# Capturing the Swirling Vortex and the Impact of Ventilation Conditions on Small-Scale Fire Whirls
## 2 Mathematical Formulation  
### Governing Equations  
The complex whirling fluid motion and heat transfer of a turbulent reacting fire whirl are predicted by the governing equations of mass, momentum, and energy. The conservation of scalar properties, such as gas chemical species, is governed by a transport equation. For the practical simulation of a weak fire whirl, several assumptions are made, including (i) low-Mach-number flow equations are considered, (ii) thermal-physical properties are constant, (iii) the ratio between mass and thermal diffusivity (Lewis number) is unity, (iv) the variable density effect is only included in the buoyancy term in the momentum equation, where Boussinesq approximation is applied, (v) no external body force and heat source are applied towards the flow, and (vi) non-adiabatic non-premixed combustion of the energy equation is adopted. Accordingly, the following form of the governing equations were utilised in this simulation study:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho\overline{ u}_{i})=0 \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho\overline{u}_{i})+\frac{\partial}{\partial x _{j}}\Big{(}\rho\overline{u}_{i}\overline{u}_{j}\Big{)}=\frac{\partial}{ \partial x_{j}}\Big{(}c_{ij}\Big{)}-\frac{\partial\overline{p}}{\partial x_{i} }-\frac{\partial\tau_{ij}}{\partial x_{j}} \tag{2}\]  
\[\frac{\partial}{\partial t}\Big{(}\rho\overline{H}\Big{)}+\frac{\partial}{ \partial x_{i}}\Big{(}\rho u_{i}\overline{H}\Big{)}=\frac{\partial}{\partial x _{i}}\bigg{(}\frac{k_{t}}{\rho_{x}}\frac{\partial\overline{H}}{\partial x_{i} }\bigg{)}+S_{h} \tag{3}\]  
\[\frac{\partial}{\partial t}(\rho\varphi)+\frac{\partial}{\partial x_{i}} \cdot(\rho\overline{u}_{i}\varphi)=\Gamma\frac{\partial\varphi}{\partial x_{i} }-\frac{\partial y_{j}}{\partial x_{i}}+S_{\varphi} \tag{4}\]  
where \(\rho\) is the mixture density, \(\overline{u}_{i}\) is the velocity vector, and \(p\) is the pressure. As the non-premixed combustion model was adopted, the heat of formation will be included in the definition of enthalpy \(H\); and \(k_{t}\) is the turbulent thermal conductivity which can be calculated as \(k_{t}=\mu_{t}/P\tau_{t}\). \(\mu_{t}\) is the eddy viscosity and [PERSON] et al. suggested that the Prandtl number \(P\tau_{t}\) is 0.7 [22]. \(S_{h}\) accounts for source term due to radiation and heat transfer to wall boundaries. They will be further discussed at a later stage. \(\varphi\) represents the scalar quantities involved in the flow system (e.g., mixture fraction, soot particulate density, and soot volume fraction). \(\Gamma\) is the diffusion coefficient of \(\varphi\). The mixture fraction combustion model was utilized in this study. Hence, all thermochemical scalars are uniquely related to the mixture fractions. Besides, the mean scalar values are extracted from the instantaneous mixture fractions by the probability density function method. The mean fluid density, \(\rho\), therefore, can be computed as:  
\[\frac{1}{\rho}=\int_{0}^{1}\frac{p(f)}{\rho(f)}df \tag{5}\]where \(p(f)\) is the probability density function and \(f\) is the mixture fraction. They will be further discussed at a later stage.