Energy balance analysis of nonlinear combustion instability

To be of practical value, analytical models for nonlinear rocket motor combustion instability must adequately represent: 1) steep fronted waves, 2) limit cycle operation, and 3) triggering phenomena. Inclusion of all these effects in an approximate analysis based on perturbation expansion procedures requires retention of terms to at least the third order in the perturbation parameter representing the system amplitude. In this paper, the acoustic energy balance method is extended into the nonlinear regime and combined with a simplified geometrical representation of the wave structure to greatly simplify this procedure. A practical approximate model for axial pressure fluctuations in a tubular rocket motor results. Effects of nonlinear combustion and nonisentropic energy losses in the steep wave fronts are represented. Since the relative amplitudes of the Fourier components that comprise the traveling shock waves may be assumed to remain effectively fixed near amplitude limiting conditions, great mathematical simplifications accrue. The behavior of the system is governed by a simple polynomial expression which gives the rate of change of the composite system amplitude. The coefficients are integral expressions taken over the chamber volume and its bounding surfaces. The familiar exponential growth rate model appears as the first-order (linear) limiting case. The model demonstrates all of the nonlinear characteristics observed in motor data and shows promise as an easily used diagnostic tool. It has been successfully employed to simulate actual experimental results from pulser tests in inert chambers and from sonic end-vent burner tests.