Spatiotemporal velocity-velocity correlation function in fully developed turbulence

Turbulence is a ubiquitous phenomenon in natural and industrial flows. Since the celebrated work of Kolmogorov in 1941, understanding the statistical properties of fully developed turbulence has remained a major quest. In particular, deriving the properties of turbulent flows from a mesoscopic description, that is, from the Navier-Stokes equation, has eluded most theoretical attempts. Here, we provide a theoretical prediction for the functional space and time dependence of the velocity-velocity correlation function of homogeneous and isotropic turbulence from the field theory associated to the Navier-Stokes equation with stochastic forcing. This prediction, which goes beyond Kolmogorov theory, is the analytical fixed point solution of nonperturbative renormalization group flow equations, which are exact in the limit of large wave numbers. This solution is compared to two-point two-times correlation functions computed in direct numerical simulations. We obtain a remarkable agreement both in the inertial and in the dissipative ranges.

Figure 1

Time dependence of the correlation function $C(t,k)$ in $k$ space (in dimensionless form) computed from direct numerical simulations at $R_{\lambda }=90$ and 160. (a) Correlation function $C(t,k)$ for $R_{\lambda }=160$ in log scale as a function of the dimensionless variable ${\widehat{z}}^2={\left(\epsilon \eta \right)}^{2/3}{\left(kt\right)}^2$, for different values of $k\eta$. In this representation, the curves for the different $k\eta$ are parallel lines, with the same slope $\tilde{\alpha }$, confirming the NPRG prediction. (b) Illustration of the collapse of the normalized correlation function $C(t,k)/C(0,k)$ for $R_{\lambda }=160$ for all values of $k$, on a single Gaussian curve in the dimensionless variable $\widehat{z}$. (c) Value of the parameter ${\tilde{\alpha }}_k$ estimated from Gaussian fits of the data for all the different values of $k$, for both $R_{\lambda }=90$ and 160. ${\tilde{\alpha }}_k=\tilde{\alpha }$ is perfectly constant for $R_{\lambda }=160$ (approximatively for $R_{\lambda }=90$).

Figure 2

Kinetic energy spectra in $d=3$ obtained from direct numerical simulations at different Taylor-scale Reynolds numbers $R_{\lambda }=90,160,433$ (from top to bottom), plotted in a dimensionless form: ${\epsilon }^{-2/3}E$ as a function of $k\eta$. The data for $R_{\lambda }=90$ and 160 are from this work; the data for $R_{\lambda }=433$ come from the Johns Hopkins Turbulence Database [32]. The numerical spectra display an inertial range, extending with $R_{\lambda }$, with a $k^{-5/3}$ decay (up to possible small corrections).

Figure 3

Same dimensionless kinetic energy spectra as in Fig. 2, multiplied by ${\left(k\eta \right)}^{5/3}$ and represented in log scale as a function of ${\left(k\eta \right)}^{2/3}$. The NPRG predicts a crossover from the $k^{-5/3}$ power law to a stretched exponential decay $exp[-\widehat{\mu }{\left(\lambda k\right)}^{2/3}]$ in the dissipative range (dashed lines), which is observed in the numerical data (plain lines with symbols).

Figure 4

Typical map of the modulus of the velocity field in simulated turbulence for $R_{\lambda }=160$, at different times.