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This paper investigates a predictive model that considers the impact of stretch on the dynamic response of laminar premixed conical and V- flames; the flame stretch consists of two components: the flame curvature and flow strain. The steady and perturbed flame fronts are determined via the linearized $G$-equation associated with the flame stretch model. Parameter analyses of the effects of Markstein length $\mathcal{L}$, flame radius $R$ and unstretched flame aspect ratio $\beta$ are also conducted. Results show that the flame stretch reduces the steady flame height, with this effect being more significant for larger Markstein lengths and smaller flame sizes. The effects of flame stretch on perturbed flames are evaluated by comparing the flame transfer function (FTF) considering the flame stretch and not. For flames of different sizes, the impact of flame stretch on FTF gain can be divided into three regions. When both $\beta$ and $R$ are relatively small, due to the decrease in steady flame height and the impact of flow strain, the FTF gain increases. As $\beta$ and $R$ gradually increase, the FTF gain of the conical flame oscillates periodically while the FTF gain of the V-flame decreases, primarily due to the flame curvature enhancing the flame front disturbance and the wrinkle counteracting effect. When $\beta$ and $R$ are large, a disruption in wrinkle counteracting effect ensues, leading to a significant increase in FTF gain. Furthermore, as the actual flame height is reduced, the flame stretch also reduces the FTF phase lag which is related to the disturbance propagation time from the flame root to the tip.

The two-way interaction between the unsteady flame heat release rate and acoustic waves can lead to combustion instability within combustors. To understand and quantify the flame response to oncoming acoustic waves, previous studies have typically considered the flame dynamic response to pure tone forcing and assumed a dynamically linear or weakly nonlinear response. In this study, the introduction of excitation with two distinct frequencies denoted $St_1$ and $St_2$ is considered, including the effect of excitation amplitude in order to gain more insight into the nature of flame nonlinearities and their link with combustion instabilities. The investigation considers laminar flames and combines a low-order asymptotic analysis (up to third order in normalised excitation amplitude) with numerical methods based on the model framework of the $G$-equation. The importance of the propagation speed of the disturbance and its variation with frequency on the nonlinear response of the flame is highlighted. The influence path of the disturbance at one of the forcing frequencies, say $St_2$, on the flame dynamic response at the other forcing frequency $St_1$ is studied in detail. In concrete terms, the perturbation at $St_2$ acts in conjunction with the perturbation at $St_1$ to induce third-order nonlinear interactions in the flame kinematics, significantly altering the behavior of the flame response at $St_1$, as compared to the case where the flame is only subjected to the excitation at $St_1$. Particularly, when the normalised forcing amplitudes at the two frequencies are 0.2 and 0.3 respectively, the heat release rate response at the former frequency is attenuated by over 40 \% compared to the single-frequency response. This provides important insights into how nonlinearity due to frequency interactions can act to reduce the flame response.

Predictions of thermoacoustic instabilities in annular combustors are essential but difficult. Axial variations of flow and thermal parameters increase the cost of numerical simulations and restrict the application of analytical solutions. This work aims to find approximate analytical solutions for the acoustic field in annular ducts with linear gradients of axially non-uniform cross-sectional area and mean temperature. These solutions can be applied in low-order acoustic network models and enhance the ability of analytical methods to solve thermoacoustic instability problems in real annular chambers. A modified WKB method is used to solve the wave equation for the acoustic field, and an analytical solution with a wide range of applications is derived. The derivation of the equations requires assumptions such as low Mach number, high frequency and small non-uniformity. Cases with arbitrary distributions of cross-sectional surface area and mean temperature can be solved by the piecewise method as long as the assumptions are satisfied along the entire chamber., Comment: 23 pages, 14 figures

Acoustic and entropy waves interacting in a duct with a steady heat source and mean flow are analysed using an asymptotic expansion (AE) for low frequencies. The analytical AE solutions are obtained by taking advantage of flow invariants and applying a multi-step strategy. The proposed solutions provide first-order corrections to the compact model in the form of integrals of mean flow variables. An eigenvalue system is then built to predict the thermoacoustic modes of a duct containing a distributed heat source or sink. Predictions from the AE solutions agree well with the numerical results of the linearised Euler equations for both frequencies and growth rates, as long as the low-frequency condition is satisfied. The AE solutions are able to accurately reconstruct the acoustic and entropy waves and correct the significant errors in the predicted entropy wave associated with the compact model. The analysis illustrates that the thermoacoustic system needs to account for the entropy wave generated by the interaction of acoustic wave and the distributed steady heat source, especially when density- or entropy-dependent boundary conditions are prescribed at the duct ends. Furthermore, a combination of the AE method and the modified WKB approximation method is discussed for a cooling case. The AE solutions remedy the disadvantage of the WKB solution in the low and very low-frequency domain and facilitate full-frequency theoretical analyses of sound propagation and entropy generation in inhomogeneous duct flow fields., Comment: 26 pages, 12 figures, to be published in the Journal of Sound and Vibration

The unsteady response of nozzles with steady heat transfer forced by acoustic and/or entropy waves is modelled. The approach is based on the quasi-one-dimensional linearised Euler equations. The equations are cast in terms of three variables, namely the dimensionless mass, stagnation temperature and entropy fluctuations, which are invariants of the system at zero frequency and with no heat transfer. The resulting first-order system of differential equations is then solved using the Magnus expansion method, where the perturbation parameters are the normalised frequency and the volumetric heat transfer. In this work, a measure of the flow non-isentropicity (in this case the steady heat transfer) is used for the first time as an expansion parameter. The solution method was applied to a converging-diverging nozzle with constant heat transfer for both sub-critical and super-critical flow cases, showing good agreement with numerical predictions. It was observed that the acoustic and entropy transfer functions of the nozzle strongly depend on the frequency and heat transfer., Comment: 15 Pages, 6 figures. Will be published soon in JFM Rapids

