Entrainment regimes and flame characteristics of wildland fires

CSIRO PUBLISHING International Journal of Wildland Fire 2012, 21, 127–140 http://dx.doi.org/10.1071/WF10034 Entrainment regimes and flame characteristics of wildland fires Ralph M. Nelson Jr A,D, Bret W. ButlerB and David R. WeiseC A US Forest Service, 206 Morning View Way, Leland, NC 28451, USA. [Retired] US Forest Service, Rocky Mountain Research Station, Missoula Fire Sciences Laboratory, Missoula, MT 59807, USA. C US Forest Service, Pacific Southwest Research Station, Forest Fire Laboratory, Riverside, CA 92507, USA. D Corresponding author. Email: [email protected] B Abstract. This paper reports results from a study of the flame characteristics of 22 wind-aided pine litter fires in a laboratory wind tunnel and 32 field fires in southern rough and litter–grass fuels. Flame characteristic and fire behaviour data from these fires, simple theoretical flame models and regression techniques are used to determine whether the data support the derived models. When the data do not support the models, alternative models are developed. The experimental fires are used to evaluate entrainment constants and air/fuel mass ratios in the model equations. Both the models and the experimental data are consistent with recently reported computational fluid dynamics simulations that suggest the existence of buoyancy- and convection-controlled regimes of fire behaviour. The results also suggest these regimes are delimited by a critical value of Byram’s convection number. Flame heights and air/fuel ratios behave similarly in the laboratory and field, but flame tilt angle relationships differ. Additional keywords: air/fuel mass ratio, combustion regimes, entrainment constant, flame height, flame tilt angle. Received 18 March 2010, accepted 23 February 2011, published online 24 November 2011 Introduction An important aspect of wildland fire behaviour deals with whether a surface fire will transition to crown fire, and if so, which type of crown fire will develop (Tachajapong et al. 2008; Cruz and Alexander 2010). The size and shape of the flames are significant factors in this transition because of their influence on important processes such as heat transfer to unburned fuel, scorching of trees, sustained fire due to breaching of firebreaks and fire spread in discontinuous fuels (Lozano et al. 2010). Flame characteristics have been studied in laboratory tests (Thomas et al. 1963; Thomas 1964; Van Wagner 1968; Fang 1969; Albini 1981; Nelson and Adkins 1986; Fendell et al. 1990; Weise and Biging 1996; Mendes-Lopes et al. 2003; Sun et al. 2006) and experimental field fires (Byram 1959; Thomas 1967; Nelson 1980; Nelson and Adkins 1988; Burrows 1994; Fernandes et al. 2002). Alexander (1998) used results from Fendell et al. (1990) to derive a relationship for fire plume angle from fireline intensity and wind speed in the development of a model to predict crown fire initiation. Anderson et al. (2006) tested currently available flame characteristic models with data from several sources and pointed out the need for standardised measurement methods. The flame geometry of 2-D wildland fires has been simulated with computational fluid dynamics (CFD). For example, flame characteristics predicted by Porterie et al. (2000) compared favourably with flame models in the literature. Morvan and Journal compilation Ó IAWF 2012 Dupuy (2004) related heat transfer and fire spread rate in Mediterranean shrub to a flame-length Froude number. Nmira et al. (2010) describe a physical model that produced low- and high-wind regimes of flame characteristic behaviour for stationary area and line fires; predicted values of flame height, flame length and flame tilt angle generally agreed with experimental data. Modelling studies of air flow around fires were reported for laboratory chaparral fires (Zhou et al. 2005; Lozano et al. 2010) and for grass fires in the field by Linn and Cunningham (2005). Past research has essentially neglected the processes governing movement of air into the fuel-bed combustion zone and its attached flame. We know of only one published report in which the flame air/fuel mass ratio is estimated from air flow measurements; the laboratory data are for fires in alcohol, wood crib and town gas fuels (Thomas et al. 1965). Wildland fire models characterising entrainment are those of Thomas (1963), Fang (1969) and Albini (1981). In the present paper, models of entrainment and flame characteristics are basically thermodynamic, and restricted to head fires of low-to-moderate intensity on flat ground. Numerical simulations (Porterie et al. 2000; Morvan 2007) suggest that a line of fire spreading in uniform fuel in response to a steady wind may burn in one of several combustion regimes. These regimes are related in part to the processes by which air is entrained into the flame. We hypothesise that when the mean wind speed is zero, the mass of entrained air increases as flame www.publish.csiro.au/journals/ijwf 128 Int. J. Wildland Fire height increases and the velocity of this air at a given height is proportional to the upward velocity of the flame fluid at that point (Taylor 1961; Thomas 1967; Fleeter et al. 1984). This process is herein referred to as classical entrainment. As the wind speed increases, convection begins to influence entrainment; flame tilt angle and rate of fire spread begin to increase significantly, and the flame height to depth ratio begins to decrease. The angle of flame tilt is determined by a momentum flux balance between the transverse components of the horizontally moving ambient air and the buoyant velocity of the flame fluid. This type of entrainment, associated with flame drag and buoyancy forces, is called dynamic entrainment in this paper. As the wind continues to increase, a point is reached beyond which the mass of air entering the flame disrupts the balance between drag and buoyancy forces. The tilt angle is determined by a ratio between the horizontal and vertical components of the flame fluid mass flux (Albini 1981). This process is referred to in the present paper as accretive entrainment. In some cases of accretion, there may be little suggestion of horizontal inflow at the lee edge of the flame because part of the impinging air flows through the flame, leading to an outflow of unreacted air (air not participating in combustion) at the lee edge. Beer (1991) reported that CSIRO (Australia) researchers did not observe a fire-induced wind in their laboratory and field experimental fires. The objective of this paper is to describe with mathematical models and experimental data how air entrainment and combustion regimes determine flame characteristics; we use models of flame height and tilt angle to evaluate entrainment constants and air/fuel mass ratios. First, we estimate air/fuel ratios for the combustion zone and flame. Second, we derive a dimensionless criterion that identifies three combustion regimes for head fires of low-to-moderate intensity. Third, we describe mass flow in the combustion zone and then develop flame characteristic equations from a 1-D analysis based on a simplified version of the Albini (1981) flame model. The present study extends Albini’s work with formulations