Hager (1987), Mazumder and Naresh (1988), Nettleton and McCorquodale (1983, 1989), .... Based on the study of Nettleton and McCorquodale (1989), the most.
Hydraulic Performance of Rectangular and Radial Stilling Basins Abdel-Azim M. Negm
Abstract Both rectangular and radial stilling basins are used to dissipate the excessive energy of flow downstream of hydraulic structures. Many studies compared the performances of both basins regarding the hydraulic jump characteristics with or without appurtenances. They concluded that the radial basin is more efficient than the rectangular one. The radial basin reduces the length of jump significantly and dissipates more energy compared to the rectangular basin. These studies are reviewed in this paper and the main differences between the two basins are highlighted. Up to the knowledge of the author, no studies have been published on the comparison between the performance of the control gate in each basin regarding the discharge characteristics. This study presents the results of large series of experiments conducted on a laboratory flume utilizing rectangular and radial basins with or without under-gate sill.. In case of radial stilling basins, the effects of different expansion ratios are discussed and the optimal expansion ratio that improves the flow behavior is specified. Prediction equations are presented for each basin in terms of the controlling variables such as the under-gate Froude number, the ratio of the differential head on the gate to the gate opening and the submergence ratio. Both free and submerged flow are considered. Keywords: Hydraulics, stilling basin, discharge, gate, hydraulic jump, radial flow, rectangular flow.
1 Introduction In order to insure the safety of the hydraulic structure, stilling basins are provided at the downstream side. Basins may be prismatic (rectangular) or non-prismatic (diverging, radial, abruptly expanded, ..etc.). Extensive review on hydraulic jumps and stilling basins can be found in, e.g., Hager (1992). Usually, the performance of the stilling basins is measured by analyzing the characteristics of the hydraulic jump formed in the basin. The overall performance of the stilling basin can be determined by comparing the characteristics of the radial hydraulic jump and the discharge capacity of the control gate with the characteristics of the classical hydraulic jump and the discharge characteristics of the gate in rectangular basin without any appurtenances.
2 Hydraulic Jump Characteristics The following equations for the jump characteristics are considered as basis for the comparisons: According to Belanger (Chow 1959, Rajaratnam 1967), the sequent depth ratio is given: 1 Y* = 1 + 8F12 − 1 (1) 2
(
)
_______________________________ 1 Associate Professor, Dept. of Water and Water Structures Engineering, Faculty of Engineering, Zagazig University, Zagazig, Egypt.
Eq.(1) can be approximated for F1 >2.5 as follows, Hager (1992): 1 Y * = 2 F1 − 2 The efficiency of the classical jump is approximated as, Hager (1992):
(
η* = 1 − 2 F1−1
)
(2)
2
(3)
The length of roller for classical jump is given by the following expressions, Hager et al. (1990): L*r F = −12 + 160 tanh 1 for y1 / b < 0.1 (4) y1 20 L*r F = −12 + 100 tanh 1 for < 0.1 y1 / b < 0.7 (5) y1 12.5 Equations (4) and (5) can be approximated as follows, Hager (1992) L*r = −12 + 8F1 for y1 / b < 0.7 and F1 < 6 (6) y1 The length of jump according to Hager (1992) based on Bradley and Peterka data (1957 ). L*j F = 220 tanh 1 (7) y1 22 The length of jump may be assumed as follows, Bradley and Peterka (1957): L j = 6 y*2 for 4 < F1 < 12 (8) Regarding the rectangular stilling basins with roughened beds, many studies are available on irregular and regular roughnesses. The first systematic experimental study on hydraulic jump under the effect of rough bed is due to Rajaratnam (1968). The equivalent relative roughness height varied between 0.02 to 0.43 while the supercritical Froude number, F1 ranged from 3 to 10. It was concluded that the depth ratio is function of both the supercritical Froude number and the relative roughness height. Also, the length of jump over rough bed is roughly about half that of the smooth one and the energy dissipation is increased. Later studies on rough bed include those of Leutheusser & Schiller (1975), Gill (1980) and Hughes and Flack (1983). Based on the studies conducted by Abdelsalam et al. (1985,1986, 1987), Hammad et al. (1988), Mohamed Ali (1991), Negm et al. (1993a), Alhamid & Negm (1996) and Negm et al. (1999a) on regularly roughened beds, the following roughness intensities were found to be the optimal from the hydraulics point of view: - 10% for roughness element with square sectional area arranged in staggered way. The following ratios parameters were found to be the most effective for the 10% roughness intensity, LR /s=28, Lb /y1 =4.5 and LR /y1 =15. - 12% for roughness elements with square sectional area arranged as strips in series transverse to the flow. - 13% for roughness elements with hexagonal sectional area arranged in staggered way. - 15% for roughness elements with hexagonal sectional area arranged as strips in series transverse to the flow. - 16% for roughness elements with circular sectional area arranged in staggered way. - 20% for roughness elements with circular sectional area arranged as strips in series transverse to the flow.
