Small Lewis Numbers Research Articles

ON THE FAST-TIME CELLULAR INSTABILITIES OF LIÑÁN'S DIFFUSION-FLAME REGIME

ABSTRACT The fast-time instability of the Liñán's diffusion-flame regime is investigated asymptotically and numerically by employing the fast inner-zone time and length scales, as a model problem for the cellular instability in diffusion flames with Lewis numbers far from unity by an amount of order unity. The stability analysis revealed the full spectral nature, particularly near the saddle-node bifurcation condition corresponding to the minimum reduced Damköhler number Δ. Contrary to the conventional belief, the minimum Δ condition, commonly known as the Liñán's diffusion-flame extinction condition, is not necessarily an extinction condition for flames with Lewis numbers less than unity which can survive beyond the saddle-node bifurcation condition. The cellular instability could emerge upon passing the saddle-node bifurcation condition. The cellular instability is thus observable for near-extinction diffusion flames with Lewis numbers less than unity, as predicted by the previous experimental studies and the linear stability analysis employing the NEF limit. The stable parametric regions of small wave number and Lewis number just below unity were not predicted by the fast-time instability. But these parametric regions lie in the inner parametric layer of the distinguished limit employed in this analysis, so that the leading-order behavior is not contradictory with the previous experimental and analytical results.

Roll, square, and cross-roll convection in ferrofluids

Using the Galerkin technique we calculated different kinds of convection patterns in the Rayleigh–Bénard system and investigated their stability for very small Lewis numbers and very high separation ratios as typical for colloidal systems like ferrofluids.

Comparison of computational methodologies for ignition of diffusion layers

The prediction of the autoignition delay time of fresh fuel in hot air is studied. This problem, encountered in many combustion systems, is in practice often calculated using a simple method, hereafter called homogeneous mixing ignition (HMI). This method, however, neglects transport effects and calculates local instead of global ignition properties. A second method is therefore studied called linear mixing ignition (LMI) and consists of the direct simulation of a one-dimensional mixing layer. This method is tested in a hydrogen and methane fuel configuration using detailed and reduced chemistry mechanisms. The results indicate that molecular transport increases the ignition delay for the methane fuel by about a factor three, but this effect is compensated in the hydrogen case by the small Lewis number of the fuel. Such detailed information is not provided by the HMI method and the LMI approach appears, therefore, as a minimum requirement for the correct estimation of the ignition delay time.

The structure of flame filaments in chaotic flows

The structure of flame filaments resulting from chaotic mixing within a combustion reaction is considered. The transverse profile of the filaments is investigated numerically and analytically based on a one-dimensional model that represents the effect of stirring as a convergent flow. The dependence of the steady solutions on the Damköhler number and Lewis number is treated in detail. It is found that, below a critical Damköhler number Da crit, the flame is quenched by the flow. The quenching transition appears as a result of a saddle-node bifurcation where the stable steady filament solution collides with an unstable one. The shape of the steady solutions for the concentration and temperature profiles changes with the Lewis number and the value of Da crit increases monotonically with the Lewis number. Properties of the solutions are studied analytically in the limit of large Damköhler number and for small and large Lewis number.

Cellular instabilities, sublimit structures and edge-flames in premixed counterflows

We examine twin premixed flames in a plane counterflow and uncover, in the parameter space, a hitherto unknown domain of cellular instability. This leads us to hypothesize that for small Lewis numbers a two-dimensional (2D) steady solution branch bifurcates from the one-dimensional (1D) solution branch at a neutral stability point located near the strain-induced quenching point. Solutions on this 2D branch are constructed indirectly by solving an initial-value problem in the edge-flame context defined by the multiple-valued bistable 1D solution. Three kinds of solution are found: a periodic array of flame-strings, a single isolated flame-string and a pair of interacting flame-strings. These structures can exist for values of strain greater than the 1D quenching value, corresponding to sublimit solutions.

Lewis Number Effects in Premixed Turbulent Combustion and Highly Perturbed Laminar Flames

Various perturbed laminar flames are numerically simulated by reducing combustion chemistry to a single reaction. The following flame configurations are addressed: expanding spherical, cylindrical, and symmetrical planar flames; converging spherical flame; expanding cylindrical and symmetrical planar flames affected by external steady or time-dependent strain rate; steady strained cylindrical and symmetrical planar flames. The results (1) show that a simple linear relation between the local consumption velocity and flame stretch rate is valid only for weakly perturbed laminar flames; (2) highlight the importance of transient effects; and (3) show that, in the case of a small Lewis number, the highest local combustion rate is reached in the expanding spherical flame ignited by the hot pocket of the critical radius. This highest local combustion rate is successfully used to describe the extensive Karpov's experimental data base on turbulent burning velocities for mixtures characterized by substantially different values of the Lewis number. Discussion of Karpov's experimental data showing the drastic effect of the Lewis number on turbulent burning velocity implies the important role played by highly perturbed flamelets in premixed turbulent combustion.

Convection in a slowly diffusing, weakly stratified salt field

We examine thermohaline convection in the small Lewis number limit for a background salt field with a weak gradient. As a result of small diffusivity, the problem cannot be reduced to a simple amplitude equation in the usual manner, but the salt field is described by an advection-diffusion equation coupled to an equation for the amplitude of the convective pattern.

