domingos x.viegas - adai and... · slope and wind effects on fire 145 0 30 60 90 120 150 180 03060...

CSIRO PUBLISHING

www.publish.csiro.au/journals/ijwf International Journal of Wildland Fire, 2004, 13, 143–156

Slope and wind effects on fire propagation

Domingos X. Viegas

Centro de Estudos sobre Incêndios Florestais–Associação para o Desenvolvimento da AerodinâmicaIndustrial, Universidade de Coimbra, Apartado 10131, 3031-601 Coimbra, Portugal.

Telephone: +351 239 790732; fax: +351 239 790771; email: [email protected]

Abstract. The vectoring of wind and slope effects on a flame front is considered. Mathematical methods forvectoring are presented and compared to results of laboratory experiments. The concept of multiple standard firespread directions is presented. The experimental laboratory study, included effects of variable wind velocity anddirection on point source flame fronts on a 30◦ inclined fuel bed.

Introduction

It is common to consider wind and topography, together withvegetation as the dominant factors affecting the spread of for-est fires. Quite often forest fires occur in areas of complextopography, with mountains and ridges that affect the propa-gation of the fire in various ways.The presence of wind makesthis picture even more complicated with the joint interactionof wind, slope and the fire front. In many instances this is thesituation that fire managers and fire fighters have to face.

In order to make the very complex problem of firebehaviour prediction more tractable, the various factors areusually treated separately. The majority of known studiesdeals with either wind or slope effects separately, and evenso they consider only the case in which the rate of spreadis aligned with wind velocity or with the local slope gradi-ent. To our knowledge not even the relatively simple case inwhich wind velocity and slope gradient are aligned is reason-ably studied in the literature. A survey of the literature showsthat, in spite of its great practical importance for the purposeof fire behaviour prediction, the general case of non-alignedwind and slope has not been studied in a systematic way.

Rothermel and his co-workers (cf. Rothermel 1972) pro-posed a simple additive model for wind and slope effectsboth with the same direction and orientation (favourablewind and up-slope propagation), although this situation wasnot included in their experimental program. In a later workRothermel (1983) proposed a vector addition of wind andslope effects for the general case of non-parallel wind andslope. Once again it is not apparent that this model was sup-ported on experimental data. This model with variations isextensively used in most fire behaviour prediction systems.Lopes (1994) used a modified version of the additive modelproposed by Rothermel. Details on both approaches are givenlater in the Appendix.

The present research was initially aimed to determine thedirection and rate of spread of the head fire front in the case ofarbitrary wind and slope. These are necessary inputs for theapplication of any fire spread model that is based on somecontagion algorithm. At an early stage of the experimentaltests it was found that the ‘head’ fire front is not always welldefined and it was found also that the application of simplespread models based on the Huygens algorithm (cf. Catchpoleet al. 1982) cannot describe adequately the observed reality.

Starting from the basic consideration of the existence oftwo independent convection flows—wind and slope inducedbuoyancy, respectively—and assuming an additive effect ofboth flows on the fire front, a simple model based on thevector sum of the corresponding rates of spread is presentedand analysed. An experimental device in which wind of arbi-trary velocity and direction could blow over a plane surfaceinclined at a defined angle was used in this study. The spreadof a point source fire on a homogeneous fuel-bed composedof dead needles of Pinus pinaster was registered with a videocamera and the results analysed in order to assess the validityof the concept and the accuracy of the model predictions.

In spite of the limitations of the experimental programdeveloped so far, the results of the present work show someinteresting points related to this very important problem offire behaviour prediction. As will be shown below it is foundthat the fire front can be decomposed in four main sectionsthat have their own standard fire spread vectors. Each sectionacts with some independence from its neighbours but it is ofcourse linked to them to form a continuous fire front. Thiscontinuity condition applies to all properties of the fire frontand leads to the non-uniformity of the fire spread conditionsin the general case, as will be shown.

This article is considered as a preliminary study on thistopic for the presentation of the problem and of a very

© IAWF 2004 10.1071/WF03046 1049-8001/04/020143

144 D. X. Viegas

simple mathematical model that is proposed to solve it. Theexperimental results, although interesting and original, arestill scarce and do not allow the extraction of definitiveconclusions about the proposed approach.

Theoretical analysis

In this section a physical analysis of the composite effectof wind- and slope-induced convection on the spread of asection of a flaming fire front is presented. In the discussionemphasis is put on the convection effects although it is recog-nised that, in this type of front, radiation from the flame orfrom the reaction zone inside the fuel bed may be the majorheat transfer mechanism responsible for the advance of thefire front. The reason for this lies in the fact that convectionnear the fire front determines the properties of the combus-tion process and therefore also the shape and size of the flamefront and the respective heat transfer fluxes.

Among the various parameters that are suitable for describ-ing the spread of the fire front, its rate of spread �R is the mostwidely used. In this article fire spread shall be characterisedexclusively by �R. Due to its vectorial nature the rate of spreadmust be defined both by its magnitude and direction at eachpoint and time step. As we are considering the propagation ofa surface fire, �R must be defined at each point of the fireperimeter characterised by two coordinates x and y. Thereforewe must have �R(x, y). This study is restricted to fire frontspropagating in homogeneous fuel beds on plane surfaces andsubject to stationary boundary conditions.

Composite slope and wind effects

It is assumed that the fuel bed is uniform and of constant slopenear the section of the fire front that is being considered. Itis also assumed that wind velocity vector remains constantduring the period of analysis.

