DIGITAL

Abstract: In this paper, additional experimental work on a dam-break flow through a 90°-bend is presented. Making use of a recently developed Voronoï digital imaging technique, an accurate surface velocity field could be measured. A great advantage of the technique, initially aimed at studying intense granular flows, is the non-intrusive aspect of the measurements, as well as the fact that a complete velocity field is obtained, instead of scarce point data. The reproducibility of the experiments made it possible to accumulate enough data to reconstruct a complete transient velocity field over the areas of interest. Detailed results are presented in the bend region, where a high curvature is imposed to the flow trajectories.

Following the experiments, numerical simulations of the flow were run, and are compared to the measured data. The model used here solves the two-dimensional Saint-Venant shallow-water equations by a Roe-type finite-volume scheme. Those comparisons allow to gain interesting information both about the flow behaviour and the implications of the assumptions made in the numerical model.

1 INTRODUCTION

Some results concerning dam-break flow in a channel with a 90° bend have already been presented, as a contribution to the 1998-1999 European concerted action CADAM (Soares et al., 1999; Soares and Zech, 1999; Soares et al., 2000). In those experiments, the measured data only consisted in the water-level evolution with time at a series of gauging points. Those measurements were used to validate a wide range of numerical models, by comparing the predicted and measured water levels. However, no information regarding the velocities was available yet.

In this paper, additional experimental work is presented. Making use of a recently developed Voronoï digital imaging technique (Capart et al., 2001), it was possible to measure the surface velocity field during a dam-break flow. A great advantage of the technique is the non-intrusive aspect of the measurements, as well as the fact that a complete velocity field is obtained, instead of scarce point data. The reproducibility of the experiments made it possible to accumulate enough data to reconstruct a complete transient velocity field over the areas of interest.

Such new measurements can be of great value for further validation of numerical models. Comparisons with simulations of the flow by a two-dimensional Roe-type finite-volume are presented, opening promising perspectives.

2 DAM-BREAK EXPERIMENTS

Figure 1 shows the experimental set-up. The channel is located in the laboratory of the Civil Engineering Department of the Université catholique de Louvain, Belgium. The upstream reservoir has dimensions of 2.44 m × 2.39 m, the channel cross-section is rectangular, 0.495 m wide, the upstream reach is about 4 m long and the downstream reach, after the bend, is about 3 m long. The channel bed level is 0.33 m above the reservoir bed level. The downstream end of the channel is open. The initial water level in the reservoir is 0.25 m above the channel bed.

The sudden raise of the gate separating the upstream reservoir from the channel simulates the dam break. The water then flows rapidly into the channel and reflects against the bend. There, a bore forms and starts to travel back in the upstream direction, until reaching the reservoir.

3 FLOW-MEASUREMENT TECHNIQUES

The flow was imaged by cameras above the flume. High-speed digital cameras were used, acquiring grey-scale images at a rate of 200 frames per second, with a resolution of 256 × 256 pixels. With reference to Figure 2, use is made of a recently developed Voronoï Particle-Tracking-Velocimetry (PTV) technique (Capart et al., 2001) to gather information about the flow velocity by tracking small tracers (white wooden floaters about 1 cm in size) distributed over the free surface of the flow.

Several steps can be distinguished in the process of analysis : at first (figures 2a and 2b), particle centres are localised on the image by filtering the latter with a Mexican Hat filter sized to the mean particle diameter, and finding peak values of brightness on the resulting image. Sub-pixel accuracy is achieved by interpolating between the brightness values of the few pixels available per particle. The second step (figure 2c) seeks at matching positions of identical particles on two successive frames. The method, described in details by Capart et al. (2001) is based on pattern templates which remain stable over a few successive frames, namely the Voronoï polygons, and has been found more robust than simpler tracking methods such as minimum displacement or trajectory-based techniques.

Finally (figure 2d), trajectories are reconstructed from frame to frame, and positions as well as velocities are transformed into real units (m and m/s respectively) by a simple procedure for camera calibration.

To represent accurately the flow behaviour, one has to prevent interactions (collisions, …) between nearby tracers, which could lead to discrepancies between water velocity and particle velocity. Special care has to be taken for homogeneous seeding of the tracers to avoid clogging effects. Consequently, the amount of particles, and hence the amount of velocity data available per experiment, is quite limited. Furthermore, as the flow is transient, one can not perform consistent temporal averages as could be done for steady flows. However, due to its excellent reproducibility, the experiment can be carried out several times in order to accumulate enough data to reconstruct a complete transient-flow velocity field over the areas of interest.

As the experimental data are randomly distributed, one has to interpolate on a regular grid, corresponding for example with the cells used in the numerical scheme, in order to be able to conveniently compare experimental and numerical data. This is done by organising the data into cells, assigning all individual velocity vectors to the centre of the nearest cell. The median value is taken as representative for the whole cell; indeed, taking the mean value would cause outliers (mismatches with wrong velocity values resulting from possible errors in the matching algorithm) to pollute the data set and affect the average unrealistically.

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Fig. 2 Process of analysis by Voronoï digital imaging technique : (a) rough image, (b) particle identification, (c) particles tracking and velocities and (d) reconstructed trajectories

4 FINITE-VOLUME NUMERICAL SCHEME

A Roe-type finite-volume scheme (Soares et al, 1999) is used to solve the two-dimensional Saint-Venant shallow-water equations. Those equations read, in conservative vector form

U is the vector of conserved hydraulic variables (mass and momentum in the x- and y-directions), F and G are the vectors of fluxes in the x- and y-direction respectively and S contains the topographical and frictional source terms.

