## 3 Simulation Tests and Analysis
### Simulation Model
(4) Species conservation equation:  
\[\frac{\partial(pm_{i})}{\partial t}+\
abla\cdot(\rho Vm_{i})=-\
abla\cdot J_{i}+R _{i}+S_{i}, \tag{20}\]  
(5) Ideal gas state equation:  
\[p=\sum_{i}m_{i}\rho RT/M_{i}, \tag{21}\]  
where \(\rho\) is the density gas mixture, \(t\) is time, \(V\) is the velocity of the gas mixture, \(p\) is the static pressure, \(\tau\) is the stress tensor, \(\rho g\) is the gravitational body force, \(F\) is the external body forces, \(E\) is the energy content per unit mass of the gas mixture, \(k_{\rm eff}\) is the effective conductivity, \(T\) is the temperature of the gas mixture, \(h_{i}\) is the enthalpy of the species i gas, \(J_{i}\) is the diffusion flux of the species i gas, \(S_{\rm h}\) is the heat of other sources, \(m_{i}\) is the mass fraction of the species i gas, \(R_{i}\) is the net rate of production of the species i gas in chemical reactions, \(S_{i}\) is the rate of creation with the addition of the dispersed phase plus any other sources, \(R\) is the universal gas constant, and \(M_{i}\) is the molecular weight of the species i gas. In the simulations, the standard \(k-\varepsilon\) turbulence model and the second-order upwind difference scheme were used. The other model constants were set to the default values. The species model was set to \"Species Transport\", and the mixture material was set to \"propane-air\". The radiation model was set to \"P1\" and the absorption coefficient was set to \"wsggm-domain-based\". The \"Eddy-Dissipation\" model was used as the turbulence-chemistry interaction model. The solution method was \"SIMPLE\". The gradient was set to \"Least Squares Cell-Based\", the pressure was set to \"Second Order\", and the momentum and turbulent kinetic energy were set to \"Second-Order Upwind\". The \"energy\" residual convergence was set to \(10^{-5}\), and the others were set to \(10^{-3}\). Observation points temperature and mass balance were also important factors in the determination of convergence. For boundary conditions, the outlet was set to \"outflow\". The fuel inlet boundary was set to \"velocity-inlet\", and the air inlet boundary was set to \"velocity-inlet\". The boundary conditions of the walls were set as \"Stationary Wall\" and \"No Slip\". The near-wall treatment was set as \"Standard Wall Functions\". Considering the water-cooling jacket that is used in real applications, for convenience, the wall temperature was set as 343.15 K.  
As the dataset shows features of the temperature field, it means a lot to the proposed method and directly influences the accuracy of the reconstruction results. Therefore, a wide range of boundary conditions were used to carry out a sufficient number of simulation results. A sample dataset was established by the CFD results whose boundary conditions were fuel velocities of 0.44, 0.54,..., 1.14 m/s, with the air velocities being determined by the excess air coefficient \(\alpha\) (\(\alpha=0.75,1.25,1.40\)). Six test cases were also set up to test the reconstruction results. The boundary conditions of the test cases are shown in Table 2.  
In order to test the proposed method in different conditions, firstly, a small fuel velocity 0.58m/s was used in this study. This was in the scope of 0.44 to 1.14 m/s, which was used for setting up the dataset. After that, 1.23 m/s was used to show the performance of the proposed method in the high-speed situation; 1.23 m/s was out of the range of 0.44 to 1.14 m/s, the reconstruction results in this \begin{table}
\begin{tabular}{c c c} \hline
**Test Case Number** & **Fuel Velocities (m/s)** & **Excess Air Coefficient \(\alpha\)** \\ \hline
1 & 0.58 & 0.75 \\
2 & 0.58 & 1.25 \\
3 & 0.58 & 1.40 \\
4 & 1.23 & 0.75 \\
5 & 1.23 & 1.25 \\
6 & 1.23 & 1.40 \\ \hline \end{tabular}
\end{table} 
Table 2: Boundary conditions of six test cases. boundary condition test whether the application range of the proposed method would be limited by the dataset boundary conditions.