# Multi-Region OpenFOAM Solver Development for Compact Toroid Transport in Drift Tube
## 4 Numerical Simulation of CT Magnetohydrodynamics
### Geometric Structure and Simulation Parameter Settings  
In the CT injection device, the acceleration region adopts a cylindrical metal conductor as the inner electrode. To investigate the influence of the inner electrode end geometry on the MHD behavior of the CT, this paper innovatively introduces a conical transition section at the end of the cylinder. To systematically study this problem and simultaneously consider the effect of the finite resistive wall, four typical cases are designed. The simulation domains can be categorized into three types, as shown in Figure 8.  
* Case 1: the internal metal electrode is a pure cylinder (hereinafter referred to as the cylindrical inner electrode);
* Case 2: a conical transition section is added to the end of the cylinder (hereinafter referred to as the conical inner electrode);
* Cases 3 and 4: based on the geometry of Case 2, an outer finite resistivity wall region is added, with different wall resistivity values in each case. In terms of simulation parameters, the inner electrode radius, \(r_{in}=1.75L_{r}\), and the outer radius from the central axis to the wall, \(r_{out}=5L_{r}\), are based on the EAST-CT device parameters [15], where \(L_{r}=0.01\) m and the axial length \(L_{z}=0.1\) m. The initial magnetic field distribution of the CT is calculated using the analytical solution by [PERSON] [33] under perfectly conducting wall conditions. The magnetic field is distributed within the region of inner radius \(1.75L_{r}\), outer radius \(5L_{r}\), and axial length \(L_{z}\), that is, the region from \(z=-0.2\) m to \(z=-0.1\) m, and the characteristic value is set as the minimum value. The initial magnetic field in other regions is zero. The initial density of the CT is given by the poloidal flux function \(\psi_{pol}=rA_{\theta}\), satisfying the following expression [21]:  
\[\rho=\rho_{\min}+(\rho_{\max}-\rho_{\min})\frac{|\psi_{\rm pol}|}{|\psi_{\rm axis }|} \tag{36}\]  
Figure 7: Comparing the analytical and numerical solutions for the validation case of the mhdMRF solver. Here, the peak density and background plasma density of the CT are \(\rho_{\rm max}=2\times 10^{-5}\) kg/m\({}^{3}\) (corresponding to \(n_{\rm CT}=10^{22}\) m\({}^{-3}\)) and \(\rho_{\rm min}=2\times 10^{-7}\) kg/m\({}^{3}\), respectively, and \(\psi_{\rm axis}\) is the poloidal flux at the axis. The background density equals 1% of the peak density of the CT. Experimentally measured CT temperatures typically range within 1 eV-10 eV. Here, the initial temperature at the location of the maximum CT density is taken as 5 eV. The initial pressure across the entire domain is kept equal, calculated via the ideal gas equation of state, yielding approximately \(2.4\times 10^{3}\) Pa. The initial velocity of the CT and other regions is set to a typical experimental value of 100 km/s, in the positive z-direction. Magnetic permeability is set to vacuum permeability, dynamic viscosity is \(\mu=0.001\) kg/(m\(\cdot\)s), and thermal conductivity is \(\kappa=1\)w/(m\(\cdot\)K). The resistivity of the CT is estimated using the Spitzer resistivity formula as \(\eta_{\rm CT}=1\times 10^{-5}\)\(\Omega\cdot\)m, and the finite resistive wall resistivity in Cases 3 and 4 are set as \(\eta_{w}=1\times 10^{-7}\)\(\Omega\cdot\)m and \(\eta_{w}=1\times 10^{-5}\)\(\Omega\cdot\)m, respectively. In Cases 1 and 2, the surfaces of the inner electrode and the outer wall use zero-gradient conditions for temperature and pressure and perfectly conducting wall conditions for magnetic flux density. Since viscous effects are largely neglected in the simulation,  
Figure 8: Schematic diagrams of the simulation domains for each case, where (**a**–**c**) correspond to Case 1, Case 2, and Cases 3–4, respectively. (**a**) The internal metal electrode is a pure cylinder. (**b**) A conical transition section is added to the end of the cylinder. (**c**) Based on the geometry of (**b**), an outer finite resistivity wall region is added. slip boundary conditions are used for velocity at the wall.