Effect of two-way coupling on the calculation of forest firespread: model development

A. M. G. LopesA,B, L. M. RibeiroA, D. X. ViegasA and J. R. RaposoA

AADAI (Associacao para o Desenvolvimento da Aerodinamica Indistrial), DEM (Department of

Mechanical Engineering), University of Coimbra, PT-3030 Coimbra, Portugal.BCorresponding author. Email: antonio.gameiro@dem.uc.pt

Abstract. The present work addresses the problem of how wind should be taken into account in fire spread simulations.The studywas based on the software systemFireStation, which incorporates a surface fire spreadmodel and a solver for the

fluid flow (Navier–Stokes) equations. The standard procedure takes the wind field computed from a single simulation inthe absence of fire, but this may not be the best option, especially for large fires. The two-way coupling method, however,considers the buoyancy effects caused by the fire heat release. Fire rate of spread is computed with the semi-empiricalRothermel model, which takes as input local terrain slope, fuels properties and wind speed and direction. Wind field is

obtained by solving themass, momentum and energy equations. Effects of turbulence on themean flow field are taken intoaccount with the k – e turbulence model. The calculation procedure consists of an interchange between the fire spreadmodel and the wind model through a dynamic interaction. The present work describes the first part of this research,

presenting the underlying models and a qualitative sensitivity analysis. It is shown that the update frequency for thedynamic interactionmarkedly influences the total calculation time. The best strategy for updating thewind field during thefire progression is presented. The dependence of results on mesh size is also described.

Additional keywords: fire spread simulation, fire–wind interaction, surface fire, wind simulation.

Received 19 March 2016, accepted 31 May 2017, published online 17 August 2017

Introduction

Prediction of forest fire spread has been a challenge toresearchers and managers for several decades, but, in spite ofthe various models and fire behaviour prediction systems that

have been developed as described by Sullivan (2009a, 2009b,2009c), there is not yet a commonly accepted fire behavioursimulator that can be applied in operational conditions for large

and complex fires. This is due to the complexity of the physicaland chemical processes that are involved in large-scale firesrequiring a large number of input parameters that are not easy toobtain and physical models that are not yet fully developed.

Physical models have been formulated mathematically andimplemented in numerical codes like Grishin (1994), Albini(1996), Linn et al. (2002), Sero-Guillaume and Margerit (2002)

and Mell et al. (2007). Nonetheless, practical applications ofthese models are not yet possible owing to large computationalrequirements and uncertainty regarding the description of key

physical processes. Evaluation of some of these models showederrors of up to an order of magnitude (e.g. Marino et al. 2012).

Simpler semi-empirical models to predict fire spread like the

one developed by Rothermel (1972, 1983) have been of moreeffective use in fire behaviour prediction both in scientific and inoperational work (Finney 1998; Andrews et al. 2008). In spite ofits limitations, like the fairly small number of fire spread

regimes that it considers and the limited range of conditions

for which it was developed and validated, Rothermel’s modeldeveloped for surface fire spread prediction is still one of mostwidely applied in fire simulators. In the present study, we shall

assume this scope of application as well, although other firespread modes like crown fires and spot fires can be added to thefire simulator using adequate models to link surface to crown

and spot fires.It is commonly accepted that the fuel bed properties, terrain

slope and wind velocity are the most important factors affectingfire behaviour, namely its rate of spread (ROS). Many studies

have shown the particular importance of wind velocity near thefire front (see Clark et al. 1996a, 1996b; Viegas 2004a; Linn andCunningham 2005) and the role of fire-induced wind in modi-

fying fire spread (see Viegas 2004a, 2006). The present workwill be dedicated mostly to this parameter and to its role on firebehaviour. Atmospheric wind flow is a boundary layer-type

turbulent flow of air that is modified by the local topography, thelevel of instability of the atmosphere and its interaction withroughness elements like vegetation, houses, rocks and other

ground-cover obstacles upwind from the fire front. In order toestimate fire behaviour correctly, it is necessary to predictchanges in wind velocity components induced by the presenceof the fire in the portion of the atmospheric boundary layer

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International Journal of Wildland Fire 2017, 26, 829–843

https://doi.org/10.1071/WF16045

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surrounding the area of the fire. A forest fire affects local windconditions considerably. The wind patterns that would exist inthe absence of a fire are replaced by localised flow structures

resulting from the interaction between the atmospheric wind andthe convection induced by the flaming combustion front. Severalauthors have analysed the coupling between the fire and the

atmospheric flow (Clark et al. 2004; Viegas 2004b, 2006; Coenet al. 2013) but this has not yet been applied in fire simulators thatcan be used operationally.

Models for simulation of wind over complex topographymay be classified into two main categories: prognostic anddiagnostic models. Prognostic models operate at large scales,with mesh sizes typically larger than 2–4 km. They describe the

dynamics of the atmospheric boundary layer, including predictivecapabilities for the variables field, according to the diurnal cycle.Weather Research and Forecasting (WRF) (Coen et al. 2013) and

(Mesoscale Model (MEMO) (Flassak and Moussiopoulos 1988)fall into this category. As pointed out in Forthofer et al. (2014),owing to the poor mesh resolution, prognostic models are not able

to capture flow effects from topography with significant interestfor fire simulation.Nevertheless, operatingwith a large simulationdomain, prognostic models constitute a good source of input data

for finer-mesh simulations. Diagnostic models provide a steady-state simulation and may be divided into two types. Mass-consistent models (also known as kinematic models) take as inputa first guess of the velocity field obtained from interpolation of the

input data, and modify it so as to achieve a divergence-freesolution. The stability of the atmosphere and other thermal effectsare simulated by adjusting the horizontal and vertical velocity

components. The Nuatmos (Ross et al. 1988) and WindNinja(Forthofer et al. 2009) models fall into this category. Kinematicmodels require very little computational effort.As shown inLopes

(2003) and in Forthofer et al. (2014), they provide rather goodsolutions upwind and above terrain features, but often fail on thelee side, especially if flow separation is present. Navier–Stokessolvers, such as WindWizard (Forthofer et al. 2014) and Wind-

Station (Lopes 2003), representing the second type of diagnosticmodel, constitute a more realistic approach, solving for mass andmomentum (Navier–Stokes equations) conservation, and often

include some form of turbulence closure as well. Application ofsuch models to atmospheric flows has been successfully done inthe past, with the Askervein experiment representing one of best-

known benchmark cases (Raithby and Stubley 1987; Castro et al.2003; Lopes 2003; Forthofer et al. 2014).

