It is shown that the two remarkable properties of turbulence, the Extended Self-Similarity (ESS) [R. Benzi et al., Phy. Rev. E 48, R29, (1993)] and the She-Leveque Hierarchical Structure (SLHS) [Z.S. She and E. Leveque, Phy. Rev. Lett. 72, 336, (1994)] are related to each other. In particular, we have shown that a generalized hierarchical structure together with the most intense structures being shock-like give rise to ESS. Our analysis thus suggests that the ESS measured in turbulent flows is an indication of the shock-like intense structures. Results of analysis of velocity measurements in a pipe-flow turbulence support our conjecture.PACS numbers: 47.27.-i Fully developed turbulence is characterized by powerlaw dependence of the moments of velocity fluctuations. It was suggested by Kolmogorov in 1941 (K41)[1] that there is a constant rate of energy transfer from large to small scales and that the statistical properties of the velocity difference across a separation r, δv r ≡ v(x+r)−v(x), depend only on the mean energy transfer or equivalently the mean energy dissipation rate ǫ and the scale r when r is within an inertial range. Dimensional considerations then lead to the prediction that the velocity structure functions, which are moments of the magnitude of the velocity difference, have simple power-law dependence on r within the inertial range:(1)Experiments [2] have indicated that there is indeed power law scaling in the inertial range but the scaling exponents are different from p/3:where ζ p has a nonlinear dependence on p. Such a deviation implies that the functional form of the probability density function (pdf) of δv r depends on r, that is, the velocity fluctuations have scale-dependent statistics. Understanding this deviation from K41 is essential to our fundamental understanding of the small scale statistical properties of turbulence.Recently, Benzi et al. [3] have discovered a remarkable new scaling property: S p (r) has a power-law dependence on S 3 (r) over a range substantially longer than the scaling range obtained by plotting S p (r) as a function of r. This behavior was named Extended Self-Similarity (ESS); its discovery has enabled more accurate determination of the scaling exponents ζ p , particularly at moderately high Reynolds numbers assessible experimentally and numerically, It was later reported that ESS is invalid for anisotropic turbulent flows such as atmospheric boundary layer and channel flow [4,5,6]. This inspires the study of a generalized ESS (GESS), a scaling behavior of the normalized structure functions when plotted against each other [7,8], which is still valid in these anisotropic flows. The validity of ESS suggests that the different order structure functions have the same dependence on r when r is near the dissipative range [9,10,11]. Very recently, Yakhot argued that some mean-field approximation of the pressure contributions in the Navier-Stokes equation would lead to ESS [12].A number of phenomenological models have been proposed to explain the anomalous scaling expon...