-
40% for roughness elements with semi-circular sectional area arranged as strips in series transverse to the flow.
The latest findings regarding the regular roughness with square cross section and height hb and arranged in staggered way was due to Alhamed (1994a,b) It was found that the optimal roughness intensity is about 12% for horizontal bed and about 11% for sloping bed based on experimental investigations. But these results are not confirmed by other studies. On the other hand, the hydraulic jump characteristics in gradually diverging channels are investigated by many authors, Riegel and Beebe (1917), Rubatta (1963), Arbhabhirama and Abella (1971), Khalifa and McCorquodale (1979), France (1981), McCorquodale (1986), Hager (1987), Mazumder and Naresh (1988), Nettleton and McCorquodale (1983, 1989), Wafaie (1992), Abdel-Aal (1995), Abdel-Aal et al. (1998), Rageh (1999) and Abdel-Aal (1999,2000). A review and evaluation of some of these studies can be found in Hager (1992). Hager (1992) differentiated between the studies dealing with the hydraulic jump characteristics and those dealing with the stilling basins configuration and specifications. The following equations can be adopted: Based on Rubatta (1963) data, the length of jump ratio in diverging channel is given by: θ θ = 9.5( F1 − 1) exp − 2.5 tan + 19 tan + 1.5 with θ < 22.6o (9) y1 2 2
Lj
According to Koloseus and Ahmad (1969), the length of roller is given by: Lr = 4.5 − 0.75( r − 1) (10) y2 where r is the radii ratio, r==r2 /r1 , r2 is the radius to the end of jump and r1 is the radius to the beginning of the jump. Based on Koloseus and Ahmad data, an approximated generalized equation for depth ratio, Y=y2/y1, is given, Hager (1992): Y = 2r −0 .38 F1 − 0.25( 3 − r )
(11)
According to Khalifa and McCorquodale (1979), the length of roller for radial jump is given by: Lr 4.2 = 4.75 − y2 F1
with θ = 13.5o , 1.2 ≤ r =
r2 ≤ 1.85, F1 ≤ 9.0 r1
(12)
According to Rageh (1999), the length of jump is given by: L j = 6hj − 4.9
with θ = 13.43o , β = 0.5, 2.5 ≤ F1 ≤ 6.5
(13)
Abdel-Aal (1995) investigated experimentally and theoretically the radial hydraulic jump on horizontal and sloping beds using an angle of divergence of θ=4.0934o and different expansion ratios. It was found that the optimal expansion ratio that reduces the length of jump
ratio and increases the energy loss ratio is about 0.5 ( β = 0 .477 ). Based on Abdel-Aal data, Negm et al. (2000a) provided the following equations for the radial jump characteristics at the optimal expansion ratio: r = 1.293 − 0.016 F1 y2 = −5.452 + 1.036 F1 + 4.584 r y1 Lj = −51.646 + 3.526 F1 + 41.599 r y1 EL = 0.228 + 0.4 Ln( F1 ) − 0.244r E1
2.8 ≤ F1 ≤ 7.4
(14)
2.8 ≤ F1 ≤ 7.4 , 1.0
