Context. The formation of astrophysical structures, such as stars, compact objects, and also galaxies, entail an enhancement of densities by many orders of magnitude, which occurs through gravitational collapse. Aims. The role played by turbulence during this process is important. Turbulence generates density fluctuations, exerts a support against gravity, and possibly delivers angular momentum. How exactly turbulence behaves and is amplified during the collapse remains a matter of investigation and is the aim of the present paper. Methods. We carried out spherical averaging of the fluid equations, leading to 1D fluid equations that describe the evolution of mean quantities in particular the mean radial velocity as well as the mean radial and transverse turbulent velocities. These equations differ from the ones usually employed in the literature. We then performed a series of 3D numerical simulations of collapsing clouds for a wide range of thermal and turbulent supports with two polytropic equations of state, P ∝ ρ^Γ, with Γ = 1 and 1.25. For each 3D simulation, we performed a series of 1D simulations using the spherically averaged equations and the same initial conditions. Results. By performing a detailed comparison between 3D and 1D simulations, we can analyse the observed behaviours in great detail. Altogether, we find that the two approaches agree remarkably well, demonstrating the validity of the inferred equations; although, when turbulence is initially strong, major deviations from spherical geometry certainly preclude quantitative comparisons. The detailed comparisons lead us to an estimate of the turbulent dissipation parameter, which, when the turbulence is initially low, is found to be in good agreement with previous estimates of non self-gravitating supersonic turbulence. When turbulence is initially dynamically significant, larger values of the dissipation appear necessary for the 1D simulations to match the 3D ones. We find that the behaviour of turbulence depends on the cloud thermal support. If it is high, initial turbulence is amplified, as proposed earlier in the literature. However, if thermal support is low, turbulence is also generated by the development of local non-axisymmetric gravitational instabilities reaching values several times larger and in equipartition with gravitational energy. Conclusions. The inferred 1D equations offer an easy way to estimate the level reached by turbulence during gravitational collapse. Depending on the cloud thermal support, turbulence is either amplified or locally generated.

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