An equation for the curvature probability density function of instantaneous flame fronts

The flame front curvature distribution is critically important for premixed flames, particularly for flames with non-unity Lewis numbers, such as lean hydrogen flames, where differential diffusion effects are strongly correlated with flame curvature. Although the statistics and evolution of the curvature of premixed flames have garnered considerable interest in recent literature, a rigorous mathematical framework for quantitatively studying the evolution of the curvature probability density function (PDF) has yet to be developed. This paper presents a rigorous derivation of the equation governing the curvature PDF of instantaneous flame fronts. The derived theory is applied to analyze a premixed developing turbulent planar flame. It is revealed that the evolution of the curvature PDF is governed by (i) a drift in curvature space caused by the curvature evolution, and (ii) the non-uniform surface area evolution associated with various curvatures. A focus in the analysis is placed on the initial transition of the planar flame to a fully developed turbulent surface. During and after this transition, distinctly different effects of flow and flame propagation on the evolution of curvature and curvature PDF are identified. These findings enhance the understanding of curvature dynamics and offer new perspectives for modeling turbulent premixed combustion based on the curvature PDF. Furthermore, the derived PDF equation can equivalently be written for other scalar quantities defined on a moving surface, offering a general framework for analyzing scalar statistics on evolving surfaces, such as displacement speed, which plays a critical role in scalar mixing and turbulent combustion. Novelty and significance statement This work presents a novel equation for the evolution of the flame curvature probability density function (PDF) and provides the first quantitative analysis of the mechanisms governing the curvature PDF evolution of instantaneous flame fronts. The significance lies in two main aspects. First, the study advances the understanding of curvature dynamics and offers new insights for modeling curvature effects in turbulent premixed combustion. Second, the derived PDF equation can be equivalently formulated for other scalar quantities defined on a moving surface. Hence, the proposed framework holds strong potential for analyzing scalar statistics on evolving surfaces, such as displacement speed, thereby improving the understanding of turbulent combustion and scalar mixing in turbulent flows.