Statistics of single and multiple floaters in experiments of surface wave turbulence

We present laboratory experiments of surface wave turbulence excited by paddles in the deep water regime. The free surface is seeded with buoyant particles that are advected and dispersed by the flow. Positions and velocities of the floaters are measured using particle tracking velocimetry. We study the statistics of velocity and acceleration of the particles, mean vertical displacements, single-particle horizontal dispersion, and the phenomenon of preferential concentration. Using a simple model together with the experimental data, we show that the time evolution of the particles has three characteristic processes that dominate the dynamics at different times: drag by surface waves at early times, trapping by short-lived horizontal eddies at intermediate times, and advection by a large-scale mean circulation at late times.

Figure 1

Experimental setup. (a) The water tank with the paddles to generate surface waves, and the two cameras used for PTV. (b) A picture of a floater, with a cut showing its cross section.

Figure 2

Eulerian energy spectra of surface deformation obtained using the fringe projection profilometry technique. (a) Spatiotemporal spectrum of the potential energy $E(k,\omega )$ (or, equivalently, power spectrum of the surface deformation amplitude) for the experiments with $A=10$ cm. Darker colors correspond to larger energy densities. The green solid line indicates the theoretical dispersion relation for the system. (b) Same for the experiments with $A=20$ cm. (c) Eulerian wave number spectrum $E\left(k\right)$ of the surface deformation amplitude for all forcing functions $A$ considered in the study. An $E\left(k\right)\sim k^{-5/2}$ power law is indicated as a reference.

Figure 3

(Top) PDFs of the Cartesian components of the fluctuating velocity of the floaters for (a) experiment A10, with maximum spatial displacement of the paddles of $A=10$ mm, and (b) for experiment A20, with maximum spatial displacement of the paddles of $A=20$ mm. Velocities are normalized by their dispersions, and thus are dimensionless. (Bottom) PDFs of the Cartesian components of the fluctuating acceleration of the floaters in the same experiments, for (c) $A=10$ mm, and (d) $A=20$ mm. As for the velocity, the accelerations are normalized by their corresponding dispersions. Labels for the three Cartesian components are given in the insets. In all cases, we also show as a reference a normal distribution centered around zero with dispersion of unity, represented by the solid black curve and denoted as $N(0,1)$.

Figure 4

Kinetic energy spectra of the floaters, for each Cartesian component of the velocity, for experiments A10 (a) and A20 (b). A power law $E\left(f\right)\sim f^{-4.5}$ for frequencies larger than 4 Hz is shown as a reference.

Figure 5

(a) Instantaneous power $dE/dt$ gained or lost by a particle in experiments A10 (with maximum forcing amplitude $A=10$ mm). The inset shows a detail of the time series. (b) PDF of the instantaneous power gained or lost by the particles, in experiments A10 and A20. The shape of the PDF is the same for both experiments. For completeness, it is worth mentioning that the ${\sigma }_{\dot{E}}$ value associated with the two experiments is ${\sigma }_{\dot{E}}^{\text{A10}}=0.14$ mW and ${\sigma }_{\dot{E}}^{\text{A20}}=0.18$ mW, respectively. A normal distribution is shown as a reference by the black solid curve, and labeled as $N(0,1)$.

Figure 6

(a) Three-dimensional trajectories of 52 particles in experiment A10, in light gray and blue. The trajectory of one particle is highlighted in black, and projected into the $x\text{−}y$, $x\text{−}z$, and $y\text{−}z$ planes. Note, in the projection onto the $x\text{−}y$ plane, the coexistence of three phenomena: fast oscillations associated with waves, circular trajectories associated with eddy trapping, and a slow drift. (b) Vertical position $z$ of a particle as a function of time in the same experiment. All other particles follow a similar vertical behavior. (c) Horizontal trajectories of two particles in the same experiment. Note the circular motions in the particle trajectories.

Figure 7

Temporal average, in an Eulerian grid, of the horizontal velocities of all particles in the observed region of experiment A10. Note the mean large-scale circulation that develops in the experiment.

Figure 8

(a) Isocontours of the joint probability distribution $P(r,t)$ of particles being displaced a horizontal distance $r=\delta r$ at a time $t$, in experiment A10. The two vertical lines separate three different regimes as explained in the text. (b) PDFs of the horizontal displacements $P(r,t)$ at a fixed time and as a function of the displacement $r$, for all the experiments.

Figure 9

(a) Mean quadratic vertical and horizontal displacement of the floaters in experiment A10, as indicated in the inset. Note the three regimes, separated as a reference by the two vertical lines. These lines indicate, from left to right, the inverse of half the frequency of the most energetic waves, and the particle velocity correlation time. (b) Mean quadratic vertical and horizontal displacement of the floaters in all experiments.

Figure 10

Mean quadratic displacements in the vertical (solid lines) and horizontal directions (dashed lines), for forcing amplitudes $A=10$, 15, and 20 mm (from bottom to top), as obtained from the experiments, and multiplied by an arbitrary factor to separate them vertically (color labels are as in Fig. 9). In black lines, we show the mean quadratic displacements in the vertical (solid) and horizontal directions (dashed) obtained from the model. The case with $A=5$ is similar and not shown for simplicity. The gray dash-dotted lines indicate a ballistic displacement of the particles predicted solely from the general circulation in each of the experiments. Note that for all times these ballistic displacements are insufficient to explain the observations, and only at late times the observed displacements approach the predictions from the mean flow.

Figure 11

Delaunay tessellation for an instantaneous configuration of floaters, using a small number of particles in experiment A10.

Figure 12

PDFs of the areas of the triangular cells resulting from the Delaunay tessellation, in experiment A10 (a) and A20 (b). As a reference, a best fit using a Gamma distribution function is shown in both panels.