of flame characteristic equations and entrainment velocity for low-wind fires. Finally, we perform regressions using the model equations and experimental data to determine air/fuel ratios for the combustion zone and flame. Combustion regimes Evidence for multiple regimes Albini (1981) stated that his flame model must fail at low wind speeds. In his model, the entrainment velocity is proportional to the horizontal wind speed. Porterie et al. (2000) simulated the laboratory pine litter fires of Mendes-Lopes et al. (1998) and reported that as wind speed increases from 1 to 2 m s 1, a transition from buoyancy-dominated to wind-dominated flow occurs. Beer (1993) observed differing functional relationships between fire spread rates and wind speed at a critical speed of 2.5 m s 1. Roberts (1979) used dimensional analysis to describe flow of effluent from a line of ocean outfall diffusers into a current of ambient water. These long pipeline diffusers have uniformly spaced ports through which effluent is forced by the water head. Dilution of the buoyant wastefield is closely R. M. Nelson et al. approximated by treating the effluent source as a line plume (Roberts et al. 1989). When applied to wildland fires, the work of Roberts suggests that if differing head fire burning regimes are related to mixing differences, these regimes can be described in terms of wind speed and fireline intensity. A combustion regime criterion We utilise the work of Roberts (1979) on the basis that lowReynolds-number ocean flows should provide a more realistic analogy for describing our low-Reynolds-number flames than models from other disciplines – for example, turbulent atmospheric plume models. When effluent issues vertically from a line diffuser into horizontally moving ocean water, the density difference between effluent and water induces a buoyancy flux that interacts with the water (Roberts 1979). This scenario differs from that for the wind-aided 2-D fire in at least two ways: (1) the flame may be more strongly buoyant than the effluent plume; (2) diffusers are straight, but wind-aided fires often exhibit one or more heads. We neglect these differences in the present study. For convenience of the reader, the ‘Symbols used in mathematical models’ section presents a list of symbols used in the mathematical modelling that follows. The buoyancy flux per unit length of diffuser introduced by Roberts (1979) is bR ¼ gDrq re ð1Þ with units of cubic metres per second cubed. The corresponding buoyancy flux bF for a line fire is bF ¼ ff gIB gIB u3 Nc ¼ a ¼ ra Hc ra cp Ta 2 ð2Þ where the first equality of Eqn 2 is obtained by analogy with Eqn 1 and ff is the mean air/fuel ratio (mass of air per mass of original fuel burned) associated with lateral movement of air into the flame. The Nc criterion is Nc ¼ 2gIB ra cp Ta u3a ð3Þ and often is called the convection number (Nelson 1993). In Eqn 3, we assume that ambient wind speed (ua) is much greater than the fire spread rate. For a given ocean current of speed (uc), Roberts defines a Froude number (FR) as FR ¼ u3c bR ð4Þ and discusses three separate mixing regimes delineated by FR. For FR , 0.2, the flow forms a plume with a strong vertical component. In the intermediate region, 0.2 , FR , 1 and the plume is unable to contain all of the incoming flow; it contacts the lower boundary for some distance downstream. For FR . 1, the flow is in full contact with the lower boundary and the upper edge of the plume forms a planar interface with ocean water. If similar behaviour occurs when a 2-D fire burns in air of Flame characteristics of wildland fires Int. J. Wildland Fire horizontal speed ua, then a fire Froude number (FF) may be obtained from Eqns 2 as FF ¼ u3a ¼ 2Nc bF 1 ð5Þ We apply the FR criteria identified by Roberts (1979) to FF so that three combustion regimes are defined by the critical values Nc ¼ 2 and 10. For a given fireline intensity, the region Nc . 10 should correspond to weak wind speeds, whereas Nc , 2 would imply strong winds; the intermediate regime should apply to moderate winds. A theoretical air/fuel mass ratio ff for the free flame may be obtained from Eqns 2. If Hc ¼ 15 000 kJ kg 1, cP ¼ 1 kJ kg 1 K 1 and Ta ¼ 300 K, then ff ¼ Hc ¼ 50 cp Ta ð6Þ with units of kilogram per kilogram. Support for Eqn 6 comes from the following considerations. Byram and Nelson (1974) found that the steady burning of 1 kg of solid wood expands the atmosphere by 41.8 m3. Suppose a mixture of combustion products and unreacted air at mean temperature To ¼ 1000 K enters the flame from the combustion zone along with air entrained laterally from the atmosphere at temperature Ta ¼ 300 K. Mass Me of the entrained air has initial volume Vo and receives heat from the combustion zone fluid. Thus Me expands to a larger volume V, causing a drop in the mean temperature of the fluid. The mixture of combustion products and entrained air exits the flame tip at temperature Tt ¼ 500 K. If the atmospheric density is 1.2 kg m 3 and air flows in steadily to replace all air leaving the visible flame volume, then the effective air/fuel mass ratio is ff ¼ (1.2  41.8) ¼ 50.2 kg air kg 1 fuel burned, in agreement with Eqn 6. in the combustion zone, air (reacted and unreacted) present in the combustion zone, and water released from the fuel owing to combustion and evaporation. Thus Eqn 7 describes the stream of combustion products, unreacted air and water vapour entering the flame from the combustion zone. Fireline intensity is defined as IB ¼ Hc Xb Wa R Combustion zone relationships The mixture flowing into the flame consists of burned and unburned volatiles, reacted air (air participating in combustion), unreacted air, water vapour formed in combustion, and water not lost during fuel preheating. An approximate mass flow rate through the fuel bed surface per unit length of fireline, mo (kg m 1 s 1), is   1 Z fM þ Nv þ þ 0:56 þ mo ¼ Xb Wa R Xb eXb eXb ð7Þ where Xb is the fraction of volatilised fuel that burns. From left to right, the terms in brackets denote the mass of volatiles produced ð8Þ where Xb is now interpreted as the ratio of the heat release rate IBcz in the combustion zone to the heat release rate IB of the entire fire. When Xb ¼ 1, Eqn 8 agrees with the widely accepted definition of fireline intensity, IB ¼ HcWaR. A different approximation of IB calculates the heat required to raise the temperature of the fluid entering the base of the flame from ambient temperature (i.e. before the production of heat by chemical reaction) to the mean temperature at the base of the flame (Albini et al. 1995). Thus when Xb ¼ 1, IB may be written as IB ¼ mo cp ðTo Ta Þ ð9Þ where cp is assumed equal for flame fluid and air. In the combustion zone of a wildland fire, Xb , 1 and the rate of heat release, from Eqns 7–9, is  cp ðTo  1 Z fM þ Nv þ þ 0:56 þ Xb eXb eXb   1 þ fcz cp ðTo Ta Þ Ta Þ  Xb Wa R Xb IBcz ¼ Hc Xb Wa R ¼ Xb Wa R  ð10Þ where fcz is the air/fuel ratio of the combustion zone given by the sum of Nv kg of reacted air and Z/eXb kg of unreacted air. If the simplified estimate of IBcz in Eqn 10 is reasonable, fcz becomes Theory of flame characteristics Consider a line head fire that burns steadily in response to horizontal wind speed ua through fuel distributed uniformly on flat terrain. Modelling of entrainment and flame characteristics requires consideration of both the combustion and flame zones; the overall rates of mass flow associated with these zones are derived below. 