Combustion, Explosion, and Shock Waves, Vol. 34, No. 4, 1998 Mathematical Modeling of the Ignition of Tree Crowns A. M. Grishin 1 and V. A. Perminov I UDC 533.6.11.6 Translated from Fizika Goreniya i Vzryva, Vol. 34, No. 4, pp. 13-22, July-August 1998. Original article submitted October 8, 1995; revision submitted March 18, 1998. 9 Numerical calculations of the transition of a downstream (surface) forest fire into an upstream (crown) forest fire based on a general mathematical model of forest fires are presented. It is found that the ignition of the forest canopy is a gas-phase phenomenon. Critical conditions for the transition of a downstream into an upstream forest fire are determined. The numerical calculations are compared with experimental data. PHYSICAL-MATHEMATICAL MODEL Mathematical models of the transition of a downstream (surface) forest fire into an upstream (crown) forest fire have been published. Thus, Fomin [1] examined the heating of the forest canopy including the vaporization of moisture but neglect- ing the pyrolysis and ignition of combustible forest materials. With this approach it is impossible to de- termine whether ignition takes place. We have an- alyzed the pyrolysis of combustible forest material and the ignition of its volatile combustible products previously [2]. The source of the downstream forest fire was modeled as a zone with an elevated temper- ature from which the inert components of the gas phase had been blown out, but this is not realistic. In fact, gaseous and dispersed pyrolysis and combus- tion products from the combustible forest materials are released into the surroundings from the source of a downstream forest fire. In this paper, we describe the interaction of the source of a downstream forest fire with the forest canopy in more detail. The physical statement of the problem is illus- trated in Fig. 1. In this scheme, we consider a ref- erence volume of continuous, compact, multiphase, reactive medium in the atmospheric boundary layer. Pyrolysis products from the combustible forest ma- terial in the ground cover enter the space above it, ignite, and form a flame flare which heats the com- bustible forest material in the forest canopy. The lat- ter releases volatile combustible pyrolysis products which subsequently ignite, providing evidence of the transition of a downstream forest fire into an up- stream fire. 1Tomsk State University, Tomsk 634050. In this paper, we use a general mathematical model for forest fires [3] in which all the above pro- cesses can be taken into account. The origin of the downstream forest fire is a source of heat and mass in the form of a circle located on the ground cover. The source temperature is specified by a functional dependence on time and the horizontal coordinate r. All the remaining functions of the process are de- termined by the conservation laws for mass and mo- mentum. We assume, in addition, that: (1) the flow has a developed turbulent charac- ter and we neglect molecular transport compared to turbulent transport, (2) the density of the gas phase is independent of pressure because the flow velocity is low compared to the speed of sound, (3) the forest canopy is assumed to be an unde- formable medium. The forest canopy is modeled as a homoge- neous, two-temperature, multiphase porous reacting medium [3]. The following components are present in the gas-dispersed phase: 02, CO, CO2, H20, N2, C (soot particles), and ash particles. The gaseous com- bustible pyrolysis products consist mainly of carbon monoxide [4]. Thus, we may assume that oxygen and carbon monoxide are the components which limit the heat-release process. We shall assume that the molec- ular weight of the gaseous mixture is the same as that of-air, since the concentration of the gaseous combus- tion products is low compared to that of nitrogen, oxygen, and carbon monoxide. These assumptions 378 0010-5082/98/3404-0378 $20.00 ~) 1999 KluwerAcademic/PlenumPublishers