J. Stat. Mech. (2014) P05002 ournal of Statistical Mechanics: An IOP and SISSA journal J Theory and Experiment Two-scale correlation and energy cascade in three-dimensional turbulent flows Y X Huang 1 , F G Schmitt 2,3,4 and Y Gagne 5 1 Shanghai Institute of Applied Mathematics and Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, People’s Republic of China 2 Universit´ e Lille Nord de France, F-59000 Lille, France 3 USTL, LOG, F-62930 Wimereux, France 4 CNRS, UMR 8187, F-62930 Wimereux, France 5 LEGI, CNRS/UJF/INPG, UMR 5519, F-38041 Grenoble, France E-mail: yongxianghuang@gmail.com, francois.schmitt@univ-lille1.fr and yves.gagne@hmg.inpg.fr Received 22 October 2013 Accepted for publication 13 March 2014 Published 9 May 2014 Online at stacks.iop.org/JSTAT/2014/P05002 doi:10.1088/1742-5468/2014/05/P05002 Abstract. In this paper, we propose a high-order harmonic-free methodology, namely arbitrary-order Hilbert spectral analysis, to estimate the two-scale correlation (TSC). When applied to fully developed turbulent velocity, we find that the scale-dependent Hilbert energy satisfies a lognormal distribution on both the inertial and dissipation ranges. The maximum probability density function of the logarithm of the Hilbert energy obeys a power law with a scaling exponent γ ’ 0.33 in the inertial range. For the measured TSC, we observe a logarithmic correlation law with an experimental exponent α ’ 0.37 on both the inertial and dissipation ranges. The correlation itself is found to be self-similar with respect to the distance between the two considered scales and a central frequency ω c in the logarithm space. An empirical nonlinear and nonlocal triad- scale interaction formula is proposed to describe the observed TSC. This triadic interaction can be interpreted as experimental evidence of a small-scale nonlinear and nonlocal coupling inside the self-similarity of the Richardson–Kolmogorov phenomenological cascade picture. c  2014 IOP Publishing Ltd and SISSA Medialab srl 1742-5468/14/P05002+15$33.00