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.

We investigate self-excited axisymmetric oscillations of a lean premixed methane–air V-flame in a laminar annular jet. The flame is anchored near the rim of the centrebody, forming an inverted cone, while the strongest vorticity is concentrated along the outer shear layer of the annular jet. Consequently, the reaction and vorticity dynamics are largely separated, except where they coalesce near the flame tip. The global eigenmodes corresponding to the linearised reacting flow equations around the steady base state are computed in an axisymmetric setting. We identify an arc branch of eigenmodes exhibiting strong oscillations at the flame tip. The associated eigenvalues are robust with respect to domain truncation and numerical discretisation, and they become destabilised as the Reynolds number increases. The frequency of the leading eigenmode is found to correspond to the Lagrangian disturbance advection time from the nozzle outlet to the flame tip. The essential role of this convective mechanism is also supported by resolvent analysis, which finds that the same flame-tip disturbance structure and frequency are optimally amplified when the flame is subjected to external white noise forcing. Strong non-modal effects in the form of pseudo-resonance are not found. Nonlinear time-resolved simulation further reveals notable hysteresis phenomena in the subcritical regime prior to instability. Hence, even when the flame is linearly stable, perturbations of sufficient amplitude can trigger limit-cycle oscillations and higher-dimensional dynamics sustained by nonlinear feedback. A Monte Carlo simulation of passive tracers in the unsteady flame suggests a nonlinear non-local instability mechanism. Notably, linear analysis of the subcritical time-averaged limit-cycle state yields eigenvalues that do not match the nonlinear periodic oscillation frequencies. This mismatch is attributed to the fundamentally nonlinear dynamics of the subcritical V-flame instability, where the dichromatic, non-local interaction between the heat release rate along the flame surface and the vortex dynamics in the jet shear layer cannot be approximated as a simple distortion of the mean flow.

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.

We use Bayesian inference, accelerated by adjoint methods, to construct a quantitatively accurate model of the thermoacoustic behaviour of a conical flame in a duct. We first perform a series of automated experiments on a ducted flame rig. Next, we propose several candidate models of the rig's components and assimilate data into each model to find the most probable parameters for that model. We rank the candidate models based on their marginal likelihood (evidence) and select the most likely model for each component. We begin this process by rigorously characterizing the acoustics of the cold rig. When the flame is introduced, we propose several candidate models for the fluctuating heat release rate. We find that the most likely flame model considers velocity perturbations in both the burner feed tube and the outer duct, even though studies in the literature typically neglect either one of these. Using the most likely model, we infer the flame transfer functions for 24 flames and quantify their uncertainties. We do this with the flames in situ, using only pressure measurements. We find that the inferred flame transfer functions render the model quantitatively accurate, and, where comparable, broadly consistent with direct measurements from several studies in the literature.

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.

Direct numerical simulation data obtained from four pairs of turbulent, lean hydrogen–air, complex-chemistry flames are analysed to explore the influence of molecular diffusion on flame surface density, displacement speed $S_d$ and the flame surface density transport equation terms. Each pair involves (i) a flame where mixture-averaged molecular diffusivities are adopted and Lewis number $Le$ is significantly less than unity and (ii) an equidiffusive flame where all molecular diffusivities are set equal to molecular heat diffusivity of the mixture and $Le=1$, with other things being equal. Reported results show that significantly higher turbulent burning rates simulated in the former flames result mainly from an increase in the local fuel consumption rate, whereas an increase in flame surface area plays a secondary role, especially in more intense turbulence. The rate increase stems from (i) an increase in the peak local fuel consumption rate and (ii) an increase in a width of a zone where the rate is significant. The latter phenomenon is of more importance in richer flames and both phenomena are most pronounced in the vicinity of the flame leading edges, thus indicating a crucial role played by the leading edge of a premixed turbulent flame in its propagation. Moreover, mean displacement speed differs significantly from the laminar flame speed even in the equidiffusive flames, varies substantially across flame brush and may be negative at the leading edges of highly turbulent flames.

