Application of dynamical systems theory to nonlinear combustion instabilities

Abstract

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.

Influences of combustion dynamics on linear and nonlinear unsteady motions in solid propellant rockets

“…Substitution into the conservation and state equations yields the governing equations for the 3 fluctuating quantities. Spatial averaging is applied to those equations followed by a modal expansion of the unsteady pressure and velocity fields in order to get a reduced order model of the system.…”

Influence of Random Excitations on Acoustic Instabilities in Combustion Chambers

“…Common terminology used to describe this type of oscillationincludestriggeredinstability, hard excitation, and subcritical bifurcation. Previous works, for example, Yang et al, 1 Paparizos and Culick, 2 and Jahnke and Culick, 3 have convincinglyshown that nonlineargasdynamicsalone does not contain the possibility of pulsed oscillations. Part of the purpose of the present investigation is to determine if noise processes could be responsible for the occurrence of pulsed oscillations.…”

Modeling and Dynamics of Nonlinear Acoustic Waves in a Combustion Chamber

“…This information is usually represented in the form of a plot of steady state values (peak amplitude in case of a limit cycle) against a suitable parameter in a bifurcation diagram. For longitudinal acoustic modes in a combustion chamber of uniform cross section, Jahnke and Culick [18] showed that results from a two-mode approximation were qualitatively dissimilar to those from a four-or six-mode approximation. However, they could draw no conclusion about the number of acoustic modes that need to be retained in their analysis, nor did their computations with second order gasdynamics alone display triggered limit cycles.…”

“…However, they could draw no conclusion about the number of acoustic modes that need to be retained in their analysis, nor did their computations with second order gasdynamics alone display triggered limit cycles. Culick et al [19] extended the work in [18] by including the nonlinear combustion model of Levine and Baum [17] in addition to the second order gasdynamic nonlinearities. They confirmed that the nonlinear velocity coupling term in the Levine-Baum model did induce triggered limit cycles.…”

“…Also, it was not easy to arrive at qualitative conclusions regarding the conditions for onset of triggered instability from numerical simulations alone. A significant step forward in the investigations came with the introduction of the methods of modern dynamical systems theory by Jahnke and Culick [18] to the analysis of nonlinear combustion instabilities. By using a continuation algorithm, it became possible to systematically and efficiently compute all steady state and limit cycle solutions over a range of parameter values.…”