129 fcz ¼ Hc cp ðTo Ta Þ 1  20 Xb ð11Þ When Xb ranges from 0.5 to 1, fcz ranges from 19.4 to 20.4 kg kg 1 and a mean theoretical fcz may be taken as 20 kg kg 1. The approximation in Eqn 10 requires (1/Xb þ fcz) .. (0.56 þ fM/eXb); we make the reasonable assumptions e . 0.5 and Xb . 0.5 (Albini 1980). For the ordinary laboratory fire (0.56 þ fM/eXb) , 2, so Eqn 10 is acceptable. For the green vegetation layers of crown fires, the estimate is less accurate because the moisture term (0.56 þ fM/eXb) could be as large as 9. Flame zone relationships In the Albini (1981) flame model, air enters the flame by accretion, in which a fraction of the impinging air of speed ua becomes incorporated into the flame. Because chemical reactions are neglected in our model, flame temperature is a maximum at the fuel bed surface and decreases upward as air entrainment increases towards the flame tip. 130 Int. J. Wildland Fire R. M. Nelson et al. Z A H Fla m e FD A FDcos A v m⫹dm FB A w z ⫹ dz ρa,cp,Ta,ua,ue u m D z FBsin A ρ,cp,T ρo,cp,To X Docos A Fuel bed Do Fig. 1. A time-averaged visible flame showing mass and energy flow variables. The transverse component of the horizontal drag force FD balances the transverse component of the vertical buoyancy force FB to determine flame tilt angle A. We rewrite the Albini (1981) model equations as follows: mass flow : m ¼ rwD ð12aÞ lateral entrainment : dm ¼ ra ue dz ð12bÞ horizontal momentum : dðmuÞ ¼ ua dm ð12cÞ vertical momentum : dðmwÞ ¼ rgD  ðT Ta  Ta Þ sensible energy : dðmcp T Þ ¼ cp Ta dm flame tilt angle : A ¼ tan 1 u w dz ð12dÞ ð12eÞ ðTo Tf Ta Þ
¼ mo þ ra ue H Ta ra ue H ¼ 2:5mo ¼ 2:5ð1 þ Xb fcz ÞWa R ¼ ff Xb Wa R ¼ ð13Þ ff Xb IB Hc ð14Þ where Xb may be taken as unity. The horizontal momentum equation is integrated with limits u ¼ uo and m ¼ mo when z ¼ 0 to obtain mu ¼ mo uo þ ua ðm ð12f Þ The flame represented by Eqns 12a–f is presented in Fig. 1. All dependent variables in these equations are regarded as timeaveraged values. Eqns 12a–f may be solved analytically by assuming that entrainment velocity ue represents a constant velocity obtained by averaging over flame height H. Quantities ra, cp, Ta and ua are assumed constant. When the entrainment and sensible energy relations Eqns 12b and 12e are integrated from the flame base (z ¼ 0) to the flame tip (z ¼ H), the flame tip mass flux (mt) becomes mt ¼ mo where mo (= ro woDo) is the combustion zone mass flux at z ¼ 0. If values of 1000, 500 and 300 K are assigned to To, Tt and Ta, then mt ¼ 3.5mo and Eqns 10 and 13 give mo Þ ð15Þ The vertical momentum equation may be rewritten by multiplying both sides by mw and substituting the entrainment and integrated sensible energy equations to obtain d ðmwÞ2 ¼  2gmo ra ue     cp ðTo Ta Þ 2gIB dm2 dm2 ¼ ra cp Ta ue cp Ta ð16Þ where IB is from Eqn 9. Integration of Eqn 16 with (mw)2 ¼ (mowo)2 at z ¼ 0 leads to ðmwÞ2 ¼ ðmo wo Þ2 þ w3c m2 2ue m2o
ð17Þ Flame characteristics of wildland fires Int. J. Wildland Fire where the quantity wc is a characteristic buoyant velocity (Nelson 2003) representative of the whole fire given by  2gIB ra cp Ta wc ¼ 1=3 Boundary velocities uo and wo depend strongly on wind speed, fuel type and fuel load. Anderson et al. (2010) have shown that at higher speeds, uo is a moderate fraction of the free stream speed ua; for small ua values, uo ranges from ua to 2ua. However, wo is instrumental in the development of model equations for flame height H and tangent of the tilt angle, tan A. Entrainment velocity equations Albini (1981) assumed that entrainment velocities for head fires in moderate winds can be described as ue ¼ Zua ð19Þ where Z is the fraction of impinging air entering the flame. Because Eqn 19 is not valid as ua approaches zero, we require an expression for ue in terms of entrainment constant a that is applicable for ua  0. On the basis of exploratory data plots (R. M. Nelson Jr, Missoula Fire Sciences Laboratory, unpubl. data), and because wc in Eqn 18 is proportional to IB1/3 (a function of Wa and ua), we infer ue ¼ awc ð20Þ an equation identical in form to the zero-wind entrainment equation (Taylor 1961). For single fires, Eqn 20 applies when ua . 0; the equation ue ¼ ao wco applies when ua ¼ 0. In general, we expect a 6¼ ao. Entrainment constants Z and a are quantified in the next section, which compares flame characteristic model equations with our experimental data. Flame tilt angle relationships The tangent of flame tilt angle A may be obtained from Eqn 15 and the square root of Eqn 17 in the form tan A ¼ u ½mo uo þ ua ðm mo ފ ¼ i1=2
w h ðmo wo Þ2 þ w3c m2 m2o =2ue ð21Þ If we invoke the assumptions used by Albini (1981) and assume little variation in angle A from z ¼ 0 to z ¼ H, then at the flame tip mt .. mo, mtua .. mouo, mt2(wc3/2ue) .. (mowo)2 and tan A may be written as 1 tan A ¼ 2 =2  ua wc  ue wc 1= 2 ð22Þ Eqns 19 and 22 yield tan A ¼ 1 1 2 =2 Z =2  ua wc 3=2 where (ua/wc)3 ¼ Nc 1. The alternative formulation for tan A using Eqn 20 in Eqn 22 leads to 1 ð18Þ 1 ¼ 1:414Z =2 Nc 1=2 ð23Þ 131 1 tan A ¼ 2 =2 a =2  ua wc  1 ¼ 1:414a =2 Nc 1=3 ð24Þ Eqns 23 and 24 are suitable for evaluating entrainment constants Z and a because the air/fuel ratio ff is missing from the two equations. Flame height relationships Anderson et al. (2006) noted that use of Froude number FH in flame tilt angle models is problematic from the standpoint of prediction because flame height H is unknown. Moreover, one can infer from Albini (1981) that FH is inversely proportional to convection number Nc. Two additional relationships for H in terms of fireline intensity IB have been applied in various studies (Albini 1981; Anderson et al. 2006), but not in the context of combustion regimes delimited by Nc ¼ 10. We explore these three flame height relationships using Eqn 14. First, Eqns 2, 14, and 19 with Xb ¼ 1 lead to ! u2a 2Hc Z Z FH ¼ ¼ 100 Nc ¼ ff gH cp Ta ff N c 1 ð25Þ where Z is obtained from a plot of Eqn 23. Second, a dimensional equation for H comes from combining Eqns 14 and 19 to yield H¼   f f IB ff IB ¼ 0:0000556 Z ua ra Hc Zua ð26Þ where ra ¼ 1.2. Finally, Eqns 14, 18, and 20 combine to give cp Ta ff w2c H¼ ¼ 2gHc a f3f cp Ta 2gr2a a3 Hc3 !1=3 2= IB 3 ¼ 0:000147   ff 2=3 I a B ð27Þ where a is evaluated using a plot of Eqn 24. We note that Eqn 27 is commonly used when ua ¼ 0; in such cases H, wc, a and IB should be written as Ho, wco, ao and IBo. Comparison of flame characteristics data with model equations The laboratory data are from fires in slash pine litter (Pinus elliottii Engelm.) and saw palmetto fronds (Serenoa repens (Bartram) Small) burned in the US Forest Service’s Southern Forest Fire Laboratory (SFFL) wind tunnel in Macon, GA (Nelson and Adkins 1986). The February 1988 field measurements were made in 1-, 2- and 4-year roughs during experimental burns in southern rough fuels of northern Florida (Osceola National Forest) and longleaf pine (Pinus palustrisMill.) litter–grass fuels of coastal South Carolina (Francis Marion National Forest). The field data, heretofore unpublished, are presented in Appendix A. 132 Int. J. Wildland Fire (a) 4 R. M. Nelson et al. (b) 2.0 Laboratory fires 3 Field fires 1.5 2 tan A tan A ⬘Outlier⬘ tan A ⫽ 1.19Nc⫺(1/2) 1.0 tan A ⫽ 0.655Nc⫺(0.03) 1 0.5 tan A ⫽ 3.931Nc⫺(2/3) tan A ⫽ 1.041Nc⫺(1/3) tan A ⫽ 1.044Nc⫺(1/3) 0 0 10 20 tan A ⫽ 4.119Nc⫺(2/3) 0.0 30 40 Nc 0 10 20 30 40 50 60 70 Nc Fig. 2. Southern Forest Fire Laboratory (SFFL) laboratory (a) and field (b) data showing two behaviour regimes for tan A separating at a value of Nc ¼ 10 for the laboratory data. For the field data, regressions (dashed lines) based on Eqn 24 for Nc , 10 and on Eqn 28 for Nc . 