In thin-layer cells, the flow of electrochemical current can induce the low-voltage electrohydrodynamic (EHD) convection. Following previous experimental work with the electrolysis of rubrene in 1,2-dimethoxyethane and theoretical calculations of the formation of driving force for the observed EHD convection, a further development of the numerical model is presented. The onset of convection, caused by an enhancement of local fluctuations of the electric current density, is modelled in terms of coupled equations of electrochemical transports and of the Navier-Stokes equations. The model system consists of the two-dimensional layer of the fluid electrolyzed between two planar electrodes, with the interelectrode distance 100 mu m. Simulated velocity distributions prove that the inhomogeneous distribution of local current densities across the solution can give rise to the electroconvection of the fluid. The comparison of numerical results with experimental data is sketched.