# Influence of Eddy-Generation Mechanism on the Characteristic of On-Source Fire Whirl
## 2 Numerical Details  
A numerical domain of the baseline model replicating the test rig of the previous experimental study [43], as well as that with an additional flow channelling slit, was constructed accordingly. The geometric features of the two models are shown in Figure 1. The fuel pan is converted from circular to square configuration with the same cross-section area as in the experimental setup to achieve a fully structured mesh, as this significantly improves the numerical accuracy and computational efficiency. Despite the inevitable \"flame wander\" phenomenon, the domain is discretised to about 800,000 elements based on a uniform division algorithm, to limit 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:  
\[\frac{\partial\overline{\rho}}{\partial t}+\frac{\partial}{\partial x_{i}} \overline{\rho}\overline{U}_{i}=0 \tag{1}\]  
\[\frac{\partial}{\partial}\overline{\rho}\overline{U}_{i}+\frac{\partial}{ \partial}\overline{U}_{i}\overline{U}_{j}+\frac{\partial}{\partial}\overline{ \rho}\overline{u_{i}}\overline{u}_{j}=-\frac{\partial\overline{p}}{\partial x_ {i}}+(v+v_{t})\frac{\partial^{2}\overline{\rho}\overline{U}_{i}}{\partial x_{j }\partial x_{j}} \tag{2}\]  
\[\frac{\partial}{\partial}\overline{\rho}\overline{\partial}_{\alpha}+\frac{ \partial}{\partial}\overline{\rho}\overline{\partial}_{\alpha}\overline{U}_{i }+\frac{\partial}{\partial}\overline{\rho}\overline{\phi}_{\alpha}u_{i}=( \Gamma+\Gamma_{t})\frac{\partial^{2}\overline{\rho}\overline{\phi}_{\alpha}} {\partial x_{i}\partial x_{i}}+\widetilde{S(\mathcal{O})} \tag{3}\]  
where \(\overline{\rho}\) and \(\overline{p}\) are the mean density and pressure, \(v\) and \(v_{t}\) are the kinematic and turbulent viscosity, \(\overline{U}_{i}\) and \(u_{i}\) are the Favre-averaged and fluctuation value of the mean fluid velocity, \(\overline{\phi}_{\alpha}\) and \(\phi_{\alpha}\) are the mean and fluctuation value of the \(\alpha\)th scalar, \(\Gamma\) and \(\Gamma_{t}\) are molecular and turbulent diffusivity, and \(\widehat{S(\Phi)}\) is the Favre-averaged chemical reaction source term. Large eddy simulation (LES) with Wall-Adapting Local Eddy-viscosity (WALE) function, which accounts for both the strain and rotation rate of the turbulent structure is employed 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 [8; 39; 44; 45; 46; 47; 48]. The chemical reaction source term in the transport equations of the involved reacting scalars is evaluated via the strained laminar flamelet approach. The involved chemical phenomenon is described as the pre-assumed probability density function (pre-PDF) of two quantities, namely the mixture fraction (\(f\)) and the scalar dissipation (\(\chi\)). Essentially, the mixture fraction governs the proportion of the fuel mixture in each control volume element in the discretised domain, and the first two moments of the mixture fraction, namely the mean mixture fraction, \(\overline{f}\), and the mixture fraction variance, \(\overline{\mathcal{S}}\), are evaluated by resolving their conservation equations, respectively. Scalar dissipation is introduced to describe the strain and extinction of the flame, in which larger values of this quantity depict its departure from chemical equilibrium [49].