According to the paraxial theory, we analyzed the linear momentum density and orbital angular momentum (OAM) the propagation characteristics of a Gaussian vortex beam in free space. The propagation characteristics are simulated and analyzed in the visualization. Further, we study the variation of the propagation characteristics with different topological charges. In addition, we also analyzed the position of momentum in the transverse profile, where the momentum density of the spot will be broadened with propagation distance. This study can provide guidance for using vortex beams in optical communication and manipulation.