Asymptotics for turbulent flame speeds of the viscous G-equation enhanced by cellular and shear flows

TL;DR

This paper investigates the asymptotic behavior of viscous G-equation flame speeds in turbulent flows, revealing how diffusion and flow type influence the speed scaling as flow amplitude increases.

Contribution

It provides new asymptotic estimates for turbulent flame speeds in cellular and shear flows, highlighting the impact of diffusion on front propagation.

Findings

Diffusion slows down front speeds in cellular flows.

In shear flows, flame speed scales linearly with flow amplitude.

Diffusion decreases the turbulent flame speed as flow amplitude grows.

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 $$ \bar H(A,d)\leq O(\sqrt {\mathrm {log}A}) \quad \text{for all $d>0$}. $$ Compared with the inviscid G-equation ($d=0$), the diffusion dramatically slows down the front propagation. For the shear flow, the limit \nit $\lim_{A\to +\infty}{\bar H(A,d)\over A} = \lambda (d) >0$ where $\lambda (d)$ 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.