  This paper is a numerical study of the condensation of the warm neutral
medium (WNM) into cold neutral medium (CNM) structures under the effect of
turbulence and thermal instability. Using low resolution simulations we
explored the impact of the WNM initial density and properties of the turbulence
(stirring in Fourier with a varying mix of solenoidal and compressive modes) on
the cold gas formation. Two sets of initial conditions which match the
observations were selected to produce high resolution simulations (1024^3)
allowing to study in details the properties of the produced dense structures.
For typical values of the density, pressure and velocity dispersion of the WNM
in the solar neighborhood, the turbulent motions of the HI can not provoque the
phase transition from WNM to CNM, whatever their amplitude and their
distribution in solenoidal and compressive modes. On the other hand we show
that a quasi-isothermal increase in WNM density of a factor of 2 to 4 is enough
to induce the phase transition, leading to the transition of about 40 percent
of the gas to the cold phase within 1 Myr. Given the observed properties of the
HI in the local ISM, the WNM and individual CNM structures in the local ISM are
sub or transsonic and their dynamics are tightly interwoven. The velocity field
bears the evidence of subsonic turbulence with a 2D power spectrum following
the Kolmogorov law as P(k) \sim k^{-8/3} while the density is highly contrasted
with a singificantly shallower power spectrum, reminiscent of what is observed
in the cold ISM. Supra-thermal line width observed for CNM might be the result
of relative velocity between cold structures. Finally, the cold structures
denser than 5 cm^{-3} reproduce well the laws M \sim L^{2.25-2.28} and sigma(v)
\sim 0.5-0.8L^{1/3} generally observed in molecular clouds.
