This paper introduces a numerical model developed based upon the arbitrary Lagrangian-Eulerian/finite element scheme to analyze bubble plumes in a periodic domain. A spherical air bubble is immersed into a large pool of quiescent water to act as an unconfined domain. While the buoyancy force is the main physical mechanism driving the upward bubble motion, an opposite pressure gradient is imposed over the flow to balance the bulk liquid region. Periodic boundary conditions (PBC) are used together with a moving-frame technique to provide good overall analysis of the flow over a reduced computational mesh, thus reducing the computational effort. The hypotheses of a fully developed regime as well as a decomposition of the pressure field are embedded into a "one-fluid" formulation of the incompressible Navier-Stokes, which deals with the two-phase flow dynamics. Such an approach integrates a PBC-corrected semi-lagrangian treatment of the advection and a robust technique to handle the interface region in a zero-thickness fashion. At last, full three-dimensional simulations comparing the model to both numerical and experimental results concerning bubble shape factors and path oscillation for different bubble diameters, Archimedes, and Eotvos numbers are presented and discussed.