SEMI-EMPIRICAL SOLUTION OF ARCH MESH
BARRIER STILLING BASIN
Yang Min Lian Jijian
Civil Engineering School of Tianjin University, Tianjin 300072,
China
E-mail:ljj£Àpublic.tjuc.com.cn
Tel. +86-22-27406390
Abstract: The arch mesh barrier is a stilling-basin appurtenance situated in
the stilling basin, which has good hydraulic characteristics. Several physical
hydraulic model investigations were conducted to evaluate the performance of the
arch mesh barrier. The results of the model test indicate that the application
of arch mesh barrier in the stilling basins can greatly decrease the scale of
the energy dissipator at the toe of the dam. Based on the experiments, the
resistance coefficient of the barrier was studied and the momentum equation of
arch mesh barrier-stilling basin was solved. An empirical formula for calculates
the conjugate depths were given in this paper. The semi-empirical method of
designing an arch mesh barrier for hydraulic jumping stilling basin was
presented in this paper.
Keywords: arch mesh barrier, hydraulic characteristics, resistance
coefficient, conjugate depth
1 INTRODUCTION
The jumping stilling basin for energy dissipation of outlet
structures is widely used in the middle-lower head hydraulic engineering. The
damping ratio of hydraulic jump stilling basin varies with the Froude Number of
contracted section
. When
is less than 4.5, the damping
ratio is very low. Thus, engineers especially focus on the problem of energy
dissipation about lower Froude Number. Varies kinds of stilling-basin
appurtenance, such as Chute Blocks, Baffle Piers, Tail Sill, T type Piers and so
on, were used in the practical engineering. U. S. Department of the Interior
Bureau of Reclamation designed five kinds of stilling basins with stilling-basin
appurtenance, which are suitable to the outlet structures of different Froude
Number. But the problem of energy dissipation on lower Froude Number has not
been solved perfectly.
The arch mesh barrier established in stilling basin is suitable to
the lower Froude Number outlet. The flow passing through arch mesh barrier (or
semi-circular mesh barrier) has special hydraulic characteristics. In case of
proper condition, the forced hydraulic jump appears. Compared with the free
jump, the existence of arch mesh barrier makes the sequent depth of hydraulic
jump decrease. Because of the action of the arch, the velocity distribution of
the flow passing through the mesh is much different from that of free jump, and
the damping ratio of it is higher than that of free jump. Based on the hydraulic
model investigation, the resistance coefficient of arch mesh barrier was studied
and an empirical formula was given in the present paper. The momentum equation
of arch mesh barrier-stilling basin was solved by using resistance coefficient
and the method of designing an arch mesh barrier for hydraulic jumping stilling
basin was presented in this paper.
2 EXPERIMENTAL SETUP
The experiments were conducted in a glass flume, which is 12 m long and 0.4 m wide. A plate gate was established in the flume. The model was
made of organic glass. The maximum test discharge rate of the model was 45 L/s. The measurement of discharge Q is
performed by means of triangular weir and the determination of upstream and
downstream water level is performed by means of point gage.
Two different mesh barrier heights h and three different mesh barrier void ratios K were used in the tests. After optimizing void ratio K among 0.3, 0.4, and 0.5, K=0.4 was selected. Fig. 1 shows the
general layout of the model.
Fig.
1 General layout of the model
3 SOLVING PROCESS OF
MOMENTUM EQUATION
The momentum equation between contracted section
and tail water section can be written as
(1)
where
force per unit width of contracted section
(
);
force per unit width of tail water section
(
); R= resistance force per unit width
of barrier (
);
=unit weight of water=9800(
); q=discharge per unit width(
); g=acceleration of gravity(
);
= flow velocity at contracted section =
(
);
= flow velocity at tail water section =
(
);
=momentum correction.
According to Rajaratnam (1964), the resistance
force of barrier can be expressed as
(2)
where
resistance coefficient of barrier; h=height of barrier.
Let
, and substitute Eq.2 into Eq.1, the Eq.1 can be rewrite as
(3)
where
(4)
The solution of Eq.3 is
(5)
where
Froude Number of flow at the contracted section.
