  We present the results of smoothed particle hydrodynamic simulations
investigating the evolution and fragmentation of filaments that are accreting
from a turbulent medium. We show that the presence of turbulence, and the
resulting inhomogeneities in the accretion flow, play a significant role in the
fragmentation process. Filaments which experience a weakly turbulent accretion
flow fragment in a two-tier hierarchical fashion, similar to the fragmentation
pattern seen in the Orion Integral Shaped Filament. Increasing the energy in
the turbulent velocity field results in more sub-structure within the
filaments, and one sees a shift from gravity-dominated fragmentation to
turbulence-dominated fragmentation. The sub-structure formed in the filaments
is elongated and roughly parallel to the longitudinal axis of the filament,
similar to the fibres seen in observations of Taurus, and suggests that the
fray and fragment scenario is a possible mechanism for the production of
fibres. We show that the formation of these fibre-like structures is linked to
the vorticity of the velocity field inside the filament and the filament's
accretion from an inhomogeneous medium. Moreover, we find that accretion is
able to drive and sustain roughly sonic levels of turbulence inside the
filaments, but is not able to prevent radial collapse once the filaments become
supercritical. However, the supercritical filaments which contain fibre-like
structures do not collapse radially, suggesting that fibrous filaments may not
necessarily become radially unstable once they reach the critical line-density.
