## 2 Numerical Details  
A numerical domain of the baseline model replicating the test rig of one previous experimental study of the fire whirl [48], as well as that with an additional flow channelling slit, was constructed accordingly. The geometric features of the two domains are shown in Figure 2. The detailed description of the numerical configuration of the model has been detailed in our previous work of the characterisation of the fire whirl formulated in an enclosed chamber with different entrainment schemes [49], for the sake of brevity, such information would not be repeated herein. In summary, the geometrical features of the model include a fuel pan located in the domain centreline. The fuel pan is converted from circular to square configuration with the same cross-section area to match the experimental setup to achieve a fully structured mesh, as it significantly enhances the numerical accuracy and computational efficiency. The domain is discretised to about \(800,000\) elements using a uniform division algorithm, to eliminate the numerical uncertainty associated with the spatial resolution of the discretised control volume. A doubling of the number of elements to \(1,600,000\) resulted in only a \(5\%\) difference in the centreline temperature, thus achieving a mesh-independent solution. The fluid flow and heat transfer within the compartment is described through the conservation equations of continuity, Navier-Stokes, and scalar quantities. A general form of transport equation can be expressed as:  
\[\frac{\partial}{\partial t}(\overline{\rho}\Phi)+\frac{\partial}{\partial x_{i}} \big{(}\overline{\rho}\Phi\overline{U}_{i}\big{)}=(\Gamma_{\Phi})\frac{\partial ^{2}\overline{\rho}\Phi}{\partial x_{i}\partial x_{i}}+\widetilde{S(\Phi)} \tag{1}\]  
where \(\Phi\) is the general field variable dependent on space and time, \(\overline{\rho}\) is the mean density, \(\overline{U}_{i}\) is the fluid velocity, \(\Gamma_{\Phi}\) is the diffusion term, and \(\widetilde{S(\Phi)}\) is the source term of the general variable. According to the general formula, the transport equations are tabulated in Table 1 as:  
Large eddy simulation (LES) and the wall-adapting local eddy-viscosity (WALE) function, is adopted as the turbulence model to describe the turbulent reaction flow behaviour. This turbulence model has been extensively validated and proved to be a valid approach for resolving various wall-bounded turbulence flow applications with a reasonable computational course [8, 43, 50, 51, 52, 53, 54]. The chemical reaction source term in the transport equations of the involved reacting scalars is determined by strained laminar flamelet approach, in which the chemistry is a pre-assumed probability \begin{table}
\begin{tabular}{c c c} \hline \hline \(\Phi\) & \(\Gamma_{\Phi}\) & \(\widetilde{S(\Phi)}\) \\ \hline \(1\) & \(0\) & \(0\) \\ \(\overline{U}_{j}\) & \(\mu+\mu_{T}\) & \(\frac{\partial\overline{\rho}}{\partial x_{j}}+\frac{\partial}{\partial x_{j} }\big{(}\mu\frac{\partial\overline{U}_{i}}{\partial x_{j}}-\frac{2}{3}\mu \frac{\partial\overline{U}_{i}}{\partial x_{j}}\big{)}-\frac{\partial x_{j}}{ \partial x_{j}}+\big{(}\overline{\rho}-\rho_{ref}\big{)}g_{j}+\widetilde{S(U)}\) \\ \(c_{p}\overline{T}\) & \(\frac{\mu}{\overline{\rho}}+\frac{\mu_{T}}{\overline{\rho}_{T}}\) & \(\widetilde{\omega}_{T}+\widetilde{S(rad)}\) \\ \(\Phi\) & \(\frac{\mu}{Sc}+\frac{\mu_{T}}{Sc_{T}}\) & \(\widetilde{S(\Phi)}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: General transport equation variable, diffusion and source terms. Figure 2: Geometric features of the computational domain. density function (pre-PDF) of two quantities including mixture fraction (f) and the scalar dissipation (x). In essence, the mixture fraction governs the amount of the fuel mixture in each control volume element in the simulation domain. The scalar dissipation is a term introduced to describe the strain and extinction of the flame, in which the larger this quantity depicts its departure from its chemical equilibrium [55].