Université d'Ottawa University of Ottawa

The Hydraulics of Fiow on Stepped Ogee-Pmfile Spillways

Mehdi Azhdary Moghadàam

A Dissertation Submitted to School of Graduate Studies

mder the supervision of Dr. RD. Townsend

in partial fiilnlment of the requirements for the degree of Doctor of Philosophy in Civil Engineering

Department of Civil Engineering Faculty of Engineering University of ûttawa

Ottawa, Canada, KIN 6N5

The Doctor of Phdosophy in Civil Engineering is a joint program between Carleton University and the University of Ottawa, which is administrated

by the Ottawa-Carleton Institute for Civil Engineering

%&di Azhdary Moghaddam, ûttawa, Canada, 1997

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The author retains ownership of the L'auteur conserve la propriété du copyngbt in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thése ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits saus son permission. autorisation.

Because of its simple form, its eSectiveness as an energy dissipator, and its

contribution to dam stability, the stepped spillway is one of the oldest h p

structure used in the construction of small dams. With the relatively recent

introâuction of Rouer Compacted Concrete (RCC) and patticuiasly (rockfill)

gabions as dam construction materiais, the stepped spiiiways continues ta be an

attractive option among the various iraditionai f o m of &op structure for smaü

dams.

The sleps on the spiliway's downstream face signincantly increase the energy

dissipation thereon over that would occur on the equivalent smoorh (ogee-prome)

version. As a conseQuence stiiiïng basin structures, usually introduced at the toe

of spillways (îo reduce excess energy in the spiiiway floor) can be either reduced

in size or elimïnated entireiy .

Alhough srepped spillways were fi.rst inîroduced around 694 B.C., in most previous

studies dealing with the hydraulic performance of stepped spillways are site-

specific. This research, which includes both mathematical and physical elements,

investigates the effect of different step configurations, in conîribution with different

spiliway dopes, on the hydraulic performance of (ogee-profile) stepped spillways.

The main goal of the research was to develop improved critena for the design of

stepped spillways.

The numerical study, a 2-D nnite element mode1 was developed to estimate, based

on a trial-enor procedure, the srepped spiilway m e r surface for a given design

head In the supporting experirnentd programme duee different spillway dopes

(45; 50; and 603 were investigated and in each case the experiments were

rweated for four Merent srep configurations. This provides data set of 12

combinations of qiUway dope and step arrangement

Improved criteria were developed to (i) distinguish between the possible flow

regimes on depped spillways, (ii) estimate both stepped spiiiway fiction factors

and energy dissipation rates, and (iii) determine optimum step configurations for

a given set of boundary conditions.

iii

In the name of God

The author wishes to express his sincece appreciation to his supervisor, Dr. RD-

Townsend, for his guidence, advice, and encouragement during the course of thïs

research.

The schoiarship pmvided by the Ministry of Cuiture and Higher Education of the

I K of Iran is gratefdiy acknowledgd

The author is gratefid to Mr. R Moore, the Hydrauiics Laboratory technician, and

ML B- Cotter, of the workshop staff, for construciing the experimental facility.

The author dso wishes to thank bis parents, parents-in-law, brothers and sisters for

their continuous encouragement. The patience of the author's daughter, Yeganeh,

has been appreciated. This study would not have been posi'ble without the

encouragement, paîience, and ovenwhelming support of the authors d e , T o m

- - Abstract ................................ u

Achowledgements . . + . . . . . - - - . . . . . . - . . . . . . . . . . . - . - . . . . - - . . iv

Tableofcontent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Symbols . . - . . . . . . . . . . . . . . . . - - . . . . . . - - . . . . . . . . . - - . . . ix . . * List of Figures . - . . .. . - . . . . . . - . . . . - . . . . . . . . . . . . . . . . . . . - - - - - xiu

-- - ListofPIates . . . . . . . . . . . - . . . . - . - - . . . . . . . - - - - . . . - . . . . . - - . . mi

ListofTables . . . . . . . . .+ . - . . . . . . . - - - . . . . . . . . - - - . . . . . . . - . . - xk

