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Interaction of rarefaction waves with area reductions in ducts

Published online by Cambridge University Press:  20 April 2006

J. J. Gottlieb
Affiliation:
Institute for Aerospace Studies, University of Toronto, Downsview, Ontario, Canada
O. Igra
Affiliation:
Institute for Aerospace Studies, University of Toronto, Downsview, Ontario, Canada Present address: Ben-Gurion University of the Negev, Beer-Sheva, Israel.

Abstract

The interaction of a rarefaction wave with a gradual monotonic area reduction of finite length in a duct, which produces transmitted and reflected rarefaction waves and other possible rarefaction and shock waves, was studied both analytically and numerically. A quasi-steady flow analysis which is analytical for an inviscid flow of a perfect gas was used first to determine the domains of and boundaries between four different wave patterns that occur at late times, after all local transient disturbances from the interaction process have subsided. These boundaries and the final constant strengths of the transmitted, reflected and other waves are shown as a function of both the incident rarefaction-wave strength and area-reduction ratio, for the case of diatomic gases and air with a specific-heat ratio of 7/5. The random-choice method was then used to solve numerically the conservation equations governing the one-dimensional non-stationary gas flow for many different combinations of rarefaction-wave strengths and area-reduction ratios. These numerical results show clearly how the transmitted, reflected and other waves develop and evolve with time, until they eventually attain constant strengths, in agreement with quasi-steady flow predictions for the asymptotic wave patterns. Note that in all of this work the gas in the area reduction is initially at rest.

Type
Research Article
Copyright
© 1983 Cambridge University Press

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References

Bannister, F. K. & Mucklow, G. F. 1948 Wave action following sudden release of compressed gas from a cylinder Proc. Indust. Mech. Engng 159, 269.Google Scholar
Bremner, G. F., Dukowicz, J. K. & Glass, I. I. 1961 One-dimensional overtaking of a rarefaction wave by a shock wave. ARS J. 1455. (Also University of Toronto Inst. for Aerophys. UTIA Tech. Note no. 33 (1960).)Google Scholar
Chester, W. 1960 The propagation of shock waves along ducts of varying cross section Adv. Appl. Mech. 6, 119.Google Scholar
Chorin, A. J. 1976 Random choice solution of hyperbolic systems J. Comp. Phys. 22, 517.Google Scholar
Glass, I. I., Heuckroth, L. E. & Molder, S. 1961 One-dimensional overtaking of a shock wave by a rarefaction wave. ARS J. 1453. (Also University of Toronto Inst. for Aerophys. UTIA Tech. Note no. 30 (1959).)Google Scholar
Glimm, J. 1965 Solution in the large for nonlinear hyperbolic systems of equations Commun. Pure Appl. Maths 18, 697.Google Scholar
Gottlieb, J. J. & Saito, T. 1983 An analytical and numerical study of the interaction of rarefaction waves with area changes in ducts - part 1: area reductions. University of Toronto Inst. for Aerospace Studies UTIAS Rep. no. 272.Google Scholar
Greatrix, D. R. & Gottlieb, J. J. 1982 An analytical and numerical study of a shock wave interaction with an area change. University of Toronto Inst. for Aerospace Studies UTIAS Rep. no. 268.Google Scholar
Igra, O., Gottlieb, J. J. & Saito, T. 1983 An analytical and numerical study of the interaction of rarefaction waves with area changes in ducts - part 2: area enlargements. University of Toronto Institute for Aerospace Studies UTIAS Rep. no. 273.Google Scholar
Jones, A. D. & Brown, G. L. 1982 Determination of two-stroke engine exhaust noise by the method of characteristics J. Sound Vib. 82, 305.Google Scholar
Kahane, A., Warren, W. R., Griffith, W. C. & Marino, A. A. 1954 A theoretical and experimental study of finite wave interactions with channels of varying area J. Aero. Sci. 21, 505.Google Scholar
Rudinger, G. 1955 Wave Diagrams for Nonstationary Flow in Ducts. Van Nostrand. (Also Nonsteady Duct Flow: Wave Diagram Analysis. Dover (1969).)
Russell, D. A. 1967 Shock-wave strengthening by area convergence J. Fluid Mech. 27, 306.Google Scholar
Saito, T. & Glass, I. I. 1979 Application of random-choice method to problems in shock and detonation-wave dynamics. University of Toronto Inst. for Aerospace Studies UTIAS Rep. no. 240.Google Scholar
Schultz-Grunow, F. 1943 Nichtstationäre, kugelsymmetrische Gasbewegung und nichtstationäre Gasströmung in Dusen und Diffusoren Ing. Arch. 14, 21.Google Scholar
Sod, G. A. 1977 A numerical study of a converging cylindrical shock J. Fluid Mech. 83, 785.Google Scholar
Warming, R. F. & Beam, R. M. 1977 On the construction and application of implicit factored schemes for conservation laws. In Proc. SIAM-AMS Symp. on Computational Fluid Dynamics, New York.