Some rotational corrections to the acoustic energy equation in injection-driven enclosures

Abstract

This article presents rotational corrections to the energy stability equation in injection-driven porous enclosures used to simulate solid rocket motors. The evaluation of stability growth rate factors is carried out both numerically and asymptotically. Analytical expressions for the energy stability factors are obtained over a spectrum of physical parameters encompassing solid rocket motor operation. For all representative motors under investigation, the analytical estimates are shown to exhibit negligible errors compared to their numerical values. Both numerics and asymptotics converge in predicting less stable systems than projected by purely irrotational stability theory. The differences can be ascribed to the dismissal of time-dependent rotational coupling in some past formulations. The current study unravels the details of six additional growth rate corrections not accounted for previously. These include the rotational flow, inviscid vortical, viscous, pseudoacoustical, pseudorotational, 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 porous enclosure. This study enables us to explain the influence of distinct flow variables on stability. Based on asymptotic approximations for individual growth rates, explicit criteria are presented in the form of critical Mach numbers, penetration numbers, or motor lengths that must not be exceeded in prevention of system instability. The net rotational corrections have been recently appended to the widely used Standard Stability Prediction code.