Fully developed isotropic turbulence: Nonperturbative renormalization group formalism and fixed-point solution

We investigate the regime of fully developed homogeneous and isotropic turbulence of the Navier-Stokes (NS) equation in the presence of a stochastic forcing, using the nonperturbative (functional) renormalization group (NPRG). Within a simple approximation based on symmetries, we obtain the fixed-point solution of the NPRG flow equations that corresponds to fully developed turbulence both in $d=2$ and 3 dimensions. Deviations to the dimensional scalings (Kolmogorov in $d=3$ or Kraichnan-Batchelor in $d=2$) are found for the two-point functions. To further analyze these deviations, we derive exact flow equations in the large wave-number limit, and show that the fixed point does not entail the usual scale invariance, thereby identifying the mechanism for the emergence of intermittency within the NPRG framework. The purpose of this work is to provide a detailed basis for NPRG studies of NS turbulence; the determination of the ensuing intermittency exponents is left for future work.

Figure 1

RG evolution of the dimensionless running function ${\widehat{h}}^{\nu }\left(\widehat{p}\right)$ in dimensions $d=3$ (upper panel) and $d=2$ (lower panel) starting from the initial condition ${\widehat{h}}^{\nu }\left(\widehat{p}\right)=1$ at the microscopic scale $\kappa ={\eta }^{-1}$. The red arrow corresponds to decreasing RG scales $\kappa$ and the black thick line to the fixed-point function.

Figure 2

Fixed-point functions ${\widehat{h}}^D\left(\widehat{p}\right)$ (lower panel) and ${\widehat{h}}^{\nu }\left(\widehat{p}\right)$ (upper panel) in dimensions $d=3$ (black curves) and $d=2$ (red curves). Both horizontal and vertical axes are in logarithmic scales.

Figure 3

Energy spectrum (multiplied by the appropriate power ${\kappa }^{-5/3}$ in $d=3$ and ${\kappa }^{-3}$ in $d=2$) of the turbulent flow in dimensions $d=3$ and 2 as a function of the dimensionless wave number $\widehat{p}=\left|\vec{p}\right|/\kappa$. The dashed lines are guidelines for the eyes.

Figure 4

Diagrammatic representation of the exact flow equation of ${\mathrm{\Gamma }}_{\perp }^{(1,1)}(\nu ,\vec{p})$, with ${\tilde{\partial }}_s\equiv {\partial }_s{R}_{\kappa }\frac{\partial }{\partial R_{\kappa }}+{\partial }_s{N}_{\kappa }\frac{\partial }{\partial N_{\kappa }}$. The combinatorial factors are not explicitly written, and diagrams involving ${\mathrm{\Gamma }}_{\kappa }^{(n,0)}$ vertices are omitted since they are vanishing.

Figure 5

Diagrammatic representation of the exact flow equation of ${\mathrm{\Gamma }}_{\perp }^{(0,2)}(\nu ,\vec{p})$, with ${\tilde{\partial }}_s\equiv {\partial }_s{R}_{\kappa }\frac{\partial }{\partial R_{\kappa }}+{\partial }_s{N}_{\kappa }\frac{\partial }{\partial N_{\kappa }}$. The combinatorial factors are not explicitly written, and diagrams involving ${\mathrm{\Gamma }}_{\kappa }^{(n,0)}$ vertices are omitted since they are vanishing.