CAREER: Analysis of G-equations in the modeling of turbulent flame speed and comparison with other math models

本项目旨在对湍流火焰速度模拟中的G方程(非胁迫性哈密顿-雅可比方程)、应变G-方程(非胁迫性非凸哈密顿-雅可比方程)和曲率G-方程(平均曲率型方程)的一些关键问题进行解析研究。粗略地说，湍流火焰速度是在流动(湍流)作用下火焰传播的平均速度。该项目的中心目标是了解湍流火焰速度与湍流强度的关系。一个特别重要的问题是弄清楚流动拉伸效应如何导致火焰速度的强烈弯曲和熄灭。该项目的另一个目标是比较各种G方程和数学文献中的其他模型预测的湍流火焰速度，例如Majda-Souganidis模型和研究得很好的标量反应-扩散-平流方程模型。所提出的一些问题与Aubry-Mather理论和弱Kam理论有着重要的联系，这两个理论对于理解不可积动力系统具有重要意义。到目前为止，很少有关于二维以上流动中湍流火焰速度的严格分析结果。这个项目的一个长期目标是了解三维流动(例如Arnold-Beltrami-柴尔德里斯流动)中的混沌结构如何影响湍流火焰速度。湍流燃烧在能源生产和发动机设计等重大工业问题中起着重要的作用。所谓的G方程及其变种是湍流燃烧中常用的模型，因为它们具有简单、高效和在拟合实验中的稳健性。该项目的所有研究部分都与湍流燃烧中最重要的悬而未决的问题之一密切相关；即预测湍流火焰速度，特别是了解它如何取决于湍流强度(例如，考虑野火传播速度与风强度之间的关系)。该项目的一个非常重要的部分是它的教育部分。除了指导研究生，首席研究员还计划参加加州大学欧文分校几个久负盛名的教育项目，从K-12到本科：加州大学数学竞赛，6-8年级的数学竞赛；欧文地区数学建模者(IAMM)，这是一个培训计划，为当地高中生准备全国高中数学建模竞赛(HiMCM)；SURF计划，这是本科生的暑期研究计划；新生研讨会计划，为加州大学欧文分校的本科生提供研究入门讲座。该项目将对燃烧科学和工程产生广泛影响，并对清洁能源以及进一步教育下一代STEM科学家产生影响。

This project aims to study analytically some key problems regarding the G-equation (a noncoercive Hamilton-Jacobi equation), the strain G-equation (a noncoercive and nonconvex Hamilton-Jacobi equation), and the curvature G-equation (a mean-curvature- type equation) in the modeling of turbulent flame speed. Roughly speaking, turbulent flame speed is the averaged flame propagation speed under the effect of the flow (turbulence). The central goal of the project is to understand the dependence of the turbulent flame speed on the turbulence intensity. A particularly important problem is to figure out how the flow-stretching effect contributes to the strong bending and quenching of flame speeds. Another goal of the project is to compare turbulent flame speeds predicted by various G-equations and other models in the mathematics literature, such as the Majda-Souganidis model and the well-studied scalar reaction-diffusion-advection equation model. Some proposed problems have significant connections with the Aubry-Mather theory and weak KAM theory, which are important for understanding nonintegrable dynamical systems. So far there are very few rigorous analytical results on turbulent flame speeds in flows of more than two dimensions. A long-term goal of this project is to understand how chaotic structures in three-dimensional flows (e.g., the Arnold-Beltrami-Childress flow) affect turbulent flame speeds. Turbulent combustion plays the main role in important industrial issues such as energy production and engine design. The so-called G-equation and its variants are popular models in turbulent combustion due to their simplicity, efficiency, and robustness in fitting experiments. All the research components of the project are closely related to one of the most important unsolved problems in turbulent combustion; namely, to predict the turbulent flame speed and, in particular, to understand how it depends on the turbulence intensity (e.g., think of the relation between the spreading velocity of a wild fire and the strength of the wind). A very important part of the project is its educational component. Besides supervision of graduate students, the principal investigator also plans to participate in several well-established educational programs at UC-Irvine, ranging from the K-12 to the undergraduate levels: California MathCounts, a 6-8th grade math competition; the Irvine Area Math Modelers (IAMM), which is a training program that prepares local high school students for the National High School Mathematical Contest in Modeling (HiMCM)); the SURF program, which is an undergraduate summer research program; the Freshman Seminar Program, which provides introductory lectures on research to undergraduate students at UC-Irvine. The project will generate broad impact on combustion science and engineering, with implications for clean energy, as well as further the education of the next generation of STEM scientists.