In order to study crown fires initiation and spread it is

developed a coupled boundary layer atmosphere – crown

forest fire behavior model that is based on laws of

conservation of mass, momentum and energy. It is useful to

apply this approach because the processes of evaporation,

pyrolysis and combustion in crown and processes of transfer

in atmosphere have different spatial scales. The conjugate

formulation of this problem let us to get solution distinctly in

different regions and take into account their influence. This

approach to solve like problems using conjugate formulation

was proposed by A.Grishin [1-4]. A coupled atmosphere fire

model, HIGRAD/FIRETEC, developed at the Los Alamos

National Laboratory [5,6], was employed to examine the

effects of the atmospheric potential temperature profiles on

the rate of spread of fire, in addition to the potential

temperature and velocity fields in the domain. The physical

multiphase model used in [8,9] may be considered as a

development and extension of the formulation proposed by

Grishin [2]. This paper presents a mathematical model of the

conjugate heat and mass transfer at crown forest fire. A

mathematical model of forest fires was obtained there based

on an analysis of known and original experimental data [2,7],

and using concepts and methods from reactive media

mechanics.

The basic assumptions adopted during the deduction of

equations, and boundary and initial conditions: 1) the forest

represents a multiphase, multistoried, spatially heterogeneous

medium; 2) in the fire zone the forest is a porous-dispersed,

two-temperature, single-velocity, reactive medium; 3) the

forest canopy is supposed to be non - deformed medium

(trunks, large branches, small twigs and needles), affects only

the magnitude of the force of resistance in the equation of

conservation of momentum in the gas phase, i.e., the medium

is assumed to be quasi-solid (almost non-deformable during

wind gusts); 4) let there be a so-called “ventilated” forest

massif, in which the volume of fractions of condensed forest

fuel phases, consisting of dry organic matter, water in liquid

state, solid pyrolysis products, and ash, can be neglected

compared to the volume fraction of gas phase (components of

air and gaseous pyrolysis products); 5) the flow has a

developed turbulent nature and molecular transfer is

neglected; 6) 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. Let the coordinate

reference point x1, x2 , x3= 0 be situated at the center of the

crown forest fire source at the height of the roughness level,

axis 0x1 directed parallel to the Earth’s surface to the right in

the direction of the unperturbed wind speed, axis 0x2 directed

perpendicular to 0x1 and axis 0x3 directed upward (Fig. 1).

Figure 1.

Using the results of [1-4] and known experimental data

[2,7] problem formulated above reduces to the solution of

systems of equations (1)-(7):

( ) , 1,2,3, 1,2,3;

v m j i

   



   

(1)

( ) | | ;

ii j d i i i

dv Pv v sc v v g mv

dt x x



   





      

(2)

);4(

)()(

TcUk

TTRqTvvc

xdt

svijp

















(3)

Numerical Solution of Problem for Forest Fire Initiation and Spread

*Note: Sub-titles are not captured in Xplore and should not be used

VALERIY PERMINOV, TATIANA BELKOVA

Department of Control and Diagnostics, Tomsk Polytechnic University

Tomsk, RUSSIA

Abstract: The theoretical study of the problems of forest fire initiation and spread were carried out in this paper.

Mathematical model of forest fire was based on an analysis of experimental data and using concept and methods

from reactive media mechanics. The research was based on numerical solution of three-dimensional Reynolds

equations for boundary layer of atmosphere and forest vegetation. The boundary-value problem is solved

numerically using the method of splitting according to physical processes.

Keywords: forest fire, mathematical model, control volume, discrete analogue, evaporation, pyrolysis,

combustion, crown fire

Received: September 9, 2021. Revised: June 12, 2022. Accepted: July 7, 2022. Published: August 3, 2022.