(6)
If the resistance coefficient
and the related height of barrier
were obtained, we can calculate
the conjugate depths of arch mesh barrier- stilling basin using Eq.5.
4 EXPERIMENTAL RESULTS
Experiment indicates that the related
height of barrier
varies with the Froude Number
. The empirical formula of
with
is
(7)
where
the base of natural logarithms = 2.7183.
The resistance coefficient of barrier
varies with the Froude Number of flow at contracted section
, the related height of barrier
and the related distance
. Where L=distance from contracted
section to barrier,
=the length of free jump=6.9(
,
= the conjugate depth of free jump. A dimensionless parameter is defined as
(8)
Table 1 lists the experimental results for various test
conditions. The relationship of
with
is plotted in Fig.2. Based on the
data of tests, an empirical formula is fitted as
(9)
The
Fig.3 illustrates the relationship of ratio of conjugate depths
with
. The squarer points indicate the experimental values
and the calculated values using Eq.5 indicated by the solid line. Because of
exist of the barrier, the flow is subjected to additional force and decreases
the requirement of the downstream water level and consequently the conjugate
depth of arch mesh barrier-stilling basin is smaller than that of free hydraulic
jump. The range of decreasing is about 30%. The length of the arch mesh
barrier-stilling basin (from contracted section to barrier) is about 50% that of
free jump. Although the distance from the contracted section to barrier L/Lj ranges form 0.4 to 0.6
conducted in this paper, L/Lj=0.5 is recommended. Because smaller value of L/Lj will result in worsen of
flow condition in front of the barrier. On the other hand, the value of L/Lj should not be more than
0.6, because the action of barrier will be weaken if the distance is larger.
Fig. 2 Dependence
of
on
Fig.
3 Theoretical
results compared with experimental
results
Table 1 Model test data
|
¡í
|
|
Resistance
Coefficient
(Cd)
|
Ratio of
Conjugate Depth
(¦Ç)
|
|
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
|
1.69
1.12
0.85
0.66
20.25
8.12
3.29
2.01
1.36
1.05
0.85
24.29
9.82
3.9
2.49
1.67
1.3
28.5
11.37
4.53
1.95
1.5
1.16
|
0.419
0.490
0.515
0.543
0.141
0.188
0.309
0.377
0.517
0.571
0.540
0.145
0.181
0.289
0.349
0.425
0.463
0.110
0.142
0.253
0.375
0.451
0.488
|
2.99
2.75
2.73
2.65
5.11
4.65
3.68
3.23
2.92
2.75
2.67
5.55
4.69
3.79
3.37
3.05
2.93
2.75
5.67
5.1
3.56
3.21
2.95
|
5 DESIGN METHOD
Under the circumstance of hydraulic
condition (such as flow Froude number at the contracted section
, discharge per unit width q) is already knows, one
can design an arch mesh barrier-stilling basin with Eq.5. The process of
designing as follow
Calculation of free hydraulic jump
Define the related distance L/Lj
The range of L/Lj is from 0.4 to 0.6, and L/Lj =0.5 is recommended.
Calculation of related height
The related height of barrier is calculated by
using Eq.7. This value calculated is the minimum height barrier and one should
use a little bigger value.
Calculation of resistance coefficient
The resistance coefficient is calculated by
using Eq.9.
Calculation of conjugate depth
The conjugate depth is calculated by using Eq.5
and Eq.6.
Note: The range of Froude Number
used in all above calculations is
from 2.5 to 5.5.
6 CONCLUSIONS
The arch mesh barrier is a new energy dissipator
suitable to lower Froude Number outlet engineering, especially used in canal
system structures. Model tests indicate that it has good hydraulic
characteristics. Compared with the general hydraulic jumping stilling basin, the
length of the stilling basin decreases about 50%, the depth decreases about 30%.
Eq.5, Eq.6, Eq.7, and Eq.9 can be used to design the arch mesh barrier- stilling
basin for small outlet engineering.
References
[1]
Rajaratnam, N., The Forced Hydraulic Jump, Part 1 and Part 2, Water Power, Vol.
16,No.1 and No.2, January and February, 1964.