# Multi-Region OpenFOAM Solver Development for Compact Toroid Transport in Drift Tube
## 4 Numerical Simulation of CT Magnetohydrodynamics
### Influence of Finite-Resistivity Walls on CTs  
Cases 3 and 4 are solved using the mhdMRF solver. Figure 11a,b show the distributions of poloidal magnetic flux at \(t=0,\ 2,\ 5\ \upmu\)s for the two cases with different resistivities of the finite resistivity wall: \(\eta_{w}=1\times 10^{-7}\ \Omega\cdot\)m (small resistivity) and \(\eta_{w}=1\times 10^{-5}\ \Omega\cdot\)m (large resistivity), respectively. The depth of color reflects the magnitude of the poloidal magnetic flux, while the closure of contour lines indicates the integrity of the magnetic field structure. By observing Figure 11a,b, it is evident that closed magnetic field structures near the wall surfaces gradually disappear over time. In the case of small wall resistivity, the magnetic field structure is more complete compared with the large-resistivity case. The poloidal magnetic flux values are also larger in the lower resistivity case. This is because the higher the wall resistivity, the faster the magnetic field of the CT diffuses into the wall and the more the magnetic structure at the wall surface is affected. Figure 10: Radial distributions of four physical quantities: (**a**) density \(\rho\), (**b**) magnetic flux density magnitude \(B_{magnitude}\), (**c**) radial velocity component \(U_{r}\), and (**d**) axial velocity component \(U_{z}\). Figure 12a,b compare the spatial density distributions of two cases at \(t=0,\ 2,\ 5\ \upmu\)s. Darker colors indicate higher densities, which also reflect the confinement effectiveness. In this study, the density evolution is primarily driven by the diffusion effects of the CT itself, as thermal conduction effects of the plasma are included. However, compared with the lower-resistivity case, the density distribution in the higher-resistivity case shows a significant shift toward the finite-resistivity walls. At \(t=2,5\ \upmu\)s, the darker colors near the walls in Figure 12b compared with Figure 12a suggest enhanced particle accumulation in the vicinity of the walls under higher resistivity conditions. Figure 11: Comparison of poloidal magnetic flux spatial distribution at \(t=0,\ 2,\ 5\ \upmu\)s for finite-resistivity walls with resistivity values: (**a**) \(1\times 10^{-7}\ \Omega\cdot\text{m}\) and (**b**) \(1\times 10^{-5}\ \Omega\cdot\text{m}\). By comparing the radial distributions of physical quantities under different conditions, the influence of finite-resistivity walls on the CT can be visualized more directly. Figure 13 presents the radial distributions of magnetic flux density magnitude and density at \(t=2,\ 5\ \upmu\)s under three configurations: perfectly conducting wall (Case 2) and finite-resistivity walls with resistivity values of \(1\times 10^{-7}\ \Omega\cdot\)m and \(1\times 10^{-5}\ \Omega\cdot\)m. The z-coordinates selected correspond to the locations where the magnetic flux density magnitude or density reaches its maximum value. The influence of finite-resistivity walls on the CT magnetic field exhibits regional variations in the radial direction. The influence of wall resistivity on the magnetic field is primarily concentrated within the range of \(0.036\ \text{m}\leq r\leq 0.05\ \text{m}\). In contrast, within the core region \(0\ \text{m}\leq r\leq 0.012\ \text{m}\), the effect is minimal, with virtually no observable difference in the central region. Under the large-resistivity case, there is obvious density accumulation near the wall, and the peak density is relatively low. In contrast, in the small-resistivity case and the perfectly conducting wall case, the density distribution remains well peaked. Figure 12: Comparison of density spatial distribution at \(t=0,\ 2,\ 5\ \upmu\)s for finite-resistivity walls with resistivity values: (**a**) \(1\times 10^{-7}\ \Omega\cdot\)m and (**b**) \(1\times 10^{-5}\ \Omega\cdot\)m.  
This type of radial transport of density is likely caused by differences in magnetic field diffusion rates at the wall surface when the compact torus interacts with the boundary.