The very high gas temperatures that occur near or at theflame front are accompanied by differences of density thatinduce a flow designated here as natural convection. Thisnatural convection flow caused by buoyancy exists even infires spreading on horizontal surfaces but it is enhanced bythe presence of a slope. In the absence of disturbances likenearby obstacles, other flame fronts, fuel bed discontinuitiesor edge effects, the flow inside and near the flame front isessentially parallel to a plane defined by the local slope gra-dient �S (direction of maximum slope) and by the verticaldirection. The OY axis in the reference system considered atthe fire front shall be taken as parallel to the maximum slopevector �S previously defined (see Fig. 1). Let us assume thatthe overall effect of natural convection on the rate of spreadof the fire front can be uniquely related to a rate of spreadvector �Rs that is parallel to the slope gradient. Its magnitudeand orientation shall be discussed later.

On the other hand wind can blow from any arbitrary direc-tion in relation to the gradient slope. The convection induced

b

d

R

→

→

→Rs

Rw

X

Y

Fig. 1. Schematic view of the wind and slope effects composition,showing the reference system and the vectorial sum of the slope andwind produced rates of spread.

by the wind is designated here as forced convection and itsmagnitude depends directly on the velocity field near thefuel bed surface. In a similar way we assume that the overalleffect of forced convection on the fire front spread can bedefined by a vector �Rw that is parallel to the wind direction,its magnitude and orientation being discussed later.

We designate the angle between wind-induced rate ofspread vector and the slope-induced rate of spread vector byβ, this angle being measured clockwise. This angle can varybetween 0◦ and 360◦. Given the symmetry of the problem inour discussion we consider only positive values of β and inthe range 0◦ < β < 180◦. As is shown later, in the same fireperimeter multiple wind and slope induced rates of spreadcan be defined. In order to avoid confusion the designationof β0 will be attributed to the angle between the positive windinduced rate of spread and the upslope induced rate of spread.

Considering that �Rw and �Rs, the vectors representing therates of spread derived respectively from wind and slopeeffects, are known and assuming an additive effect of bothfactors, the local rate of spread due to wind and slope isgiven by:

�R = �Rw + �Rs. (1)

In graphical form this equation is represented by the triangleshown in Fig. 1, resulting from the vectorial sum of �Rw and�Rs. In this figure the angle between �R and �Rs is designatedby δ and denoted as the deflection (or deviation) angle.

Slope and wind effects on fire 145

0

30

60

90

120

150

180

0 30 60 90 120 150 180

b (º)

d (

º)

� � 0.1

� � 0.25 � � 0.5

� � 0.75 � � 1.0

� � 1.5 � � 2

� � 3 � � 5

Y

� � 0

Fig. 2. Variation of deflection angle δ as a function of β, for variousvalues of ε.

Using the wind induced rate of spread �Rw as a reference,the following non-dimensional parameters can be defined:

ε = RS

RW

(2)

ξ = R

RW

. (3)

Using these parameters the following non-dimensionalequations can de derived:

tan δ = sin β

ε + cos β(4)

ξ2 = (ε + cos β)2 + sin2 β. (5)

The functions described by equations (4) and (5) can becomputed easily and their general behaviour is shown in Figs2 and 3, respectively.

In Fig. 2 the different variation of δ with β depending onwhether ε < 1 or ε > 1 can be seen. In the limiting case ofε = 1 the angle δ is equal to β/2. When ε < 1 the value of δ

increases monotonically and tends to 180◦ as β increases. Onthe contrary, when ε > 1, δ has a maximum and then tends tozero when β increases.

In Fig. 3 the monotonic decrease of ξ with β can be seen inall cases. For ε < 2 the value of ξ becomes quite small whenβ approaches 180◦.

Non-dimensional forms

Although equations (2) and (3) are already expressed innon-dimensional form, it is convenient to present new non-dimensional forms in order to allow the use of empiricalresults obtained in slightly different conditions and to providelaws that can be applied to other fuel bed situations.

0

2

4

6

8

0 30 60 90 120 150 180b (º)

� � 0 � � 0.1 � � 0.25� � 0.5 � � 0.75 � � 1�

� � 1.5 � � 2

� � 5

� � 3

j

Fig. 3. Variation of the relative rate of spread modulus ξ as a functionof β, for various values of ε.

If we divide both parts of the second member of thoseequations by R0, the so-called basic rate of spread—obtainedfor the same fuel bed in the absence of slope and wind—wehave respectively:

ε = RS/R0

RW/R0= fS

fW

(6)

ξ = R/R0

RW/R0= f

fW

. (7)

In these equations fW , fS and f represent respectively thewind, slope and wind-slope effect on the rate of spread.

For one given fuel bed the non-dimensional functions fW ,fS should depend respectively only on the wind velocity andon the fuel bed slope angle. It is found however that there maybe some dependence on the moisture content and on otheruncontrolled fuel bed properties. In order to cope with thisvariation and also to compare results obtained in laboratory orfield experiments in which fuel moisture content is not a fixedparameter, it is more convenient to use the non-dimensionalform given by equations (6) and (7).

Curvature effects

Up to now we have been considering infinite linear fire fronts.It is well known that, if the fire line has a finite width ora curvature radius that is of the order of magnitude of theflame length, the spread properties of the fire front are quitedifferent from the previous ones (Weber 1989). In principlethe rate of spread is lower in the case of finite width and/orcurved flame fronts. As the functions fW or fS are differentfor straight and for curved fire fronts it is necessary to discusswhich ones we must consider.

In his report Rothermel (1983) considers only straight firefronts and therefore no other conditions can be obtained fromhis fire spread model. On the other hand Rothermel’s model is

146 D. X. Viegas

applied extensively to curved fire fronts. This may be accept-able for practical purposes if the curvature radius of the firefront is greater than 5–10 times the flame length, which is thecase in many instances.

