On Lagrangian single-particle statistics

In turbulence, ideas of energy cascade and energy flux, substantiated by the exact Kolmogorov relation, lead to the determination of scaling laws for the velocity spatial correlation function. Here we ask whether similar ideas can be applied to temporal correlations. We critically review the relevant theoretical and experimental results concerning the velocity statistics of a single fluid particle in the inertial range of statistically homogeneous, stationary and isotropic turbulence. We stress that the widely used relations for the second structure function, D₂(t) ≡ ([v(t) - v(0))]²∝ εt, relies on dimensional arguments only: no relation of D2(t) to the energy cascade is known, neither in two- nor in three-dimensional turbulence. State of the art experimental and numerical results demonstrate that at high Reynolds numbers, the derivative )d D(t) dt2t has a finite non-zero slope starting from t ≈ 2τν. The analysis of the acceleration spectrum Φ_A(ε) indicates a possible small correction with respect to the dimensional expectation Φ_A(ε;) ~ ε⁰ but present data are unable to discriminate between anomalous scaling and finite Reynolds effects in the second order moment of velocity Lagrangian statistics.