The experimental run examined until now was a run conducted with 2 particles only, and attention was focused on the motions of a single one of them. To demonstrate the wider applicability of the technique and highlight certain additional features, this last subsection presents results for a run involving 10 simultaneously orbiting particles. The corresponding Reynolds number is slightly higher at Re=VS/ν =1030. All 10 particles were initially places in the right half of the cavity, and stayed there for the entire run. Other runs conducted with particles on both sides showed a high degree of symmetry.
Figure 12 shows the orbits of the 10 particles, as tracked over 500 successive frames (a segment of 17 seconds in the asymptotic regime after the start-up transient has died out).
Panel a of Fig. 12 shows a stereoscopic long exposure image, while panels b to d show the corresponding 3D trajectories. Like the image of Fig. 2a, the long exposure view of Fig. 12a is produced directly from the video frames and does not rely on the positioning algorithms.
By contrast, the curves plotted on panels b to d represent stereo measurements, after full processing by the forward-backward Kalman filters. The juxtaposition of these two independently produced results can be used as a qualitative check of the methods.
Encouragingly, jitter due to measurement noise is attenuated by the Kalman filters, without significantly distorting the shapes of the orbits. Less systematic filtering procedures tried in the initial stages of the present work, by contrast, produced unacceptable distortion.
The results also highlight the most intriguing feature of the experiments. Despite starting from arbitrary positions and being allowed to wander freely within the cavity, the neutrally buoyant particles do not explore all regions of the flow. Instead, they tend to cluster along preferential pathways of the internal circulation: an inner coil along which particles spiral inwards from the sidewall toward the centre plane, and an outer coil along which particles drift back from the centre plane toward the sidewall. Conversely, there are various zones in which the particles do not venture: 1) the core of the main vortex, near the centre of the cavity, 2) the corner eddies located at the four corners of the bottom face of the cavity; 3) a toroidal zone located between the inner and outer coils along which particles spiral inward and outward.
While the origin of these preferential pathways requires further scrutiny, we interpret them as resulting from particle migrations toward certain belt zones of the flow. Assuming that the underlying mechanism is similar to the one observed in Poiseuille flows, these migrations would be due to the finite size of the solid particles relative to the local variations in the liquid shear.
6
Conclusions
In the present work, stereo imaging and signal processing techniques were combined to monitor the three-dimensional orbits of individual particles in a lid-driven cavity flow. Using a digital camera, video sequences of long duration were acquired, continuously recording the particle motions under two different viewpoints. By tracing calibrated rays into the viewing volume, the 3D particle positions and corresponding measurement errors were estimated. To attenuate the effect of these errors, the position signals were then processed by Kalman filters, based on a simple stochastic model of the kinematics. Kalman filters with forward and
backward passes were found to give good results, attenuating the noise without shifting the signals in time or significantly distorting the orbital shapes.
The measured particle trajectories present a number of interesting characteristics. Overall, the particles are observed to undergo spiral motions modulated by sideways excursions from the centre plane to the side wall and back. These spiral motions are found to exhibit slightly different patterns during the start-up phase and the subsequent regime. Observed over long times, particle trajectories are seen to cluster along certain pathways of the internal circulation, and avoid altogether certain regions of the cavity flow. Such preferential migrations appear due to finite size effects, similar to those observed for neutrally buoyant solid particles in laminar Poiseuille flows. Here however the viscous flow in which the particle motions are embedded features a more complicated, fully three-dimensional internal structure.
The research suggests various avenues for further work. The measurement methods could benefit from a number of improvements. Instead of using the Kalman filters at the post-processing stage only, they could be incorporated already at the particle tracking stage.
This could make the methods more robust, allowing them to deal with more difficult illumination conditions or larger numbers of simultaneously orbiting particles. More rigorous ways of tuning Kalman filters would also be desirable. On the other hand, further work is needed to enhance our understanding of the motions of solid particles suspended in three-dimensional viscous flows. For the lid-driven cavity, we are currently engaged in efforts aimed to characterise particle trajectories over a wider range of Reynolds numbers, and to probe the relationship between the Lagrangian particle trajectories and the Eulerian viscous flow field.