Analysis of flamelet leading point dynamics in an inhomogeneous flow

Several studies have utilized “leading points” concepts to explain the augmentation of burning rates in turbulent flames by flow fluctuations. These ideas have been particularly utilized to explain the strong sensitivity of turbulent burning rates to fuel composition. Leading point concepts suggest that the burning velocity is controlled by the velocity of the points on the flame that propagate farthest out into the reactants – thus, they de-emphasize the classical idea that burning velocity enhancement is due to increases in flame surface area. Rather, within this interpretation, flame area creation is the effect, not the cause, of augmented turbulent burning velocities. However, the theory behind the implementation of leading point concepts in turbulent combustion modeling needs further development and the definition of “leading point” has not been fully clarified. For a certain class of steady shear flows, it is straightforward to demonstrate the leading point concept in an intuitive manner, but the problem becomes more complex when the leading points themselves evolve in time. In this paper, we use the G-equation to describe the flame dynamics and, utilizing results for Hamilton–Jacobi equations from the Aubry–Mather theory, demonstrate both the utility and limitations of leading points interpretations for front propagation, at least for deterministic problems. Specifically, we show how the large-time behavior of the solutions is controlled by discrete points on the flame under certain conditions and is, therefore, independent of the rest of the flow field details – a key hypothesis of leading points theories. However, it is possible to find other conditions where the large time behavior of the flame is not controlled by discrete points on the flame, but rather by the velocity field over its entire surface. Moreover, we also show that even in cases where the burning rate is controlled by discrete points, these points are not necessarily the most forward lying points in the flame front. Finally, we consider the case where the laminar flame speed is a function of flame front curvature and derive exact results for the sensitivity of the front speed to the Markstein length, ℓ, for ℓ > 0. These solutions explicitly illustrate how the reduction of front displacement speed for increasing ℓ can be interpreted in terms of leading points dynamics in some cases.