HYDRAULIC RESEARCH ON SIDE-CHANNEL SPILLWAYS BASED ON
PHYSICAL MODELING AND OPTIMIZATION
MARIANA MARADJIEVA
BOGDAN KAZAKOV
University of Architecture, Civil Engineering and Geodesy
Sofia 1046 Bulgaria 1 “Chr. Smirnenski” blvd.
tel. +359 2 660 830 fax: + 359 2 963 1796
E mail: marmar_fhe@uacg.bg
Abstract: The hydraulic operation of side-channel spillways has been studied. The most
relevant hydraulic parameters are investigated by physical modeling and optimization
analysis. An effective measurement is proposed on the basis of variational principles and
numerical modeling. Two main cases have been considered: a) extremum problem that leads
to the minimum of lengthwise area of the channel; b) analysis of the maximum water
discharge overflowing above the crest of weir. The last case leads to the optimal design under
risk and emergency. Some specific examples for practical engineering needs have been solved
and formulas for velocity and water depth have been obtained for overflow by one, two or
three sides of the trough. Theoretical results are compared with experiments and some
standard engineering methods.
Keywords: side-channel spillway, weir, variational principle, functional, linear momentum,
Euler-Lagrange equation, differential equation, Reech-Froude criteria.
1. INTRODUCTION
Side-channel spillways are commonly used to release waterflow from a reservoir in places
where the sides are steep and have a considerable height above the dam. A trapezoidal crosssection is the most commonly used along the length of the channel ( figure 1).
a) Side-Trough profile
b) Side-Trough section
Fig. 1. One-sided collector
Besides, some variants of the above mentioned construction are considered – for instance
spillways constructed on two or on three sides of the trough. In this paper the first case as
shown in figure 1 is closely examined.
2. PHYSICAL MODELING
Experimental results were collected with a hydraulic model of the side-channel spillway for
releasing the peak overflow of dam Markieh, Syria. The total length of the trough is 150 m
with elliptical profile of ogee crest (a slight vacuum profile). The ogee crest and a downstream
chute slope of 2:1 are shown in figure 5. The general layout of the trough (figure 2) illustrates
the bottom width that increases from 10m to 30m.
A
67,50
150,00
1
63,00
63,00
64,50
63,00
10,00 1,75 1,00
2,1
spillway crest
63,00
63,00
64,50
64,50
64,50
drainage
30,00
axes of trough
I=3%
56,50
1
4,0
2,1
I=0,8%
n=0,014
10,00
A
Fig. 2. Scheme of non-prismatic trough spillway
The cross-section is trapezoidal (2:1) along the trough with a bottom slope ib = 3% . The total
depth (H) is 8 m at the end of the trough. Model-prototype similarity is performed according
to the Reech-Froude criteria (Carlier 1972):
V2
Fr =
(1)
where Frp = Frm
gh
and indices p, m denote the Froude number for the prototype and the model.
In practice the weir models are scaled with a Froude similarity and viscous scale effects must
be minimized. The model flow must be turbulent and the turbulence level should be the same
in the prototype and in the model. This condition is guaranteed by exchanging scale effects,
namely:
Re m ≥ Re lim
(2)
where Re lim is the recommended limit value of the Reynolds number Re .
Usually this number is Re lim = 1.10 4 ÷ 1,3.10 4 for side-channel spillways (Sliskii 1986). The
linear scale was determined by (2) for the discharge Q0.1% = 665m 3 s −1 as:
2/3
⎛ Re rp ⎞
⎟
M =⎜
(3)
⎜ Re ⎟
lim
⎝
⎠
where Re rp is a representative Reynolds number for a prototype determined for a typical
min
L
section. Having in mind the hydraulic structure of the side-channel the spillway crest could be
accepted as such section and then for M Lmin follows:
2/3
⎛ 4,03.10 6 ⎞
⎟⎟ = 54,7
M = ⎜⎜
4
⎠
⎝ 10
The model scale was selected to
M Lmin = 50 (1 : 50)
min
L
(4)
and with a resistance scaling:
λ p = λm and C p = C m
(5)
where λ and C are friction coefficient and Chezy’s velocity factor.
