# Comparative Assessment of Wildland Fire Rate of Spread Models: Effects of Wind Velocity

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Semi-Empirical, Laboratory-Developed ROS Models

_{P}), over the heat per unit volume required by the fuel volume to ignite (L

_{ig}).

_{R}) which corresponds to the heat flux generated from the fire front; in this case, the propagating heat flux (I

_{P}) is a fraction of I

_{R}, defined as the propagating heat flux ratio (ξ). Through mathematical manipulation, assumptions and theoretical reasoning, the model expresses the ROS under quiescent and horizontal conditions (R

_{0}) as a function of physical properties that can be measured, calculated or correlated to measured fuel parameters through laboratory experimentation (c.f. Table A1).

_{R}) only originates from the flaming combustion of the fire front, and, on the other hand, fuel ignition initiates when the fuel pyrolysis is completed. In addition, Wilson proposed different functional forms and property dependencies on I

_{R}and ξ (c.f. Table A2).

_{P}). The latter, in this case, is not expressed as the product of I

_{R}·ξ, but it is directly empirically correlated to the fuel characteristics, as well as to wind velocity. It is noted that the functional form for the wind velocity correction can also yield results for the case of quiescent conditions. Moreover, the L3 model keeps the proposal of Wilson [11] on the onset of fuel ignition after the completion of pyrolysis. Finally, this model considers the effect of fuel bed width (W) by means of a proportional multiplier of R for fuel beds that have a width lower than 1 m.

#### 2.2. Empirical, Laboratory-Developed ROS Models

_{p}).

#### 2.3. Wind Correction Empirical Sub-Models

_{0}) models, aiming to “correct” their values for wind effects. The wind correction sub-model of Rothermel [10] (W1) makes use of a correction factor (Φ

_{W}), which is estimated from laboratory experiments and is empirically correlated to the wind velocity (U) and to other fuel and fuel bed parameters (c.f. Table A3). The dependence of Φ

_{W}on the fuel characteristics is associated with the physical observation that when the fuel bed is less compact, heat transfer from the burning front to the fuel bed is more efficient and vice versa in the case of a closely packed fuel bed.

_{B}), which is an alternative expression for the energy generated, similar to I

_{R}. Subsequently, an increase in I

_{B}implies the increasing momentum of the buoyant plume behind the fire front, which tends to pull the flame away from the unburned fuel bed, thus decreasing the above-mentioned flame geometry characteristics (height and angle) and the contribution of flame radiation heating. Thus, external wind initiates a “competition” between the pushing of the flaming front towards the fuel bed and the pulling of the flaming front by the buoyant plume whose strength is increased, indirectly, by the wind. By assuming that the momentum of the buoyant plume is analogous to the rate that the fuel is added to the flame (${\dot{m}}^{\prime}$), the W2 model introduces a correction factor (R

_{U}) that is empirically correlated to the wind velocity (U), ${\dot{m}}^{\prime}$ and the ratio of fuel bed to flame height (δ/H

_{f})

_{0}; the latter parameter quantifies the flame extension above the fuel bed. The model assumes that ${\dot{m}}^{\prime}$= m

^{″}·R

_{0}, while the (δ/H

_{f})

_{0}parameter is empirically estimated as a function of FMC (c.f. Table A4), based on a previous work of Rossa and Fernandes [21].

_{B}= h·m

^{″}·R

#### 2.4. Empirical, Field-Based Models and Wind Adjustment Factor

_{z}). The Wind Adjustment Factor is estimated using Equation (3) [23] and is formulated under the assumption that the flame height above the fuel bed is considered to be approximately equal to the height of the fuel bed [24].

_{z}) is based on the additional assumption that the wind tunnel nominal velocity, due to the scale of the laboratory fires, is approximately equal to $\overline{\overline{U}}$, an assumption that is commonly made in studies of similar nature [6].

