Asymptotic Growth Rates and Strong Bending of Turbulent Flame Speeds of G-Equation in Steady Two-Dimensional Incompressible Periodic Flows

The study of turbulent flame speeds (the large time front speeds) is a fundamental problem in turbulent combustion theory. A significant project is to understand how the turbulent flame speed (sT ) depends on the flow intensity (A). G-equation is a very popular level set flame propagation model in turbulent combustion community. The main purpose of this paper is to study properties of limA→+∞ sT A and limA→+∞ sT (if finite, or strong bending) in the G-equation model for two dimensional (2d) divergence free periodic flows. Our analysis is based on the invariant measures and rotation vectors of the 2d flows and the travel times of the associated flow trajectories under control. Optimal linear/sublinear growth and strong bending conditions are precisely given in terms of rotation vectors and periodic orbits. A strong bending formula of sT in the cat’s-eye flow is discovered by averaging the controlled characteristics of the G-equation. The growth rate of sT and that of the related front speeds of reaction-diffusion-advection equations (with Kolmogorov-Petrovsky-Piskunov nonlinearity) are shown to be zero or nonzero simultaneously in 2d flows, yet they differ in the 3d Roberts cell flows that depend on two spatial variables. A future program is to extend our analysis to more complex fluid flows, such as unsteady 2d flows and 3d flows with chaotic structures. AMS subject classifications. 70H20, 76F25, 76M50, 76M30.