  We extend our previous work on simulations with the code RAMSES on accretion
driven turbulence by including self-gravity and study the effects of core
formation and collapse. We show that radial accretion onto filaments drives
turbulent motions which are not isotropic but radially dominated. In contrast
to filaments without gravity, the velocity dispersion of self-gravitating
filaments does not settle in an equilibrium. Despite showing similar amounts of
driven turbulence, they continually dissipate their velocity dispersion until
the onset of core formation. This difference is connected to the evolution of
the radius as it determines the dissipation rate. In the non-gravitational case
filament growth is not limited and its radius grows linearly with time. In
contrast, there is a maximum extent in the self-gravitational case resulting in
an increased dissipation rate. Furthermore, accretion driven turbulence shows a
radial profile which is anti-correlated with density. This leads to a constant
turbulent pressure throughout the filament. As the additional turbulent
pressure does not have a radial gradient it does not contribute to the
stability of filaments and does not increase the critical line-mass. However,
this radial turbulence does affect the radius of a filament, adding to the
extent and setting its maximum value. Moreover, the radius evolution also
affects the growth timescale of cores which compared to the timescale of
collapse of an accreting filament limits core formation to high line-masses.