1. Introduction

2. Physical and Mathematical Model

EQUATIONS

DOI: 10.37394/232021.2022.2.20

Valeriy Perminov, Tatiana Belkova

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131

Volume 2, 2022

;5,1,)( 5













cmRcv

xdt



(4)

4 4 0,

;

RR S S g

ckcU k T k T

x k x

k k k









   







(5)

;)(

)4( 4

2233

SRs

iipi

TcUkRqRq

















(6)

,,,































(7)

),0,0(,),,(,,1 32

ggvvvv

RTpc e  









1 2 3 54 55

(1 ) ,

m R R R R R



     

51 3 5 52 1 5

53 54 4 1 55 3

3 3*

, (1 ) ,

0, , .

R R R R R R

R R R R R





     

   

Reaction rates of these various contributions (pyrolysis,

evaporation, combustion of coke and volatile combustible

products of pyrolysis) are approximated by Arrhenius laws

whose parameters (pre-exponential constant ki and activation

energy Ei) are evaluated using data for mathematical models

[2-4].

0.5

1 1 1 1 2 2 2 2

0.25

2.25

3 3 3 1 5 5 2

exp , exp ,

exp , exp .

R k R k T

RT RT

c M c M

R k s c R k M T

RT M M RT



   





   

   

   

   

  

   

  



The system of equations (1)–(7) must be solved taking into

account the initial and boundary conditions:

,,,0,0,0:0 321

ieieS

ccTTvvvt









(8)

1 1 1 1

: 0, 0, 0, 0,

0, 0 ;

xx x x x x

T c cU

x k x





   





    

  

(9)

2 20

2 2 2 2

: 0, 0, 0, 0,

0, 0 ;

xx x x x x

T c cU

x k x





   





    

   

(10)

2 2 2 2

: 0, 0, 0, 0,

0, 0 .

xx x x x x

T c cU

x k x





   





    

  

(11)

3 1 2

3 30 1 2

3 1 2

0 : 0, 0, 0, 0 ,

, , ,

0, , , ;

x v v U

x k x

v v T T x x

v T T x x





      

     

     

(12)

3 3 3 3

: 0, 0, 0, 0,

0, 0 .

xx x x x x

T c cU

x k x





   





    

  

(13)

Here and above

is the symbol of the total (substantial)

derivative; v is the coefficient of phase exchange;



- density

of gas – dispersed phase, t is time; vi - the velocity

components; T, TS, - temperatures of gas and solid phases, UR

- density of radiation energy, k - coefficient of radiation

attenuation, P - pressure; cp – constant pressure specific heat

of the gas phase, cpi,





i – specific heat, density and volume

of fraction of condensed phase (1 – dry organic substance, 2

– moisture, 3 – condensed pyrolysis products, 4 – mineral part

of forest fuel), Ri – the mass rates of chemical reactions, qi –

thermal effects of chemical reactions; kg , kS - radiation

absorption coefficients for gas and condensed phases;

the ambient temperature; c



- mass concentrations of



component of gas - dispersed medium, index



=1,2,...,5,

where 1 corresponds to the density of oxygen, 2 - to carbon

monoxide CO, 3 - to carbon dioxide and inert components of

air, 4 - to particles of black, 5 - to particles of smoke; R –

universal gas constant; M



, MC, and M molecular mass of 

-components of the gas phase, carbon and air mixture; g is the

gravity acceleration; cd is an empirical coefficient of the

resistance of the vegetation, s is the specific surface of the

forest fuel in the given forest stratum, g – mass fraction of gas

combustible products of pyrolysis,



4 and



5 – empirical

constants. To define source terms which characterize inflow

(outflow of mass) in a volume unit of the gas-dispersed phase

were used the following formulae for the rate of formulation

of the gas-dispersed mixture



, outflow of oxygen

changing carbon monoxide

, generation of black

and

smoke particles

. Coefficients of multiphase (gas and

solid phase) heat and mass exchange are defined

SSPV dSmCS /4,



 

. Here α = Nuλ/dS – coefficient of

heat exchange for sample of forest combustible material (for

example needle), Nu – Nusselt number for cylinder, λ –

coefficient of heat conductivity for pine needle; γ –

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Valeriy Perminov, Tatiana Belkova

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132

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parameter, which characterize relation between molecular

masses of ambient and inflow gases.

