DAM-BREAK WAVES OVER MOVABLE BED CHANNELS

Abstract: His paper presents the results of one experimental study focused on dam-break wave propagation over movable beds. Tests consisted in the sudden opening of a vertical lift-gate, which separated the initial water and sediments levels upstream and downstream of the gate. They allowed the simulation of the following initial conditions: with or without initial bed step at the gate cross-section; with or without water downstream of the gate; with or without sediments downstream of the gate. The results obtained are used to discuss the influence of the movable bed on the celerity of the wave fronts upstream and downstream of the gate, as well as on the depth of the downstream wave front; the total volume of dislodged sediments is also assessed.

1 INTRODUCTION

During the last century, the propagation of dam-break flood waves has been the object of intense scientific and technical activity. This problem was initially approached by finding analytical solutions for the shallow-water equations in schematic situations featuring fixed bed and nil flow resistance. An example of this approach is found in Stoker (1957), pp. 333-341.

The vertiginous increase in the potentiality of automatic calculus, which occurred during the last decades, has made it possible to achieve numerical solutions for more realistic situations, that include the flow resistance (see, e.g., the studies by Alcrudo (1992) or Franco (1996)). Sometimes these numerical studies have been accompanied by experimental work, being the objective to obtain data for the validation of the developed models.

Recently, Lauber and Hager (1998) carried out a study highlighting the potentialities of the experimental approach to comprehend the phenomena involved in dam-break flood waves, namely in the determination of the variables such as the water height of the downstream wave front or the celerity of the upstream and downstream wave fronts.

The studies mentioned above refer to fixed beds. The study of movable beds is rather complex, and has only recently been considered. The studies carried out by Capart and Young (1998) and Leal (1999) are particularly relevant on the topic. Capart and Young used a lightweight granular bed material with uniform grain while Leal used natural sand. It should be noted that the phenomena involved are, for these two studies, substantially different. The first case deals with a debris flow whilst the second deals with a bed-load dominated flow.

This paper is based on the study of Leal (1999), in which the test results regarding the propagation of upstream and downstream wave fronts are presented and discussed as well as the total volume of dislodged sediments.

2 DESCRIPTON OF THE INSTALLATION AND EXPERIMENTAL PROCEDURES

The study was carried out in a rectangular flume whose lateral walls are made of glass and whose floor is metallic. The flume is horizontal, 0.70m wide, 1.00 m high and 20.00 m long. In order to be able and simulate a dam, a vertical PVC lift-gate was installed 8 m from the flume upstream end. The lift-gate, which is 60 cm high, slides on two Perspex runs glued to the flume walls and is opened by means of a metallic counter-weight structure which was specially dimensioned for this effect, and two pulleys (Fig. 1). The mechanism is set in motion manually by pulling the cable.

When the gate is opened, the difference of level of the water surface in the gate cross-section induces a wave that breaks immediately, causing a downstream propagating bore. As the flume is interrupted by a vertical wall, the downstream wave tends to be reflected upstream, altering the configuration of previously modelled bed. With the objective of minimising this alteration, a sloping layer of gravel was laid at the flume downstream end.

The duration of each test, defined as the time taken for the downstream wave to reach the wall at the end of the flume, is around 7.0 s. A set of video-cameras was used to record water and bed level evolution. Actual measurements were obtained by direct video inspection.

Concerning the bed type, two groups of tests were carried out: with a fixed bed and with a movable bed. The movable bed was made of sand with a grain size distribution defined by the median grain size, D₅₀ = 1 mm, and by the gradation coefficient, s_D = 1.7.

The test procedures include the following stages: (1) for the movable bed tests, sand was placed both upstream and downstream of the lift-gate, at a height of hs_u and hs_d, respectively; (2) for all the tests, water was introduced both upstream and downstream of the lift-gate at a height of h_u and h_d, respectively; (3) commencement of video images recording; (4) opening of the lift gate.

3 RESULTS AND DISCUSSION

3.1 Characterisation of the tests

Three tests with fixed bed and twenty-two with movable bed were carried out. Their main features are presented in Table 1. In this table, a is the ratio h_d/h_u and t_omax = (2h_u/g)^1/2 is the maximum time for which the raising of the lift-gate is considered instantaneous, according to Lauber and Hager (1998). The gate opening-times achieved for the lift mechanism varied between 0.1 s and 0.2 s. Therefore, observing the values of t_omax in Table 1, the raising of the gate can be considered instantaneous.

