  Predicting the star formation rate (SFR) in galaxies is crucial to understand
their evolution and morphology. To do so requires a fine understanding of how
dense structures of gas are created and collapse. In that, turbulence and
gravity play a major role. Within the gravo-turbulent framework, we assume that
turbulence shapes the ISM, creating density fluctuations that, if
gravitationally unstable, will collapse and form stars. The goal of this work
is to quantify how different regimes of turbulence, characterized by the
strength and compressibility of the driving, shape the density field. We are
interested in the outcome in terms of SFR and how it compares with existing
analytical models for the SFR. We run a series of hydrodynamical simulations of
turbulent gas. The simulations are first conducted without gravity, so that the
density and velocity are shaped by the turbulence driving. Gravity is then
switched on, and the SFR is measured and compared with analytical models. The
physics included in these simulations is very close to the one assumed in the
classical gravo-turbulent SFR analytical models, which makes the comparison
straightforward. We found that the existing analytical models convincingly
agree with simulations at low Mach number, but we measure a much lower SFR in
the simulation with a high Mach number. We develop, in a companion paper, an
updated physically-motivated SFR model that reproduces well the inefficient
high Mach regime of the simulations. Our work demonstrates that accurate
estimations of the turbulent-driven replenishment time of dense structures and
the dense gas spatial distribution are necessary to correctly predict the SFR
in the high Mach regime. The inefficient high-Mach regime is a possible
explanation for the low SFR found in dense and turbulent environments such as
the centers of our Milky Way and other galaxies.
