Cascade of energy in turbulent flows

A starting point for the conventional theory of turbulence [12-14] is the n otion that, on average, kinetic energy is transferred from low wave number modes to high wave number modes [19]. Such a transfer of energy occurs in a spectral range beyond that of injection of energy, and it underlies the so -called cascade of energy, a fundamental mechanism used to explain the Kolm ogorov spectrum in three-dimensional turbulent flows. The aim of this Note is to prove this transfer of energy to higher modes in a mathematically rig orous manner, by working directly with the Navier-Stokes equations and stat ionary statistical solutions obtained through time averages. To the best of our knowledge, this result has not been proved previously; however, some d iscussions and partly intuitive proofs appear in the literature. See, e.g., [1,2,10,11,16,17,21], and [22]. It is noteworthy that a mathematical frame work can be devised where this result can be completely proved, despite the well-known limitations of the mathematical theory of the three-dimensional Navier-Stokes equations. A similar result concerning the transfer of energ y is valid in space dimension two. Here, however, due to vorticity constrai nts not present in the three-dimensional case, such energy transfer is acco mpanied by a similar transfer of enstrophy to higher modes. Moreover, at lo w wave numbers, in the spectral region below that of injection of energy, a n inverse (from high to low modes) transfer of energy (as well as enstrophy ) takes place. These results are directly related to the mechanisms of dire ct enstrophy cascade and inverse energy cascade which occur, respectively, in a certain spectral range above and below that of injection of energy [1, 15]. In a forthcoming article [9] we will discuss conditions for the actual existence of the inertial range in dimension three. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.