Flame curvature

FIGURE 10.7. Schematic illustrations of influence of flame stretch and of flame curvature, through Lewis-number effects, on rates of heat release per unit area for adiabatic, one-reactant flames with one-step chemistry of large overall activation energy. 

As mentioned above a phenomenological dependence of on flame curvature was anticipated by Markstein [8] and a phenomenological... 

The 0(6) jump in x n results from unbalanced tangential stresses. It is proportional to the vorticity [Vx ] produced at the flame and to the rotation V x n of a surface element. We note in particular the dependence on both the heat release q, and the Prandtl number Pr. Finally, equation (13) states that a net normal momentum flux across the flame, results from unbalanced pressure gradients, flame curvature and from the momentum associated with the net excess mass given by (11). 

For a given conduction-diffusion system, therefore, the balance between these two effects determines the overall local stability to spatial perturbation. If the conductive influences are dominant, the planar front will be stable. This arises if the thermal diffusivity is greater than the molecular diffusion coefficient, i.e., if the Lewis number (Le) is less than unity. Instability and the growth of flame curvature occurs under the opposite conditions, when molecular diffusion is dominant and (Le) > 1. This latter situation can arise with light, mobile fuels such as H2 or if light, mobile chain carriers such as H-atoms are produced (lean hydrocarbon flames have (Le) < 1 rich flames have (Le) >1). The effect of this instability is to produce cellular flames. (It should also be mentioned that a different instability, leading to oscillatory flame speeds, can arise for (Le) < 1.)... 

The characteristic dimensional parameter Lm takes into account the flame curvature effect on the burning velocity. The higher its absolute value, the stronger the curvature effect is. The Markstein length relation to the laminar flame thickness S = dSu, where - the laminar flame velocity, is known as the Markstein number Ma = Lyild. Table 1.1 [15] presents the Markstein length for hydrogen-air mixtures at 298 K and 0.1 MPa... 

It should be noted that the extrapolation method is complicated and time consuming. An absolutely different approach has been offered in [27, 47, 48]. It is based on a well-known technique for the investigation of a developing spherical flame using Schlieren photography. The idea of this method is to obtain a flame velocity correction relying on the Markstein assumption that such a velocity is a linear function of the flame curvature. The phenomenological ratio... 

A somewhat different method has been used in [30, 49, 50]. In this method, firstly, a visible flame velocity Sf, = drfjdt is found by differentiating the experimental flame radius - time function and calculating the = SiJo value. A stretch correction has been performed by the linear extrapolation to zero flame curvature. Numerical experiments in [43] proved that the same value of can be obtained by use of either of the mentioned methods. 

R.M. Fristrom, Definition of burning velocity and a geometric interpretation of the effects of flame curvature. Phys. Fluids 8(2), 273-280 (1965)... 

At the anticipated laminar flame velocity 2.39 m/s and combustion products expansion factor a = 7.26 (no turbulent effect or flame curvature) a visible velocity of 17.35 m/s was predicted. 

Markstein length Characteristic dimensional parameter Lm taking into account a flame curvature effect on a combustion velocity. The greater is an absolute value of this parameter, the more the curvature effect is. A ratio of a Markstein length Lm to a laminar flame thickness d = alS, where -combustion laminar velocity, is called the Markstein number Ma = Lm/S. 

Bogen, m. bow, arc, bend, curve, curvature, arch (Elec.) tc sheet (of paper). >fiamme, /. arc flame,... 

The shape of the stretch rate distribution curve for the lean limit propane flame, shown in Figure 3.1.9b, is very similar. However, for this flame, the estimated contribution of the curvature is more important than with the methane flame, reaching about 40% of the maximum stretch rate. 

V. Nayagam and F. A. Williams, Curvature effects on edge-flame propagation in the premixed-flame regime, Combust. Sci. Tech. 176 2125-2142,2004. 

R is the radius of curvature of the flame is a characteristic length of the order of the flame... 

The Lewis number, Le, is that of the deficient species (fuel or oxidant) in the mixture. In their analysis, Clavin and Williams used the simplifying approximation that the shear viscosity, the Lewis number, and the Prandtl numbers are all temperature-independent. They also showed that, at least for weak flame stretch and curvature, the change in local flame speed due to stretch and curvature is described by the same Markstein number ... 

The cooling effect of the channel walls on flame parameters is effective for narrow channels. This influence is illustrated in Figure 6.1.3, in the form of the dead-space curve. When the walls are <4 mm apart, the dead space becomes rapidly wider. This is accompanied by falling laminar burning velocity and probably lowering of the local reaction temperature. For wider charmels, the propagation velocity w is proportional to the effective flame-front area, which can be readily calculated. On analysis of Figures 6.1.2b and 6.1.3, it is evident that the curvature of the flame is a function of... 

Dead space and the radius of curvature of flame as functions of the distance between channel walls for flames shown in Figure 6.1.2. 

The radius of curvature of flame is shown in Figure 6.1.7 as a function of the quenching distance (Figure 6.1.7a) and of the equivalence ratio (Figure 6.1.7b). The radius was determined from the flame pictures. For lean mixtures, the radius increases linearly with the channel width, both for the downward and upward propagating flames. For rich mixtures and downward propagation, the increase is linear for quenching distances up to Dq = 7 mm, but the increase is not as steep as that of lean mixtures. However, the increase accelerates. For rich... 

To examine the details of the structure of flames in channels under quenching conditions, numerical methods were used. Two-dimensional CFD simulation of a propane flame approaching a channel between parallel plates was carried out using the FLUENT code [25]. The model reproduced the geometry of the real channels investigated experimentally. Close to the quenching limit, the burning velocity, dead space, and radius of curvature of the flames were all close to the experimental values. 

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