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

TL;DR

The paper addresses how the turbulent flame speed $s_{\rm T}(p,A)$ in the inviscid G-equation grows with flow intensity $A$, testing Yakhot's $O\left(\frac{A}{\sqrt{\log A}}\right)$ prediction. It develops a rigorous 2D analysis based on the control representation of the Hamilton-Jacobi equation and a detailed streamline-cell structure, proving there is no intermediate growth between $O\left(\frac{A}{\log A}\right)$ and $O(A)$ for Lipschitz incompressible periodic flows with bounded swirls, while allowing degeneracies in critical points. The work also discusses unsteady flows and lower-regularity regimes, showing $O\left(\frac{A}{\log A}\right)$ or $O(A)$ growth can occur, and that half-log-Lipschitz regularity could yield $O\left(\frac{A}{\sqrt{\log A}}\right)$ under certain constructions. In 3D, especially for ABC and Kolmogorov-type flows, linear growth in $A$ is observed in relevant regimes, while unsteady cellular flows can maintain $O(A/\log A)$ bounds. Together, the results delineate how regularity, chaos, and multi-scale cell structures influence front-speed scaling in turbulent combustion within the G-equation framework.

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.

Figures (7)

• Figure 1: G-equation model.
• Figure 4: New coordinate system near $x$.
• Figure 5: Possible relations between $R_{\xi_y}$ (region enclosed by the orbit $\xi_y$ which is partially shown in blue color) and $U^\pm_x(H(y))$.
• Figure 6: Picture of $R_\xi$ and a cell $S$
• Figure 7: Construction of a cell close to an orbit that is asymptotic to $\Gamma$

Theorems & Definitions (30)

• Theorem 1.1
• Remark 1.1
• Remark 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• Definition 2.1
• Lemma 2.4