Software tools for simulating electrochemical processes (e.g. COMSOL Multiphysics, ELSYCA) are mostly of commercial type. Besides, three-dimensional simulations in complex cell geometries are known to become resource-expensive, as typically thin concentration boundary layers need to be resolved. This work presents a simulation framework for electrochemical processes based on the open source platform OpenFOAM. The finite volume method used and combined with domain decomposition is able to efficiently benefit from multi-core computer architectures. Our framework takes into account electrolyte flow, which is well known to affect mass transfer, and allows to consider multi-species electrolytes and forcing of the electrolyte. The stability and fast convergence of the method presented is found to rely on the linearization of the Butler-Volmer condition in the iterative solver. The framework is validated against an analytical solution valid for simplified conditions and an electrodeposition process at a conically shaped electrode in an external magnetic field. The latter exhibits transient departure of the concentration boundary layer from the cathode, and excellent agreement with COMSOL simulation results is found.