A three-phase Eulerian model is proposed to investigate the induced flow due to the generation of gas bubbles between two parallel plates without forced convection with application to alkaline water electrolysis (AWE). Earlier models, assuming a laminar regime, accurately predicted the multiphase flow near electrodes but struggled to calculate bulk liquid electrolyte flow away from them. Herein, we study the influences of electric current density distribution, turbulence effects, and the interaction between flow and the magnetic field known as magnetohydrodynamics (MHD). Based on our modeling results, the traditional method using an averaged uniform current density along electrodes (e.g. here 2000 A m−2) is feasible, as incorporating calculated non-uniform current distribution minimally affects the multiphase velocity field. The Lorentz force, originating from flow interaction with the (self-induced) magnetic field, is negligible compared to forces like drag or bubble dispersion. Consequently, MHD effects only become relevant upon introducing an external magnetic field. Including turbulence in the model, being minor in magnitude but non-negligible, significantly improves the predicted velocity profile. Modeling results are validated against an experiment.