Reduced-Order Modeling of Thermoacoustic Systems: From the Helmholtz Equation to a Single Algebraic Equation
Motivation
The Helmholtz equation, a partial differential equation that represents the wave equation in the frequency domain, is a cornerstone of thermoacoustic modeling. Combined with a representation of the flame response, it is widely used to analyze the stability behavior of thermoacoustic systems. While solving the associated eigenvalue problem is computationally inexpensive compared to high-fidelity numerical simulations, it remains resource-intensive for parametric analyses.
Moreover, the contributions of the flame and other subsystems (e.g., resonators or acoustic boundaries) to the overall stability of the system are often obscured. For these reasons, reduced-order models (ROMs) offer an appealing alternative, enabling efficient parametric studies and enhancing system interpretability. Such models establish a direct connection between acoustic subsystems and the eigenfrequencies that govern system stability across distinct frequency ranges.
The one-mode Galerkin expansion continues to be a viable approach for deriving ROMs. However, its applicability is limited to regions near the eigenfrequency selected for the analysis, typically corresponding to the natural frequencies of the thermoacoustic system under study.
Objectives and Strategy
In a recent study by our group, a modified version of the one-mode Galerkin expansion has been introduced, extending the model's validity across a broader frequency range. Unlike the classical approach, this new variant also accounts for intrinsic thermoacoustic modes. Consequently, the novel Galerkin expansion—expressed mathematically as a simple algebraic equation—emerges as a powerful tool for enhancing the understanding of how the flame and other subsystems influence the overall stability of the thermoacoustic system. Ongoing research aims to adapt this model to the time domain, enabling the investigation of limit cycles, a defining characteristic of unstable thermoacoustic systems