On the asymptotic behavior of average energy and enstrophy in 3D turbulent flows

On Energy Cascades in the Forced 3D Navier–Stokes Equations

“…We conclude with several remarks. Overall the results of Section 5, obtained for the physical scales, mirror closely the ones obtained in [12] in the framework of Fourier scales and statistical solutions (see Remark 5). Recall, [12] shows that Kolmogorov's dissipation law (defined there in terms of generalized limits) induces the same restrictive scaling on energy and its dissipation rate as the one obtained here.…”

“…Namely, ε scales like Gr 3/2 , while the kinetic energy scales like Gr, where Gr is the Grashof Number, representing non-dimensional magnitude of the force (2.17), (2.23). These types of scaling laws were discovered in the context of statistical solutions in [12], and the fact that the same scaling is manifested in the framework of (K 1 , K 2 )-averages points to a remarkable consistency between the two approaches. We exploit this scaling to obtain a saturation property for Cauchy-Schwarz-type inequality for averages of f · u, which, in turn, allows us to formulate a better sufficient condition for energy cascades, as well as to estimate the width of the inertial range in terms of Grashof number (see Theorem 6 and the remarks after thereafter)…”

“…In this section our goal is to obtain the lower bound on the kinetic energy e 0 in periodic setting described above. The idea behind the bounds presented here is similar to [12], where the lower bound on energy was obtained in a different context.…”

“…In that particular framework, Gr ≫ 1 would automatically imply energy cascade in Fourier scales on the inertial range [τ 0 , r f ], where r f is the scale of the force. The reason for this is that the aforementioned Gr-scaling implies τ 0 (= κ −1 τ ) ∼ Gr −1/4 ≪ 1 (see [12]), and thus the sufficient condition for cascades given in [22,23] holds.…”

Dissipation Length Scale Estimates for Turbulent Flows: A Wiener Algebra Approach

“…One can make an argument similar to the one presented in Section 3.1 that would justify the corresponding assumption on the initial data, but working on the 3D weak attractor. For background on the weak attractor, see [7] or [17].…”