The Ruhrstahl-Heraeus (RH) vacuum system is vitally important in the secondary refining process since it is highly effective on decarburization and degassing, which involves complex multiphase flow and transport phenomena. Many investigations on the flow field in the RH refining process have been reported. However, only several investigations are focused on the bubble expansion in the up-leg snorkel. In this work, the combined mathematical model and physical model were employed to simulate the fluid flow and mixing phenomena in the RH reactor. A water model for a practical 210 t RH reactor was established according to similitude principle, and the flow field on the center section of the physical model was captured by PIV (particle image velocimetry) system. The coupled VOF (volume of fluid) model and DPM (discrete phase model) were used to simulate the multiphase fluid flow in the RH reactor. Both the k-ε model and LES (large eddy simulation) model were performed to describe the turbulent characteristics during the RH refining process. The mathematical model was validated by the water model with the same experimental conditions. It suggests that the calculated results show a good agreement with the measured one. Based on the LES model, the instantaneous velocity distribution and the generate and the dissipate of vortex were computed. Also, the mixing time of different position in the ladle was measured and calculated. The results show that the mixing time near the up-leg snorkel is larger than that near the down-leg snorkel. A model for bubble expansion was developed and used to simulate the bubble behavior in the steel-argon system. The results show that the bubble expansion has a strong impact on the flow field in the RH reactor.