EFFECTS OF A SHARP BEND ON DAM-BREAK FLOW

 

S. SOARES FRAZÃO and Y. ZECH

 

Civil Engineering Department, Université catholique de Louvain

Vinci - Place du Levant 1, B-1348 Louvain-la-Neuve

Tel : +32-10-47 21 24, Fax : +32-10-47 21 79, soares@gc.ucl.ac.be

 

 

abstract

Forecasting the exact velocity and thus the arrival time of a dam-break wave is a key issue of risk assessment, and quite difficult to achieve especially in natural valleys with complex geometry. In this paper, two dam-break experiments are presented, whereby the effects of a sharp bend on the dam-break flow were studied. The laboratory experiments consisted of a dam break in a channel with a respectively 90° an 45° bend and were carried out in the Civil Engineering Department Laboratory of the Université catholique de Louvain (UCL, Belgium). The measured flow is compared with a numerical simulation by a Boltzmann finite-volume model, and good agreement is obtained in both the 90° and 45° bend cases. It is then shown how the bend influences the flow, as well upstream and downstream, by means of the dam-break front characteristic. Upstream, the formation of a bore travelling back to the reservoir leads to an important rise of the water level. Downstream, the local slowing of the wave clearly appears, as well as the dependence upon the sharpness of the bend.

 

Keywords: dam break, bend, physical model, hydraulic jump, bore

 

Introduction

Natural rivers mostly have an irregular shape, the cross section varies along the flow-path and the flow-path itself shows a lot of bends and curves. When a dam breaks and the water storms in such a river, the shape of the valley has a great influence on the flow and on the arrival time of the wave. Especially, bores can form in the bends, mostly in the sharp ones, they can travel upward, increasing the water level, but also, they will affect the front propagation downstream. Those effects are studied here by means of laboratory experiments and numerical simulations.

 

Experimental facilities

In order to produce a set of reliable data to validate numerical schemes, several experiments were carried out in the Civil Engineering Laboratory of the Université catholique de Louvain (UCL, Belgium). The channel used has a rectangular cross section of 0,5m width and is 7,6m long. It was build in such a way that it can be separated in two parts, which allows to change the straight shape by inserting for example a bend element between the upstream and downstream reaches. The upstream reservoir has dimensions of 2,44m × 2,39m and the dam is represented by a lift gate. The bottom level of the channel is 33cm higher than the bottom level of the reservoir, which means that there is a step at the entrance of the channel (Fig.1c). To simulate a dam break, the gate is pulled up rapidly so that the failure can been considered as instantaneous.

 

 

Fig.1: Plane view of a) 90° bend and b) 45° bend, and c) profile of the entrance (dimensions in cm) - origin of the x-y axes in the lower left corner

 

The aim of the experiments was to validate a wide range of numerical schemes on 2-D problems, in the frame of the European Concerted Action on Dam-Break Modelling (CADAM), so two different bend elements were used : a 90° bend and a 45° bend (see Figs.1a and 1b)

The channel is equipped with a set of measurement devices, located at a fixed number of gauging points (see Fig. 1 for their location). During the experiment, records of the water level are made at a frequency of 10 Hz. Snapshots of the flow are also taken, as a more qualitative mean of validation to compare with the shape of the water surface calculated by the numerical models.

The following Manning friction coefficients were calculated : 0,0095 s.m^-1/3 for the bottom friction (steel plates) and 0,0195 s.m^-1/3 for the wall friction (glass plates). The rather high wall roughness is due to the joints between the glass plates.

 

Dam break in a channel with a 90° bend (L-Shaped channel)

The shape of the channel is shown in Fig.1a. The initial water level in the upstream reservoir is 20 cm above the channel bed level. The channel downstream is dry in a first set of measurements and wet (1 cm initial water depth) in a second one.

