Nonlinear equation for curved nonstationary flames and flame stability

A time-dependent nonlinear equation for a nonstationary curved flame front of an arbitrary expansion coefficient is derived under the assumptions of a small but finite flame thickness and weak nonlinearity. On the basis of the derived equation, stability of two-dimensional curved stationary flames propagating in tubes with ideally adiabatic and slip walls is studied. The stability analysis shows that curved stationary flames become unstable for sufficiently wide tubes. The obtained stability limits are in a good agreement with the results of numerical simulations of flame dynamics and with semiqualitative stability analysis of curved stationary flames. Possible outcomes of the obtained instability at the nonlinear stage are discussed. The instability may result in extra wrinkles at a flame front close to the stability limits and in self-turbulization of the flame far from the limits. The self-turbulization can also be interpreted as a fractal structure. The fractal dimension of a flame front and velocity of a self-turbulized flame are evaluated.     

Numerical studies of curved stationary flames in wide tubes

The nonlinear problem of the propagation of curved stationary flames in tubes of different widths is studied by means of direct numerical simulation of the complete system of hydrodynamic equations including thermal conduction, viscosity, fuel diffusion and chemical kinetics. While only a planar flame can propagate in a narrow tube of width smaller than half of the cut–off wavelength determined by the linear theory of the hydrodynamic instability of a flame front, in wider tubes stationary curved flames propagate with velocities considerably larger than the corresponding velocity of a planar flame. It is shown that only simple ‘single-hump’ slanted stationary flames are possible in wide tubes, and ‘multi–hump’ flames are possible in wide tubes only as a nonstationary mode of flame propagation. The stability limits of curved stationary flames in wider tubes and the secondary Landau–Darrieus instability are investigated. The dependence of the velocity of the stationary flame on the tube width is studied. The analytical theory describes the flame reasonably well when the tube width does not exceed some critical value. The dynamics of the flame in wider tubes is shown to be governed by a large–scale stability mechanism resulting in a highly slanted flame front. In wide tubes, the skirt of the slanted flame remains smooth with the length of the skirt and the flame velocity increasing progressively with the increase of the tube width above the second critical value. Results of the analytical theory and numerical simulations are discussed and compared with the experimental data for laminar flames in wide tubes.     

Effect of geometrical parameters on thermo-acoustic instability of downward propagating flames in tubes

Experiments and theoretical analysis are presented to clarify the effect of geometrical parameters on thermo-acoustic instability of downward propagating flames in tubes. The experiments reveal that the longer tubes have higher instability compared to shorter tubes and the lower diameter tubes have higher instability compared to higher diameter tubes. The secondary instability leading to turbulent burning is found to be more sensitive to change in geometrical parameters compared to primary instability (oscillating flat flame). The secondary instability is re-stabilized for some intermediate burning velocity conditions even though lower and higher burning velocity conditions show secondary instability. The appearance of such re-stabilization is only observed for some specific lengths of the tube. Present experimental observations pertaining to the effect of geometrical parameters is found to be contradicting the theoretical predictions based on pressure coupling mechanism. To clear the underlying mechanism, analytical growth rate is computed considering velocity coupling mechanism. The computed growth rates correctly predict the effect of geometrical parameters on thermo-acoustic instability of downward propagating flames. This work provides further evidence to believe that the flame -acoustic coupling in downward propagating flames is due to flame area modulation (leading to heat release modulation) through action of acoustic acceleration.     

Effect of vorticity production on the structure and velocity of curved flames

A nonlinear equation describing curved stationary flames with arbitrary gas expansion theta = rho(fuel)/rho(burnt) is obtained in a closed form without an assumption of weak nonlinearity. The equation respects all conservation laws and takes into account vorticity production in the flame. In the scope of the asymptotic expansion for theta -->1, the new equation solves the problem of stationary flame propagation with accuracy of the sixth order in theta - 1. Its analytical solutions give the flame velocity in tubes of arbitrary width, which agrees with available results of direct numerical modeling.     

Characteristics of microjet methane diffusion flames

Characteristics of microjet methane diffusion flames stabilized on top of the vertically oriented, stainless-steel tubes with an inner diameter ranging from 186 to 778 μ m are investigated experimentally, theoretically and numerically. Of particular interest are the flame shape, flame length and quenching limit, as they may be related to the minimum size and power of the devices in which such flames would be used for future micro-power generation. Experimental measurements of the flame shape, flame length and quenching velocity are compared with theoretical predictions as well as detailed numerical simulations. Comparisons of the theoretical predictions with measured results show that only Roper's model can satisfactorily predict the flame height and quenching velocity of microjet methane flames. Detailed numerical simulations, using skeletal chemical kinetic mechanism, of the flames stabilized at the tip of d = 186, 324 and 529 μ m tubes are performed to investigate the flame structures and the effects of burner materials on the standoff distance near extinction limit. The computed flame shape and flame length for the d = 186 μm flame are in excellent agreement with experimental results. Numerical predictions of the flame structures strongly suggest that the flame burns in a diffusion mode near the extinction limit. The calculated OH mass fraction isopleths indicate that different tube materials have a minor effect on the standoff distance, but influence the quenching gap between the flame and the tube.     

