Thermoacoustic instability in a solid rocket motor: non-normality and nonlinear instabilities

Abstract

An analytical framework is developed to understand and predict the thermoacoustic instability in solid rocket motors, taking into account the non-orthogonality of the eigenmodes of the unsteady coupled system. The coupled system comprises the dynamics of the acoustic field and the propellant burn rate. In general, thermoacoustic systems are non-normal leading to non-orthogonality of the eigenmodes. For such systems, the classical linear stability predicted from the eigenvalue analysis is valid in the asymptotic (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 an initial transient growth for suitable initial perturbations even when the system is stable according to the classical linear stability theory. The terms contributing to the non-normality in the acoustic field and unsteady burn rate equations are identified. These terms, which were neglected in the earlier analyses, are incorporated in this analysis. Furthermore, the short-term dynamics are analysed using a system of differential equations that couples the acoustic field and the burn rate, rather than usingad hocresponse functions which were used in earlier analyses. In this paper, a solid rocket motor with homogeneous propellant grain has been analysed. Modelling the evolution of the unsteady burn rate using a differential equation increases the degrees of freedom of the thermoacoustic system. Hence, a new generalized disturbance energy is defined which measures the growth and decay of the oscillations. This disturbance energy includes both acoustic energy and unsteady energy in the propellant and is used to quantify the transient growth in the system. Nonlinearities in the system are incorporated by including second-order acoustics and a physics-based nonlinear unsteady burn rate model. Nonlinear instabilities are analysed with special attention given to ‘pulsed instability’. Pulsed instability is shown to occur with pressure coupling for burn rate response. Transient growth is shown to play an important role in pulsed instability.

Thermoacoustic interplay between intrinsic thermoacoustic and acoustic modes: non-normality and high sensitivities

“…In summary, previous analyses of the nonnormal nature of the Rijke tube conclude that transient growth of acoustic modes is to be expected to contribute to energy amplification by at most a factor of ten; nonlinearities will prevent further growth, as a limit-cycle or other non-linear dynamics (Subramanian et al 2013;Mariappan & Sujith 2010) is reached and self-sustained oscillations prevail. As these amplification factors are not considered large, recent research efforts have abandoned the study of triggering induced by transient growth.…”

“…This type of analysis, however, provides information about the long-time (Lyapunov) stability of the systems which may differ significantly from the short-time dynamics; in other words, thermoacoustic systems could exhibit transient growth (Schmid & Henningson 2001;Schmid 2007). This behaviour is typical of nonnormal system, or systems whose eigenvectors are not orthogonal, and over the past decade, thermoacoustic systems have been found both theoretically (Nicoud et al 2007;Balasubramanian & Sujith 2008a,b;Mariappan & Sujith 2010;Juniper 2011;Mangesius & Polifke 2011;Mariappan et al 2011;Sujith et al 2016) and experimentally (Wieczorek et al 2011;Zhao 2012;Mariappan et al 2015) to exhibit non-normal characteristics. These findings are important for the design of combustion systems: firstly, transient growth phenomena may cause significant direct stress to the engine.…”

Adjoint Methods as Design Tools in Thermoacoustics

“…In general, thermoacoustic systems are non-normal because their eigenfunctions are not orthogonal to each other [40]. As reviewed by Sujith et al [39], non-normality in thermoacoustics was investigated in ducted diffusion flames [36,37] and heat sources [34,35,[41][42][43][44][45]; solid rocket motors [46,47]; and premixed flames [48]. Later on, the authors of [38] showed that thermoacoustic non-normality in ducted diffusion flames is not as influential as it was thought to be.…”

“…When working in complex spaces, instead of a bilinear form a sesquilinear form 55 that defines an inner product is commonly used 45 . Therefore, adjoint equations do depend on the definition of the bilinear/sesquilinear form, which means that an adjoint model is not a physical model per se 46 . See Sec.…”

“…Some systems may be self-adjoint with respect to an inner product but may not be with respect to other inner products. 46 This does not imply that no physical information can be extracted from the adjoint model. 47 By (i) projecting the direct equations onto the adjoint function and the adjoint equations onto the direct function, and (ii) combining these two projections; it follows that d(q + (t) · q(t))/dt = 0. higher order.…”