wildland fire modelling including turbulence and fire spotting · inderpreet kaur modelling random...

Wildland Fire Modellingincluding

Turbulence and Fire spotting

Inderpreet Kaur1

in collaboration withAndrea Mentrelli1,2, Frederic Bosseur3, Jean Baptiste Filippi3,

Gianni Pagnini1,4

1BCAM, Bilbao, Basque Country – Spain2Department of Mathematics & AM2, University of Bologna, Italy

3SPE–CNRS/University of Corsica, Corte, Corsica – France4Ikerbasque, Bilbao, Basque Country – Spain

Inderpreet KAUR Modelling random processes in wildfires 1 / 37

Introduction

Introduction

Wildland fires are one of the natural phenomena that can causehazardous situations for people and property. They also constitute a

very serious environmental problem.


Introduction

Random Character of Wildland FireTURBULENCE

Moisture and Heat release from fuel combustion alters localthermal structure of the lower Atmospheric Boundary layer (ABL)and induces turbulent circulation in vicinity of fire.Vertical scale : 1-1000 m; Horizontal Scale : 1-1000 m ( Sullivan2013, Intl. J. Wildland Fire)


Introduction

Random Character of Wildland FireFIRE-SPOTTING

Behaviour of a fire producing sparks or embers that are carried by the windand start new spot fires beyond the zone of direct ignition by the main fire

The random effects cause a rapid increase in the rate of spread of thefire.

New isolated fires can develop across fire break zones like rivers/roads.

Vertical scale : 1-3000 m; Horizontal Scale : 1-10000 m ( Sullivan 2013,Intl. J. Wildland Fire)


Model Description Notation

Model DescriptionNotation

Let Γ be a simple closed curve , or an ensemble of simplenon-intersecting closed curves, representing a propagating interface intwo dimensions. Let S ⊆ R2 be the domain of interest.The subset of the domain S corresponding to the region Ω enclosed byΓ may be conveniently identified as the region selected by the indicatorfunction IΩ : S × [0,+∞[→ 0,1 defined as

IΩ(x, t) =

1 , x ∈ Ω ,

0 , elsewhere .(1)


Model Description Random Front Formulation

Random Front Formulation

Let Xω(t ,x0) = x(t ,x0) + χω + ξω be the ω-realization of a randomtrajectory driven by the random noises χ and ξ corresponding toturbulence and fire spotting, respectively.

For every realization, the initial condition is Xω(0,x0) = x0.

Average turbulent fluctuations: 〈χ〉 = 0.

Fire spotting is assumed to be independent of turbulence and to be adownwind phenomenon, then, in the leeward sector only, it is stated:ξω = `ω nU , where ` is the landing distance from the main fireline suchthat 〈`〉 > 0 and nU is the mean wind direction.




The trajectory of an active burning point can be marked out in terms ofδ- function. Using the property of δ-function: g(x) =

∫g(x) δ(x− x)dx,

the evolution in time of the ω-realization of a random front contourγω(x, t) is given by

γω(x, t) =

∫Sγ(x0) δ(x− Xω(t ,x0)) dx0 , (2)

which in terms of the random indicator IΩω(x, t) reads

IΩω(x, t) =

∫S

IΩ0(x0) δ(x− Xω(t ,x0)) dx0

=

∫Ω0

δ(x− Xω(t ,x0)) dx0

=

∫Ω(t)

δ(x− Xω(t ,x)) dx , (3)

where J =dx0

dx= 1 is assumed.




Let ϕe(x, t) : S × [0,+∞[→ [0,1] be an effective indicator and it maybe defined as

ϕe(x, t) = 〈IΩω(x, t)〉 =

⟨∫Ω(t)

δ(x− Xω(t ,x))dx

⟩

=

∫Ω(t)〈δ(x− Xω(t ,x))〉dx

=

∫Ω(t)

f (x; t |x) dx , (4)

where f (x; t |x) = 〈δ(x− Xω(t ,x))〉 is the probability density function(PDF) of fluctuations of the fireline perimeter around the deterministicpart contour Γ(t) = x ∈ S|γ(x, t) = γ∗.



Burning Criteria

Function ϕe(x, t) continuosly ranges from 0 to 1.

Q: How to mark a point as burned?

A: For example, points such that ϕe(x, t) > 0.5 can be marked asburned, i.e. Ωe(t) = x : ϕe(x, t) > 0.5.

Q: How to take into account the effects due to turbulence and firespotting which act ahead the fireline?

A: For example, with a model for a heating-before-burning law.