Flame fingers and interactions of hydrodynamic and thermodiffusive instabilities in laminar lean hydrogen flames

Lean hydrogen/air flames are prone to hydrodynamic and thermodiffusive instabilities. In this work, the contribution of each instability mechanism is quantified separately by performing detailed simulations of laminar planar lean hydrogen/air flames with different diffusivity models and equations of state to selectively suppress the hydrodynamic or thermodiffusive instability mechanism.From the analysis of the initial phase of the simulations, the thermodiffusive instability is shown to dominate the flame dynamics. If differential diffusion and, hence, the thermodiffusive instability is suppressed, the flame features a strong reduction of the instability growth rates, whereas if present, a wide range of unstable wave numbers is observed due to the strong destabilizing nature of differential diffusion. When instabilities are fully developed, lean hydrogen/air flames feature the formation of small-scale cellular structures and large-scale flame fingers. While the size of the former is known to be close to the most unstable wave length of a linear stability analysis, this work shows that flame fingers also originate from the thermodiffusive instability and most noteworthy, are not linked to an interaction of the two instability mechanisms. They are stable with respect to external perturbations and feature an enhanced flame propagation as the formation of a central cusp at their tip enables the co-existence of two strongly curved leading edges with high reactivity. The thermodiffusive instability is shown to significantly affect the flames’ consumption speed, while the consumption speed enhancement caused by the hydrodynamic instability is significantly smaller. Further, the surface area increase due to wrinkling is strongly diminished if one of the two instability mechanisms is missing. This is linked to a synergistic interaction between the two mechanisms, as the propagation of flame fingers is enhanced by the presence of the hydrodynamic instability due to a widening of the streamlines ahead of the flame fingers.     

Stationary combustion regimes and extinction limits of one-dimensional stretched premixed flames in a gap between two heat conducting plates

Stationary combustion regimes, their linear stability and extinction limits of stretched premixed flames in a narrow gap between two heat conducting plates are studied by means of numerical simulations in the framework of one-dimensional thermal-diffusion model with overall one-step reaction. Various stationary combustion modes including normal flame (NF), near-stagnation plane flame (NSF), weak flame (WF) and distant flame (DF) are detected and found to be analogous to the same-named regimes of conventional counterflow flames. For the flames stabilized in the vicinity of stagnation plane at moderate and large stretch rates (which are NF, NSF and WF) the effect of channel walls is basically reduced to additional heat loss. For distant flame characterized by large flame separation distance and small stretch rates intensive interphase heat transfer and heat recirculation are typical. It is shown that in mixture content / stretch rate plane the extinction limit curve has ε-shape, while for conventional counterflow flames it is known to be C-shaped. This result is quite in line with recent experimental findings and is explained by extension of extinction limits at small stretch rates at the expense of heat recirculation. Analysis of the numerical results makes possible to reveal prime mechanisms of flame quenching on different branches of ε-shaped extinction limit curve. Namely, two upper limits are caused by stretch and heat loss. These limits are direct analogs of the upper and lower limits on conventional C-shaped curve. Two other limits are related with weakening of heat recirculation and heat dissipation to the burner. Thus, the present study provides a satisfactory explanation for the recent experimental observations of stretched flames in narrow channel.     

Large scale effects in weakly turbulent premixed flames

In this study we numerically investigate large scale premixed flames in weakly turbulent flow fields. A large scale flame is classified as such based on a reference hydrodynamic lengthscale being larger than a neutral (cutoff) lengthscale for which the hydrodynamic or Darrieus–Landau (DL) instability is balanced by stabilizing diffusive effects. As a result, DL instability can develop for large scale flames and is inhibited otherwise. Direct numerical simulations of both large scale and small scale three-dimensional, weakly turbulent flames are performed at constant Karlovitz and turbulent Reynolds number, using two paradigmatic configurations, namely a statistically planar flame and a slot Bunsen flame. As expected from linear stability analysis, DL instability induces its characteristic cusp-like corrugation only on large scale flames. We therefore observe significant morphological and topological differences as well as DL-enhanced turbulent flame speeds in large scale flames. Furthermore, we investigate issues related to reaction rate modeling in the context of flame surface density closure. Thicker flame brushes are observed for large scale flames resulting in smaller flame surface densities and overall larger wrinkling factors.     

