racsam - math.unice.frjsimon/pubs/simon-i1.pdf · vol. 95 (1), 2001, pp. 13–27 matematica...

RACSAMRev. R. Acad. Cien. Serie A. Mat.VOL. 95 (1), 2001, pp. 13–27Matematica Aplicada / Applied MathematicsComunicacion Preliminar / Preliminary Communication

Modelling of convective phenomena in forest fire

M. I. Asensio, L. Ferragut and J. Simon

Abstract. We present a model coupling the fire propagation equations in a bidimensional domainrepresenting the surface, and the air movement equations in a three dimensional domain representing anair layer. As the air layer thickness is small compared with its length, an asymptotic analysis gives a threedimensional convective model governed by a bidimensional equation verified by a stream function. Wealso present the numerical simulations of these equations.

Modelizaci on de la convecci on en incendios forestales

Resumen. Presentamos un modelo que acopla las ecuaciones de propagacion de incendios en un do-minio bidimensional que representa la superficie y las ecuaciones de movimiento del aire en un dominiotridimensional que representan una capa de aire. Como el grosor de la capa de aire es pequeno com-parado con su longitud, un analisis asintotico da un modelo convectivo tridimensional gobernado por unaecuacion bidimensional sobre una funcion de corriente. Ademas se presentan simulaciones numericas deestas ecuaciones.

1. Introduction

We present a model coupling the fire propagation equations in a bidimensional domain representing thesurface, and the air movement equations in a three dimensional domain representing an air layer. As theair layer thickness is small compared with its length, an asymptotic analysis gives a three dimensionalconvective model governed by a bidimensional equation verified by a stream functionk. We also presentthe numerical simulations of these equations.

More precisely, we compute explicitly the three dimensional air velocity as a function of the verticalcoordinatez, the stream functionk(t, x), the surface temperatureq(t, x) and the surface heighth(x), wherex = (x1, x2) are the horizontal coordinates.

The combustion model presented here is based on the model proposed inASENSIO andFERRAGUT,[1] incorporating solid phase and gaseous phase. The convection model is an adaptation to the forest firesconvective phenomena of the shallow water models proposed inBRESCHet al., [3].

2. Combustion equations

Lets d ⊂ IR2 be a bidimensional bounded domain, representing the projection of the three dimensionalgeographical surface,x be any of its points andt be the time. We use small letters for quantities concerning

Presentado por ***Recibido: . Revisado. Aceptado: ***.Palabras clave / Keywords: convection, forest fire modellingMathematics Subject Classifications: 76D05, 80A25c© 2001 Real Academia de Ciencias, Espana.

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the bidimensional problem, and capital letters for the tridimensional problem.Fire development is described by the amount of solid fuel, gaseous fuel and oxygen, and also by the

temperature. We now give the corresponding nondimensional equations.The solid fuel amountys is driven by

∂tys = −βsyse−γs

q (1)

whereq is the temperature and∂t = ∂/∂t. The right-hand side represents the solid fuel transform intogaseous fuel due to the pyrolysis.

The gaseous fuel amountyg is driven by

∂tyg + v .∇xyg − κg∆xyg = −αgygyoe−γg

q + βsyse−γs

q (2)

wherev is the wind. The first term of the right-hand side represents the amount of gaseous fuel burned withthe oxygen. The oxygen amountyo is driven by

∂tyo + v .∇xyo − κo∆xyo = −αoygyoe−γg

q . (3)

Finally, the temperatureq is driven by the equation

∂tq + v .∇xq −∇x . (κqq3∇xq) = µygyoe

−γg

q , (4)

where radiation is modelled as inWEBER, [7].We complete these equations with the following initial conditions

ys|t=0= y0

s , yg|t=0= 0, yo|t=0

= χ, q|t=0= q0, (5)

wherey0s is the initial solid fuel amount,q0 the initial temperature, that we suppose known, andχ is the

initial oxygen amount, that is constant; and with the boundary conditions

yg|∂d= 0, yo|∂d

= χ, q|∂d= q0|∂d

. (6)

Greek letters represent several strictly positives coefficients, non depending ont or x.

Remark.There is no boundary condition onys because (1) is an ordinary differential equation int, for fixedx andq.

