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We study the renormalization group flow of the average action of thestochastic Navier--Stokes equation with power-law forcing. Using Galileaninvariance we introduce a non-perturbative approximation adapted to the zerofrequency sector of the theory in the parametric range of the H\"older exponent$4-2\,\varepsilon$ of the forcing where real-space local interactions arerelevant. In any spatial dimension $d$, we observe the convergence of theresulting renormalization group flow to a unique fixed point which yields akinetic energy spectrum scaling in agreement with canonical dimension analysis.Kolmogorov's -5/3 law is, thus, recovered for $\varepsilon=2$ as also predictedby perturbative renormalization. At variance with the perturbative prediction,the -5/3 law emerges in the presence of a \emph{saturation} in the$\varepsilon$-dependence of the scaling dimension of the eddy diffusivity at$\varepsilon=3/2$ when, according to perturbative renormalization, the velocityfield becomes infra-red relevant.

A non-perturbative renormalization group study of the stochastic Navier--Stokes equation