Elsevier

Physics Reports

Volume 325, Issues 4–5, March 2000, Pages 115-237
Physics Reports

Dynamics and stability of premixed flames

https://doi.org/10.1016/S0370-1573(99)00081-2Get rights and content

Abstract

The latest achievements in the theory of premixed flames including both the analytical theory and the numerical simulations are reviewed. Gas dynamics of curved flames and flame-induced flows as well as flames propagating under confinement in closed burning chambers is considered. Much attention is paid to the nonlinear stage of the hydrodynamic instabilities inherent to any premixed flame in a gaseous fuel, such as the Darrieus–Landau instability and the Rayleigh–Taylor instability of a flame in a gravitational field. Influence of compressibility, flame generated acoustic waves and shock waves on flame dynamics is considered. Development of a fractal structure of a flame front is discussed.

Introduction

People have always been mesmerised by flame and fire since ancient times and until nowadays. Flame possesses an almost mystic appeal, being one of the most important things for people from the very beginning of human civilization. Today we value not just a pleasure of warming near fireplace and watching flames but also industrial applications of the burning process.

The industrial revolution of the 19th century was largely fuelled by coal and, as industrialization developed, the close relationship between economic growth and increased demand for primary energy sources was established. Since the late 19th century petroleum demand has also steadily increased, with transport, both road and air, the biggest user. Few inventions have had as great an impact on society, the economy, and the environment as the reciprocating internal combustion (IC) engine. Yet for decades, IC-engine design and improvement remains largely a cut-and-try experiential process. Engineers develop new combustion systems by making variations in previously successful configurations. However, today the automotive industry has faced numerous challenges. One of the most compelling has been reducing exhaust emissions and reductions in fuel consumption.

During intake in a spark ignition (SI) engine, the fuel–air mixture enters the combustion chamber and mixes with the gas remaining from the previous burn cycle (residual gas). A flame, initiated by a spark discharge, propagates into the fuel–air mixture, converting unburned reactants into combustion products. Most of the combustion occurs during the expansion stroke. By the time of ignition, the composition is approximately uniform; hence, the term “premixed” is used to describe this type of combustion. Fuel consumption in SI-engines can be reduced by high compression ratios, but increase of compression ratio is limited by “knock”. Knock is a phenomenon that occurs when the unburned reactants spontaneously ignite ahead of the flame. To reduce fuel consumption and NOx emissions some exhaust may be recycled back into the intake manifold for the next burn cycle. Contrary to SI-engines, in diesel engines combustion is initiated by autoignition, in which compression of the air increases combustion chamber temperature enough that burning begins as soon as fuel vaporises and mixes with air. Diesel combustion occurs mostly in a “diffusion mode”, in which fuel and air are still mixing during burning. In diesel engines, the combination of high compression ratio, which is not limited by knock, and a high overall ratio of air to fuel results in low fuel consumption. However, diesel engines pose several problems, including particularly harmful emissions and limited power density. The complete combustion of a hydrocarbon fuel–air mixture yields carbon dioxide and water. At high temperature, NOx is formed by oxidation of nitrogen in the air. Strategies for reducing NOx typically involve lowering the combustion temperature in various ways, such as by using exhaust-gas recirculation. Other unwanted products of combustion are carbon monoxide, unburned hydrocarbons, etc. The ideal IC-engine would produce high power and operate with no emissions and with low fuel consumption.

The shapes of the intake ports and chamber result in a turbulent flow within the cylinder. Turbulence can also be caused or enhanced by intrinsic flame instabilities. Fluctuations range in size from a large fraction of the bore diameter to submillimeter scale. Turbulence promotes the mixing of fuel and air, and changes heat transfer and flame propagation rates. Because of turbulence the combustion duration in a SI-engine corresponds to about one-sixth of a crank rotation and is largely independent of engine speed. Note, that in the absence of turbulence, a flame could not propagate from the ignition site to the combustion chamber walls at high engine speed and this would make the engine inoperable. In cylinder turbulence includes motion at length scales as small as 10−3 cm. This is a factor of about 104 smaller than the largest flow scales, which are of the size of the bore diameter. Computers do not exist, and will not exist in the foreseeable future, that can store all the numbers required to fully resolve phenomena over such a wide range. At present, it is impossible to extract quantity of fuel consumption, and emission numbers directly from the models. Instead, the engineers must look at established “figures of merit”, such as burn-rate curves, and draw inferences regarding engine performance from them. Therefore, theoretical models of combustion processes and turbulent combustion flow are required. The fundamental understanding of the flame propagation is essential to the success of the combustion model, and for development both novel analytical methods as well as numerical simulations.

