Directional change of fluid particles in two-dimensional turbulence and of football players

Multiscale directional statistics are investigated in two-dimensional incompressible turbulence. It is shown that the short-time behavior of the mean angle of directional change of fluid particles is linearly dependent on the time lag and that no inertial range behavior is observed in the directional change associated with the enstrophy-cascade range. In simulations of the inverse-cascade range, the directional change shows a power law behavior at inertial range time scales. By comparing the directional change in space-periodic and wall-bounded flow, it is shown that the probability density function of the directional change at long times carries the signature of the confinement. The geometrical origin of this effect is validated by Monte Carlo simulations. The same effect is also observed in the directional statistics computed from the trajectories of football players (soccer players in American English).

Figure 1

Mean absolute angle of directional change as a function of the time lag $\tau$, in statistically stationary 2D turbulence, forced at large scales, in periodic and circular-wall-bounded geometry.

Figure 2

Probability density functions of the directional change of fluid particles in 2D turbulent flows. (a) Periodic boundary conditions. An equipartition of the angles is expected at long times. (b) Circular no-slip walls. The asymptotic shape of the PDF, corresponding to the angle between two line segments connected by three randomly placed points in a circle, is represented by a thick black dashed line.

Figure 3

Probability density functions of $1-cos\left(\mathrm{\Theta }\right)$, where $\mathrm{\Theta }$ corresponds to the directional change of fluid particles in 2D turbulent flow (forward cascade) in a periodic domain. Rescaled and fitted by a theoretical model.

Figure 4

Sketch illustrating the effect of confinement on the directional change at long times in (a) a circular domain and (b) rectangular domains with a large aspect ratio.

Figure 5

PDFs for the angle $\mathrm{\Theta }$, between two connected line segments defined by three randomly positioned points in a rectangular domain. The aspect ratio of the domain is varied between 1 and 400.

Figure 6

Monte Carlo results for (a) the mean-absolute angle and (b) the variance (second centered moment), in a circular domain, and in rectangular domains with different aspect ratios.

Figure 7

Statistics in the inverse cascade range of small-scale forced, 2D turbulence. (a) energy spectrum; (b) absolute average directional change as a function of the time lag.

Figure 8

Statistics in the inverse cascade range of small-scale forced, 2D turbulence: (a) PDF of the instantaneous angle $\mathrm{\Theta }$; (b) normalized PDF of $1-cos\left(\mathrm{\Theta }\right)$ and model, assuming Gaussianity and independence.

Figure 9

Trajectories of four players during a 5 min interval of a football match.

Figure 10

Angular statistics of football players: (a) Lagrangian velocity spectrum; (b) mean absolute angle for the two halftimes of the match; (c) PDFs of the instantaneous angle $\mathrm{\Theta }(\tau ,t)$; (d) normalized PDFs.