Stability characteristics and flowfields of turbulent non-premixed swirling flames

A simple, yet representative, burner geometry is used for the investigation of highly swirling turbulent unconfined, non-premixed, flames of natural gas. The burner configuration comprises a ceramic faced bluff-body with a central fuel jet. The bluff-body is surrounded by an annulus that delivers a swirling primary flow of air. The entire burner assembly is housed in a wind tunnel providing a secondary co-flowing stream of air. This hybrid bluff-body/swirl burner configuration stabilizes complex turbulent flames not unlike those found in practical combustors, yet is amenable to modelling because of its well-defined boundary conditions. Full stability characteristics including blow-off limits and comprehensive maps of flame shapes are presented for swirling flames of three different fuel mixtures: compressed natural gas (CNG), CNG–air (1:2 by volume) and CNG–H₂ (1:1 by volume).

It is found that with increased fuel flow, flame blow-off mode may change with swirl number, S_g. At low swirl, the flame remains stable at the base but blows off in the neck region further downstream. At higher swirl numbers, the flames peel off completely from the burner's base. Swirling CNG–air flames are distinct in that they only undergo base blow-off. In the low range of swirl number, increasing S_g causes limited improvement in the blow-off limits of the flames investigated and (for a few cases) can even lead to some deterioration over a small intermediate range of S_g. It is only above a certain threshold of swirl that significant improvements in blow-off limits appear. Six flames are selected for further detailed flowfield and composition measurements and these differ in the combination of swirl number, primary axial velocity through the annulus, U_s, and bulk fuel jet velocity, U_j. Only velocity field measurements are presented in this paper. A number of flow features are resolved in these flames, which resemble those already associated with non-reacting swirling flows of equivalent swirl obtained with the present burner configuration. Additionally, asymmetric flowfields inherent to some flames are revealed where the fluidic centreline of the flow (defined in the two-dimensional (U–W velocity pair) velocity field by the ?ω? = 0 tangential velocity contour), meanders strongly on either side of the geometric centreline downstream by about one bluff-body diameter. Flow structures revealed by the velocity data are correlated to flame shapes to yield a better understanding of how the velocity field influences the flames physical characteristics.     

Impact of the gravity field on stability of premixed flames propagating between two closely spaced parallel plates

The stability of a planar flame front propagating between two parallel adiabatic plates inclined at an arbitrary angle is investigated in the frame of narrow-channel approximation. It is demonstrated that buoyancy forces can suppress the hydrodynamic (Darrieus–Landau) and cellular (diffusive-thermal) instabilities for sufficiently large value of the gravity parameter for the case of downward-propagating flames. The stability analysis reveals that in the case of oscillatory diffusive-thermal instability, the flame front cannot be stabilized in the similar way. Finally, the stability results are compared satisfactorily with unsteady numerical simulations.     

Modeling curvature effects in turbulent autoigniting non-premixed flames using tabulated chemistry

Turbulent flames are intrinsically curved. In the presence of preferential diffusion, curvature effects either enhance or suppress molecular diffusion, depending on the diffusivity of the species and the direction of the flame curvature. When a tabulated chemistry type of modeling is employed, curvature-preferential diffusion interactions have to be taken into consideration in the construction of manifolds. In this study, we employ multistage stage flamelet generated manifolds (MuSt-FGM) method to model autoigniting non-premixed turbulent flames with preferential diffusion effects included. The conditions for the modeled flame are in MILD combustion regime. To model the above-mentioned curvature-preferential diffusion interactions, a new mixture fraction which has a non-unity Lewis number is defined and used as a new control variable in the manifold generation. 1D curved flames are simulated to create the necessary flamelets. The resulting MuSt-FGM tables are used in the simulation of 1D laminar flames, and then also applied to turbulent flames using 2D direct numerical simulations (DNS). It was observed that when the curvature effects are included in the manifold, the MuSt-FGM results agree well with the detailed chemistry results; whereas the results become unsatisfactory when the curvature effects are ignored.     

