Combustion instability analysis in annular systems often relies on reduced-order models that represent the complexity of combustion dynamics in a framework in which the flame is represented by a ‘flame describing function’ (FDF), portraying its heat release rate response to acoustic disturbances. However, in most cases, FDFs are only available for a limited range of disturbance amplitudes, complicating the description of the saturation process at high oscillation levels leading to the establishment of a limit cycle. This article shows that this difficulty may be overcome using a novel experimental scheme, relying on injector staging and in which the oscillation amplitude at limit cycle can be controlled, enabling us to measure FDFs from simultaneous pressure and heat release rate recordings. These data are then exploited to replace the standard modelling, in which the heat release rate is expressed as a third-order polynomial of pressure fluctuations, by a function of the modulation amplitude, allowing an easier adaptation to experimental data. The FDF is then used in a dynamical framework to analyse a set of staging configurations in an annular combustor, where two families of injectors are mixed and form different patterns. The limit-cycle amplitudes and the coupling modes observed experimentally are suitably retrieved. Finally, an expression for the growth rate is derived from the slow-flow variable equations defining the modal amplitudes and phase functions, which is shown to exactly agree with that obtained previously by using acoustic energy principles, providing a theoretical link between growth rates and limit-cycle amplitudes.

Two-dimensional gaseous detonations near critical propagation state were studied numerically in a channel with stoichiometric H$_2$/air and H$_2$/O$_2$ mixtures. Detonation waves exhibit a mode-locking effect (MLE) in a single-headed mode regime. Increasing the channel width alters the strength and propagation period of the single transverse wave. This leads to MLE failure and the occurrence of the single-dual-headed critical mode, featuring the emergence of a new transverse wave. For a stoichiometric H$_2$/air mixture, generation of the new transverse wave is due to interactions between the detonation front and the local explosion wave originating from interactions between the transverse wave and unreacted gas pocket downstream. Whereas, for a stoichiometric H$_2$/O$_2$ mixture, a transverse wave interacting with the wall produces Mach reflection bifurcation, causing MLE failure and generation of the new transverse wave. Our results show that all transverse waves manifest as strong transverse wave (STW) structures, with most belonging to the second kind, and an acoustic coupling exists between the typical second kind of STW structure and the acoustic wave in the induction zone behind the Chapman–Jouguet detonation front. A high-pressure region close to the STW structure plays a crucial role in exploring the transverse dynamics of this structure. Shock polars with rational assumptions are adopted to predict flow states in this region. The roles of pivotal factors in influencing the flow states and wave structure are clarified, and characteristic pressure values derived adequately represent the STW structure’s transverse dynamic behaviours. Lastly, the relationship between the kinematics and kinds of STW structures is unveiled.

Premixed hydrogen flames are prone to thermodiffusive instabilities due to strong differential diffusion effects. Reproducing these instabilities in large eddy simulations (LES), where their effects are only partially resolved, is challenging. Combustion models that account for differential diffusion effects have been developed for laminar flames, but to use them in LES, models for the turbulence/flame subfilter interactions are required. Modelling of the subfilter interactions is particularly challenging as instabilities synergistically interact with turbulence resulting in a strong enhancement of the turbulent flame speed. In this work, a combustion model for LES, which accounts for thermodiffusive instabilities and their interactions with turbulence, is presented. In the first part, an a priori analysis based on a direct numerical simulation (DNS) of a turbulent hydrogen/air jet flame is discussed. Progress variable, progress variable variance and mixture fraction are rigorously identified as suitable model input parameters, and an LES combustion model based on pre-tabulated unstretched premixed flamelets with varying equivalence ratio is formulated. Subfilter closure is achieved via a presumed probability density function and a significant reduction of modelling errors is achieved with the presented model. In the second part, LES of the DNS configuration are performed for an a posteriori analysis. The presented combustion model shows significant improvements in predicting the flame length and local phenomena, such as super-adiabatic temperature, compared with combustion models that either neglect differential diffusion effects or consider these effects but neglect the subfilter closure. Two variants of the model formulation with a water- or hydrogen-based progress variable have been tested, yielding overall similar predictions.

