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

Imaging methods developed to characterise the oscillatory free surface of rapid flows are presented and applied to torrential currents over sediment antidunes. The aim is to obtain high-resolution relief maps of the free surface topography. Two measurement principles are tested, both based on the imaging of floating tracers dispersed on the rapidly flowing surface. The first technique involves direct stereoscopic measurements. The second technique is indirect, and exploits a Bernoulli relation to derive surface elevations from the horizontal velocity field acquired using a single camera. Special attention is paid to error estimation and control. Relief maps obtained for various bedform patterns are presented, allowing comparison between the two techniques.

1. Introduction

Free surface measurements constitute an important challenge in experimental fluid mechanics. The free surface is often of interest because of the key role it plays in the flow phenomena under consideration (e.g. Hammack et al. 1989; Jähne et al. 1994; Lang and Gharib 2000). Moreover, it may be the only flow feature which is readily accessible to measurements by imaging or remote sensing methods (e.g. Stilwell and Pilon 1974; Nicolas et al. 1997; Craeye et al. 1999). Both circumstances are encountered in the present study of antidune flows.

Antidunes are trains of bed waves (Gilbert 1914; Kennedy 1963; Reynolds 1965; Allen 1968) which appear when rapid, shallow currents flow over coarse granular material, and are characterised by in-phase coupling between the oscillatory sediment bed and the water free surface. As illustrated on Fig. 1, they can be observed both in the field (Alexander and Fielding 1997; Blair 2000) and in the laboratory (Middleton 1965). Along with other types of fluvial bedforms, they are of interest to geomorphologists and hydraulic engineers (Shaw and Kellerhals 1977). Antidunes are notable in particular for their evanescent character: they vanish rapidly once the flow wanes, and leave few lasting traces aside from bedding and grain sorting effects. As a result, their geometrical configuration is best studied when the flow is active.

Figure 1. Antidune flows on a beach of Eastern Taiwan (top) and in the Louvain laboratory flume (bottom). Photographs by B. Spinewine and H. Capart.

Long- or short-crested, arranged in regular arrays or in narrow trains of peaks and troughs, antidunes come in a variety of patterns. In the present study, a characterisation of such patterns is sought through measurements of the water free surface topography. Reflection and refraction by the rough free surface impedes visual access to the underlying sediment bed, hence no direct measurement of the bottom topography is possible. Since the two surfaces are locked in phase with each other, however, the water surface topography provides an indirect image of the bedform pattern.

Measurements of water surface topography most often involve point sensors, placed in multi-sensor arrays at fixed locations or scanned across the surface (e.g. Hammack et al. 1989; Wessels et al. 1989). Sensors used to measure water elevation include resistive or capacitive gauges, pressure transducers and acoustic beams. Resistive gauges are inapplicable in the present case because of the high sensitivity of antidune flows to intruding objects. More generally, the spatial and temporal resolution of point sensors is limited by the number of available devices or the time required to scan a sensor across the field of interest. This makes them unsuitable for the transient (on a time scale of a few seconds), spatially varied surfaces of flows over evolving bedforms.

Because they permit non-intrusive, fast whole-field measurements, imaging techniques constitute attractive alternatives to point gauges and sensors. Described in a review by Jähne et al. (1994), a variety of imaging principles have been proposed to characterise water surface shapes. Based on photogrammetric techniques widely applied to land topography, one approach is to measure water surface heights using stereo photography (Banner et al. 1989; Shemdin and Tran 1992). This requires corresponding features or regions on the surface to be matched between distinct views. For water surfaces, this correspondence problem is difficult to solve reliably because of the specular nature of light reflection at the free surface (Jähne et al. 1994).

For this reason, most investigators have turned to measuring surface slope rather than surface height. Approaches based on shading, reflection and refraction have been developed for this purpose, and are reviewed in Jähne et al. (1994). Recent measurements based on reflection are documented in Craeye et al. (1999) and Dabiri and Gharib (2001), while recent studies relying on refraction (shadowgraph) include Weigand (1996) and Lang and Gharib (2000). These techniques, however, are subject to various limitations. Shape-from-shading does not work for transparent, specularly reflecting surfaces. On the other hand, shape-from-reflection is restricted to a narrow slope range. Finally, shape-from-refraction requires light rays to pass through the top and the bottom of the water layer. These limitations make the techniques inapplicable to the present water flows featuring rough and highly varied free surfaces, and propagating over opaque sediment beds.

Like the earlier stereo techniques, the approach explored in the present work relies on the matching and tracking of surface features. To address the correspondence problem, floating particles are dispersed on the water free surface, furnishing point-like features that look the same on distinct frames. Images of these floating tracers are then exploited using two different reconstruction principles. The first is a classical stereoscopic principle based on the matching of particles on image pairs acquired from two cameras. The second is an original velocimetric principle requiring only a single vertical camera: the surface velocity field is first acquired by particle tracking velocimetry, then converted into a vertical elevation map using a Bernoulli relation derived from the fluid mechanics of the water free surface.

The parallel development of two distinct techniques was motivated by the following considerations. First, it makes it possible to weigh the respective advantages of the methods and evaluate their applicability to more complex situations. Secondly, an assessment of the validity of both techniques can be obtained by comparing their results. To permit cross-validation, the two methods are applied to the same experiments, while each relies on its own separate camera footage and data analysis pipeline. Preliminary results of the present research effort were reported in Devriendt et al. (1998) and Douxchamps et al. (2000).

The paper is organised as follows. In the next section, the experimental set-up and camera systems used for the measurements are presented. Then, basic particle imaging algorithms used as building blocks by the two reconstruction methods are outlined. The next two sections are devoted to a detailed presentation of each of the two reconstruction methods. This includes special procedures for the estimation and filtering of measurement errors at each step of the analysis. For the stereo principle, the surface reconstruction and error estimation procedures are validated using a solid surface of known shape. Finally, the methods are applied to the free surfaces of water flows over antidunes. Results from the stereo and velocity-based methods are shown, compared and discussed in the last section preceding the overall conclusions of the work.