If n p is equal to 0,015, nm would be:
0,015
(6)
= 0,008
501 / 6
The surface of the channel was of very smooth cement mixture and painted with latex, so that
the coefficient of roughness for the model was approximately the same as that calculated by
equation (6).
nm =
3. OPTIMIZATION ANALYSIS
Consider a one-sided spillway and the co-ordinate system xy (figure 3).
xi
LSP
xi +1
+x
Δy
O
y
hi
hi+1
Δz '
z i'
+y
Δx
hn
i0
h
A = const
i
z i' +1
z'
O ' Datum
Fig. 3. Hydraulic scheme of the trough
The equation describing steady flow is (Kurganoff and Dupljak 1982):
⎛ αv 2 ⎞
(1 − θ / v ) QdQ
⎟⎟ + i f dx + α 0
dy = d ⎜⎜
gw 2
⎝ 2g ⎠
(7)
where α ,α 0 are kinetic and momentum energy coefficients; i f is the friction factor; h = p / γ
is the flow depth of section; v and Q are the velocity and discharge; θ is the velocity
component by the additional discharge in the flow direction and w is a cross-section area
(figure 3).
Equation (7) is considered after neglect of the variables i f and θ which are of small order.
Then this equation can be rewritten in the form:
⎛ αv 2 ⎞ α 0 v 2
⎟⎟ +
dy = d ⎜⎜
dQ
⎝ 2 g ⎠ gQ
The functional can be defined as:
(8)
x1
I [(v )] = ∫ H ( x )dx
x0
where
(9)
H ( x ) = h( x ) + y ( x ) = h ( x ) +
αv 2
2g
+
α 0 v 2 (s )q(s )
x1
g
∫
Q
x0
(10)
ds
Q( x )
(11)
b( x )v( x )
Further the following value will be accepted α ≈ α 0 . The variable y (x ) is shown in figure 3
and b(x ) is the width of the bottom having in mind a rectangular cross section that is
equivalent to the trapezoidal one (with the same discharge and a bottom width). The
approximate assumptions in (11) and (8) will be verify by experiments and by some wellknown methods. If a linear variation of water discharge is accepted, the following formulas
are valid:
Q( x ) = qx and q = Qsp / Lsp
(12)
h=
where Q(x ) is a linear function of a specific discharge q calculated from the maximum
discharge Qsp and the length of the trough Lsp . By means of (8) the functional (9) has the
form:
⎡ Q( x)
α 2
α
v 2 ( x )q ⎤
I [v ( x )] = ∫ ⎢
v ( x ) + ( x1 − x )
dx
+
b( x )v ( x ) 2 g
g
Q ( x ) ⎥⎦
x0 ⎣
x1
(13)
x1
or I [v ( x )] = ∫ G[x, v ( x ), v ' ( x )]dx
(14)
x0
The Euler-Lagrange equations are:
Ω = {v = v( x) ∈ c'| v( x0 ) = A, v( x1 ) = B} Gv −
d
(15)
Gv ' = 0
dx
Because the values A, B are not known a problem with free boundaries arises. As Gv ' = 0 the
final result is:
⎡g
v( x) = ⎢
⎣α
1
⎤3
Q 2 ( x)
(Q( x)b( x) + 2( x1 − x)b( x)q )⎥⎦
(16)
For a linear widening bottom with a given widths bn , b0 (figure 2) the result is:
1
⎛ g
⎞3
v(0) = 0; v(Lsp ) = ⎜⎜
qLsp ⎟⎟
⎝ αbn
⎠
Equation (16) can be modified for:
• overflow through two sides of the trough:
(17)
1
2
⎡g
⎤3
(
Qsp x + Q0 L' sp )
Qsp
; v( x) = ⎢
Q ( x ) = qx + Q0 ; q =
⎥
L' sp
⎣⎢α L' sp (b0 + tgβx )(2Qsp L' sp +Q0 L' sp −Qsp x )⎦⎥
(18)
where Q0 , Qsp are the maximum discharges along the short and long side respectively and L'sp
is the length of the long side.
