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

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.