  We use Direct Numerical Simulations (DNS) of the forced Navier-Stokes
equation for a 3-dimensional incompressible fluid in order to test recent
theoretical predictions. We study the two- and three-point spatio-temporal
correlation functions of the velocity field in stationary, isotropic and
homogeneous turbulence. We compare our numerical results to the predictions
from the Functional Renormalization Group (FRG) which were obtained in the
large wavenumber limit. DNS are performed at various Reynolds numbers and the
correlations are analyzed in different time regimes focusing on the large
wavenumbers. At small time delays, we find that the two-point correlation
function decays as a Gaussian in the variable $kt$ where $k$ is the wavenumber
and $t$ the time delay. The three-point correlation function, determined from
the time-dependent advection-velocity correlations, also follows a Gaussian
decay at small $t$ with the same prefactor as the one of the two-point
function. These behaviors are in precise agreement with the FRG results, and
can be simply understood as a consequence of sweeping. At large time delays,
the FRG predicts a crossover to an exponential in $k^2 t$, which we were not
able to resolve in our simulations. However, we analyze the two-point
spatio-temporal correlations of the modulus of the velocity, and show that they
exhibit this crossover from a Gaussian to an exponential decay, although we
lack of a theoretical understanding in this case. This intriguing phenomenon
calls for further theoretical investigation.