3. Convection equations

Lets consider now the three dimensional domain

D = {(x, z) : x ∈ d, h(x) < z < δ}

representing the air layer. We assume that the ratio of the height to the width,δ, is very small, and the heightof the surface at pointx, h(x), is smaller thanδ. We denote by an indexxz the three dimensional operators,that is,∇xz = (∂x1 , ∂x2 , ∂z) and∆xz = ∂2

x1x1+ ∂2

x2x2+ ∂2

zz.The air velocityU = (U1, U2, U3) and the potentialP satisfy the Navier–Stokes equations. On one

hand, the momentum equation, reads

∂tU + U .∇xzU − 1Re

∆xzU +∇xzP = ϕQe3 (7)

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whereQ is the temperature ande3 = (0, 0, 1). The right-hand side represents the Archimedes force dueto the expansion under the effect of heat, which has the formc(Q − Q0). Notice that the part of the forcecorresponding to the reference temperatureQ0 is ϕQ0e3 = ∇xz(ϕQ0z), that allows to include it into thepotential term∇xzP . The density variations due to the temperature have been neglected into the otherterms of the equation. On the other hand, the air compressibility is also neglected, so that

∇xz . U = 0. (8)

In order to specify the boundary conditions, we decompose the boundary into∂D = S ∪A∪L, where the

��

��

��

��

��

��

��

��

z

δ

h(x)

d

S

L

A

D

x

x 1

x 2

x 2

x 1

Figure 1. Domains in 2D and 3D.

surfaceS, the air lateral boundaryA and the air side boundaryL are respectively

S = {(x, z) : x ∈ d, z = h(x)}, A = {(x, z) : x ∈ d, z = δ},L = {(x, z) : x ∈ ∂d, h(x) < z < δ}.

On surfaceS,

U . N = 0,∂U

∂N

∣∣∣tang

= ζU, (9)

whereN is the inner unit normal vector field to∂D, and the subscripttangdenotes the tangential component,that is,ftang = f − (f . N)N for any vector fieldf . The first condition in (9) express that the air does notcross the surface, and the second condition express that the traction force is proportional to the tangentialvelocity. Indeed, the tangential stressσN |tang is equal toc ∂U

∂N |tangwhenU . N = 0.On the air upper boundaryA,

U . N = 0,∂U

∂N

∣∣∣tang

= 0, (10)

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that is, the air does not cross this boundary and there is no traction forces. Since this part of the boundary ishorizontal, this condition readsU3 = 0 and∇zU1 = ∇zU2 = 0.

On the air side boundaryL,U

∣∣L

= (vm, 0) (11)

more preciselyU(t, x, z) = (vm(t, x), 0) wherevm is the meteorological wind, that we assumed to beknown, horizontal, non depending onz and with null flux, that is,

∂zvm = 0,

∫

∂d

(δ − h) vm . nds = 0 (12)

wheren = (n1, n2) is the inner unit normal vector field to∂d.We complete these equations with the initial condition

U |t=0= U0 (13)

whereU0 is the initial velocity, that we assume to be known.

Remark.The equations (7) to (13) are well posed: for givenQ, vm andU0, there exists a solution(U,P ) ifthe data are “smooth enough”, that is, if the data are given in appropriate functional spaces; the solution isunique (up to an additive constant forP ) if the data are “small enough ”. This can be proved as inBRESCH

et al., [3], where a similar problem is considered.

Remark.We prove in§ 4. that hypothesis (12) is needed when the condition onL is given by (11). Never-theless, it is not necessary that the meteorological wind be neither horizontal, nor independent onz ; thatis, we replace (11) byU |L = um with um such that

∫L

um . N dS = 0.But, in the asymptotic convection model of§ 5. the condition onL will only concern the normal compo-

nent of the vertical average of the horizontal velocity,cf. (24) (because equation (7) degenerates, on verticaldirection into (19) and (20)). The vertical component and vertical variations onL of the meteorologicalwind have no influence on the model to be solved and therefore they can be chosen nulls.

4. Horizontal flux

An important property of such a velocity field is that its horizontal flux is incompressible. More precisely,we distinguish the vertical velocity from the horizontal one denoting

V = (U1, U2), W = U3

and we define the horizontal flux on a pointx ∈ d at timet by

V (t, x) =∫ δ

h(x)

V (t, x, z) dz. (14)

The incompressibility and the fact that the air does not crossS andL involve

∇x . V = 0 (15)

that is, this flux is incompressible. Indeed, (8) reads∇x . V + ∂zW = 0, and thus

∇x .∫ δ

h(x)

V (t, x, z) dz =∫ δ

h(x)

∇x . V (t, x, z) dz − V (t, x, h(x)) .∇xh(x)

= −W (t, x, δ) + W (t, x, h(x))− V (t, x, h(x)) .∇xh(x)

= (U . N)(t, x, δ) + c(x)(U . N)(t, x, h(x))

that is null due to (10), that shows (15).

