Lagrangian Statistical Theory of Fully Developed Hydrodynamical Turbulence

Abstract

The Lagrangian velocity structure functions in the inertial range of fully developed fluid turbulence are for the first time derived based on the Navier-Stokes equation. For time tau much smaller than the correlation time, the structure functions are shown to obey the scaling relations K_{n}(tau) proportional, varianttau;{zeta_{n}}. The scaling exponents zeta_{n} are calculated analytically without any fitting parameters. The obtained values are in amazing agreement with the experimental results of the Bodenschatz group. A new relation connecting the Lagrangian structure functions of different orders analogously to the extended self-similarity ansatz is found.

Anomalous scaling for Lagrangian velocity structure functions in fully developed turbulence

“…The Lagrangian scaling exponents thus obtained are in good agreement with the experimental measurements for lower orders but less good agreement with them for high orders [7]. Recently, Beck [10] has developed a superstatistical model for Lagrangian scaling exponents, and Zybin et al [11] have calculated the Lagrangian scaling exponents using the vortex filaments. Using the multifractal formalism, the She and Leveque (SL) model [12] introduces a Eulerian hierarchical structure to describe the Eulerian scaling exponents with its successes.…”

“…The sweeping model is The experimental data support the straining model rather than the sweeping model. The straining model indicates that the Lagrangian scaling exponents increase monotonously with its orders and saturate at infinitely large p. Zybin et al's model [11] predicts that the Lagrangian scaling exponents decrease for p > 7, and Beck's model [10] predicts that the Lagrangian scaling exponents could decrease to be negative for larger p. The decrease of Lagrangian scaling exponents at higher orders need to be further justified, while the decrement observed in the experiments is due to insufficient data samples [5].…”

Inertial range Eulerian and Lagrangian statistics from numerical simulations of isotropic turbulence

“…(9). Such effect is probably the highest difficulty to overcome in both stochastic modeling (Sawford (2001)) and theory of Lagrangian turbulence (Yakhot (2009);Zybin et al (2008)). …”

Fully Developed Turbulence and the Multifractal Conjecture

“…The suggested road map should be the following: (i) first measure the Eulerian scaling exponents ζ(p); (ii) then via an inverse Legendre transform extract the F (h)-spectrum; (iii) finally, apply the relation (27) and calculate the Lagrangian scaling. Such procedure is working well, at least within the statistical limitation and the Reynolds number limitations allowed by numerical and experimental state-of-the-art techniques [5] (see also [41] for a recent theoretical attempt). Even more interesting, in [5], the same argument, leading to the spatial dissipative fluctuating scale (7), has been extended into the Lagrangian domain to obtain an expression for the fluctuating dissipative time scale [42]:…”