The finite-volume integration of (1) over a non-Cartesian grid yields the following numerical scheme

where A_i is the cell area, nb the number of cell interfaces, T_j the rotation matrix corresponding to the local axis system attached to the considered interface, F^* the normal numerical flux through the interface and L_j the interface length. The numerical fluxes in the direction normal to each interface are evaluated by Roe's scheme (Glaister, 1988).

The computations presented in this paper were run with a 4 cm square mesh, and convergence was successfully checked on the one hand with mesh sizes from 16 to 1 cm and on the other hand with non-Cartesian orthogonal meshes. All computations were run with a 0.9 CFL number.

5 MEASURED AND COMPUTED VELOCITY FIELD

Some characteristic results of the comparison between experimental and numerical data are presented on figure 3, corresponding to the situation 7 s after the gate opening. The computed water level is shown on figure 3a : the wave front has already reached the end of the channel, and the bore formed by the reflection in the bend is travelling in the upstream direction, back to the reservoir. Measured velocity vectors and magnitudes are presented on figures 3b and 3d respectively. White regions on the plots are regions where no tracers were identified. Corresponding computed results are shown on figures 3c and 3e.

Before comparing more deeply the numerical simulation to the experiments, it is important to remind that the numerical model considers depth-averaged velocities, while the experimental measurements concern the surface velocity field. Both values are assumed to be sufficiently close to allow comparison, at least partially.

A generally good agreement is observed both for the velocity direction and amplitude. The acceleration zone downstream from the bend is particularly well reproduced by the numerical model.

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Fig. 3 Comparison between the measured velocity field and computed results (7 s after gate opening): (a) picture of the flow from the numerical simulation, (b) (c) measured and computed velocity field and (d) (e) measured and computed velocity magnitude

In the upstream reach, the location of the receding bore can be clearly identified as well in the measurements as in the simulation. However, in the numerical simulation, the bore appears as a sharp discontinuity in the variables (water depth and velocity). In the reality, the bore is like a breaking wave with a strong re-circulation region in the vertical plane, spread over a certain distance (low surface velocities measured between approximately 5.5 m and 5.8 m on figure 3d). Also the actual bore seems to be curved : its recession is faster along the walls due to lower velocities of the near-wall supercritical flow at the toe of the bore. One can also observe that the numerical bore is slightly delayed comparing to the actual one. The probable explanation is to be found in the overestimation of the upstream supercritical flow in the numerical model since the wall friction is not explicitly taken into consideration. The recession of the bore against this overestimated flow is thus artificially delayed.

In the bend, the experimental measurements clearly show that the flow attempts to smoothen the high curvature imposed by the sharp bend. This is particularly clear on figure 3d : the outer corner and the near-wall region just downstream from the inner corner do not participate actively to the flow. Moreover, the surface velocity measurements even allow to identify a slow re-circulation in both regions.

Looking at the outer corner of the bend (see insets on figures 3b and 3c), a significantly different behaviour can be identified. Even if the slower velocities are globally well represented by the numerical model, the latter was unable to reproduce the re-circulation region observed experimentally. The underlying assumptions of the Saint-Venant shallow water equations are such that the water is considered as an ideal fluid, without internal friction excepted the bottom friction diffused vertically by the velocity distribution. This assumption has proven to be adequate in most cases. However, the here-presented experiments clearly illustrate the consequences of such an assumption. The re-circulation region in figure 3b is due to the presence of non-negligible horizontal shear stresses in vertical planes, due to turbulent exchanges of momentum. The numerical model is based on the Saint-Venant equations, which do not contain such terms in their common formulation. The model thus only reproduces the typical ideal-fluid behaviour in a corner.

6 CONCLUSIONS

Additional experimental work on a dam-break flow through a 90°-bend has been presented. Thanks to a recently developed Voronoï digital-imaging technique, an accurate surface velocity field could be measured. The technique, initially aimed at studying intense granular flows, has proven to be of high interest for the characterisation of severe transient hydrodynamic flows where particles are used only as tracers.

The measurements, compared to a numerical simulation, allowed to gain interesting information both about the flow behaviour and the implications of the assumptions made in the numerical model.

Work is still under progress to gather the maximum information from the experiments. Yet it can already be considered that such new measurements are of great value and give attractive perspectives in further validation and improvement of flow prediction models.

Capart H., Young D.L.; Zech Y. (2001), Voronoï imaging methods for the measurement of granular flows, accepted for publication in “Experiments in Fluids”.

Glaister P. (1988), Approximate Riemann solutions of the shallow water equations, J. Hydr. Research, Vol. 26(3), pp. 293-306.

Soares Frazão S., Zech Y. (1999), Effects of a sharp bend on dam-break flow, Proceedings 28th IAHR Congress, Graz, Austria, published on CD-ROM.

Soares Frazão S., Sillen X., Zech Y. (1999), Dam-break Flow through Sharp Bends - Physical Model and 2D Boltzmann Model Validation, Proceedings of the CADAM meeting Wallingford, United Kingdom, 2 and 3 March 1998, European Comission, Brussels, pp 151-169.