Numerical solution of differential and integral equations
Considering that natural investigations of large forest fire initiation problems are merely impossible, methods of mathematical modeling are urgent. The forest canopy is considered as a homogeneous two temperatures, reacting, non - deformed medium. Temperatures of condensed (solid) and gaseous phases are separated out. The first includes a dry organic substance, moisture (water in the liquid-drop state), condensed pyrolysis and combustion products (coke, ash), and mineral part of forest fuels. In the gaseous phase we separate out only the components necessary to describe reactions of combustion (oxygen, combustible products of pyrolysis of forest fuels and the rest inert components). The solid phase constituting forest fuels has no intrinsic velocity, and its volumetric fractions, as compared to the gaseous phase, can be neglected in appropriate equations. It is considered that 1) the flow has a developed turbulent nature, molecular transfer being neglected, 2) gaseous phase density doesn't depend on the pressure because of the low velocities of the flow in comparison with the velocity of the sound, 3) forest canopy is supposed to be non-deformed porous medium. To describe the transfer of energy by radiation diffusion approximation is used, while to describe convective transfer controlled by the wind and gravity, we use Reynolds equations. To close the this system of equations, the components of the tensor of turbulent stresses, and the turbulent heat and mass fluxes are determined using the local-equilibrium model of turbulence (Grishin, 1997). It should be noted that this system of equations describes processes of transfer within the entire region of the forest massif, which includes the space between the underlying surface and the base of the forest canopy, the forest canopy and the space above it. The research is done by means of mathematical modeling of physical processes. To obtain discrete analogies a method of control volume is used. Calculation method and program have been check. The boundary-value problem is solved numerically using the method of splitting according to physical processes. In the first stage, the hydrodynamic pattern of flow and distribution of scalar functions are calculated. The system of ordinary differential equations of chemical kinetics obtained as a result of splitting is then integrated. A discrete analog for equations is obtained by means of the control volume method using the SIMPLE - like algorithm. As a result of mathematical modeling the fields of temperatures, mass concentrations of components of gaseous phase, volume fractions of components of solid phase, as well as vectorial fields of velocity at different instants of time will be obtained. It allows to investigate dynamics of forest fire spread under influence of various external conditions: a) meteorology conditions (air temperature, wind velocity etc.), b) terrain, c) type (various kinds of forest combustible materials) and their state (load, moisture etc.). A great deal of final and intermediate gaseous and dispersed combustion products of forest fuels is known to be exhausted into the atmosphere during forest fires: carbon monoxide, carbon dioxide, soot, smoke and etc. The knowledge of these kinds of ejections enables a full estimate of the damage from forest fires to be made. In this paper attention is given to questions of description of the initial stage in the development of a mass forest fires initiated by high altitude radiant energy source.
Note. Abstracts are published in author's edition
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