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Remark.Lets verify that the hypothesis (12) is necessary for the equations (8) to (11) have a solution. Bythe Stokes formula and (15),

∫∂d

V . nds =∫

d∇x . V dx = 0. Then it follows (12) asV = (δ − h) vm on

∂d due to (11).

Remark. As the meteorological wind is almost constant on a fire scale (it is an expected wind, not ameasured wind) we can take it in general in the form

vm(t, x) =cDm(t)δ − h(x)

(16)

wherem is the “global” expected wind in the area andcD is the average value of1/(δ − h) on ∂d, that iscD =

∫∂d

1/(δ−h) ds/ ∫

∂dds, and socD/(δ−h) has an average1. Thenvm is the “local” corresponding

wind, adapted to the surface ; Moreover, it verifies the hypothesis (12) as∫

∂dnds = 0.

Particularly, the hypothesis (12) is satisfied for all meteorological constant windvm when the surface isflat, that is whenh = 0, and thencD = 1 andm = vm.

5. Convection asymptotic model

We now use the fact that the layer thickness where the convective air motion due to the fire takes place , issmall in relation to its width, that is

δ ¿ 1. (17)

We also assume that the wind in not too strong, more precisely

δ2Re ¿ 1. (18)

Preserving only the dominant terms and rescalingP , then equations (7) and (8) give, as we see on nextsection,

−∂2zzV +∇xP = 0, (19)∂zP = λQ, (20)

∇x . V + ∂zW = 0. (21)

whereλ = ϕRe. The conditions (9), (10) and (11) give, particularly,

∂zV = ζV, (V, W ) . N = 0 onS, (22)∂zV = 0, W = 0 onA, (23)

V . n = vm . n on∂d. (24)

Remark.Equations (19) to (24) are well posed : for givenQ andvm, there exists a unique solution(V, W,P )(except forP which is unique up to an additive constant) that we compute on sections§ 8. and 9.

6. Justification of the asymptotic analysis

Obtaining the incompressibility equation (21)

Notice thatx ∼ 1 (as the characteristic length is the one ofd) andz ∼ δ, so

∇x ∼ 1, ∂z ∼ 1/δ. (25)

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For an appropriate choice of the characteristic velocity(V,W ) ∼ 1, so the incompressibility condition (8),which reads∇x . V + ∂zW = 0, gives

V ∼ 1, W ∼ δ, (26)

and (21) holds.

Obtaining the vertical diffusion equation (19)

We now consider the operators appearing in the momentum equation (7). Firstt ∼ 1 (for an appropriatechoice of the characteristic time), so∂t ∼ 1. Second, with (25) and (26),U .∇xz = V .∇x + W∂z ∼ 1.Moreover, (25) implies,∆xz = ∆x + ∂2

zz ∼ ∂2zz so with (18),∆xz/Re ∼ 1/(δ2Re) À 1. All this gives

∂t + U .∇xz − 1Re

∆xz ∼ − 1Re

∂2zz (27)

As the horizontal component ofQe3 is null, then the horizontal component of (7), in the limit, is

− 1Re

∂2zzV +∇xP = 0. (28)

Here, the term∇xP cannot be neglected because on the contrary we would have∂2zzV = 0, andV would

be determined by the conditions onS andA, so it would not depend on either meteorological wind ortemperature, and this is absurd. The term1Re∂

2zzV cannot be neglected either because on the contrary,V

would be solely governed by the incompressibility equation ant it would be partly indeterminate. As noterm of (28) is negligible, this equation can be written as (19) usingReP a the new potential.

Obtaining the hydrostatic equilibrium equation (20)

From (27), and the above rescaling ofP , the vertical component of (7) in the limit gives

−∂2zzW + ∂zP = λQ. (29)

From (25) and (26),∂2zzW ∼ 1/δ. From (19),P ∼ 1/δ2 and then∂zP ∼ 1/δ3. As this term controls the

previous one, (29) reduces to∂zP = λQ, that is (20).