Several authors have coupled fire spread algorithms with

weather models (e.g. Clark et al. 2004, Coen et al. 2013). TheWRF-Fire model (Coen et al. 2013) is based on the WRF

software, which is a prognostic weather model developed forlarge-spatial-scale simulations. The WRF-Fire version includes

a model for coupling fire and wind. Coen et al. (2013) provide arather detailed description of this model, with application to atest case on flat terrain. WRF is configured in LES (Large Eddy

Simulation) mode, in order to be applicable to smaller scales(,40-mmesh spacing). On the simulations reported in the latterwork, zero constant slope should be considered, in order to use

periodic boundary conditions. It is not clear how themodelwouldperform in a real complex terrain situation with a real firesituation and what the limits are as concerns spatial resolutionand associated numerical errors and computational effort. The

WRF model has also been used for providing initial conditionsand boundary conditions for theCAWFE (CoupledAtmosphere–Wildland Fire Environment) predictionmodel (Coen 2013). This

model was applied to a specific wildland fire (Coen and Riggan2014) and, according to the authors, the main distinguishing firefeatures were correctly depicted by the simulation. Numerical

weather prediction models, such as WRF, may also be usedfor improving the accuracy of simpler kinematic modelsthrough a downscaling process, as demonstrated in the work

by Wagenbrenner et al. (2016).The present study describes an evolution of the fire simulator

(FireStation) presented in Lopes et al. (2002), now incorporat-ing a wind calculation module that takes into account feedback

from the convective flow induced by the fire on the atmosphericflow and its effect on the spread of the fire. This wind calculationengine is based on the original WindStation model (Lopes

2003), including additional features such as, among others, theinclusion of the energy equation and more flexibility in theassignment of incident velocity profiles. The present paper

describes the underlying physical and numerical models, aswell as simplified simulation scenarios. The fundamental equa-tions for fluid flow and turbulence modelling are presented,

along with a brief description of the numerical methodology fortheir solution. Next, models for fire spread are described,including fire ROS, fire growth and interaction between windand fire. A qualitative sensitivity analysis of this methodology is

undertaken, where the influence of the update frequency for thedynamic interaction between the fire and the wind is studied.The influence of mesh resolution for wind calculation is also

analysed.

Theory background

Wind field modelling

Governing equations

In the present approach, the Boussinesq approximation isadopted, considering that density is constant except in its

contribution to buoyancy forces. As the rate of change inboundary conditions due to fire is very low, the flow is solvedusing a steady-state approach. In fact, air flow velocity is much

greater than fire spread velocity. This means that, during thetime needed for a fluid parcel to travel through the fire-affectedarea, the heat release rate distribution at the ground due to fireremains essentially unchanged – thus, the fluid flow may be

regarded as behaving in a steady-state manner with respect tofire.

The Navier–Stokes equations represent momentum conser-

vation for a fluid flow. Considering the simplifications justdescribed, the formulation in a Cartesian coordinate system (seee.g. White 2011, p. 247), for a steady-state condition and

constant density, is:

@

@xjruiuj� �

¼ @

@xjG@ui@xj

� �þ @

@xjG@uj@xi

� �� @p

@xiþ Bi ð1Þ

where mi ¼ (u, v, w), xi ¼ (x, y, z) for i ¼ (1, 2, 3), r is the fluiddensity, p is pressure, and G ¼ meff ¼ m þ mt is the diffusion

coefficient for momentum, i.e. the effective viscosity, which is a

830 Int. J. Wildland Fire A. M. G. Lopes et al.

function of the turbulence variables, as described below. Buoy-ancy forces are present only for the vertical momentum compo-nent (i 5 3) and are given by:

B3 ¼ rref gb T � yð Þ ð2Þ

whereb is the thermal expansion coefficient, g is gravity,T is the

temperature and y is the temperature corresponding to the dryadiabatic lapse rate (�108C km�1). Thus, in the absence of otherthermal effects, the neutral atmosphere will produce no buoy-

ancy forces.Mass conservation is described by the continuity equation,

for which the steady-state formulation is:

@

@xiruið Þ ¼ 0 ð3Þ

The temperature field, necessary for the computation of

buoyancy forces, is obtained through the solution of the energyequation:

@

@xircpuiT� �

¼ @

@xiG@T

@xi

� �þ ST ð4Þ

In this case, the diffusion coefficient G is obtained as:

G ¼ mPr

þ mPrt

� �cp ð5Þ

with cp as the air specific heat and Pr and Prt as the laminar and

the turbulent Prandtl numbers respectively. The source term STin Eqn 4 accounts for energy exchanged with the ground, whichincludes heat release by the fire. Although not considered in thepresent work, this term may include modelling other heat

sources through raster mapping to take into account, for exam-ple, solar radiation heating. Alternatively, a specified tempera-ture field may be imposed at the ground.