In the present experiments point ignition fires were usedin order to determine the main fire spread direction and themagnitude of the rate of spread vector. Given the scale of theexperiments the condition of quasi-straight fire lines—highvalues of the curvature radius in comparison with the flamelength—is not met; therefore, we shall use the functions fW

and fS corresponding to curved fire fronts obtained in similarconditions, with point ignition fires.

Reference rates of spread

For a given slope angle of the fuel bed we may have eitherup-slope or down-slope propagating fire. It is well knownthat the up-slope rate of spread designated as �Rs1 is muchlarger that the down-slope rate of spread �Rs2, which is usuallyof the order of R0. Depending on fuel-bed properties differentauthors found that �Rs2 can be either slightly higher or lowerthan R0 (Fang 1969; Dupuy 1995; Weise and Biging 1996;Viegas 2004). For most practical purposes we can assume,for simplicity, that �Rs2 ≈ R0.

In order to make our discussion more specific we introducethe following symbols fS1 and fS2 to designate respectivelythe multiplying factors to determine the up-slope and down-slope effects respectively, from R0.The values of these factorsmust be determined either from models or from experiments.As mentioned above fS2 ≈ 1, for practical purposes. The ori-entation of �Rs1 is that of the slope gradient (pointing towardsthe top of the slope) and that of �Rs2 is the contrary.

Similarly for wind effects we must retain two referencerate of spread values: �Rw1 and �Rw2, to define respectively thedown-wind (favourable wind) or up-wind (contrary wind)propagation conditions. Consequently the symbols fW1 andfW2 are used to define the respective multiplying factors.Following the considerations that were made above for con-trary slope effect we can also assume that fW2 ≈ 1. Similarlythe orientation of �Rw1 and �Rw2 is respectively positive andnegative in relation to the wind velocity orientation.

Multiple spread directions

In the case of a fire spreading in a relatively large area wemay have sections of the fire front that are more influencedby the local slope and wind conditions than by the pres-ence of another section of the fire. This will occur if thedistance between two sections of the fire front—for exampleone flank and the other—is larger than 5–10 times the flamelength. In this case, quite common in real fires, each sec-tion of the fire front will act independently from each otherand we may find multiple spread directions dictated by localeffects. This assertion is not against a previous finding of the

(a)

A

B

C

D

(b)

R11

R12

R21

R22

Fig. 4. (a) Fire perimeter showing the four sections and their cor-responding limiting points. Slope gradient is from bottom to top offigure. Wind direction is indicated by vectors tangent at points B and D.(b) Standard spread directions indicated by the dark arrows.

strong interaction between adjacent sections of the fire frontas demonstrated in Viegas et al. (1994, 1998).

Let us consider again the simpler case of a plane sur-face fuel bed under uniform and permanent wind conditions.Under these conditions the fire perimeter at any time stepwill be represented by a convex closed line, i.e. a line with-out kinks or fingers. The normal vector at any point of thisline will always be pointing towards unburned fuel.

If the independent propagation state is reached for givensections of the fire front, we may consider the perimeter ofthe fire divided into various sections, according to the relativeorientation of the wind velocity and slope gradient vectors,as it is shown in Fig. 4a. In this figure points A and C arethose tangent to the slope gradient and B and D are tangentto the wind direction, as shown. Therefore the perimeter of


Table 1. Standard fire spread directions for favourable or unfavourable windand slope conditions along the fire perimeter

Ref. Section Slope Wind Standard rate Non-dimensionalof spread factors

11 a BC FavourableA Favourable R11 ε11 ξ1112 d AB Favourable Contrary R12 ε12 ξ1221 b CD Contrary Favourable R21 ε21 ξ2122 c DA Contrary Contrary R22 ε22 ξ22

AUpslope propagating fire.

the fire is divided in four sections, a, b, c and d, which havethe following cases of standard local fire spread directionspresented in Table 1 and in graphical form in Fig. 4b.

It is obvious that when the wind velocity and slope gradientvectors are parallel, �U//�S, then points A and B coincide andthe same happens to points C and D. In this simpler case thereare only two standard fire spread directions. In the remainderof this article the more general case of non-parallel wind andslope will be considered.

It may happen that one or more of these four standardspread directions is not realistic in physical terms. Such iscertainly the case if a resultant rate of spread vector pointstowards the already burned area. This case must be looked atmore carefully.

If the vector sum of a pair of reference velocities givesa negative result, i.e. a vector pointing towards the alreadyburned fuel, we have to consider two possibilities: either thefire is extinguished or it is still spreading. If the fire is extin-guished the rate of spread is null. If the fire is spreading, asthe dominant effect is contrary to fire spread, the situationwill be similar to a down-slope or contrary wind fire. Con-sequently the rate of spread of the fire front will be of theorder of the basic rate of spread for that fuel-bed, i.e. R0. Themodel does not indicate if the fire will be extinguished or ifit will propagate in a particular case; this input must comefrom another source.

Usually the standard spread direction defined by �R11 cor-responds to the head of the fire and it is the most relevant interms of fire spread and hazard. This spread direction shallbe referred to from now on as the ‘main fire’ or ‘head fire’.Accordingly �R22 correspond to the ‘back fire’ and �R12 and�R21 to the ‘flank fires’. As will be shown later there aresituations in which the maximum rate of spread does notcorrespond to �R11, but to one of the other standard direc-tions. The designation ‘head fire’ must then be interpretedadequately.