#### 2.5. Laboratory Experimental Data

_{v}) needed in the L1 and L2 models, respectively, were assumed to be 18,608 and 12,306 kJ/kg, respectively, based on information presented in references [6,10]. The heat of pyrolysis (Q

_{P}), the moisture damping constant (k) and the equivalent characteristic moisture content (M

_{c}) required for models L2 and L3 were taken from [12], based on the specific fuel type of each fire test. The moisture of extinction (M

_{x}), the total mineral content (S

_{T}) and the effective mineral content (S

_{e}), needed for the calculation of I

_{R,1}of model L1, were assumed to be equal to 0.3, 0.0555 and 0.01, respectively, based on [10]. The parameter ${\overline{n}}_{x}$ required for the L2 model was assumed to be equal to 3, according to [11]. For the wind correction model W2, (f

_{il}) was assumed to be 2.42 [14]. In terms of the F1 model, the fuel cover parameter (CF) was assumed to be equal to 1, since the fuel beds used in the experiments had a “continuous” fuel distribution. For certain fuel properties, i.e., h, S

_{t}, S

_{e}, M

_{x}, h

_{v}and f

_{il}, a sensitivity analysis was performed, aiming to investigate the impact of their assumed values on the respective model performance.

_{p}parameters of the Pinus Pinaster needles that were used in [25] were not reported, they were assumed to be equal to 3057 m

^{−1}and 511 kg/m

^{3}, respectively; these values were taken from laboratory tests performed on the same fuel [30,31]. The assumptions that were made in order to be able to use all the investigated models for all the considered fire tests (c.f. Table A6) may have affected, to a certain degree, the accuracy of the predicted ROS values.

## 3. Results

_{pred}and R

_{exp}are the predicted and measured ROS values, respectively, and n is the total number of experiments.

#### 3.1. Laboratory-Developed Models in Quiescent Conditions

^{″}) for all the considered test cases. It is evident that the scatter of the predicted values is reduced with increasing fuel load, thus suggesting high sensitivity of the ROS models to relatively low fuel load values. Among the five ROS models, the L3 model exhibits higher sensitivity in low m

^{″}values, thus resulting in the relatively high error values reported in Table 1.

#### 3.2. Laboratory-Developed Models, Combined with Wind-Correction Sub-Models, against External Wind Conditions

#### 3.3. Field-Developed Models against External Wind Conditions

#### 3.4. Sensitivity Analysis

_{t}) and effective (S

_{e}) mineral content and the moisture of extinction (M

_{x}) of model L1; the fuel’s pyrolysis gas lower calorific value (h

_{v}) for model L2; and the ignition line length factor (f

_{il}) used in wind correction W2. Towards this end, the value of each parameter was varied by ±25%; these variation limits were selected as an adequate representation of the “common” value range of the corresponding specific fuel properties. The results of the sensitivity analysis are presented in Table 6, in the form of the change in the overall RMSE values when each parameter value was changed by ±25%. It is evident that variations in the fuel heat content (h) used in model L1, the fuel’s pyrolysis gas lower calorific value (h

_{v}), used in model L2, as well as the ignition line length factor (f

_{il}) used in model W2 (the model combination L5-W2 was used for the sensitivity analysis), result in moderate sensitivity of the obtained prediction accuracy, whereas the rest of the fuel parameters have less significant impacts. The relatively higher sensitivity on the h, h

_{v}and f

_{il}parameter values is expected, since the h and f

_{il}parameters are essentially “multipliers” in the respective ROS equations (c.f. Table 1), whereas h

_{v}is an additional input through a more complicated functional form. Overall, the calculated sensitivity of the investigated parameters is considered to be reasonable, since their assumed values do not have an unexpected strong influence on the obtained results.

## 4. Discussion

_{7}or B

_{7}(c.f. Table 1); thus, there is less focus on the potential changes in convective heat transfer. This may be one reason for the increasing discrepancies exhibited by W2’s predictions with increasing wind velocity values (c.f. Figure 4). On the other hand, in the very low wind velocity region, the observed W2 discrepancies may be attributed to the fact that when the wind correction parameter R

_{U}tends to zero, the predicted ROS value is largely affected by the parameter f

_{il}(c.f. Table 1), thus leading to the potential over-prediction of the experimentally determined values. The effects of packing ratio are only taken into account by sub-model W1. The packing ratio value affects both radiation and convection; closed-packed beds are both blocking the radiation to the fuel bed’s bottom layer and decreasing the ventilation through the fuel bed. Thus, the packing ratio may be an important parameter that has to be included in generic ROS models that account for the effects of wind.