As turbulence effects on fluid flowmay occur on a very largerange of time and length scales, obtaining the solution of thetransport equations for the instantaneous variables field is not

feasible for most practical applications, including atmosphericflows, where length scales may range from centimetres up tokilometres. As awork-around, eddy viscosity turbulencemodels

simulate the effects of turbulence on the mean flow field byincreasing the effective viscosity. One the most widely used isthe k – e turbulence model, proposed by Launder and Spalding(1972, 1974). The turbulence kinetic energy (k), which is a

measure of the flow velocity fluctuations, is related to turbu-lence intensity TI as:

TI ¼

ffiffiffiffiffi23k

qV

) k ¼ 3

2TIVð Þ2 ð6Þ

whereV is the local average velocitymagnitude. The dissipation

rate of turbulence kinetic energy (e) is related to the dissipationlength scale L as follows:

L ¼ Cmk3=2

e) e ¼ Cm

k3=2

Lð7Þ

Turbulence kinetic energy and its dissipation rate are used to

compute turbulence viscosity with the following equation:

mt ¼ Cmrk2

eð8Þ

In the original formulation of the k – emodel, the ‘constant’Cm

was adjusted to the value Cm ¼ 0.09, based on laboratoryexperiments for flow over a smooth plate. As stated in Raithbyand Stubley (1987), this value is to be corrected for the case ofatmospheric flows. Data analysis from several atmospheric stud-

ies byPanofsky andDutton (1984) providedCm¼ 0.033 for awiderange of surface conditions and scales. This is the value adopted inthe present work. The turbulence kinetic energy and its dissipation

rate are described by the following transport equations:

@

@xjrujk� �

¼ @

@xjmþ mt

sk

� �@k

@xj

� �þ P1 þ G � re ð9Þ

@

@xjruje� �

¼ @

@xjmþ mt

se

� �@e@xj

� �þ ek

C1P1 þ C1C3G � C2reð Þ

ð10Þ

where sk and se are turbulence Prandtl numbers. The term P1

represents the production rate of k as the result of the velocity

gradients:

P1 ¼ �ru0iu

0j

@ui@xj

¼ mt@ui@xj

þ @uj@xi

� �@ui@xj

ð11Þ

The term G (Markatos et al. 1982; Henkes et al. 1991)represents the production or destruction of turbulence due to

buoyancy forces and has the following formulation:

G ¼ �bgmtst

@T

@zð12Þ

The model constants are:

sk ¼ 1:0; se ¼ 1:3; C1 ¼ 1:45; C2 ¼ 1:9; C3 ¼ tanhwffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

u2 þ v2p

ð13Þ

Effects of buoyancy on turbulence dissipation are less

understood than the effects on turbulence kinetic energy andthere is no consensual formulation for this. The term C3, asformulated above (Henkes et al. 1991), cuts buoyancy effects on

dissipation for horizontal shear layers and progressively allowstheir influence as the velocity vector approaches the verticaldirection. The remaining constants are given in Launder andSpalding (1972, 1974).

Boundary conditions and initialisation

At the ground boundary, a logarithmic law is assumed forvelocity variation with height, z, above the ground:

V

V�¼ 1

kln

z

z0

� �ð14Þ

Two-way coupling: model development Int. J. Wildland Fire 831

where V is the velocity magnitude, k ¼ 0.41 is the von Karmanconstant and z0 is known as the roughness parameter, which is

taken as a fraction f of a typical roughness height (h0) through theequation z0 ¼ fh0. Values for f range from 0.03 to 0.25,depending on the layout and shape of the roughness elements

(ground cover elements such as vegetation). A conservativevalue of f ¼ 0.15 is recommended by Plate (1982) for mostnatural surfaces. The ratio z/z0must always be higher than unity,

which means that the height of the first calculation level (centreof the control volume adjacent to the ground) should be higherthan the roughness parameter z0. The friction velocity V* is ameasure of the ground shear stress, tw ¼ rV 2

� and provides the

boundary condition for the momentum equations:

tw ¼ r Vk= lnz

z0

� �� �2

ð15Þ

Eqn 15 is evaluated at the first computational node near theground.

Boundary conditions for the turbulence quantities are obtainedbased on the hypothesis of local equilibrium. It can be provedthat, based on this assumption, turbulence kinetic energy is givenby (Raithby and Stubley 1987; Castro et al. 2003):

k ¼ V 2�ffiffiffiffiffiffiCm

p ð16Þ

which constitutes the boundary condition for k. The dissipationrate near the ground is given by:

e ¼ V 3�

kdð17Þ

with d as the distance of the first node to the ground.

For the energy equation, the ground boundary condition is theheat flux rate exchange with the ground. If a local fire exists, thepropagation model provides this value, to be included in the termST, inEqn 4. This is computedwithEqn 30, to be presented below.

If no fire exists, an adiabatic boundary is a reasonable assumption.At downstream and lateral boundaries, a null gradient condi-

tion is imposed for all variables (zero first derivative). One

should be aware that boundaries are always subject to someuncertainty; consequently, the wind calculation domain shouldalways be larger than the area of interest, in order to minimise

the effect of these boundaries on the region of interest.For the top boundary, null gradients are assigned to all

variables.Initial values must be assigned to the whole 3-D wind

calculation domain – this includes velocity magnitude anddirection, as well as turbulence quantities and temperature.Boundaries where inflow is verified are considered as inlet

boundaries, with conditions that remain unchanged during thecalculation. Temperature values are obtained from a specifiedprofile or computed from a selected class of atmospheric

stability. For velocity and turbulence quantities, fully developedprofiles typical of atmospheric flows are internally available inthe software, such as the example shown in Fig. 1. Velocity field

initialisation may also be done based on data available frommeteorological stations. In this case, wind speed and direction ateach meteorological station are employed to compute the shapeof the local boundary layer, using a power law equation of the

following form:

V=V1 ¼ z=dð Þ1=a ð18Þ

where VN is the velocity at the top of the boundary layer, d is theboundary layer height and a is an exponent characterising theboundary layer shape (Plate 1971). The profile, characterised by

the values VN, d and a, is determined from knowledge of thevelocity at two heights, and an estimated value of the boundary

(a) (b) (c)

1

10

100

1000

0 10 20

Z (

m)

V (m s�1)

1

10

100

1000

0 0.125 0.25 0.375 0.50

TI

1

10

100

1000

0 5 10 15 20 25

L (m)

Fig. 1. Example of fully developed profiles for wind field initialisation: (a) velocity; (b) turbulence intensity; (c) turbulence length scale.

