The effect of gravity and thermal expansion on the propagation of a triple flame in a horizontal channel

We investigate the effect of thermal expansion and gravity on the propagation of a triple flame in a horizontal channel with porous walls, where the fuel and oxidiser concentrations are prescribed. The triple flame therefore propagates in a direction perpendicular to the direction of gravity, a configuration that does not seem to have received any dedicated investigation in the literature. In particular, we examine the effect of the non-dimensional flame-front thickness on the propagation speed of the triple flame for different values of the thermal expansion coefficient a and the Rayleigh number Ra. When gravity is not accounted for (Ra = 0), and for small values of , the numerically calculated propagation speed is found to agree with predictions made in previous studies based on scaling laws [1]. We show that the well known monotonic relationship between U and , which exists in the constant density case when the Lewis numbers are of order unity or larger, persists for triple flames undergoing thermal expansion. Under strong enough gravitational effects (Ra) 1), however, the relationship is no longer found to be monotonic. For a fixed value of , the relationship between the Rayleigh number and the propagation speed is shown to vary qualitatively depending on the value of chosen, exhibiting hysteresis if is small enough and displaying local maxima, local minima or monotonic behaviour for other values of. All of the steady solutions presented in the paper have been found to be stable, except for those on the middle branches of the hysteresis curves. The study of triple flames, which consist of two premixed branches and a trailing diffusion flame, has been extensive since their first experimental observation by Phillips [2]. Early analytical studies were carried out by Ohki and Tsuge [3], followed by Dold and collaborators [4,5]. These initial studies utilised the constant density approximation, thus decoupling the underlying hydrody-namics of the system from the equations of heat and mass. Most analytical studies since have used this approximation while investigating several practical aspects affecting triple flames. These aspects include preferential diffusion [6,7], heat losses [8–10], reversibility of the chemical reaction [11,12] and the presence of a parallel flow [13]. For further references see the review papers [14,15]. In this paper we dispense with the constant density approximation in order to describe the coupled effect of thermal expansion and gravity on the propagation of the triple flame. To this end, it …