1. Simple Uniform Bores (no obstacle)

We examine the bores generated in the present experimental setup with no obstacle placed in the tank. The initial water depth is kept constant at h0 = 20 mm, while the impoundment behind the gate is varied from 100 to 300 mm. Analytical predictions for the generated bores are made based on fully nonlinear shallow-water theory by the method of characteristics, and the results are shown in Table 1 with the definition sketch in Fig. 1.1. Figure 1.2 shows the temporal variations of the water-surface profiles at x = 5.2 m measured by a wave gage. As expected, the time of arrival, and therefore the averaged propagation speed c is in good agreement with the predictions for all cases. The agreement in water level is also good; however, the measured data show a gradual increase in depth, whereas the theory indicates a constant depth behind the bore front. This discrepancy emerges when the impoundment h1 ≥ 150 mm and becomes prominent as the bore strength increases. The timing of the negative wave reflected from the upstream end wall (marked with * in the figures) is also predicted well by the method of characteristics.

 

Table 1. Bore heights h2, propagation speeds c, flow velocities u2 and arrival times at x = 5.2 m of the bore t0 and any effects of the back wall t* for different impoundments h1. Froude number in the moving coordinates, F0, and in the laboratory coordinates, F2.

h1

(mm)

h0

(mm)

h2

(mm)

c

(m/s)

u2

(m/s)

t0

(sec)

t*

(sec)

 

 

 

 

 

 

 

 

 

 

 

100

20

50.8

0.94

0.57

5.54

14.15

2.12

0.81

125

20

58.5

1.06

0.70

4.90

12.54

2.39

0.92

150

20

65.8

1.18

0.82

4.42

11.39

2.66

1.02

175

20

72.7

1.29

0.93

4.05

10.51

2.91

1.10

200

20

79.2

1.39

1.04

3.74

9.82

3.14

1.18

225

20

85.5

1.49

1.14

3.50

9.26

3.36

1.25

250

20

91.5

1.58

1.24

3.29

8.79

3.57

1.31

275

20

97.4

1.67

1.33

3.11

8.39

3.77

1.36

300

20

103.0

1.76

1.42

2.95

8.05

3.97

1.41

 

Figure 1.1. Definition Sketch

 

Figure 1.3 shows how the predicted velocity histories at x = 5.2 m compare with the measurements by LDV and DPIV. In each figure, the dots represent DPIV and the solid line LDV measurements. The straight solid line segments are the predictions based on the method of characteristics. As in the water level comparisons, the arrival of the negative wave reflected from the upstream end wall is marked with an asterisk (*). Note that the DPIV measurements were made at z = 35 mm from the bottom, except for the cases of h1 = 250 mm and 300 mm; for those two cases, the measurements were made at z = 15, 35, and 75 mm, and the depth averaged velocities are computed in addition to the span-wise average over the DPIV view area (160 x 160 mm) at x = 5.2 m. The PIV measurements were made at the center of the tank y = 0, x = 5.2 m, and the multiple elevation points from the tank floor: the maximum of 9 points for the cases of h1 = 250 and 300 mm to the minimum of 3 points for the case of h1 = 100 mm (see Experimental Data c.1). The flow velocities were computed by depth averaging the multiple LDV data. 

Just as with the water level, slight discrepancies result as the bore strength increases, h1 ≥ 150 mm: the velocity decreases during the passing of the bore, and the calculations overestimate the velocity. Note that the measured flow velocity does not exceed the predicted value near the immediate passage of the bore front – in fact the agreement is good near the bore front. On the other hand, the water-surface elevation is lower than the prediction near the front, then gradually increases and eventually exceeds the prediction. The product of the depth averaged velocity and the depth (i.e. the discharge rate) slightly increases during the passing of the bore: the maximum of 9% increase for the case of h1 = 250 mm.

The discrepancies observed in the cases of the larger bores (h1 150 mm) might be caused by several factors. The assumptions of the shallow-water theory are those of 1) hydrostatic pressure field, 2) incompressible fluid, and 3) uniform velocity over the depth, consequently no viscous effect. It is noted that the condition h1 ≥ 150 mm implies the bore becomes supercritical, i.e. F2 > 1.0 as indicated in Table 1. As the bore becomes larger, these assumptions start to break down near the bore front. Another possibility is that the removal of the gate is not instantaneous but takes a finite time, although the timing is repeatable.

Recommendation: perform numerical simulations with a 3-D turbulence model for a simple dam-break problem with F2 > 1.0.

 

a)

 

b)

 

c)

 

d)

 

e)

 

f)

 

g)

 

h)

 

i)

Figure 1.2.  Comparison of measured water surface level with the theory for a) h1 = 100 mm; b) 125 mm; c) 150 mm; d) 175 mm; e) 200 mm; f) 225 mm; g) 250 mm; h) 275 mm; i) 300 mm.

 

a)

 

b)

 

c)

 

d)

 

e)

 

f)

 

g)

 

h)

 

i)

Figure 1.3. Comparison of measured velocity with theory for for a) h1 = 100 mm; b) 125 mm; c) 150 mm; d) 175 mm; e) 200 mm; f) 225 mm; g) 250 mm; h) 275 mm; i) 300 mm.