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An Application of Dynamical Systems Theory to Nonlinear Combustion Instabilities

Jahnke, Craig C. and Culick, F. E. C. (1993) An Application of Dynamical Systems Theory to Nonlinear Combustion Instabilities. AIAA Papers, 93-0114. American InstHute of Aeronautics and Astronautics .

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Two important approximations have been incorporated in much of the work with approximate analysis of unsteady motions in combustion chambers: truncation of the series expansion to a finite number of modes, and time averaging. A major purpose of the analysis reported in this paper has been to investigate the limitations of those approximations. In particular two fundamental problems of nonlinear behavior are discussed: the conditions under which stable limit cycles of a linearly unstable system may exist; and conditions under which bifurcations of the limit cycle may occur. A continuation method is used to determine the limit cycle behavior of the equations representing the time dependent amplitudes of the longitudinal acoustic modes in a cylindrical combustion chamber. The system includes all linear processes and second-order nonlinear gas dynamics. The results presented show that time averaging works well only when the system is, in some sense, only slightly unstable. In addition, the stability boundaries predicted by the two-mode approximation are shown to be artifacts of the truncation of the system. Systems of two, four, and six modes are analyzed and show that more modes are needed to analyze more unstable systems. For the six-mode approximation with an unstable second mode two bifurcations are found to exist. A pitchfork bifurcation causes a new branch of limit cycles to exist in which the odd acoustic modes are excited. This new branch of limit cycles then undergoes a torus bifurcation that causes the system to exhibit stable quasi-periodic motions.

Item Type:Report or Paper (Report)
Additional Information:©1993 by the American Institute of Aeronautics and Astronautics, Inc. For permission to copy or republish, contact the American InstHute of Aeronautics and Astronautics.
Group:Guggenheim Jet Propulsion Center, GALCIT
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Guggenheim Jet Propulsion CenterUNSPECIFIED
Series Name:AIAA Papers
Issue or Number:93-0114
Record Number:CaltechAUTHORS:20101122-100414304
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:20938
Deposited By: Ruth Sustaita
Deposited On:10 Dec 2010 00:12
Last Modified:03 Oct 2019 02:17

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