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 the Caffarelli-Kohn-Nirenberg (CKN) theory for the Navier-Stokes equations analytically and numerically. The key quantity is written in standard notations \documentclass[12pt]{minimal}\begin{document}$\delta (r)=1/(\nu r)\int _{Q_r}\left|\nabla {\bm u}\right|^2\,d{{\bm x}} \,dt$\end{document}δ(r)=1/(νr)∫Qr∇u2dxdt, which can be regarded as a local Reynolds number over a parabolic cylinder Qr. First, by re-examining the CKN integral, we identify a cross-over scale \documentclass[12pt]{minimal}\begin{document}$r_* \propto L\left( \frac{ \overline{\Vert \nabla \bm {u} \Vert ^2_{L^2}} }{\Vert \nabla \bm {u} \Vert ^2_{L^\infty }} \right)^{1/3},$\end{document}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 rmin in turbulence: \documentclass[12pt]{minimal}\begin{document}$r_{\rm min}^2 \Vert \nabla {\bm u}\Vert _\infty \propto \nu ,$\end{document}r min 2‖∇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* ∝ νa with \documentclass[12pt]{minimal}\begin{document}$a=\frac{4}{3(3-q)}-1$\end{document}a=43(3−q)−1. Parametric representations are then obtained as \documentclass[12pt]{minimal}\begin{document}$\Vert \nabla {\bm u}\Vert _\infty \propto \nu ^{-(1+3a)/2}$\end{document}‖∇u‖∞∝ν−(1+3a)/2 and rmin ∝ ν3(a+1)/4. By the assumptions of the regularity and finite energy dissipation rate in the inviscid limit, we derive \documentclass[12pt]{minimal}\begin{document}$\lim _{p \rightarrow \infty }\frac{\zeta _p}{p}=1 - \zeta _2$\end{document}limp→∞ζpp=1−ζ2 for any phenomenological models on intermittency, where ζp is the exponent of pth order (longitudinal) velocity structure function. It follows that ζp ⩽ (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 Rλ ≈ 100 starting from random initial conditions, we have found that δ(r) ∝ r4throughout 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 \documentclass[12pt]{minimal}\begin{document}$\mu =\frac{4a}{1+a} \approx 0.8,$\end{document}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.