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We reconsider the functional renormalization-group (FRG) approach to decaying Burgers turbulence, and extend it to de- caying Navier-Stokes turbulence. The method is based on a renormalized small-time expansion, equivalent to a loop expansion, and naturally produces a dissipative anomaly and a cascade after a finite time. We explicitly calculate and analyze the one-loop FRG equa- tions in the zero-viscosity limit as a function of the dimension. For Burgers they reproduce the FRG equation obtained in the context of random manifolds. Breakdown of energy conservation due to shocks and the appearance of a direct energy cascade corresponds to failure of dimensional reduction in the context of disordered systems. For Navier-Stokes in three dimensions, the velocity-velocity correlation function acquires a linear dependence on the distance, ζ2 = 1, in the inertial range, instead of Kolmogorov’s ζ2 = 2/3; however the possibility remains for corrections at two- or higher-loop order. In two dimensions, we obtain a numerical solution which conserves energy and exhibits an inverse cascade, with explicit analytical results both for large and small distances, in agreement with the scaling proposed by Batchelor. In large dimensions, the one-loop FRG equation for Navier-Stokes converges to that of Burgers.
Author(s):
Andrei Fedorenko
Laboratoire de Physique de l'ENS de Lyon
France
Pierre Le Doussal
LPT ENS
France
Kay Wiese
LPT ENS
France