832 Int. J. Wildland Fire A. M. G. Lopes et al.

layer height. It is assumed that wind direction is constant withheight at the station location. Values at each computational

mesh node are calculated through bi-linear interpolation fromeach available station location, using the inverse of the distanceas the weighting factor.

Numerical solution

The mesh where the flow solution takes place is of a

structured type (regular arrangement of parallel epipedic ele-ments), corresponding to a rectangular computational domain.Themesh isCartesianwith uniform spacing in the horizontal (x, y)

plane, running from west to east and from south to north, asexemplified in Fig. 2a. In order to properly describe the topo-graphic characteristics, a terrain-following (boundary-fitted) coor-

dinate system is adopted in the vertical direction, as depicted inFig. 2b. The mesh is generated algebraically, using a gradualvariable expansion factor in the vertical direction, computed so asto satisfy the specified number of vertical levels, the near-ground

first-level height and total domain height, ensuring a smoothtransition in the mesh spacing.

The transport equations are transformed from their original

Cartesian form into a generalised coordinate form using thechain rule:

@f@xi

¼ @x@xi

@f@x

þ @Z@xi

@f@Z

þ @x@xi

@f@x

¼ xxi@f@x

þ Zxi@f@Z

þ zxi@f@zð19Þ

where terms like xxi are the contravariant metrics of the

transformation, relating the computational coordinates (x, Z, z)and the physical coordinates xi (x1¼ x, x2¼ y, x3¼ z), andf is ageneric variable (u, v, w, k, e or T). After some mathematical

manipulation, the transport equation for a generic variable f is(see Lopes et al. 1995 for additional information):

@

@xjJrUjf� �

¼ @

@xjJ G gij

@f@xj

� �� �þ b ð20Þ

where Uj is a generic contravariant velocity, given by:

U1 ¼ xxuþ xyvþ xzw; U2 ¼ Zxuþ Zyvþ Zzw;

U3 ¼ zxuþ zyvþ zzwð21Þ

and J is the Jacobian of the transformation:

J ¼xx xZ xz

yx yZ yz

zx zZ zz

ð22Þ

where the quantities in the determinant are the co-variant

metrics, which are a function of the contravariant metrics. Inturn, b includes all terms not fitting in the diffusion term (crossderivatives, momentum sources due to temperature, pressure

gradients and other terms, depending on the particular concre-tisation for f). The generalised coordinate form of the massconservation equation is:

@

@xjJrUj

� �¼ 0 ð23Þ

Owing to fact that the mesh is Cartesian in the (x, y) plane and

the vertical mesh lines are straight, some of the metrics are null(e.g. xy, xz, xz). This allows some simplifications in the final formof the generalised equations, with benefits in terms of computa-

tional time and memory requirements. After discretisation andintegration, the equations are cast in the general algebraic form:

aPfP ¼Xnb

anbfnb þ b ð24Þ

This equation relates the value of the generic variable f(velocity components, turbulence quantities and temperature) at

location P with its neighbour (nb) values. Coefficients aP andanb account for diffusion and advection fluxes.

(a) (b)

10.

Height(m)

52.94.136.178.220.262.304.346.388.430.472.514.556.598.640.682.724.766.808.850.

Fig. 2. Example of mesh for wind calculation. (a) Top view, showing the Cartesian mesh component and terrain topography coloured according to local

height (area of 22.5 � 18.5 km). (b) Lateral view for a small region from (a), showing the terrain-following lines in a resampled mesh.

Two-way coupling: model development Int. J. Wildland Fire 833

A control volume approach is adopted for the integration ofthe transport equations. The SIMPLECalgorithm (VanDoormaaland Raithby 1984), which is a modification of the original

SIMPLE algorithm proposed by Patankar (1980), is employedas a means for the segregated solution of the primitive Cartesianvelocity components (u, v, w) and pressure. To account for the

non-staggered mesh arrangement, where all variables are locatedat the control volume (CV) centres, the Rhie–Chow interpolationprocedure (Rhie and Chow 1983) is implemented.

The iterative process is considered to be converged when allthe normalised residuals are lower than a predefined valueRconv.

Max Ru;Rv;Rw;Rm;Rk ;Re;RTð ÞoRconv ð25Þ

The total normalised residual for the transport equations isdetermined as follows:

Rf ¼Xall

aPfP �Pnb

anbunb þ b

uaP fmax � fminð Þ

0BB@1CCA=Xall

u ð26Þ

where (fmax –fmin) quantifies the amplitude of the variablef in

the calculation domain and u is the volume of each computa-tional cell.

