Thermoacoustic Instability in Solid Rocket Motor: Non-Normality and Nonlinear Instabilities

Abstract

An analytical frame work is developed to understand and predict the thermoacoustic instability in solid rocket motors, taking into acc ount the non-orthogonality of the eigenmodes of the unsteady coupled system that comprises of the acoustic field and the propellant burn rate dynamics. In general, thermoa coustic systems are non-normal leading to non-orthogonality of the eigenmodes. For such systems, classical linear stability predicted from the eigenvalue analysis is valid only in the a symptotic (large time) limit. However, the short term dynamics can be completely different and a generalized stability theory is needed to predict the linear stability for all times. Non -normal systems show initial transient growth for suitable initial perturbations even when the sy stem is stable according to classical linear stability theory. The terms contributing to the no n-normality in the acoustic field and unsteady burn rate equations are identified. These terms, which were neglected in the earlier analyses, are incorporated in the present a nalysis. Furthermore, the short term dynamics is analysed using a system of differential equations that couples the acoustic field and the burn rate, rather than using ad hoc respons e functions which were used in earlier analyses. In the present case, an SRM with homogeneous propellant grain is analysed. Nonlinearities in the system are incorporated by in cluding second order acoustics and a physics based nonlinear unsteady burn rate model. Nonlinear instabilities are analysed with special attention given to ‘pulsed instability’. P ulsed instability is shown to occur with pressure coupling for burn rate response. Transien t growth is shown to play an important role in pulsed instability.