ASCI/Alliances Center for Astrophysical
Thermonuclear Flashes
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.
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.
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Comparison of the three models
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