The phenomenon of a succession of upstream-advancing solitary waves generated by underwater disturbances moving steadily with a transcritical velocity in two- dimensional shallow water channels is investigated. The two-dimensional Navier–Stokes (NS) equations with the complete set of viscous boundary conditions are solved numerically by the finite-difference method to simulate the phenomenon. The overall features of the phenomenon illustrated by the present numerical results are unanimous with observations in nature as well as in laboratories. The relations between amplitude and celerity, and between amplitude and period of generation of solitary waves can be accurately simulated by the present numerical method, and are in good agreement with predictions of theoretical formulae. The dependence of solitary wave radiation on the blockage and on the body shape is investigated. It furnishes collateral evidence of the experimental findings that the blockage plays a key role in the generation of solitary waves. The amplitude increases while the period of generation decreases as the blockage coefficient increases. It is found that in a viscous flow the shape of an underwater object has a significant effect on the generation of solitary waves owing to the viscous effect in the boundary layer. If a change in body shape results in increasing the region of the viscous boundary layer, it enhances the viscous effect and so does the disturbance forcing; therefore the amplitudes of solitary waves increase. In addition, detailed information of the flow, such as the pressure distribution, velocity and vorticity fields, are given by the present NS solutions.