Modeling and analysis of premixed flame dynamics by means of distributed time delays

The unsteady response of a flame to acoustic or flow perturbations plays a crucial role in thermoacoustic combustion instability. The majority of studies on this subject presents and analyzes flame dynamics in the frequency domain by means of a flame transfer function or a flame describing function. The present review concentrates on work that adopts a time-domain perspective. In such a framework, the linear dynamics of an acoustically compact flame is completely characterized by its impulse response. The concept of distributed time delays emerges as an appropriate description of the convective transport of flow and flame perturbations. A time-domain perspective facilitates the physics-based interpretation of important features of the flame response and supports the development of passive or active means of stability control. The present review first provides mathematical background on linear time-invariant systems and introduces the impulse response as a quantity that fully characterizes the dynamics of such systems. It will then be shown by way of example how typical features of the frequency response of premixed flames can be generated in a very natural, physically intuitive manner from time delay distributions. Analytical results for the impulse response of laminar premixed flames to modulations of velocity or equivalence ratio are presented in a unified framework. The next chapter discusses low-order parametric models, which exploit prior knowledge on the underlying convective processes that govern the flame dynamics, but nevertheless require input from experiment or high-fidelity simulation to fix parameter values. Next, a variety of approaches devised to derive distributed time delay models of flame dynamics from simulation data are reviewed. The most recent developments, which combine large eddy simulation of turbulent combustion with system identification, have demonstrated that it is possible to estimate reduced-order models of flame dynamics that are quantitatively accurate even for complex, swirling flame in geometries of technical interest. The last chapter reviews work on acoustically non-compact flames, strategies for passive control of thermoacoustic instabilities that exploit distributed delays, and the effect of convective dispersion on the time delay distribution and strength of entropy waves.