10 fit the data poorly. We select the regression for all Nc (solid line) as representative of tan A for the field data. Fuel consumption in the field was estimated by weighing oven-dried pre- and post-burn fuels. In some cases, this led to overestimation of the available fuel load. Wind speeds were measured just behind the fireline with a hand-held digital wind meter at mid-flame height. Fire spread rate and flame characteristics were measured using the video methods of Nelson and Adkins (1986). Theoretically, the data for tan A and H should pass through zero when the independent variable X ¼ 0, so we set the intercept term to zero and fitted models of the form Y ¼ g1Xl where l is an exponent determined analytically and g1 was estimated statistically by simple linear regression with unweighted least-squares for the model relationships in Eqns 23–27. Student’s t-statistic tested significance of the parameter estimate. Overall quality of the regression models was evaluated using root mean squared error (RMSE) and mean absolute error (MAE). We used the Akaike Information Criterion (AICc), adjusted for small sample size (Burnham and Anderson 2004), to compare the different model formulations. Because the commonly used coefficient of determination (R2) can provide spurious information when the intercept term is set zero (Eisenhauer 2003), we calculated R2 as R2 ¼ P P to 2 Y^i = Yi2 to measure how much of the total variation of the dependent variable (also known as the uncorrected sum of squares) was described by our regression through the origin models. The models and their fit statistics are presented in Appendix B. SFFL laboratory fires – flame tilt angle The wind tunnel fires in beds of slash pine litter and slash litter under palmetto fronds were treated as coming from a single fuel type (Nelson and Adkins 1986). Initial fuel loads ranged from 0.5 to 1.1 kg m 2, the dead fuel moisture content fraction from 0.09 to 0.13, and wind speed from 0.6 to 2.3 m s 1. Palmetto frond fractional moisture content at the time of burning ranged from 0.9 to 1.25. Fig. 2a is a plot of tan A according to Eqn 23 for all Nc. The data separate into two regimes at Nc ¼ 10. The slope estimate of the line for Nc , 10 is 1.190, yielding Z ¼ 0.71. The data also are plotted according to Eqn 24 in Fig. 2a. The slope estimate of 1.044 for Nc , 10 implies a ¼ 0.55. These values of Z and a are independent of ff, and are used in all subsequent evaluations of ff with the tan A and H equations. Thus we are assuming these Z and a values also apply to the field fires, but only for Nc , 10. Eqns 23 and 24 describe the data well for Nc , 10, but different behaviour is observed for Nc . 10. The theory appears valid only when an accretion mechanism is operative (Nc , 10). This is not surprising, as Eqns 12 and 19 were written to describe flames in moderately strong winds (Albini 1981). An alternative theory is needed to describe tan A for Nc . 10. We assume that tan A for low winds (dynamic entrainment) is determined by a balance between transverse components of the drag force exerted on the flame by the impinging ambient air and the vertical buoyancy force resulting from combustion. Details of a model for tan A based on this approach are available in an Accessory publication (see http://www.publish.csiro.au/?act= view_file&file_id=WF10034_AC.pdf). It is shown that tan A for Nc . 10 is given by tan A ¼ 3:85Z2 Nc 2=3 ð28Þ Fig. 2a also includes a plot of the regression of tan A on Nc 2/3 for Nc . 10, tan A ¼ 3.931Nc 2/3, and suggests that Eqn 28 is a good description of the data. Though not significantly different from zero at the 0.05 level, the slope term was significantly different at the 0.077 level; thus Z ¼ (3.93/3.85)1/2 ¼ 1.01. The Z ¼ 1 estimate for Nc . 10 implies that fires in light winds entrain a larger fraction of the impinging air (the total amount Flame characteristics of wildland fires (a) 2.0 Int. J. Wildland Fire (b) 2.0 Laboratory fires Field fires 1.5 FH FH 1.5 133 FH ⫽ 1.676Nc⫺1 1.0 0.5 FH ⫽ 1.362Nc⫺1 1.0 0.5 FH ⫽ 2.421Nc⫺1 FH ⫽ 0.878Nc⫺0.62 FH ⫽ 1.726Nc⫺1.04 0.0 0 10 20 Nc FH ⫽ 2.31Nc⫺1 0.0 30 40 0 20 40 Nc 60 80 Fig. 3. Southern Forest Fire Laboratory (SFFL) laboratory (a) and field (b) data showing that the relationship between FH and Nc is similar for both datasets and that regression suggests a slight difference at approximately Nc ¼ 10. Solid lines denote fitted equations based on the Nc criterion; dashed lines illustrate fitted regressions using all Nc data. is relatively small) than fires in stronger winds (Nc , 10) for which the value Z ¼ 0.71 was estimated earlier. It was not possible to determine a by using Eqn 20 to derive an equation similar to Eqn 28 because the result would imply tan A ¼ constant. Fig. 2a shows that tan A for Nc . 10 is not constant, but described well by Eqn 28. Thus, only Eqn 19 describes the entrainment velocity and flame tilt angle for Nc . 10. This result may be due to suppression of vertical flow in the SFFL tunnel. We believe a result approximating tan A ¼ constant is representative of fires in large wind tunnels. SFFL field fires – flame tilt angle For the palmetto–gallberry fuels, flame heights ranged from 0.4 m in 1-year roughs to 5 m in the 4-year roughs; in the litter– grass fuels, flame heights exhibited intermediate values. Fractional moisture content of the dead grass was 0.18; the L and F layers ranged from 0.2 to 0.5, and the live palmetto fronds and gallberry leaves from 1 to 1.4. Eqns 23–24 describe the tan A laboratory data for Nc , 10, but not the corresponding field data (Fig. 2b); a regression according to Eqn 24 leads to tan A ¼ 1.041Nc 1/3, an extremely poor fit (a brief discussion of flame tilt angle in the wind tunnel and field is available in an Accessory publication, see www. publish.csiro.au/?act=view_file&file_id=WF10034_AC.pdf). The linear increase in tan A is physically questionable and disagrees with the numerical modelling results of Nmira et al. (2010) and the experimental data of Fendell et al. (1990) discussed by Alexander (1998); these investigators show that tan A should be proportional to a reciprocal power of Nc smaller than unity. Thus we consider the four outermost data points for Nc , 10 as outliers due to errors in measurement of tan A and available fuel load Wa; ignoring these points suggests tan A ¼ constant. To study Eqn 28 for Nc . 10, we plotted tan A v. Nc in Fig. 2b. An outlying point initially was neglected in both the plot and regression; the result, tan A ¼ 4.119Nc 2/3, yields Z ¼ (4.12/ 3.85)1/2 ¼ 1.03 and supports the earlier result for the laboratory data, Z ¼ 1 when Nc . 