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 $-α(εL)^{2/3}|\sum t_i \vec p_i|^2$, where $α$ 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.
Eulerian spectral closures for isotropic turbulence using a time-ordered fluctuation-dissipation relation
协方差函数的时间排序过程，如前一篇论文（K。Kiyani和W. D. McComb Phys.Rev.E70，066303（2004））的方法进行了扩展，并用于表明与Kraichnan-Wyld扰动级数相关的二阶响应函数可以由局部（波数）能量平衡确定。这些时间排序程序也允许两次制定减少到时间无关的形式，通过指数近似，它被验证的响应方程没有红外发散在无限雷诺数。最后，单次马尔可夫封闭方程（在前面的论文中所述）的推导和证明是兼容的Kolmogorov分布，而不需要引入一个特设常数。
