We investigate the evolution of the cosmic web since z. =. 5 in grid-based cosmological hydrodynamical simulations, focusing on the mass and velocity fields of both baryonic and cold dark matter. The tidal tensor of density is used as the main method for web identification, with lambda(th). =. 0.2-1.2. The evolution trends in baryonic and dark matter are similar, although moderate differences are observed. Sheets appear early, and their large-scale pattern may have been set up by z. =. 3. In terms of mass, filaments supersede sheets as the primary collapsing structures from z. similar to 2-3. Tenuous filaments assembled with each other to form prominent ones at z. <.. 2. In accordance with the construction of the frame of the sheets, the cosmic divergence velocity, vdiv, was already welldeveloped above 2-3 Mpc by z. =. 3. Afterwards, the curl velocity, vcurl, grew dramatically along with the rising of filaments, becoming comparable to vdiv, for < 2-3 Mpc at z. =. 0. The scaling of vcurl can be described by the hierarchical turbulence model. The alignment between the vorticity and the eigenvectors of the shear tensor in the baryonic matter field resembles that in the dark matter field, and is even moderately stronger between w and e1, and. and e3. Compared with dark matter, there is slightly less baryonic matter found residing in filaments and clusters, and its vorticity developed more significantly below 2-3 Mpc. These differences may be underestimated because of the limited resolution and lack of star formation in our simulation. The impact of the change of dominant structures in overdense regions at z. similar to 2-3 on galaxy formation and evolution is shortly discussed.