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 and 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 will study the behavior of $\bar{H}=\bar{H}(A,d)$ as $A\to +\infty$ at any fixed diffusion constant $d>0$. For the cellular flow, we show that $$\bar{H}(A,d)\le O(\sqrt{\log A})\text{for all d>0}.$$ Compared with the inviscid G-equation ($d=0$), the diffusion dramatically slows down the front propagation. For the shear flow, the limit \nit ${\lim }_{A\to +\infty }\frac{\bar{H}(A,d)}{A}=\hat{I}»(d)>0$ where $\hat{I}»(d)$ 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.

27 pages. We improve the upper bound from no power growth to square root of log growth