When the gate is opened, the water flows rapidly into the channel and reaches the bend after approximately 3 seconds. There, the water reflects against the wall, a bore forms and begins to travel in the upstream direction, back to the reservoir. For the water flowing further downstream after the bend, multiple reflections on the walls can be observed. In the upstream reach of the channel, the flow is mainly 1D, while it is clearly 2D in the downstream reach.

Due to the contraction at the entrance of the channel, there are some local 2D features near the gate, but ±70cm downstream from the gate, these effects vanish. In the upstream reservoir, non symmetric features can also be observed, as a consequence of the non centred position of the gate : the circular emptying front does not reach all the walls at the same time, producing asymmetric reflections and oscillations. However, this does not seem to have a great influence on the flow in the channel itself, especially during the first seconds when the strongest phenomenon consists of the dam break wave.

After 15 s, the receding bore formed by the reflection in the bend reaches the reservoir and disappears. The flow then becomes almost permanent, i.e. the general shape of the water surface is preserved but the water level decreases progressively.

The channel was equipped with six gauges (see Table 1) recording the time evolution of the water level. Several measurements campaigns showed a very good reproducibility in both the dry and wet bed cases.

 

Dam break in a channel with a 45° bend

The shape of the channel is shown in Fig. 1b. The initial water level in the upstream reservoir is 25 cm. The channel downstream is dry in a first set of measurements and wet (1 cm) in a second one.

When the gate is suddenly opened, the water flows rapidly into the channel like in the previous case and reaches the bend after approximately 3s. There, the water reflects against the wall, but the reflection is not so strong as in the 90° case. The bore first forms parallel to the oblique wall after the bend, then becomes perpendicular to the upstream reach axis and begins to travel back to the reservoir. Again, multiple left and right reflections occur in the downstream reach, leading to a 2-D flow downstream from the bend.

The bore takes 20 s to reach the reservoir and disappear, which is longer than in the previous case. The absolute velocity of the bore is a = c - U where c is the relative celerity and U the base-flow velocity. The main reason for the smaller absolute velocity of the bore lies in the origin of the bore itself which is due to a partial reflection against an oblique wall leading to a weaker front than in the 90° case. Another reason is that in this case, the water level initially assigned in the reservoir is higher than in the 90° case, which leads to a greater value of U and a slower bore.

The channel was equipped with nine gauges (see Fig. 1) recording the time evolution of the water level. Again, several measurements campaigns showed a very good reproducibility in both the dry and wet bed cases.

 

Comparison with experimental results

In this section, a comparison between a Boltzmann numerical model results and the measured data is presented. The Boltzmann model is an explicit finite volume integration of the shallow equations (Sillen, to be published, and Soares et al., 1998). The numerical fluxes between the cells are calculated by means of a mathematical analogy with the gas dynamics Boltzmann equation. This leads to an upwinding of the fluxes, in way similar to the propagation of information within the flow described by the characterisics.

To compare with the 90° bend, the Boltzmann model is computed on a rectangular mesh, and second-order accuracy in space is achieved using the MUSCL approach (Van Leer, 1974). The 45° bend case was computed on an unstructured triangular mesh, allowing to represent the shape of the channel exactly. The latter scheme is only first order accurate.

 

L-shaped channel (90° bend)

The case chosen for comparison consists of a dam break simulation with an initial water level in the upstream reservoir of 20 cm. The bed of the channel is initially dry and the downstream boundary condition is an open boundary. The time evolution of the water level at the six gauging points is shown on Fig. 2.

 

a) G1	b) G2
c) G3	d) G4
e) G5	f) G6

 

Fig. 2 : Time evolution of the water level at the six gauging points, (----) experiments and (---) Boltzmann model

 

Gauge G1 (Fig. 2a) represents the emptying curve of the reservoir. The water level evolution is correct, which means that the numerical model computes the right discharge coming into the channel. On Figs. 2d, 2c and 2b we can see the arrival times of the receding reflected front (abrupt increase of the water level) at gauge G4, G3 and G2 respectively. After the bore has reached these gauges, the time evolution of the water level is similar to the one in the reservoir, and represents the progressive emptying of the system. The main differences between the numerical model results and the measurements are in the arrival times of the bore : too early at G2 and too late at G3 and G4, the maximum difference being of 1 s.

