## 3 Numerical Calculation  
The computational fluid dynamics software Fluent was used to calculate the unified model. The computational domain is firstly discretized by a very dense and unstructured grid system. To ensure an accurate calculation of the electric-magnetic-thermal-hydrodynamical effect in plasma arc, grid size within the torch is the smallest, which is just 0.1 mm. A smaller grid size is applied in the area beneath the welding torch, and the size is 0.2 mm. In the front of the torch, a relatively bigger grid size is used, while behind the torch, a very big grid size is applied. In this way, both the computational accuracy and the efficiency are considered. Figure 4 shows the detailed grid system of the model.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Boundary** & \(V\) **(m/s)** & \(T\) **(K)** & \(\Phi\) **(V)** & \(A\) **(T-em)** \\ \hline Plasma gas inlet & 3 L/min & 300 & \(\partial\phi/\partial n=0\) & \(\partial A/\partial n=0\) \\ Shielding gas inlet & 15 L/min & 300 & \(\partial\phi/\partial n=0\) & \(\partial A/\partial n=0\) \\ Pressure outlet & — & 300 & \(\partial\phi/\partial n=0\) & \(\partial A/\partial n=0\) \\ Tungsten tip & — & 3000 & \(j=I/\pi r^{2}\) & \(\partial A/\partial n=0\) \\ Nozzle wall & — & 1000 & \(\partial\phi/\partial n=0\) & \(\partial A/\partial n=0\) \\ Workpiece inlet & \(-\)120 mm/min & 300 & 0 & 0 \\ Workpiece outlet & \(-\)120 mm/min & 500 & 0 & 0 \\ Side workpiece wall & — & \(h_{\text{core}}\Delta T+\varepsilon a\big{(}T^{4}-T_{\infty}^{4}\big{)}\) & 0 & 0 \\ Top surface & — & 300 & \(\partial\phi/\partial n=0\) & \(\partial A/\partial n=0\) \\ Bottom surface & — & 300 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Boundary conditions of the unified model.  
Figure 4: Grid system of the unified model. (**a**) Overall grid map; (**b**) Front-view grid (_x-z_ plane); (**c**) Side-view grid (_y-z_ plane).  
The current conservation equation and magnetic potential vector equation were established by the User-defined Scalars (UDS) module in Fluent. All the source terms and physical parameters, such as magnetic induction intensity, current density, Joule heat, electron transfer energy, radiative heat loss, electromagnetic force, plasma arc drag force, Marangoni force and buoyancy force, are calculated by a self-developed code with the User-Defined Function (UDF) module in Fluent.  
The governing equations are solved by the Pressure-Implicit with Splitting of Operators (PISO). Momentum conservation equations, energy conservation equation and UDS equations are all calculated with the second-order upwind discrete scheme. A small time step \(10^{-3}\) s was used in calculation, and the sum of the normalized residues for the mass and momentum equations, energy equation and UDS equations is set to be below \(10^{-3}\), \(10^{-6}\) and \(10^{-4}\), respectively.  
Both plasma gas and shielding gas are pure argon in the PAW process, and the thermophysical properties are only a function of temperature. All the parameters used in the model can refer to literature [30]. The workpiece is stainless-steel 304, and its thermophysical properties such as density, specific heat, viscosity and thermal conductivity change with temperature, as shown in reference [8]. The process parameters are all set equal to the experimental conditions as in Part 4.