A Numerical Study of Turbulent Flame Speeds of Curvature and Strain G-equations in Cellular Flows

Abstract

We study front speeds of curvature and strain G-equations arising in turbulent combustion. These G-equations are Hamilton-Jacobi type level set partial differential equations (PDEs) with non-coercive Hamiltonians and degenerate nonlinear second order diffusion. The Hamiltonian of strain G-equation is also non-convex. Numerical computation is performed based on monotone discretization and weighted essentially nonoscillatory (WENO) approximation of transformed G-equations on a fixed periodic domain. The advection field in the computation is a two dimensional Hamiltonian flow consisting of a periodic array of counter-rotating vortices, or cellular flows. Depending on whether the evolution is predominantly in the hyperbolic or parabolic regimes, suitable explicit and semi-implicit time stepping methods are chosen. The turbulent flame speeds are computed as the linear growth rates of large time solutions. A new nonlinear parabolic PDE is proposed for the reinitialization of level set functions to prevent piling up of multiple bundles of level sets on the periodic domain. We found that the turbulent flame speed $s_T$ of the curvature G-equation is enhanced as the intensity $A$ of cellular flows increases, at a rate between those of the inviscid and viscous G-equations. The $s_T$ of the strain G-equation increases in small $A$, decreases in larger $A$, then drops down to zero at a large enough but finite value $A_{*}$. The flame front ceases to propagate at this critical intensity $A_*$, and is quenched by the cellular flow.