Fire spread modelling

Fire rate of spread

The fire behaviour model implemented in the present work isbasedonRothermel’s surface fire spreadmodel (Rothermel 1972),

mainly owing to the latter’s well-established character and versa-tility of application. This is a semi-empirical model developed tobeusedunder awide rangeof conditions.Rothermel’smodel takes

as input fuel characteristics, wind speed and direction and terrainslope, and provides as output the fire ROS along the main spreaddirection. The fundamental equation of this model expresses an

energy balancewithin a unit volume of the fuel aheadof the flame.It illustrates the concept that the ROS is governed by the ratiobetween the rate of heating of the fuel and the energy required forbringing that same fuel to ignition. The fundamental equation in

the original proposal is modified to assume a vectorial character(Lopes et al. 1995), allowing for different directions betweenwindand slope:

~R ¼ Rb~1þ~jw þ~js

�) ~R ¼

Irxf ~1þ~jw þ~js

�rbe f Qi

ð27Þ

In Eqn 27, Rb is the rate of spread with zero wind speed andzero terrain slope, Ir is the reaction intensity, which representsheat release per unit area of the flame front (Wm�2) and xf is thepropagating flux ratio, i.e. the fraction of heat release that is

responsible for fuel heating and subsequent ignition. The sym-bol~1 represents a unit vectorwith the same direction as~jw þ~js,which are both vectorial quantities. The denominator of this

equation represents the heat sink, with rb as the bulk density,which is the mass of fuel per unit volume, ef is the effectiveheating number, quantifying the ratio between the bulk density

and the mass of fuel involved in the ignition process, andQi, theso-called heat of pre-ignition, is the heat required to bring a unitmass of fuel to ignition including the energy to dehydrate thefuel. The basic rate of spread Rb is affected by the presence of

wind and terrain slope through the factors~jw and~js respectively.Both these factors act directionally as a conditioning factor on theROS. As may be seen, each contribution (wind and slope) has its

own direction, which is added to the omnidirectional contributionof the zero-slope, zero-wind, basic ROS. Wind velocity forapplication of the previous equation should be taken at a vertical

distance from the ground dependent on the midflame height.Albini and Baughman (1979) considered the average valueresulting from the integration of the logarithmic law between

the vegetation top and the flame height as follows:

Vf

V1

¼1þ 0:36 hr=hf

� �ln z1þ 0:36hr

0:13hr

� lnhf =hr þ 0:36

0:13� 1

� �ð28Þ

where Vf is the computed average wind velocity value, V1 is thewind velocity computed at the first vertical level, hf stands forflame height and hr is the vegetation height.

All quantities in Eqn 27 are computed using fuel character-istics as defined by the fuel model concept (Albini 1976;Anderson 1982). These characteristics include fuel load, fuel

particle heat content, surface-to-volume ratio or fuel bed depth.The environment characteristics are also taken into account withparameters related to fuel moisture. Most of fuel properties are

supplied for different fuel size classes and dead–live classifica-tion. Because different size classes are allowed, fuel load,

(a) (b)

Fig. 3. Fire shape predicted by two ellipse models. The black dot

represents the ignition point: (a) double-ellipse model; (b) single ellipse

model.

Burning region

Smouldering region after firefront passage

Non-burning region

Fig. 4. Representation of coarse (thicker lines) and fine meshes used in

wind and fire calculation respectively. Burning and smouldering regions are

represented by shaded areas. In this example, there are 25 fire cells to each

wind cell.

834 Int. J. Wildland Fire A. M. G. Lopes et al.

surface-to-volume ratio andmoisture content are averaged usingthe total surface area as the weighting factor, following thesuggestion of Rothermel (1972). Dead and live components are

treated separately, each being used to compute its own reactionintensity. The final value for the reaction intensity is obtained asthe sum of the corresponding values for dead and live fuels.

Apart from fire ROS, Rothermel’s model provides severaloutput quantities concerning fire behaviour, such as firelineintensity (Wm�1), flame length (m), residence time (s) and rate

of heat release, also referred as reaction intensity Ir (W m�2).This last variable describes the fire energy source for the wind–fire interaction calculation. Residence time multiplied by thefire ROS allows determination of the flame front depth (m),

which, along with the fire perimeter, defines the burning region.

When a flame extinguishes, which occurs after the local burningtime exceeds the residence time, an exponential decay isassigned to Ir, corresponding to the smouldering phase. This is

computed according to the following equation:

Ird tð Þ ¼ fdIre�t=td ð29Þ

where t represents time, Ird is the reaction intensity during thedecay phase, fd is an initial reaction intensity cut-off due to flame

extinguishment and td is a relaxation timecharacteristic of the fuel.

Fire shape

Although Rothermel’s model describes fire ROS along themaximum spread direction, it does not provide information

8.788.347.907.457.016.576.135.685.244.804.353.913.473.022.582.141.701.250.810.37

Wind speed(m s�1)

(a) (b)

(c)

(d)

34.9737.9840.9843.9946.9950.00

31.9728.9625.9622.9519.94

Temp.(�C)

Fig. 5. Velocity field 10 m above ground and burned area shape. (a) No fire situation; (b) 450 ha of burned area; (c) 1000 ha

burned area; (d) vertical plane visualisation along the fire central line depicted in (c). Domain height is 2000 m.

Two-way coupling: model development Int. J. Wildland Fire 835

about ROS along other directions. This information is, never-theless, necessary to compute fire shape. Several models existfor fire shape, mostly based on ellipses whose geometrical

parameters are given as a function of the wind speed (e.g.Green et al. 1983). The effect of topography is modelledthrough the addition of a slope-equivalent wind speed that

would produce the same effect as the actual slope on the fireROS. The ‘effective’ wind speed, to be considered on the fireshape model, is obtained through the vectorial summation of

the actual wind velocity and the equivalent wind velocity dueto slope.

The model proposed by Anderson (1983), describing fireshape through a double ellipse, as depicted in Fig. 3a, represents,

according to the authors, the best practical realistic approxim-ation for observed fire shapes under a variety of circumstances.This model, nevertheless, does not cope with zero-wind situa-

tions. In the present implementation, when the local equivalentwind speed at midflame height falls below 0.2 m s�1, the doubleellipse formulation is replaced by a single ellipse (Fig. 3b), as

described in Alexander (1985).

