Reduced-Order Modeling and Dynamics of Nonlinear Acoustic Waves in a Combustion Chamber

Abstract

For understanding the fundamental properties of unsteady motions in combustion chambers, and for applications of active feedback control, reduced-order models occupy a uniquely important position. A framework exists for transforming the representation of general behavior by a set of infinite-dimensional partial differential equations to a finite set of nonlinear second-order ordinary . differential equations in time. The procedure rests on an expansion of the is a criterion for deciding how many modes must be retained to give accurate results. Addressing that problem is the principal purpose of this paper. Our analysis shows that, in case of longitudinal modes, a first mode instability problem requires a minimum of four modes in the modal truncation whereas, for a second mode instability, one needs to retain at least the first eight modes. A second important problem concerns the conditions under which a linearly stable system becomes unstable to sufficiently large disturbances. Previous work has given a partial answer, suggesting that nonlinear gasdynamics alone cannot produce pulsed or 'triggered' true nonlinear instabilities; that suggestion is now theoretically established. Also, a certain form of the nonlinear energy addition by combustion processes is known to lead to stable limit cycles in a linearly stable system. A second form of nonlinear combustion dynamics with a new velocity coupling function that naturally displays a threshold character is shown here also to produce triggered limit cycle behavior.

Transient Growth and Triggering in the Horizontal Rijke Tube

“…It provides a clearer link between work on linear transient growth [1] [2] and work on non-linear triggering [4]. A linear analysis shows that the governing equations are non-normal and that this causes transient growth.…”

“…When eqns (4)(5) are substituted into eqns (1-3), they produce a system of linear ordinary differential equations (ODEs), which in this paper are called the fullylinearized system.…”

Triggering in a Thermoacoustic System with Stochastic Noise

“…This is known as triggering, or pulsed oscillation [2,3]. An analogy has been suggested with bypass transition to turbulence in hydrodynamics [1,Ch1 §IV] [4].…”

Bifurcation and nonlinear analysis of a time-delayed thermoacoustic system

“…Ananthkrishnan et al [60] obtained the reduced-order models to capture the global behavior of chamber dynamics via truncating the modal expansions and determined the number of modes required for accurate results. Heckl and Howe [61] conducted stability analysis of the Rijke tube by making use of a Green's function.…”