Research Article Optimum design of stepped spillways … … · · 2014-02-08dissipater. During the last three decades, research in the hydraulics of stepped spillways has been

International Journal of Civil and Environmental Research (IJCER) 1 (1): 19-31, 2013 ISSN xxxx-xxxx © Academic Research Online Publisher

Research Article

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Optimum design of stepped spillways with skimming flow

Farzin Salmasi

Department of Water Engineering, Faculty of Agriculture, Tabriz University, Tabriz-Iran.

* Corresponding author. Tel.: +984113392786;

E-mail address: [email protected]

A b s t r a c t

Keywords:

Stepped spillway,

Energy dissipation,

Optimum,

Nnumber of step,

Skimming flow.

On a spillway chute, a stepped design increases the rate of energy dissipation on

the chute itself and reduces the size of a downstream energy dissipater. A study

was conducted herein to gain a better understanding of the flow properties in

stepped chutes with skimming flow. The work was based upon wide variety

experimental facilities in large-size models obtained previously by other

researchers. The present results yield a new and practical design procedure of

stepped spillways with skimming flow. The design method includes some key

issues not foreseen in prior studies. The outcomes are valid for a wide range of

chute geometry and flow conditions typical of embankment chutes. A hydraulic-

design chart for a stepped channel has been proposed. From these charts, the

optimum number of step for maximum energy dissipation is determined and as

results it can reduces the size of a downstream energy dissipater. This procedure is

easy for designer to determine all parameter required like: optimum number of

step for maximum energy dissipation, the representative flow depth, the averaged

velocity at the end of the stepped channel, height of the sidewall and stilling basin

type.

Accepted:15 November2013 © Academic Research Online Publisher. All rights reserved.

1. Introduction

Spillway is a major part of a dam, which is built to release flood flow. Design and contraction of

spillways are very expensive and include a major part of the dams contraction cost. For large dams it

is about 20% of total dams construction cost, and for small dams about 80 % [1]. Depend on the

hydraulic conditions of flow and the geologic characteristics of the dam's site; spillways can be built

in different types and shapes.

Due to the high flow discharge over the spillways, their design and construction are very complicated

and usually faced with difficulties such as cavitation and high flow kinetic energy. For example, the

Farzin Salmasi / International Journal of Civil and Environmental Research (IJCER) 1 (1): 19-31, 2013

20 | P a g e

amount of energy which should be dissipated at the toe of the Tarbella dam's spillway is about 40000

MW which is about 20 times of the constructed power station capacity [2].

A stepped spillway is an energy dissipator having profile made up of steps. Stepped spillways allow

continuously dissipating a considerable amount of the flow kinetic energy, such that the downstream

stilling basin, where the residual energy is dissipated by hydraulic jump, can be largely reduced in

dimensions. Stepped channels have been used for more than 3500 years. The Greek engineers were

possibly the first to design an overflow stepped spillway [3]. The stepped design increases the rate of

energy dissipation on the chute above the steps and reduces the size of the downstream energy

dissipater. During the last three decades, research in the hydraulics of stepped spillways has been very

active [4].

The embankment dams are more prone to overtopping failure than other types of dams because of

breaching or erosion of the downstream face of the embankment. Despite the catastrophic effects of

failure, dam overtopping constitutes the majority of identified dam failures. Before the 1980s,

overtopping counter-measures consisted mainly of increasing the reservoir storage or spillway

capacity. Lately overtopping protection systems have gained acceptance because they safely allow

controlled flows over the dam wall during large flood events (Fig. 1).

a


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Fig. 1: Embankment dam stepped spillways. (a): Stepped spillway of the Opuha embankment dam, (b): Melton

dam secondary spillway in Australia [5].

For a given stepped chute, the water flows as a succession of free-falling nappes (nappe flow regime)

at small discharges. For an intermediate range of flow rates, a transition flow regime is observed.

Most prototype spillways operate at large discharges per unit width for which the waters skim as a

coherent stream over the pseudo-bottom formed by step edges. The skimming flows are characterized

by significant form losses and momentum transfer from the main stream to the recirculation zones

(e.g. [6,7]).

Modern stepped spillways are typically designed for large discharge capacities corresponding to a

skimming flow regime as illustrated in Fig. 2.

b


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Fig. 2: Skimming flow on steep (1 upon 0.6) stepped spillway: two-phase flow has air in water, and water in air;

two zones are separated by connected surface.

