Acoustic Instability of the Slab Rocket Motor

Abstract

This article provides rigorous mathematical details leading to an improved formulation of acoustic instability in solid rocket motors. The evaluation of stability growth rate factors is carried out both numerically and asymptotically. Analytical expressions for the stability factors are obtained over a broad spectrum of operating parameters. For all representative rocket motors under investigation, the analytical estimates exhibit an error of 5% or less. Both numerics and asymptotics converge in predicting markedly less stable systems than projected by classic stability theory. The dramatic differences can be ascribed to the dismissal of time-dependent rotational coupling in the previous formulation. The current study unravels the details of six additional growth rate corrections not accounted for previously. These include the rotational flow, mean vorticity, viscosity, pseudo acoustic, pseudo vorticity and unsteady nozzle growth rate factors. The fourth and fifth terms are due to acoustical and vortical interactions with the often neglected pseudopressure. The sixth is due to the energy associated with the unsteady rotational flow exiting the nozzle. This study enables us to isolate the impact of various flow attributes on stability. These involve the motor aspect ratio, surface Mach number, viscous parameter, oscillation mode shape number, and surface admittance. Based on the slab motor geometry, we find that the flow turning correction is cancelled identically by another rotational term not accounted for previously. We also find that the unsteady nozzle damping effect is offset by another source of instability due to the pseudopressure.