Time-Evolution of a Fractal Distribution: Particle Concentrations in Free-Surface Turbulence

TL;DR

This study investigates how floating particles in surface turbulence evolve from a uniform distribution to a steady-state fractal pattern, revealing exponential convergence in their correlation dimension within a turbulent water tank.

Contribution

It provides the first detailed measurement of the time evolution of fractal dimensions of particles in free-surface turbulence, highlighting the exponential approach to steady state.

Findings

Particles form a fractal distribution in turbulence.

Steady state reached in about 1 second.

Correlation dimension approaches steady state exponentially.

Abstract

Steady-state turbulence is generated in a tank of water and the trajectories of particles forming a compressible system on the surface are tracked in time. The initial uniformly distributed floating particles coagulate and form a fractal distribution, a rare manifestation of a fractal object observable in real-space. The surface pattern reaches a steady state in approximately 1 s. Measurements are made of the fractal dimensions $D_q(t)$ ($q=1$ to $6$) of the floating particles starting with the uniform distribution $D_q(0)$ = 2 for Taylor Microscale Reynolds number $Re_{\lambda} \simeq 160$. Focus is on the the time-evolution of the correlation dimension $D_2(t)$ as the steady state is approached. This steady state is reached in several large eddy turnover times and does so at an exponential rate.