Flow resistance is predominantly form drag. The flow is non-aerated at the upstream end. Free-

surface instabilities are however observed and strong air–water mixing occurs downstream of the

inception point of free-surface aeration. Detailed air–water flow measurements highlighted large

amounts of entrained air further downstream and very-strong interactions between main stream

turbulence, step cavity recirculation zones and free-surface (e.g. [7,8]).

Stepped chute hydraulics is not simple because of the different flow regimes but importantly because

of strong free-surface aeration, flow turbulence, and interactions between entrained air and turbulence

[7-9]. Chanson [3,4] reviewed basic stepped spillway design considerations.

A spillway designer considering the stepped alternative will have the following basic data available:

- Height of spillway

- Slope of spillway and

- Unit flow

From these data the designer will want to ascertain the following structural components:


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- Optimum sizes of steps

- Estimate the terminal energy at the toe of the spillway and size of the stilling basin at the base of the

spillway

- Estimate the air concentration in the two-phase flow down the spillway and hence the d90 level (i.e.

that level which consists of 90% air and 10% water) above which are basically small droplets. This

level is suggested as the minimum level for the training walls to prevent overflow and erosion.

This paper presents the more detail results of the skimming flow characteristics under a wide range of

experimental conditions. These experiments have been performed previously by Boes and Minor [10];

Matos [11]; Ohtsu et al. [12]; Chanson et al. [13] and a systematic investigation by Ohtsu et al. [12].

By reanalysis of these experiment data, some useful graphs, showing relationship among different

involved dimensionless parameter, were provided. By using these graphs, an engineer can design

proper and optimum stepped spillway. Based on this research, for any stepped spillway, one can

determine optimum step height for maximum energy dissipation, so the downstream stilling basin can

be largely reduced in dimensions. Determination of optimum step height has not foreseen in prior

studies.

2. Materials and methods

As discussed in previous section, hydraulic of stepped chutes is very complicated and so several

researchers attempted to investigate its properties. Based on several experiments, Ohtsu et al. [12]

developed systematically investigated under a wide range of channel slopes (00 557.5 ).

Rajaratnam [6] assumed that Reynolds stress between the skimming stream and the recirculation

stream can be expressed by the equation 2/2VC f where = mass density of the fluid; V=

mean velocity of flow; and fC = coefficient of friction (Darcy friction factor f, is equal to fC4 ).

Based on the experimental results of Sorensen [14], fC was found to be approximately 0.18.

Based on Ohtsu et al. [12] works, the upper limit of the relative step height Scdh )/( for the

formation of skimming flows depends on the channel slope )( as:

oo

Sc

ford

h557.5tan

6

7 6/1

(1)

Where h is step height, dc is critical depth and is spillway slope. By measuring the air-concentration

ratio in the aerated-flow region, a clear-water depth (d) is obtained from Eq. 2.

9.0

9.0

01)1( yCdyCd mean

y

y

(2)


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In Eq. 2, C=air concentration; Cmean=averaged air concentration for the range of 0<y<y0.9 , y=normal

coordinate from the bottom and y0.9=y at C=0.9. In practice averaged air concentration can be

determine by Eq. 3 with having cdh / and :

cc

meand

h

d

hDC 45exp3.0

2

(3)

oo

oo

forD

forD

55190357.00214.000024.0

197.53.0

2

Friction factor of a skimming flow is given by Eq. 4 (Ohtsu et al., [12]).

Scc

cc

d

h

d

hforff

d

hfor

d

hAff

5.0

5.01.05.0

max

2

max

(4)

Where:

oofor

f

A

197.5

10*2.310*6.110*2.4

10*5.110*4.610*7.1

2224

max

1223

oofor

f

A

5519

10*31.210*75.210*32.2

452.0

1325

max

Drop height required to form quasi-uniform flow (He) is given by Eq. 5 (Ohtsu et al., [12]).

oo

c

c

e

ford

h

d

H

557.55.6exp7.67.5

*30.110*13.710*60.110*21.1122335

(5)

where: c

e

c

dam

d

H

d

H


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Residual energy of skimming flows (Eres) is given by Eq. 6., which Eres is residual energy at spillway

toe and subscript 'U' refer to uniform flow (Chanson, [3]).

2

32

2cos.