The meaning of these standard spread directions is thefollowing: each one of the four sections of the fire front willbe governed by the corresponding standard rate of spread.This does not mean that all the points of each section arepropagating at this rate of spread, of course, but it shows thetrend of each section. The various sections of the fire frontare linked together and therefore there are some restrictions

to the movement of the points of each section. In light ofthe fire line rotation concept (Viegas et al. 1994, 1998), theconvective flow in the vicinity of the fire front will transportheat across it and will induce a non-uniformity of the rateof spread vector along the fire line for each section, as wasobserved for linear fire fronts and for point source fires ina slope. Accordingly one can anticipate that those sectionswith a flame front inclined towards the unburned fuel due tothe transverse transport of heat fire line will rotate and tendto become parallel to the standard rate of spread. This canbe observed in the experimental results of Catchpole et al.(1982). The ‘backward’ spreading sections of the fire frontwill propagate remaining essentially parallel to their previousposition, with the restrictions imposed by the linkage to theneighbour sections.

Methodology

Experimental procedure

The experimental program was carried out at the Fire Lab-oratory of the University of Coimbra using the CombustionTable MC II, described in Viegas et al. (1998). This table hasa platform of 1.6 × 1.6 m that can be inclined in relation tothe horizontal up to 40◦ by 5◦ steps. This platform can alsobe rotated around an axis that is perpendicular to the fuelbed surface creating therefore an arbitrary inclination of theedges of the table.

A flow producing device was attached to one of the edgesof the table producing a uniform flow essentially parallel tothe table surface and to two of its edges. For a given slopeangle α the flow direction could be varied independently fromthe slope gradient direction, by rotating the table (change ofβ0). Both angles α and β0 could be varied independently at 5◦steps in the following ranges, (0–40◦) and (0–180◦), respec-tively. Although other values of α were tested, the resultsreported in this article refer to a slope angle α = 30◦. Theorientation angle β0 of the wind flow was varied at 30◦ steps.

The flow velocity can be adjusted at will so a wide rangeof conditions can be considered in these experiments. Twodevices were used in this program. The first was a set of lowpower axial fans attached to the wall of a box with an exitsection of 1.6 × 0.25 m, producing a maximum velocity U1

of 5 m/s; the second device consisted of a single cross flow

148 D. X. Viegas

u

s

Fig. 5. Combustion Table MC II with the axial flow fans duringexperiment SW 108 (α = 30◦: β0 = 30◦; U1).

fan with an exit section of 0.9 × 0.1 m, producing a maximumvelocity U2 of 11 m/s. In a set of preliminary experiments theflow field above the table was measured using a wind vaneand it was found that the flow was reasonably symmetricaland uniform for the purpose of the present work. In Fig. 5 theCombustion Table MC II with the axial fan flow producingdevice is shown.

Although both fan systems could provide a range of flowvelocities between zero and the maximum value mentionedabove, only the results obtained for the maximum flow veloc-ity values are reported here. In order to avoid damage by thefire front to the fans, the maximum value of β0 was restrictedto 90◦ and to 150◦, respectively for the first (U1) and thesecond (U2) flow producing systems.

The experiments described in this paper were performedin fuel beds made with dead needles of Pinus pinaster with afuel load of 1.0 kg/m2, on a dry basis. The moisture contentof the particles was determined using an electronic mois-ture balance. This device provided a quick measurement ofthe fuel moisture in order to correct the fuel load; as it wasfound that this method is not very accurate, samples of par-ticles were also oven dried for 24 h at 60◦C in each test.The present experiments were performed with fuel moisturecontent (FMC) values in the range of 10–15%, the majoritybeing with 11 < FMC < 12%.The basic rate of spread R0 wasmeasured several times during the experimental program; itsaverage value was 0.20 cm/s.

In each experiment the fire was ignited at a single pointwith the assistance of a piece of hydrophyl cotton soakedin a mixture of petrol and kerosene. Some time was left tolet the surrounding fuel bed start burning before the fanswere switched on. Experiments with a linear ignition linewere also performed in this research program. In this case awool thread soaked in the same mixture was used instead;a single match produced a practically instantaneous anduniform linear starting flame.

A video camera placed above the combustion table with itsoptical axis always perpendicular to the surface of the tableregistered the entire sequence of the experiment. The timeintervals required by the fire front to cut line threads placedparallel to the OX axis at 20 cm intervals were registeredduring each test with the help of a digital stop watch.

Data analysis

Data analysis was based mainly on the images registeredby the video camera placed above the fuel bed. Using acommon video acquisition system, images corresponding topre-selected time steps were digitised to form a file for eachtest. An example is given in Fig. 6 for the test ref. SW 111, forwhich β0 = 90◦; U1; the time step of each frame is indicatedin the figure (see Table 2).

In the sequence of photos shown in this figure it can beseen that the overall behaviour of the fire front changes withtime. In the initial stages the fire acts as a whole; after sometime the fire front sections are sufficiently separate and beginto act relatively independently from each other, as can beseen at frames of 60′′ and 90′′ in Fig. 6. The development ofthe composite effects of wind and slope that were describedabove can be observed at different sections of the fire perime-ter. The top left section of the fire front is dominated by theslope effect whereas the top right section is dominated by thewind effect (cf. frames of 120′′, 150′′ and 180′′ in Fig. 6).This type of behaviour was observed in all experiments. Therelatively small size of the combustion table did not allow foran adequate development of the fire front in some cases.

Using standard drawing software the contour of the firefront was drawn point by point for each time step. The wholeset of contours was then grouped to form a general pictureof the evolution of the fire front in each particular test. Anexample is shown in Fig. 7 for the same test ref. SW 111.

The main or first standard spread direction, correspondingusually to the head section of the fire front was easy to iden-tify. Therefore the corresponding deviation angle δ11 and rateof spread R11 were determined graphically from the contourlines.

The identification of the second and third standard spreaddirections is not so easy as for the head fire; therefore, a differ-ent methodology was used to analyse these spread directions.A graphical construction of the vector addition was made ineach case in order to determine the respective standard rate ofspread direction. A straight line starting from the fire originwas drawn parallel to the standard spread direction and therate of spread of the fire front along this line was determined.As a result the deviation angle was the same as was givenby the model and only the corresponding rate of spread wasdetermined experimentally.