Premixed turbulent flames, encountered in power generation and propulsion engines, are an archetype of a randomly advected, self-propagating surface. While such a flame is known to exhibit large-scale intermittent flapping, the possible intermittency of its small-scale fluctuations has been largely disregarded. Here, we experimentally reveal the inner intermittency of a premixed turbulent V-flame, while clearly distinguishing this small-scale feature from large-scale outer intermittency. From temporal measurements of the fluctuations of the flame, we find a frequency spectrum that has a power-law subrange with an exponent close to $-2$, which is shown to follow from Kolmogorov phenomenology. Crucially, however, the moments of the temporal increment of the flame position are found to scale anomalously, with exponents that saturate at higher orders. This signature of small-scale inner intermittency is shown to originate from high-curvature, cusp-like structures on the flame surface, which have significance for modelling the heat release rate and other key properties of premixed turbulent flames.

This study investigates the linear stability of a laminar premixed flame, anchored on a square cylinder and confined inside a channel. Many modern linear analysis concepts have been developed and validated around non-reacting bluff-body wake flows, and the objective of this paper is to explore whether those tools can be applied with the same success to the study of reacting flows in similar configurations. It is found that linear instability analysis of steady reacting flow states accurately predicts critical flow parameters for the onset of limit-cycle oscillations, when compared to direct numerical simulation performed with a simple one-step reaction scheme in the low Mach number limit. Furthermore, the linear analysis predicts a strong stabilising effect of flame ignition, consistent with documented experiments and numerical simulations. Instability in ignited wake flows is, however, found to set in at sufficiently high Reynolds number, and a linear wavemaker analysis characterises this instability as being driven by hydrodynamic mechanisms of a similar nature as in non-reacting wake flows. The frequency of nonlinear limit-cycle flame oscillations in this unstable regime is retrieved accurately by linear eigenmode analysis performed on the time-averaged mean flow, under the condition that the full set of the reacting flow equations is linearised. If, on the contrary, unsteadiness in the density and in the reaction rate are excluded from the linear model, then the congruence between linear and nonlinear dynamics is lost.

A high-fidelity large-eddy simulation and unsteady flamelet combustion model construct is deployed to numerically investigate the effects of crosswind magnitude and pool fire shape on large-scale pool fire attributes. These include general flame dynamics, flame shape and radiative flux magnitude in and around the fire. Three pool fire shapes at a nominal length scale of 10 m are subjected to four crosswind magnitudes between 0 and 20 m s$^{-1}$. The pool shapes studied are circular, square and rectangular. The study includes the sensitivity of parameters to mesh and time step refinement. Results demonstrate that the rectangular shape, under crosswind, has low-levels of vertical velocity induction, resulting in a plume that is closer to the ground. In the quiescent regime, under-resolved meshes provide a higher radiative heat flux prediction compared with the most refined mesh. However, as crosswind increases, low mesh resolutions underpredicted radiative flux. This is due to the coarse mesh resolution not capturing small-scale vortical features that increased mixing and combustion efficiency. A transition of peak radiative flux with respect to crosswind occurs from the leeward- to windward-side of the pool, while sharp pool features result in larger radiative heat fluxes concentrated in regions of high scalar dissipation rate.

The objective of this work is to provide physical insight into the mechanisms governing flame–turbulence interactions and explore the impact of the ubiquitous Darrieus–Landau instability on the propagation. It is based on the hydrodynamic theory of premixed flames that considers the flame thickness much smaller than all other fluidynamical length scales. In this asymptotic limit, the flame is thus confined to a surface whilst the diffusion and reaction processes occurring inside the flame zone are accounted for by two parameters: the unburned-to-burned density ratio and the Markstein length. The robust model, which is free of phenomenology and turbulence modelling assumptions, makes transparent the mutual interactions between the flame and the fluid flow, and permits examining trends in flame and flow characteristics while varying the turbulence intensity and mixture properties. It is used in this study to examine the morphological changes of the flame surface that result from the intertwined effects of the turbulence and instability, as demonstrated by the local displacement and curvature of the flame front, the extent of wrinkling and folding of the flame surface, and the overall flame brush thickness. It also provides a direct evaluation of the turbulent flame speed and its dependence on the mean flame curvature and on the hydrodynamic strain that it experiences. Also discussed are the effects of the flame on the flow by examining the various mechanisms of enstrophy and scalar gradient production/destruction, the degree of anisotropy created in the burned gas, and the restructuring of the vortical motion beyond the flame.