Fire propagation

The local model for fire ROS, together with the globalmodel for fire shape, defines the physical behaviour of the fire.As input parameters are usually space-dependent owing to

terrain topography and non-uniform fuels distribution, not tomention time dependence due to changing atmospheric condi-tions, simulations must be carried out on a step-by-step basis in

order to account for the abovementioned spatial and temporalvariations. In terms of implementation, the fire time evolutionmay be regarded either as vector- or as raster-based. In the

former approach, the application of Huygens’ principle(Richards 1990) leads to the definition of the new fire frontas the envelope of small ellipses computed locally from theprevious fire front. For the raster approach, the one adopted in

the present work, the calculation area is divided into rectangu-lar- or other-shaped cells, where conditions are assumed aslocally uniform. Following Dijkstra’s dynamic programming

algorithm (Kourtz and O’Regan 1971), for each cell along thefire front, a contagion process is computed by assigningignition instants for a predefined set of neighbour cells.

Examination of all potential ignition candidates allows theidentification of the next ignited cell – the correspondingignition instant makes the solution advance in time. Thus, the

time progression is not constant, depending on the ignitioninstant of each cell.

Interaction between wind and fire

Wind and fire calculations are carried out with differentmeshes. Each computational cell for wind calculation (wind

cell) is defined by joining together several finer cells (fire cells)where fire propagation is calculated. Owing to the differentresolution of wind and fire meshes, the fire front may occupy a

fraction of a (coarse) wind cell, as exemplified in Fig. 4. Thetotal heat release rate per unit volume SF at the cells adjacent tothe ground, to be included in the source term ST in the energyconservation Eqn 4, is computed based on the reaction intensity

given by the fire spreadmodel in each fine cell located inside thecoarse cell as follows:

SF ¼Xntn¼1

IrnAn

!=v ð30Þ

where v is the coarse cell volume, An is the surface area of eachof the total nt fine cells located inside the coarse cell and Irnrepresents the reaction intensity Ir due to the fire in each fine celln. Total heat released at each coarse cell is used to compute anew temperature distribution within the air, by solving Eqn 4,

which affects buoyancy and turbulence production or destruc-tion, through the source terms in Eqns 1, 9 and 10.

The calculation procedure consists in an alternation between

the fire spread calculation and the wind calculation. After thefire area increases by a certain amount, the new heat release ratedistribution at the ground is passed into the wind calculationmodule. After convergence of the fluid flow and energy equa-

tions, the updated wind field is used as input to continue the firespread calculation.

(a)

(b)

0

100

200

300

400

500

600

700

800

900

1000

0 50 100 150 200 250 300

Are

a (h

a)

Static

Updt 250 (ha)

Updt 150 (ha)

Updt 100 (ha)

Updt 50 (ha)

0

100

200

300

400

500

600

700

800

900

1000

0 50 100 150 200 250 300

Time (min)

Updt 50 (ha); 3x

Updt 50 (ha); 2x

Updt 50 (ha)

Fig. 6. Influence of wind update criterion on fire area as function of time:

(a) update (Updt) for constant area increase; (b) update for constant area

fraction increase.

836 Int. J. Wildland Fire A. M. G. Lopes et al.

Results

Herein, we present a sensitivity analysis of the effect of somenumerical parameters on the calculation of the interaction

between wind and fire. These include the criterion for windupdate and the resolution adopted for the wind field calculation.In the companion paper (A. M. G. Lopes, M. G. Cruz and D. X.

Viegas, unpubl. data), application of the models described to alaboratory fire and to a large-scale real fire are reported.

Simulations were carried out for a digital terrain model incentral Portugal, as depicted in Fig. 2a, encompassing an area

of 22.5� 18.5 km (41 625 ha). The topography is fairly smoothwith a few isolated hills – terrain height is between 40 and500m above sea level, with localisedmaximum inclinations of

308. The size of the fire mesh for fire simulation is 50 m in bothdirections. For fire spread, an idealised uniform fuel bed was

adopted, with a load of 0.7 kg m�2 for 1-h fuels, 0.3 kg m�2 for10-h fuels and 0.6 kg m�2 for 100-h fuels, with fuel depth

0.6 m (the same value was adopted as terrain roughness forwind calculation). Surface area-to-volume ratio is 6000 m�1

and moisture content is set constant at 10% for dead fuels and

60% for live fuels. Reaction intensity cut-off and relaxationtime (fd and td in Eqn 29) are 0.5 and 30 min respectively. Theincident wind field is from the north, with a velocity of

15 m s�1 at the boundary layer height of 500 m above surface,with 1/a ¼ 0.28 (cf. Eqn 18). This corresponds to a windvelocity of 5 m s�1 at 10 m from the ground. The verticaltemperature profile is set to the adiabatic profile, thus corre-

sponding to neutral stability conditions. The domain top islocated at an altitude of 2000 m. Along the vertical direction,15 to 30 mesh levels were employed, with the first calculation

node at a distance of 2 m from the ground. On the horizontalplane, different resolutions were tested, ranging from 750 to50 m.

Fig. 5a–c shows visualisations of the wind field 10 m aboveground and fire shape at different stages. From these figures,one can easily see the influence of topography on wind, as well

as the air in-draft towards the fire region due to the convectiveplume from the fire. This is clearly visualised in Fig. 5d, whereinformation on the air temperature field is included in a lateralview. In this representation, temperature is corrected for alti-

tude, thus corresponding to potential temperature. Computedmaximum temperatures in the fire region are,240 and 1808C atvertical distances of 5 and 10 m above ground respectively.