The fire front movement analysis derived from the videocamera images was subject to some error as part of the firefront was hidden by the flame. This happened at the headsections of the fire front and might induce some error in the


10 30

90

150

210

60

120

180

Fig. 6. Sequence of photos taken at different time steps of the test SW 111 (α = 30◦: β = 90◦; U1). Wind is blowingfrom left to right, slope gradient is from bottom to top of the figure. Time from test start is indicated in seconds near eachframe.

150 D. X. Viegas

Table 2. Parameters of two sets of experiments

Wind velocity U1 Wind velocity U2

Ref. β0 Ref. β0

SW 108 30◦ SW 208 0◦SW 110 60◦ SW 209 30◦SW 111 90◦ SW 210 60◦SW 120 30◦ SW 211 90◦SW 124 60◦ SW 212 120◦

SW 213 150◦

S

Fig. 7. Schematic representation of the contours of the fire line atdifferent time steps of the test SW 111 (α = 30◦: β0 = 90◦; U1). Theblack square corresponds to the limit of the fuel bed. The conditions arethe same as for the previous figure. Wind is blowing from left to rightand the slope gradient direction is shown in the figure.

definition of the exact location of the fire front. For this reasonthe images were analysed always by the same person using thesame evaluation criteria in order to minimise the errors. Theauthor believes that the major conclusions of this paper arenot hampered by this fact. In further experiments an infraredcamera is being used in order to avoid this source of error.

Test parameters

A very extensive program of experiments was carried outduring this study. A detailed list can be found in Pires andSousa (1999). With the exception of very few exploratoryexperiments all the others were analysed in detail using themethodology described above.

Due to space limitations only a selection of the resultsobtained shall be presented in this article. The parameterscharacterising the selected experiments are given in Table 2.In all these experiments the slope angle was α = 30◦.

Table 3. Reference non-dimensional parameters of two sets ofexperiments

Flow α Velocity fs1 fw1 ε11 ε21 ε12 ε22device

1 30◦ U1 4.13 1.38 3.0 0.725 4.13 12 30◦ U2 4.13 7.30 0.568 0.138 4.13 1

Results and discussion

Reference conditions

A series of experiments was carried out in the combustiontable in order to determine the reference spread conditions,namely Rw1, Rs1 and R0. From the range of available dataonly some cases belonging to two sets of experiments will bepresented here.

As can be seen in Table 3 the choice of the control param-eters in both sets of experiments was such to have values ofε11 > 1 in one of them and ε11 < 1 in the other. The rate ofspread Rw1 produced by the flow device U1 was only 38%above the basic rate of spread, while the corresponding valueof Rw1 for flow device U2 was 7.3 times the basic rate ofspread. As a result in the first set of experiments slope wasdominating wind for the main or head fire front, while thecontrary happened in the second set.

Multiple spread directions

The contour lines corresponding to the following cases: SW209, 210, 211, 212 and 213 are shown in Figs 8–12, respec-tively. Time steps between these lines are indicated in thelegends. In these figures the head, back and the flank firepropagation directions can be easily identified. Using theknown values of fw1 and fs1 for these cases (cf. Table 3),the corresponding spread vectors are also depicted in thesefigures. The rate of spread vectors are represented in units ofR0. The graphical scale is the same for all this set of figures.Dotted line vectors represent wind- and slope-induced ratesof spread at the corresponding section, while the resultingstandard rate of spread is drawn in solid lines.

In Fig. 8 the results for β0 = 30◦ are shown. The domi-nating effect of wind is clearly shown in this case, with �R11

being the major value among the standard rates of spread.Theevolution of the most advanced point of the contour lines is inaccordance with the predicted direction of �R11. The left flankof the fire (top right of the figure) is practically parallel to thisvector. It can also be seen that the right flank (lower right ofthe figure) follows a different direction and tends to becomeparallel to �R21. The sections corresponding to �R12 (top left)and �R22 (lower left) can also be identified and the generalagreement with the model can be assessed in the figure.

In Fig. 9 the case for β0 = 60◦ is shown.The considerationsmade for the previous figure can be applied to this case aswell. It is worth noting that wind effect appears to dominatethe main fire spread after some 40 s after ignition.


SlopeWind

R11R21

R22

R12

Fig. 8. Fire spread contours for test SW 209 (α = 30◦: β0 = 30◦; U2).Time steps (seconds since fire start) of the contour lines are: 10, 20, 30,40, 50, 60, 70, 80.

Wind

R11

R12

R22

R21

Slope

Fig. 9. Fire spread contours for test SW 210 (α = 30◦: β0 = 60◦; U2).The open lines on the left side of the figure correspond to cases in whichpart of the fire front was extinguished. Time steps (seconds since firestart) of the contour lines are: 10, 20, 30, 40, 50, 60, 70, 80.

In Fig. 10 wind and slope gradient make an angle β0 = 90◦.It is interesting to compare the contour lines of Figs 7 and10, which are obtained for the same values of α and β0, butwith different values of U. The difference between them is

Wind

Slope

R21

R22

R11

R12

Fig. 10. Fire spread contours for test SW 211 (α = 30◦: β0 = 90◦; U2).Time steps (seconds since fire start) of the contour lines are: 10, 20, 30,40, 50, 60, 70.

of course due to the wind velocity that is much higher in thesecond case. The section corresponding to �R12 is not formedin Fig. 10. It seems unlikely that the fire line would becomeparallel to �R12 in this case; at most it should become parallelto the slope gradient.

