The Generalized Rayleigh Criterion and other Entropic Contributions to Combustion Instability

Abstract

The Rayleigh Criterion is a common measure used to quantify the stability of thermal systems. Typically the Rayleigh Criterion is expressed as the correlation between unsteady pressure and unsteady heat release where an in-phase combustion response leads to driving of instabilities. This expression however, is defined under the guise of an isentropic flow where entropy generation due to irreversibilities in combustion processes or viscous and thermal dissipation are neglected. This is largely untrue in reacting flows. The impact of entropy generation due to unsteady combustion is shown to be significant. The implications of a non-negligible unsteady entropy field leads to a generalization of the Rayleigh Criterion as well as the emergence of several combustion related driving mechanisms commonly omitted from stability analysis. Nomenclature p c = specific heat = heat release m = tangential wave number p = pressure Q = heat release per unit temperature, / HT s = entropy T = temperature t = time u = velocity Symbols  = density  = tangential coordinate  = phase  = frequency of oscillation Superscripts and Subscripts i = (superscript) denotes the imaginary part of a complex number r = (superscript) denotes the real part of a complex number 0 = (subscript) denotes a steady field variable 1 = (subscript) denotes an unsteady field variable 1 Stability Engineer, AIAA Member 2 Stability Engineer, AIAA Member Downloaded by CORNELL UNIVERSITY on July 19, 2015 | http://arc.aiaa.org |

On the Reduction of Unsteady CFD to Determine the Sources and Sinks of Unsteady Energy

“…More information regarding these details is in the paper by Jacob and Batterson. 11 They show that entropy terms can be very large compared to other sources and are the largest sources of instability for this simulation.…”

“…5 Since this wave is isentropic and irrotational, simple thermodynamic relationships can be used to express the acoustic temperature and density; 6 namely, 11 …”