The Model: Heating-before-burning Law

The presence and persistence of ϕe(x, t) can be understood as anaccumulation in time of the igniting heat and, since the amount of heatψ(x, t) is proportional to the increasing of the fuel temperature T (x, t),it follows

ψ(x, t) =

∫ t

0ϕe(x, s)

dsτ∝ T (x, t)− T (x,0)

Tign − T (x,0), (5)

where τ is an ignition delay time-scale and Tign is the ignitiontemperature. The evolution of T (x, t) is

∂T∂t∝ ϕe(x, t)

Tign − T (x,0)

τ, T ≤ Tign . (6)

Hence, when for example ψ(x, t) = 1, it is stated that IΩ(x, t) = 1 andan ignition occurs in (x, t) starting a new fire.



The Model: Heating-before-burning Law

Electrical resistance analogy can be considered in order to theestimate the ignition delay τ , which is due to the combined actions byheat transfer and fire spotting.

Because of this two pathways of ignition, heating and firebrand act asresistances in parallel giving

1τ

=1τh

+1τf

=τh + τf

τh τf. (7)


Model Description Model for Random Processes : Turbulence

Model for Turbulence

A Gaussian distribution is chosen as the most simple model forturbulent dispersion of hot air

G(x− x; t) =1

2π σ2(t)exp

−(x − x)2 + (y − y)2

2σ2(t)

, (8)

σ2(t) = 〈(x − x)2〉 = 〈(y − y)2〉 = 2D t . (9)


Model Description Model for Random Processes : Fire-spotting

Model for Random Processes : Fire-spotting

A numerical study by Sardoy et al., Combust. Flame, 2008 proposedthat the firebrands follow a log-normal distribution

q(`; t) =1√

2π s(t) `exp

−(ln `− µ(t))2

2 s2(t)

, (10)

where µ(t) = 〈ln `〉 and s(t) = 〈(ln `− µ(t))2〉 are the mean and thestandard deviation of ln `.


Model Description Model for Random Processes : Fire-spotting

The f (x; t |x) probability density function

The PDF of the sum of two independent random variables is given bythe convolution of the corresponding PDFs.

Then in the leeward sector, i.e. 0 ≤ θ < π/2 with θ the angle betweenn and nU ,

f (x; t |x) =

∫ ∞0

G(x− x− ` nU ; t) q(`; t) d` , (11)

otherwise in the windward sector

f (x; t |x) = G(x− x; t) . (12)


Approaches to Fire Modelling

Approaches to Fire Modelling

Numerical simulation of wildfires is a two fold process”Fire Spread Model” : Rate of Spread Model provides the velocityof the front driven by the fuel characteristics, atmosphericconditions and topography

Rothermel model (Rothermel, 1971)Balbi Model (Balbi et al., 2005)

”Fire-front tracking Method” : Simulates the advancement of thefire-front.

Level Set Method (Mandel et al., 2011)Discrete EVents System Specification (DEVS; Filippi et al., 2010)Huygens Principle (Rios et al., 2014; Anderson et al., 1971)

Both models are developed independently of each other.


Approaches to Fire Modelling Level Set Method

The Level-Set Method

Let γ(x, t) be a function such that a certain isolineγ(x, t) = γ0 = constant corresponds to the interface Γ(t).Then the motion of the interface Γ(t) is determined by the evolutionequation of the isoline

DγDt

=∂γ

∂t+

dxdt· ∇γ = 0 , (13)

dxdt

= V(x, t) = V(x, t) n , n = − ∇γ||∇γ||

, (14)

∂γ

∂t= V(x, t) ||∇γ|| , (15)

where V(x, t) is the Rate of Spread (ROS) of the fireline. A hugeliterature exists for its theoretical and phenomenological determination.


Approaches to Fire Modelling Level Set Method

Fire Propagation MethodsLevel Set Method (Mandel et al., 2011 )

Eulerian time-driven techniqueRepresents the burning region on a simple cartesian grid.Temporal resolution constrained by the global time step restrictionimposed by the Courant-Friedrichs-Levy (CFL; ∆t < ∆x/Vmax )criteria.Direction of propagation is given by the normal to the front.


Approaches to Fire Modelling ForeFire

Fire Propagation MethodsForeFire (Filippi et al., 2009)

Lagrangian specification of the fire interface.Fire interface is discretized by a set of points/markers.No global time step is defined.Each marker moves asynchronously according to its speed anddirection function and the CFL condition (∆t < ∆x/Vmax ) isapplied locally.The asynchronous movement is managed by Discrete EVentsystems Specification (DEVS).