In Fig. 11 the results for β0 = 120◦ are shown. In this case�R11 and �R21 are of the same order and indeed in Fig. 11 a sortof bifurcation of the main head can be observed. In this ratherinteresting case a practically linear fire front is formed at thehead of the fire. Size limitations of the combustion table donot allow the verification of the stability of this configuration.

The bifurcation effect mentioned above is shown evenmore clearly in Fig. 12, for the case of β0 = 150◦. In this con-figuration �R11 is smaller than �R12 or �R21, which become themain drivers of two important sections of the fire front. Onceagain a practically linear fire front is formed in the sectionBC (cf. Fig. 4) that in this case is practically perpendicularto �R11.

Main spread direction R11

The results obtained for the deflection angle δ11 and thefire front spread (R11) of the head fire are analysed in thissection, given the relative importance of the head fire for

152 D. X. Viegas

WindSlope

R11

R21

R22R12

Fig. 11. Fire spread contours for test SW 212 (α = 30◦: β0 = 120◦;U2). Time steps (seconds since fire start) of the contour lines are: 10,20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120.

practical purposes. The analysis of the other standard firespread directions is made in the next sections.

The deflection angle δ11 of the head fire was determinedin each test looking at the fire spread contours searching forthe main fire front development. The results are shown inFig. 13 as a function of β, for the two sets of experiments.In this figure the theoretical lines according to equation (4)are represented for ε = 0.57, ε = 1.8 and ε = 3. Results of thesecond set of experiments (U2) fit quite well to the theoreticalcurve for ε = 0.57. According to Table 3, the results of thefirst set of experiments should fit to a curve correspondingto ε = 3, as can be seen in Fig. 13.

The results obtained for the non-dimensional rate of spreadξ11, of the head fire front (R11/R0) are shown in Fig. 14 as afunction of β, for the two sets of experiments. Once again thetheoretical curve for ε = 0.57 fits reasonably the data fromthe second set of experiments, with the exception of the firsttwo points. The results from the first set of experiments seemto fit better to a theoretical value of ε = 1.8, instead of theoriginal experimental value of 3. This discrepancy may beexplained by a possible inaccuracy in the determination of ε11

for this particular set of experiments. In both cases there is adeparture from the theoretical predictions for small values ofβ. It seems that the concept of adding slope and wind induced

WindSlope

R11

R21

R22R12

Fig. 12. Fire spread contours for test SW 213 (α = 30◦: β0 = 150◦;U2). Time steps (seconds since fire start) of the contour lines are: 10,30, 60, 90, 120, 150, 180, 210.

0

30

60

90

120

150

180

0 30 60 90 120 150 180

d11

b

� � 3

� � 0.57

� � 1.8

U1

U2

Fig. 13. Deviation angle δ11 of R11 as a function of β.

effects does not work well for small angles of β. The reasonfor this is not clear at this stage and further experiments mustbe made under more carefully controlled conditions to verifythese laws.

Second standard spread direction R12

Using the methodology described above for the second stan-dard spread direction, the deviation angle is estimated from


0

1

2

3

4

5

0 30 60 90 120 150 180

j 11

b

� � 3� � 0.57� � 1.8U1U2

Fig. 14. Non-dimensional rate of spread modulus ξ11 of R11 as afunction of β.

0

5

10

15

20

0 30 60 90 120 150 180

d12

b

� � 4.1U1U2


the mathematical model and therefore it is practically coin-cident with the prediction of the model, as can be seen inFig. 15.

On the contrary the predicted rate of spread is quite dif-ferent from the model prediction. The experimental resultsare generally lower than the theoretical values, as can be seenin Fig. 16 where the predicted and measured values of thenon-dimensional rate of spread modulus ξ12 are shown. Thereason for this discrepancy can be found in the concept offire line rotation proposed in Viegas (2002). According tothis concept a section of the fire front that is not perpendic-ular to the convective flow in its vicinity will not be uniformand its rate of spread will change from one point to another.The fire front movement will be composed by a translationand a rotation that will tend to make that section of the firefront either parallel or perpendicular to the convective flow.As a consequence these sections of the fire front will becomeflanks that spread with a very low rate of spread, of the orderof R0. In any case the evolution of this section of the fire front

0

1

2

3

4

5

6

0 30 60 90 120 150 180

j 12

b

� � 4.1U1U2


0

30

60

90

120

150

180

0 30 60 90 120 150 180

d21

U1

U2

b

� � 0.73

� � 0.14


is quite complex and it is not completely defined by a singletranslation rate of spread.

Third standard spread direction R21

Similar comments can be made for the third standard spreaddirection. The results are shown in Figs 17 and 18 for thedeviation angle δ21 and for the non-dimensional rate of spreadmodulus ξ21, respectively.

Rear spread direction R22

The results obtained for the deflection angle δ22 and the rateof spread (R22) of the rear or back fire are analysed in thissection.

The deflection angle δ22 of the back-fire is shown in Fig. 19as a function of β, for the two sets of experiments. In thisfigure the theoretical line according to equation (4) is rep-resented for ε = 1. As can be seen the experimental resultsfollow the trend of the theoretical model although the abso-lute values of δ do not coincide. This is partially due to the

154 D. X. Viegas

0.0

0.5

1.0

1.5

2.0

0 30 60 90 120 150 180

� � 0.73� � 0.14U1U2

j 21

b


0

30

60

90

0 30 60 90 120 150 180

d22

U1U2

� � 1

b


fact that it is not easy to determine the direction of R22 withaccuracy.