Apparent inconsistency between (i) experimental and direct numerical simulation (DNS) data that show the significant influence of differential diffusion on the turbulent burning rate and (ii) recent complex-chemistry DNS data that indicate mitigation of the influence of differential diffusion on conditioned profiles of various local flame characteristics at high Karlovitz numbers, is explored by analysing new DNS data obtained from lean hydrogen–air turbulent flames. Both aforementioned effects are observed by analysing the same DNS data provided that the conditioned profiles are sampled from the entire computational domain. On the contrary, the conditioned profiles sampled at the leading edge of the mean flame brush do not indicate the mitigation, but are significantly affected by differential diffusion phenomena, e.g. because reaction zones are highly curved at the leading edge. This observation is consistent with a significant increase in the computed turbulent burning velocity with decreasing Lewis number, with all the results considered jointly being consonant with the leading point concept of premixed turbulent combustion. The concept is further supported by comparing DNS data obtained by allowing for preferential diffusion solely for a single species, either atomic or molecular hydrogen.

The ignition of a self-sustained premixed expanding flame constitutes a crucial problem in fundamental combustion research. In this work, a transient formulation on the forced ignition of a premixed expanding spherical flame in a quiescent mixture is proposed under the framework of the thermal-diffusive model. The present theory considers the unsteady evolution of the temperature and fuel mass fraction distributions subject to finite duration central heating. It can determine both critical heating power and minimum ignition energy for successful ignition. The transient flame initiation process is found to consist of four stages, including fast establishment of the ignition kernel, ignition-energy-supported flame kernel propagation, unsteady transition of the flame kernel, and quasi-steady spherical flame propagation. The unsteady effects lead to the observation of flame kernel establishing stage and considerably affect the subsequent flame kernel development by altering the flame propagation speed. Time scale analysis indicates that the transient formulation completely degenerates to the quasi-steady theory in the limits of both stationary flame ball and planar flame. Previous quasi-steady theory shows that the critical heating power for successful ignition is proportional to the cube of the critical flame radius. However, that scaling relation shall be revised in the transient formulation due to the unsteady thermal conduction from heating centre to flame front. The memory effect that persistently supports flame propagation subsequent to switching off the central heating is examined. It is found that as the heating power grows, the memory effect becomes increasingly important, and it can greatly reduce the predicted minimum ignition energy.

Shock-flame interactions are fundamental problems in many combustion applications ranging from flame acceleration to flame control in supersonic propulsion applications. The present paper seeks to quantify the rate of deformation of the flame surface and burning velocity caused by the interaction and to clarify the underlying mechanisms. The interaction of a single shock wave with a cellular flame in a Hele-Shaw shock tube configuration was studied experimentally, numerically and theoretically. A mixture of stoichiometric hydrogen-air at sub-atmospheric pressure was chosen such that large cells can be isolated and their deformation studied with precision subsequent to the interaction. Following passage of the incident shock and vorticity deposition along the flame surface, the flame cusps are flattened and reversed backwards into the burned gas. The reversed flame then goes through four stages. At times significantly less than the characteristic flame burning time, the flame front deforms as an inert interface due to the Richtmyer–Meshkov instability with nonlinear effects becoming noticeable. At times comparable to the laminar flame time, dilatation due to chemical energy release amplifies the growth rate of the Richtmyer–Meshkov instability. This stage is abruptly terminated by the transverse burnout of the resulting flame funnels, followed by a longer front re-adjustment to a new cellular flame evolving on the cellular time scale of the flame. The proposed flame evolution model permits us to predict the evolution of the flame geometry and burning rate for arbitrary shock strength below the shock-induced auto-ignition point and flames with unit Lewis number in two dimensions.

