Nonlinear behavior of acoustic waves in combustion chambers—I

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This paper is concerned with the general problem of the nonlinear growth and limiting amplitude of acoustic waves in a combustion chamber. The analysis is intended to provide a formal framework within which practical problems can be treated with a minimum of effort and expense. There are broadly three parts. First, the general conservation equations are expanded in two small parameters, one characterizing the mean flow field and one measuring the amplitude of oscillations, and then combined to yield a nonlinear inhomogeneous wave equation. Second, the unsteady pressure and velocity fields are expressed as syntheses of the normal modes of the chamber, but with unknown time-varying amplitudes. This procedure yields a representation of a general unsteady field as a system of coupled nonlinear oscillators. Finally, the system of nonlinear equations is treated by the method of averaging to produce a set of coupled nonlinear first order differential equations for the amplitudes and phases of the modes. These must be solved numerically, but results can be obtained quite inexpensively. Subject to the approximations used, the analysis is applicable to any combustion chamber. The most interesting applications are probably to solid rockets, liquid rockets, or thrust augmentors on jet engines. The discussion of this report is oriented towards solid propellant rockets.