FIXED BED
Test	h_u [m]		h_d [m]		a = h_d/h_u [-]	t_omax = (2h_u/g)^1/2 [s]
T.1	0.30		0.00		0.00	0.25
T.2	0.49		0.03		0.06	0.31
T.3	0.49		0.21		0.43	0.32
MOVABLE BED
Test	h_u [m]	h_d [m]		hs_u [m]	hs_d [m]	t_omax = (2h_u/g)^1/2 [s]
Ts.1		0.00		0.10	0.10
Ts.2	0.40	0.05		0.10	0.10	0.29
Ts.3		0.10		0.10	0.10
Ts.4		0.20		0.10	0.10
Ts.5		0.00		0.15	0.05
Ts.6	0.35	0.05		0.15	0.05	0.27
Ts.7		0.10		0.15	0.05
Ts.8		0.00		0.10	0.05
Ts.9	0.40	0.05		0.10	0.05	0.29
Ts.10		0.10		0.10	0.05
Ts.11		0.00		0.15	0.00
Ts.12	0.35	0.05		0.15	0.00	0.27
Ts.13		0.10		0.15	0.00
Ts.14		0.00		0.10	0.00
Ts.15	0.40	0.10		0.10	0.00	0.29
Ts.16		0.05		0.10	0.00
Ts.17		0.00		0.05	0.00
Ts.18		0.05		0.05	0.00
Ts.19	0.45	0.10		0.05	0.00	0.30
Ts.20		0.00		0.05	0.05
Ts.21		0.05		0.05	0.05
Ts.22		0.10		0.05	0.05

The objective of test T.1 was to obtain data regarding an extreme situation (nil water depth downstream). Tests T.2 and T.3 were performed to obtain data regarding, respectively, the situation in which the flow in the lift-gate cross-section is critical (a £ 0.1384, according to the analytic solution (Stoker (1957), pages. 338-339)) and the situation in which the flow in the same section is subcritical (a > 0.1384). The movable bed tests simulated the six types of initial situations defined in Table 2.

3.2 Upstream wave front celerity

According to the analytical solution (Stoker (1957), page 340) for fixed downstream dry bed, the celerity of the upstream wave front (UF subscript), defined in the most upstream section where the water depth remains unaltered, is

. The corresponding non-dimensional parameter

takes the value of –1.

In Figure 1, the variation of the parameter

with the parameter

and the relation a = h_ed/h_u, is presented for all the tests carried out. In these parameters, t is the time for which the upstream wave propagates the distance x_UF, measured from the lift-gate cross-section, and h_ed is the effective initial water depth downstream the gate as defined in Figure 2. Figure 1 illustrates that, for a < 0.1384,

, i.e., the upstream wave front has a celerity

, which is not affected by the variation of h_d. The celerity obtained in the tests is, in absolute value, greater than the celerity of the analytical solution, in which

. Lauber and Hager (1998) have already referred to this result for fixed bed situations. The test results presented in Figure 1 for a < 0.1384 cover tests with both a fixed bed and a movable bed. From these tests it may be concluded that, for the type of sediment used, the movable bed does not effect the upstream wave front propagation.

Fig. 2 Initial conditions: (a) without bed step;
(b) with bed step superior to the downstream water depth;
(c) with bed step inferior to the downstream water depth.

Figure 3 presents the variation of the parameter

with a,

being calculated by regression over the test data obtained from all the tests. It may be concluded that for a £ 0.1384, the dimensional value of celerity,

, is constant and equal to

. From this value,

appears to approach –1 when a approach 1. These results are coherent since, for a £ 0.1384, the flow at the lift-gate cross-section is critical and, therefore, it is not influenced by the effective initial water depth downstream, h_ed. Similarly, for a > 0.1384, the flow at that cross-section is no longer critical, becoming subcritical throughout the entire flume and depending on the downstream initial water depth.

3.3 Downstream wave front celerity

In the study of the downstream wave front propagation, the ratio a = h_d/ h_u was determined measuring h_d from the existing bed downstream of the lift-gate and not upstream, rendering the previous definition of a invalid. This option is justified by the fact that the existence of water downstream influences the downstream wave front propagation, even when its level is inferior to that of the movable bed upstream.

