## 2 Design and Simulations
### Mathematical Models
The turbulent stress, \(\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\), in Equation (2) and the turbulent heat flux, \(\rho\overline{T^{\prime}u_{j}^{\prime}}\), in Equation (3) can be calculated as follows:  
\[-\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}=\mu_{t}\left(\frac{\partial u_{i }}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)-\frac{2}{3} \left(\rho k+\mu_{t}\frac{\partial u_{i}}{\partial x_{i}}\right)\delta_{ij} \tag{5}\]  
\[-\rho\overline{T^{\prime}u_{j}^{\prime}}=\frac{\mu_{t}}{Pr_{t}}\left(\frac{ \partial T}{\partial x_{j}}\right) \tag{6}\]  
In the standard \(k\)-\(\epsilon\) model, when the effects of buoyancy, as well as the flow compressibility (small March number) on the turbulence model are neglected, the turbulent kinetic energy, \(k\), and its dissipation rate, \(\epsilon\), are described as follows:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku_{i})}{\partial x_{ i}}=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}} \right)\alpha_{k}\frac{\partial k}{\partial x_{j}}\right]+G_{k}-\rho\epsilon \tag{7}\]  
\[\frac{\partial(\rho\epsilon u_{i})}{\partial x_{i}}=\frac{\partial}{\partial x _{j}}\left[\left(\mu+\frac{\mu_{t}}{\sigma_{\epsilon}}\right)\alpha_{k}\frac{ \partial\epsilon}{\partial x_{j}}\right]+C_{1\epsilon}\frac{\epsilon}{k}G_{k}- C_{2\rho}\frac{\epsilon^{2}}{k} \tag{8}\]  
\[\mu_{t}=\frac{\rho C_{\mu}k^{2}}{\epsilon} \tag{9}\]  
\(G_{k}\) is the production of the turbulence kinetic energy, which is defined as follows:  
\[\frac{\partial}{\partial x_{j}}\big{(}\rho u_{i}u_{j}\big{)}=-\frac{\partial p }{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\Bigg{[}\mu\Bigg{(}\frac{ \partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\Bigg{)}- \rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\Bigg{]} \tag{2}\]0  
In the simulation, the model constants are as follows: \(\alpha_{k}=1.0\), \(\alpha_{\epsilon}=1.3\), \(C_{1\epsilon}=1.44\), \(C_{2\epsilon}=1.92\), \(C_{\mu}=0.09\), \(\sigma_{\text{k}}=1.0\), and \(\sigma_{\epsilon}=1.3\). The nitrogen is considered to be incompressible, whose gravity is ignored. The material of the heating resistance is 20 Cr80 Ni. The boundary conditions are shown in \[\frac{\partial}{\partial x_{j}}\big{(}\rho u_{i}u_{j}\big{)}=-\frac{\partial p }{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\Bigg{[}\mu\Bigg{(}\frac{ \partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\Bigg{)}- \rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\Bigg{]} \tag{2}\]1 The velocity inlet and pressure outlet are applied. The heating surfaces are set to be heating walls, where the surface heat flux is calculated from the heating power of the electric heater with a constant current of 100 A. The remaining walls are assumed to be adiabatic. The operation pressure is 10 MPa. Considering the temperature variation of the physical parameters of nitrogen, the polynomial fitting of the relevant physical parameters, including the density, thermal conductivity, viscosity, and specific heat, are applied, as shown in \[\frac{\partial}{\partial x_{j}}\big{(}\rho u_{i}u_{j}\big{)}=-\frac{\partial p }{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\Bigg{[}\mu\Bigg{(}\frac{ \partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\Bigg{)}- \rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\Bigg{]} \tag{2}\]2  
\[\frac{\partial}{\partial x_{j}}\big{(}\rho u_{i}u_{j}\big{)}=-\frac{\partial p }{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\Bigg{[}\mu\Bigg{(}\frac{ \partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\Bigg{)}- \rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\Bigg{]} \tag{2}\]3 Thermal boundary conditions of the electric heater.