Mechanical Disturbances Arising From Thermal Power Deposition in a Gas with Application to Supercritical Liquid Rocket Engine Combustion Instability

Combustion instability in a liquid rocket engine (LRE) is facilitated by a relatively large thermal energy release rate per unit mass. The localized thermal energy deposition transient can amplify existing acoustic oscillations and/or be the immediate source of propagating mechanical disturbances (acoustics, shocks and blast waves) [1-4]. Unstable combustion chamber phenomena is characterized by the presence of growing pressure oscillations (mechanical disturbances) as well as mean pressure excursions from design values, concomitant thrust variations leading to rocket mission failure, damaging system vibrations and enhanced localized heat transfer. Operational instabilities are often unpredictable, given the scientific database available to rocket designers and modelers. The uncertainty is likely a measure of the community’s insufficient knowledge of the diverse multi-scale physico-chemical phenomena occurring within the complexity of elements that comprise a LRE. Although research on this topic has gone on since the 1950-60’s, and many innovative ideas have resulted from original and creative research studies [e.g., 5], the instability problem is something of an enigma. There remains the need to develop a more comprehensive understanding of the fundamental physics and chemistry occurring in an LRE chamber. The objective must be to create fully predictive models of instability phenomena along with computational methodologies capable of resolving the crucial physical phenomena on a diversity of time and length scales. A substantial portion of the existing modeling literature [6 for an early survey, 7 for an insightful discussion of fundamentals] can be roughly categorized as a thermoacoustics approach to instability. The objective in these wave equation-based models is to predict how and when longitudinal and transverse disturbances (oscillations) are amplified and/or altered by spatially distributed, transient, combustion-generated heat release occurring throughout a combustion chamber [e.g., 8,9]. These models are closely allied with Rayleigh’s criterion; oscillation amplitudes grow (decay) when heat addition occurs at pressure maxima (minima) [18]. Wave equation-based models resolve hyperbolic phenomena. The description of non-hyperbolic phenomena that may be the immediate source of mechanical disturbances [1], discussed later in this preprint, requires a new mathematical formulation. In contrast to thermo-acoustics, the primary objective of thermo-mechanical modeling [1-4] is to explain how the deposition of localized, spatially distributed, transient heat release initiates a mechanical disturbance (or how thermal energy is converted to kinetic energy). In the context of a turbulent, nonuniformly distributed supercritical mixture of fuel/oxidizer in a LRE chamber, the conceptual perspective is that best-mixed finite micro-volumes ignite first, with substantial localized heat release occurring on a dimensional (primes) chemical time-scale tH , to be compared with the acoustic time-scale tA′ of the volume. The volumes are anticipated to be small relative to the characteristic dimension of the chamber. Rigorous, asymptotic mathematical methods have been used [1] to show that when tH′ << tA′, and the energy addition is less than a specific threshold value, localized heating of an inert gas occurs at nearly constant volume conditions, leading to a local pressure rise with temperature (high pressure, hot spots). The small, but finite, density decrease is a result of localized gas expansion. The characteristic internal expansion Mach number, Mi , based on the local high temperature sound speed, is very small. In contrast, the corresponding Mach number at the expanding volume edge, Me , based on the lower temperature sound speed there, ranges from small to large depending on the energy addition to the volume. The “piston-like effect” of the fluid expelled through the original volume surface is the source of mechanical disturbances in the gas [10] external to the