Chapter 1 InaioducCoa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - 1

1.1. Preamble . . . . . - . . . . . . . . . - . . . . . . . . . . - - - - . . . - . . . . . . - . . - 1

1.2. Statement of the problem . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . 3

1.3. Study Objectives . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . - 4

1.4. Scope of the Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . 5

1.5. Thesis Description . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . . - . . . 6

Chapter 2 ~~ht 'Dgt!eVtmmt spurpw . . . . . . . . . - . . . . . . . . - . . . . . . . . . . . . . . - 7

2.1.Introduction . . . . . . . . . . . . . . . - . . . . . . . . - . - . . . - - . - . - - - - - . . 7

2.2. Spiiiway Types . . . . . . . . - . . . . . . . . . . . . . - . . . . . . - - . - . . . - - . . 8

2.3. The Ogee Spillway . . . . . . . . . . . . . . . . . . . . . . - . * . . . . . . . . . . . - 9

2.3.1.CrestDetail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.2. Dow~~stceam Face . - . - . . . . . . . . . . . . . . . . . . . . . . . . . . . - 1 1

2.3.3. Energy Dissipators . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . - 15

Chapter 6 ExptiimeablRedts ....................................... 115

.......................................... 6.1. Introduction 115

................................... 6.2. Discharge Coefficient 115

........................ 63 . Friction Factor for Stepped Spillways 116

....................... 6.4. Energy Dissipation for Slnmming Flow 118

Chapter 7 .......................... Nmneiid nlodelliag of Spilhvay Elow 157

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Introduction 157

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Literaîure Review 158

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Proposed Numerical Mode1 162

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1. Boundiuy Conditions 163

................................... 7.4. Finite Element Method 165

. . . . . . . . . . . . . . . . . . . . . . . . 7.4.1. Finite Element Disnetîzation 166

. . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2. Weigtited-Residuai Form 167

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3. Finite Element Mode1 170

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.4. Interpolation Fuuctions 171

. . . . . . . . . . . . . . . . . . . . . . . . . 7.4.5. Coordinaîe Transformation 174

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.6. Numerical Integration 176

. . . . . . . . . . . . . . . . . . . . . . 7.4.7. Assembly of Element Equations 176

. . . . . . . . . . . . . . . . . . . 7.4.8. Imposition of Bomdary Conditions 177

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.9. Solution of Equations 178