The reactive Navier–Stokes equations with adaptive mesh refinement and a detailed chemical reactive mechanism (11 species, 27 steps) were adopted to investigate a detonation engine considering the injection and supersonic mixing processes. Flame acceleration and deflagration-to-detonation transition (DDT) in a premixed/inhomogeneous supersonic hydrogen–air mixture with and without transverse jet obstacles were addressed. Results demonstrate the difficulty in undergoing DDT in the premixed/inhomogeneous supersonic mixture within a smooth chamber. By contrast, multiple transverse jets injected into the chamber aid detonation transition by introducing perturbed vortices, shock waves and a suitable blockage ratio. Increasing distance between the leading shock and the flame tip impedes detonation transition due to an insufficient blockage ratio. The extremely perturbed distributions of fuel-lean and fuel-rich mixtures lead to more complicated flame structures. Also, a larger flame thickness appears in the inhomogeneous mixture compared with the premixed mixture, resulting in a lower combustion temperature. The key findings are that the DDT, detonation quenching and reinitiation are generated in the inhomogeneous supersonic mixture, but both DDT mechanisms are ascribed to a strong Mach stem with the Zel'dovich gradient mechanism. Additionally, the obtained results demonstrate that an intensely fuel-lean mixture (equivalence ratio = 0.15) results in a partially decoupled flame front. However, detonation reinitiation and subsequent self-sustained detonation occur when a fierce shock wave propagates through a highly sensitive mixture, even within a smaller and elongated area. Moreover, the inhomogeneous mixture also augments the propagation speed and detonation cell structure instabilities and delays the sonic point resulting from the extending non-equilibrium reaction.

The combined effects of heater position, mean flow parameters and flame models on thermoacoustic instability in a one-dimensional Rijke tube are studied systematically by classic linear stability analysis (LSA) and lattice Boltzmann method (LBM) simulation. In the former, the stability range of the linear flame model under low Mach number assumption is solved analytically, while in the more general case, it is obtained by numerically solving the dispersion relation. Both the linear and nonlinear flame model cases are studied using the LBM with a spectral multiple-relaxation-time collision model and a newly developed heat source term. With the linear flame model, the LBM is in good agreement with LSA in predicting the transition point and growth rates, while with the nonlinear flame model, LBM simulations are consistent with solutions of limit cycle theory in the fully developed state. These results demonstrate the applicability of the LBM in solving complex thermoacoustic problems.

This study comprehensively investigates the response of a combusting droplet during its interaction with a high-speed transient flow imposed by a coaxially propagating blast wave. The blast wave is generated using a specially designed miniature shock generator that produces blast waves using the wire-explosion technique, facilitating a wide range of Mach numbers (1.03 < Ms < 1.8). The experiments are performed in two configurations: open field and focused blast wave. The charging voltage and the configuration determine the Mach number (Ms) and flow characteristics. The flame is found to exhibit two major response patterns: partial extinction followed by reignition and full extinction. Increasing the Mach number (Ms > 1.1) makes the droplet flame more vulnerable to extinction. Additionally, the flame exhibits stretching and shedding, followed by reignition at lower Mach numbers (Ms < 1.06). In all cases, the flame base lifts off in response to the imposed flow, and the advection of the flame base interacting with the flame tip results in flame extinction. The entire interaction occurs in two stages: (i) interaction with the blast wave and the decaying velocity profile associated with it, and (ii) interaction with the induced flow behind the blast wave as a result of the entrainment (delayed response). Alongside the flame's response, the droplet also interacts with the flow imposed by the blast wave, exhibiting different response modes including pure deformation, Rayleigh–Taylor piercing bag breakup and shear-induced stripping.

