Taylor dispersion, Turing instability and a lubrication theory for flames

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.

Planned Impact
The project main beneficiaries are combustion scientists, chemical and mechanical engineers, and applied mathematicians interested in instabilities and mixing in reaction-diffusion systems.
The models of the lubrication theory for flames which will be developed, and the results of the novel investigations on flame-flow interactions and flame instabilities to be undertaken will lead to a significant increase in our ability to correctly model flame behaviour in applications involving premixed combustion. This can lead to, for example, more reliable computational approaches based on rational subgrid-models for Large Eddy Simulation in turbulent combustion. More generally, the acquired knowledge and know-how will provide new modelling tools to engine designers, and engineers and scientists working in the automotive, aerospace defense and other industries. These tools should be helpful in tackling industrial and environmental challenges related to energy production, both associated with longstanding scientific problems (turbulent combustion, efficiency of burners and engines) and more contemporary problems (clean combustion and pollution concerns, alternative fuels such as bio-fuels, developing field of combustion-based micropower generation, etc.). Beyond the academic and industrial sectors involving combustion, the findings of the project could also have an impact in other fields of chemical engineering, mathematical biology, and environmental science applications involving reaction-diffusion propagating fronts.

In order to ensure maximum impact, we will ensure that our findings are communicated to a wide audience (journals, conferences, etc.) and an interdisciplinary workshop will be organised in Manchester to ring together various experts; in addition we will develop collaborations with experimentalists (USA, Europe).