Characteristic patterns of thermodiffusively unstable premixed lean hydrogen flames

Large-scale two-dimensional numerical simulations of thermodiffusively unstable, lean, premixed hydrogen flames have been performed using detailed finite rate chemistry to analyze flame intrinsic scales. The simulations feature a long integration time and large domain sizes to rule out effects of confinement on the dynamics of the flame front. For sufficiently large domain sizes, the total consumption speed of the flame is found to become independent of the domain size. An assessment of the characteristic scales of the flame front corrugation reveals the existence of a smallest and a largest flame intrinsic length scale. The smallest length manifests itself by local cusps, which lead to the formation of characteristic cells along the flame front. Their size is remarkably close to the most unstable wavelength predicted by a linear stability analysis of the flame front evolution in the linear regime. Independently of the domain size, a specific largest flame intrinsic structure, here referred to as flame finger, emerges from the interaction of multiple small-scale cusps. Thermodiffusively unstable flames are found to periodically form and destroy these flame fingers, but the formation of a global cusp that is known to emerge for purely hydrodynamically unstable flames is suppressed. The finite size of the largest scale fingers is explained by an instability in their movement. As they proceed towards the unburnt mixture, they tend to tilt and move laterally, thereby eventually being incorporated again by the rest of the flame. This behavior arises from the interaction of the flame fingers and the diverging velocity field ahead of them. Finally, the effect of equivalence ratio and unburnt gas temperature is investigated showing that flame fingers are found to develop only in case of a thermodiffusively unstable flame.     

Pressure disturbances and strained premixed flames

A mathematical model is presented for pressure interactions with premixed flames in a prescribed strained velocity field. A stability analysis is carried out including pressure disturbances and an approximate frequency condition obtained. For positive strain the unsteady analysis indicates that the pulsating instability is suppressed. However, for a converging flow (negative strain), the instability is encouraged. Furthermore, the change of the neutral stability boundary in parameter space is explored, showing that a sharp pressure reduction makes the pulsating instability much more accessible.     

Numerical simulation of finite disturbances interacting with laminar premixed flames

Compression waves can be generated during combustion processes and subsequently interact with flames to augment their behaviour. The study of these interactions thus far has been limited to shock and expansion waves only. In this study, the interaction of finite compression waves with a perturbed laminar flame is investigated using numerical simulations of the compressible Navier–Stokes equations with single-step chemical kinetics. The interaction is characterised using three independent parameters: the compression wavelength, the pressure ratio of the disturbance, and the perturbation amplitude of the flame interface. The results reveal a wide range of behaviours in terms of flame length and heat release rate that could occur during such an interaction. The results are compared to the classical reactive Richtmyer–Meshkov instability and the role of baroclinic torque and vorticity generation are shown to be primary drivers of the flow instability.     

Characteristic boundary conditions for direct simulations of turbulent counterflow flames

Improved Navier–Stokes characteristic boundary conditions (NSCBC) are formulated for the direct numerical simulations (DNS) of laminar and turbulent counterflow flame configurations with a compressible flow formulation. The new boundary scheme properly accounts for multi-dimensional flow effects and provides nonreflecting inflow and outflow conditions that maintain the mean imposed velocity and pressure, while substantially eliminating spurious acoustic wave reflections. Applications to various counterflow configurations demonstrate that the proposed boundary conditions yield accurate and robust solutions over a wide range of flow and scalar variables, allowing high fidelity in detailed numerical studies of turbulent counterflow flames.     

Velocity of weakly turbulent flames of finite thickness

The velocity increase of a weakly turbulent flame of finite thickness is investigated using analytical theory developed in previous papers. The obtained velocity increase depends on the flow parameters: on the turbulent intensity, on the turbulent spectrum and on the characteristic length scale. It also depends on the thermal and chemical properties of the burning matter: thermal expansion, the Markstein number and the temperature dependence of transport coefficients. It is shown that the influence of the finite flame thickness is especially strong close to the resonance point, when the wavelength of the turbulent harmonic is equal to the cut off wavelength of the Darrieus–Landau instability. The velocity increase is almost independent of the Prandtl number. On the contrary, the Markstein number is one of the most important parameters controlling the velocity increase. The relative role of the external turbulence and the Darrieus–Landau instability for the velocity increase is studied for different parameters of the flow and the burning matter. The velocity increase for turbulent flames in methane and propane fuel mixtures is calculated for different values of the equivalence ratio. The present theoretical results are compared with previous experiments on turbulent flames. In order to perform the comparison, the theoretical results of the present paper are extrapolated to the case of a strongly corrugated flame front using the ideas of self-similar flame dynamics. The obtained theoretical results are in a reasonable agreement with the experimental data, taking into account the uncertainties of both the theory and the experiments. It is shown that in many experiments on turbulent flames the Darrieus–Landau instability is more important for the flame velocity than the external turbulence.     

