  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.