Approaches to Fire Modelling ForeFire

Fire Propagation MethodsForeFire (Filippi et al., 2009)

The maximum distance covered by each marker is fixed,Quantum Distance (∆q). Details smaller than this distance mightnot be considered.Perimeter Resolution (∆c) restricts the maximum distanceallowed between the two markers.Measure of ∆c is used to decompose/coalesce the markers.The choice of ∆q and ∆c is dependent on the type of problem.The advection scheme is first order Euler scheme in space:

xn+1i = xn

i + ∆q · nbi (16)

tn+1i = tn

i +∆qvb

i(17)

The direction of propagation is given by the bisector of the anglemade by each marker with its immediate neighbours.


Simulation Setup

Simulation Setup

The formulation of random processes in both Eulerain andLagragian front tracking methods is studied.It is tried to parametrize the models in an identical set-up.The grid size in LSM is chosen to be 20 m in x and y direction.∆q = 4m,∆c = 18m for zero wind; and ∆q = 0.3m,∆c = 8m inthe presence of wind.To avoid instability in the presence of wind, ∆q is chosen to be ofa much higher resolution than the wind data (20 m× 20 m in thissetup).No particular type of vegetation is defined and work withpre-defined value of ROS.ROS is assumed to be constant throughout the terrian.These are simplified and idealised test cases and no attempt ismade to choose the parameters for a realistic setup


Results

LSM & DEVS (without random phenomena)

1000 1500 2000 2500 3000 3500 4000X (m)

1000

1500

2000

2500

3000

3500

4000

Y (m

)

040

80

120

160

(a)

1000 1500 2000 2500 3000 3500 40001000

1500

2000

2500

3000

3500

4000

X (m)

Y (

m)

(b)

04080120160

Time [min]

Evolution in time of the fire perimeter in no wind conditions with an initialcircular profile of radius R = 300 m and ROS = 0.05 ms−1

by using the LSM approach (a) and the DEVS-based ForeFire simulator (b).


Results

LSM & DEVS

1000 1500 2000 2500 3000 3500 40001000

1500

2000

2500

3000

3500

4000

X (m)

Y (

m)

(a)

020406080100120140

Time [min]

1000 1500 2000 2500 3000 3500 40001000

1500

2000

2500

3000

3500

4000

X (m)

Y (

m)

(b)

020406080100120140

Time [min]

Evolution in time of the fire perimeter in no wind conditions with an initialcircular profile of radius R = 300 m and ROS = 0.05 ms−1 in presence of afirebreak line by using the LSM approach (a) and the DEVS-based ForeFiresimulator (b).


Results


0 1000 2000 3000 4000 5000 6000 7000X (m)

2000

3000

4000

5000

6000

7000

8000

Y (m

)

20 40

60

80

100

120140

160180

(a)

0 1000 2000 3000 4000 5000 6000 70002000

3000

4000

5000

6000

7000

8000

X (m)Y

(m)

(b)

020406080100120140160180

Time [min]

Evolution in time of the fire perimeter in no wind conditions with an initialcircular profile of radius R = 30 m and ROS = 0.3 ms−1; 0.2 ms−1; 0.1 ms−1 inthe upper left, upper right and lower quadrants respectively, by using the LSMapproach (a) and the DEVS-based ForeFire simulator (b).


Results


2000 2500 3000 3500 4000X (m)

1500

2000

2500

3000

3500

Y (m

)

020 40

60 80

100

120

140

160

(a)

2000 2500 3000 3500 40001500

2000

2500

3000

3500

X (m)

Y (

m)

(b)

020406080100120140160

Time [min]

Evolution in time of the fire perimeter with an initial circular profile of radiusR = 300 m, ROS = 0.03 U · n and mean wind U = 3 ms−1



Results


2000 2500 3000 3500 4000X (m)

1500

2000

2500

3000

3500

Y (m

)

0

20

40 60

80

100

120 140

160

(a)

2000 2500 3000 3500 40001500

2000

2500

3000

3500

X (m)

Y (

m)

(b)

020406080100120140160

Time [min]

Evolution in time of the fire perimeter with an initial square profile of sideR = 600 m, ROS = 0.03 U · n and mean wind U = 3 ms−1



Results

LSM & DEVS (without random phenomena)With prescription of head, flank and rear ROS:

ROS =

εo(1 + a

√U cosn θ), if |θ| ≤ π

2εo(α+ (1− α)| sin θ|), if |θ| > π

2 , (Mallet et al., Comput. Math. Appl., 2009)

