Abstract : This work is motivated by the recent international effort to create an experimentally self-sustained dynamo. The dynamo effect, whose existence was proposed by Larmor at the beginning of the 20th century, is believed to be the explanation for the magnetic field of Earth and other celestial bodies due to the flow of a conducting fluid. In order to numerically study the von Kármán flow, which models the configuration of the dynamo experiment implemented at Cadarache, we have developed a new numerical approach for solving the magnetohydrodynamic equations in potential formulation in a finite cylindrical geometry. The poloidal-toroidal decomposition has been used to ensure the solenoidal character of the velocity and magnetic fields. We use the influence matrix technique to impose the boundary conditions for the velocity and the continuity between the internal and external magnetic fields. The computational power of the code, which is the result of the MPI-based parallelization, enabled us to investigate two problems concerning turbulence in cylindrical geometry: axisymmetric turbulence and a bifurcation between turbulent flows.