The results obtained for the non-dimensional rate of spreadξ22, of the back-fire front (R22/R0) are shown in Fig. 20 asa function of β, for the two sets of experiments. The experi-mental values of ξ22 are practically constant and they do notfollow the trend predicted by the theoretical curve for ε = 1that is shown in that figure. Our results confirm that, in thepresence of wind and slope, the modulus of the rate of spreadof the back-fire tends to be of the order of R0 (ξ22 ≈ 1).

Discussion and conclusion

The theoretical model presented in this article describes theinteraction of the convection effects induced by wind andtopography in the general case of a surface fire spreading ina uniform slope with arbitrary wind. A vector sum of thoseeffects was presented and discussed.

The existence of four main sections at the fire perimeterwhen slope gradient and wind velocity are not aligned was

0.0

0.5

1.0

1.5

2.0

2.5

0 30 60 90 120 150 180

j 22

� � 1U1U2

b


shown. These sections behave with a certain degree of inde-pendence of each other and are driven by the four standardfire spread directions in the general case, as was shown in thisstudy. The theoretical model based on the additive effect ofslope and wind induced convection at the fire front seems toproduce reasonably good predictions for the fire front spreaddirection and at a lesser degree for its rate of spread. Essen-tially the additive concept proposed by Rothermel (1983)seems to work well, although it is not yet demonstrated whichreference values of �Rw or �Rs should be taken in each casein order to obtain a good agreement with the experimentalresults.

As was said above the standard rates of spread do notcorrespond exactly to the rate of spread at each section of thefire front. Although it seems that in a developed fire thesesections may have a situation of quasi-independence in termsof local conditions, due to fireline continuity they are notentirely free to move. Each section must remain ‘anchored’by their extremities to their neighbour sections. The conceptof fireline rotation brings some light to interpret the evolutionof the elements of the various sections of the fireline.

The existence of four standard spread directions in thegeneral case causes a non-symmetric evolution of the fireperimeter shape. This fact implies that the use of simplesymmetrical ellipses with their axis parallel to the mainspread direction to represent the fire shape can only be anapproximation of the actual shape of the fire front.

The effect of the curvature radius of the fire front on theconjugate effect of wind and slope must be analysed morecarefully. The present study covers only a very limited rangeof values of the ratio between the curvature radius of the firefront and the flame front length. Larger scale experiments arerequired for this purpose.

The present study was also carried out in a homogeneousand plane fuel bed under uniform wind and slope condi-tions. The real world situation is usually quite different from


this. The author believes that a better understanding of thesimpler case that is considered here is a first step that willenable a generalisation for the more complex situations ofterrain curvature, fuel bed heterogeneity and the existenceof non-uniform or non-permanent wind conditions. In a firstapproximation the case of a large fire in complex terrain canbe analysed by sections and for short periods of time for whichthe simplified conditions considered here may be applied.The existence of multiple fire fronts, fire fingers and otherfeatures may be explained under the context of the presentresearch.

This work shall be extended using a larger experimentalfacility with a better-controlled and more-uniform wind flow.A search of real fires in which the effects of wind and slope arepresent shall be made. Whenever possible these cases shall bedocumented and analysed in order to validate the findings ofthis research. Using the fire line rotation concept and laws amathematical model for the prediction of the fire front undernon-aligned wind and slope shall be attempted.

Acknowledgements

The author acknowledges the help given by his studentsMr E. Pires and Mr H. Sousa and by Mr Nuno Luis in theperformance of the laboratory experiments and in the prelim-inary analysis of the data. The author thanks his colleaguesDr A. R. Figueiredo and Dr A. G. Lopes for their criticalrevision of the manuscript. The comments and suggestionsmade by anonymous reviewers of this article are also grate-fully acknowledged. This work was carried out as part ofthe work program of the research projects INFLAME andSPREAD supported by the EU respectively under contractsENV4-CT98–0700 and EVG1-CT-2001–00043. The supportgiven by the Portuguese Science and Technology Foundationto this research program is also acknowledged.

Appendix. Rothermel’s formulation and its modifiedversion by Lopes

In this appendix we look at two previous studies on this sub-ject. These are the formulation proposed by Rothermel andits modified version proposed by Lopes (1994).

Rothermel (1972) proposed the following formulation toevaluate the rate of spread Rs and Rw, induced by wind orslope respectively, from the basic rate of spread R0:

Rs = R0(1 + φs) (A1)

Rw = R0(1 + φw). (A2)

According to this formulation the modified rate of spreadcan be considered as the sum of the basic rate of spread anda variation induced either by slope or by wind:

Rs = R0 + �Rs = R0 + φsR0 (A3)

Rw = R0 + �Rw = R0 + φwR0. (A4)

For φw and φs Rothermel presented analytical equationsbased on laboratory and on field experiments. The functionφs depended on the slope angle α and on the porosity of thefuel bed; φw depended on a reference wind velocity and onthe surface-to-volume ratio of the particles and on the fuelbed porosity as well. Both functions φw and φs are equal tozero when either wind velocity or slope angle is null.

It is easy to see that:

fs = 1 + φs (A5)

fw = 1 + φw, (A6)

in which fs and fw are the corresponding functions definedby equation (6) in the present study.

The above equations (A3) and (A4) were applicable wheneither wind or slope acted independently from each other. Forthe case in which wind and slope were present simultaneously,but parallel to each other, Rothermel proposed the followingformulation for the rate of spread modulus:

R = R0(1 + φs + φw). (A7)

This equation is a generalisation of the previous ones andrespects the following condition:

U = 0 R = Rs (A8)

α = 0 R = Rw, (A9)

Equation (A7) does not consider that the rate of spread isgiven by the addition of the elementary rates of spread Rw

and Rs. Instead this equation considers the additive effect ofthe variations �Rw and �Rs.