The goal is to develop modelling that can be used to guide the design of new engines before prototypes are built and that becomes an integral part of engine design. Progress in the combustion research is needed that will change situation from a research technology to a practical design tool. In an ideal way, the numerical modelling should provide a way of examining the various trade-off that must be made to move current designs toward the optimum. An experience alone is not sufficient to create the significant improvements now sought: The method of cut-and-try is still used, but we need newer approaches to achieve today's performance targets. However, though the basis equations governing the combustion process are well known, turbulent burning in a real engine is too complicated process for direct numerical simulation. The numerical scheme can be created in the right way only if the basic elementary properties of the flame are well known: the typical length scale of the flame instabilities, characteristic increase of the flame velocity due to the curved flame shape, outcome of the flame interaction with acoustic and shock waves, influence of external turbulence on flame dynamics, etc. In this point, the combustion theory is far behind the industrial necessities.

Another task in the development of industrial combustion is reduction of the level of dangerous exhaust from internal combustion engines and search for the new automotive fuels for car engines. Among a number of alternative automotive fuels, such as alcohol, gasoline, and others, hydrogen may be the most promising one. In principle, usage of hydrogen in IC engines satisfies the most stringent requirements for ultra-low-emission vehicle. On the other hand, the wide flammability and explosion limits of hydrogen in the air and its low ignition energy are well known. In addition, its rapid diffusion mitigates the formation of explosive mixtures. Therefore, we face not only the problem of combustion processes optimization but also safety issues, so that the problem of transition from slow combustion regime to a detonation regime becomes a key issue of the research.

Flame hydrodynamics will be of the first priority in this paper and we shall not go into details of chemical aspects of the burning process. Realistic burning in a flame may involve up to several hundreds elementary reactions (Zeldovich et al., 1985; Williams, 1985). Still for most of the hydrodynamical flame phenomena the details of chemical reactions are not very important so that the whole burning process may be described satisfactory by use of one simplified irreversible reaction that transfer the fresh fuel mixture into the products of burning. On the other hand, a flame front is not a purely mathematical construction for us, we are always interested in realistic physical outcome of the obtained results applicable to laboratory flames or flames of some other kind. When discussing flames we mean usual chemical flames first of all, the flames that one can observe in laboratory or in industry. Still there are many other physical phenomena that possess similar properties and may be called flames in a sense. These are, for example, a laser-induced deflagration in inertial confinement fusion (Manheimer et al., 1982; Bychkov et al., 1994), thermonuclear flames in Supernova events (Nomoto, 1984; Bychkov and Liberman 1995a, Bychkov and Liberman 1995b, Bychkov and Liberman 1995c, Bychkov and Liberman 1995d; Niemeyer and Woosley, 1997), ionizing waves in gas discharges (Ebert et al., 1997), evaporation wave in an overheated fluid (Shepherd and Sturtevant, 1982; Frost, 1988) and many other phenomena.

When we consider a flame in a gaseous fuel mixture, we assume that the burning process does not involve any phase transition: both the matter before the burning and the matter after the burning are in the gaseous state. This is a typical situation that takes place in car engines or aircraft gas turbines. From the theoretical point of view the absence of phase transitions makes easier analysis of the flames, besides, the equation of state of a perfect gas describing the burning gaseous mixture is probably the simplest known equation of state. The other important point inherent to the gaseous burning is that the burning mixture undergoes strong expansion. Because of the expansion the burning process becomes strongly coupled to the fuel flow; the coupling leads to different flame instabilities, generation of pressure waves, etc. Particularly, one of the most important flame parameters that will be considered often in this paper is the expansion coefficient of a flame Θ defined as the ratio of the fuel density ρf and the density of the burnt matter ρb: Θ=ρf/ρb. For most of laboratory flames this coefficient is rather large, being about Θ=5–10.

