Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-02-11T09:32:43.688Z Has data issue: false hasContentIssue false
  • English
  • Français

Resonant acoustic oscillations with damping: small rate theory

Published online by Cambridge University Press:  29 March 2006

Brian R. Seymour  and
Michael P. Mortell
Brian R. Seymour
Affiliation:
Department of Mathematics, New York University
Michael P. Mortell
Affiliation:
Center for the Application of Mathematics, Lehigh University Present address: Department of Mathematical Physics, University College, Cork, Ireland.
Get access Rights & Permissions [Opens in a new window]

Abstract

A gas in a tube is excited by a reciprocating piston operating at or near a resonant frequency. Damping is introduced into the system by two means: radiation of energy from one end of the tube and rate dependence of the gas. These define a lumped damping coefficient. It is shown that in the small rate limit the signal in the periodic state suffers negligible distortion in one travel time, and hence its propagation according to acoustic theory is valid. The shape of the signal is determined by a nonlinear ordinary differential equation. The small rate condition provides a test of the applicability of the theory to given experimental conditions. When there is no damping, shocks are a feature of the flow for frequencies in the resonant band. For a given amount of damping an upper bound on the piston acceleration which ensures shockless motion is given. The resonant band is analysed for both damped and undamped cases. The predictions of the theory are compared with experiment.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 58 , Issue 2 , 17 April 1973 , pp. 353 - 373
Copyright
© 1973 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

Amerio, L. 1949 Determinazione delle condizioni di stabilityà per gli integrali di un'equazione interessante l'elettrotecnica. Ann. Mat. Pura & Appl. 30, 75-90.Google Scholar
Amerio, L. 1950 Studio asintotico del moto di un punto su una linea chiusa per azione di forze indipendenti dal tempo. Ann. Scuola Norm. Sup. Pisa, 3, 17–57.Google Scholar
Betchov, R. 1958 Nonlinear oscillations of a column of gas. Phys. PZuids, 1, 205–212.Google Scholar
Blythe, P. A. 1969 Non-linear wave propagation in a relaxing gas. J. Fluid Mech. 37, 31–50.Google Scholar
Böhm, C. 1953 Nuovi criteri di esistenza di soluzioni periodiche di una nota equazione differenziale non lineare. Ann. Mat. Pura & Appl. 35, 343–352.Google Scholar
Chester, W. 1964 Resonant oscillations in closed tubes. J. Fluid Mech. 18, 44–64.Google Scholar
Chu, B. T. & Ying, S. J. 1963 Thermally driven nonlinear oscillations in a pipe with travelling shock waves. Phys. Fluids, 6, 1625–1637.Google Scholar
Collins, W. D. 1971 Forced oscillations of systems governed by one-dimensional non-linear wave equations. Quart. J. Mech. Appl. Math. 24, 129–153.Google Scholar
Eninger, J. 1971 Nonlinear resonant wave motion of a radiating gas. Ph.D. thesis, Stanford University.
Hayes, W. D. 1953 On the equation for a damped pendulum under constant torque. Z. angew. Math. Phys. 4, 398–401.Google Scholar
Lillo, J. C. & Xeifert, G. 1955 On the conditions for stability of solutions of pendulum type equations. Z. angew. Math. Phys. 6, 239–243.Google Scholar
Minorshy, N. 1962 Nonlinear Oscillations. Van Nostrand.
Mortell, M. P. 1971 A Resonant thermal-acoustic oscillations. Int. J. Engng Sci. 9, 175–192.Google Scholar
Mortell, M. P. 1971b Resonant oscillations: a regular perturbation approach. J. Math. Phys. 12, 1069–1075.Google Scholar
Mortell, M. P. & Seymour, B. R. 1972a Pulse propagation in a nonlinear viscoelastic rod of finite length. Siam J. Appl. Math. 22, 209–224.Google Scholar
Mortell, M. P. & Seymour, B. R. 1972b The evolution of a self-sustained oscillation in a nonlinear continuous system. J. Appl. Mech. to appear.Google Scholar
Mortell, M. P. & Vbrley, E. 1970 Finite amplitude waves in a bounded media: non-linear free vibrations of an elastic panel. Proc. Roy. Soc. A 318, 169–196.Google Scholar
Rayleigh, Lord 1945 The Theory of Sound vol. 11, p. 319. Dover.
Saenger, R. A. & Hudson, G. E. 1960 Periodic shock waves in resonating gas columns. J. Acoust. Soc. Am. 32, 961–970.Google Scholar
Sansone, G. & Conti, R. 1964 Nonlinear Differential Equations (trans. A, H. Diamond). Pergamon.
Seymour, B. R. & Mortell, M. P. 1972 Resonant oscillations with damping in open tubes. Rep. to Ballistic Research Laboratories, Aberdeen Proving Ground, Md., August 1972.
Seymour, B. R. & Varley, E. 1970 High frequency periodic disturbances in dissipative systems. I. Small amplitude finite rate theory. Proc. Roy. Soc. A 314, 387–;415.Google Scholar
Stoeer, J. J. 1950 Nonlinear Vibrations in Mechanical and Electrical Systems. Interacience.
Whitham, G. B. 1952 The flow pattern of a supersonic projectile. Gomm. Pure Appl. Math. 5, 301–348.Google Scholar