In this work, we investigate both numerically and theoretically the sound generated by entropy waves passing through sudden area expansions. The numerical approach is based on a triple decomposition of the flow variables into a steady mean, a small-amplitude coherent part, and a stochastic turbulent part. The mean flow is obtained as the solution of the Reynolds-Averaged Navier-Stokes (RANS) equations. The equations governing the coherent perturbations are linearised and solved in the frequency domain. To account for the effect of turbulence on the coherent perturbations, a frozen eddy viscosity model is employed. When entropy fluctuations pass through the area expansion, the generated entropy noise behaves as a low-pass filter. The numerical predictions of the noise at low frequencies are compared to the predictions of compact, quasi-one-dimensional, and isentropic theory and large discrepancies are observed. An alternative model for the generated entropy noise tailored for area expansions is then proposed. Such model is based on the conservation of mass, momentum, and energy written in integral form. The model assumes zero frequency and the one-dimensionality of the flow variables far upstream and downstream of the expansion. The predictions of this model agree well with the numerical simulations across a range of finite subsonic Mach numbers including low, intermediate, and high Mach numbers., Comment: 35 pages, 16 figures

In ducts with varying cross-sectional area and sustaining a subsonic non-isentropic mean flow, the axially varying flow conditions affect the acoustic energy balance of the system. This is significant in understanding and controlling thermo-acoustic phenomena, particularly in combustors. This work aims at quantifying the acoustic energy change in such configurations, using the acoustic absorption coefficient, $\Delta$. The acoustic response of the duct to acoustic forcing is determined using an analytical model, neglecting the effect of entropy fluctuations on the acoustic field, and subsequently, $\Delta$ is estimated. The model predictions of $\Delta$ are validated using a linearised Euler equations (LEEs) solver. The model was found to be accurate for Mach numbers below $0.25$, provided the lower frequency limit set by the analytical solution is satisfied. For conically varying area ducts with linear mean temperature gradient, it was observed that $\Delta$ showed very little dependence on frequency, and that the absolute value of $\Delta$ tended to be maximised when the upstream boundary was anechoic rather than non-anechoic. More importantly, $\Delta$ was also observed to show stronger dependence on the mean temperature gradient than area gradient variation for such configurations. Further parametric and optimisation studies for $\Delta$ revealed a crucial finding that a positive mean temperature gradient, representing a heated duct caused acoustic energy absorption. Similarly, a negative mean temperature gradient, representing a cooled duct caused acoustic energy generation -- a key result of this analysis. This behaviour was shown to be consistent with a simplified analysis of the acoustic energy balance. Based on this finding, a linearly proportional reduction in acoustic energy generation was achieved by changing the mean temperature gradient.

Hydrogen-enrichment of conventional natural gas mixtures is an actively-explored strategy for reducing pollutant emissions from combustion. This study investigates the effect of hydrogen enrichment on the unsteady flame response to perturbations, with a view to understanding the implications for thermoacoustic stability. The Level Set Approach for kinematically tracking the flame front was applied to a laminar conical premixed methane / hydrogen / air flame subjected to 2D incompressible velocity perturbations. For hydrogen enrichment levels ranging from 0% to 80% by volume, the resulting unsteady heat release rate of the flame was used to generate the Flame Describing Functions (FDFs). This was performed across a range of perturbation frequencies and levels at ambient pressure. The mean heat release rate of the flame was fixed at $\overline{\dot{Q}} = 2.69\ \textrm{kW}$ and the equivalence ratio was set to $\varphi=1.08$ for all hydrogen enrichment levels. Hydrogen-enrichment was found to shift the FDF gain drop-off to higher frequencies, which will increase propensity to thermoacoustic instability. It also reduced the effective flame time delay. Sensitivity analyses at $\varphi = 0.8$ revealed that the changes in FDF were driven predominantly by the flame burning speed, and were insensitive to changes in Markstein length., Comment: Accepted for publication in International Journal of Hydrogen Energy

In this article, we show that the projection-free, snapshot-based, balanced truncation method can be applied directly to unstable systems. We prove that even for unstable systems, the unmodified balanced proper orthogonal decomposition algorithm theoretically yields a converged transformation that balances the Gramians (including the unstable subspace). We then apply the method to a spatially developing unstable system and show that it results in reduced-order models of similar quality to the ones obtained with existing methods. Due to the unbounded growth of unstable modes, a practical restriction on the final impulse response simulation time appears, which can be adjusted depending on the desired order of the reduced-order model. Recommendations are given to further reduce the cost of the method if the system is large and to improve the performance of the method if it does not yield acceptable results in its unmodified form. Finally, the method is applied to the linearized flow around a cylinder at Re = 100 to show that it actually is able to accurately reproduce impulse responses for more realistic unstable large-scale systems in practice. The well-established approximate balanced truncation numerical framework can therefore be safely applied to unstable systems without any modifications. Additionally, balanced reduced-order models can readily be obtained even for large systems, where the computational cost of existing methods is prohibitive.