Filaments, forming in the context of cosmological structure formation, are not only supposed to host the majority of the baryons at low redshifts in the form of the warm-hot intergalactic medium (WHIM), but also to supply forming galaxies at higher redshifts with a substantial amount of cold gas via cold steams. In order to get insight into the hydro- and thermodynamical characteristics of these structures, we performed a series of hydrodynamical simulations. Instead of analysing extensive simulations of cosmological structure formation, we simulate certain well-defined structures and study the impact of different physical processes as well as of the scale dependencies. In this paper, we continue our work, and extend our simulations into three dimensions. Instead of a pancake structure, we now obtain a configuration consisting of well-defined sheets, filaments and a gaseous halo. We use a set of simulations, parametrized by the length of the initial perturbation L, to obtain detailed information on the state of the gas and its evolution inside the filament. For L > 4 Mpc, we obtain filaments which are fully confined by an accretion shock. Additionally, they exhibit an isothermal core, in which temperature is balanced by radiative cooling and heating due to the extragalactic ultraviolet background. This indicates on a multiphase structure for the medium temperature WHIM. We obtain scaling relations for the main quantities of this core. After its formation, the core becomes shielded against further infall of gas on to the filament, and its mass content decreases with time. In the vicinity of the halo, the filament’s core can be attributed to the cold streams found in cosmological hydrosimulations. They are constricted by the outwards moving accretion shock of the halo. Thermal conduction can lead to a complete evaporation of the cold stream for L > 6 Mpc h−1. This corresponds to haloes more massive than Mhalo= 1013 M⊙, and implies that star formation in more massive galaxies cannot be supplied by cold streams.