy
Meng Wang
Affiliation:
Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309–0427, USA
D. R. Kassoy
Affiliation:
Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309–0427, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Small-amplitude wave motion in an inert gas confined between a moving piston and a fixed cylinder endwall is studied using the unsteady Euler equations. The waves, generated by either initial disturbances or piston motion, reflect back and forth in the cylinder on the acoustic timescale. The accumulated effect of these waves controls the bulk variations of velocity and thermodynamic variables on the longer piston timescale. Perturbation methods, based on the small ratio of acoustic to piston time, are employed to formulate the gasdynamic problem. The application of multiple timescaling allows the gasdynamic wave field to be separated from the bulk response of the gas. The evolution of the wave phenomena, including nonlinear wave deformation and weak shock formation during the piston passage time, is described in terms of time-dependent Fourier series solutions, whose coefficients are computed from a truncated system of coupled nonlinear ordinary differential equations. The long-time asymptotic flow field after shock formation is sawtooth-like, in which case the Fourier modes become decoupled. A remarkably simple relation between the shock amplitude and piston velocity is discovered. It is demonstrated that (i) the wave amplitude and frequency strongly depend on the piston motion; (ii) shock waves can be damped in a significant way by internal dissipation; and (iii) the mathematical approach developed in this study possesses certain advantages over the more traditional method of characteristics.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 221 , December 1990 , pp. 23 - 52
Copyright
© 1990 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Betchov, R.: 1958 Nonlinear oscillations of a column of gas. Phys. Fluids 1, 205–212.Google Scholar
Chester, W.: 1964 Resonant oscillations in closed tubes. J. Fluid Mech. 18, 44–64.Google Scholar
Chu, B. T.: 1963 Analysis of a self-sustained nonlinear vibration in a pipe containing a heater. Phys. Fluids 6, 1638–1644.Google Scholar
Gear, C. W.: 1971 Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall.
Hindmarsh, A. C.: 1974 GEAR: Ordinary Differential Equation Solver. Lawrence, Livermore Laboratory, Rep. UCID-30001 (Revision 3).Google Scholar
Kassoy, D. R.: 1979 The response of a confined gas to a thermal disturbance: Slow transients. SIAM J. Appl. Maths 36, 624–634.Google Scholar
Klein, R. & Peters, N., 1988 Cumulative effects of weak pressure waves during the induction period of a thermal explosion in a closed cylinder. J. Fluid Mech. 187, 197–230.Google Scholar
Kluwick, A.: 1981 The analytical method of characteristics. Prog. Aerospace Sci. 19, 197–313.Google Scholar
Kufner, A. & Kadlec, J., 1971 Fourier Series. London: Iliffe Books.
Landau, L. D. & Lifshitz, E. M., 1959 Fluid Mechanics. Pergamon.
Lax, P. D.: 1973 Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM Regional Conference Series in Applied Mathematics, Vol. 11.Google Scholar
Lighthill, M. J.: 1949 A technique for rendering approximate solutions to physical problems uniformly valid. Phil. Mag. 40, 1179–1201.Google Scholar
Lighthill, M. J.: 1978 Waves in Fluids. Cambridge University Press.
Lin, C. C.: 1954 On a perturbation theory based on the method of characteristics. J. Maths Phys. 33, 117–134.Google Scholar
Morse, P. M. & Ingard, U. I., 1968 Theoretical Acoustics. McGraw-Hill.
Ochmann, M.: 1985 Nonlinear resonant oscillations in a closed tube - An application to the averaging method. J. Acoust. Soc. Am. 77, 61–66.Google Scholar
Radhwan, A. M. & Kassoy, D. R., 1984 The response of a confined gas to a thermal disturbance: II Rapid boundary heating. J. Engng Maths 18, 133–156.Google Scholar
Rott, N.: 1980 Thermoacoustics. Adv. Appl. Mech. 20, 135–174.Google Scholar
Schneider, G. H.: 1981 Kompression und Expansion eines Gases in einem Zylinder als Störproblem. Acta Mech. 41, 157–184.Google Scholar
Smoller, J.: 1983 Shock Waves and Reaction-Diffusion Equations. Springer.
Temkin, S.: 1969 Propagating and standing sawtooth waves. J. Acoust. Soc. Am. 45, 224–227.Google Scholar
Wang, M.: 1989 Piston generated dynamic compression and expansion of an inert gas in a cylinder. Ph.D thesis, University of Colorado, Boulder.
Wang, M. & Kassoy, D. R., 1990a Dynamic compression and weak shock formation in an inert gas due to fast piston acceleration. J. Fluid Mech. 220, 267–292.Google Scholar
Wang, M. & Kassoy, D. R., 1990b Dynamic response of an inert gas to slow piston acceleration. J. Acoust. Soc. Am. 87, 1466–1471.Google Scholar
Whitham, G. B.: 1974 Linear and Nonlinear Waves. John Wiley and Sons.