# Capturing the Swirling Vortex and the Impact of Ventilation Conditions on Small-Scale Fire Whirls
## 2 Mathematical Formulation
### Subgrid-Scale Combustion Modelling  
For the combustion system, as the non-premixed mixture fraction combustion model was applied in this study, the basic assumption of the mixture fraction model is that all the species mass fraction are only functions of the mixture fraction [30]. Therefore, the thermochemical state of the combustion reactions can be governed by the mixture fraction \(\overline{f}\) and the mixture fraction variance \(\overline{f^{\prime 2}}\).  
\[\frac{\partial}{\partial t}\Big{(}\rho\overline{f}\Big{)}+\
abla\cdot\Big{(} \rho\overline{u_{i}}\overline{f}\Big{)}=\
abla\Bigg{(}\Bigg{(}\frac{k}{C_{p}} +\frac{\mu_{t}}{Pr_{t}}\Bigg{)}\
abla\overline{f}\Bigg{)} \tag{14}\]  
where \(k\) is the laminar thermal conductivity of the mixture and \(C_{p}\) is the mixture-specific heat.  
For large eddy simulation, the transport equation is not solved for the mixture fraction variance. Instead, it is modelled as:  
\[\overline{f^{\prime 2}}=C_{\text{tr}}L_{S}^{2}\Big{|}\
abla\overline{f}\Big{|} \tag{15}\]  
where \(C_{\text{tr}}\) is the dynamic stress constant that equals to 0.5 and \(L_{S}\) is the subgrid length scale.  
In LES (large eddy simulation), the flame is typically not resolved by the computational grid [24]. Therefore, the chemical reaction is assumed to occur at the flame sheet, where a state relationship exists for all reactants. Under the mixture fraction assumption, the state relationship could be represented by the chemistry equilibrium assumption and experimental state relationships established by [PERSON] and [PERSON] [31]. The instantaneous temperature, species mass fraction, and density can be parameterized by the instantaneous enthalpy and mixture fraction \(\phi_{i}=\phi_{i}(f,H)\)[32]. While, in the non-adiabatic simulation, changes in enthalpy due to heat loss must be considered. In this study, heat loss was assumed to not impact the turbulent enthalpy fluctuations. Therefore, the density-weighted scalars can be calculated as:  
\[\overline{\phi}_{i}=\int\limits_{0}^{1}\phi_{i}\Big{(}f,\overline{H}\Big{)}p( f)df \tag{16}\]  
where \(\overline{H}\) is the mean enthalpy and \(p(f)\) is the shape of the assumed probability density function (PDF). Furthermore, in the simulation, the shape of the assumed PDF could be described by the \(\beta\) function, and this PDF shape is given by the following function of mean mixture fraction \(\overline{f}\) and mixture fraction variance \(\overline{f^{\prime 2}}\).  
\[p(f)=\frac{f^{\alpha-1}(1-f)^{\beta-1}}{\int f^{\alpha-1}(1-f)^{\beta-1}df} \tag{17}\]  
where \(\alpha=\overline{f}\Big{[}\frac{\overline{f}(1-\overline{f})}{\overline{f^{ \prime 2}}}-1\Big{]}\) and \(\beta=\big{(}1-\overline{f}\Big{)}\Big{[}\frac{\overline{f}(1-\overline{f})}{ \overline{f^{\prime 2}}}-1\Big{]}\).  
The mathematical result of Equation (17) for the non-adiabatic system was generated and stored in a 3D look-up table for the non-adiabatic system. The mean value of each mass fraction, density, and temperature are the function of \(\overline{f}\), \(\overline{f^{\prime 2}}\), and \(\overline{H}\), respectively. In the flow domain, these values can be obtained by table interpolation.