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

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In this experiment we study the transient evolution of a fractal topology in a laboratory setting, namely, the clustering of floating particles at the surface of a turbulent fluid. If neutrally buoyant, non-inertial particles are introduced into a turbulent flow, they quickly distribute themselves throughout the volume of the fluid. Further mixing due to the turbulent flow leaves the particle density distribution uniform. The buoyant particles used in this experiment have a density much less than that of the fluid on which they move. Hence their behavior is entirely different. Now, their motion is confined to the surface of the turbulent fluid.

If the spatial particle distribution is initially uniform at t = 0, continuous motion of the underlying fluid evolves this spatial distribution into string-like structures. Ultimately a steady state is reached, at which time the floaters occupy a fractal dimension much less than 2. This time evolution into a fractal is a generic effect. It occurs even if the interactions between the floaters is negligibly small. A common manifestation of this phenomenon is the coagulation of scum on the surface of the sea, as is often seen in an ocean harbor.

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A range of values of the fractal dimensions Dq are measured for various times in the experiment. These results indicate a multi-fractal distribution of the particles for the inertial range of the flow after the particles evolve from a homogeneous one at t = 0 s. The system evolves in a time of the order of the lifetime of the largest eddies of the turbulent flow to a steady state where the measured Dq(t) approach a value that is slightly greater than 1, implying the formation of the string-like structures. The correlation dimension D2(t) evolves exponentially as the steady state is approached.

It is not possible, so far, to deduce these observations from the Navier-Stokes equations for the compressible flow studied here. A compressible system of particles in free-surface turbulence represents an instance of a chaotic attractor in real space. The analysis performed in this work uses the tools developed for chaotic and dynamical systems to study this complex coagulation phenomenon.