The study of transient pressure waves in both low- and high-frequency domains has become a new research area to provide potentially high-resolution pipe fault detection methods. In previous research works, radial pressure waves were evidently observed after stopping the laminar pipe flows by valve closures, but the generation mechanism and components of these radial pressure waves are unclear. This paper intends to clarify this phenomenon. To this end, this study first addresses the inefficiencies of the current numerical scheme for the full two-dimensional (full-2D) water hammer model. The modified efficient full-2D model is then implemented into a practical reservoir-pipeline-valve (RPV) system, which is validated by the well-established analytical solutions. The generation mechanism and components of the radial pressure waves, caused by different flow perturbations from valve operations, in transient laminar flows are investigated systematically using this efficient full-2D model. The results indicate that nonuniform changes in the initial velocity profile form pressure gradients along the pipe radius. The existence of these radial pressure gradients is the driving force of the formation of radial flux and radial pressure waves. In addition, high radial modes can be excited, and the frequency of flow perturbations by valve oscillation can redistribute the energy entrapped in each high radial mode.