0 10002000300040005000600070008000X (m)

0

1000

2000

3000

4000

5000

6000

7000

8000

Y (m

)

0

10

20

30

40

50 60

70

(a)

0 1000 2000 3000 4000 5000 6000 7000 80000

1000

2000

3000

4000

5000

6000

7000

8000

X (m)

Y (

m)

(b)

010203040506070

Time [min]

Evolution in time of the fire perimeter with initial circular profile of radius R = 30 m according to

Mallet ROS, where n = 3, U = 3 ms−1, a = 0.5 sm−1, ε = 0.2, α = 0.8 and θ is the angle

between the line joining a contour point and the center of the initial profile and the mean wind

direction, by using the LSM approach (a) and the DEVS-based ForeFire simulator (b).


Results

LSM & DEVS (without random phenomena)With prescription of head, flank and rear ROS:

ROS =

εo(1 + a


2εo(α+ (1− α)| sin θ|), if |θ| > π


0 10002000300040005000600070008000X (m)

0

1000

2000

3000

4000

5000

6000

7000

8000

Y (m

)

0

10

20

30

40

50 60

70

(a)

0 1000 2000 3000 4000 5000 6000 7000 80000

1000

2000

3000

4000

5000

6000

7000

8000

X (m)

Y (

m)

(b)

010203040506070

Time [min]

Evolution in time of the fire perimeter with initial circular profile of radius R = 30 m according to

Mallet ROS, where n = 3, U = 3 ms−1, a = 0.5 sm−1, ε = 0.2, α = 0.8 and θ is the angle

between the line joining a contour point and the center of the initial profile and the mean wind

direction, by using the LSM approach (a) and the DEVS-based ForeFire simulator (b).Inderpreet KAUR Modelling random processes in wildfires 26 / 37

Results


0 10002000300040005000600070008000X (m)

0

1000

2000

3000

4000

5000

6000

7000

8000

Y (m

) 0

10

20

30

40

50

60 70

(a)

0 1000 2000 3000 4000 5000 6000 7000 80000

1000

2000

3000

4000

5000

6000

7000

8000

X (m)

Y (

m)

(b)

010203040506070

Time [min]

Evolution in time of the fire perimeter with an initial circular profile of radius R = 30 m and

ROS =

εo(1 + a


2εo(α+ (1− α)| sin θ|), if |θ| > π


where n = 3, U = 3 ms−1, a = 0.5 sm−1, ε = 0.2, α = 0.8 and θ is the angle between the

outward normal in a contour point and the mean wind direction, by using the LSM approach (a)

and the DEVS-based ForeFire simulator (b).


Results

Turbulence and Fire Spotting Set Up

Turbulence is modelled with a Gaussian with diffusion coefficient D.

The jump-length of embers is modelled with a stationary log-normaldistribution with (Perryman et al., Int. J. Wildland Fire, 2013):

mean value µ = 1.32 I0.26f U0.11 − 0.02,

standard deviation s = 4.95 I−0.01f U−0.02 − 3.48,

where U is the modulus of the mean wind and If = I + It withI = 10000kWm−1 is the fire intensity and It = 0.015 kWm−1 is the treetorching intensity.

ROS = 0.05 ms−1 with no windROS = 0.03 U · n with wind.

Ignition delay of hot air τh = 600 s,ignition delay of firebrands τf = 60 s.


Results

LSM & DEVS (with turbulence)

0 1000 2000 3000 4000 50000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

X (m)

Y (

m)

(a)

020406080100120140

Time [min]

0 1000 2000 3000 4000 50000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

X (m)Y

(m

)

(b)

020406080100120140

Time [min]

Evolution in time of the fire perimeter with an initial circular profile of radiusR = 300 m for zero wind and diffusion coefficient D = 0.15 m2s−1, by usingthe LSM approach (a) and the DEVS-based ForeFire simulator (b).


Results


0 1000 2000 3000 4000 50000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

X (m)

Y (

m)

(a)

020406080100120140

Time [min]

0 1000 2000 3000 4000 50000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

X (m)Y

(m

)

(b)

020406080100120140

Time [min]

Evolution in time of the fire perimeter with an initial circular profile of radiusR = 300 m, for zero wind and diffusion coefficient D = 0.30 m2s−1, by usingthe LSM approach (a) and the DEVS-based ForeFire simulator (b).