 

Fig. 3 : Water level at gauge 2, (----) experiments and (---) Boltzmann model

 

A sensitivity analysis showed that the bore velocity is highly dependent upon the friction coefficient introduced in the numerical model. This appears clearly on Fig. 3 where three computations with different global friction coefficients were run. A higher friction coefficient makes the water travel slower downward and thus increases the absolute bore velocity a = c - U. Here, a @ 0,33 m/s, and as it is small, a little error on the velocity a results in great differences in the time needed for the bore to travel from the bend back to the reservoir.

 

Channel with 45° bend

The case chosen for comparison consists of a dam break simulation with an initial water level in the upstream reservoir of 25 cm above the flume bottom. The bed of the channel is initially dry and the downstream boundary condition is a chute. The time evolution of the water level at the nine gauging points is shown on Fig. 4.

a) G1	b) G2
c) G3	d) G4
e) G5	f) G6
g) G7	h) G8
	Fig.4 : time evolution of the water level at the nine gauging points, (----) experiments and (---) Boltzmann model
i) G9

 

The conclusions for gauges G1 to G4 are the same as in the previous case. The gauges located in the bend are G5 (outer gauge), G6 (centreline gauge) and G7 (inner gauge). Their time evolution is similar, but the water level is always higher at G5 than at G7. After the shock against the wall (near G5) and the formation of the bore, the flow can be considered as almost steady.

 

propagation of the dam-break wave

The front velocity is a good indicator to point out the effects of those sharp bends on the dam-break wave. The front characteristics are plotted on Figure 5 in the x-t plane, for the following cases : a straight channel, the channel with a 90° bend and the channel with a 45° bend. Even in the straight channel case, the velocity slowly decreases due to the friction.

The experimental values are taken from the gauges measurements. The numerical model reproduces quite well the measured arrival times, what could already be observed in the previous comparison plots.

 

Fig. 5 : front characteristics

 

Clearly, the effect of the bend is to slow down the front locally. After passing the bend, the velocity increases, but does not reach its initial value. Energy losses occur in the bend, and their importance depends on the sharpness of the bend, as can be seen when comparing the front characteristics for the 90° and 45° cases. It should also be noted that, even in the straight channel case, the front characteristic shows a continuous slowing of the front, due to the bottom friction.

 

CONCLUSIONS

Two dam-break experiments were presented, and the measurements were compared to numerical simulations by a Boltzmann finite-volume model. Despite the quite simple geometry of the channel compared to real valleys, it was possible to point out the effects of a sharp bend on dam-break flows. Especially, they showed how the front is slowed down by the bend, and that the sharper the bend is, the more important its effects are. However, if this slowing and the associated energy losses are positive effects of the bend on the dam-break wave, the important rise in water level due to the upward travelling bore will enlarge the flooded area in the upper reach.

 

acknowledgements

The authors wish to thank X. Sillen for his contribution to the mathematical modelling, D. Bousmar and T. de Béthune for their help in doing the measurements.

 

rEfErences

Sillen X., (to be published in 1998), Simulation numérique et physique d'écoulements transitoires extrêmes en hydraulique à surface libre, PhD thesis, Université Catholique de Louvain, Belgium

Soares Frazão S., Sillen X., Zech Y. (1998), Dam-break Flow through Sharp Bends - Physical Model and 2D Boltzmann Model Validation, to be published in CADAM Proceedings Wallingford Meeting - 2nd / 3rd March 1998-08-03

Van Leer B. (1974), Towards the ultimate conservation difference scheme II, monotonicity and conservation combined in a second order scheme, Journal of Computational Physics, Vol. 14, 361-370