## 2 Experimental Methods  
The PMIF system used in this study consists of a burner, the flow output and control system of fuel and air, and a water-cooled alumina plate, as shown in Figure 1. The burner comprises a diverging tube, three equalizing chambers, and a converging nozzle. Methane (purity 99.95%) and dry air are controlled by two mass flow controllers (MFC, DPC-47, Aalborg Instruments & Control Int., New York, NY, USA) with an accuracy of \(\pm 0.01\) SLPM. Methane and air are mixed in a mixing chamber, and the mixture is introduced into the burner. Five layers of stainless steel mesh are installed in the burner to reduce the flow fluctuations of the mixture, ensure uniform flow, and prevent flame flashback. Gaskets are utilized to prevent methane and air leaks. Finally, the mixture flows to the nozzle exit to produce a laminar premixed methane-air flame. The volume flow rate of methane and air is determined based on different equivalence ratios and Reynolds number operating conditions. The equivalence ratio (\(\phi\)) and Reynolds number (\(Re\)) of the PMIFJ are defined as follows:  
\[\phi=\frac{\left(Q_{CH_{4}}/Q_{air}\right)_{act}}{\left(Q_{CH_{4}}/Q_{air} \right)_{st}} \tag{1}\]  
where \(Q_{CH_{4}}\) and \(Q_{air}\) are the volume flow rates of methane and air, respectively.  
\[Re=\frac{\rho_{m}V_{e}d}{\mu_{m}} \tag{2}\]  
where \(\rho_{m}\) is the density of the mixture, \(V_{e}\) is the velocity of nozzle exit, \(d\) is the nozzle diameter, and \(\mu_{m}\) is the dynamic viscosity of the mixture.  
\[\rho_{m}=\sum Y_{i}\rho_{i} \tag{3}\]  
where \(Y_{i}\) is the mass fraction of species \(i\), and \(\rho_{i}\) is the density of species \(i\).  
\[\mu_{m}=\frac{\sum\mu_{i}X_{i}\sqrt{M_{i}}}{\sum X_{i}\sqrt{M_{i}}} \tag{4}\]  
where \(\mu_{i}\) is the dynamic viscosity of species \(i\), \(X_{i}\) is the molar fraction of species \(i\), and \(M_{i}\) is the molar mass of species \(i\).  
In order to observe the flame shape and characteristics of PMIFJ, luminous images of the flame were taken by a 4K CMOS camera. In addition, the cone's leading edge defines the height of the premixed cone to provide validation for numerical simulations. To verify the heat transfer performance of the simulated PMIFJ, a water-cooled alumina plate was used as a heat exchanger to measure its total heat transfer rate. Aluminum sticks fixed the heat exchanger and adjusted the distance of nozzle-to-plate (\(H\)). A water pump and water flow meter supply the cooling water at a constant room temperature and flow rate to the heat exchanger. Two K-type thermocouples were used to measure the temperature of the water flowing into and out of the heat exchanger. After the mixture is ignited, the inlet and outlet water temperatures are recorded through the temperature acquisition card (NI USB-9162, National Instruments, Austin, TX, USA) once the outlet water temperature reaches a steady state. According to the energy balance equation for cooling water under steady-state conditions [44], Equation (5) calculates the total heat transfer rate at different conditions.  
\[\dot{Q}_{T}=\dot{m}C_{P}(T_{out}-T_{in}) \tag{5}\]  
where \(\dot{Q}_{T}\) is the total heat transfer rate, \(\dot{m}\) is the mass flow rate of water, \(C_{P}\) is the specific heat of water, and \(T_{in}\) and \(T_{out}\) are the water temperatures at the inlet and outlet of the heat exchanger, respectively.