The formation of astrophysical structures, such as stars, compact objects but also galaxies, entail an,enhancement of densities by many orders of magnitude which occurs through gravitational collapse. 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 turbulence exactly behave during the collapse and get amplified remains a matter of investigation. Spherical averaging of the fluid equations is carried out, 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 perform a series of 3D numerical simulations of collapsing clouds for a wide range of thermal and turbulent supports with two polytropic equation of state, $P propto rho^Gamma$, with $Gamma=1$ and 1.25. For each 3D simulations we perform a series of 1D simulations using the spherically averaged equations and with the same initial conditions. By performing a detailed comparison between 3D and 1D simulations, we can analyse in great details the observed behaviours. 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 that when the turbulence is initially low, is found to be in good agreement with previous estimate of non self-gravitating supersonic turbulence. abridged.
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