Obtaining the conditions on the surface (22)

The slide condition in (9) readsU . N = V . Nx + WNz = 0 whereNx = −∇xh/C, Nz = −1/C andC = (1 + |∇xh|2)−1/2. As h ∼ δ, we have

Nx ∼ δ, Nz ∼ 1.

From (25) and (26), it follows thatV . Nx ∼ WNz, so the slide condition is preserved. It can be written(V, W ) . N = 0.

Lets consider the traction condition in (9), where

∂U

∂N

∣∣∣tang

=( ∂V

∂N−

( ∂U

∂N. N

)Nx,

∂W

∂N−

( ∂U

∂N. N

)Nz

).

As ∂/∂N = N .∇xz = Nx .∇x + Nz∂z ∼ ∂z then∂U/∂N . N ∼ ∂zU . N = ∂zV . Nx + ∂zWNz ∼ 1and then∂U/∂N |tang ∼ (∂zV, 0). MoreoverU ∼ (V, 0), so the traction condition∂U/∂N |tang = ζUgives to the limit∂zV = ζV .

Obtaining the equilibrium conditions on the air (23)

It is exactly condition (10) sinceA is an horizontal plane.

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Obtaining the lateral flux condition (24)

The equality with the meteorological wind (11) particularly shows thatV = vm on L. Integrating overzbetweenh(x) andδ, for a fixedx ∈ ∂d, it follows V = (δ − h) vm, and soV . n = (δ − h) vm . n, that is(24).

We cannot impose any other conditions onL because, for fixedQ and x, equations (19) and (20)determineV (x, z) up to a polynomial functionaz2 + bz + c, and conditions (22), (23) and (24) determinea, b andc. Likewise, (21) and (23) determineW (x, z).

The approximation of the solution of the convective equations (7) – (13) by the solution of (19) – (24) isonly possible up to a limit layer onL. This limit layer has no physical meaning. As an evidence notice thatgiven a domaind′ ⊃ d such that∂d′ 6= ∂d, there is no limit layer ond : it has been “displaced” ond′. Thelimit layer appears because, on a thin air layer, the vertical wind variations are asymptotically determinedby the conditions “on the upper and lower boundaries”. In more mathematic terms, the operator degeneratesinto the vertical direction, and this involves a loss of boundary conditions on the vertical boundary. Thelimit operator does not contain the horizontal diffusion term that allowed to fit the initial problem solutionto any given boundary condition.

Remark.The previous asymptotic analysis can be mathematically justified whenδRe ¿ 1, which is a morerestrictive condition that (18). The idea of the proof is to use a vertical scaling with rateδ in order to get aproblem in a fixed domain independent ofδ with coefficients proportional to some powers ofδ, for whichconvergence follows from some energy estimations.

This study was made inBRESCHet al., [3] for a similar model to the one described here. In fact, theproblem described there was a model of a lake, where the depth (δ − h in the present model) was null on∂d (the lake shore). This involves to use weighted spaces, so this part of demonstration was more complexthan in the present case. On the contrary, there is no limit layer, since depth is null on∂d, so this part ofdemonstration was more simple than in the present case.

This is also solved inBAYADA et al., [2] for a similar model of lubrication.

7. Coupling with temperature

The temperatureq considered in the combustion equations (1) to (3) is obviously the value ofQ on thesurface, that isq(t, x) = Q(t, x, h(x)). We assume that

Q(t, x, z) = q(t, x)δ − z

δ − h(x)(30)

that is, the air temperature linearly decreases with the height and vanishes on the upper boundary of the airlayer.

The velocityv considered in the combustion equations is the value ofV on the surface, that is thehorizontal component of the wind velocityU on the surface ; in other words,

v(t, x) = V (t, x, h(x)).

Indeedd is the projection of the surfaceS (that is aIR3 surface) on thex-plane, so the propagation velocityv in d, at a pointx at timet, is the projection onS of the propagation velocityU , that isV , at point(x, h(x))and at timet.

Remark.The hypothesis (30) about the linear variation of the temperature as a function of heightz couldbe justified by an asymptotic analysis of heat diffusion equations.

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8. Solution of the convection asymptotic model

Lets compute explicitlyP (t, x, z) andV (t, x, z) in terms ofz, q(t, x), h(x) and∇xp(t, x) wherep is a 2Dpotential.

