## 3 Numerical Simulation
### Numerical Models  
To obtain the most reliable simulation results for the PMIFJ, a double-precision implicit separation solver was used in Fluent, and GRI-Mech 3.0 of the detailed chemical mechanism for methane was used (53 species and 325 reactions) [46]. Additionally, each species' thermodynamic and transport properties are imported into Fluent in Chemkin format. The specific heat capacity of each species is obtained based on a temperature-dependent piecewise polynomial approximation, and its viscosity and thermal conductivity were calculated using kinetic theory. For mixtures, the density and specific heat capacity are calculated according to the incompressible ideal gas law and the mixing law, respectively. The ideal gas mixing law calculates the mixture's viscosity and thermal conductivity. The mass diffusion coefficient is calculated using kinetic theory. Since the flow state of the PMIFJ discussed in this study is laminar flow, the laminar finite rate turbulence-chemistry interaction is selected, which ignores the influence of turbulence fluctuation. The laminar  
Figure 3: Meshing process using non–uniform quadrilateral meshes.  
Figure 2: Schematic diagram of computational domain and boundary conditions.  
finite rate model is acceptable for combustion with small turbulence-chemistry interactions. The volumetric reaction uses the stiff chemistry solver and multi-component diffusion. The in situ adaptive tabulation (ISAT) algorithm of the integration method proposed by [PERSON] is adopted [PERSON] (1997) to reduce the computational load of time integration and achieve detailed chemistry. The ISAT error tolerance is a vital setting parameter that controls the numerical errors that occur when retrieving data from the ISAT table. The ISAT error tolerance is set to 5 \(\times\) 10\({}^{-5}\) to ensure that the results of interest in this study remain unchanged. In addition, the maximum storage setting of ISAT is 500 MB.  
In all cases, the coupling between pressure and velocity is accomplished using the SIM-PLE algorithm. The second-order upwind scheme is used for the spatial discretization. The volumetric reaction and energy equation are canceled in the solver to solve non-reacting flow (cold flow). It takes about 1500-2000 iterations until convergence. A small region above the nozzle exit is patched with an artificially specified temperature of 2000 K to ignite the mixture. The under-relaxation factors of all species equations are set to 0.1, and the under-relaxation factor of the energy equation is set to 0.05 to avoid computational divergence. The volumetric reaction and energy equations are then activated, and the solution of the reactive flow is calculated by solving the mass, momentum, matter, and energy conservation equations. The under-relaxation factors are gradually raised to their default values once ignition occurs. For computational convergence, the residual convergence criteria for the continuity, momentum, and species equations are set to 10\({}^{-6}\) and to 10\({}^{-8}\) for the energy equation. In addition, the total heat transfer rate at the two-sided wall is monitored to ensure that the iteration process is complete.