When a flame is premixed (which is the opposite case to diffusion flames), then it implies that all components necessary for the reaction are present in the fuel mixture from the very beginning and in order to start the active reaction one has only to heat the mixture. In the case of premixed flames the burning rate is not controlled by the diffusion process, still even for premixed flames the fuel diffusion (together with thermal conduction) is an important factor of propagation and cannot be neglected. Particularly, when the fuel diffusion is stronger than the thermal conduction, then the thermal-diffusion instability of a planar stationary flame may develop (Barenblatt et al., 1962). In the opposite case of zero diffusion (which is not typical for a gaseous fuel) a planar stationary flame cannot propagate either: the domination of thermal conduction over diffusion leads to pulsations of the reaction zone (Zeldovich et al., 1985). Both these instabilities are of diffusive nature and depend strongly on the details of the chemical reaction. Since we are interested in hydrodynamical properties of a flame, then we will try to avoid additional complications caused by the diffusion effects. We will consider mostly the situation of equally strong fuel diffusion and thermal conduction, when the hydrodynamical flame properties can be studied in the most pure form.

Though turbulent flames are encountered more often in technical applications, the theory of turbulent flames is much less developed because of its complexity. In a sense, the theory of laminar flames should be a natural basis for the turbulent flame theory in order to avoid unjustified phenomenological assumptions about turbulent flame properties. As a matter of fact the words “turbulent flames” cover a large number of quite different burning regimes from almost laminar flames to well stirred reactors as presented in Fig. 1.1. Different regimes are characterized by different velocities and different length scales of the turbulent flow Uturb and Lturb, respectively, scaled by the laminar flame velocity Uf and by the laminar flame thickness Lf. Obviously, turbulence cannot exist at small Reynolds numbers Returb=UturbLturb/ν<1, where ν is the kinematic viscosity. As it will be shown below, flame thickness and velocity satisfy the order of magnitude relation UfLfν, therefore the condition Returb<1 is equivalent to UturbLturb/UfLf<1 and presented in Fig. 1.1 by line 1. The region below this line corresponds to decay of a turbulent flow. Another obvious condition is that for a weak turbulence Uturb<Uf (line 2) the influence of the turbulent flow on flame dynamics is small or even negligible. In this case flames may be described as laminar or quasi-laminar wrinkled fronts. In the opposite case of a very strong turbulence, when the typical turbulent time Lturb/Uturb is much smaller than the reaction time Lturb/Uturb<Lf/Uf (line 4), the reaction zone is spread over the whole flow. This regime of turbulent burning is called the regime of well stirred reactors. The intermediate regimes between the regimes of very weak and very strong turbulence are the regimes of flamelets and thick flames. To separate these two regimes we compare the flame velocity Uf and the velocity of turbulent pulsations uturb on the length scale equal flame thickness uturb=uturb(λ), for λ=Lf. As is known for the well-developed (Kolmogorov) turbulence, the velocity pulsations depend on the length scale of the pulsations as uturbλ1/3 (Landau and Lifshitz, 1987). Then the velocity pulsations on the length scale equal the flame thickness are uturb=Uturb(Lf/Lturb)1/3. Line 3 corresponds to the condition Uf=Uturb(Lf/Lturb)1/3. For larger turbulent velocities the pulsations of the flow penetrate inside the flame and influence the internal flame structure. In this regime (called the regime of thick flames) the reaction front propagates due to turbulent thermal conduction. For smaller turbulent velocities in the flamelet regime the external flow is strong enough to bend the flame front considerably and make it corrugated, but the flow is too weak to change the internal structure of the flame.

Usually, the characteristic length scale of a turbulent flow in SI-engines and gas turbines is much larger than the flame thickness and the turbulence intensity is not so strong to smash the flame front: the concept of flamelets may be applied to these flames. As a result for a turbulent flame in an engine we can find certain length scales and certain time intervals, for which the flame may be considered as quasi-laminar one at least in a sense of an intermediate asymptotic. It has been pointed out above that it is impossible to resolve all necessary length scales in numerical simulations of turbulent burning in an engine taking into account the complicated geometry of a burning chamber and realistic chemical process involving up to hundred elementary reactions. Indeed, the size of the burning chamber is about 10 cm, while the typical thickness of a flame front is 10−4–10−3 cm, and the typical width of the reaction zone is even smaller being about 10−5–10−4 cm. These estimates imply that one has to resolve length scales that differ by 5–6 orders of magnitude, which is practically impossible at present. Suppose that we can resolve length scales changing by 2 orders of magnitude from 10 cm of an engine size to 0.1 cm. How can we get information about flame dynamics on scales from 10−5 to 0.1 cm? This information may come only from the flame theory.