It is supposed that the optical properties of a medium are

independent of radiation wavelength (the assumption that the

medium is “grey”), and the so-called diffusion approximation

for radiation flux density were used for a mathematical

description of radiation transport during forest fires. The

components of the tensor of turbulent stresses, as well as the

turbulent fluxes of heat and mass are written in terms of the

gradients of the average flow properties [2]. 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, while

the appropriate components of the data base are used to

calculate the specific properties of the various forest strata and

the near-ground layer of atmosphere. This approach

substantially simplifies the technology of solving problems of

predicting the state of the medium in the fire zone numerically.

The thermodynamic, thermophysical and structural

characteristics correspond to the forest fuels in the canopy of

a different (for example pine forest [2]) type of forest. The

conditions of symmetry are used because of the patterns of

flow and distributions of all scalar functions are symmetrical

relatively to the plates Ox1. Ox3.

The boundary-value problem (1)–(7) we solve numerically

using the method of splitting according to physical processes

[2]. In the first stage, the hydrodynamic pattern of flow and

distribution of scalar functions was calculated. The system of

ordinary differential equations of chemical kinetics obtained

as a result of splitting [2] was then integrated. A discrete

analog was obtained by means of the control volume method

using the SIMPLE like algorithm [10]. The accuracy of the

program was checked by the method of inserted analytical

solutions. The time step was selected automatically. Fields of

temperature, velocity, component mass fractions, and volume

fractions of phases were obtained numerically.

Figures 2 a, b and c illustrate the time dependence of

dimensionless temperatures of gas and condensed phases (a),

concentrations of components (b) and relative volume

fractions of solid phases (c) at crown base of the forest

1 1 1 2 2 1

)(1 / ,2 / , 300 ), )(1 / ,2 / ,

e s s e e e e

a T T T T T T T K b C C C C C C        

10.23)

C

1 1 1 2 2 2 3 3 3

)(1 / ,2 / ,2 / )

e e e

        

     

At the moment of ignition, the gas combustible products

of pyrolysis burn away, and the concentration of oxygen is

rapidly reduced. The temperatures of both phases reach a

maximum value at the point of ignition. The ignition processes

is of a gas - phase nature, i.e. initially heating of solid and

gaseous phases occurs, moisture is evaporated. Then

decomposition process into condensed and volatile pyrolysis

products starts, the latter being ignited in the forest canopy.

Note also that the transfer of energy from the fire source takes

place due to radiation; the value of radiation heat flux density

is small compared to that of the convective heat flux.

Figure 2.

As a result of heating of forest fuel elements, moisture

evaporates, and pyrolysis occurs accompanied by the release

of gaseous products, which then ignite. The effect of the wind

on the zone of forest fire initiation is shown in Figures 3-5

present the space distribution of field of temperature for gas

phase for different instants of time (t=3.3 sec., 3.8 sec. and

4.8 sec.) when a wind velocity Ve= 7 m/s. We can note that

the isosurfaces are deformed by the action of wind. The

isosurfaces of the temperature of gas phase 1, 2, 3 и 4

correspond to the temperatures

= 1.2., 2, 3 and 4. In the

vicinity of the source of heat and mass release, heated air

masses and products of pyrolysis and combustion float up.

The wind field in the forest canopy interacts with the gas-jet

obstacle that forms from the surface forest fire source and

from the ignited forest canopy base. Recirculating flow forms

beyond the zone of heat and mass release, while on the

windward side the movement of the air flowing past the

ignition region accelerates. Under the influence of the wind

the tilt angle of the flame is increased. As a result, this part of

the forest canopy, which is shifted in the direction of the wind

from the center of the surface forest fire source, is subjected

to a more intensive warming up. The isosurfaces of the gas

phase temperature are deformed in the direction of the wind.

Figures 4 and 5 present the distribution of the velocity and

isosurfaces of the temperature at the different instants of time

when a wind velocity Ve= 7 m/s.

3. Numerical Method and Results

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Figure 3.

Figure 4.

Figure 5.

