# Curvature effect in shear flow: slowdown of turbulent flame speeds with Markstein number

## TL;DR

This paper rigorously proves that in shear flows, the turbulent flame speed decreases as the Markstein number increases, confirming the combustion folklore that curvature effects slow down flame propagation.

## Contribution

First theoretical proof showing the turbulent flame speed decreases with Markstein number in shear flows within the G-equation model.

## Key findings

- Turbulent flame speed decreases with increasing Markstein number.
- Derived a closed-form formula for the limiting solution as Markstein number approaches zero.
- Solved the selection problem for weak solutions in the inviscid limit.

## Abstract

It is well known in the combustion community that curvature effect in general slows down flame propagation speeds because it smooths out wrinkled flames. However, such a folklore has never been justified rigorously. In this paper, as the first theoretical result in this direction, we prove that the turbulent flame speed (an effective burning velocity) is decreasing with respect to the curvature diffusivity (Markstein number) for shear flows in the well known G-equation model. Our proof involves several novel and rather sophisticated inequalities arising from the nonlinear structure of the equation. On a related fundamental issue, we solve the selection problem of weak solutions or find the "physical fluctuations" when the Markstein number goes to zero and solutions approach those of the inviscid G-equation model. The limiting solution is given by a closed form analytical formula.