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

We study space-time integrals, which appear in the 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)2dxdtδ(r)=1/(νr)∫Qr∇u2dxdt, 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 r∗∝L(∥∇u∥2L2¯¯¯¯¯¯¯¯¯¯¯¯∥∇u∥2L∞)1/3,r*∝L‖∇u‖L22¯‖∇u‖L∞21/3, at which the CKN Reynolds number δ(r) changes its scaling behavior. This reproduces a result on the minimum scale r min in turbulence:r2min∥∇u∥∞∝ν,rmin2‖∇u‖∞∝ν, consistent with a result of Henshaw et al. [“On the smallest scale for the incompressible Navier-Stokes equations,” Theor. Comput. Fluid Dyn.1, 65 (1989)10.1007/BF00272138]. For the energy spectrum E(k) ∝ k −q   (1 < q < 3), we show that r * ∝ ν a with a=43(3−q)−1a=43(3−q)−1. Parametric representations are then obtained as ∥∇u∥∞∝ν−(1+3a)/2‖∇u‖∞∝ν−(1+3a)/2 and r min ∝ ν3(a+1)/4. By the assumptions of the regularity and finite energy dissipation rate in the inviscid limit, we derive limp→∞ζpp=1−ζ2limp→∞ζpp=1−ζ2 for any phenomenological models on intermittency, where ζ p is the exponent of pth order (longitudinal) velocity structure function. It follows that ζ p ⩽ (1 − ζ2)(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 Navier-Stokes 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=4a1+a≈0.8,μ=4a1+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.