A hydrodynamic theory of premixed flame propagation within closed vessels is developed assuming the flame is much thinner than all other fluid dynamic lengths. In this limit, the flame is confined to a surface separating the unburned mixture from burned combustion products, and propagates at a speed determined from the analysis of its internal structure. Unlike freely propagating flames that propagate under nearly isobaric conditions, combustion in a closed vessel results in continuous increases in pressure, burning rate and flame temperature, and a progressive decrease in flame thickness. The flame speed is shown to depend on the voluminal stretch rate, which measures the deformation of a volume element of the flame zone, and on the rate of pressure rise. Both effects are modulated by pressure-dependent Markstein numbers that depend on heat release and mixture properties while capturing the effects of temperature-dependent transport and stoichiometry. The model applies to flames of arbitrary shape propagating in general flows, laminar or turbulent, within vessels of general configurations. The main limitation of hydrodynamic flame theories is the assumption that variations inside the flame zone due to chemistry or turbulence, which could potentially alter its internal structure, are physically unresolved. Nonetheless, the theory, deduced from physical first principles, identifies the various mechanisms involved in the combustion process as demonstrated in detailed discussions of planar flames propagating in rectangular channels and spherically expanding flames in spherical vessels. It also enables the construction of instructive models to numerically simulate the evolution of multi-dimensional and corrugated flames under confinement.

Melt-front instabilities during the combustion of a spinning polymethylmethacrylate disk in air are investigated. Mainly straight rivulet-type flow patterns were found, though under certain conditions saw-tooth patterns were observed. The measured wavelengths of the instabilities agree with earlier theoretical predictions of driven contact-line instabilities.

This paper presents a comprehensive theoretical analysis of the interaction between two quasi-steady burning droplets with differing temperatures, sizes and distances, building upon the mass-flux-potential model and flame-sheet assumption. In contrast to existing research, this study introduces a fresh perspective on droplet interactions by considering the different temperatures of the droplets. Utilizing the bispherical coordinate approach, theoretical solutions for the Stefan flow, scalar fields, droplet evaporation/burning rates, interaction coefficients and flame positions have been derived successfully. A comparison with extensive numerical simulations indicates a good agreement between the analytical and numerical results under a variety of conditions. It is revealed that proximity between the droplets causes non-uniform evaporation rates on their surfaces, and in some cases, leads to condensation on the cooler droplet. Notably, when the temperatures of the two droplets differ, this results in an uneven temperature distribution across the flame surface, and increasing the temperature of one droplet substantially elevates the temperature of the nearby flame. This study also establishes a criterion for the transition between different combustion modes, specifically between group and separated combustion. The findings of this study are crucial in deepening our understanding of evaporation and combustion processes, as well as the dynamics of flame spreading, local ignition, and extinction in systems involving multiple droplets.

Combustion instabilities in annular systems raise fundamental issues that are also of practical importance to aircraft engines and ground-based gas turbine combustors. Recent studies indicate that the injector plays a significant role in the stability of combustors by defining the flame dynamical response and setting the inlet impedance of the system. The present investigation examines the effects of combinations of injectors of two different types ($U$ and $S$) on thermoacoustic instabilities in a laboratory-scale annular combustor and compares different circumferential staging strategies. The combustor operates in a stable fashion when all injection units belong to the $S$-family, but exhibits large amplitude pressure oscillations when all these units are of the $U$-type. When the system comprises a mix of $U$- and $S$-injectors, it is possible to determine the number of $S$-injectors leading to stable operation. For a fixed proportion of $U$- and $S$-injectors, some arrangements give rise to stable operation while others do not. Results also show that introducing symmetry-breaking elements affects the system's modal dynamics. These experimental observations are interpreted in an acoustic energy balance framework used to derive an expression for the growth rate as a function of the describing functions of the flames formed by the different injectors and their respective azimuthal locations. Growth rates are determined for the different configurations and used to explain the various observations, estimate the system damping rate and predict the location of the nodal line when the standing mode prevails.