_{b}as an input. It is interesting to note that ρ

_{b}(or, more specifically, its dimensionless form (β)) was shown to be affecting the prediction accuracy of the L-W model combinations. Thus, it may be the case that the effect of the fuel bed compactness, described by means of the ρ

_{b}or β parameters, is an important descriptive quantity that can be closely connected to the impact of external wind on fire spreading.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbol | Units | Description |

A, B, …, F | - | Empirically fitted constants or functions |

a, b, c | - | Fuel parameters, used in model F5 |

FC | % | Surface fuel cover |

f_{il} | - | Ignition line length factor, used in sub-model W2 |

H_{f} | m | Flame height |

h | kJ/kg | Fuel lower calorific value |

h_{v} | kJ/kg | Fuel’s pyrolysis gas lower calorific value |

I_{B} | kW/m | Byram’s fireline intensity |

I_{P} | kW/m^{2} | Propagation heat flux |

I_{R} | kW/m^{2} | Reaction intensity |

ISI | - | Initial Spread Index, used in model F5 |

k | - | Moisture damping constant, used in model L3 |

L_{ig} | kJ/m^{3} | Heat per unit volume required for ignition |

M* | - | Fuel moisture content—FMC (dry basis) |

M_{c} | - | Characteristic moisture, used in model L2 |

M_{x} | - | Moisture of extinction, used in model L1 |

${\dot{m}}^{\prime}$ | kg/s·m | Rate of fuel added to combustion zone |

m_{n}^{″} | kg/m^{2} | Net fuel load |

m_{d}^{″} | kg/m^{2} | Dry fuel load |

n_{x} | - | Extinction index |

${\overline{n}}_{x}$ | - | Extinction adjustment factor |

n | - | Total number of experiments |

P_{f} | - | Probability function for fire extinction |

Q_{p} | kJ/kg | Heat of pyrolysis |

Q_{w} | kJ/kg | Required heat to evaporate the fuel’s moisture |

R_{0} | m/s* | “Base” ROS in quiescent and horizontal conditions |

R | m/s* | Rate of spread |

R_{exp} | m/s* | Experimentally measured rate of spread |

R_{pred} | m/s* | Predicted rate of spread |

R_{u} | - | Wind correction factor, used in sub-model W2 |

S_{e} | - | Effective mineral content |

S_{t} | - | Total mineral content |

s | m^{2}/kg | Fuel particle specific surface |

T_{a} | °C | Ambient temperature |

U | m/s* | Wind velocity |

U_{z} | m/s* | Wind velocity measured at height z |

$\overline{\overline{U}}$ | m/s* | Wind velocity at mid-flame height |

W | m | Fuel bed width |

z | ft* | Wind velocity measuring height |

β | - | Fuel bed packing ratio |

Γ′ | min^{−1} | Potential reaction velocity |

δ | m | Fuel bed height |

ε | - | Effective heating number |

η_{Μ} | - | Moisture damping coefficient |

η_{S} | - | Mineral damping coefficient |

ξ | - | Propagating heat flux ratio |

ρ_{b} | kg/m^{3} | Fuel bed density |

ρ_{p} | kg/m^{3} | Fuel particle density |

σ | m^{−1} | Surface area-to-volume (SAV) ratio |

Φ_{w} | - | Wind correction factor, used in sub-model W1 |

* Properties may, in certain cases, be expressed in different units (see text). Numerical subscripts (in Table 1, Table A1, Table A2 and Table A6) indicate different constant or function. |

## Appendix A. Detailed Forms of Equation Functions

Function | Units |
---|---|

${I}_{R,1}={\mathsf{\Gamma}}^{\prime}{{m}^{\u2033}}_{n}h{\eta}_{M,1}{\eta}_{S}$ | (kJ/min·m^{2}) |

${\mathsf{\Gamma}}^{\prime}=\frac{{\left[\frac{\beta}{0.20395{\sigma}^{-0.8189}}\mathrm{exp}\left(1-\frac{\beta}{0.20395{\sigma}^{-0.8189}}\right)\right]}^{\frac{1}{6.7229{\sigma}^{0.1}-7.27}}}{0.0591+2.926{\sigma}^{-1.5}}$ | (min^{−1}) |