The effect of the wind on the forest fire spread is shown in

Figures 6(a, b, c) present the horizontal distribution of field

of temperature for gas phase in plane 0x1 x2 for different

instants of time (a - t=4.3 sec., b - 6 sec., c - 8 sec.); when a

wind velocity Ve= 5 m/s and moisture of forest combustible

materials – 0.6. We can note that the isotherms is moved in

the forest canopy and deformed by the action of wind. Also,

the fields of component concentrations are deformed. It is

concluded that the forest fire begins to spread and the fire

front is extent.

Figure 6. Fields of velocity and isotherms for:

1 2, 2 2.5, 3 3.,4 4.; / , 300 .

T T T T T T T T K         

Figures 7 and 8 (a, b, c) present the distribution of field of

concentration of oxygen and volatile combustible products of

pyrolysis concentration for the same instants of time (a -

t=4.3 sec., b - 6 sec., c - 8 sec.), when a wind velocity Ve= 5

m/s and moisture of forest combustible materials – 0.6 (

/ , 0.23

cс c с





). The lines of equal levels of

component concentrations are deformed. It is confirmed that

the forest fire begins to spread

Figure 7. The distribution of oxygen

;

1 0.9, 2 0.8, 3 0.5,4 0.4;

1 1 1 1

c c c c       

Figure 8. The distribution of

for t=4.3 sec (a), 6

sec (b), 8 sec (c) and Ve = 5 m/sec;

1,5.0

1,1.0

1 ccc

Mathematical model and the result of the calculation give an

opportunity to evaluate critical height of the forest canopy

and carry out preventive measures, which allows preventing

initiation of crown fires.

Mathematical model and the result of the calculation give

an opportunity to evaluate critical condition of the forest fire

initiation and spread which allows applying the given model

for preventing fires. The model overestimates the rate of the

crown forest fires spread. The results obtained agree with the

laws of physics and experimental data [2,7]. This paper

represents the attempt for application of three dimensional

4. Conclusion

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models for description of crown forest fires initiation and

spread.

[1] A.M.Grishin, V.M.Fomin, Conjugate and non-stationary problems of

reacting media mechanics, Novosibirsk: Nauka, Russia, 1984.

[2] A.M. Grishin A.M., `Mathematical modelling of forest fires and new

methods for fighting them.' (Publishing House of the Tomsk State

University: Tomsk, 1997.

[3] A.M. Grishin, Heat and Mass Transfer and Modeling and Prediction of

Environmental Catastrophes. Journal of Engineering Physics and

Thermophysics, 74, 2001. 895-903.

[4] A.M. Grishin, Conjugate Problems of Heat and Mass Exchange and the

Physicomathematical Theory of Forest Fires. Journal of Engineering

Physics and Thermophysics, 74, 2001.pp. 904-911.

[5] Philip Cunningham, Rodman R. Linn, Numerical simulations of grass

fires using a coupled atmosphere-fire model: Dynamics of fire spread,

Journal of Geophysical Research. 112, pp. 1-17. 2007.

[6] Jonah J. Colman, Rodman R. Linn, Separating combustion from

pyrolysis in HIGRAD/FIRETEC, Int. Journal of Wildland Fire, 16, 4,

2007, pp. 493 – 502.

[7] E.V. Konev, The physical foundation of vegetative materials

combustion. Novosibirsk: Nauka, Russia, 1977.

[8] D. Morvan,.J.L. Dupuy, Modeling of fire spread through a forest fuel

bed using a multiphase formulation. Combustion and Flame, 127,

2001, pp. 1981-1994.

[9] D. Morvan, J.L. Dupuy, Modeling the propagation of wildfire through

a Mediterranean shrub using a multiphase formulation. Combustion

and Flame, 138, 2004. pp. 199-210.

[10] S.V. Patankar, Numerical Heat Transfer and Fluid Flow. New York:

Hemisphere Publishing Corporation, 1981.

References

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(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the Creative

Commons Attribution License 4.0

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DOI: 10.37394/232021.2022.2.20

Valeriy Perminov, Tatiana Belkova

E-ISSN: 2732-9976

135

Volume 2, 2022