In the presentation of the basis of the BEHAVE system,Rothermel (1983) deals with the evaluation of the local spreadof sections of the fire front under arbitrary wind and slopeconditions. He then proposes a vector sum of the rate of spread�Rw induced locally by wind and the rate of spread �Rs, inducedlocally by slope. Although this is not specified by the author,there are indications that the modulus of each one of thesevectors is given by equations (A1) and (A2), respectively.Introducing the unit vectors �ew and �es, defining the slope andwind direction respectively, this formulation is described by:

�R = R0(1 + φw)�ew + R0(1 + φs)�es. (A10)

It is easy to see that, when �ew//�es, this equation does notgive the same result as equation (A7). In order to over-come this discrepancy, Lopes (1994) adopted the followingformulation:

�R′ = R0 φs�es + R0 φw�ew + R0�esw. (A11)

The unit vector �esw is parallel to the vector sum of ��Rs and��Rw, as is shown in Fig. A1. This formulation is in accor-dance with the original equation (A7) proposed by Rothermelfor the simpler case of parallel wind and slope effects.

156 D. X. Viegas

b

d�

→R0

→R�

→�Rw

→�Rs

Fig. A1. Vectorial sum of slope and wind effects according to theformulation proposed by Lopes (1994).

Table A1. Evaluation of the composite effect of wind and slope bytwo formulations (β = 30◦)

φs φw |R|/R0 |R′|/R0 δ δ′(Equation (Equation (Equation (Equation

A12) A13) A14) A15)

3 4 8.70 7.77 16.7◦ 17.2◦3 10 14.6 13.7 22.1◦ 23.2◦6 4 11.6 10.7 12.4◦ 11.9◦

It is easy to see that the formulation expressed by equa-tion (A11) is not suitable to be described by non-dimensionalequations like those presented above for the initial formu-lation. Any non-dimensional form of this equation involvesexplicitly three parameters: β, φs and φw. Therefore it is notso convenient for a synthetic analysis. In order to compareboth formulations we shall use a non-dimensional form ofequations (A10) and (A11) given by:

R

R0=

√[1 + φs + (1 + φw) cos β]2 + [(1 + φw) sin β]2

(A12)

R′

R0= 1 +

√(ϕs + ϕw cos β)2 + (ϕw sin β)2. (A13)

It is easy to find that the angles δ between �R and �Rs, and δ′between �R′ and �Rs are given respectively by:

tan δ = (1 + ϕw) sin β

1 + ϕs + (1 + ϕw) cos β(A14)

http://www.publish.csiro.au/journals/ijwf

tan δ′ = ϕw sin β

ϕs + ϕw cos β. (A15)

It is obvious that equations (A10) and (A11) are not equivalentas is illustrated with a particular case. If we consider β = 30◦,for different values of the pair φs, φw, we obtain for eachformulation the results that are shown in Table A1. In orderto make the comparison easier the modulus of the rate ofspread is given with reference to the basic rate of spread R0.

As can be seen in this table the results given by bothformulations for this sample case are quite similar but never-theless they are slightly different.The accuracy attained in thepresent experimental program does not permit the extractionof conclusions about the superiority of either formulation.Therefore a detailed discussion of this point is left to a futurestudy.

References

Catchpole EA, de Mestre NJ, Gill AM (1982) Intensity of fire at itsperimeter. Australian Forest Research 12, 47–54.

Dupuy JL (1995) Slope and fuel load effects on fire behaviour: labora-tory experiments in pine needles fuel beds. International Journal ofWildland Fire 5(3), 153–164.

Fang JB (1969) ‘An investigation of the effect of controlled wind onthe rate of fire spread.’ PhD thesis, University of New Brunswick,Canada. 169 pp.

Lopes AG (1994) ‘Modelação numérica e experimental do escoa-mento turbulento tridimensional em topografia complexa:Aplicaçãoao caso de um desfiladeiro.’ PhD thesis, University of Coimbra,Portugal. 320 pp. [In Portuguese]

Pires EM, Sousa HM (1999) ‘Acção do vento e do declive na propa-gação de uma frente de chamas.’ Undergraduate Report, Universityof Coimbra. 58 pp. [In Portuguese]

Rothermel RC (1972) ‘A mathematical model for predicting fire spreadin wildland fuels.’ USDA Forest Service, Intermountain Forest andRange Experiment Station Research Paper INT-115. 40 pp. (Ogden,UT)

Rothermel RC (1983) ‘How to predict the spread and intensity of for-est and range fires.’ USDA Forest Service, Intermountain Forestand Range Experiment Station General Technical Report INT-143.161 pp. (Ogden, UT)

Viegas DX (2002) Fire line rotation as a mechanism for fire spread on auniform slope. International Journal of Wildland Fire 11(1), 11–23.doi:10.1071/WF01049

Viegas DX (2004) On the existence of a steady-state regime for slopeand wind driven fire. International Journal of Wildland Fire 13(1),101–117. doi:10.1071/WF03008

Viegas DX, Varela VM, Borges CM (1994) On the evolution of a linearfire front on a slope. In ‘Proceedings of 2nd international confer-ence on forest fire research’. Coimbra, 21–24 November 1994, B.11,pp. 301–318.

Viegas DX, Ribeiro PR, Maricato L (1998) An empirical model forthe spread of a fireline inclined in relation to the slope gradient orto wind direction. In ‘Proceedings of III international conference onforest fire research, 14th conference on fire and forest meteorology’.Coimbra, 16–20 November 1998, B.05, pp. 325–342.

Weber RO (1989) Analytical models for fire spread due to radi-ation. Combustion and Flame 78, 398–408. doi:10.1016/0010-2180(89)90027-8

Weise DR, Biging GS (1996) Effects of wind velocity and slope on flameproperties. Canadian Journal of Forest Research 26, 1849–1858.

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