10. Including the outlier in the regression tan A ¼ 4.458Nc 2/3 produces a slope estimate not significantly different from 4.119; however, the fit statistics are less desirable for Nc . 20. Although Eqn 28 is a possible descriptor of tan A for Nc . 10 and useful for estimating Z for the field fires, inspection of all data in Fig. 2b suggests that tan A is best described as constant. For example, we consider 26 of the 32 data points in the figure to approximate the horizontal line tan A ¼ 0.65. We have regressed all data as coming from a single population of tan A values and compared the results with statistics obtained from regressing the data according to Nc , 10 and Nc . 10. The statistical fit based on all data (outlier removed) is superior to the fits obtained when the data are separated into two groups (Appendix B). Thus the single-regression equation, tan A ¼ 0.655Nc 0.03, indicates that flame tilt angle for the field data is given by tan A ¼ 0.655 – a behaviour not seen in the laboratory fires. The result tan A ¼ constant for all Nc can be derived from the idea that flame tilt is determined by a balance involving rates at which work is done by rising parcels of flame and parcels of moving air. An equation based on this approach is available in an Accessory publication (see http://www.publish.csiro.au/?act= view_file&file_id=WF10034_AC.pdf). Setting tan A in this equation to 0.655, tan A ¼ CD ra a3 ¼ 3:85a3 ¼ 0:655 rc ð29Þ and a ¼ 0.55, in agreement with the laboratory fire estimate. This result supports our assumption that the laboratory fire 134 Int. J. Wildland Fire (a) 1.0 R. M. Nelson et al. (b) 6 Laboratory fires H⫽ Field fires 0.0033IBua⫺1 5 0.8 H ⫽ 0.0035IBua⫺1 4 H (m) H (m) 0.6 0.4 H⫽ 3 H ⫽ 0.0024IBua⫺1 0.0024IBua⫺1 2 0.2 1 0.0 0 0 50 100 150 IBua⫺1 200 (kJ 250 300 350 m⫺2) 0 500 1000 1500 IBua⫺1 m⫺2) (kJ 2000 2500 Fig. 4. Southern Forest Fire Laboratory (SFFL) laboratory (a) and field (b) data indicating that the H v. IB ua 1 relationship is linear; dark triangles denote Nc , 10, open triangles Nc . 10. Both the laboratory and field data are scattered, but divided into separate regions according to Nc. Three outliers in the field data are omitted from the regressions. results, Z ¼ 0.71 and a ¼ 0.55 for Nc , 10 and Z ¼ 1 for Nc . 10, can be used to calculate ff for the field fires. This constant-angle regime of burning is referred to as kinetic entrainment. SFFL laboratory and field fires – flame height The height H of wind-blown flames is described by Eqns 25–27. For the laboratory data, FH and Nc are plotted in Fig. 3a using Eqn 25; the estimated slope is 1.676 for Nc , 10. Thus for Z ¼ 0.71, ff ¼ 42.4. For Nc . 10, the slope is 2.421, so with Z ¼ 1, ff ¼ 41.3. A regression using all data yielded FH ¼ 1.726Nc 1.04, which was significant (Appendix B). Fig. 3b for the field data shows plots of FH v. Nc according to Eqn 25. For Nc , 10, FH ¼ 1.362Nc 1 and the slope estimate with Z ¼ 0.71 yields ff ¼ 52.2. For Nc . 10, FH ¼ 2.310Nc 1, giving ff ¼ 43 for Z ¼ 1. If the Nc ¼ 10 criterion is not applied and all data are considered, a model in which both the slope and exponent were estimated from the data, FH ¼ 0.878Nc 0.62, is a better fit than a model that assumed the Nc 1 formulation, FH ¼ 1.397Nc 1 (not shown in Fig. 3b). For Nc ¼ 10 in Fig. 3, FH E 0.25 for the laboratory and field fires. Pagni and Peterson (1973) and Morvan and Dupuy (2004) state that when flame-length Froude number FL , 0.25, fire spread in pine needle beds is radiation (buoyancy)-controlled and flame tilt is close to vertical; when FL . 1, the spread rate is controlled by a combination of radiation and convection. Neglecting the small difference between FL and FH for our fires, we interpret the intermediate region 0.25 , FH , 1 as one in which radiative preheating decreases as FH - 1 while the convective contribution due to wind increases. For FH . 1, fire spread becomes increasingly wind-driven. Because FH . 1 for only two of our laboratory fires, this regime requires further study. The second relationship for flame height is Eqn 26, which relates H to IBua 1. The data in Fig. 4a show two burning regimes. For Nc , 10, H ¼ 0.0033IBua 1; thus for Z ¼ 0.71, Eqn 26 yields ff ¼ 42.4 – in agreement with the corresponding laboratory value. For Nc . 10, H ¼ 0.0024IB ua 1 and ff ¼ 42.4 if Z ¼ 1. The field data in Fig. 4b show three outliers (circled), and preliminary regressions for all data points resulted in poor fits. These outliers were explained by exploratory plots that showed that Wa for the three points was overestimated by a factor of 2. Reduction of fireline intensity IB by this factor places the data points close to their respective regression lines. When the regressions were repeated with outliers omitted, the equation for Nc , 10, H ¼ 0.0035IBua 1, resulted in ff ¼ 44.3; for Nc . 10, the equation H ¼ 0.0024IB ua 1 gave ff ¼ 43.3. With outliers removed, ff for the laboratory and field fires agreed closely. The third equation, Eqn 27, describes H in terms of IB2/3. The laboratory data plotted in Fig. 5a are scattered, with a tendency for Nc . 10 data to be associated with lower IB. We accepted Eqn 27 as a descriptor of the data in Fig. 5a for two reasons. First, the fitted regression for all laboratory fire data in Fig. 5a, H ¼ 0.0142IB2/3, was not statistically different from the corresponding regression for all field data (Fig. 5b). Second, even though two points from each Nc regime overlap into the other regime, R2 values for both Nc regimes in Fig. 5a exceed 0.95. For Nc , 10, H ¼ 0.0132IB2/3 and ff ¼ 49.4 when Z ¼ 0.71. For Nc . 10, H ¼ 0.0173IB2/3, which gives ff ¼ 64.7 if Z ¼ 1. Plots of the field data according to Eqn 27 are shown in Fig. 5b. Separate regressions for Nc , 10 and Nc . 10 (not presented) produced slope estimates of 0.0138 and 0.0137 respectively, so there was no suggestion of two burning regimes dependent on Nc. A single line with a slope of 0.0137 for all data did not describe the bulk of the data; thus, the most outlying point was not included in a new regression. The slope estimate of Flame characteristics of wildland fires (a) 1.0 Int. J. Wildland Fire (b) 6 Laboratory fires H⫽ 135 Field fires 0.0142IB(2/3) 5 0.8 H⫽ 0.0173IB(2/3) H ⫽ 0.0155IB(2/3) 4 H ⫽ 0.0132IB(2/3) H (m) H (m) 0.6 3 0.4 2 0.2 1 0 0.0 0 100 200 IB (kW 300 400 500 0 2000 4000 m⫺1) IB (kW 6000 8000 m⫺1) Fig. 5. Southern Forest Fire Laboratory (SFFL) laboratory (a) and field (b) data suggesting similar trends for H v. IB; dark triangles denote Nc , 10, open triangles Nc . 10. Solid lines are regression results; dashed line for the laboratory data denotes a regression for all Nc. Ranges of H and IB in the laboratory data constitute only a small fraction of the corresponding ranges in the field data. Circled data point is an outlier. Table 1. Individual and averaged entrainment parameters and air/fuel ratios for Southern Forest Fire Laboratory (SFFL) laboratory and field fires Flame characteristics Model equations Entrainment constants Z SFFL laboratory tan A tan A FH H H Laboratory average SFFL field tan A tan A FH H H Field average Overall average Air/fuel ratios a fcz ff Nc , 10 Nc . 10 Nc , 10 Nc . 10 Nc , 10 Nc . 10 Nc , 10 Nc . 