For a fixedx, equation (20), that is∂zP = λQ, with Q given by (30) provides

P (t, x, z) = p(t, x) +λq(t, x)δ − h(x)

(δz − 1

2z2)

(31)

for somep(t, x).Equation (19), that is−∂2

zzV = ∇xP , together with conditions∂zV (t, x, δ) = 0 and∂zV (t, x, h(x)) =ζV (t, x, h(x)) included in (23) and (22), then provides

V (t, x, z) =(

12z2 − δz − 1

2h2(x) + (δ + ξ)h(x)− ξδ)∇xp(t, x)

+(− 1

24z4 + 16δz3 − 1

3δ3z + 124h4(x)

− 16h3(x)(δ + ξ) + 1

2ξδh2(x) + 13δ3h(x)− 1

3ξδ3)∇xq(t, x)

(32)

where

ξ =1ζ, q(t, x) =

λq(t, x)δ − h(x)

. (33)

Notice thatq(t, x) = (λ/δ) Q(t, x, 0), cf. (30). Up to the factorλ/δ, it is the temperature at heightz = 0.

9. Stream function 2D equations

We now assume that∂d is connected. (34)

Since it is bounded, thend is connected, simply connected (this means that it does not possess “holes” be-cause dimension is 2) and it is “inside” its boundary∂d. The hypothesis (12), that is

∫∂d

(δ−h) vm . nds =0 gives the existence of a functionv∗m such that

∇x . v∗m = 0, v∗m . n = (δ − h) vm . n on∂d. (35)

Since∇x . V = 0, cf. (15), it follows that∇x . (V − v∗m) = 0 therefore, again thanks to (34),cf. forexampleSIMON, [6], there exists a functionk such that

∇⊥x k = V − v∗m, k|∂d= 0, (36)

where the superscript⊥ denotes a rotation ofπ/2, that is(f1, f2)⊥ = (−f2, f1) for any vectorf . Integrating(32) with respect toz, from h(x) to δ, we obtain,cf. definition (14),

V = −a∇xp− b∇xq (37)

wherea(x) = 1

3δ3 + ξδ2 − δ(δ + 2ξ)h(x) + (δ + ξ)h2(x)− 13h3(x)

= 13 (δ − h(x))2(3ξ + δ − h(x)),

(38)

b(x) = 215δ5 + 1

3ξδ4 − 13δ3(δ + ξ)h(x) + 1

6δ2(δ − 3ξ)h2(x)

+ 16δ(δ + 4ξ)h3(x)− 1

6 (δ + ξ)h4(x) + 130h5(x)

= 130 (δ − h(x))2(2δ2(2δ + 5ξ)− 2δ(δ − 5ξ)h(x)− (3δ + 5ξ)h2(x) + h3(x)).

(39)

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This implies

−∇x .(1

a∇xk

)= f, k|∂d

= 0, (40)

where

f = ∇⊥x . b∇xq + v∗ma

. (41)

Indeed, (36) and (37) give∇⊥x k = −(a∇xp + b∇xq + v∗m) (42)

therefore, dividing bya, taking the orthogonal⊥, and taking into account thatf⊥⊥ = −f , we get−∇xk/a = −(∇xp + (b∇xq + v∗m)/a)⊥. Taking the divergence, this gives the first equation of (40),with f = −∇x . (∇xp + (b∇xq + v∗m)/a)⊥. It yields (41) because∇x .∇⊥x = 0 and−∇x . g⊥ = ∇⊥x . g.The second equation of (40) follows from (36).

Remark.Equation (40) is elliptic because, by (38),

a(x) ≥ η > 0

whereη = ξδ2 + 13δ3 andδ = δ − supx∈d h(x) > 0. Therefore it has a unique solutionk.