The present paper focuses on the structures and dynamics of flame edges in planar turbulent non-premixed flames bounded with a wall using direct numerical simulation (DNS). The global quenching behaviour was first examined and the flame edges were identified based on the intersections of mixture fraction and OH mass fraction iso-surfaces. For the upper branch of the planar jet flame, it is observed that the structures of flame edges change from tribrachial to monobrachial with increasing scalar dissipation rate. The flame edge speed is negatively correlated with the scalar dissipation rate in regions away from the wall, highlighting the role of turbulent mixing on the flame edge dynamics. During flame–wall interactions, the propagation speed of flame edges is mainly affected by the projection of edge flame normal in the wall-normal direction, i.e. $\boldsymbol {N}_{Z}\boldsymbol {\cdot }\boldsymbol {N}_{wall}$. In particular, the propagation speed increases with increasing $\boldsymbol {N}_{Z}\boldsymbol {\cdot }\boldsymbol {N}_{wall}$ in the near-wall region. The interactions of flame edges and turbulence bounded with a wall are characterized by the alignment between edge flame normal and principal strain rates. The normal of quenching edges has a tendency to align with the most extensive strain rate $\boldsymbol {e}_{1}$ in regions where the heat-release-induced dilatation is dominant over turbulent strain. In contrast, when the heat loss by cold wall effect is large enough to counteract the heat release induced by chemical reactions, turbulent strain is prevalent and the edge flame normal of the quenching edges preferentially aligns with the most compressive strain rate $\boldsymbol {e}_{3}$.

It has been suggested that a cellularly unstable laminar flame, which is freely propagating in unbounded space, can accelerate and evolve into a turbulent flame with the neighbouring flow exhibiting the basic characteristics of turbulence. Famously known as self-turbulization, this conceptual transition in the flow regime, which arises from local interactions between the propagating wrinkled flamefront and the flow, is critical in extreme events such as the deflagration-to-detonation transition (DDT) leading to supernova explosions. Recognizing that such a transition in the flow regime has not been conclusively demonstrated through experiments, in this work, we present experimental measurements of flow characteristics of flame-generated ‘turbulence’ for expanding cellular laminar flames. The energy spectra of such ‘turbulence’ at different stages of cellular instability are analysed. A subsequent scaling analysis points out that the observed energy spectra are driven by the fractal topology of the cellularly unstable flamefront.

Three-dimensional (3-D) measurements of flame stretch are experimentally challenging. In this paper, two-dimensional (2-D) and 3-D measurements of flame stretch and turbulence–flame interactions were examined using direct numerical simulation (DNS) data of turbulent premixed flames, and models to estimate 3-D statistics of flame stretch-related quantities by correcting 2-D measurements were developed. A variety of DNS cases were simulated, including three freely propagating planar flames without a mean shear and a slot-jet flame with a mean shear. The main findings are summarized as follows. First, the mean shear mainly influences the flame orientations. However, it does not change the flame stretch and turbulence–flame interactions qualitatively. The distributions of out-of-plane angle of all cases are nearly isotropic. Second, models were proposed to approximate the 3-D statistics of flame stretch-related quantities using 2-D measurements, the performance of which was verified by comparing modelled and actual 3-D surface averages and probability density functions of tangential strain rate, curvature and displacement velocity. Third, 2-D measurements of flame stretch capture properly the trends of the 3-D results, with flame surface area being produced in low curvature regions and destroyed in highly curved regions. However, the magnitude of flame stretch was under-estimated in 2-D measurements. Finally, 2-D and 3-D turbulence–flame interactions were examined. The flame normal vector is aligned with the most compressive strain rate in both 2-D and 3-D measurements. Meanwhile, the flame normal vector is misaligned (weakly aligned) with the most extensive strain rate in 3-D (2-D) measurements, highlighting the difference in 2-D and 3-D results of turbulence–flame interactions.

The study of a turbulent premixed flame often involves analysing quantities conditioned to different iso-surfaces of a reactive scalar field. Under the influence of turbulence, such a surface is deformed and translated. To track the surface motion, the displacement speed ($S_d$) of the scalar field respective to the local flow velocity is widely used and this quantity is currently receiving growing attention. Inspired by the apparent benefits from a simple decomposition of $S_d$ into contributions due to (i) curvature, (ii) normal diffusion and (iii) chemical reaction, this work aims at deriving and exploring new evolution equations for these three contributions averaged over the reaction surface. Together with a previously obtained $S_d$-evolution equation, the three new equations are presented in a form that emphasizes the decomposition of $S_d$ into three terms. This set of equations is also supplemented with a curvature-evolution equation, hence providing a new perspective to link the flame topology and its propagation characteristics. Using two direct numerical simulation databases obtained from constant-density and variable-density reaction waves, all the derived equations and the term-wise decomposition relations are demonstrated to hold numerically. Comparison of the simulated results indicates that the thermal expansion weakly affects the key terms in the considered evolution equations. Thermal expansion can cause variations in the averaged $S_d$ and its decomposed parts through multiple routes more than introducing a dilatation term. The flow plays a major role to influence the key terms in all equations except the curvature one, due to a cancellation between negatively and positively curved surface elements.

