40 estudio en un modelo fisico de vertederos con canal lateral

8/11/2019 40 Estudio en Un Modelo Fisico de Vertederos Con Canal Lateral

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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 1796E mail: [email protected]

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 cross-

section 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


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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.

drainage

63,00

150,00

1,7

5

10,0

0

1,0

0

63,0063,001

1

spillway crest

I=3%

63,00

64,5064,50

63,00

30,0

0

4,02

,1

10,00

A

axes of trough

n=0,014

I=0,8%

56,50

64,502,

1

64,50

67,50

A

Fig. 2. Scheme of non-prismatic trough spillway

The cross-section is trapezoidal (2:1) along the trough with a bottom slope %3=bi . 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):

wheremp FrFr =

gh

VFr

2

= (1)

and indicesp, mdenote the Froude number for the prototype and the model.

In practice the weir models are scaled with a Froude similarity and viscous scale effects mustbe 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:

limReRe m (2)

where limRe is the recommended limit value of the Reynolds number Re .

Usually this number is 44lim 10.3,110.1Re = for side-channel spillways (Sliskii 1986). The

linear scale was determined by (2) for the discharge 13%1.0 665 = smQ as:

3/2

lim

min

Re

Re

=

r

p

LM (3)

where rpRe is a representative Reynolds number for a prototype determined for a typical

section. Having in mind the hydraulic structure of the side-channel the spillway crest could be

accepted as such section and then formin

LM follows:

7,5410

10.03,43/2

4

6min =

=LM (4)

The model scale was selected to

)50:1(50min =LM


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and with a resistance scaling:

mp = and mp CC = (5)

where and Care friction coefficient and Chezys velocity factor.

If pn is equal to 0,015, mn would be:

008,0

50

015,06/1 ==mn (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).

3. OPTIMIZATION ANALYSIS

Consider a one-sided spillway and the co-ordinate systemxy(figure 3).

'O

Fig. 3. Hydraulic scheme of the trough

The equation describing steady flow is (Kurganoff and Dupljak 1982):( )

QdQgw

vdxi

g

vddy f 20

2 /1

2

++

= (7)

where 0, are kinetic and momentum energy coefficients; fi is the friction factor; /ph=

is the flow depth of section; v and Q are the velocity and discharge; 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 variablesf

i and which are of small order.

Then this equation can be rewritten in the form:

dQgQ

v

g

v

ddy

2

02

2

+

= (8)

The functional can be defined as:

( )[ ] ( )dxxHvIx

x

=1

0

(9)

where

SPL

1+ix i

x

ih 1+ih

x

'

iz '

1+iz

0i

nh

y+

y

x+

y

h

'z

i

Datum

constA =

O

'z


4/8

( ) ( ) ( ) ( ) ( ) ( )

dsQ

sqsv

gg

vxhxyxhxH

x

x

++=+=1

0

20

2

2

(10)

( )( ) ( )xvxb

xQh= (11)

Further the following value will be accepted 0 . The variable ( )xy is shown in figure 3

and ( )xb is the width of the bottom having in mind a rectangular cross section that isequivalent 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 well-

known methods. If a linear variation of water discharge is accepted, the following formulas

are valid:

( ) qxxQ = and spsp LQq /= (12)

where ( )xQ is a linear function of a specific discharge q calculated from the maximumdischarge

spQ and the length of the trough

spL . By means of (8) the functional (9) has the

form:

++=

1

0)(

)(

)()(2)()(

)(

)]([

2

1

2

x

xdxxQ

qxv

xxgxvgxvxb

xQ

xvI

(13)

or [ ]=1

0

)('),(,)]([

x

x

dxxvxvxGxvI (14)

The Euler-Lagrange equations are:

{ }BxvAxvcxvv ==== )(,)(|')( 10 0' = vv Gdx

dG (15)

Because the valuesA, Bare not known a problem with free boundaries arises. As 0' =vG the

final result is:

( )

3

1

1

2

)()(2)()(

)(

)(

+=

qxbxxxbxQ

xQg

xv (16)

For a linear widening bottom with a given widths 0, bbn (figure 2) the result is:

( )3

1

;0)0(

== sp

n

sp qLb

gLvv

(17)

Equation (16) can be modified for:

overflow through two sides of the trough:

sp

sp

L

QqQqxxQ

'

;)( 0 =+= ;( )

( )( )

3

1

00

2

0

''2'

')(

++

+=

xQLQLQxtgbL

LQxQgxv

spspspspsp

spsp

(18)

wheresp

QQ ,0 are the maximum discharges along the short and long side respectively and'

spL

is the length of the long side.

overflow through three sides:

( )3

1

00

2

0

2)0(

+=

QQb

Qgv

sp;

3

1

0)(

+=

n

sp

spb

QQgLv

(19)


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The formula for the velocity has the same form as in (18), provided that the long side is

symmetrical and ''' 2 spsp LL = . The final results for the depth at the boundaries are:

3

1

2

0

2

00

0

2

+=

b

QQQ

gh

sp;

( ) 31

2

0

2

0

+=

b

QQ

gh

sp

n

(20)

The conclusion for a one-sided spillway, is according to (17):3

1

2

= q

ghn

(21)

Therefore the depth at the down stream end is equal to the critical depth and the sameconclusion 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 areavailable, the volume of the trough would be a minimum as well. In this situation theboundary conditions are free and the Euler-Lagrange equation gives a weak extremum of the

functional. Using the second variation I2 the Legendre condition is fulfilled also. If the

water surface 'A (figure 1) is located above the weir crest, submerged conditions arise formaximum 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 underthe overflow crest, i. e. submergence is avoided. This is an isoparametric problem. After sometransformation the constants, the parameters and the boundary conditions can be obtained bymeans of variational principles with fixed boundaries.