One of the most effective tools to study flame dynamics in a turbulent flow in the flamelet regime is the renormalization group theory (Yakhot, 1988; Yakhot and Orzag 1986a, Yakhot and Orzag 1986b), the main idea of which may be formulated as follows. When the Reynolds number is large and the spectrum of velocity fluctuations is broad we can decompose the turbulent velocity field into components corresponding to different length scales (narrow quasi-monochromatic bands in the wave number space of Fourier expansion of the velocity field). Within every band of a particular length scale flame dynamics may be interpreted as the dynamics of a quasi-laminar flame with some effective thickness and velocity depending on the length scale in a weak external flow, which can be also treated as a quasi-laminar. Formulated in this way the problem becomes similar to the approach by Clavin and Williams (1979) to a flame in weak long wavelength turbulence. We go from one scale of the turbulent flow to another accumulating the effects of the flow on different length scales into a new average effective flame velocity and thickness, until we cover all length scales that we cannot resolve numerically. The above method has been used by Yakhot (1988) to calculate turbulent flame velocity for the case of strong turbulence and negligible influence of intrinsic flame properties in the model of a flame front described as a passive scalar field. As one can see, the basic step of the renormalization group theory of turbulent flames is given by the dynamics of a quasi-laminar flame front in an external quasi-laminar flow. Therefore, to understand dynamics of turbulent flames one has to understand first what is going on with laminar flames, which is the purpose of the present review.

Opposite to the Yakhot theory, in the present review we consider intrinsic dynamics of curved flames resulting from their own properties and not from an external flow. Most of the reviewed papers consider development of a curved shape of a flame front and propagation of curved flames in different combustion configurations. What is so important about a curved flame shape? The configuration of a planar flame front is much simpler, but a laminar flame almost never propagates as a planar front even in the case of negligible losses like in a tube with ideally slip and adiabatic walls. A flame front may be bent by hydrodynamic instabilities, gravity, acoustic and shock waves and many other effects. With the development of a curved shape the surface area of a flame increases, so that the flame consumes more fuel per unit time and propagates faster. The velocity of flame propagation is one of the most crucial characteristics for the numerical modeling of turbulent burning and, consequently, one of the main purposes of the present review is to show how flame velocity changes due to the curved flame shape.

A remarkable feature of premixed combustion is an outstandingly strong dependence of the reaction rate on temperature expressed in the Arrhenius law for the reaction rate ∝exp(−E/T), where E is the activation energy measured in temperature units. The activation energy of many reactions is so large, that the reaction rate at the room temperature may be taken zero. On the contrary, increase of the fuel temperature even by a factor 2 may lead to increase of the reaction rate by 10–12 orders of magnitude and to a noticeable reaction (Zeldovich et al., 1985). In the case of a strongly exothermic reaction when a considerable energy release is involved, relatively slight increase of the temperature at some region ignites the reaction, which eventually extends over the whole gas.

Once a reaction is ignited it can propagate in a self-supporting regime. In the case of a flame the physical mechanism of flame propagation may be described as follows. The burnt matter has larger temperature and thermal conduction transports energy from the hot burnt matter to the cold fuel. The temperature of the fuel close to the burnt matter increases, the reaction in this fuel goes faster until another portion of the fuel is burnt and some additional energy is released. The released energy is transported by thermal conduction to the next fuel layer and so on, resulting in propagation of the reaction front.

Flame velocity and thickness may be estimated on the basis of a simple dimensional analysis (Landau and Lifshitz, 1987). If the burning process is characterized by the typical time τb, then the only combination of velocity dimension that may be constructed out of the thermal diffusivity κ/ρfCP and the reaction time τb isUfκ/ρfCPτb,where κ is the coefficient of thermal conduction and CP is the specific heat of the fuel at constant pressure. Obvious conclusion from estimate (1.1) is that the shorter the reaction time and the stronger the thermal conduction, the faster flame propagates. Similarly, the dimensional analysis gives the formula for the flame thickness Lfκτb/(ρfCP). Since exact value of the flame thickness depends on the definition, we can adopt the equationLf=κ/ρfUfCPas the definition of the flame thickness. In the case of a thermal conduction depending on the gas temperature and density one can choose the value of the thermal conduction coefficient in the fuel κf for definition (1.2).

