In combustion and flame propagation studies, a typical flame model view the burning front as an infinitely thin sharp discontinuity, which transforms reactants into products. The interrelation between the motion of this front and the the underlying fluid mechanics is complex; the burning of the flame releases heat along the flame front, which causes as exothermic (heat release) expansion; at the same time, vorticity is generated along the flame front due to baroclinic terms arising from the cross-product of the pressure gradient with the density gradient. These additional velocity fields then update the hydrodynamics, which then require appropriate Navier-Stokes solvers, and this in turn moves the flame front.

A level set view of these combustion phenomena views the flame front as the zero level set of a level set function. The jump conditions along the flame front, which result from exothermic expansion, create an additional velocity field, as does the baroclinic term. The flame speed depends on a local burning velocity as well as the mitigating effects of curvature (this is a flame model similar to that suggested by Markstein).



Annotated References:

• Ref. 1 links paper to apply level set methods to combustion; it couples projection methods for fluid flows to the problem of tracking a cold flame in a swirling fluid. This is paper couples projection methods and Navier-Stokes solvers to level set interface techniques; in these calculations, there is no back-coupling between the flame motion and the flow dynamics.
  
  
• Ref. 2 build a more complete flame model, and studies the turbulizing effects of a flame motion in a hydrodynamic field. The model studies exothermic heat release, flame-induced vorticity generation, solid wall boundary conditions, upstream fluid turbulence, curvature-modified flame speeds, and the combined effects on flame brush and stability.

Examples:
The figure on the left is a flame attached to a flame holder in a turbulent field; many positions in time are superimposed on top of each other. The figure on the right is the same, however, the self-turbulizing effects of the flame due to vorticity, exothemicity, and flame stretch are present, causing a spreading in the flame brush. See Reference 2 below.



New Book and Resource on Level Set and Fast Marching Methods



References:

1. Projection Methods Coupled to Level Set Interface Techniques : Zhu, J., and Sethian, J.A., Journal of Computational Physics, 102, pp. 128-138, 1992.

   Abstract

   In this paper, we merge modern techniques for computing the solution to the viscous Navier-Stokes equations with modern techniques for computing the motion of interfaces propagating with curvature-dependent speeds. The resulting algorithm tracks the motion of an evolving interface in a complex flow field, and easily handles complex changes in the front, including the development of spikes and cusps, topological changes and breaking/merging. As examples, we apply the resulting algorithm to interface boundaries in a driven cavity and in a shear layer, and cold flame propagation in a hydrodynamic field.

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2. Dynamical Study of a Premixed V flame : Rhee, C., Talbot, L., and Sethian, J.A., Journal of Fluid Mechanics, 300, pp. 87-115, 1995.

   Abstract

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