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.10. Postprocessing 178

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. Discussion 179

........................................... 7.6. Conciusion 180

Chapter 8

Smmaty. Coddans, ad Reconinuniilt;ons for Fubit I l t ~ h . . . . . . 194

8.1.Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

...................................... 8.2. Main Conclusions 196

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Future Research 198

Cbapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rtfelcaces 200

List of Symbols

a = coefficient

a, = constant coficient

ali = constant coefficient

A = area of cross section

b = coefficient

cf = coefficient of fluid fiction

C = discharge coefficient

C, = locd air concentration

Cc = Chezy coefficient

CD = drag coefficient

C, = centrifuga1 acceletation factor

Cs = discharge coefncient for stepped spillway

d = location of water surface profiie

ds = arclength of au idntesimal iine element dong the boundary

DH = hydraulic dep&

E, = energy at upstream

E, = energy at section 1

E, = energy at section 2

K , = m-um energy

f = Darcy-Weisbach fiction factor

Fr, = Froude nuniber at section 1

g = gravitaiional acceleration

h = height of step

h, = static head at the inception point

H = operating head

EI,, = design head

& = theoretical head on the crest

J = lacobian matrix

15 = equivalent saad rougbness

k', = roughness of spillway d a c e

K = constant coefficient

1 = length of step

L = effective length of the crest

L,= distance f?om die wall to the position y,

L, = die distauce h m the start of the growth of the boundary layer

Lj = lengh of the jump

L, = roughness spacing

m = number of Gauss points

n = manning's constant

n, = number of nodes

n, = normal outward vector on the boundary

n, = component of unit vector in x-direction

n, = component of Mit vecîor in y-direction

N = constant coefficient

N, = number of steps

p, = the proportion of the energy loss per step

p = hydrostatic pressure

p, = order of triangle

q = discharge per unit width

Q = discharge

r = local coordinate

$ = coefficient of determination

ri = location of Gauss points

R = radius of bucket

R, = radius of curvature

s = local coordinate

& = secondaqr variable

u = velocity in the x-direction

v = velocity in y-direction

V = velocity

V, = approach velocity

V, = cntical velocity

V,, = maximum velociq

x = horizontal coordinnte

x, = distance dong the slope which the boundary layer grows

X, = the horizontal length Erom the crest to the saturation point

y = vertical coordinate

y, = depth as section 1

y, = depth at section 2

y, = flow depth corresponding to 99 percent air concentration

y, = criticai depth

ya = de* of water

y, = the depth of flow at the inception point

y,, = nomai depîh

y, = pool depth

y, = depth corresponding to a pure water colurnn

w = weight function

z = elevation above datum

ai = weight factor

~ t , = weight factor

p = the angle of the jet where it hits the bed

T, = boundary of the eIement

y = specific weight of the Quid

AJ2 = energy lost

6 = boundazy Iayer thickness

8, = Kronecker-Delta

8 = downstream dope

K = bed cwature

1L; = Lagrange interpolation fimction

p = m a s densiîy of the water

Y = Stream fiinction

Cl = fiow domain

@ = element

= vorticity

xii

List of Figutes Fig.2.1. Examples of spillway types: (a) Freesverfall, (b) Ogee, (c) Side-

Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Fig.2.2. Furiher examples of spillway types: (a) Open-chamel, (b) Shafi, (c)

Siphon (after Novak, et al, 1990) . . . . . . . . . . . . , . . . . . . . . . . , 23

Fig.2.3. Cavitation prevention cntenon for design head - ogee spillways

(afterchaudtuy, 1993). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Fig.2.4. Spülway crest shape (after U S Amy Corps of Engrs., 1977) . . . 24

Fig.2.5. Pnncipai zones of aerated flow on spillway face . . . . . . . . . . . . . 24

Fig-2.6, S e l f - d o n mechanism on spillway face (after Chanson, 1993) . . 25

Fig.2.7. Recommended radius of bucket (after Creager et al. , 1964) . . . . 25

Fig.2.8. Examples of standard stilling basins: (a) Type II, (b) Type IïI (after

Peterka, 1958) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Fig.2.9. Flip buckeî and Ski jump spillways (USBR, 1977) . . . . . . . . . . . 27

Fig.2.10. Roller bucket energy dissipators (USBR, 1977) . . . . . . . . . . . . . 27

Fig.2.11. Examples of a stepped-spillway . . . . . . . . . . . . . . . . . . . . . . . 28

Fig.2.12. Various types of step geometry . . . . . . . . . . . . . . . . . . . . . . . 28

Fig. 2.13. H-Q relationship for standard ogee spillway (after Baban,

1995) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Fig3.1. Examples of stepped spillway structures: (a) Almansa dam (Spain,

1576), (b) Tibi-Alicante dam (Spain, 1594)- (c) Pabellon dam

(Mexico, lï3O), (d) Gabion stepped spillway profile . . . . . . . . . . 46

Fig.32. Unit head 1 0 s over plain gabion steps (after Peyras, et al., 1992) . 47

Fig.3.3. Brownwaod Country Club dam (de r Hansen, 1986) (USA) . . . . 47

Fig.3.4. Spillway of the De Mist Kraal weir (after Hollingworth and D m ,

1986) (S. Afnca) . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . 48

Fig.3.5. Spillway of the Zaaihoek dam (after HoUingworth and D m ,

1986) (S. Africa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Fig.3-6. Stepped spillway energy dissipation for Upper Stillwater,

Monksviiie. and Stagecoach Dam (after Hensen and Reinhardt,

........................................... 1991) 49

Fig.3.7. Energy dissipation characteristics at the toe of stepped spillways

................................. (after Frizeli, 1992) 49

....... Fig.3.8. AE/E,, vs yD,h relationship (after Christodoulou, 1993) 50

Fig.3.9. Flow velocity at the toe of a stepped spillway (after Bindo, et al.,

1993) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Fig.3 .10 . Velocity profile on a spillway: (a) smooth, (b) Stepped (after Rice,

andKadavy, 1996) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

. . . . . . . . . . . . . Fig 4.1. Generai arrangement of the experimental flume 61

