## 3 Numerical Simulation
### Computational Domain, Boundary Conditions, and Mesh Setup  
The two-dimensional axisymmetric computational domain was established using Ansys DesignModeler(r), as shown in Figure 2. The coordinate origin is set at the center of the nozzle exit, and the axial and radial directions are represented by \(z\) and \(r\), respectively. The nozzle's diameter (\(d\)) is 13.5 mm, and its wall thickness is 0.55 mm. The heat exchanger has a diameter \((D)\) of 210 mm and a thickness of 2 mm. The radial dimension of the domain is fixed at 105 mm, which is the same size as the heat exchanger. Previous studies have demonstrated that this radial distance is long enough to allow the development of radial flow and ensure an equilibrium state of the burned gas at the outlet boundary [4; 45]. In order to calculate the nozzle-to-plate distance, change the axial dimension to \(H=20.25\)-40.5 mm (i.e., \(H/d=1.5\)-3.0). Figure 2 also shows the different boundary conditions in this domain. The axisymmetric boundary condition is imposed at the centerline of the nozzle. This study uses a converging nozzle, so the velocity distribution at the nozzle exit is a top-hat distribution, which is approximated using a sixth-power polynomial, as shown in Equation (11). An expression language is adopted to set the velocity inlet condition.  
\[V_{z}=(V_{z})_{max}\left[a_{6}\left(\frac{r}{R}\right)^{6}+a_{5}\left(\frac{r} {R}\right)^{5}+a_{4}\left(\frac{r}{R}\right)^{4}+a_{3}\left(\frac{r}{R}\right) ^{3}+a_{2}\left(\frac{r}{R}\right)^{2}+a_{1}\left(\frac{r}{R}\right)+a_{0}\right] \tag{11}\]  
where \(R\) is the radius of the nozzle, \(a_{0}\) is the constant term, and \(a_{1}\sim a_{6}\) are the coefficients of each multi-order term, respectively. Furthermore, the inlet temperature is set to 300 K. The methane-air mixture at the inlet is set according to different equivalence ratio (\(\phi\)) conditions to set the mass fraction of different gases. When \(\phi=1\), the mass fractions of methane, oxygen, and nitrogen are:  
\[Y_{\text{CH}_{4}}=0.055\;;\;Y_{\text{O}_{2}}=0.22\;,Y_{\text{N}_{2}}=0.725\; for\;air\;(21\%\;\text{O}_{2},\;79\%\;\text{N}_{2})\]  
The pressure inlet and outlet boundaries are set to ambient air under room temperature and atmospheric pressure conditions as follows:  
\[T=300\;\text{K}\;;\;P=P_{atm}\;;\;Y_{\text{O}_{2}}=0.23\;;\;Y_{\text{N}_{2}}=0.77\;\;for\;air\;(21\%\;\text{O}_{2},\;79\%\;N_{2})\]  
The nozzle edge and impingement plate are regarded as wall boundaries, and conditions of no-slip and constant temperature (300 K) are set. In this study, to consider solid heat transfer for a wall region with solid and fluid regions on each side, the \"two-sided wall\" method is adopted to calculate the internal heat transfer between the methane-air premixed flame and the solid. It is set to the \"coupled\" condition to calculate heat transfer from profiles in adjacent cells and calculate heat fluxes in fluids and solid walls according to [PERSON]'s law. The solid material is set to aluminum, with a thermal conductivity coefficient of 202.4 W/m-K.  
ANSYS Meshing(r) was used for meshing, and the overall computational domain was discretized using a quadrilateral structured mesh, as shown in Figure 3. For demonstration and clarity, the meshes in the figure are shown to be much sparser than the actual meshes. In the central flame and near-plate regions, high-resolution meshes are used to accurately analyze the fluid flow, mass transfer, and heat transfer behaviors in these regions. The mesh size in the central flame region (\(0\leq r\leq 10\) mm) is set to a non-uniform quadrilateral mesh with \(\Delta r=50\)\(\upmu\)m and \(\Delta z=200\)\(\upmu\)m. The mesh is refined in the region close to 5 mm from the plate by reducing the axial mesh size to 0.25 times. Outside the central flame and near-plate regions, the mesh size gradually increases to minimize the total mesh number.