Spatio-temporal correlations in 3D homogeneous isotropic turbulence

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.