Taylor dispersion, Turing instability and a lubrication theory for flames

在反应-扩散系统中，扰动前沿是普遍存在的，它们的不稳定性及其与流动的相互作用是具有重要实际意义的现象。例如，在预混燃烧，这是我们的主要重点，火焰不稳定性和火焰流动相互作用提出了重要的模拟燃烧装置的设计者的挑战;特别是，火焰不稳定性，特别是当火焰不稳定性与噪声相结合时，是燃烧室失效的一个主要原因，而且火焰-气流相互作用在发动机性能中起着重要的作用。尽管进行了广泛的研究，在我们对这些现象的理解上，特别是对厚火焰的理解上，仍然存在巨大的差距。在文献中可用的理论通常限于相对于典型问题的长度尺度（例如燃烧室的尺寸或通常是湍流的流场的尺度）而言较薄的火焰。这样的薄火焰理论是不令人满意的许多问题，这是很重要的应用，如火焰的传播局限于几何形状与小间隙和高纵横比，小尺度流动的有效传播速度和火焰建模的影响，在发展领域的燃烧为基础的微型发电。为了扩展我们解决这些问题和其他问题的能力，我们打算在预混燃烧中发展一种润滑理论，并将其应用于更好地理解火焰不稳定性和火焰流动相互作用。该理论适用于火焰厚度可以被认为小于或相当于火焰传播横向方向上的典型长度尺度的情况。我们的方法基于PI开创的“厚火焰渐近极限”方法，是原创的，因为它旨在揭示和利用G.I. Taylor，A. Turing和G. Damköhler分别与通常所知的Taylor色散、Turing不稳定性和Damköhler湍流燃烧假设有关。该项目的一个主要部分是解决湍流燃烧的挑战性问题（例如所谓的弯曲效应的机制），当这些问题被公式化为层流火焰时。另一个主要部分是正在发展的润滑理论的框架内的火焰不稳定性的调查。我们将开始推导这种理论在厚火焰渐近极限的数学模型，特别是泰勒色散，变密度和化学反应。由于泰勒色散修改了质量和热量的有效扩散系数，这反过来又控制了（图灵式）热扩散火焰不稳定性，我们将研究泰勒色散对热扩散不稳定性的影响，使用解析推导和数值模拟。强迫对流下的各种火焰不稳定性之间的耦合，然后将被解决，特别是采用Hele-Shaw细胞配置在最近和目前活跃的火焰实验中使用。最后，火焰流的相互作用将被解决的单向多尺度流和二维涡流，并试图综合的结果，以获得改进的公式的有效传播速度。最后一步是在新的方向上扩展调查，特别是在燃烧外遇到的反应扩散前沿，例如通过检查泰勒色散触发此类前沿的图灵不稳定性的能力。为了确保产生最大的影响，我们的研究结果将通过燃烧期刊和应用数学期刊上的出版物以及将在第三年组织的科学会议和跨学科研讨会向广大受众传达。项目。

Propagating fronts are ubiquitous in reaction-diffusion systems and their instabilities and interactions with flows are phenomena of high practical importance. For example, in premixed combustion, which is our main focus, flame instabilities and flame-flow interactions pose important modelling challenges for the designers of combustion devices; in particular, flame instabilities, especially when coupled with acoustics are a major reason for combustor failure, and flame-flow interactions play an important role in engine performance.Despite extensive investigations, there are still huge gaps in our understanding of these phenomena especially for thick flames. Theories available in the literature are generally restricted to flames that are thin relative to the length scales typical of the problems such as the size of the combustion chamber or the scales of the flow field which is often turbulent. Such thin-flame theories are not satisfactory for many problems which are important in applications such as the propagation of flames confined to geometries with small gaps and high aspect ratios, the effect of small scale flows on the effective propagation speed and flame modelling in the developing field of combustion-based micropower generation. To extend our ability to address such problems and others, we intend to develop a lubrication theory in premixed combustion and apply it to better understand flame instabilities and flame-flow interactions. The theory is applicable in situations where the flame thickness may be considered smaller than, or comparable with, a typical length scale in a direction transverse to flame propagation. Our approach, based on a methodology corresponding to the 'thick flame asymptotic limit' pioneered by the PI, is original as it aims to unveil and exploit the links between three seminal contributions by G.I. Taylor, A. Turing and G. Damköhler associated respectively with what are commonly known as Taylor dispersion, Turing instability, and Damköhler's hypotheses of turbulent combustion.One major part of the project is to tackle challenging questions from turbulent combustion (such as the mechanism of the so-called bending effect), when these questions are formulated for laminar flames. Another major part is the investigation of flame instabilities in the framework of the lubrication theory being developed. We shall begin by deriving the mathematical models of such theory in the thick flame asymptotic limit, accounting in particular for Taylor dispersion, variable density and chemical reactions. Since Taylor dispersion modifies the effective diffusion coefficients of mass and heat which in turn control a (Turing-like) thermo-diffusive flame instability, we shall investigate the effect of Taylor-dispersion on the thermo-diffusive instability using analytical derivations and numerical simulations. The coupling between various flame instabilities under forced convection will then be addressed, adopting specifically the Hele-Shaw cell configuration used in recent and currently active experiments on flames. Finally the flame-flow interactions will be addressed for unidirectional multi-scale flows and two-dimensional vortical flows, and attempts will be made to synthesize the findings to derive improved formulas for the effective propagation speed. The final step is to extend the investigations in new directions, in particular to reaction-diffusion fronts encountered outside combustion, e.g. by examining the ability of Taylor dispersion to trigger Turing-like instabilities for such fronts. In order to ensure maximum impact, our findings will be communicated to a wide audience via publications in both combustion journals and applied mathematics journals with broader focus, as well as via scientific meetings and an inter-disciplinary workshop to be organised in the third year of the project.