Taylor dispersion, Turing instability and a lubrication theory for flames

传播锋在反应扩散系统中普遍存在，其不稳定性和与流动的相互作用是具有高度实际意义的现象。例如，在预混燃烧中，这是我们的主要关注点，火焰不稳定性和火焰流相互作用对燃烧装置的设计师提出了重要的建模挑战；特别是，火焰不稳定性，特别是当与声学相结合时，是燃烧室失效的主要原因，火焰流相互作用在发动机性能中起着重要作用。尽管进行了广泛的调查，但我们对这些现象的理解仍然存在巨大的差距，特别是对厚火焰的理解。文献中可用的理论通常仅限于相对于典型问题（如燃烧室的大小或流场的大小）的长度尺度较薄的火焰，通常是湍流的。这种薄火焰理论不能满足许多重要的应用问题，如火焰在小间隙和高宽高比几何形状中的传播，小尺度流动对有效传播速度的影响以及燃烧微发电领域中火焰的建模。为了扩展我们解决此类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相互作用。该理论适用于火焰厚度可能被认为小于或与火焰传播横向方向的典型长度尺度相当的情况。我们的方法，基于一种与PI首创的“厚焰渐近极限”相对应的方法，是原创的，因为它旨在揭示和利用G.I.泰勒，a .图灵和G. Damköhler三个开创性贡献之间的联系，这些贡献分别与通常称为泰勒色散，图灵不稳定性和Damköhler的湍流燃烧假设有关。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），当这些问题被制定为层流火焰时。另一个主要部分是在正在发展的润滑理论框架内对火焰不稳定性的研究。我们将首先推导出这种理论在厚焰渐近极限下的数学模型，特别是考虑泰勒色散、变密度和化学反应。由于泰勒弥散改变了质量和热量的有效扩散系数，从而控制了（类似图灵的）热扩散火焰不稳定性，我们将使用解析推导和数值模拟来研究泰勒弥散对热扩散不稳定性的影响。然后将讨论强迫对流下各种火焰不稳定性之间的耦合，具体采用最近和目前正在进行的火焰实验中使用的Hele-Shaw单元配置。最后将讨论单向多尺度流动和二维涡旋流动的火焰-流动相互作用，并尝试综合研究结果，推导出有效传播速度的改进公式。最后一步是将研究扩展到新的方向，特别是在燃烧外遇到的反应扩散前沿，例如通过检查泰勒分散触发此类前沿的图灵不稳定性的能力。为了确保最大的影响，我们的研究结果将通过在燃烧期刊和应用数学期刊上发表的出版物，以及在项目的第三年组织的科学会议和跨学科研讨会，传达给广泛的受众。

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.