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

7. Conclusions

In the present work, two approaches to the measurement of free surface topography in high Froude number flows were presented and compared. Both rely on the imaging of floating tracers dispersed on the surface, but they resort to different reconstruction principles: stereoscopic matching of rays associated with two camera viewpoints, on the one hand, and indirect estimation of the elevations based on the horizontal velocity field measured using a single camera, on the other hand. A waveform of known shape was used to check the stereo algorithms and error estimation approach. The results of the two techniques were then compared for flows over antidunes featuring various surface patterns. Although somewhat larger than expected based on error estimates, the observed discrepancies were found to be reasonably small. The obtained relief maps vividly depict the variety of motifs that can evolve as a result of interaction between shallow flows and loose sediment beds.

Drawbacks, advantages and possible improvements can be identified for both techniques. Stereoscopic matching and velocimetric tracking both involve pairing particles on distinct images, but the stereo methods were found to generate a much greater proportion of mismatches (up to 40 % of the data points). This stems partly from the fact that the stereo technique matches isolated particles whereas the velocimetric technique tracks patterns of neighbouring particles. The higher vulnerability of the stereo technique to mismatches has various consequences: greater accuracy requirements for the camera calibration procedure; tighter limits on the density of particles which can be reliably imaged; precedence of robustness over accuracy when interpolating data. Camera synchronisation is also a paramount concern, best addressed by synchronising two sensors at the time of image acquisition (rather than a posteriori as in the present work).

A key advantage of the stereo technique is that, in principle, it applies to general flows independently of any specific assumption (occlusion being the only limitation). In practice, the stereo technique requires the flow to vary slowly in order to combine data from multiple image pairs and obtain a surface of sufficient resolution. The velocimetric technique, by contrast, fundamentally depends on certain physical hypotheses. The assumed Bernoulli equation requires the flow to be quasi-steady, and its simplified version further assumes moderate perturbations superimposed upon a quasi-uniform mean current. It also requires the mean surface level to be known by other means. When these conditions are met, however, the present results demonstrate that the velocity-based approach works well, producing reasonable topography measurements. For the present experiments, estimated error levels for the velocimetric techniques are nevertheless about twice as large as the error levels incurred by the stereo approach. This can be ascribed in part to the lower pixel resolution of the camera used for velocimetry. For both methods, camera sensors having a better resolution than the relatively limited equipment used in the present work would lead to improvements in accuracy.

From a methodological point of view, the present work underscores the importance of characterising the error formation process. Many significant experimental errors cannot be treated as Gaussian random variables of zero mean that will simply average out if measurements can be repeated a sufficient number of times. In the present experiments, non-Gaussian sources of error include: 1) stereo mismatches distributed evenly within the viewing volume as a result of epipolar ambiguity; 2) velocity errors due to position noise, which cancel each other along particle trajectories; 3) attenuation associated with binning, averaging and filtering; 4) geometric distortion associated with projection onto a non-planar surface. In all these cases, the special character of the errors had to be taken into account in order for their influence on the results to be controlled. The objective of ultimately comparing two independent sets of measurements (stereo and velocity-based) served as a powerful motivation in striving to get a grip on the errors inherent to both methods.

Provided errors can be controlled, it was found possible to use both stereoscopic and velocimetric techniques to capture the free surface topography of antidune flows, with results that compare favourably with each other. Either method can be recommended for measuring the free surface shape of quasi-steady shallow flows, especially in cases where slope-based methods are inapplicable. Issues for future work include the possible blending of the two approaches (e.g. using streamline relationships to constrain stereoscopic measurements). Comparisons with theoretical and computational descriptions of antidune flows will be sought. Also, further applications of the techniques are contemplated, including measurements of evolving antidune fields at larger scales as well as other free surface flows. On a practical note, future work should rely on synchronized high-resolution cameras and smaller particles in order to obtain more detailed and more accurate information.

Financial support for the present work was provided by the Fonds National de la Recherche Scientifique, Belgium, through a Fernand De Waele Prize awarded to H. Capart under convention 1.5.300.99 F (1). Assistance from Ni Wei-Jay of the National Taiwan University in setting up the stereo validation tests is also gratefully acknowledged.

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