Flame balls in non-uniform mixtures: existence and finite activation energy effects

Abstract

The paper's broad motivation, shared by a recent theoretical investigation [Daou and Daou, “Flame balls in mixing layers,” Combustion and Flame, Vol. 161 (2014), pp. 2015–2024], is a fundamental but apparently untouched combustion question; specifically, ‘What are the critical conditions insuring the successful ignition of a diffusion flame by means of an external energy deposit (spark), after mixing of cold reactants has occurred in a mixing layer?’ The approach is based on a generalisation of the concept of Zeldovich flame balls, well known in premixed reactive mixtures, to non-uniform mixtures. This generalisation leads to a free boundary problem (FBP) for axisymmetric flame balls in a two-dimensional mixing layer in the distinguished limit β → ∞ with ε_L = O(1); here β is the Zeldovich number and ε_L is a non-dimensional measure of the stoichiometric premixed flame thickness. The existence of such flame balls is the main object of current investigation. Several original contributions are presented. First, an analytical contribution is made by carrying out the analysis of Daou and Daou (2014) in the asymptotic limit ε_L → 0 to higher order. The results capture, in particular, the dependence of the location of the flame ball centre (argued to represent the optimal ignition location which differs from the stoichiometric location) on ε_L. Second, two detailed numerical studies of the axisymmetric flame balls are presented for arbitrary values of ε_L. The first study addresses the infinite-β FBP and the second one the original finite-β problem based on the constant density reaction–diffusion equations. In particular, it is shown that solutions to the FBP exist for arbitrary values of ε_L while actual finite-β flame balls exist in a specific domain of the β–ε_L plane, namely for ε_L less than a maximum value proportional to ; this scaling is consistent with the existence of solutions to the FBP for arbitrary ε_L. In fact, the flame ball existence domain is found to have little dependence on the stoichiometry of the reaction and to coincide, to a good approximation, with the domain of existence of the positively-propagating two-dimensional triple flames in the mixing layer. Finally, we confirm that the flame balls are typically unstable, as one expects in the absence of heat losses.

Keywords:

• flame balls
• forced ignition of diffusion flame
• ignition in non-uniform mixtures

Notes

1. It is useful to note that ε_L is in fact inversely proportional to the square root of the Damköhler number Da, ε_L∝Da^−1/2, where Da is suitably defined in Equation (6(6) ) below.

2. To avoid potential confusion, we note that the non-dimensional model (equations and boundary-conditions) formulated in this section pertains only to flame balls and not to triple or diffusion flames. A few computations of triple and diffusion flames will be needed however in Section 5; there, the necessary modifications of the model will be described succinctly.

3. The reason for this assertion will become clearer later. Indeed, as we shall see, the non-dimensional location of the flame ball centre z_c is determined by the leading order asymptotic result (12(12) ) derived in [1], and the two-term expansion (35(35) ) derived in the present paper and checked numerically in Figures 2 and 3.

4. It will be convenient below to use on occasion non-dimensional coordinates based on L, characterised by bars, namely , and . Note that (5(5) ) implies that , and , with according to (2(2) ).

5. The non-theoretically-inclined reader may find the material of Section 5 easiest to understand.

6. This formula is not to be used in the limit Δ → 0 with ε_L fixed; clearly the second term is a function of Δ, which is discontinuous at Δ = 0 with distinct right and left limits there. This discontinuity can be traced back to the (finite) discontinuity of the third order derivative appearing in (31(31) ). This occurs in the limit z₀ → 0, corresponding to Δ → 0, and only in this limit. At any other value of z₀ (the maximum of corresponding to a non-zero value of Δ) the function is in fact infinitely differentiable.

7. This trend is in agreement with that exhibited by the solutions of the FBP in Figure 6. Note that the diffusion flame seen inside the flame ball when plotting the reaction rate fields for finite β and non-small values of ε is absent in the asymptotic model which describes the exterior of the flame ball only; the reaction rate of the diffusion flame inside the ball can be shown indeed to be vanishingly small compared with that of the premixed flame envelope in the limit β → ∞ with ε_L fixed.

8. Strictly speaking, flame balls and positively propagating triple flames exist for , where ε_ign corresponds to the ignition of the one-dimensional diffusion flame. However ε_ign, which strongly decreases with β, is effectively zero; e.g. ε_ign ≈ 2 × 10⁻⁵ for β = 5 and ε_ign ≈ 10⁻¹¹ for β = 10.

9. Similar symmetry arguments can be used below, if needed. For example, a reflection with respect to the vertical axis in Figures B2 and 18 will extend the graphs to negative values of Δ; alternatively, Δ may be replaced by |Δ|.

10. In the finite-β cases, z_c is defined as the barycentre of the one-dimensional region I on the z-axis where the reaction rate is non-negligible (ω > 10⁻⁴); specifically, z_c = ∫_Iz dz/∫_Idz. This definition is used because it is consistent with that adopted in the infinite-β FBP.