The Fast-Time Stability of a Simple Deflagration

ABSTRACT Fast-time instabilities refer to dynamics that may develop within the reaction zone of a flame, where the chemical kinetic time scales are very brief. Premixed flames with order one heat losses and diffusion flames with order one reactant leakage are known to possess this instability when the Lewis number is one (Peters, 1978). Work by Stewart (1986) indicates that the instability should also arise for adiabatic flames with Lewis numbers exceeding one. Recent work in reverse smouldering combustion (Lozinski and Buckmaster, 1994) requires an examination of the combined effect of heat losses and Lewis numbers well below unity. Because of the general nature of the results, they are presented here. The numerical computations reveal that while heat losses tend to destabilize the flame, small Lewis numbers have a countervailing stabilizing effect, suppressing the instability for an order one range of heat losses. This plays a key role in selecting the physically significant solution to the reverse smouldering combustion model (Lozinski and Buckmaster, 1994). The instability is one-dimensional; an examination of higher dimensions reveals no new instabilities arising for reasonable parameter values.

On open and closed tips of bunsen burner flames

An adiabatic, constant-density reaction-diffusion-advection model for the Bunsen burner flame tip is studied numerically. It is shown that for Lewis numbers exceeding unity the reaction rate and flame speed gradually increase toward the flame tip. For small Lewis numbers the picture is quite different. The reaction rate drops near the tip. In spite of this the flame survives and, moreover, manages to consume all the fuel supplied to the reaction zone. There is no leakage of the fuel through the front. The flame speed varies nonmonotonously along the front from gradual reduction to steep increase near the tip.

Numerical Analysis of Convective Ignition and Flame Development over a Porous Sphere

Abstract This work investigated transient processes in gas-phase combustion over a porous sphere. According to a numerical model, the ignition criterion,dg/dt = 0, at the spherical surface is verified to be appropriate for the entire flow field that includes both the front and wake regions of the sphere. The flame developed stage is defined to be when the surface heat flux becomes invariant with time for all orientations. The effects of Damkohler and Lewis numbers on the ignition delay and the period of flame development are examined. For a small Lewis number or a large Damkohler number, ignition initiates near the forward stagnation point, and a stable enveloped flame is formed. Conversely, for a large Lewis number or a small Damkohler number, ignition takes place in the wake region and the developed flame is a wake flame.

A Flame-String Model and its Stability

Abstract Barely flammable mixtures with small Lewis numbers can support “flame-balls”, stationary, spherical, prcmixed flames for which the mass-averaged velocity is zero everywhere. The theory of flame-balls is on a sound footing. Recent experiments by Whaling and Ronney in NASA's KC-135 show that, for certain mixtures, g-jitter can stretch a flame-ball into a long cylinder of flame, a “flame-string”. Flame-strings display a peristaltic instability and break up into discrete flame-balls, in appearance similar to a jet of liquid breaking up into drops under the influence of surface tension. A simple theory of flame-string structure and stability is described.

Linear stability of experimental Soret convection.

Linear stability results for convection in binary fluid mixtures are given for experimental boundary conditions and parameter values. Normal $^{3}\mathrm{\ensuremath{-}}^{4}$He and ethanol-water mixtures are considered with no-slip boundary conditions on the velocity and no outward mass flux. Two cases are considered in detail: fixed temperature at top and bottom, and fixed temperature at the top and fixed thermal flux at the bottom. The role played by the Biot number is emphasized. The results are presented in a manner most useful to experimentalists. The errors incurred in determining the separation ratio S from experimental observations using scaled results with free boundaries are quantified. Neutral curves for both oscillatory and steady-state instabilities together with the corresponding critical wave numbers ${k}_{c}$ are determined as functions of S. For the latter, ${k}_{c}$ vanishes as S increases. The critical values of S for which this first occurs are determined analytically. Growth rates and frequencies for supercritical Rayleigh numbers are also given. The codimension-two point is found to be masked by steady-state instabilities that set in at smaller Rayleigh numbers, but the masking vanishes in the limit of small Lewis numbers and large Prandtl numbers. The lowest observable frequency along the Hopf curve is determined.

Extinction of interacting premixed flames: Theory and experimental comparisons

The structure and extinction states of two interacting stretched premixed flames of unequal mixture concentrations and distinct fuel, oxidizer and thermal diffusivities are analyzed. Results show that the extinction behavior depends critically on the effective Lewis numbers of the mixtures in that flames with large Lewis numbers interact weakly such that extinction mainly depends on the concentration of the stronger flame, while flames with small Lewis numbers interact strongly such that extinction depends on the concentrations of both flames. Theoretical predictions identify the various extinction modes and satisfactorily explain the experimental results on the extinction boundaries of lean and rich methane/air and butane/air flames.

Flame Stabilization in a Layered Medium

Abstract We analyze the steady-state structure of a l1ame stabilized on a burner which is fed by a nonuniform mixture. Our diffusional-thermal model is based on the assumption of large overall activation energy, and the nonuniformity in the feed is treated as a perturbation of the uniform case. The flame shape for a given inflow enthalpy profile depends on Lewis number, inflow velocity, and conductance of the burner, and for sufficiently small Lewis number, we exhibit the onset of cellular flame formation. Unlike previously encountered flame instabilities, the transition to a cellular name occurs here in a gradual, rather than an abrupt, fashion through an "imperfect bifurcation."

On the nonuniqueness and instability of the stationary combustion modes in the boundary layer, with strong injections

A mathematical model given in /1/ describing approximately the heat and mass exchange between the inert thermostat and reacting gas flow in the presence of strong injection of a chemically frozen gas, is used to analyze the influence of the force of injection and other factors on the stability of combustion in the boundary layer. The boundaries of the regions of stability and uniqueness of the stationary combustion modes are constructed in the plane of the Damköhler numbers. At small Lewis numbers a region corresponding to the self-oscillatory combustion modes is discovered, and an estimate given for the influence exerted by various factors on the frequency and amplitude of the self-oscillations. It is established that a strong injection of a chemically frozen gas stabilises the combustion process, since the region of instability narrows appreciably with increasing force of injection. On the other hand, increase in the injection rate leads to appreciable increase in the size of the region of nonuniqueness of the stationary modes.