10 23 28 25 26 27 0.71 – – – – 0.71 – 1.01 – – – 1.01 0.55 – – – – 0.55 – – – – – – – – 42.4 42.4 49.4 44.7 – – 41.3 42.4 64.7 49.5 – – 15.7 15.7 18.5 16.6 – – 15.2 15.7 24.6 18.5 28 29 25 26 27 – – – – – – 0.71 1.03 – – – – 1.03 1.02 – 0.55 – – – 0.55 0.55 – 0.55 – – – 0.55 0.55 – – 52.2 44.3 58.0 51.5 48.1 – – 43.3 43.3 58.0 48.2 48.9 – – 19.6 16.4 21.9 19.3 18.0 – – 16.0 16.0 21.9 18.0 18.3 the resulting regression, H ¼ 0.0155IB2/3, implies ff ¼ 58 if a ¼ 0.55. This equation has fit statistics similar to the corresponding regression for all laboratory data, H ¼ 0.0142IB2/3. Apparently, use of Eqn 20 for ue masks any dependence of H on ua or Nc under field conditions. Summary of results Numerical models describing wildland fire (Porterie et al. 2000; Morvan 2007; Nmira et al. 2010) identify a low-wind combustion regime where buoyant forces exceed ambient wind inertial forces, and a moderate-to-high wind regime in which the dominant force is exerted by the wind. To a large extent, these results are supported by findings of the present study. Experimental data for the SFFL laboratory and field head fires showed that the criterion Nc ¼ 10 often indicated transition from dynamic entrainment (Nc . 10) to accretive entrainment (Nc , 10) as wind speed increased. These two burning regimes are likely to appear in analyses of tan A and H involving wind speed ua – i.e. use of Eqn 19 for entrainment velocity ue. Alternately, when Eqn 20 for ue was used, tan A in the field data was essentially constant. The presence of fireline intensity IB in the analysis of 136 Int. J. Wildland Fire R. M. Nelson et al. flame height H in the field fires seemed to incorporate the effects of ua on H automatically; thus for all values of Nc, the fires burned in a single regime. This result differed from laboratory fire data for H v. IB (Fig. 5a), which separated according to Nc. The difference may be due to experimental design (three fuel groups with constant Wa within groups) and confined buoyant convection in the SFFL wind tunnel. Entrainment parameters and flame zone air/fuel mass ratios are summarised in Table 1. Accretive and dynamic regimes of entrainment are indicated by Nc , 10 and Nc . 10 respectively. The combustion zone air/fuel ratio fcz is calculated from Eqns 14 as: fcz ¼ 0:4ff  1 Xb  ¼ 0:4ff 1:3 3. 4. ð30Þ with volatile burn fraction Xb taken as 0.75. Table 1 gives an overall air/fuel ratio, fcz þ ff, equal to 67 kg kg 1 – a value within the range 60–80 kg kg 1 (Thomas et al. 1965). The earlier theoretical estimates, ff ¼ 50 and fcz ¼ 20 kg kg 1, compare favourably with the semi-empirical values in Table 1. The laboratory fires duplicated the field fires with two exceptions. First, for Nc , 10, tan A for the laboratory fires was proportional to Nc 1/2 or Nc 1/3, whereas tan A for the field fires was independent of Nc. The Albini (1981) model for flame tilt angle, tan A ¼ u/w, was descriptive of only the Nc , 10 data from the SFFL wind tunnel, requiring two additional models for describing tilt angle: (1) Eqn 28 for low-wind-speed fires in the tunnel, and (2) Eqn 29 for all data from the field experiments. The second exception was that the laboratory data for H tended to separate according to the Nc ¼ 10 criterion, whereas the field data exhibited a similar relationship, but without the Nc separation. These differences among tan A, H, IB and Nc seem related to experimental design and the fire environments, rather than to fuel or wind-speed differences. Froude number FH was proportional to Nc 1 for both burning regimes; H was proportional to IBua 1 and to IB2/3 for both the laboratory and field fires. Conclusions The objectives of this study were to: (1) develop criteria to determine whether differences in observed flame characteristics can be related to differences in air entrainment mechanisms; (2) derive equations for relating flame height and tilt angle to commonly used fire behaviour variables and entrainment parameters; and (3) develop estimates of entrainment parameters by using the model equations and regression methods to generate statistical fits of the laboratory and field data. Specific conclusions drawn from the present work are: 1. Two burning regimes are found in laboratory wind-tunnel fires in slash pine litter beds; the same regimes are present in field fires in the palmetto–gallberry and longleaf pine litter– grass fuel types. Transition from a low wind speed to a higher wind speed regime is indicated by Nc ¼ 10. 2. Equations for flame tilt angle and flame height generally describe the experimental tilt angles and heights well. For the field fires, tan A is constant rather than a power function of reciprocal Nc. Kinetic energy fluxes in the ambient air and 5. 6. flame describe the constant tilt angle regime. Laboratory data for the H v. IB2/3 relationship separate according to the Nc ¼ 10 criterion, but the field data for H do not separate. Air enters head fire flames by: (i) dynamic entrainment (Nc . 10) in which the entrainment velocity approximates the mid-flame wind speed, or (ii) accretion (Nc , 10) in which air is blown into the flame either at a velocity equal to 71% of the mid-flame wind speed, ua, or at a velocity equal to 55% of the characteristic vertical flame velocity, wc. The mean velocity of entrainment, ue, is proportional to either ambient wind speed ua with proportionality constant Z, or to the characteristic buoyant velocity wc with proportionality constant a. For moderate winds (Nc , 10), these semiempirical constants from the laboratory data are Z ¼ 0.71 and a ¼ 0.55 (assumed equal for both laboratory and field fires). For low winds (Nc . 10), Z ¼ 1.02; no value is available for a. Theoretical flame-zone air/fuel ratio ff is 50 kg kg 1; combustion-zone air/fuel ratio fcz is 20 kg kg 1. Corresponding experimental ratios (averaged for laboratory and field burns over all Nc) are 48.5 and 18.2. Thus the theoretical overall air/fuel ratio of 70 compares favourably with the semi-empirical ratio of 67. Field fires in the southern rough and longleaf litter–grass fuel types can be simulated well with laboratory wind-tunnel fires insofar as estimates of the air/fuel mass ratio and flame height are concerned, but tangent of the flame tilt angle is sensitive to environmental conditions. Symbols used in mathematical models Roman symbols A, flame tilt angle from vertical (degrees of angle) Af, flow area of flame (m2) Ap, projected flame area (m2) bF, fireline buoyancy flux (m3 m 3) bR, ocean plume buoyancy flux (m3 m 3) CD, flame drag coefficient cos A, cosine of angle A cp, constant-pressure specific heat of burned and unburned volatiles, flame fluid and air (kJ kg 1 K 1) D, horizontal width of flame at z (m) Do, flame depth at z ¼ 0 (m) FB, flame buoyant force (kg m s 2) FD, horizontal drag force on flame (kg m s 2) FH, flame height Froude number FF, fire Froude number FL, flame length Froude number FR, effluent plume Froude number f, fraction of original moisture remaining after preheating g, acceleration of gravity (m s 2) H, flame height (m) Hc, convective low heat of combustion (kJ kg 1) IB, overall fireline intensity (kW m 1) IBcz, combustion zone contribution to IB (kW m 1) L, unit length of fireline (m) M, fractional moisture content Me, mass of an entrained air parcel (kg) m, vertical mass flow rate at z (kg m 1 s 1) mo, vertical mass flow rate at z ¼ 0 (kg m 1 s 1) Flame characteristics of wildland fires mt, vertical mass flow rate at z ¼ H (kg m 1 s 1) Nc, convection number Nv, stoichiometric air/fuel mass ratio of volatiles (kg kg 1) Dp, pressure drop in flame (kg m 1 s 2) q, effluent volumetric discharge rate per unit length of diffuser (m2 s 1) R, rate of fire spread (m s 1) sec A, secant of angle A T, mean flame temperature at z (K) Ta, ambient air temperature (K) To, mean flame temperature at z ¼ 0 (K) Tt, mean flame temperature at z ¼ H (K) t, time (s) u, horizontal component of flame velocity at z (m s 1) ua, mid-flame ambient wind speed (m s 1) uc, horizontal ocean current speed (m s 1) ue, mean entrainment velocity (m s 1) uo, mean value of u at z ¼ 0 (m s 1) v, mean axial flame velocity at z (m s 1) V, volume of heated air parcel of mass Me (m3) Vo, volume of ambient air parcel of mass Me (m3) W, mean work done by parcels of air or flame (kg m2 s 2) Wa, available fuel loading (kg m 2) w, vertical component of flame velocity at z (m s 1) wc, characteristic buoyant velocity (m s 1) wco, wc for zero-wind fires (m s 1) wo, mean value of w at z ¼ 0 (m s 1) Xb, fraction of volatiles produced that burns Yi ; Y^i , observed and predicted value of dependent variable respectively Z, mass of unreacted air in the combustion zone per mass of original fuel z, vertical distance above fuel bed surface (m) Greek symbols a, entrainment constant ao, entrainment constant when ua ¼ 0 g1 ; ^g1 , analytically derived and statistically estimated values of power function coefficient l, analytically derived exponent Dr, density difference between diffuser effluent and ambient water (kg m 3) e, combustion efficiency Z, entrainment constant r, flame mass density at z (kg m 3) ra, ambient air mass density (kg m 3) rc, flame mean mass density (kg m 3) re, diffuser effluent mass density (kg m 3) ro, flame mass density at T ¼ To (kg m 3) fcz, combustion zone mean air/fuel mass ratio ff, free flame mean air/fuel ratio Acknowledgements We thank Dale Wade, Ted Ach, Wayne Adkins and Hilliard Gibbs, all formerly of the Southern Forest Fire Laboratory, Macon, GA, for their aid in burn plot preparation and collection of fire behaviour data during the 1988 Int. 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In ‘Proceedings of the Tenth Symposium (International) on Combustion’, 14–20 July 1968, Poitiers, France. pp. 983–996. (The Combustion Institute: Pittsburgh, PA) Van Wagner CE (1968) Fire behavior mechanisms in a red pine plantation: field and laboratory evidence. Canadian Department of Forestry and Rural Development, Forestry Branch Publication 1229m. (Ottawa, ON) Weise DR, Biging GS (1996) Effects of wind velocity and slope on flame properties. Canadian Journal of Forest Research 26, 1849–1858. doi:10.1139/X26-210 Zhou X, Mahalingham S, Weise D (2005) Modeling of marginal burning state of fire spread in live chaparral shrub fuel bed. Combustion and Flame 143, 183–198. doi:10.1016/J.COMBUSTFLAME.2005. 05.013 www.publish.csiro.au/journals/ijwf Flame characteristics of wildland fires Int. J. Wildland Fire 139 Appendix A. Southern Forest Fire Laboratory (SFFL) 1988 field data Fire number A OS1A1 OS1A2 OS1B1 OS1C1 OS1D1 OS1D2 OS1E1 OS1E2 OS1F1 OS1F2 OS2A1 OS2B1 OS2C1 OS2D1 OS2D2 OS2E1 OS2E2 OS2F1 OS4B1 OS4B2 OS4C1 OS4C2 OS4D1 OS4D2 FM1A2B FM1D1 FM2A1 FM2A2 FM2C1 FM2D1 FM4A1 FM4D1 A R (m s 1) Wa (kg m 2) ua (m s 1) Do (m) H (m) IB (kW m 1) Nc tan A 0.063 0.076 0.043 0.073 0.063 0.092 0.065 0.019 0.036 0.053 0.043 0.122 0.125 0.100 0.046 0.085 0.144 0.048 0.226 0.120 0.259 0.136 0.301 0.341 0.102 0.087 0.154 0.168 0.054 0.052 0.298 0.138 0.534 0.552 0.370 0.527 0.279 0.605 0.397 0.366 0.226 0.643 0.972 1.100 0.536 0.765 0.983 0.755 1.350 0.444 0.483 0.600 2.010 1.480 1.050 0.945 0.272 0.620 0.466 0.889 0.575 0.459 0.370 0.578 2.03 1.74 0.91 2.68 1.31 1.31 1.37 0.51 0.90 1.12 2.46 2.70 2.24 1.22 2.00 1.34 1.34 0.90 1.79 2.00 3.59 1.34 3.59 3.59 1.12 1.57 1.79 2.46 1.57 1.79 2.24 1.52 0.68 0.53 0.56 0.68 0.52 0.91 0.66 0.40 0.41 0.56 0.68 1.35 1.81 1.13 0.57 0.92 1.24 0.75 1.39 1.23 1.45 1.10 2.66 3.55 0.62 0.59 0.95 1.02 0.61 0.67 1.73 0.74 0.57 1.06 0.78 1.40 0.61 1.16 0.83 0.61 0.37 0.58 0.94 2.50 2.58 1.85 1.11 1.93 2.15 1.04 2.85 1.64 3.30 2.07 4.60 4.97 0.98 1.33 1.64 1.70 1.11 1.29 3.30 1.63 505 634 239 577 264 834 387 102 122 511 627 1980 1004 1147 678 964 2915 323 1638 1084 7824 3016 4728 4818 416 809 1076 2240 466 355 1654 1196 3.32 6.72 16.86 1.63 6.35 19.74 8.00 41.00 9.04 19.74 2.28 5.40 4.87 33.57 4.63 21.43 64.00 23.32 15.63 7.54 9.04 72.34 5.69 5.69 15.63 11.74 10.27 8.00 6.72 3.32 8.00 18.22 0.727 0.577 0.675 0.649 0.649 0.601 0.810 0.727 0.933 1.235 0.601 0.649 0.577 0.554 0.424 0.727 0.325 0.404 0.554 0.601 0.649 0.532 0.649 0.601 0.781 0.510 0.754 0.649 0.625 0.325 0.554 0.488 OS1A1 denotes a fire in the Osceola National Forest, 1-year rough, plot A1. FM1A2 denotes a fire in the Francis Marion National Forest, 1-year rough, plot A2. B 140 Int. J. Wildland Fire R. M. Nelson et al. Appendix B. Regression equations and statistical fits of the model equations to experimental data t-test results show if a parameter estimate ¼ 0. Y indicates that the estimate is significantly different from zero (rejected null hypothesis). N indicates that the null hypothesis was not rejected. Probability value of t-value # 0.05 defined as significant. Fit statistics are: P 2 residual sum of squares deviance Y^ ¼1 R2 ¼ P i2 ¼ 1 uncorrected sum of squares ðUSSÞ USS Yi 2KðK þ 1Þ AICc ¼ AIC þ n K 1 where K is the number of parameters in a model and n is the number of observations Figure Model t-test results Nc b1 2a 2b 3a 3b 4a 4b 5a 5b tan A ¼ 1.190Nc 1/2 tan A ¼ 1.044Nc 1/3 tan A ¼ 3.931Nc 2/3 tan A ¼ 4.458Nc 2/3 tan A ¼ 4.119Nc 2/3, outlier removed tan A ¼ 1.041Nc 1/3, outlier removed tan A ¼ 0.655Nc 0.03, outlier removed FH ¼ 2.421Nc 1 FH ¼ 1.676Nc 1 FH ¼ 1.726Nc 1.04 FH ¼ 2.310Nc 1 FH ¼ 1.362Nc 1 FH ¼ 1.397Nc 1 FH ¼ 0.878Nc 0.62 H ¼ 0.0024IB ua 1 H ¼ 0.0033IB ua 1 H ¼ 0.0024IB ua 1, outliers removed H ¼ 0.0035IB ua 1, outliers removed H ¼ 0.0173IB2/3 H ¼ 0.0132IB2/3 H ¼ 0.0142IB2/3 H ¼ 0.0155IB2/3, outlier removed ,10 ,10 .10 .10 .10 ,10 All Nc .10 ,10 All Nc .10 ,10 All Nc All Nc .10 ,10 .10 ,10 .10 ,10 All Nc All Nc Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y b2 N Y Y Fit statistics RMSE Error d.f. MAE AICc 0.1665 0.1352 0.0839 0.2578 0.1927 0.1896 0.1341 0.0157 0.1464 0.1159 0.0408 0.1480 0.1178 0.0932 0.0947 0.1076 0.3342 0.5514 0.1223 0.1248 0.1405 0.5445 12 12 8 13 12 17 29 8 12 20 13 17 31 30 8 12 11 16 8 12 21 26 0.097 0.094 0.060 0.189 0.155 0.139 0.095 0.013 0.092 0.071 0.031 0.106 0.082 0.060 0.073 0.0784 0.270 0.362 0.096 0.094 0.105 0.392 5.55 10.98 14.12 5.83 1.77 5.02 31.77 44.31 8.90 27.16 45.22 13.94 42.65 56.30 11.95 12.75 12.04 31.83 7.34 13.06 20.32 47.27 R2 0.956 0.971 0.977 0.862 0.906 0.916 0.956 0.985 0.966 0.965 0.913 0.845 0.837 0.901 0.973 0.968 0.955 0.943 0.954 0.958 0.941 0.931 International Journal of Wildland Fire doi:10.1071/WF10034_AC ©IAWF 2012 Accessory publication Entrainment regimes and flame characteristics of wildland fires Ralph M. Nelson JrA,D, Bret W. ButlerB and David R. WeiseC A US Forest