Diffusive-thermal instabilities of diffusion flames: onset of cells and oscillations

A comprehensive stability analysis of planar diffusion flames is presented within the context of a constant-density model. The analysis provides a complete characterization of the possible patterns that are likely to be observed as a result of differential and preferential diffusion when a planar flame becomes unstable. A whole range of physical parameters is considered, including the Lewis numbers associated with the fuel and the oxidizer, the initial mixture fraction, and the flow conditions. The two main forms of instability are cellular flames, obtained primarily in fuel-lean systems when the Lewis numbers are generally less than one, and planar pulsations, obtained in fuel-rich systems when the Lewis numbers are generally larger than one. The cellular instability is predominantly characterized by stationary cells of characteristic dimension comparable to the diffusion length, but smaller cells that scale on the reaction zone thickness are also possible near extinction conditions. The pulsating instability is characterized by planar oscillations normal to the flame sheet with a well-defined frequency comparable to the reciprocal of the diffusion time; high-frequency modes are also possible just prior to extinction. The analysis also alludes to other possible patterns, such as oscillating cellular structures, which result from competing modes of instability of comparable and/or disparate scales. The expected pattern depends of course on the underlying physical parameters. Consequently, stability boundaries have been identified for the onset of one or another form of the instability. The conditions for the onset of cellular and pulsating flames, as well as the predicted cell size and the frequency of oscillations, compare well with the experimental record.     

Thermal expansion effect on the propagation of premixed flames in narrow channels of circular cross-section: Multiplicity of solutions,axisymmetry and non-axisymmetry

The present study examines, in presence of thermal expansion effects, the existence of the multiplicity of solutions previously reported within the context of diffusive-thermal modeling in [15], for lean premixed flames with low Lewis number (Le?<?1) propagating in narrow circular adiabatic channels subject to a Poiseuille flow. For this, direct numerical simulations have been carried out within the framework of variable-density Navier–Stokes equations and zero-Mach-number approximation. The simulations, conducted for both axisymmetric and three-dimensional cylindrical geometries, confirm the coexistence of multiple steady flame structures for a given flow rate. They show that axisymmetric flames concave towards the upstream are more unstable to three-dimensional perturbations than convex (toward the upstream) flames. This result evinces earlier findings obtained from stability analysis. The non-axisymmetry property of the flame is also found to push back the critical flashback limits at larger flow rate when compared to those predicted under the assumption of flame axisymmetry.     

Analytical study of stretched ultra-lean premixed flames within porous inert media

It has been shown both theoretically and experimentally that combustion within porous inert media can extend the flammability limits of reactant mixtures for unstretched stationary premixed flames. However little attention has been given to flames within porous media submitted to stretch conditions. This work presents a closed form approximate analytical solution for the problem of ultra-lean premixed flames within porous inert media subjected to small stretch rates in an impinging flow configuration against a constant temperature wall. The solution is obtained using the method of matched asymptotic expansions taking advantage of the large difference between the solid- and gas-phase thermal conductivities. The model allows for thermal nonequilibrium between the phases and is able to predict the flame temperature, velocity and position as function of the stretch rate. The results show that within porous media low stretch rates may increase the flame temperature, further extending the lean flammability limit of the reactant mixture when compared to planar flames. The model is restricted to low porosities, low stretch rates, low heat losses and intense interphase heat transfer.     

Blowout limits of inclined nonpremixed turbulent jet flames

The blowout behavior of inclined nonpremixed turbulent jet flames is investigated by varying the jet inclined angle in the range of -90° to 90° The critical jet velocity at blow-out limit is quantified experimentally for various nozzle diameters, different fuels and inclined angles. Numerical simulations are performed to emphasize the flow field difference for the positive and negative inclined angles. Physical modeling is conducted to incorporate the effect of the inclined angle on blow-out behavior. Major findings include: (1) The negatively inclined jet flames show more intense yellow luminosity with larger sooting zones than the positively inclined jet flames; (2) The blowout limit decreases appreciably with the jet inclined angle for the negatively inclined flames, while for the positively inclined jet flames, this decrease is relatively small; (3) Physical analysis of the flow development of inclined jets is conducted, indicating the centerline velocity along the jet trajectory decreases faster for the flame with smaller inclined angle. And the decrease rate is relatively larger for the negatively inclined jet flames; (4) Based on the analysis of the flow development as well as the characteristic velocity with the inclined angle variation, a model based on the Damköhler number (Da) accounting for the effect of jet inclined angle is developed to characterize the blowout limits of inclined jet flames. The proposed model successfully correlates the experimental data. The present findings provide new data and a basic scaling law for the blowout limit of nonpremixed inclined turbulent jet flames, revealing the effect of the relative angle between the jet momentum and buoyancy.