2cos.

d

dd

g

VdE c

res

3/23/12

sin82

1cos

sin82

1cos

ff

d

d

d

d

d

E c

cUc

res (6)

For nonuniform-flow regions, Eres /dc is empirically formulated by:

425

115.15.1

m

H

H

d

E

d

Em

e

dam

Uc

res

c

res

(7)

Relative energy dissipation is computed by Eq. 8, which Emax is maximum energy upstream of

spillway )5.1( max cdam dHE .

cdam

resres

dH

E

E

EE

E

E

5.11

max

max

max

(8)

3. Results and discussion

Several factors such as: design discharge, spillway width, number of steps, spillway slope, height of

step and total spillway height influence on energy dissipation on stepped spillway. In this study, try is

to show these dependences by non-dimensional figures and demonstration about existence of

optimum step number. With using Eqs. 3- 8, several data point in Excel spreadsheet was produced and

then some practical graphs among different parameter were drawn. For example relation between

relative energy dissipation )/( maxEE as a dependent variable and two independent parameters, i.e.,

step number (N) and cdam dH / has been drawn in Fig. 3. Based on Fig. 3, for a specified value for

cdam dH / , energy dissipation increases with increasing in step number (N) until a limit, and then

energy dissipation decreases with increasing in step number (N). Increasing part has more slope than

decreasing part. So in Fig. 3, there is a optimum step number for occurring maximum energy

dissipation. This optimum step number can be seen in the Fig. 4 also, which presents variation of

friction factor (f) vs. cdh / and spillway slope )( . In fact, maximum value of f for each cdh / and

)( demonstrates maximum value of )/( maxEE that take places at specified step height (h). Such

graphs give better understanding about relationship among different parameters and their interactive


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with each other. With these, an engineer can select best dimension for proposed stepped spillway that

energy dissipation become maximum.

Fig. 3: Variation of relative energy dissipation )/( maxEE vs. step number (N) and cdam dH /

0.89

0.90

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

50 70 90 110 130 150 170 190 210 230 250

E

/ E

max

N

Hdam / dc=124.2

94.78

78.24

67.43

59.71


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Fig. 4: Variation of friction factor (f) vs. cdh / and spillway slope (a)

Figs. 5-7 show another relationship among different parameter involved in stepped spillway. In Fig. 6,

hw refers to representative flow depth and in Fig. 7, Fr refers to Froude number defined as:

wwr gdVF / where Vw is averaged velocity: ww dqV / .

To demonstrate application of Figs. 3-7 for design of stepped spillway, in the following an example

was presented.

0.05

0.07

0.09

0.11

0.13

0.15

0.17

0.19

0.21

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

f

h / d c

a=10 Deg.

15

20

25

30

35


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Fig. 5: Variation of air bubble concentration (C) vs. cdh / and spillway slope (a)

Fig. 6: Variation of friction factor (f) vs. cw dh / and spillway slope (a)

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

C

h / d c

a=10 Deg.

15

20

25

30

35

0.05

0.07

0.09

0.11

0.13

0.15

0.17

0.19

0.21

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

f

h w / d c

a=10 Deg.

15

20

25

30

35


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Fig. 7: Variation of relative energy dissipation vs. Froude number (Fr1) and spillway slope (a)

Example: Assume for a design of a stepped spillway, the following data are given. Determine energy

dissipation of flow over spillway (or residual energy dissipation at the toe of spillway) and optimum

step height:

- Design flow (Q)= 5 m3/s

- Height of dam (Hdam)=50 m

- Spillway slope )( = 22 Deg. (based on natural earth topography in dam site).

Solution:

- By assuming spillway width=2.5 m, we have q=2 m2/s and then dc=(q

2/g)

1/3=0.74 m.

Then we can find other optimum geometry and hydraulic condition of stepped spillway by using Figs.

3 to 7 as:

- From Fig. 3 based on cdam dH / =67.43 and its representative curve, there is optimum number of

step (N) that energy dissipation become maximum. This number of step is N=147, then

h=Hdam/N=0.34 m (optimum step height) and %8.94/ max EE .

- From Fig. 4 and havingo

c anddh 2246.0/ , we have f=0.181

- From Fig. 5 and havingo

c anddh 2246.0/ , we have C=0.32

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1 3 5 7 9

E

/ E

ma

x

Fr1

a=10 Deg.

15

20

25

30

35


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- From Fig. 6 and havingoandf 22181.0 , we have hw/h=1.26, then hw=0.43 m and height

of stilling basin wall (Hw)= 0.43*1.4=0.6 m.

- From Fig. 7 and having %8.94/ max EE oand 22 , we will have Fr1=3.34, then Vw=6.87

m/s so type of stilling basin can be determine.

In table 1, other examples for optimum number of step were presented.