Results


0 1000 2000 3000 4000 50000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

X (m)

Y (

m)

(a)

020406080100120140160

Time [min]

0 1000 2000 3000 4000 50000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

X (m)

Y (

m)

(b)

020406080100120140160

Time [min]

Evolution in time of the fire perimeter with an initial circular profile of radiusR = 300 m , with a mean wind U = 3 ms−1 and diffusion coefficientD = 0.15 m2s−1, by using the LSM approach (a) and the DEVS-basedForeFire simulator (b).


Results

LSM & DEVS (with turbulence and fire spotting)

0 1000 2000 3000 4000 50000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

X (m)

Y (

m)

(a)

020406080100120140160

Time [min]

0 1000 2000 3000 4000 50000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

X (m)

Y (

m)

(b)

020406080100120140160

Time [min]

Evolution in time of the fire perimeter with an initial circular profile of radiusR = 300 m , a mean wind U = 3 ms−1 and diffusion coefficientD = 0.15 m2s−1, by using the LSM approach (a) and the DEVS-basedForeFire simulator (b).


Results


0 1000 2000 3000 4000 50001500

2000

2500

3000

3500

4000

4500

5000

X (m)

Y (

m)

(a)

020406080100120140160180200220

Time [min]

0 1000 2000 3000 4000 50001500

2000

2500

3000

3500

4000

4500

5000

X (m)

Y (

m)

(b)

020406080100120140160180200220

Time [min]

Evolution in time of the fire perimeter with an initial circular profile of radiusR = 300 m in presence of two firebreak zones with a mean wind U = 3 ms−1

and diffusion coefficient D = 0.07 m2s−1, by using the LSM approach (a) andthe DEVS-based ForeFire simulator (b).


Results

LSM & DEVS (with turbulence and fire spotting)

0 1000 2000 3000 4000 50001500

2000

2500

3000

3500

4000

4500

5000

X (m)

Y (

m)

(a)

020406080100120140160180200220

Time [min]

0 1000 2000 3000 4000 50001500

2000

2500

3000

3500

4000

4500

5000

X (m)

Y (

m)

(b)

020406080100120140160180200220

Time [min]

Evolution in time of the fire perimeter with initial circular profile of radiusR = 300 m in presence of two firebreak zones with a mean wind U = 3 ms−1

and diffusion coefficient D = 0.07 m2s−1, by using the LSM approach (a) andthe DEVS-based ForeFire simulator (b).


Results

LSM & DEVS

1500 2000 2500 3000 3500 4000 4500X (m)

1000

1500

2000

2500

3000

3500

4000

Y (m

)

0

20

40 6080

100

120

140

LSM based model

1500 2000 2500 3000 3500 4000 45001000

1500

2000

2500

3000

3500

4000

X (m)

Y (

m)

LSM based model + Turbulence

020406080100120

Time [min]

1500 2000 2500 3000 3500 4000 45001000

1500

2000

2500

3000

3500

4000

X (m)

Y (

m)

LSM based model + Turbulence + Fire spotting

020406080100120

Time [min]

1500 2000 2500 3000 3500 4000 45001000

1500

2000

2500

3000

3500

4000

X (m)

Y (

m)

DEVS based model

020406080100120140

Time [min]

1500 2000 2500 3000 3500 4000 45001000

1500

2000

2500

3000

3500

4000

X (m)

Y (

m)

DEVS based model + Turbulence

020406080100120140

Time [min]

1500 2000 2500 3000 3500 4000 45001000

1500

2000

2500

3000

3500

4000

X (m)

Y (

m)

DEVS based model + Turbulence + Fire spotting

020406080100120140

Time [min]


Conclusions

Conclusions

This formulation emerges to be suitable for both Eulerian (LSM) and Lagrangian(DEVS) approaches to model dangerous situations as:

the increase in the ROS as a consequence of the pre-heating of the fuel bythe hot air and generation of new fires due to fire spotting phenomenon.the spread of the fire in the direction normal to the mean wind (flanking fire)and opposite to the mean wind direction (backing fire).the propagation of fire across the roads, rivers, firebreak lines etc. whereboth LSM and DEVS fail due to null ROS in the absence of the ground fuel.

DEVS uses the bisector of the angle between the marker and its neighbours aspropagation direction of the fire perimeter which generates differences with theLSM. Such differences result in ”additional” flanking fires, which however providea more ”realistic” fire contour.

The LSM and DEVS based models show a variety of behaviours in response tothe random phenomena so that it is not possible to provide a unique conclusionabout their effects on the models’ differences.


Conclusions

Thank You for your attention !


wildland fire modelling including turbulence and fire spotting · inderpreet kaur modelling random...

Documents