Intermittency and local Reynolds number in Navier-Stokes turbulence: A cross-over scale in the Caffarelli-Kohn-Nirenberg integral

Abstract

We study space-time integrals which appear in Caffarelli-Kohn-Nirenberg (CKN) theory for the Navier-Stokes equations analytically and numerically. The key quantity is written in standard notations δ(r) = 1/(νr) Qr |∇u| 2 dx dt, which can be regarded as a local Reynolds number over a parabolic cylinder Q r .First, by re-examining the CKN integral we identify a cross-over scale, at which the CKN Reynolds number δ(r) changes its scaling behavior. This reproduces a result on the minimum scale r min in turbulence: r 2 min ∇u ∞ ∝ ν, consistent with a result of Henshaw et al.(1989). For the energy spectrum E(k) ∝ k −q (1 < q < 3), we show that r * ∝ ν a with a = 4 3(3−q) −1. Parametric representations are then obtained as ∇u ∞ ∝ ν −(1+3a)/2 and r min ∝ ν 3(a+1)/4 . By the assumptions of the regularity and finite energy dissipation rate in the inviscid limit, we derive lim p→∞ ζp p = 1 − ζ 2 for any phenomenological models on intermittency, where ζ p is the exponent of p-th order (longitudinal) velocity structure function. It follows that ζ p ≤ (1 − ζ 2 )(p − 3) + 1 for any p ≥ 3 without invoking fractal energy cascade.Second, we determine the scaling behavior of δ(r) in direct numerical simulations of the NavierStokes equations. In isotropic turbulence around R λ ≈ 100 starting from random initial conditions, we have found that δ(r) ∝ r 4 throughout the inertial range. This can be explained by the smallness of a ≈ 0.26, with a result that r * is in the energy-containing range. If the β-model is perfectly correct, the intermittency parameter a must be related to the dissipation correlation exponent µ as µ = 4a 1+a ≈ 0.8 which is larger than the observed µ ≈ 0.20. Furthermore, corresponding integrals are studied using the Burgers vortex and the Burgers equation. In those single-scale phenomena, the cross-over scale lies in the dissipative range. The scale r * offers a practical method of quantifying intermittency. This paper also sorts out a number of existing mathematical bounds and phenomenological models on the basis of the CKN Reynolds number.

Gradual wavelet reconstruction of the velocity increments for turbulent wakes

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Study of the Navier–Stokes regularity problem with critical norms

“…Let us assume Q(t)/ν 2 ≥ 1, because we are not interested in small initial data, for which global existence is known. Multiplying (12) by this factor, we have…”

Characterization of blowup for the Navier-Stokes equations using vector potentials

“…which reproduces the right-hand side of (8). However, the trial ansatz (10) is incomplete as its left-hand side vanishes when i = k. ii) To compensate for the inconsistency, we add a diagonal element…”

“…18. Lots of progress has been made since then, including 2,3,[5][6][7][8][9][10][11][12]15,19,21,[23][24][25][26]28 and various kinds of blowup or regularity criteria have been developed. We define the vector potential A by u = ∇ × A in three dimensions, where ∇ · A = 0 and the stream function in two dimensions by u = (∂ 2 ψ, −∂ 1 ψ).…”