ASCI/Alliances Center for Astrophysical Thermonuclear Flashes An Interface Propagation Model for Reaction-Diffusion Advection Adam Oberman An Interface Propagation Model for Reaction-Diffusion Advection Adam Oberman Discussion of current work Existing models represent flames as infinitely thin interfaces, which propagate according to a kinematical rule. The interface propagates in the normal direction according to Huygen’s Principle, (Huygen’s principle) which is often solved numerically as a partial differential equation, the G (geometric) equation (G-equation) Our current work is a generalization of the G-equation, which takes into account a small but finite flame thickness. (G-equation) where K is the mean curvature of the front, and D is the diffusivity of the progress variable. The new equation is derived as an asymptotic limit of the reaction diffusion equation, It reduces to the G-equation in the limit of infinitely thin flames. The derivation indicates that the model will break down when the scales of variation of the velocity field are on the order of the flame thickness. This is reasonable, since then the thin flame assumption breaks down, and we are in a different regime. Numerical tests were performed which compared the three models. The results show that the model is a significant improvement: in the limit of thin flames, the models agree, when the flame thickness approached the small scales of the velocity field, the G-equation breaks down, but the G-K equation agrees for a wide range parameters with the full reaction diffusion equations. Discussion of current work Existing models represent flames as infinitely thin interfaces, which propagate according to a kinematical rule. The interface propagates in the normal direction according to Huygen’s Principle, (Huygen’s principle) which is often solved numerically as a partial differential equation, the G (geometric) equation (G-equation) Our current work is a generalization of the G-equation, which takes into account a small but finite flame thickness. (G-equation) where K is the mean curvature of the front, and D is the diffusivity of the progress variable. The new equation is derived as an asymptotic limit of the reaction diffusion equation, It reduces to the G-equation in the limit of infinitely thin flames. The derivation indicates that the model will break down when the scales of variation of the velocity field are on the order of the flame thickness. This is reasonable, since then the thin flame assumption breaks down, and we are in a different regime. Numerical tests were performed which compared the three models. The results show that the model is a significant improvement: in the limit of thin flames, the models agree, when the flame thickness approached the small scales of the velocity field, the G-equation breaks down, but the G-K equation agrees for a wide range parameters with the full reaction diffusion equations. Introduction The problem of turbulent combustion is a big challenge. Flame chemistry, fluid turbulence, and the interaction of flames and the turbulence are all very hard problems, both scientifically and computationally. At any level, model simplifications must be made. The approach we have taken in previous work is to simplify the chemistry as much as possible: we represent the flame by a reacting and diffusing progress variable, which is advected by the fluid. We assume that we are given a proscribed velocity field, and make it our mission to determine important flame features (flame velocity, mass consumption) from the dynamics of the system. Previous work has been successful in determining analytically flame speeds as a function of fluid properties in the case of structurally simple, non-turbulent velocity fields. Even at this level of simplification, the wide range of spatial and temporal scales make it too costly to resolve the rich inner structure of the flame in a fully turbulent velocity field. Introduction The problem of turbulent combustion is a big challenge. Flame chemistry, fluid turbulence, and the interaction of flames and the turbulence are all very hard problems, both scientifically and computationally. At any level, model simplifications must be made. The approach we have taken in previous work is to simplify the chemistry as much as possible: we represent the flame by a reacting and diffusing progress variable, which is advected by the fluid. We assume that we are given a proscribed velocity field, and make it our mission to determine important flame features (flame velocity, mass consumption) from the dynamics of the system. Previous work has been successful in determining analytically flame speeds as a function of fluid properties in the case of structurally simple, non-turbulent velocity fields. Even at this level of simplification, the wide range of spatial and temporal scales make it too costly to resolve the rich inner structure of the flame in a fully turbulent velocity field. Discussion of Future Work A closer look at the relevance of spatial scales and diffusion is underway. Joint work with Alan Kerstein has produced a phase diagram which classifies burning regimes and classical predictions of flame speeds according to spatial scales and the Schmidt number. One dimensional turbulence models have been used as numerical validation of the results. A future project is to examine flame-fluid coupling. There are local effects are a result of vorticity production due to density differences across a flame front. Global coupling arises in the presence of gravity through buoyancy effects. Discussion of Future Work A closer look at the relevance of spatial scales and diffusion is underway. Joint work with Alan Kerstein has produced a phase diagram which classifies burning regimes and classical predictions of flame speeds according to spatial scales and the Schmidt number. One dimensional turbulence models have been used as numerical validation of the results. A future project is to examine flame-fluid coupling. There are local effects are a result of vorticity production due to density differences across a flame front. Global coupling arises in the presence of gravity through buoyancy effects.