A study of Helmholtz resonators to stabilize thermoacoustically driven pressure oscillations.

This paper studies passive control of thermoacoustic instabilities from an unconventional mathematical perspective. These instabilities are notoriously known to result from the complex dynamic exchange between the unsteady heat release and the acoustic waves within a finite volume such as a combustor. One possible passive control strategy is to utilize Helmholtz resonators. Under certain simplifications, the ensemble combustion dynamics including the resonators reduces to a linear-time invariant-multiple time-delayed system (LTI-MTDS). As the main contribution of the paper, an exact analytical procedure is proposed to determine the placement of the resonators to avoid instabilities. A unique mathematical paradigm, called the cluster treatment of characteristic roots, is used to accomplish this task. It declares exactly the necessary and sufficient stability conditions for an LTI-MTDS in the space of the system parameters. This concept paper is written with the mindset that this analytical tool can invite yet unexplored design capabilities for similar noise control applications where acoustic dampers are used.

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