Numerical studies of flames in wide tubes: stability limits of curved stationary flames

Flame dynamics in wide tubes with ideally adiabatical and slip walls is studied by means of direct numerical simulations of the complete set of hydrodynamical equations including thermal conduction, fuel diffusion, viscosity, and chemical kinetics. Stability limits of curved stationary flames in wide tubes and the hydrodynamic instability of these flames (the secondary Darrieus-Landau instability) are investigated. The stability limits found in the present numerical simulations are in a very good agreement with the previous theoretical predictions. It is obtained that close to the stability limits the secondary Darrieus-Landau instability results in an extra cusp at the flame front. It is shown that the curved flames subject to the secondary Darrieus-Landau instability propagate with velocity considerably larger than the velocity of the stationary flames.     

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.     

Exact equation for curved stationary flames with arbitrary gas expansion

An exact equation describing freely propagating stationary flames with arbitrary values of the gas expansion coefficient is obtained. This equation respects all conservation laws at the flame front, and provides a consistent nonperturbative account of the effect of vorticity produced by the curved flame on the front structure. It is verified that the new equation is in agreement with the approximate equations derived previously in the case of weak gas expansion.     

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.     

圆柱火焰的可燃极限及稳定性分析

本文从理论上分析了有辐射热损失和曲率的圆柱火焰,推导出了关于火焰位置、火焰温度同热损失和来流速度之间的关系式。并在此基础上对圆柱火焰的可燃极限进行了研究,结果表明热损失对可燃极限产生很大的影响。另外,作者运用线性稳定分析法对有辐射热损失的圆柱火焰作了稳定性分析,得出了判断圆柱火焰稳定与否的通用表达式。     

Turbulent propagation of premixed flames in the presence of Darrieus–Landau instability

We investigate the role played by hydrodynamic instability in the wrinkled flamelet regime of turbulent combustion, where the intensity of turbulence is small compared to the laminar flame speed and the scale large compared to the flame thickness. To this end the Michelson–Sivashinsky (MS) equation for flame front propagation in one and two spatial dimensions is studied in the presence of uncorrelated and correlated noise representing a turbulent flow field. The combined effect of turbulence intensity, integral scale, and an instability parameter related to the Markstein length are examined and turbulent propagation speed monitored for both stable planar flames and corrugated flames for which the planar conformation is unstable. For planar flames a particularly simple scaling law emerges, involving quadratic dependence on intensity and a linear dependence on the degree of instability. For corrugated flames we find the dependence on intensity to be substantially weaker than quadratic, revealing that corrugated flames are more resilient to turbulence than planar flames. The existence of a threshold turbulence intensity is also observed, below which the corrugated flame in the presence of turbulence behaves like a laminar flame. We also analyze the conformation of the flame surface in the presence of turbulence, revealing primary, large-scale wrinkles of a size comparable to the main corrugation. When the integral scale is much smaller than the characteristic corrugation length we observe, in addition to primary wrinkles, secondary small-scale wrinkles contaminating the surface. The flame then acquires a multi-scale, self-similar conformation, with a fractal dimension, for one-dimensional flames, plateauing at 1.23 for large intensities. The existence of an intermediate integral scale is also found at which the turbulent speed is maximized. When two-dimensional flames are subject to turbulence, the primary wrinkling patterns give rise to polyhedral-cellular structures which bear a very close resemblance to those observed in experiments on hydrodynamically unstable expanding spherical flames.     

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.     

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.     

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.     

A theoretical derivation of cellular structures of flames

Using an equation for the formation of flame fronts derived by Sivashinsky and augmented by an additional term describing buoyancy effects we present an analytical treatment of the formation of cellular structures of flames formed by plane burners. In particular we find rectangular and square patterns. We first study the stability of the plane flame front by linear stability analysis and then transform the basic equation into a set of equations for the amplitudes of the stable and unstable modes. The amplitudes of the stable modes can be eliminated by the slaving principle so that generalized Ginzburg-Landau equations result which in a general frame were previously derived by one of us (H.H.). These equations are then solved explicitly and the stability of the resulting pattern is proven.     

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.     

