Lagrangian dispersion in turbulent channel flow and its relationship to Eulerian statistics

Abstract

Lagrangian and Eulerian statistics were obtained from the direct numerical simulation of a turbulent channel flow. The Eulerian integral time scales and time microscales were compared to their Lagrangian equivalents. It is found that the Lagrangian time scales equal the Eulerian time scales at the wall. Otherwise, these proved to be consistently larger than the Eulerian time scales, and the latter are also found to be scaled by the propagation velocity rather than the mean velocity. The near-wall behavior of the ratio of the Lagrangian to Eulerian integral time scales is explained by the difference between the mean velocity and the propagation velocity of the turbulent structure. The ratio is proportional to the inverse of the turbulence intensity ( T L/ T E = βU/ σ). β for the streamwise component is nearly a constant value of 0.6 (20 < y + < 70), in agreement with atmospheric data and analyses. The ratio of the Lagrangian to the Eulerian time microscales is fairly constant away from the wall ( y + > 40). After period over the integral time scale elapsed, the effect of shear on turbulent diffusion appears and the mean-square dispersion is almost proportional to t 3 in agreement with the predictions of Riley and Corrsin [The relation of turbulent diffusivities to Lagrangian velocity statistics for the simplest shear flow. J. Geophys. Res. 79 (1974) 1768–1771] for homogeneous shear flow. After particles are distributed fairly uniformly between two walls, the mean-square dispersion becomes proportional to t and in agreement with theory of Taylor’s longitudinal dispersion.