Forward and Inverse Modelling in Thermoacoustics

Thermoacoustic instabilities are self-sustained oscillations that arise in combustion systems such as domestic boilers, gas turbines, and rocket engines. These instabilities result from the feedback between unsteady combustion, unsteady flow, and the structural dynamics of the system, leading to undesirable vibrations that range from irritating noise to severe oscillations capable of causing structural damage and safety risks. Due to the complexity of the underlying physics, the accurate prediction, prevention, and mitigation of thermoacoustic instabilities remain a challenge in the combustion industry. This research work addresses this problem from three key perspectives: (1) the modelling and prediction of thermoacoustic instabilities, (2) the impact of mean-flow uncertainties and feedback forcings on a thermoacoustic system, and (3) the acoustic properties of passive control devices such as porous materials and acoustic liners. For the first two perspectives, we demonstrate the application of adjoint approaches for stability analysis and sensitivity analysis in thermoacoustics. In the last part, we show the application of the homogenization approach in acoustic modelling. First, a theoretical framework based on the adjoint Green’s function (AGF) is developed to predict thermoacoustic instabilities in systems with mean flow. The adjoint method aids in determining the AGF system of non-self-adjoint systems. This approach converts the acoustic analogy equation with convection into an integral equation, yielding sensitivity functions that quantify the system's response to initial conditions, boundary conditions, and external forcings. The framework is applied to two types of resonators with different boundary conditions: a Rijke tube and a matrix burner, where control parameters include heat source position, heater power, and tube length. The results demonstrate the model's ability to capture key nonlinear behaviours observed in experiments, including limit cycles, triggering phenomena, hysteresis, and Hopf bifurcations. The analysis further reveals that mean flow velocity plays a crucial role in stabilizing the system and alteration of bistability. Additionally, the AGF framework is extended to thermoacoustic systems with non-uniform temperature fields and stochastic forcing. The study highlights the influence of mean temperature differences on system stability and examines the effects of white noise and pink noise. The impact of noise on the transient phase of the instability, before saturation is reached, is examined, and noise-induced triggering phenomena are found near stability margins. Pink noise is found to be more effective than white noise in triggering instabilities, while the presence of mean flow exhibits a hindering effect. Second, a comprehensive adjoint-based sensitivity analysis framework is demonstrated for a low-order lumped model of a gas turbine combustor. Three types of sensitivity analyses are carried out. The base-state sensitivity reveals how uncertainties in base flow properties, such as the density of cold gas in the premixer, significantly impact the growth rates of the eigenmodes. The structural sensitivity to steady feedback forcings uncovers steady control strategies such as local mass suction and heating as effective stabilizing mechanisms. Furthermore, the structural perturbation sensitivity examines the effect of localized feedback forcings on the perturbation flow. The results identify Helmholtz resonators as effective stabilizers and pinpoint their optimal placement. These insights broaden the scope of uncertainty quantification and control strategies for premixed combustion systems. Finally, the acoustic properties of passive control devices, such as porous materials and acoustic liners with periodic microstructures, are examined. While classical homogenization methods effectively describe wave refraction inside the bulk region of porous media, they fail to capture the reflection of acoustic waves coming from the free field at material surface faces. To address this limitation, a homogenization approach combined with asymptotic matching is employed, yielding an effective interface jump condition for acoustic propagation in porous materials with rigid scatterers. The results show that this interface condition effectively models the reflection of the acoustic waves at the free-field-porous material interface for porous materials of porosity $\phi \geq 0.8$. In another case, a homogenized model is developed to describe acoustic propagation through a single array of rigid cylinders. The model introduces an effective domain represented as a slab of specified thickness. Reflection and transmission coefficients are computed using the upscaled model and compared with direct numerical results and experimental measurements. The results demonstrate good agreement at lower frequencies, while discrepancies increase at higher frequencies due to the decreasing wavelength, which affects the validity of the length scale separation assumption.