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This paper concerns the influence of the phase of the heat release response on thermoacoustic systems. We focus on one pair of degenerate azimuthal acoustic modes, with frequency ω 0 . The same results apply for an axial acoustic mode. We show how the value ϕ 0 and the slope − τ of the flame phase at the frequency ω 0 affects the boundary of stability, the frequency and amplitude of oscillation, and the phase ϕ qp between heat release rate and acoustic pressure. This effect depends on ϕ 0 and on the nondimensional number τω 0 , which can be quickly calculated. We find for example that systems with large values of τω 0 are more prone to oscillate, i.e. they are more likely to have larger growth rates, and that at very large values of τω 0 the value ϕ 0 of the flame phase at ω 0 does not play a role in determining the system’s stability. Moreover for a fixed flame gain, a flame whose phase changes rapidly with frequency is more likely to excite an acoustic mode. We propose ranges for typical values of nondimensional acoustic damping rates, frequency shifts and growth rates based on a literature review. We study the system in the nonlinear regime by applying the method of averaging and of multiple scales. We show how to account in the time domain for a varying frequency of oscillation as a function of amplitude, and validate these results with extensive numerical simulations for the parameters in the proposed ranges. We show that the frequency of oscillation ω B and the flame phase ϕ qp at the limit cycle match the respective values on the boundary of stability. We find good agreement between the model and thermoacoustic experiments, both in terms of the ratio ω B / ω 0 and of the phase ϕ qp , and provide an interpretation of the transition between different thermoacoustic states of an experiment. We discuss the effect of neglecting the component of heat release rate not in phase with the pressure p as assumed in previous studies. We show that this component should not be neglected when making a prediction of the system’s stability and amplitudes, but we present some evidence that it may be neglected when identifying a system that is unstable and is already oscillating [ABSTRACT FROM AUTHOR]

The Flame Describing Function (FDF) is a useful and relatively cheap approximation of a flame’s nonlinearity with respect to harmonic velocity fluctuations. When embedded into a linear acoustic network, it is able to predict the amplitude and stability of harmonic thermoacoustic oscillations through the harmonic balance procedure. However, situations exist in which these oscillations are not periodic, but their spectrum contains peaks at several incommensurate frequencies. If one assumes that two frequencies dominate the spectrum, these oscillations are quasiperiodic, and the FDF concept can be extended by forcing the flame with two amplitudes and two frequencies. The nonlinearity is then approximated by a Flame Double Input Describing Function (FDIDF), which is a more expensive object to calculate than the FDF, but contains more information about the nonlinear response. In this study, we present the calculation of a non-static flame’s FDIDF. We use a G -equation-based laminar conical flame model. We embed the FDIDF into a thermoacoustic network and we predict the nature and amplitude of thermoacoustic oscillations through the harmonic balance method. A criterion for the stability of these oscillations is outlined. We compare our results with a classical FDF analysis and self-excited time domain simulations of the same system. We show how the FDIDF improves the stability prediction provided by the FDF. At a numerical cost roughly equivalent to that of two FDFs, the FDIDF is capable to predict the onset of Neimark–Sacker bifurcations and to identify the frequency of oscillations around unstable limit cycles. At a higher cost, it can also saturate in amplitude these oscillations and predict the amplitude and stability of quasiperiodic oscillations. [ABSTRACT FROM AUTHOR]

We analyse the linear response of laminar conical premixed flames modelled with the linearised front-track kinematic G -equation. We start by considering the case in which the flame speed is fixed, and travelling wave velocity perturbations are advected at a speed different from the mean flow velocity. A previous study of this case contains a small error in the Flame Transfer Function (FTF), which we correct. We then allow the flame speed to depend on curvature. No analytical solutions for the FTF exist for this case so the FTF has to be calculated numerically as its parameters – aspect ratio, convection speed and Markstein length – are varied. Then we consider the stability and sensitivity of thermoacoustic systems containing these flames. Traditionally, the stability of a thermoacoustic system is found by embedding the FTF within an acoustic network model. This can be expensive, however, because the FTF must be re-calculated whenever a flame parameter is varied. Instead, we couple the linearised G -equation directly with an acoustic network model, creating a linear eigenvalue problem without explicit knowledge of the FTF. This provides a simple and quick way to analyse the stability of thermoacoustic networks. It also allows us to use adjoint sensitivity analysis to examine, at little extra cost, how the system’s stability is affected by every parameter of the system. [ABSTRACT FROM AUTHOR]