${{m}^{\u2033}}_{n}={m}^{\u2033}\left(1-{S}_{T}\right)$ | (kg/m^{2}) |

${\eta}_{M,1}=1-2.59\frac{M}{{M}_{X}}+5.11{\left(\frac{M}{{M}_{X}}\right)}^{2}-3.52{\left(\frac{M}{{M}_{X}}\right)}^{3}$ | |

${\eta}_{S}=0.174{S}_{e}^{-0.19}$ | |

${\xi}_{1}=\frac{{\mathrm{e}}^{\left(0.792+3.7597{\sigma}^{0.5}\right)\left(\beta +0.1\right)}}{192+7.9095\sigma}$ | |

${L}_{1}={\rho}_{b}\epsilon \left(581+2594M\right)$ | (kJ/m^{3}) |

$\epsilon ={\mathrm{e}}^{-\frac{4.528}{\sigma}}$ | |

σ | (cm^{−1}) |

Function | Units |
---|---|

${\xi}_{2}=1-{\mathrm{e}}^{-0.17\sigma \beta}$ | |

${L}_{2}={\rho}_{b}\epsilon \left({Q}_{p}+M{Q}_{w}\right)$ | (kJ/m^{3}) |

${I}_{R,2}={{\mathsf{\Gamma}}^{\prime}}_{2}{m}^{\u2033}{h}_{v}{\eta}_{M,2}$ | (kJ/min·m^{2}) |

${{\mathsf{\Gamma}}^{\prime}}_{2}=0.34\sigma {\left(\sigma \beta \delta \right)}^{-\frac{1}{2}}{\mathrm{e}}^{-\frac{\sigma \beta}{3}}{P}_{f}\left({n}_{x}\right)$ | (min^{−1}) |

${P}_{f}\left({n}_{x}\right)=\frac{1}{1+{e}^{-\pi \frac{{n}_{x}-{\overline{n}}_{x}}{1.2\sqrt{3}}}}$ | |

${n}_{x}=\frac{\mathrm{ln}\left(\sigma \beta \delta {h}_{v}/{Q}_{w}\right)}{M+\frac{{Q}_{p}}{{Q}_{w}}}$ | |

${\eta}_{M,2}={\mathrm{e}}^{-kM}$ | |

${I}_{P,3}=\left(495.5+1934{U}^{0.91}\right){\mathrm{e}}^{-\frac{800}{\sigma}}{\beta}^{0.501}{\eta}_{{\rm M},2}$ | (kW/m^{2}) |

σ | (cm^{−1}) |

Function | Units |

${\mathsf{\Phi}}_{W}=7.47{\mathrm{e}}^{-0.8711{\sigma}^{0.55}}{3.281}^{0.15988{\sigma}^{0.54}}{\left(\frac{\beta}{0.20395{\sigma}^{-0.8189}}\right)}^{-0.715{\mathrm{e}}^{-0.01094\sigma}}{U}^{0.15988{\sigma}^{0.54}}$ |

Function | Units |

${\left(\frac{\delta}{{H}_{f}}\right)}_{0}=0.1779+3.713\cdot {10}^{-3}M$ |

Function | Units |

$ISI={A}_{12}{e}^{{B}_{12}{U}_{z}}\left\{{C}_{12}\cdot {e}^{{D}_{12}M}\left[1+{E}_{12}{M}^{{F}_{12}}\right]\right\}$ | |

