Generalized flame balls

Abstract

We consider generalized flame balls which correspond to stationary spherical flames with a flow of hot inert gas, either a source or a sink, at the origin. Depending on the flow, these flames can have positive, zero, or negative burning speeds, with zero speeds characterizing the Zeldovich flame balls. A full analytical description of these structures and their stability to radial perturbations is provided, using a large activation energy asymptotic approach and a thermo-diffusive approximation. The results are also complemented by a numerical study. The number and stability of the generalized flame balls are identified in various regions of the l-M-h ₀ space, where l is the (reduced) Lewis number, and M and h ₀ the flow rate and its enthalpy at the origin, respectively. It is typically found that, when the flow is a source, there is a maximum value of the flow rate M _ax depending on l and h ₀, above which no stationary solutions exist, and below which there are two solutions characterizing a small stable flame ball and a large unstable flame ball; the implications of these results to the problem of ignition by a hot inert gas stream are discussed. When the flow is a sink, however, there is typically a single unstable solution, except for sufficiently large values of the Lewis number and large negative values of M, where three flame balls exist, the medium one being stable. Finally, the relation between the flame speed, positive or negative, and the flame curvature, small or large, is discussed.

Keywords:

• flame balls
• ignition by a hot stream
• flame stability

Notes

¹Indeed, since θ⁰ + y ⁰ _F =1 everywhere, and y_F =0 in the burnt gas, the left-hand side of (9c) reduces to , which is the (non-dimensional) mass of fuel reaching the reaction sheet per unit area of the latter and unit time, in the limit β→∞.

²Using the location of the reaction sheet to define the flame speed is a convenient and unambiguous choice in the limit β→∞, although other choices are possible [16].

³See e.g. [15, p. 530].

⁴More explicitly, Taylor expansions to yield the relations ]+, where the brackets designate jumps evaluated at r=R. These lead to the jump conditions (28) once the stationary solutions and given by (10) and (11) are used.

⁵This conclusion follows, e.g., from a plot of the simple explicit relation (29) of l versus the variable assumed real and positive. Also, the assumption Re which has just been made is consistent with this conclusion.

⁶The notation ‖·‖ stands here for the maximum norm.

⁷We have checked that this small neighbourhood of the lower turning point with unstable solutions shrinks as β is increased (not shown here), improving thus the agreement with the asymptotics.