Integral Methods for Thermoacoustic Combustion Instability

Abstract

Thermoacoustic instabilities pose significant challenges to the development of modern combustors. These instabilities arise from the interaction between heat and sound in propulsion systems, potentially leading to limit cycle pressure oscillations or, in extreme cases, engine failure. Thus, it is crucial to study and quantify the acoustic waveforms in realistic combustion devices. One effective approach is to investigate simplified models, such as one-dimensional wave propagation in a Rijke tube. A three-pronged approach is introduced, which comprises of theoretical and numerical investigations that leverage experimentally available data in order to explore the time-dependent field within a Rijke tube. A perturbation expansion method is presented, which uses a naturally occurring small parameter within the context of a one-dimensional tube, both with and without a spatially variable heat source. This approach yields accurate predictions for the pressure and velocity modal shapes and frequencies for any specified temperature distribution, along with other acoustic properties. Furthermore, this method results in a set of linear partial differential equations that can be solved using a Green’s function to describe thermoacoustic pressure, velocity, heat oscillations, temperature, density, acoustic intensity, energy density, entropy, boundary-layer heat flux, and velocity under arbitrary endpoint conditions. Our findings highlight the significant axial dependence of the waveforms and nodal locations on the thermal profile and respective thermal gradient. The analysis further reveals that the peak value of the energy-flux vector modulus—representing the product of acoustic velocity and pressure—occurs at the heat source location and increases with the heat power input. Moreover, for a finite heat source, we derive the acoustic frequencies analytically. These frequencies are shown to increase monotonically with successive increases in temperature gain across the heat source, retraction of the heat source, reductions in heat source length, and smoothing of the temperature gain profile. Lastly, we introduce the dual reciprocity boundary element method to extend our formulation to two-dimensional (and three-dimensional) thermoacoustic wave propagation. Results for the 2-D Rijke tube and a cyclonic rocket engine are presented, demonstrating the applicability of our approach to more complex systems.