10. Velocity on the surface

The velocity on the surface isv(t, x) = V (t, x, h(x)), therefore (32) gives

v(t, x) = −ξ (δ − h(x))(∇xp(t, x) +

(13δ2 + 1

3δh(x)− 16h2(x)

)∇xq)

and with (42) it follows

v(t, x) =ξ

c(x)(∇⊥x k(t, x) + e(x)∇xq(t, x) + v∗m(t, x)) (43)

wherec(x) = 1

3 (δ − h(x)) (δ + 3ξ − h(x)), e(x) = 145 (δ − h(x))5. (44)

Remark. For a given meteorological windvm on ∂d, verifying∫

∂d(δ − h) vm . nds = 0, there exist

infinitely many solutionsv∗m of (35) however all of them give the same value tov.Indeed, the general solution of (35) isv∗m = v0 + ∇⊥g wherev0 is a particular solution of (35) and

whereg is any function null on∂d. Then (40) – (41) can be written

−∇x .(1

a∇x(k + g)

)= ∇⊥x . b∇xq + v0

a, (k + g)|∂d

= 0,

and thenk + g does not depend on the choice ofg. In other words, allv∗m give the same value tok + g, andalso to∇⊥x (k + g) + v0, that is to∇⊥x k + v∗m, and also to∇xp due to (42), and also to the velocityV thankto (32), and particularly of the fluxV and of the velocity on the surfacev, cf. (43).

The same conclusion could be obtained observing that conditions∇x . V = 0 andV . n = (δ−h)vm . non∂d give, with (37),

−∇x . (a∇xp) = ∇x . (b∇xq) = 0,∂p

∂n= − b

a

∂q

∂n+

δ − h

avm . n

that determine∇xp in a unique way, that is, independently of the choice ofv∗m.

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11. Resume of the convection model

For a fixed timet, to any temperature fieldq defined ond, we associate a velocity fieldv defined ond by

v =ξ

c(∇⊥x k + e∇xq + v∗m)

wherek is the unique solution of

−∇x .(1

a∇xk

)= ∇⊥x . b∇xq + v∗m

a, k|∂d

= 0,

and whereq = qλ/(δ − h) is the normalized temperature. Functionsa(x), b(x), c(x) ande(x), that modelthe influence of the surface, are defined by (38), (39) and (44).

The meteorological windvm is a data on∂d such that∫

∂d(δ− h) vm . nds = 0, andv∗m is any velocity

field defined ind such that

∇x . v∗m = 0, v∗m . n = (δ − h) vm . n on∂d.

This givesv = L(q, vm, h)

whereL is a timet independent operator, affine with respect toq andvm, and non linear with respect toh.

Remark.It can be shown that, givenh in C(d), the application(q, vm) 7→ v is continuous fromH1(d) ×(L2(∂d))2 into (L2(d)2. If the boundary conditions are defined “in the sense ofH1

0 ”, it is enough thatd bean open bounded set.

12. About the validity of the asymptotic model

Condition (18), that isδ2Re ¿ 1, makes negligible the nonlinear terms as seen in§ 6. This is the Reynoldsapproximation, classic in lubrification, seeBAYADA et al, [2].

If Reynolds number is not small enough for that, it can be lowered as it is made in oceanography whenthe molecular viscosity is replaced by the turbulent viscosity. This is equivalent to replace the nonlineartransport terms by linear diffusion terms. Briefly, we can say that, in oceanography, the vortex of size lowerthan mesh size are removed, whereas here, are removed the vortex of size lower than the height of the airlayer perturbed by the fire. In oceanography, this is justified by the insufficient of the actual computer ; here,the aim is to obtain a velocity field explicitly computable in terms ofz. In both cases, a part of the effectsdue to the turbulence (that can be interpreted as instabilities or multiple bifurcations due to the nonlinearity)are lost.

The obtained approximation is reasonable because it takes into account, on one hand, Archimedes forcedue to the air expansion under the effect of heat, and on the other hand, the local movement transmissiongenerated in the fluid set under the effect of the incompressibility. These two phenomena roughly determinethe air movement.

13. Numerical method

The numerical method employed here is an adaptive finite element method combined with a time-steppingsplitting technique. Each time step is subdivided in three sub-steps, correspondingly equations (1)-(4) aresplit into a radiation equation, a convection equation and a diffusion equation. The radiation step is solvedby an implicit Euler method, the convection step by a characteristic method and the diffusion step using P1-Lagrange finite element approximation. Previously, the velocity at the surface has been computed solving

22


the stream function problem (40) and using (43). At each step, a mesh adaptation is performed. Thisnumerical algorithm has been implemented using freeFEM+, a finite element package of O. Pironneau andF.Hecht, [5].

14. Numerical simulation

As an application, we present the numerical results of a simulated combustion in a normalized rectangle ofsize2× 1, for an initial fire focus and uniform wind given on the boundary. There is a hill in the center ofthe rectangle.