Premixed flame propagation is a rich interface problem. Instabilities and nonlinearities lead to the formation of cusps pointing toward the burnt gas. These cusps, which undergo complex dynamics, enhance the reaction rate by increasing the flame surface. These crests can be interpreted as pole solutions of the Michelson–Sivashinsky equation that evolve according to ordinary differential equations. Thanks to a quasi-bidimensional experimental facility (a Hele-Shaw burner) we evaluate the accuracy of the description of flame dynamics by elementary interaction between cusps. In particular, we address the time for which a direct comparison between experiments and numerical integration is feasible. The sensitivity to initial conditions and noise is discussed. We demonstrate that at any time of evolution, interesting features can be recovered by describing the flame surface evolution as pole dynamics.

This paper investigates the steady axisymmetric structure of the cold boundary-layer flow surrounding fire whirls developing over localized fuel sources lying on a horizontal surface. The inviscid swirling motion found outside the boundary layer, driven by the entrainment of the buoyant turbulent plume of hot combustion products that develops above the fire, is described by an irrotational solution, obtained by combining Taylor's self-similar solution for the motion in the axial plane with the azimuthal motion induced by a line vortex of circulation $2 {\rm \pi}\Gamma$. The development of the boundary layer from a prescribed radial location is determined by numerical integration for different swirl levels, measured by the value of the radial-to-azimuthal velocity ratio $\sigma$ at the initial radial location. As in the case $\sigma =0$, treated in the seminal boundary-layer analysis of Burggraf et al. (Phys. Fluids, vol. 14, 1971, pp. 1821–1833), the pressure gradient associated with the centripetal acceleration of the inviscid flow is seen to generate a pronounced radial inflow. Specific attention is given to the terminal shape of the boundary-layer velocity near the axis, which displays a three-layered structure that is described by matched asymptotic expansions. The resulting composite expansion, dependent on the level of ambient swirl through the parameter $\sigma$, is employed as boundary condition to describe the deflection of the boundary-layer flow near the axis to form a vertical swirl jet. Numerical solutions of the resulting non-slender collision region for different values of $\sigma$ are presented both for inviscid flow and for viscous flow with moderately large values of the controlling Reynolds number $\Gamma /\nu$. The velocity description provided is useful in mathematical formulations of localized fire-whirl flows, providing consistent boundary conditions accounting for the ambient swirl level.

This paper considers the relationship between nonlinearly interacting helical flow disturbances and flame area response in a swirling premixed flame. The present study was performed to determine whether there are nonlinear mechanisms through which helical modes ($m_{u}\neq 0$) can lead to non-zero unsteady heat release rate oscillations. The results show that for single frequency content (at $\unicode[STIX]{x1D714}_{0}$), helical modes excite unsteady heat release rate response of $O(\unicode[STIX]{x1D716}^{3})$ and that two-frequency excitation (e.g. at $\unicode[STIX]{x1D714}_{0}$ and $2\unicode[STIX]{x1D714}_{0}$), leads to a response of $O(\unicode[STIX]{x1D716}^{2})$ at $\unicode[STIX]{x1D714}_{0}$. There are two mechanisms through which this can occur: First, helical flow disturbances can distort the time-averaged flame shape to have an azimuthal component that matches that of the incident disturbance, $\exp (im_{u}\unicode[STIX]{x1D703})$. Second, multiple helical modes can nonlinearly interact to cause axisymmetric unsteady flame wrinkling. The paper derives the various modal contributions in the incident velocity disturbance that satisfy these criteria. These results suggest that it is only the $m_{u}=0$ mode which controls the linear dynamics (e.g. instability inception conditions) of these flames (where $\unicode[STIX]{x1D716}\ll 1$), but that their nonlinear dynamics is also controlled by the $m_{u}\neq 0$ helical modes.