A counterflow diffusion flame between gaseous hydrogen and liquid oxygen (LOx) is studied numerically at 1 and 2 bar pressures. Conditions at the liquid interface are modelled using the Clausius-Clapeyron relation and the species profiles are evaluated with a one-dimensional numerical code. Complex chemistry and multicomponent transport are employed. Thermodynamic and transport properties are taken from Chemkin and the corresponding Transport packages. Typical species and temperature profiles are presented. The extinction strain rate is evaluated as a function of the inlet hydrogen temperature. This varies from 1.2 × 105 s−1 at a hydrogen temperature of 20K to 6.0 × 105 s−1 at a temperature of 310K, indicating that hydrogen/LOx flames are extremely resistant to strain rate. The effect of the temperature gradient on the liquid side of the interface is examined and found to be negligible. When applied to one aspect of the flame-holding in cryogenic rocket motors, these results may be used to infer that extinction by strain rate is improbable in the injector near-field, even for very low hydrogen stream temperatures. [Copyright &y& Elsevier]

Abstract: When a premixed flame is placed within a duct, acoustic waves induce velocity perturbations at the flame’s base. These travel down the flame, distorting its surface and modulating its heat release. This can induce self-sustained thermoacoustic oscillations. Although the phase speed of these perturbations is often assumed to equal the mean flow speed, experiments conducted in other studies and Direct Numerical Simulation (DNS) conducted in this study show that it varies with the acoustic frequency. In this paper, we examine how these variations affect the nonlinear thermoacoustic behaviour. We model the heat release with a nonlinear kinematic G-equation, in which the velocity perturbation is modelled on DNS results. The acoustics are governed by linearised momentum and energy equations. We calculate the flame describing function (FDF) using harmonic forcing at several frequencies and amplitudes. Then we calculate thermoacoustic limit cycles and explain their existence and stability by examining the amplitude-dependence of the gain and phase of the FDF. We find that, when the phase speed equals the mean flow speed, the system has only one stable state. When the phase speed does not equal the mean flow speed, however, the system supports multiple limit cycles because the phase of the FDF changes significantly with oscillation amplitude. This shows that the phase speed of velocity perturbations has a strong influence on the nonlinear thermoacoustic behaviour of ducted premixed flames. [Copyright &y& Elsevier]

We create a general model for the tracking of a premixed flame sheet. This model is based on a Lagrangian formulation of the interface motion, which is more computationally efficient than the level-set formulation solved on a grid. We apply this model to a conical Bunsen flame in which flame-base oscillations and flame stretch effects are also considered. We demonstrate that the flame-front dynamics, the heat release rate perturbations, and the flame transfer function (FTF) are affected by three features: (i) the velocity field locally perturbing the flame-front, (ii) the flame-base oscillations normal to the steady flame-front propagating along the flame and (iii) the flame-base oscillations tangential to the flame-front reducing and increasing the flame-length. We provide a physical explanation for the saturation of the FTF phase shift with increasing forcing frequency, which is found experimentally. We show why a n − τ model with fixed τ cannot model this simple flame and that, when building a quantitatively-accurate model of a thermoacoustic system containing this flame, the flame dynamics need to be simulated directly or modelled more accurately than with the n − τ model. The flame model used in this paper has been designed to be differentiable so that it can be used for rapid data assimilation using Laplace's method. Code is provided so that the user can generate the FTF of a conical premixed flame with any set of parameters and any flame-base trajectory. [ABSTRACT FROM AUTHOR]