4. VERIFICATION TESTS AND NUMERICAL MODELING

Experimental studies were carried out for two water discharges namely 13%1.0 665 = smQ and

13

%01.0 1000 = smQ . The measured depths along the trough are given in figure 4 for five

typical cross-sections of trapezoidal shape (2:1) and uniform slope %3=bedi . First the results

are compared with optimization analysis given above. Numerical modeling of theoptimization problem is realized through inverse variational principle of the form:

fAu= (22)

whereAis a linear differential operator andfis a given function.An important special case arises when A is symmetrical and positive defined operator(Fletcher 1984):

( ) ( )AvuvAu ,, = and ( ) 0, uAu (23)If equation (22) has a solution, it provides a minimum of the functional:

( ) ( ) ( )fuuAuuJ ,2, = (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 thedepth of the water respectively, are obtained by numerical simulation. The solution is receivedby Simpsons formula for numerical integration of (10). Second the results are compared withexperimental study figure 4.


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damlake level

5,0

0

Distance0+000

0+000

Q=665 m/s

Right wall

Channel Axess

Left wall

3

4,6

0

4,8

5

Right wall

Channel Axess

Distance

Left wall

5,1

6

5,1

2

4,8

6

0+037

0+075

0+037

4,5

0

4,9

5

4,8

0

5,5

8

5,6

05,3

2

0+075

4,2

5

4,8

0

4,5

0

6,3

0

6,0

55,6

5

Cross-section1

3Q=1000 m/s

1

61,00

1

Cross-section2

Cross-section3

64,50level of crest

3%

water level of trough

4,5

0

0+112,5

0+150

0+112,5

4,3

0

4,9

0

4,5

0

6,0

3

5,7

85,9

8

0+150

3,5

0

4,0

5

5,9

5

5,6

8

4,3

0

Cross-section4

Cross-section5

A

Q=1000m3/s

57,70

A

Fig. 4. Depths of the trough along the length

In figure 5 the water levels for maximum discharge 13max 1000 = smQ are shown, both in the

beginning and in the end of the trough.

60,1751

2

2:1

67,50

65,7565,7865,50

64,50

15,00

4,95

4,80

4,502

:1

50

40

50

110

63,00

66,13

66,80

30

R2,77

Cross section 2

3,5

0

67,50

R2,77

66,1363,6563,38

62,00

4,5

0`

1

257,70

64,50

63,00

66,80

4,0

5

30,00

2:1

2:1

54

,48

50

40

1,5

0

50

30

Cross section 5

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.


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5. RESULTSThe final results point to the fact that the spillway remains unsubmerged for a discharge

13

%1.0 665 = smQ according to numerical methods, mentioned above, and experiments. For

the maximum discharge 13%01.0 1000 = smQ the crest has submerged part (figure 4) according

to experiments and has no submergence according to numerical methods. The same effect canbe 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 methodprovides a smaller depth in the last part of the trough in comparison with experiments and thenumerical methods.

Comparative table of hydraulic parameters in the trough Table 1.

N

Dista

nce

[m]

Q

[m3/s]

Bottomb(x)

[m]

( )xh [m]

( )xH [m]

( )xh [m]

( )xH [m]

( )xh [m]

( )xH [m]

( )xh [m]

( )xH [m]

1 10 66.7 11.33 3.45 3.53 4.68 4.79 3.90 3.90 5.30 3.80

2 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

%77,4=bi %99,1=bi %6,2=bi %3=bi

Hinds optimization Kurganoff experiments

-8

-7

-6

-5

-4

-3

-2

-1

0

10 m. 30 m. 60 m. 90 m. 120 m. 150 m.

H(x) [m]

y

6-. Method of Kurganoff


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-10.000

-8.000

-6.000

-4.000

-2.000

0.000

2.000

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.

H(x) [m]

y

6-b. Experimental results

-8.000

-7.000

-6.000

-5.000

-4.000

-3.000

-2.000

-1.000

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.

H(x) [m]

y

6-c. Method of optimization

Fig. 6. Comparative results

6. CONCLUSION

The velocity component by the additional discharge in the flow direction can be given an

account only just by experiments. All considered numerical methods provide nosubmergence for the maximum discharge. The effect of submergence of spillway observedby experimental study is reduced in the last part of the trough. The influence of theapproximate assumptions in equations (8), (11) points to a bigger lengthwise area and aglobal 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, (inRussian).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).

40 estudio en un modelo fisico de vertederos con canal lateral

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