  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 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 analyzing
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 from
Klar & M\"ucket (2010), 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, which
temperature is balanced by radiative cooling and heating due to the UV
background. This indicates on a multiphase structure for the medium temperature
WHIM. We obtain scaling relations for the main quantities of this core. In the
vicinity of the halo, the filament's core can be attributed to the cold streams
found in cosmological hydro-simulations. Thermal conduction can lead to a
complete evaporation of the cold stream for L > 6 Mpc/h. This corresponds to
halos more massive than M_halo = 10^13 M_Sun, and implies that star-formation
in more massive galaxies can not be supplied by cold streams. For perturbations
on scales L > 6 Mpc/h the filament does not longer exhibit a cold core.