The nondimensional activation energy for gas combustion isγg = 33 and for solid pyrolysisγs = 20.We also have taken for diffusionκg = κo = 0.1 andκq = 0.1 for heat radiation coefficient in (4). Theother parameters of the combustion equations has been taken equal to unit.

Concerning convection model, the velocity on the boundary isvm = (220, 0) (which roughly corre-sponds to a real velocity between2 − 5m/s) with a friction coefficientζ = 10 andλ = 3 in equation (20).All this values has been chosen in order to have reasonable results.

The initial fire focus is given by the expression

q0(x1, x2) = 30e−200((x1−0.25)2+(x2−0.5)2) (45)

After 1000 time steps we obtain the temperature contours shown in figure 2. After this time, a stablefire front has been obtained. Figure 3 shows the contour plot of thex-component of velocity. Observe thatthe velocity near the fire front and the meteorological wind have opposite direction. Comparing this figurewith figure 4, where the fire front is near the top of the hill, we can appreciate the effect of the hill, so thatthe combined effect of the meteorologic wind and the slope of the terrain surface is bigger than the effectof the gradient of the temperature in the fire front. We can appreciate the burned area with the contourlines corresponding to the solid fuel in figure 5. Figures 6 and 7, show the contour lines of the gas fuel andoxygen respectively. In figure 8, a typical adapted mesh is shown.

34.820732.98831.155429.322727.4925.657423.824721.99220.159418.326716.49414.661312.828710.9969.163347.330675.498013.665341.832670

Figure 2. Temperature.

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69.706966.037162.367258.697455.027651.357747.687944.018140.348336.678433.008629.338825.668921.999118.329314.659410.98967.319793.64996-0.0198745

Figure 3. x-component of velocity after 1000 time steps.

63.678760.756657.834554.912551.990449.068346.146343.224240.302137.3834.45831.535928.613825.691822.769719.847616.925514.003511.08148.15932

Figure 4. x-component of velocity after 600 time steps.

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10.9473680.8947370.8421050.7894740.7368420.6842110.6315790.5789470.5263160.4736840.4210530.3684210.3157890.2631580.2105260.1578950.1052630.05263160

Figure 5. Solid fuel.

0.4968250.4706760.4445280.4183790.392230.3660820.3399330.3137840.2876360.2614870.2353380.2091890.1830410.1568920.1307430.1045950.07844610.05229740.02614870

Figure 6. Gaseous fuel.

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10.9473680.8947370.8421050.7894740.7368420.6842110.6315790.5789470.5263160.4736840.4210530.3684210.3157890.2631580.2105260.1578950.1052630.0526315-7.0e-08

Figure 7. Oxygen.

Figure 8. Mesh.

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15. Acknowledgement

This work is supported by the Research Projects REN2001-0925-C03-03 from the Science and TechnologyMinistry (Spain) and SA089/01 from the Castilla and Leon regional government (Spain).

References

[1] M.I. Asensio & L. Ferragut, On a wildland fire model with radiation,Int. J. for Num. Meth. in Eng., 54(2002), 137–157.

[2] G. Bayada & M. Chambat, The transition between the Stokes equations and the Reynolds equation: Amathematical proof,Appl. Math. Optim.14 (1986), 73-93.

[3] D. Bresch, J. Lemoine & J. Simon, Nonstationary models for shallow lakes,Asymptotic Anal., 22(2000), 15–38.

[4] G. N. Mercer & R. O. Weber, Radiation enhanced combustion wave speeds,Proc. R. Soc. Lond. A, 453(1997), 1543–1549.

[5] O. Pironneau & F. Hecht, FreeFEM (software file), On the web http://www-rocq.inria.fr/.

[6] J. Simon, Demonstration constructive d’un theoreme de G. de Rham,C. R. Acad. Sci. Paris, Ser. I, 316(1993), 1167–1172.

[7] R.O. Weber, Towards a comprehensive wildfire spread model,Int. J. of Wildland Fire., 1(4) (1991),245–248.

M. I. Asensio, L. Ferragut J. SimonDepartamento de Matematica Aplicada CNRS, Laboratoire de Mathematiques AppliqueesUniversidad de Salamanca Universite Blaise PascalPlaza de la Merced s/n, 37008, Salamanca 63177 Aubiere cedexSpain France{mas,ferragut }@usal.es [email protected]

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