Does Yakhot's growth law for turbulent burning velocity hold?

TL;DR

This paper rigorously investigates Yakhot's proposed growth law for turbulent flame speed, proving the absence of intermediate growth laws in certain flow regimes and discussing related flow phenomena.

Contribution

It provides the first rigorous mathematical validation regarding the growth law of turbulent flame speed, specifically ruling out intermediate growth rates in 2D flows.

Findings

No intermediate growth law between O(A/log A) and O(A) for certain flows.

Proves Yakhot's growth law does not hold in the specified flow regimes.

Discusses flows with lower regularity and chaotic behavior.

Abstract

Using formal renormalization theory, Yakhot derived in ([32], 1988) an $O\left(\frac{A}{\sqrt{\log A}}\right)$ growth law of the turbulent flame speed with respect to large flow intensity $A$ based on the inviscid G-equation. Although this growth law is widely cited in combustion literature, there has been no rigorous mathematical discussion to date about its validity. As a first step towards unveiling the mystery, we prove that there is no intermediate growth law between $O\left(\frac{A}{\log A}\right)$ and $O(A)$ for two dimensional incompressible Lipschitz continuous periodic flows with bounded swirl sizes. In particular, we do not assume the non-degeneracy of critical points. Additionally, other examples of flows with lower regularity, Lagrangian chaos, and related phenomena are also discussed.