2.2. Governing Equations
FDS uses computational fluid dynamics multiphase modeling techniques to solve governing conservation equations for buoyant flow, combustion rate, energy, and species transport in a low Mach number approximation. The Large Eddy Simulation (LES) technique is used to account for the turbulences [
17]. The filtered instantaneous continuity and momentum equations used in FDS are [
17,
18,
19]:
where
ρ,
u,
x,
P,
τ,
g,
f denote the density, velocity, coordinate, pressure, viscous stress, gravitational acceleration, and force term respectively. The
i,
j subscripts are direction indices and bar and tiled denote the Reynolds filter and Favre filtered quantities. FDS solves energy conservation by coupling the ideal gas law equation and the Poisson equation for pressure in the flow field. When the Navier–Stokes form resolves the velocity field, it is used the computed pressure term in the Poisson equation given below [
17,
18]:
noting the stagnation energy per unit mass
H,
F is the momentum flux,
is the vorticity,
is the instantaneous density and
is the external force vector excluding gravity. The pressure
P, that spatially and temporally resolves for low-speed applications such as fire is decomposed into background pressure
(z, t) and perturbation
(x, y, z, t) where only the background pressure retains in the ideal gas equation:
Here
R is the gas constant and
W is the molecular weight. The relationship of internal energy, e and enthalpy,
h, is described in terms of thermodynamic (background) pressure:
h = e +
to use in energy conservation equation [
17] as below:
where
is the sensible enthalpy,
is the heat release rate per unit volume from chemical reactions and
is the energy transferred to sub-grid-scale (SGS) particles. The conductive, radiative, and diffusive heat flux are represented by the term
. This term is computed by solving the heat transport equations. In FDS hydrodynamic solver guarantees that Equation (6) is satisfied for the energy conservation of the model. Fire is an inefficient combustion process of many fuel gases containing C, H and many other atoms that produce various products [
17]. Tracking all the species is computationally expensive and FDS implements lumps species approach (a mixture of gas species transport together) to solve transport equations efficiently by reducing the number of equations. The species transport equation [
17] that is solved in FDS given as:
where
Z is the mass fraction of
species,
D is the diffusion coefficient,
source terms represent the additions of mass from evaporating droplets or other SGS particles such as vegetation and firebrands. FDS assumes the composition of gas in a cell space is either completely mixed or completely unmixed. At any point in time, the composition of a computational cell is determined by the mixed and unmixed portions as follows:
where
denote as cell mean mass fraction, initial cell mean mass fraction, and mass fraction of the species in the mixed reactor zone respectively.
is the unmixed fraction of mass within the cell. Time differential form of Equation (8) provides the solution of the chemical source term
to use in the species equation (Equation (7)) as below.
where,
is the mixing time. The heat release rate per unit volume
depends on the combustion model. This quantity is fundamentally important in fire physics as it contributes significantly to the velocity divergence and heat transfer. In general, the total heat release rate [
17] is calculated as the summation of the products of
(in Equation (9)) of each species and their heats of formation Δ
h, given as:
FDS consists of a special model for vegetation pyrolysis. The model contains three reactions to represent the solid phase thermal degradation for endothermic moisture evaporation (Equation (11)) [
20], endothermic pyrolysis of dry vegetation (Equation (12)) [
20], and exothermic char oxidization (Equation (13)) [
20] as follows:
where
M is the dry basis vegetation moisture content,
is the mass fraction of dry vegetation that is converted into char during pyrolysis. A significant amount of char oxidization only occurs at the vegetation material temperature is much higher than the temperature achieved in our simulations. Therefore, char oxidization is not accounted for below tree burning and forest fire modeling.
The Lagrangian particles are used in FDS to represent a wide variety of SGS objects (fuel particles, firebrands, etc.) along with their thermo-physical properties and geometric parameters [
20]. They are introduced to the domain as stationary or dynamic particles. Vegetation is represented by stationary particles to randomly distribute through a given volume. In the current work, we specify the number of vegetation particles in a certain volume given by sextuplet in the form of N_PARTICLES and MASS_PER_VOLUME (kg/m
3) [
20]. The default shapes of the particles can be set as SPHERE, CUBE, and CYLINDER for both stationary and dynamic particles [
20]. The firebrands are also introduced into the domain by a Lagrangian particle-based transport scheme to map their distribution and trajectories. This Lagrangian framework describes the physical variables for a solid element that passes through a flow where explains by the Eulerian framework to identify the location specified properties within the considering space [
21]. Each Lagrangian particle interacts with the carrier fluid individually (two-way coupling). Due to this interaction (drag), a momentum loss of the particle is added to the fluid and vice versa. The total momentum
fb exchanged between Lagrangian particles and gas phase within a cell space is expressed as [
17]:
where,
Cd is the drag coefficient,
Ap,c is the cross-sectional area of the particle,
up is the particle velocity,
mp is the particle mass, and
u is the gas velocity. The momentum transfer on the particle [
17] results in acceleration which is given by:
The position of the particle can be determined by [
17]:
The drag coefficient is a function of Reynolds number that is subjected to relative internal movement due to different fluid velocities. The default solution of the drag coefficient [
17] in FDS is expressed for sphere particles where the Reynolds number
Re is based on particle diameter
D,
denoting
is the dynamic viscosity of air at temperature
T. Wadhwani et al. [
15] found the Haider and Levenspiel drag model [
16] showing good agreement for cylinder shape particles [
15] of the short-range firebrand transport. In Haider and Levenspiel drag model, a few empirical correlations are accounted to represent the shapes of particles according to their sphericity. Therefore, the original FDS source code was modified including the Haider and Levenspiel drag model to use in the current simulations. The Haider and Levenspiel drag coefficient [
16]:
where
AHa,
BHa,
CHa,
DHa =
f(
ψ), are the empirical correlations expressed as a function of sphericity,
ψ. The objective of applying the Haider and Levenspiel drag model is to replicate the drag forces acting on the firebrands and their movement with much realism and accuracy. The breakage of firebrands from vegetation is a complex combination of various parameters. Therefore, the mechanisms of firebrand production/tear-off are not investigated in this model.