Results depicted in Fig. 5 were obtained using a fire areagrowth of 150 ha as wind update criterion, i.e. wind field is

Updt 50 (ha)

Updt 100 (ha)

Updt 150 (ha)

Updt 250 (ha)

Static

(b)

(a)

0

2

4

6

8

10

12

0 50 100 150 200 250 300

Tim

e (m

in)

Area (ha)

Updt 50 (ha); 3x

Updt 50 (ha); 2x

Updt 50 (ha)

Fig. 7. Influence of wind update criterion on: (a) calculation time; (b) fire size and shape 4 h after ignition. Gridlines in the graph are

1000 m apart. Red dot represents ignition location.

Table 1. Calculation domain and mesh data

Horizontal

resolution (m)

Horizontal

domain size (km)

Total number

of nodes

1250 22.5� 18.5 20� 16� 15¼ 4800

750 22.5� 18.5 31� 26� 15¼ 12 090

500 22.5� 18.5 46� 38� 15¼ 26 220

250 22.5� 18.5 91� 76� 15¼ 103 740

150 22.5� 18.5 151� 126� 20¼ 380 520

100 8.6� 9.4 87� 95� 20¼ 165 300

50 8.6� 9.4 173� 189� 20¼ 653 940

Two-way coupling: model development Int. J. Wildland Fire 837

updatedwhen the fire area increases 150 ha since the last update.

A spatial resolution of 500 m for the wind mesh with 15 verticallevels was used. The influence of these parameters will now beinvestigated.

Study on dynamic interaction update frequency

For the first set of tests, to be presented next, wind is updated

every time the fire area increases by a specific constant valuesince the last wind update. Fig. 6a shows the fire area evolutionwith time for different values of fire area increase. Theseresults are compared with the ‘static’ case (no two-way cou-

pling), for which the initial wind field, computed with no heattransfer effects, is considered at all stages of fire growth. Thefirst noticeable characteristic is that fire spread is much slower

when dynamic interaction is taken into account. This behaviourcan be anticipated, because the buoyancy induced by the firecreates an inflow wind towards the fire region, which is

opposed to fire growth direction, a clearly evident feature inFig. 5b, c. The effect of wind update frequency influences firedevelopment mainly in the early stages after fire start. As

Fig. 6a shows, as soon as thewind is first updated, the change in

the wind magnitude and direction causes the fire ROS todecrease. Further updates do not change the wind by as large anamount as the first update. This suggests that it is not that useful

to update the wind based on relatively small amounts of firearea increase – a fractional increase would be a wiser criterion,mainly in terms of computational effort. In Fig. 6b, the results

obtained using various values of fractional increase are shown,starting from the first wind update (Updt) at 50 ha. Thus, forexample, for the case ‘Updt[50ha];2�’, the wind is updated for

fire areas of 50, 100, 200, 400 and 800 ha, which leads to fivewind calculation runs, instead of 20 runs for a constant areaincrease of 50 ha. As may be seen, even for the 3� case, firegrowth does not differ markedly from the constant 50-ha sit-

uation in the simulated temporal region. This reflects, never-theless, on computational time, because wind calculation isresponsible for more than 99% of the computational effort, as

may be appreciated from Fig. 7a. This emphasises the conve-nience of using a non-constant area increase as a wind updatecriterion. Fig. 7b, in turn, represents the fire shape 4 h after

ignition, for several update modes, including the static (nointeraction) case.

Study on wind calculation resolution

The choice of the spatial resolution adopted for the wind cal-culation mesh should be done with care. Higher mesh resolu-

tions reduce the numerical discretisation errors of the equations.In the case of simulations for topography, the mesh size alsodictates how correctly terrain features are described. It is a

general rule that each geometrical feature should be meshedwith at least 10 nodes. Nevertheless, owing to the fractaldimension of terrain characteristics, this criterion is virtually

impossible to satisfy and, thus, truly mesh-independent resultsare not obtainable. In the present work, several spatial resolu-tions in the horizontal plane for the wind mesh were adopted,ranging from 50 m, which corresponds to the fire mesh resolu-

tion for fire spread calculation, up to 1250 m. For the coarserwind meshes, a larger domain was considered (depicted inFig. 2a), in order to keep boundaries sufficiently far from fire in

terms of calculation nodes. A constant number of 15 verticallevels was considered in all situations. Table 1 synthesisespertinent data for the different wind meshes. For these simula-

tions, the dynamic interaction was updated starting at 50 ha ofburned area, and using an increment factor of 2�.

Results concerning rate of fire area growth are presented in

Fig. 8a. It may be seen that a considerable dependence onhorizontal resolution exists, though, as expected and desirable, itis attenuated for finer wind meshes; in fact, it may be stated thatresults for 100- and 50-m resolution are reasonably similar.

Fig. 8b shows that computation time increases in a strong non-linear manner with mesh resolution – in the interpretation of thisgraphic, one should keep in mind that the two finer-mesh data

sets correspond to a smaller domain.Results for fire shape 4 h after ignition are depicted in Fig. 9.

Results for the three coarsest meshes show a marked depen-

dency on mesh spacing. However, fire shapes for the two finermeshes (100 and 50 m) are rather similar. Visualisation of thewind flow, framed on the whole wind calculation domain for the50-m mesh, is presented in Fig. 10. For better visualisation, a

(b)

(a)

0

100

200

300

400

500

600

700

800

900

1000

Are

a (h

a)

Time (min)

Mesh size 1250 (m)Mesh size 750 (m)Mesh size 500 (m)Mesh size 250 (m)Mesh size 150 (m)Mesh size 100 (m)Mesh size 50 (m)

0.1

1

10

100

1000

0 50 100 150 200 250 300 350

0 200 400 600 800 1000 1200 1400

Tim

e (m

in)

Mesh size (m)

Large domain

Small domain

Fig. 8. Influence of mesh resolution for wind calculation: (a) fire area as

function of time; (b) computation time.

