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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.