Recent observations of protostellar cores suggest that most of the material in the protostellar phase is accreted along streamers. Streamers in this context are defined as velocity coherent funnels of denser material potentially connecting the large scale environment to the small scales of the forming accretion disk. Using simulations which simultaneously resolve the driving of turbulence on the filament scale as well as the collapse of the core down to protostellar disk scales, we aim to understand the effect of the turbulent velocity field on the formation of overdensities in the accretion flow. We perform a three-dimensional numerical study on a core collapse within a turbulent filament using the RAMSES code and analyse the properties of overdensities in the accretion flow. We find that overdensities are formed naturally by the initial turbulent velocity field inherited from the filament and subsequent gravitational collimation. This leads to streams which are not really filamentary but show a sheet-like morphology. Moreover, they have the same radial infall velocities as the low density material. As a main consequence of the turbulent initial condition, the mass accretion onto the disk does not follow the predictions for solid body rotation. Instead, most of the mass is funneled by the overdensities to intermediate disk radii.