## 3 Research Methods
### Numerical Method  
The two-dimensional axisymmetric viscous compressible N-S equations and a thermochemical non-equilibrium model are used to compute the nozzle starting process. The solving equation is as follows:  
\[\frac{\partial Q}{\partial t}+\frac{\partial}{\partial x}+(F_{x}+G_{x})+\frac{ \partial}{\partial r}(F_{r}+G_{r})=0 \tag{1}\]  
Among them:  
\[Q=r[\rho,\rho\sigma_{1},\rho\sigma_{2},\cdots\rho\sigma_{7},\rho u,\rho v,e] \tag{2}\]  
\[F_{x}=r[\rho u,\rho u\sigma_{1},\cdots,\rho u\sigma_{7},\rho u^{2}+p,\rho uv,u (e+p)]^{T} \tag{3}\]  
\[G_{x}=r[0,\rho D\frac{\partial\sigma_{1}}{x},\cdots,\rho D\frac{\partial \sigma_{7}}{x},-\tau_{xx},-\tau_{xx},\dot{q}_{x}-u\tau_{xx}-v\tau_{xx}]^{T} \tag{4}\]  
\[F_{r}=r[\rho v,\rho v\sigma_{1},\cdots,\rho v\sigma_{7},\rho v\mu,\rho v^{2}+ p,v(e+p)]^{T} \tag{5}\]  
\[G_{r}=r[0,\rho D\frac{\partial\sigma_{1}}{r},\cdots,\rho D\frac{\partial \sigma_{7}}{r},-\tau_{xx},-\tau_{rr},\dot{q}_{r}-u\tau_{xx}-v\tau_{rr}]^{T} \tag{6}\]  
where \(\rho\), \(p\), \(u\), \(v\) are the density, pressure, flow velocity components in the \(x\) and \(r\) directions, respectively; \(\sigma_{i}\) is the mass fraction of component \(i\); \(\tau\) is the viscous stress tensor. \(G_{(x,r)}\) is calculated from the shear stress relationship under the Stokes assumption of the N-S equations, [PERSON]'s law of heat conduction, and [PERSON]'s law of mass diffusion. The internal energy per unit mass, \(e\), is defined as follows:  
\[e=\frac{P}{\gamma-1}+\rho\frac{(u^{2}+v^{2})}{2}+\sum_{i=1}^{N}\rho\sigma_{i} \bigtriangleup Hf_{i} \tag{7}\]  
Each component of the mixture satisfies the ideal gas equation, and using [PERSON]'s law, we can write:  
\[P=\sum_{i=1}^{7}P_{i}=\rho TR_{0}\sum\frac{\sigma_{i}}{M_{i}} \tag{8}\]  
where \(M_{i}\) is the molecular weight of component \(i\), and \(R_{0}\) is the universal gas constant. These equations are discretized by the two-order precision finite volume method, using the TVD-minmod high-resolution shock capture format. The thermodynamic property parameters of the gas use the polynomials of [PERSON] et al. (2016). The Sutherland-Wilke model is used for the viscosity and heat transfer coefficient. Seven components (\(O_{2}\), \(N_{2}\), \(O\), \(N\), \(NO\), \(e^{-}\), \(NO^{+}\)), two temperatures (translational temperature and vibration temperature) are used in the chemical reaction model, according to the [PERSON] model, which is described in detail in the literature ([PERSON], 2016), and the reaction temperature uses the [PERSON]'s dual temperature model ([PERSON], 2016; [PERSON], 2016; [PERSON], 2016). The initial condition of the flow field is the air at room temperature (\(T\) = 300 K) at a given pressure, the wall boundary condition is the non-slip and non-catalytic wall, and the temperature of the wall is 300 K. The gas at the inlet boundary condition is the equilibrium gas component at the corresponding temperature and pressure. The length of the nozzle is 2.6 m, and the outlet diameter is 510 mm. Considering that the nozzle is a rotating body, the calculated region is a 1/2 section, as shown in Figure 2a At a total temperature of 4119 K, a total pressure of 1.333 MPa, and an initial pressure of 50 Pa, the nozzle startup process is calculated by using 260 \(\times\) 30, 520 \(\times\) 50 and 1040 \(\times\) 80 mesh node models, respectively. The results show that when the grid node number of 520 \(\times\) 50 or 1040 \(\times\) 50 is used, the startup duration is the same, and the temperature and pressure distributions are more similar, the difference is not more than 3%. In order to reduce the calculation time, the 520 \(\times\) 50 grid node model is selected. The finite volume method is used to discretize the governing equations. Additionally, the four-stage Runge-Kutta method is employed for time integration. The CFL number is 0.8. The computationalcriteria are the flow field parameters, including temperature, pressure, velocity and mass fraction of each component converging to \(1.0\times 10^{-6}\).