Triggering of Longitudinal Pressure Oscillations in Combustion Chambers. I: Nonlinear Gasdynamics

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Abstract
Based on spatial and temporal averaging of the conservation equations, an approximate analysis provides a convenient vehicle for analyzing the nonlinear behavior of unsteady motions in combustion chambers. Recent work has been concerned with the conditions for existence and stability of limit cycles, and in particular their dependence on the order of nonlinear terms contained in the equations of motion. It seems to be generally true that if only nonlinear gasdynamic terms to second order are accounted for, a stable nontrivial limit cycle is unique. This implies that no initial disturbance will cause a linearly stable system to execute a limit cycle, a result contrary to prior experience with, for example, solid propellant rocket motors. In this paper, we study some conditions under which true nonlinear instabilities may be found, with special attention focused on the effect of nonlinear gasdynamics. Our results indicate that third-order terms in the acoustic motions do not lead to triggering to either a stable or an unstable limit cycle; they only modify the stability domain of the system. However, the interactions between mean flow and nonlinear acoustic waves may trigger a linearly stable system to an unstable limit cycle. The influence of nonlinear combustion response may have quite different consequences and will be discussed in Part II of this work.