Let us compare the flame velocity to the sound speed. If we use the estimate for thermal diffusivity coefficient expressed through the sound speed cs (that is about the thermal velocity of molecules) and the mean free time τcoll: κ/ρfCPcs2τcoll, then we obtain Uf/csτcollb. Since only a very small fraction of colliding molecules react because of the large potential barrier of a reaction (because of a large activation energy), then τcollτR and flame velocity is much smaller than the sound speed. Typical velocities of the flame are shown in Table 1.1. We can point out the hydrogen–oxygen flame as one of the fastest, which has the velocity about 9 m/s, and the flame in the mixture 6% CH4 as one of the slowest ones with the velocity 5 cm/s. However such slow flames are usually close to extinction limits because of some inevitable losses and it is difficult to observe them. The velocity of most of laboratory flames ranges from 15 cm/s to 1 m/s.

With representative values for κ, CP and ρ for gas mixtures and with the velocities from Table 1.1 we find that the typical thickness of combustion zone ranges from 5×10−2 to 5×10−4 cm. It follows from Eq. (1.2) that flame thickness exceeds considerably the molecular mean free path (about 10−5 cm) and therefore the continuum equations of fluid dynamics are valid for description of flames.

If we consider another type of a flame instead of chemical one, then the energy release may be caused by other reasons such as, for example, thermonuclear reactions in Supernova flames (Timmes and Woosley, 1992) or the laser radiation absorbed by plasma layers close to the critical surface of a target in inertial confined fusion (Manheimer et al., 1982). However for any kind of a flame the released energy is transported by thermal conduction and flame propagates relatively slow in comparison with the sound speed. Therefore a flame may be also defined as a subsonic regime of reaction propagation.

Flame is not the only possible self-supporting regime of reaction propagation. A reaction can also propagate in a fast supersonic regime of detonation (Landau and Lifshitz, 1987). In the case of a detonation the reaction is induced by a shock wave compressing and heating the fuel. The burning mixture expands and acts like a piston pushing a leading shock and supporting the detonation. From the technical point of view detonation is a very undesirable process that can damage engines. Transitions from the slow regime of flame propagation to a detonation regime are observed quite often in experiments (Shelkin 1940, Shelkin 1966; Zeldovich et al., 1985). In industrial applications the transition of a flame to a detonation is related to the problem of knock in car engines, was encountered from the very beginning of the car industry.

Sometimes a third regime of burning is distinguished, which is the regime of spontaneous reaction (Zeldovich, 1980). A spontaneous reaction corresponds to the configuration of a fuel with a nonuniform initial temperature distribution. In this case subsequent (though independent) development of the reaction in the neighboring fuel layers may be interpreted as propagation of a reaction front with the front velocity depending on the initial temperature distribution. Though the regime of spontaneous reaction is much more specific than the regimes of flame and detonation, it may be also interesting from the point of view of flame/detonation ignition (Zeldovich et al., 1970).

Any theory starts from the simplest problem and then goes to more and more complicated problems. The simplest configuration of a slow combustion is a planar stationary flame front. In a certain sense the modern flame theory started from the paper by Zeldovich and Frank-Kamenetski (1938), where the analytical formulas for the velocity and the structure of a planar stationary flame supported by an Arrhenius reaction have been found. It does not mean that the obtained formulas could be used to calculate the velocity of a real flame with a complicated many step reaction, but the paper (Zeldovich and Frank-Kamenetski, 1938) proposed the methodology that made up the basis for the whole flame theory. The Zeldovich–Frank-Kamenetski theory will be discussed briefly in Section 2. More detailed discussion of the theory of planar flame propagation may be found, for example, in Zeldovich et al. (1985).

The next important step of the theory was the Darrieus–Landau (DL) solution for the flame stability problem (Darrieus, 1939; Landau, 1944), where it has been shown that an infinitely thin planar flame front is absolutely unstable against two- and three-dimensional perturbations bending the flame. Nowadays the DL theory of the hydrodynamic flame instability is a subject of textbooks (Landau and Lifshitz, 1987; Zeldovich et al., 1985; Williams, 1985); in the present review it will be considered in Section 3.1. The main conclusion of the DL theory is that a flame cannot propagate as a planar stationary front, but instead it becomes curved, sometimes non-stationary and probably even turbulent.

