Abstract An underwater object moving at a near-critical speed in a shallow-water domain had been observed to generate a sequence of upstream propagating solitary waves with an elongated depression of water surface and a train of dispersive waves followed in the downstream. This study presents the development of a two-dimensional stream function–vorticity based viscous fluid model with satisfied nonlinear free-surface conditions to study the generation of solitary waves and the induced vortex motion under the forcing of a moving object. A combined finite analytic and finite difference method is adopted to solve the flow field equations and free-surface boundary conditions in a transient curvilinear coordinate system. The model is shown to produce free-surface elevations in fairly good agreements with published results for a test case of a moving smooth bump. Other tests for the generation of recirculation zone behind a body of square shape in a confined fluid domain are also conducted to further verify the model performance. The results showing the generation of upstream advancing solitary waves and downstream vortex pattern by a blunt rectangular body moving at a critical speed along the bottom in a domain with free surface are presented. Comparisons of results from potential flow and viscous flow conditions are made to demonstrate the importance of viscosity to the wave generation. Different from the relatively regular vortex pattern occurred under the case of Re = 3500, the transition of the vortex motion for a larger Reynolds number (e.g. Re = 35,000) evolves without a regular pattern throughout the generation process of the advancing solitons. The effects of the size and bluntness of a moving object on the generated flow field and free-surface elevations are also analyzed and discussed.