U_{z} | (km/h) |

## Appendix B. Values of Empirical Parameters

Model | Parameter | Value | Model | Parameter | Value |

L4, L5 | A_{4} | 0.2859 | F1 | A_{8} | 40.982 |

A_{5} | 0.1557 | B_{8} | 1.399 | ||

B_{4,5} | −0.7734 | C_{8} | 1.201 | ||

C_{4,5} | 0.9440 | D_{8} | 1.699 | ||

D_{5} | 0.8173 | F2, F3 | A_{9} | 5.6715 | |

W2 | A_{7} | 2.143 × 10^{−5} | B_{9} | 0.9102 | |

B_{7} | 1.710 | C_{9} | 0.2227 | ||

C_{7} | −1.169 | D_{9} | 0.0762 | ||

D_{7} | −1.166 | A_{10} | 3.8320 | ||

F5 | α | 45 | B_{10} | 1.0927 | |

b | 0.0305 | C_{10} | −0.2098 | ||

c | 2 | D_{10} | 0.0721 | ||

A_{12} | 0.208 | E_{9,10} | 9 | ||

B_{12} | 0.05039 | F_{9,10} | 0.00316 | ||

C_{12} | 91.9 | F4 | A_{11} | 0.773 | |

D_{12} | −0.1386 | B_{11} | 0.707 | ||

E_{12} | 4.93 × 10^{−7} | C_{11} | −0.039 | ||

F_{12} | 5.31 | D_{11} | 0.188 |

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**Figure 1.**Comparison of measured ROS values for the 37 no-wind tests against predicted ROS values using the five laboratory-developed models.

**Figure 2.**Relative error of laboratory-developed models as a function of (dry) fuel load for the 37 no-wind fire tests.

**Figure 3.**Comparison of measured ROS values for the 166 wind tests against predicted ROS values using the two “wind correction” sub-models (W1:

**left**; W2:

**right**), applied to the five laboratory-developed models.

**Figure 4.**Relative error as a function of wind velocity (

**top**) and packing ratio (

**bottom**) for the ten combinations of laboratory-developed ROS models and wind corrections.

**Figure 5.**Comparison of measured ROS values for the 166 wind tests against predicted ROS values using the five field-developed models.

Model Type | Model | General Form | Ref. |
---|---|---|---|

ROS models developed using laboratory tests | L1 | ${R}_{0}=\frac{{I}_{R,1}\left(\sigma ,\beta ,h,M,{{m}^{\u2033}}_{n},{S}_{e}\right){\xi}_{1}\left(\sigma ,\beta \right)}{{L}_{1}(\sigma ,{\rm M},{\rho}_{b})}$ | [10] |

L2 | ${R}_{0}=\frac{{I}_{R,2}\left(\sigma ,\beta ,\delta ,{h}_{v},M,{{m}^{\u2033}}_{d},{Q}_{p,}{Q}_{w}\right){\xi}_{2}\left(\sigma ,\beta \right)}{{L}_{2}(\sigma ,{\rm M},{\rho}_{b},{Q}_{p,}{Q}_{w})}$ | [11] | |

L3 | $R=\frac{{I}_{P,3}\left(\sigma ,\beta ,M,U\right)}{{L}_{2}(\sigma ,M,{\rho}_{b},{Q}_{p,}{Q}_{w})}$ | [12] | |

L4 | ${R}_{0}={A}_{4}{M}^{{B}_{4}}{\delta}^{{C}_{4}}$ | [13] | |

L5 | ${R}_{0}={A}_{5}{M}^{{B}_{5}}{\delta}^{{C}_{5}}\mathrm{ln}\left({D}_{5}s\right)$ | [13] | |

“Wind correction” sub-models | W1 | $R={R}_{0}\left(1+{\mathsf{\Phi}}_{W}\right),{\mathsf{\Phi}}_{W}={A}_{6}\left(\sigma ,\beta \right){U}^{{B}_{6}\left(\sigma \right)}$ | [10] |

W2 | $R={f}_{il}{R}_{0}{R}_{u},{R}_{u}=1+{A}_{7}{U}^{{B}_{7}}{\dot{m}}^{\prime {C}_{7}}{\left(\frac{\delta}{{H}_{f}}\right)}_{0}^{{D}_{7}}$ | [14] | |

ROS models developed using field tests | F1 | $R={A}_{8}\frac{{U}_{z}^{{B}_{8}}F{C}^{{C}_{8}}}{{M}^{{D}_{8}}}$ | [15] |

F2 | $R=\frac{{A}_{9}{U}_{z}^{{B}_{9}}{\delta}^{{C}_{9}}{e}^{{D}_{9}M}}{1+{E}_{9}{e}^{{F}_{9}{W}^{2}}}$ | [16] | |

F3 | $R=\frac{{A}_{10}{U}_{z}^{{B}_{10}}{\rho}_{b}{}^{{C}_{10}}{e}^{{D}_{10}M}}{1+{E}_{10}{e}^{{F}_{10}{W}^{2}}}$ | [16] | |