• overflow through three sides:
1
3
1
⎡ g Qsp + Q0 ⎤ 3
⎡g
⎤
Q
v ( 0) = ⎢
⎥
⎥ ; v ( Lsp ) = ⎢
bn ⎦
⎢⎣ α b0 (2Q sp + Q0 )⎥⎦
⎣α
2
0
(19)
The formula for the velocity has the same form as in (18), provided that the long side is
symmetrical and L'sp' = 2 L'sp . The final results for the depth at the boundaries are:
1
1
⎡ α (Q0 + Qsp )2 ⎤ 3
⎡α 2Q0Qsp + Q02 ⎤ 3
h0 = ⎢
h
=
;
⎢
⎥
⎥
n
b02
b02
⎢⎣ g
⎥⎦
⎣g
⎦
The conclusion for a one-sided spillway, is according to (17):
(20)
1
⎡α ⎤ 3
(21)
hn = ⎢ q 2 ⎥
⎣g ⎦
Therefore the depth at the down stream end is equal to the critical depth and the same
conclusion follows for the cases of two or three sides. Obviously the problem defined by (9)
leads to a minimum lengthwise area of the trough. If one-dimensional conditions are
available, the volume of the trough would be a minimum as well. In this situation the
boundary conditions are free and the Euler-Lagrange equation gives a weak extremum of the
functional. Using the second variation δ 2 I the Legendre condition is fulfilled also. If the
water surface A' (figure 1) is located above the weir crest, submerged conditions arise for
maximum discharge. This is a risk under emergency discussed in the next section. Let us
consider the intermediate profile C ' (figure 1). The depth and the velocity at the control
section (end of the trough) are determined in advance so that the water surface would be under
the overflow crest, i. e. submergence is avoided. This is an isoparametric problem. After some
transformation the constants, the parameters and the boundary conditions can be obtained by
means of variational principles with fixed boundaries.
4. VERIFICATION TESTS AND NUMERICAL MODELING
Experimental studies were carried out for two water discharges namely Q0.1% = 665m 3 s −1 and
Q0.01% = 1000m 3 s −1 . The measured depths along the trough are given in figure 4 for five
typical cross-sections of trapezoidal shape (2:1) and uniform slope ibed = 3% . First the results
are compared with optimization analysis given above. Numerical modeling of the
optimization problem is realized through inverse variational principle of the form:
Au = f
(22)
where A is a linear differential operator and f is a given function.
An important special case arises when A is symmetrical and positive defined operator
(Fletcher 1984):
( Au, v ) = (u, Av ) and ( Au, u ) ≥ 0
(23)
If equation (22) has a solution, it provides a minimum of the functional:
J (u ) = ( Au , u ) − 2(u , f )
(24)
The inverse statement is true also: If an element exists with a minimum of the functional (24),
this element is a solution of (22). Here the inverse part was used. After substitution of the
formula for velocity (16) in equations (10), (11) the expressions for the total depth and the
depth of the water respectively, are obtained by numerical simulation. The solution is received
by Simpson’s formula for numerical integration of (10). Second the results are compared with
experimental study – figure 4.
A
1
level of crest 64,50
water level of trough
Q=1000m3/s
damlake level
61,00
3%
57,70
1
Cross-section 5
Cross-section 3
Cross-section 4
0+150 5,95 5,68 4,30
Distance
0+112,5 6,03 5,78 5,98
Right wall
0+037 5,58 5,60 5,32
Channel Axess
0+000 5,16 5,12 4,86
Left wall
0+075 6,30 6,05 5,65
3
Q=1000 m /s
Cross-section 2
Cross-section 1
A
3
0+150 4,50 4,05 3,50
Distance
0+112,5 4,90 4,50 4,30
Right wall
0+075 4,80 4,50 4,25
Channel Axess
0+037 4,95 4,80 4,50
Left wall
0+000 5,00 4,85 4,60
Q=665 m/s
Fig. 4. Depths of the trough along the length
In figure 5 the water levels for maximum discharge Qmax = 1000m 3 s −1 are shown, both in the
beginning and in the end of the trough.
