An analytical model of the initial expansion of a fireball is presented.The model is based on an exact solution of the low Mach number combustion equations in the form initially proposed by the authors. The equations consist of the conservation of mass, momentum, and energy with an isobaric equation of state. The heat release rate is a prescribed spherically symmetric function characterized by a flame expansion velocity, a flame brush thickness that increases with time, and a heat release rate per unit surface area. The introduction of a prescribed heat release rate obviates the need for an explicit turbulence model. Thus, the inviscid forms of the conservation equations can be used in the analysis. The velocity field is decomposed into a spherically symmetric expansion field and a solenoidal component determined by the buoyancy induced vorticity field. The expansion field together with the induced pressure rise and temperature fields are spherically symmetric. However, the buoyancy forces induce vorticity where the temperature changes rapidly and break the spherical symmetry of the velocity field. The solution remains internally consistent because the vorticity field is confined to a narrow region. Thus, even though the vorticity is intense, the induced solenoidal velocity is initially much smaller than the expansion field. The solutions obtained are valid until either all the fuel is exhausted or the solenoidal velocity becomes comparable with the expansion field, whichever comes first.
International Seminar on Fire and Explosion Hazards