The DL theory raised many questions. The most important of them were the questions about the typical length scale of the DL instability and the final outcome of the instability at the nonlinear stage of the perturbation growth. To answer the former question one has to take into account the finite flame thickness and the transport processes that determine the flame thickness such as thermal conduction, fuel diffusion, etc. This is a rather difficult problem from the mathematical point of view, therefore at the beginning phenomenological methods were developed to solve the problem of flame stability. The phenomenological approaches to the linear theory of flame stability have been summarized in the review paper by Markstein (1964). In the present review we will not consider the phenomenological theories, since at present there is a rigorous solution of the stability problem for a flame of a finite thickness. The first attempt to describe rigorously the stabilization of the DL instability due to the finite flame thickness was undertaken in the paper (Istratov and Librovich, 1966), but the correct solution has been obtained much later by Pelce and Clavin (1982). Particularly, it was found (Pelce and Clavin, 1982) that the typical length scale of the DL instability exceeds the flame thickness by two orders of magnitude. Long before the rigorous theory was developed the effect of the unexpectedly large length scale of the DL instability has been observed experimentally (Zeldovich et al., 1985). The difference between the experimentally observed length scales of the instability and the predictions of the DL theory has been a puzzle for theoreticians for a long time. The theory of thermal stabilization of the DL instability will be briefly considered in Section 3.2; more detailed description of the theory as well as other results developing the approach of the paper (Pelce and Clavin, 1982) may be found in the reviews (Sivashinsky, 1983; Clavin 1985, Clavin 1994).

The other question raised by the DL theory was the outcome of the instability at the nonlinear stage. At the beginning it was supposed that the DL instability leads to turbulization of the flame front (Landau, 1944), since there was a good deal of experimental results on spontaneous transition from a laminar regime of flame propagation to turbulent one and then to detonation (Zeldovich et al., 1985). However later in many semi-qualitative papers (see Markstein, 1964; Zeldovich, 1966 and references therein) it was shown that the self-turbulization of a flame is unlikely. As an alternative it was proposed that the flame front perturbations develop into a smooth curved and, probably, stationary flame shape. In this sense the question about the nonlinear stage of the DL instability may be formulated as the problem of flame propagation in a tube with ideally slip and adiabatic walls. The configuration of a flame in an ideal tube is equivalent to the configuration of a freely propagating flame with a periodic structure at the front, since the ideal tube walls may be considered as symmetry axes.

Some progress in understanding the nonlinear stage of the DL instability has been achieved due to the nonlinear equation for a curved flame front derived by Sivashinsky (1977) in the limit of small expansion coefficients Θ−1⪡1, when the density of the products of burning differs only slightly from the fuel density. The Sivashinsky equation describes qualitatively nonlinear stabilization of the DL instability as well as many properties of curved flames. The equation became even more popular when an analytical solution was obtained for the two-dimensional version of the equation (Thual et al., 1985). Still many important questions remained without an answer in scope of the Sivashinsky theory, since the peculiar limit of a small expansion coefficient Θ−1⪡1 is quite far from the situation of realistic laboratory flames with Θ=5–10. Particularly, it was unclear even by order of magnitude what is the increase of the flame velocity due to the curved flame shape: different extrapolations of the theory (Sivashinsky, 1977) into the domain of realistic expansion coefficients could give quite different velocity amplification from few percent to the factor of about 10. Besides, it remained questionable if a flame front always acquires a smooth curved shape or at certain conditions (say, for the expansion coefficients larger than some critical value) self-turbulization of the flame happens.

An interesting approach to the problem of curved flames has been started in the papers (Matalon and Matkowsky, 1982; Clavin and Joulin, 1983), where the evolution equation for a flame front of a finite thickness in some unknown upstream flow of the fuel has been obtained together with the conservation laws at the front. These equations will be considered in Section 4.

Among different attempts to solve the problem of flame propagation in an ideal tube we would like to mention the paper by Zeldovich and co-authors (1980), where the problem was formulated in the most clear way. Unfortunately, the authors failed to solve the formulated problem rigorously and instead they presented only qualitative evaluation of the curved flame velocity.