Table. 1: Examples for optimum number of step in stepped spillway

1 2 3 4 5 6 7 8 9 10 11 Q (CMS) H dam (m) W (m) q (m2/s) dc (m) h (m) h/dc (h/dc)s He/dc Hdam/dc 2 50 22 2.5 0.8 0.40 0.25 0.62 1.0 15.43 124.20 3 50 22 2.5 1.2 0.53 0.28 0.53 1.0 15.69 94.78 5 50 22 2.5 2 0.74 0.41 0.55 1.0 15.61 67.43 7 50 22 2.5 2.8 0.93 0.52 0.56 1.0 15.59 53.88 8 50 22 2.5 3.2 1.01 0.58 0.57 1.0 15.55 49.29

12 13 14 15 16 17 18 19 20 21 fmax f dw (m) Vw (m/s) Cmean y90 (m) Hw (m) N Eres/dc E/Emax 0.182 0.175 0.16 5.12 0.33 0.23 0.33 200.0 3.680 0.971 0.182 0.181 0.21 5.79 0.33 0.31 0.43 178.6 3.608 0.963 0.182 0.180 0.29 6.88 0.33 0.43 0.61 122.0 3.618 0.948 0.182 0.180 0.36 7.70 0.33 0.54 0.76 96.2 3.622 0.935 0.182 0.179 0.40 8.06 0.33 0.59 0.83 86.2 3.630 0.929

4. Conclusions

The flow characteristics of skimming flows have been investigated systematically. A spillway

designer considering the stepped alternative will have the following basic data available: Height of

spillway, slope of spillway and unit flow.

From these data the designer will want to ascertain the structural components such as: optimum sizes

of steps and size of the stilling basin at the base of the spillway.

A hydraulic-design chart for a stepped channel has been proposed (Figs. 3-7). From these charts, the

optimum number of step for maximum energy dissipation is determined and as results it can reduces

the size of a downstream energy dissipater. Also the representative flow depth (dw) and the averaged

velocity (Vw) at the end of the stepped channel can be predicted easily. Then height of the sidewall

(Hw) and stilling basin type can be determined from these data.

References

[1] Yousefian H. Hydraulic design of Labyrinth spillways based on the physical model. M.Sc. Thesis,

Isfahan University of Technology, 1996; Iran.


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[2] Novak P, Maffat AIB, Narayanan R, Hydraulic structures. Published by the academic division of

uniwin Ltd., 1990; 546.

[3] Chanson H. Hydraulic Design of Stepped Cascades, Channels, Weirs and Spillways. Pergamon,

Oxford, UK, January, 1995; 292.

[4] Chanson H. The Hydraulics of Stepped Chutes and Spillways. Balkema, Lisse, The Netherlands,

2001; 418

[5] Gonzalez CA, Chanson H. Hydraulic Design of Stepped Spillways and Downstream Energy

Dissipators for Embankment Dams. Dam Engineering, 2008; Vol. XVVII, Issue 4.

[6] Rajaratnam N. Skimming Flow in Stepped Spillways. J. Hydraul. Engng. ASCE, 1990; 116(4),

587–591.

[7] Chanson H, Toombes L. Air-Water flows down stepped chutes: Turbulence and flow structure

observations. Intl J. Multiphase Flow, 2002; 28(11), 1737–1761.

[8] Yasuda Y, Chanson H. Micro- and Macroscopic Study of Two-Phase Flow on a Stepped Chute.

In: Ganoulis, J. and Prinos, P. (eds.), Proceedings of the 30th IAHR Biennial Congress,

Thessaloniki, Greece, 2003; D: 695–702.

[9] Chanson H, Toombes L. Strong Interactions between Free-Surface Aeration and Turbulence in an

Open Channel Flow. Exp. Ther. Fluid Sci. 2003; 27(5), 525–535.

[10] Boes RM, Minor HE. Guidelines for the hydraulic design of stepped spillways. Proc., Int.

Workshop on Hydraulics of Stepped Spillways, Zurich, Switzerland, Balkema, Rotterdam,

The Netherlands, 2000; 163–170.

[11] Matos J. Discussion of ‘characteristics of skimming flow over stepped spillways. J. Hydraul.

Engng., ASCE, 2000; 126(11), 865–869.

[12] Ohtsu I, Yasuda Y, Takahashi M. Flow characteristics of skimming flows in stepped channels. J.

Hydraul. Engng., ASCE, 2004; 130(9): 860-869.

[13] Chanson H, Yasuda Y, Ohtsu I. Flow Resistance in Skimming Flows and its Modeling. Can. J.

Civ. Eng. 2002; 29(6), 809–819.

[14] Sorensen RM. Stepped spillway hydraulic model investigation. J. Hydraul. Engng., ASCE, 1985;

111 (12): 1461-1472.

Documents

Research Article Optimum design of stepped spillways … … · · 2014-02-08dissipater. During the last three decades, research in the hydraulics of stepped spillways has been