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

In this paper, we present theoretical results on the statistical properties of stationary, homogeneous and isotropic turbulence in incompressible flows in three dimensions. Within the framework of the Non-Perturbative Renormalization Group, we derive a closed renormalization flow equation for a generic $n$-point correlation (and response) function for large wave-numbers with respect to the inverse integral scale. The closure is obtained from a controlled expansion and relies on extended symmetries of the Navier-Stokes field theory. It yields the exact 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 point equation, which yields the analytical form of the leading wave-number and time dependence of $n$-point correlation functions, for large wave-numbers and both for small $t_i$ and in the limit $t_i\to \infty$. At small $t_i$, the leading contribution at large wave-number is logarithmically equivalent to $-\alpha (\epsilon L)^{2/3}|\sum t_i \vec p_i|^2$, where $\alpha$ is a nonuniversal constant, $L$ the integral scale and $\varepsilon$ the mean energy injection rate. For the 2-point function, the $(t p)^2$ dependence is known to originate from the sweeping effect. The derived formula embodies the generalization of the effect of sweeping to $n-$point correlation functions. At large wave-number and large $t_i$, we show that the $t_i^2$ dependence in the leading order contribution crosses over to a $|t_i|$ dependence. The expression of the correlation functions in this regime was not derived before, even for the 2-point function. Both predictions can be tested in direct numerical simulations and in experiments.