Cross section 2
67,50
65,78
50
65,75
64,50
4,95
4,80
2:1
,77
R2
4,50
40
63,00
50
30
66,13
110
65,50
2:1
66,80
2
1
60,175
15,00
67,50
66,80
Cross section 5
40
R2
,77
63,65
63,38
3,50
2:1
2:1
62,00
57,70
4,50`
1,50
64,50
54,48°
50
30
63,00
4,05
50
66,13
2
1
30,00
Fig. 5. Two typical cross-sections
The experimental results and three numerical methods - by Kurganoff (Kurganoff 1982),
Hinds (Hinds 1926) and optimization analysis are presented in table 1 and in figure 6.
5. RESULTS
The final results point to the fact that the spillway remains unsubmerged for a discharge
Q0.1% = 665m 3 s −1 according to numerical methods, mentioned above, and experiments. For
the maximum discharge Q0.01% = 1000m 3 s −1 the crest has submerged part (figure 4) according
to experiments and has no submergence according to numerical methods. The same effect can
be seen in figure 6. The Kurganoff method gives small deviations, concerning the depths,
because of correction factors received by experiments (figure 6-a). The optimization method
provides a smaller depth in the last part of the trough in comparison with experiments and the
numerical methods.
Comparative table of hydraulic parameters in the trough
Q
[m3/s]
1
Dista
nce
[m]
10
2
Table 1.
h( x )
H (x )
h( x )
H (x )
h( x )
H (x )
h( x )
H (x )
66.7
Bottom
b(x)
[m]
11.33
[m]
3.45
[m]
3.53
[m]
4.68
[m]
4.79
[m]
3.90
[m]
3.90
[m]
5.30
[m]
3.80
30
200.0
14.00
4. 17
4.59
5.73
6.09
4.55
4.65
5.60
4.40
3
90
600
22
4.74
7. 17
5.62
6.96
5.90
6.20
6.10
6.20
4
120
800
26.00
4. 82
8.68
5.26
7.15
6.05
6.80
6.03
7.10
5
130
866.7
27.33
4.83
9.22
5.13
7.22
6.10
7.35
5.92
7.40
6
150
1000
30
4.85
10.10
4.85
7.41
6.20
7.60
5.65
8.00
N
ib = 4,77%
ib = 1,99%
ib = 2,6%
ib = 3%
Hinds
optimization
Kurganoff
experiments
0
-1
10 m.
30 m.
60 m.
90 m.
120 m.
150 m.
-2
-3
H(x) [m]
-4
y
-5
-6
-7
-8
6-а. Method of Kurganoff
2.000
0.000
10 m. 20 m.
30 m.
40 m.
50 m.
60 m.
70 m.
80 m
90 m. 100 m. 110 m. 120 m. 130 m. 140 m. 150 m.
-2.000
H(x) [m]
-4.000
y
-6.000
-8.000
-10.000
6-b. Experimental results
0.000
10 m.
20 m.
30 m.
40 m.
50 m.
60 m.
70 m.
80 m
90 m.
100 m.
110 m.
120 m.
130 m.
140 m.
150 m.
-1.000
-2.000
-3.000
H(x) [m]
-4.000
y
-5.000
-6.000
-7.000
-8.000
6-c. Method of optimization
Fig. 6. Comparative results
6. CONCLUSION
The velocity component θ by the additional discharge in the flow direction can be given an
account only just by experiments. All considered numerical methods provide no
submergence for the maximum discharge. The effect of submergence of spillway observed
by experimental study is reduced in the last part of the trough. The influence of the
approximate assumptions in equations (8), (11) points to a bigger lengthwise area and a
global volume in comparison with experiments.
REFERENCES
M. Carlier (1972), Hydraulique generale et appliquee, Eyrolles, Paris.
S. Sliskii (1986), Hydraulic estimation of high-pressure hydraulic structures, Moskow, (in
Russian).
A. Kurganoff and V.D. Dupljak (1982), Hydraulic of Spillways, Kiev, (in Russian).
C. Fletcher (1984), Computational Galerkin Methods, Springer-Verlag.
J. Hinds (1926), Side-Channel Spillays, Trans. ASCE, 89(881).