Considerable progress in the theory of curved flames has been achieved recently and the authors of the present review contributed to the success too. For the most part the success was provided by active use of direct numerical simulations of the hydrodynamical equations describing flames, still the analytical theory of curved flames has been also developed. Particularly, the problem of flame propagation in an ideal tube has been solved numerically in Bychkov et al. (1996) and then the rigorous analytical solution of this problem has been obtained. The developed analytical theory and the obtained numerical results are discussed in Section 5.

In Section 6 we consider influence of a gravitational field on flame dynamics and velocity. The section starts with the theory of bubble rising, since in the case of large gravitational acceleration and sufficiently wide tubes the bubble motion of the products of burning is the dominant process for flames propagating both in vertical and horizontal tubes. This section also includes the results of numerical simulations of flame dynamics in a gravitational field.

In 7 Interaction of a flame front with acoustic waves, 8 Interaction of curved flames and weak shocks interaction of curved flames with pressure waves (acoustic waves and weak shock waves) is considered. The possibility of flame stabilization by acoustic waves due to the oscillating effective “gravity” acceleration is discussed as well as the parametric flame instability induced by acoustic waves of a sufficiently large amplitude. It is shown that weak shock waves colliding with curved flames may also lead both to flame stabilization and destabilization depending on the shock intensity and the expansion coefficient of the flame. The effect of inversion of the flame shape by a weak shock is discussed, when the convex parts of the front become concave and vice versa.

Section 9 is devoted to flame propagation in closed burning chambers, which involve all physical effects mentioned above: the hydrodynamical instabilities, formation of curved flames (which are only quasi-stationary in the configuration of a closed tube), generation of acoustic waves and weak shock waves by a flame under confinement, interaction of the flame with the generated pressure waves, etc. Besides, flame dynamics in a closed chamber involves many other interesting phenomena, such as pre-compression of the unburnt fuel, possibility of flame acceleration or deceleration depending on the flame parameters, detonation ignition ahead of the flame front and other effects.

The influence of compressibility on dynamics of curved flames is considered in Section 10. It is shown that the effects of compressibility may strongly destabilize a flame front and increase the velocity of curved flames even in the case of small Mach numbers (the Mach number is defined as the ratio of the flame velocity to the sound speed in the fuel).

In the last Section 11 we consider the problem of stability of curved stationary flames and development of the fractal structure at a flame front. This problem is not solved at present, so that in this section we review the results on fractal flames obtained up to now and discuss prospects of the future work. Among the papers devoted to fractal flames we would like to mention the experimental paper by Gostintsev et al. (1988) and resent theoretical papers by Procaccia and co-authors. Interesting results on the stability of curved stationary flames have been obtained in the recent paper by the authors of the present review (Bychkov et al., 1999). In Section 11 we discuss stability of curved stationary flames in ideal tubes and flames propagating from a center. The experimental evidence of spherical flames propagating in a self-similar accelerating regime (fractal flames) is discussed, as well as numerical simulations of fractal flames on the basis of simplified nonlinear equations for a curved flame. Theoretical ideas on the fractal structure of a flame front are presented.

Section snippets

Complete system of equations

Dynamics of a flame front in a gravitational field g is described by the hydrodynamical equations of mass, momentum and energy conservation with the account of reaction kinetics and transport processes of thermal conduction, fuel diffusion and viscosity. For the sake of simplicity a single irreversible reaction is admitted, so that the governing equations are the following:(/t)ρ+(/xi)(ρvi)=0,(/t)(ρvi)+(/xj)(ρvivjijP−τij)=ρgi,(/t)(ρe+12ρvjvj)+(/xi)(ρvih+12ρvivjvj+qi−vjτji)=ρgivi,t

Darrieus–Landau instability of an infinitely thin flame front

As it was observed experimentally, a flame front in a tube propagates rather seldom as a planar stationary front. Usually, the flame acquires a curved shape (Uberoi, 1959; Maxworthy, 1962) and sometimes transition to a turbulent regime of propagation happens (Shelkin 1940, Shelkin 1966; Zeldovich et al., 1985), which is accompanied by considerable amplification of the flame velocity. While the observed transition to the turbulent regime may be accounted for the interaction of the flow and rough

