Stereoscopic and velocimetric reconstructions of the free surface topography of antidune flows

6. Results and discussion

Examples of free surface topography measurements obtained using both the stereoscopic and velocimetric methods are presented on Figures 9 through 14. Taken from a series of 10 runs, the three experimental runs shown (runs 3, 5, and 6) exhibit a range of different patterns: wide-crested rolls for run 3 (Fig. 9), a zig-zag pattern of staggered peaks and troughs for run 5 (Fig. 10), and a narrow wave train with sharply-peaked crests on one side of the channel for run 6 (Fig. 11). The gray-coded surface elevation maps of Figures 9-11 provide full-field pictures of these different patterns. Since the velocimetric method cannot determine the mean water level , the latter has been subtracted from the stereo results to obtain for both methods the perturbed topography .

Figure 9. Reconstructed free surface topography for run 3 (broad crested rolls): a stereoscopic method; b velocimetric method.

Figure 10. Reconstructed free surface topography for run 5 (zig-zag pattern): a stereoscopic method; b velocimetric method.

Figure 11. Reconstructed free surface topography for run 6 (narrow wave train on one side of the channel): a stereoscopic method; b velocimetric method. Circled: crater-like region where stereo measurements break down.

These maps allow qualitative comparisons between the two methods. Some superficial differences between the stereoscopic and velocimetric results can be noted. First, due to the different binning procedures used, the spatial coverage of the two methods do not fully coincide. Because the corresponding binning assigns an elevation even to areas where few particles were identified, the stereo results extend over the whole domain. The velocimetric results, on the other hand, cover only the restricted domain where particle trajectories were retrieved. Secondly, the surfaces exhibit distinct textures that reflect the peculiarities of each of the two methods. Because the density of the cloud of three-dimensional points can vary abruptly, the maximum-likelihood fitting used in the stereo method leads to staircase-like surfaces. On the other hand, the velocimetric maps are smoother, but feature longitudinal stripes due to the streamlines along which velocities are measured and averaged. These are fine-grained effects, however, which do not affect the overall surface elevation maps.

Qualitatively, the agreement between the two methods is seen to be quite good for all three of the patterns examined. The magnitudes, locations and arrangements of the peaks and troughs are in good correspondence. Local regions of discrepancy are nonetheless present, for instance the spurious crater-like feature exhibited by the stereo results of Fig. 11a near position , but absent from the velocimetric results of Fig. 11b. This is a region of wave-breaking where the stereo method appears to have failed, likely due to an insufficient number of successfully matched stereo pairs in that zone.

To facilitate precise comparisons, the same results are plotted again in a different format on Figures 12 through 14. Here profiles reconstructed by the two methods are plotted together for various positions y across the width of the channel. The stereo profiles (thin lines) and velocimetric profiles (thick lines) are found to be in reasonably close qualitative and quantitative agreement. Results in the central region of the viewing volume are generally better than the results near the edges. Especially poor results are recorded for the stereo profile of Fig. 14a, which passes through the “crater” zone of Fig. 11a. For this run (run 6), the stereo cameras were placed lower with respect to the surface than for runs 3 and 5, leading to near-occlusion in the troughs in addition to the difficulties associated with wave-breaking at the peaks.

Figure 12. Comparison of free surface profiles for run 3 obtained by the stereoscopic (thin lines) and velocimetric (thick lines) methods at selected sections: a y = 400 mm; b y = 300 mm; c y = 200 mm; d y = 100 mm.

Figure 13. Comparison of free surface profiles for run 5 obtained by the stereoscopic (thin lines) and velocimetric (thick lines) methods at selected sections: a y = 400 mm; b y = 300 mm; c y = 200 mm; d y = 100 mm.

Figure 14. Comparison of free surface profiles for run 6 obtained by the stereoscopic (thin lines) and velocimetric (thick lines) methods at selected sections: a y = 400 mm; b y = 300 mm; c y = 200 mm; d y = 100 mm.

Computation of the root-mean-squared discrepancy between the results of the two methods (averaged over a square domain of 500 mm × 500 mm) for the three runs 3, 5, and 6 yields rms errors of 3.8, 5.1, and 4.7 mm, respectively. Relative to elevation range η'_max-η'_min≈40 mm, this amounts to relative errors of the order of 10 to 15 %, which is comparable to the error level encountered in the stereo validation tests of section 4.4 (see also Table 1). If instead one gauges errors against the wavelength, taken as representative macroscopic length scale, relative errors of the order of 1 to 1.5 % are obtained. The following rule of thumb thus appears applicable here: to obtain 10 % accuracy for the perturbed velocities and elevations, 1 % accuracy relative to the mean flow quantities (mean velocity and wavelength) is required.

The detailed values of the predicted error contributions are listed in Table 1 for the three different runs. To estimate the stereo reconstruction error, the surface obtained by the velocimetric technique was used as reference surface. The observed discrepancies (of the order of 4-5 mm) turn out to be somewhat higher than the predicted discrepancies (of the order of 2-3 mm), obtained by summing the squares of the predicted error levels in both techniques, then taking the square root. This slight underestimation of the error levels may be ascribed to three factors: 1) the approximate nature of the error analysis procedures; 2) model errors (for instance deviations of the image formation process from a perfect perspective projection, or departures of actual streamline dynamics from the assumed Bernoulli relation); 3) the presence of regions where one of the two methods performs particularly poorly, as for instance in the “crater zone” of Fig. 11a mentioned previously.

Overall, the quality of the comparison is encouraging. Errors are not negligible, but remain within reasonable bounds considering that both methods yield whole-field measurements rather than point values. Furthermore, the order of magnitude of the incurred errors appears consistent with the error estimates derived using the techniques of sections 4 and 5. Errors not included in the analysis can therefore be ascertained to be small. Despite incipient wave breaking, for instance, streamline dynamics appear well approximated by the Bernoulli relation on the scale of a few wavelengths. Note that this implies only that the flow is nearly inviscid, since the Bernoulli equation applied on a streamline-by-streamline basis does not require the flow to be irrotational (see e.g. Batchelor 1967).

Most importantly, the spatial patterns associated with flows over antidunes are successfully captured by both methods. The results vividly depict a variety of motifs: broad crested rolls, zigzags, and narrow train of peaks and troughs. Due to the complex way in which light interacts with the rough water surface, these organized patterns could not be so easily grasped by pure visual inspection of the laboratory flows.