Identifying the governing parameters of self-sustained oscillation is crucial for the diagnosis, prediction and control of thermoacoustic instabilities. In this paper, we propose and validate a novel method for computing the parameters of thermoacoustic oscillation in a stochastic environment, which exploits a physics-informed neural network (PINN). Specifically, we introduce a negative log-likelihood loss function that integrates the stochastic samples and the solution of the Fokker–Planck equation. The proposed framework is validated using the numerically generated signal and the experimental data obtained from an annular combustor, both before and after the supercritical Hopf bifurcation. The results of PINN-based system identification show good agreement with the actual system parameters and the original stochastic signal, with improved accuracy compared to established methods. To the best of our knowledge, this study constitutes the first demonstration of the PINN-inverse approach that uses the noise-induced dynamics of thermoacoustic systems, opening up new pathways for diagnosing and predicting the thermoacoustic behaviour of various combustion systems.

The unresolved scalar variance in large-eddy simulations of turbulent flows is a fundamental physical and modelling parameter. Despite its importance, relatively few algebraic models have been developed for this important variable with the most prominent models to date being the classic scale-similarity and gradient models. In this work a new generalized modelling framework based on reconstruction has been developed, which in contrast to classic modelling approaches allows the construction of base static variance models of arbitrary accuracy. It is demonstrated that higher-order reconstructions naturally lead to base static variance models of increased accuracy, and that the classic scale-similarity and gradient models are subsets of more general and higher-order models. The classic scale-similarity assumption for developing dynamic models is also revisited, and it is demonstrated that this can essentially be reinterpreted as a two-level reconstruction approach. Based on this result, a new general methodology is proposed that allows the construction of dynamic models for any given base static model, and a corresponding general reconstruction operator, algebraic or iterative. Consequently, improved static and dynamic models for the scalar variance are developed. The newly developed models are then thoroughly tested a priori using two high-fidelity direct numerical simulation databases corresponding to two substantially different flame and flow configurations, and are shown to outperform classic algebraic models for the variance.

Can-annular combustors feature clusters of thermoacoustic eigenvalues, which originate from the weak acoustic coupling between $N$ identical cans at the downstream end. When instabilities occur, one needs to consider the nonlinear interaction between all $N$ modes in the unstable cluster in order to predict the steady-state behaviour. A nonlinear reduced-order model for the analysis of this phenomenon is developed, based on the balance equations for acoustic mass, momentum and energy. Its linearisation yields explicit expressions for the $N$ complex-valued eigenfrequencies that form a cluster. To treat the nonlinear equations semianalytically, a Galerkin projection is performed, resulting in a nonlinear system of $N$ coupled oscillators. Each oscillator represents the dynamics of a global mode that oscillates in the whole can-annular combustor. The analytical expressions of the equations reveal how the geometrical and thermofluid parameters affect the thermoacoustic response of the system. To gain further insights, the method of averaging is applied to obtain equations for the slow-time dynamics of the amplitude and phase of each mode. The averaged system, whose solutions compare very well with those of the full oscillator equations, is shown to be able to predict complex transient dynamics. A variety of dynamical states are identified in the steady-state oscillatory regime, including push–push (in-phase) and spinning oscillations. Notably, the averaged equations are able to predict the existence of synchronised states. These states occur when the frequencies of two (or more) unstable modes with nominally different frequencies lock onto a common frequency as a result of nonlinear interactions.

The tangential strain rate in premixed flames impacts significantly the flame surface area generation and thus the combustion process. Studies on incompressible isotropic turbulence have revealed that the mean tangential strain rate at material and iso-scalar surfaces is positive and exhibits a universal value when normalized by the Kolmogorov time. This is associated with the preferential alignment of the surface normal with the most compressive principal strain rate. The present study investigates such effects in premixed hydrogen and iso-octane flame kernels using direct numerical simulations. It is shown that the normalized mean tangential strain rate of the investigated flames has a very similar value compared with the incompressible flows. However, in the reaction zone, the flame surface normal aligns preferentially with the most extensive principal strain rate. Furthermore, this alignment depends on the reaction progress variable and the Lewis number, while the tangential strain rate remains independent of these parameters. Such counter-intuitive behaviour is systematically investigated by decomposing the effects of dilatation and residual solenoidal turbulence. It is found that the solenoidal turbulence influences significantly the tangential strain rate. A general effect of turbulence on the tangential strain rate is identified, which is consistent with incompressible flows and independent of the Lewis number and the reaction progress variable. This is a remarkable finding indicating that models of the tangential strain rate developed based on incompressible flows apply also to premixed flames with different Lewis numbers, and, for the modelling, only the solenoidal turbulence should be considered.

