Asymptotics for Turbulent Flame Speeds of the Viscous G-Equation Enhanced by Cellular and Shear Flows

Abstract

G-equations are well-known front propagation models in turbulent combustion which describe the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are Hamilton–Jacobi equations with convex (L ¹ type) but non-coercive Hamiltonians. Viscous G-equations arise from either numerical approximations or regularizations by small diffusion. The nonlinear eigenvalue ${\bar H}$ from the cell problem of the viscous G-equation can be viewed as an approximation of the inviscid turbulent flame speed s _T. An important problem in turbulent combustion theory is to study properties of s _T, in particular how s _T depends on the flow amplitude A. In this paper, we study the behavior of ${\bar H=\bar H(A,d)}$ as A → + ∞ at any fixed diffusion constant d > 0. For cellular flow, we show that

$$\bar H(A,d)\leqq C(d) \quad \text{for all}\ d >0 ,$$

where C(d) is a constant depending on d, but independent of A. Compared with ${\bar H(A,0)= O(A/\log A), A\gg 1}$, of the inviscid G-equation (d = 0), presence of diffusion dramatically slows down front propagation. For shear flow, ${\lim_{A\to +\infty}\frac{\bar H(A,d)}{A} = \lambda (d) >0 }$ where λ (d) is strictly decreasing in d, and has zero derivative at d = 0. The linear growth law is also valid for s _T of the curvature dependent G-equation in shear flows.