In this study, a single Taylor bubble and a train of Taylor bubbles rising in a vertical tube were simulated numerically. A finite difference method was used to solve the mass and momentum equations for the liquid-gas region. The liquid-gas interface was captured by a level set function which is defined a signed distance from the interface. For a train of Taylor bubbles repeated periodically in space, the periodic conditions were imposed at the boundaries normal to the gravitational direction and the pressure boundary conditions were iteratively determined so that the computed flow rate should be equal to a given flow rate. Based on the numerical simulation, the calculated shape and rise velocity of a Taylor bubble were found to be in good agreement with the experimental data reported in the literature.