Finite flame thickness effects on Kolmogorov-Petrovsky-Piskunov turbulent burning velocities

KPP (Kolmogorov-Petrovsky-Piskunov) solutions of the reaction-diffusion equation have application in various physical phenomena occurring in biology, ecology, and reacting flows. In particular, these solutions are commonly used in turbulent combustion to scale turbulent burning velocities. Subject to certain conditions on reaction rate profile through the flame brush and turbulent diffusivity, this theory relates the turbulent burning velocity to the derivative of the reaction rate (ω[over ̃]) at the leading edge of the flame brush (dω[over ̃]/dc[over ̃]|_{c[over ̃]=0}). Such waves are often referred to as "pulled fronts." However, turbulent flames never actually satisfy the KPP conditions for a pulled front, as the turbulent flame brush, parametrized here by the thickness δ_{t}, consists of an ensemble of laminar flamelets of thickness δ, where ε=δ/δ_{t}≪1 is very small, but nonzero, and dω[over ̃]/dc[over ̃] tends to zero at the brush leading edge for high activation energy, combustion-type kinetics. This paper analyzes these effects on KPP wave solutions, parametrized by ε=δ/δ_{t} and Zeldovich number Ze focusing on whether turbulent flames retain their pulled front character and what the correction to the KPP wave speed is. Variational solutions of the reaction-diffusion equation show that the solution can be expanded in powers of 1/|lnε|. Both numerical and asymptotic results are presented, showing that the wave still exhibits pulled front solutions but with significant corrections to the KPP result. The leading order correction is of the form |lnε|^{-2} and independent of Ze. Higher order corrections are function of both ε and Ze. However, the dominant factor influencing the wave speed correction is due to the finite ε, with Ze exhibiting a weaker effect.