Derivation of the evolution equation for an infinitely thin flame front

Propagation of a flame is described by the system of hydrodynamic equations, which takes into account thermal conduction and the energy release. This is a rather complicated problem even for numerical solution and any simplification of the problem is of great interest. The problem may be simplified considerably if the region of thermal conduction and energy release can be replaced by a surface of discontinuity, separating the fuel and the products of burning. In this case one must specify the

Physical mechanism of the nonlinear stabilization of the flame instability

As it was pointed out in Section 3 one of the main reasons for a flame to lose its planar configuration is the DL instability of a planar flame front. Because of the DL instability small perturbations of a flame front grow and bend the front. Development of perturbations at the nonlinear stage of the DL instability has been an important issue of many discussions for a long time since the classical works by Landau and Darrieus (Landau, 1944; Darrieus, 1938). First it was a general belief

Theory of rising bubbles

Velocity of curved stationary flames resulting from development of the DL instability may be amplified by factor 1.2–1.3 in comparison with the planar flame velocity in the case of 2D flames and by factor 1.6–1.8 in the case of 3D flames like curved flames in cylindrical tubes. However experiments on flame propagation in tubes show that velocities of curved stationary flames may exceed these values considerably (Coward and Hartwell, 1932; Levy, 1965; Bregeon et al., 1978; von Lavante and

Experimental results on flame–acoustic interaction

In the previous sections we have considered flame dynamics in uniform unbounded fuel mixtures, however in industrial applications the configuration of a closed burning chamber is much more common (Ramos, 1995). When a flame propagates in a closed burning chamber, then many additional effects influence flame dynamics and development of the DL instability (Gonzalez et al., 1992; N'konga et al., 1992; McGrevy and Matalon 1992, McGrevy and Matalon 1994; Liberman et al., 1998). One of these effects

Linear analytical theory of flame-shock interaction

It is well known that a flame propagating in a closed burning chamber always generates pressure waves (Landau and Lifshitz, 1989; Zeldovich et al., 1985). These pressure waves in turn influence the flame shape and velocity, which may lead under certain conditions to considerable flame acceleration. Besides, the pressure waves heat the unburnt fuel and make easier detonation triggering ahead of the flame front. When the typical time scale of flame dynamics in a closed burning chamber is larger

Planar flames in closed tubes and scalings for the hydrodynamic flame instability

Flame dynamics in closed tubes is a very important subject of combustion science since it models the burning process in a typical industrial configurations. Particularly, flame propagation in a closed chamber is a key point for understanding of the “knocking” effect in engines, when autoignition or detonation is triggered ahead of the flame front due to the flame–acoustic interaction. Though effects that influence flame behaviour under confinement have been widely discussed in a number of books

The Darrieus–Landau instability of a flame with a nonzero Mach number

As one can see in Table 1.1 most of the laboratory flames propagate with a velocity well below the sound speed and may be considered in the isobarical approximation. However, there are situations when compressibility can play significant role like, for example, the case of a flame propagating in a closed burning chamber. A flame travelling from a closed end of a tube can get considerably accelerated and reach rather high speeds so that compressibility effects become the principal ones (

Stability of a curved flame in a tube

Solution , , of the stationary nonlinear equation (5.45) describes dynamics of curved stationary flames in tubes of an arbitrary width R including very wide tubes of a width much larger than the cut-off wavelength of the DL instability R/λc⪢1. However, in very wide tubes the stationary curved flames like those shown in Fig. 5.3, Fig. 5.8 do not happen in reality. The radius of curvature close to the hump of these flames goes to infinity with increase of the tube width, so that such flames

Conclusion

We have reviewed the latest results on dynamics and stability of curved laminar premixed flames, when the curved flame shape develops because of the hydrodynamic instabilities of a flame front such as the DL instability and the RT instability. Investigation of flame stability taking into account a finite flame thickness shows that a flame front becomes unstable against long-wavelength perturbations λ>λc, where the cut-off wavelength λc exceeds the flame thickness considerably, by a factor of

Acknowledgements

This work was supported in part by the Swedish National Board for Industrial and Technical Development (NUTEK), by the Swedish Natural Science Research Council (NFR) and by the Swedish Royal Academy of Sciences.

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