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Abstract
We consider the propagation of laminar premixed flames in the presence of a parallel flow whose scale is smaller than the laminar flame thickness. The study addresses fundamental aspects with relevance to flame propagation in narrow channels, to the emerging micro-combustion technology, and to the understanding of the effect of small scales in a (turbulent) flow on the flame structure. In part, the study extends the results of a previous analytical study carried out in the thick flame asymptotic limit which has in particular addressed the validity of Damköhler's second hypothesis in the context of laminar steady parallel flows. Several new contributions are made here.
Analytical contributions include the derivation of an explicit formula for the effective speed of a premixed flame U
T
in the presence of an oscillatory parallel flow whose scale ℓ (measured with the laminar flame thickness δ
L
) is small and amplitude A (measured with the laminar flame speed U
L
) is 
(1). The formula shows a quadratic dependence on both the amplitude and the scale of the flow. The validity of the formula is established analytically in two distinguished limits corresponding to (1) frequencies of oscillations (measured with the natural frequency of the flame U
L
/δ
L
), and to higher frequencies of (A/ℓ) (the natural frequency of the flow). The analytical study yields partial support of Damköhler's second hypothesis in that it shows that the flame behaves as a planar flame (to leading order) with an increased propagation speed which depends on both the scale and amplitude of the velocity fluctuation. However our formula for U
T
contradicts the formula given by Damköhler in his original paper where U
T
has a square root dependence on the scale and amplitude.
Numerical contributions include a significant set of two-dimensional calculations which determine the range of validity of the asymptotic findings. In particular, these account for volumetric heat loss and differential diffusion effects. Good agreement between the numerics and asymptotics is found in all cases, both for steady and oscillatory flows, at least in the expected range of validity of the asymptotics. The effect of the frequency of oscillation is also discussed. Additional related aspects such as the difference in the response of thin and thick flames to the combined effect of heat loss and fluid flow are also addressed. It is found for example that the sensitivity of thick flames to volumetric heat loss is negligibly affected by the parallel flow intensity, in marked contrast to the sensitivity of thin flames. Interestingly, and somewhat surprisingly, thin flames are found to be more resistant to heat loss when a flow is present, even for unit Lewis number; this ceases to be the case, however, when the Lewis number is large enough.
Notes
∗Here and below, we have normalized the numerical values of U T by the corresponding numerical value of the planar, unstretched flame speed. This allows a fair comparison between the asymptotics (based on an infinite β assumption) and the numerics, without having to use excessively large values of β.