Service, 206 Morning View Way, Leland, NC 28451, USA. [Retired]. B US Forest Service, Rocky Mountain Research Station, Missoula Fire Sciences Laboratory, Missoula, MT 59807, USA. C US Forest Service, Southwest Research Station, Forest Fire Laboratory, Pacific Riverside, CA 92507, USA. D Corresponding author. Email: [email protected] Herein we report details of the derivation of two supplementary flame characteristic models and a discussion of flame tilt angle in the laboratory and field for which space was not available in the published text. Background In the published text, equations for entrainment parameters and flame characteristics of steadily burning 2-D head fires in uniform wildland fuels are derived. The text suggests three separate regimes of flow above such fires, with two of these regimes delineated by a critical value of the Byram convection number Nc = 10. The starting point for the flame characteristic derivations is a simplified version of the Albini (1981) flame model. The model equations are tested with fire behaviour data from laboratory wind tunnel burns in slash pine litter fuels (Nelson and Adkins 1986) and field data reported in Appendix A of the text. It is shown that flame characteristics derived from the Albini model are descriptive of flame tilt angle only in the laboratory fires and, as expected, only when Nc < 10. The authors wish to present alternative flame angle models for the Nc > 10 regime to give the reader a complete report of our work and provide modeling approaches that bring the models into agreement with the experimental data. tanA in laboratory and field fires for Nc > 10 The sketch in Fig. 1 of the text depicts a time-averaged visible flame of height H tilted at mean angle A from vertical; the flame shape approximates a rectangular solid with flow area Af (thickness DocosA by unit width L of fireline into the page) and length HsecA. A mixture of burning volatiles and combustion-zone air flows steadily along the flame axis with a velocity whose ‘whole fire’ mean vertical component (rather than vertical velocity w at z) is the characteristic velocity wc (Eqn 18 of the text). The mean flame temperature of 750 K ((1000 + Page 1 of 4 International Journal of Wildland Fire doi:10.1071/WF10034_AC ©IAWF 2012 500)/2) is computed from previously assumed values for To and Tt. We assume that viscous forces are negligible and the fluid is incompressible (mean density ρc = 0.48 kg m–3); thus, the integrated form of the Euler equation (Lay 1964) may be used to write the vertical buoyant force as ⎛ ρc wc2 ⎞ FB = − Af Δp = gDo cos A( ρ a − ρc ) HL sec A = ⎜ ⎟ HL (A1) ⎝ 2 ⎠ where Δp is the pressure drop in the flame due to buoyancy. The horizontal drag force on the flame, using Eqn 19 of the text, is FD = C D ρ a ue2 Ap 2 = CD ρ aη 2ua2 HL (A2) 2 where CD is the drag coefficient for the inclined flame and Ap is the projected area (the area normal to the direction of air flow). The balance of transverse forces that determines angle A is FBsinA = FDcosA and leads to tan A = −2 FD CD ρ aη 2ua2 = = 3.85η 2 N c 3 (A3) 2 ρc wc FB where CD = 1.54 (Fang 1969). Differences in tanA data for laboratory and field fires Fig. 2 of the text indicates that tanA relationships for the laboratory and field fires differ significantly. For the laboratory fires, tanA is proportional to either Nc–1/2 or Nc–1/3 when Nc < 10, and follows Eqn A3 when Nc > 10. In the field, tanA is constant for all Nc. These differing results may be related to hindered v. freely moving combustion products in and above the flame for the laboratory and field fires respectively. We expect smaller tilt angles and reciprocal Nc values in field measurements than would be observed for the same fire in a wind tunnel. In the field, the reduced influence of wind speed and tilt angle should combine with generally greater fuel loads and an increased rate of spread due to greater fireline length (Cheney and Sullivan 1997) to drive tanA toward a constant value. The dependence of tanA on powers of Nc close to –1/3 seems associated with fires in wind tunnels with fixed ceilings (Taylor 1961; Nelson and Adkins 1986); an exception is the study of Weise and Biging (1996) who found a dependence close to Nc–1/3 even though their relatively small tunnel was operated with a moving ceiling. However, a tendency toward Nc independence, or at most a weak dependence, seems to occur in relatively large wind tunnels (Anderson et al. 2006) and in tunnels that allow free convection (Fendell et al. 1990). Page 2 of 4 International Journal of Wildland Fire doi:10.1071/WF10034_AC ©IAWF 2012 tanA in the field fires based on kinetic energy flux We assume the flame tilt angle is determined by a balance between the transverse components of the kinetic energy flux of ambient air approaching the flame and the vertical flame fluid kinetic energy flux due to buoyancy. This balance is given by: (dW/dt)drag = (dW/dt)buoyancy = FDuecosA = FBwcsinA where W is work done and t is time. With this interpretation, rates at which parcels of air and flame fluid do work apparently govern flame tilt angle for moderate winds in the field, whereas a mass flux balance is operative in wind tunnels such as the SFFL tunnel in which the steady winds are more unidirectional because convection is confined. Use of Eqns 20 of the text and A1 and A2 above leads to tanA = CDρaα3/ρc = 3.85α3 (A4) This equation gives an estimate of entrainment constant α identical to that derived for the lab fires from Eqn 23 of the text. References Albini FA (1981) A model for the wind-blown flame from a line fire. Combustion and Flame 43, 155–174. doi:10.1016/0010-2180(81)90014-6 Anderson W, Pastor E, Butler B, Catchpole E, Dupuy JL, Fernandes P, Guijarro M, Mendes-Lopes JM, Ventura J (2006) Evaluating models to estimate flame characteristics for free-burning fires using laboratory and field data. In ‘Proceedings, V International Conference on Forest Fire Research’, 27–30 November 2006, Figueira da Foz, Portugal. (Ed. DX Viegas). (CD-ROM) (Elsevier BV: Amsterdam) Cheney P, Sullivan A (1997) ‘Grassfires: fuel, weather and fire behaviour.’ (CSIRO Publishing: Melbourne) Fang JB (1969) An investigation of the effect of controlled wind on the rate of fire spread. PhD thesis, University of New Brunswick, Fredericton, NB. Fendell FE, Carrier GF, Wolff MF (1990) Wind-aided fire spread across arrays of discrete fuel elements. US Department of Defense, Defense Nuclear Agency, Technical Report DNA-TR-89–193. (Alexandria, VA) Lay JE (1964) ‘Thermodynamics: a macroscopic-microscopic treatment.’ (Charles E Merrill Books, Inc.: Columbus, OH) Nelson RM Jr, Adkins CW (1988) A dimensionless correlation for the spread of wind-driven fires. Canadian Journal of Forest Research 18, 391–397. doi:10.1139/x88-058 Page 3 of 4 International Journal of Wildland Fire doi:10.1071/WF10034_AC ©IAWF 2012 Taylor GI (1961) Fire under the influence of natural convection. In ‘The use of models in fire research’. (Ed. WG Berl) National Academy of Science, National Research Council Publication 786, pp. 10–32. (Washington, DC) Weise DR, Biging GS (1996) Effects of wind velocity and slope on flame properties. Canadian Journal of Forest Research 26, 1849–1858. doi:10.1139/x26-210 Page 4 of 4