838 Int. J. Wildland Fire A. M. G. Lopes et al.

sampling of one out of four vectors was performed in eachdirection, so the wind field is presented on a 200-m-spacedmesh. One may note the strong in-draft all around the fire

perimeter. Fig. 11a, b shows a visualisation in a vertical planeperpendicular to themainwind direction (transect represented inFig. 10), with colour contours for temperature and velocity

magnitude.As the burning region assumes a doughnut-like shape,the central part, characterised by a lower heat release rate, issubject to air down-draft. The buoyant region on the flaming

front, however, produces a rising plume, with a vertical windcomponent that is even stronger than the horizontal component,creating a large vortex surrounding the fire. In this figure, one

may identify a typical dimension of ,500 m for the buoyantplume. Taking into consideration that a mesh size of 50 m wasfound to be appropriate for this case, this is consistent with ameshing requirement of 10 nodes for typical flow features.

Dependence of the results on the number of vertical levels ismuch weaker, as may be appreciated from Fig. 12, where allresults were computed using a 100-m horizontal resolution,

keeping the first vertical level 2 m above ground.

Conclusions

We propose a two-way coupling method for fire behaviourprediction, where the buoyancy effects caused by the fire heatrelease are fully simulated. The present paper describes the

underlying models for wind field and fire spread calculation.The wind field calculation consists of a Navier–Stokes solver

(a) (b)

Mesh size 250 (m)

Mesh size 500 (m)

Mesh size 750 (m)

Mesh size 1250 (m)

Mesh size 50 (m)

Mesh size 100 (m)

Mesh size 150 (m)

Mesh size 250 (m)

Fig. 9. Fire shape 4 h after ignition. Gridlines in the graph are 1000 m apart. Red dot represents ignition

location. (a) Results for coarser meshes; (b) results for finer meshes.

20.0019.0518.1017.1516.2015.2514.3013.3512.4011.4510.509.558.607.656.705.754.803.852.901.951.00

200 m Wind speed(m s�1)

Fig. 10. Fire shape and wind field 10m above ground computed on a 50-m

grid. Framing line corresponds to calculation domain boundaries. For ease of

visualisation, wind vectors are displayed on a 200-m grid. Fire region is

coloured according to rate of heat release. White line represents the transect

used in Fig. 11.

Two-way coupling: model development Int. J. Wildland Fire 839

applied to a boundary fitted mesh. For simplicity, the mesh isCartesian in the horizontal plane and terrain-following in thevertical direction. This considerably simplifies the calculation

procedure, leading to quite modest calculation times for aNavier–Stokes solver. Fire spread is computed with Rothermel’smodel for surface fire, with a cell-based approach, where fire

progression takes place through a contagion process. Fire shapeis given by ellipse-type models. The wind field is computed on amesh made by agglomerating cells from the fire calculation

mesh. The reaction intensity given by the fire spread modelprovides the heat release rate to be used for computing buoyancyfor the wind calculation.

We conducted a sensitivity analysis studying the influence of

numerical parameters on the calculated wind field and fire shapefor a simplified scenario. It is shown that the two-way couplingis most important in the early stages of fire development, but its

importance decreases as the fire area gets larger. As the windcalculation represents themajority of the computational effort, itis advisable to rationalise the number of wind calculations. It is

shown that updating the wind field at constant values for the firearea increase does not lead to significant differences on resultswhen compared with an update based on an area fractionincrease of 2� or even 3�. Furthermore, calculation times are

reduced to 1/5�with the latter option. Studies of the influence ofwindmesh resolution clearly indicate that the number of vertical

Temp.(�C)

Wind speed(m s�1)

47.4851.4555.4359.4063.3867.35

43.5039.5335.5531.5827.6023.6319.65

22.5024.00

21.0019.5018.0016.5015.0013.5012.0010.509.00

6.007.50

500 m

600 m

(a)

(b)

Fig. 11. Vertical plane perpendicular to incident wind direction, passing though fire shape centre (transect depicted in

Fig. 10): (a) temperature field; (b) velocity magnitude field.

20 levels

15 levels

30 levels

Fig. 12. Dependence of fire shape on number of vertical levels, using a

100-m mesh.

840 Int. J. Wildland Fire A. M. G. Lopes et al.

calculation levels affects the solution less than the horizontalresolution. The classical concept of mesh-independent resultson computational fluid dynamics is difficult to apply to wind

over complex topography, owing to the fractal characteristics ofterrain shape. Results indicate that, at least for the simulatedcases, horizontal wind mesh spacing should not be larger than,

typically, 50 to 100 m.

Conflicts of interest

The authors declare no conflicts of interest.

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Variable Definition

ap, anb coefficients in the general discretised equation

Ac area of the coarse cell (m2)

An area of the fine cell (m2)

b generic source term

cp specific heat (J kg�1 K�1)

Bi buoyancy force per unit volume, along direction i

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(Continued)

Table 2. (Continued)

Variable Definition

V1 wind velocity at the first vertical level (m s�1)

Vf average wind velocity (m s�1)

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G generic diffusion coefficient

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b thermal expansion coefficient (unitless)

d boundary layer height (m)

ef effective heating number (unitless)

e rate of dissipation of turbulence kinetic energy (m2 s�3)

f generic variable

m, mt laminar and turbulent dynamic viscosity (N sm�2)

xf propagation flux ratio (unitless)

y temperature for the vertical adiabatic lapse rate (K)

r density (kgm�3)

rb bulk density (kgm�3)

sk, se Prandtl number for turbulence kinetic energy and turbu-

lence dissipation rate (unitless)

x, Z, z computational coordinates (m)

~js slope factor for fire spread (unitless)

k von Karman constant (unitless)

u volume (m3)

~jw wind factor for fire spread (unitless)

tw ground shear stress

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