The present study investigates the flame dynamics of a contactless burning fuel droplet under free fall subjected to a co-flow. The dynamic external relative flow established due to co-flow and droplet acceleration results in a series of droplet flame transitions. Different flame structures were observed, including a wake flame, reversed wake flame and enveloped flame. Following ignition, the droplet is allowed to fall through the central tube of a co-flow arrangement, and, at its exit, the droplet flame encounters the co-flow. The wake flame, which was established based on the droplet's instantaneous velocity of descent, encounters the abrupt relative velocity jump due to the co-flow. This causes the droplet flame to go through various transitions as it approaches equilibrium with the surrounding flow. Once it equilibrates, the droplet flame evolves in response to the instantaneous relative flow velocity. The droplet flame evolves by altering both its shape and the stabilization mechanism. Two stabilization mechanisms were identified for the droplet wake flame: edge-flame stabilization and bluff-body stabilization. The stabilization mechanism for different flame structures and the transition events have been theoretically analysed, and the relation between flame shape evolution and flow velocity has been determined based on the flow-field characteristics at the corresponding Re (Reynolds number) range. Furthermore, these correlations are employed in a mathematical formulation based on the spring–mass analogy, which predicts the droplet flame evolution after encountering the co-flow, including all the transition events.

Sample materials such as sediments and soils contain complex mixtures of different carbon-containing compounds. These bulk samples can be split into individual fractions, based on the temperature of thermal decomposition of their components. When coupled with radiocarbon (14C) measurement of the isolated fractions, this approach offers the advantage of directly investigating the residence time, turnover time, source, or age of the different components within a mixed sample, providing important insights to better understand the cycling of carbon in the environment. Several laboratories have previously reported different approaches to separate radiocarbon samples based on temperature in what is a growing area of interest within the research community. Here, we report the design and operation of a new ramped oxidation facility for separation of sample carbon on the basis of thermal resistance at the NEIF Radiocarbon Laboratory in East Kilbride, UK. Our new instrumentation shares some characteristics with the previously-reported systems applying ramped oxidation and/or ramped pyrolysis for radiocarbon measurement, but also has several differences which we describe and discuss. We also present the results of a thorough program of testing of the new system, which demonstrates both the reproducibility of the thermograms generated during sample combustion, and the reliability of the radiocarbon measurements obtained on individual sample fractions. This is achieved through quantification of the radiocarbon background and analysis of multiple standards of known 14C content during standard operation of the instrumentation.

The present experimental study investigated the dynamics of single- and multi-port gaseous jet diffusion flames exposed to acoustic excitation via a standing wave situated in a closed waveguide at atmospheric pressure. High-speed imaging of the oscillatory flame was analysed via proper orthogonal decomposition (POD), revealing distinct signatures in both mode shapes and phase portraits for transitions in the acoustically coupled combustion process. For Reynolds numbers between 20 and 100, and for low to moderate forcing amplitudes, the flame exhibited sustained oscillatory combustion (SOC) that was highly coupled to the acoustic forcing. Frequency analysis of the temporal POD modes accurately recovered the forcing frequency and its higher harmonics. At higher forcing amplitudes, a multi-frequency response was observed, resulting from a combination of the forcing frequency and much lower frequency oscillations due to periodic lift-off and reattachment (PLOR) of the flame, preceding a transition to flame blow-off (BO). For both single- and triple-jet flames, transitions from SOC to PLOR to BO were characterized by significant alterations in primary modal energetic content, deflection and eventual smearing in phase portraits, and the development of additional frequencies in modal spectra, although transitional behaviour for the triple jet flames involved additional complexity in the dynamics due to its structure. These features provide the potential for the development of reduced-order models that can characterize and predict acoustically coupled combustion behaviour.