微重力环境下V型层流预混火焰锋面不稳定性分析

本章试图寻求描述火焰锋面动态特性的方法，以解释微重力环境下出现的V型火焰锋面的涟漪现象。采用线性稳定性理论从经典的G方程中导出了描述火焰锋面动态结构的一阶偏微分方程。采用该方程计算了声波扰动后，不同时刻的V型火焰锋面的动态结构．对于谐波扰动，其频率与波数的关系是分析固有火焰锋面不稳定性的基础。因此，微重力环境下V型火焰锋面的不稳定性可能是声波与谐波相耦合的结果。     

Structure and stability characteristics of turbulent planar flames with inhomogeneous jet in a concentric flow slot burner

Turbulent flames with compositionally inhomogeneous mixtures are common in many combustion systems. Turbulent jet flames with a circular nozzle burner were used earlier to study the impact of inhomogeneous mixtures, and these studies showed that the nozzle radius affects the flame stability. Accordingly, planar turbulent flames with inhomogeneous turbulent jet are created in a concentric flow slot burner (CFSB) to avoid this effect in the present study. The stability characteristics, the mixing field structure, and the flame front structure were measured, and the correlations between stability and the mixing field structure were investigated. The mixture fraction field was measured in non-reacting jets at the nozzle exit using highly resolved Rayleigh scattering technique, and the flame front was measured in some selected turbulent flames using high-speed Planar Laser-Induced Fluorescence (PLIF) of OH technique. The data show strong correlations between flame stability and the range of mixture fraction fluctuations. The flames are highly stabilized within a mixing field environment with the range of fluctuation in mixture fraction close to the range of the flammability limits. The mixing field structure is also illustrated and discussed using a mixing regime diagram and showed that the scatter of the data of the different cases is consistent with the classified mixing regimes. Lean flames are stabilized in the current slot burner. The flame front structure topology varies consistently from thin, small curvature at the low level of turbulence and higher equivalence ratio to more wrinkled, larger curvature, but a thicker structure at a higher level of turbulence and lower equivalence ratio.     

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.     

On polyhedral structures of lean methane/hydrogen Bunsen flames: Combined experimental and numerical analysis

In premixed flame propagation of lean hydrogen or hydrogen-enriched blends, both hydrodynamic and thermo-diffusive instabilities are governing the flame front shape and affect its propagation velocity. As a result, different types of cellular patterns can occur along the flame front in a laminar scenario. In this context, an interesting phenomenon is the formation of polyhedral flames which can be observed in a Bunsen burner. It is the objective of this work to systematically characterize the polyhedral structures of premixed methane/hydrogen Bunsen flames in a combined experimental and numerical study. A series of lean flames with hydrogen content varying between 20 and 85% at two equivalence ratios is investigated. The experiments encompass chemiluminescence imaging together with Planar Laser-induced Fluorescence (PLIF) measurements of the OH radical. Characteristic cell sizes are quantified from the experiments and related to the characteristic length scales obtained from a linear stability analysis. In the experiments, it is observed that the cell sizes at the base of the polyhedral Bunsen flames decrease almost linearly with hydrogen addition and only a weak dependence on the equivalence ratio is noted. These trends are well reflected in the numerical results and the length scale comparison further shows that the wavelength with the maximum growth rate predicted by the linear stability analysis is comparable to the cell size obtained from the experiment. The correlation between the experimental findings and the linear stability analysis is discussed from multiple perspectives considering the governing time and length scales, furthermore drawing relations to previous studies on cellular flames.     

预混气体燃烧火焰闪烁现象分析

在低速射流的预混火焰和扩散火焰中都存在火焰闪烁现象。对扩散火焰,其机理已比较明确,是由于浮力诱导引起的一种水力学不稳定性。而对预混火焰闪烁现象则存在水力学不稳定性和热驱动不稳定性两种观点。本文根据水力学不不稳定性观点,把预混火焰的闪烁现象看成是包围火焰锋面的已燃混气层中内、外区间在垂直方向上的相对脉动,应用Ｋｅｌｖｉｎ－Ｈｅｌｍｈｏｌｔｚ不稳定性机理进行了分析,获得了火焰闪烁频率与重力和压力的关系式,并与已有的结果作了对比。     

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.     

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.     

Extinction and flame bifurcations of stretched dimethyl ether premixed flames

Extinction limits and flame bifurcation of lean premixed dimethyl ether–air flames are numerically investigated using the counterflow flame with a reduced chemistry. Emphasis is paid to the combined effect of radiation and flame stretch on the extinction and flammability limits. A method based on the reaction front is presented to predict the Markstein length. The predicted positive Markstein length agrees well with the experimental data. The results show that flow stretch significantly reduces the flame speed and narrows the flammability limit of the stretched dimethyl ether–air flame. It is found that the combined effect of radiation and flow stretch results in a new flame bifurcation and multiple flame regimes. At an equivalence ratio slightly higher than the flammability limit of the planar flame, the distant flame regime appears at low stretch rates. With an increase in the equivalence ratio, in addition to the distant flame, a weak flame isola emerges at moderate stretch rates. With a further increase in the equivalence ratio, the distant flame and the weak flame branches merge together, resulting in the splitting of the weak flame branch into two weak flame branches, one at low stretch and the other at high stretch. Flame stability analysis demonstrates that the high stretch weak flame is also stable. Furthermore, a K-shaped flammability limit diagram showing various flame regimes and their extinction limits is obtained.