Turbulence in the scrape-off layer (SOL) is investigated using a 2D fluid model for the interchange instability as a paradigm. A constant driving flux governs the dynamics of both the equilibrium and fluctuating parts of the density and electric potential. The turbulent flux exhibits intermittent bursts, called avalanches. These events account for a significant part of the total transport, and are manifested as poloidally localized density fingers, extending towards the far SOL. The time averaged density profile looks exponential, and the SOL width increases weakly with the driving source (scaling exponent 2/9). Viscosity is found to govern the characteristic radial size of convective cells, which in turn control the transport magnitude. The larger ν, the larger the turbulent transport. Finally, the impact on turbulence of local biasing is investigated, possibly modeling Langmuir probe measurements. For a too large extent of the theoretical probe, the density drops by factors at the probe, due to the local build up of a screening vortex. The ambient density is recovered for a sufficiently small probe. In this case, fluctuations exhibit a similar Fourier spectrum at and next to the probe, though the probe still misses a significant number of large bursts. Finally, the experimental probe characteristics are recovered qualitatively when varying the biasing potential.