# Multi-Region OpenFOAM Solver Development for Compact Toroid Transport in Drift Tube
## 1 Introduction  
Core fueling plays an important role in maintaining the steady operation of fusion reactors [1], enhancing fusion power, and reducing the requirement for tritium breeding rate [2, 3]. Due to the high-temperature and high-density characteristics of fusion reactors, the commonly used fueling technologies (for example, supersonic molecular beam injection (SMBI) [4], cryogenic pellet injection [5], and gas puffing [6]) have difficulty achieving core fueling. Compact toroids (CTs), which possess high plasma densities (typically up to ten times those of target fusion reactor plasma densities) and stable axisymmetric poloidal and toroidal magnetic field structures, can be accelerated to hundreds of kilometers per second using a CT injection device. Thus, CT injection is considered one of the most promising methods for core fueling. The concept of CT was first proposed by [PERSON] in 1958. The ring accelerator experiment at Lawrence Livermore National Laboratory successfully accelerated CT to approximately 2500 km/s [7,8]. In 1988, [PERSON] and Perkins first proposed the application of CT for fueling in fusion devices [9, 10]. Over the past decades, CT fuelingsystems have been widely employed in experimental reactors, including the ENCORE device at the California Institute of Technology [11], the STOR-M device at the University of Saskatchewan in Canada [12], and the JFT-2M device in Japan [13]. China's first CT injection device, KTX-CTI [14], was developed for fueling experiments on the reversed field pinch (RFP) device KTX at the University of Science and Technology of China. Another device, EAST-CTI, serves as a CT injection system for the Experimental Advanced Superconducting Tokamak (EAST) [15], capable of achieving continuous fueling through multi-pulse, repetitive operation. Although extensive theoretical [9] and experimental [16; 17; 18; 19] studies have been conducted on the formation and injection processes of CT, the dynamic processes of CT within the injection devices and after injection into the target reactor remain poorly understood due to limitations in diagnostic techniques. MHD simulations under experimental parameter conditions can provide an accurate depiction of CT injection dynamics and offer critical guidance for experimental efforts [20; 21; 22]. However, numerical simulation studies have been scarce in the past few decades, and most existing numerical simulations have considered the direct injection of an ideal CT into the target plasma, neglecting the variations during the transport process of the CT in the drift tube. For example, [PERSON] investigated the effects of penetration depth and injection angle of the CT using three-dimensional MHD numerical simulations employing an ideal CT model [20]. Similarly, [PERSON] explored the influence of different parameter ranges of the CT on injection performance and its evolution in the target plasma using ideal MHD simulations, also assuming an ideal CT model at the initial moment of injection into the target plasma [22]. In recent years, the only numerical simulation study on the transport process of CT in the drift tube was conducted by [PERSON], who examined the deceleration effect of stray magnetic fields from fusion devices on the CT and demonstrated that longer inner electrode lengths lead to poorer confinement [21]. To enable the numerical investigation of CT dynamics within drift tubes, this work develops a multi-region coupled MHD solver named mhdMRF, based on the OpenFOAM platform. OpenFOAM, short for Open-Source Field Operation and Manipulation, is a computational fluid dynamics (CFD) platform written in C++ and based on the finite volume method [23]. As an open-source framework, it supports dozens of solvers for different physical processes, including compressible and incompressible flows, multiphase flows, electromagnetics, heat transfer, and chemical reactions. Its open architecture allows users to modify or add physical models as needed to meet their simulation requirements.