Predictions of unsteady flame spread and burning processes by the vorticity‐stream function formulation of the compressible navier‐stokes equations

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Abstract
A two‐dimensional mathematical model of flame spread and solid burning is presented. For the gas phase, it consists of variable density, fully elliptic Navier‐Stokes momentum, energy and chemical species mass equations. Combustion processes are treated according to a one‐step, finite‐rate, reaction. The solid phase model describes a porous cellulosic fuel for a range of thicknesses from the thermally thin to the thermally thick limit. Conductive and convective heat transfer takes place as the solid degrades, by two first order Arrhenius reactions, to volatiles and chars. Variations of solid phase densities account for fuel burn‐out. Effects of gas phase and surface radiation are also included. A steady formulation of gas phase equations with respect to the unsteady solid phase mathematical model is proposed, gas phase characteristic times being much shorter than those of the solid phase. The non‐constant density Navier‐Stokes equations are formulated in terms of vorticity and stream function, avoiding the pressure‐velocity coupling and, at the same time, the adoption of a sample‐fixed coordinate system allows unsteady flame spread processes to be simulated. The solution is computed numerically by means of an iterative, operator‐splitting method based on implicit finite‐difference approximations. Numerical simulations of the dynamics of flame spread over cellulosic solids are presented and extinction limits as a consequence of reduced rates of fuel generation are determined.