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In this paper, we present theoretical results on the statistical propertiesof stationary, homogeneous and isotropic turbulence in incompressible flows inthree dimensions. Within the framework of the Non-Perturbative RenormalizationGroup, we derive a closed renormalization flow equation for a generic $n$-pointcorrelation (and response) function for large wave-numbers with respect to theinverse integral scale. The closure is obtained from a controlled expansion andrelies on extended symmetries of the Navier-Stokes field theory. It yields theexact leading behavior of the flow equation at large wave-numbers $|\vec p_i|$,and for arbitrary time differences $t_i$ in the stationary state. Furthermore,we obtain the form of the general solution of the corresponding fixed pointequation, which yields the analytical form of the leading wave-number and timedependence of $n$-point correlation functions, for large wave-numbers and bothfor small $t_i$ and in the limit $t_i\to \infty$. At small $t_i$, the leadingcontribution at large wave-number is logarithmically equivalent to $-\alpha(\epsilon L)^{2/3}|\sum t_i \vec p_i|^2$, where $\alpha$ is a nonuniversalconstant, $L$ the integral scale and $\varepsilon$ the mean energy injectionrate. For the 2-point function, the $(t p)^2$ dependence is known to originatefrom the sweeping effect. The derived formula embodies the generalization ofthe effect of sweeping to $n-$point correlation functions. At large wave-numberand large $t_i$, we show that the $t_i^2$ dependence in the leading ordercontribution crosses over to a $|t_i|$ dependence. The expression of thecorrelation functions in this regime was not derived before, even for the2-point function. Both predictions can be tested in direct numericalsimulations and in experiments.

Breaking of scale invariance in the time dependence of correlation functions in isotropic and homogeneous turbulence