According to the analytical solution for a downstream dry bed, the celerity of the downstream wave front (DF subscript) is

(see Stoker (1957), page 340), i.e.,

Figure 4 presents, for the fixed bed tests and for the tests with a movable bed, and no initial bed step at the lift-gate cross-section, the variation of

with

, in which x_DF is the downstream wave front position measured from the lift-gate, at the instant t. Analysing Figure 4a) for a = 0, it is concluded that the downstream wave front propagates at a celerity which is smaller than the theoretical celerity found in the analytical solution. This is contrary to what happens with the upstream wave front, where the observed absolute value of celerity is bigger than the theoretical value. It is noted that when water exists downstream(a > 0), the celerity obtained experimentally is bigger than that obtained theoretically, approximating the theoretical value for a » 0.43. In fact, in test T.3 (a » 0.43), the difference between the observed and the computed wave front celerities is less pronounced than in the other cases (a = 0; a = 0.06). It is also noted that, for both values of a smaller than 0.1384, the wave front celerity is practically the same, i.e., the celerity is not influenced by the value h_d. From the analysis of Figure 4b) it is concluded that an increase in the initial water depth downstream from the lift gate induces the decrease of the downstream wave celerity even for a < 0.1384. It is illustrated that, according with the fixed bed results (Figure 4a)), the measured downstream wave front celerity is smaller than the dry bed analytical solution.

Figure 5 presents the variation

with a,

resulting from regression of the experimental data presented in Figure 4. It is shown that the downstream wave front celerity is the highest in the case of fixed bed, for a < »0.5. Besides this, regardless of the type of bed, the celerity also appears to decrease lineally with a, for a < »0.3, appearing to approach the unit asymptotically, further on. In fact, for a = 1,

should be equal to 1.0. The fact that the downstream wave front celerity is smaller for a movable bed could be explained by the increase in roughness due to the presence of sediments. Increasing a reduces this effect, since the relative roughness also diminishes.

Fig. 4 Variation of

with

: (a) tests with fixed bed;
(b) tests with movable bed and without initial bed step.

Fig. 5 Variation of

with a, tests with fixed bed and tests
with movable bed and without initial bed step.

Figure 6 present the variation of

with

for the tests with initial bed step at the lift-gate cross-section. This bed step is equal to 0.05 m, 0.10 m and 0.15 m respectively. From the analysis of Figure 6, it is verified that, even though an initial bed step exists, the downstream wave front celerity continues to be influenced by a.

Fig. 6 Variation of

with

for tests with movable bed:
(a) initial bed step of 0.05 m; (b) initial bed step of 0.10 m; (c) initial bed step of 0.15 m.

Observing Figure 7, where the values of

obtained by regression over the test data are presented, it is also concluded that for a given a the wave front celerity increases with the increase in the initial bed step. This occurs especially for DZ_b = (hs_u - hs_d)/h_u greater than approximately 0.25. As for Figure 5, Figure 7 allows to conclude that, for a values smaller than approximately 0.3, the downstream wave front celerity decreases linearly with a, whether or not, there exists an initial bed step, regardless of its amplitude.

4 CONCLUSIONS

In accordance with the analysis presented above and within the range of tests carried out, the following conclusions were reached:

(1) there is no influence of the movable bed on the upstream wave front celerity;

(2) the downstream wave front propagation is influenced by the presence of sediments, especially when an initial bed step exists at the lift-gate cross-section;

(3) the presence of water downstream of the lift-gate only alters the upstream wave front propagation when the relation a is greater than 0.1384, while the downstream wave propagation is influenced by h_d for any value of a;

(4) the upstream wave front celerity obtained experimentally is, in absolute value, greater than that obtained analytically;

(5) the downstream wave front celerity obtained experimentally is, with the exception of a values close to zero, greater than that obtained analytically, approximating the analytical result when a » 0.43.

This study was partly financed by projects PRAXIS/3/3.2/CEG/32/94 and PRAXIS/C/ECM/12040/98 from the Fundação para a Ciência e Tecnologia (Foundation for Science and Technology), an institution we would like to thank for the referred financing.

Capart, H.; Young, D. L. (1998) – Formation of a jump by the dam-break wave over a granular bed. Journal of Fluid Mechanics, Vol. 372, pp. 165-187.

Franco, A. B. (1996) – Modelação computacional e experimental de escoamentos provocados por rotura de barragens. PhD. Thesis. Universidade Técnica de Lisboa, Instituto Superior Técnico, Lisbon.

Lauber, G.; Hager, W. H. (1998) – Experiments to dambreak wave: horizontal channel. Journal of Hydraulic Research, Vol. 36, Nº 3.

Leal, J. B. (1999) – Modelação matemática da propagação de ondas de cheia provocadas por ruptura de barragens em canais de leito móvel. MSc. Thesis. Universidade Técnica de Lisboa, Instituto Superior Técnico, Lisbon.

Stoker, J. J. (1957) – Water Waves. The mathematical theory with applications. Wiley-Interscience Publication (reprinted in 1992).