F4 | $R={A}_{11}{U}_{z}^{{B}_{11}}{e}^{{C}_{11}M}{\delta}^{{D}_{11}}$ | [17] | |

F5 | $R=a{(1-{e}^{-b\cdot ISI})}^{c}$ | [18] |

Ref. | Fire Tests | No-Wind/Wind Tests | Fuel Type |
---|---|---|---|

[25] | 9 | 2/7 | Pine needles (Pinus Pinaster) |

[26] | 6 | 0/6 | Bamboo sticks |

[27] | 163 | 30/133 | Pine needles (Pinus Ponderosa)/Excelsior |

[28] | 7 | 1/6 | Pine needles (Pinus Sibirica) |

[29] | 18 | 4/14 | Pine needles (Pinus Sibirica) |

**Table 3.**Error metric values of the laboratory-developed ROS model predictions for the 37 no-wind fire tests.

Model | RMSE | MAPE (%) | MBE |
---|---|---|---|

L1 | 3.5 | 42.8 | −2.8 |

L2 | 3.2 | 59.3 | 1.0 |

L3 | 4.7 | 71.6 | 1.5 |

L4 | 3.1 | 45.8 | −0.2 |

L5 | 2.9 | 50.9 | 0.3 |

**Table 4.**Error metric values of predictions using the laboratory-developed ROS models/wind correction combinations for the 166 external wind fire tests.

Models | RMSE | MAPE (%) | MBE |
---|---|---|---|

L1-W1 | 40.1 | 46.5 | −25.3 |

L2-W1 | 51.5 | 59.2 | 14.8 |

L3-W1 | 91.8 | 49.3 | 30.8 |

L4-W1 | 40.2 | 52.0 | −6.7 |

L5-W1 | 42.1 | 43.1 | 1.2 |

L1-W2 | 143.8 | 78.0 | 34.4 |

L2-W2 | 127.8 | 105.3 | 35.2 |

L3-W2 | 101.0 | 67.1 | 26.8 |

L4-W2 | 131.3 | 94.9 | 33.7 |

L5-W2 | 124.5 | 90.1 | 33.0 |

**Table 5.**Error metric values of the field-developed ROS model predictions for the 166 wind fire tests.

Model | RMSE | MAPE (%) | MBE |
---|---|---|---|

F1 | 59.2 | 119.0 | −24.0 |

F2 | 44.4 | 67.5 | −21.2 |

F3 | 30.7 | 63.1 | −3.9 |

F4 | 51.5 | 231.3 | 35.2 |

F5 | 280.0 | 1278.3 | 251.6 |

**Table 6.**Absolute and relative change in the RMSE values, resulting from the ±25% relative change in the values of six model parameters.

Parameter | Model | −25% Change | Assumed Value | +25% Change |
---|---|---|---|---|

h | L1 | 4.42 (25.9%) | 3.51 | 2.67 (−23.8%) |

S_{t} | L1 | 3.46 (−1.5%) | 3.51 | 3.56 (1.5%) |

S_{e} | L1 | 3.31 (−5.6%) | 3.51 | 3.65 (4.2%) |

M_{x} | L1 | 3.80 (8.3%) | 3.51 | 3.31 (−5.7%) |

h_{v} | L2 | 2.82 (−13.0%) | 3.24 | 4.70 (45.3%) |

f_{il} | W2 (L5-W2) | 85.70 (−31.2%) | 124.50 | 165.84 (33.2%) |

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## Share and Cite

**MDPI and ACS Style**

Kolaitis, D.I.; Pallikarakis, C.; Founti, M.A.
Comparative Assessment of Wildland Fire Rate of Spread Models: Effects of Wind Velocity. *Fire* **2023**, *6*, 188.
https://doi.org/10.3390/fire6050188

**AMA Style**

Kolaitis DI, Pallikarakis C, Founti MA.
Comparative Assessment of Wildland Fire Rate of Spread Models: Effects of Wind Velocity. *Fire*. 2023; 6(5):188.
https://doi.org/10.3390/fire6050188

**Chicago/Turabian Style**

Kolaitis, Dionysios I., Christos Pallikarakis, and Maria A. Founti.
2023. "Comparative Assessment of Wildland Fire Rate of Spread Models: Effects of Wind Velocity" *Fire* 6, no. 5: 188.
https://doi.org/10.3390/fire6050188