Fig.42. General profile@) of smooth spïiiway model(s) (45; 50; 60" ) . . 62

. . . . . . . . Fig.4.3. Water sucface profiles for smooth 45' spillway mode1 62

. . . . . . . . Fig.4.4. Water d a c e profiles for smooth 50' spiIlway mode1 63

. . . . . . . . Fig.4.5. Water surface profiles for smooth 60' spiUway mode1 63

Fig-4.6. Observed water surhce profiles for (a) 45-CL-BL model, (b) 45-

CS-BL mode1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Fig.4.7. Observed water d a c e profiies for (a) 45-CL-BS model, (b) 45-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CS-BSmodel 65

Fig.4.8. Observed mer d a c e profles for (a) 50-CL-BL model, (b) SO-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CS-BL mode1 66

Fig.4.9. Observed water d a c e profiles for (a) 50-CL-BS model, (b) 50-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CS-BSmodel 67

Fig.4.10. Observed water surface profiles for (a) 60-CL-BL model, @) 60-

CS-BLmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Fig.4.11. Observed water d a c e profiles for (a) 60-CL-BS model, (b) 60-

CS-BS mode1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

. . . . . . . . . . . . . Fig.5.1. Simple &op structures (a) vertical, @) inclineci 98

Fig.5.2. Cornparison of eq(5.26) with experimentd data from various

SOUfCeS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Fig.5.3. Cornparison of eq(5.27) with experimental data from various

Sowces .........................................

Fig.5.4. Comparison of eq(528) with Rajaratnam, and Charnani's (1995)

data ...........................................

Fig.5.5. Comparison of eq(529) with experimentll data nom various

sources .........................................

Fig.5.6. Observeci y, /h versus computed y , /h ....................

.................... Fig.5.7. Obsenred y@ v e m computed y$

Fig.5.8. Observed case versus computed cosp ...................

. . . . . . . . . . . . . . . . . Fig.5.9. Obsemed AElE,, versus computed ME,,

Fig.S.lO. Characteristic flows over a stepped spillway: (a) nappe flow with

M y developed hydraulic jump y @) nappe flow with partially

. . . . . . . . . . . . . . developed hydrauiic jump, (c) s b i n g flow

Fig.5.11. Nappe flow profile with Wy-developed hydraulic jump on a

siagiestep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig.5.12. Cornparison of eq(5.38) with various criteria devdoped by

........................................... ohem

. . . . . . . . . . . Fig . 5.1 3 . Cornphson of eq(S.3 8) with experimental data

Fig.5.14. Comparison of eq.(5.4S),and (5.43), with experimentai data fiom

V ~ ~ ~ Q U S S O W C ~ S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig.5. 1 5 . Cornparison of eq.(5.46), and (5.44). with experimental data nom

variou~sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fig.5.16. Friction factor for stepped channeis on flat slopes (8 c 12' )

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (after Chansen, 1994)

Fig.5.17. Friction fricar of stepped channels on steep slopes (8 > 27' )

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (after Chansen, 1994)

...... Fig.5.18. Friction f m r for stepped d a c e s (after Tozzi, 1994)

Fig.5.19. Cornparison of eq(5.71) with experimentai data nom various

sources .........................................

Fig.6.1. H-Q nlaîionships for 45 O models ......................

Fig.6.2. H-Q relationskps for 50' models . . . . . . . . . . . . . . . . . . . . . .

Fig.6.3. H-Q reiatioaships for 60' models ......................

Fig.6.4. Cornparisan of eq(6-4) with CLBL mode1 experimentd chta ...

Fig.6.5. Cornparison of eq(6.4) with CLBS mode1 experimental data ...

Fig.6.6. Cornparison of eq(6.4) with CSBL mode1 experimentai data ...

Fig.6.7. Cornparisan of eq.(6.4) wiîh CSBS mode1 experimentd data ...

Fig-6.8, Cornparison of eq(6.4) with 45' stepped mode1 experimental

data ............................................

Fig.6.9. Comparison of eq(6.4) with 50' stepped mode1 experirnental

data ............................................

Fig6.10. Comparison of eq(6.4) with 60' stepped mode1 experùnentai

data ............................................

Fig.6. I 1- Observed f versus computed f . . . . . . . . . . . . . . . . . . . . . . .

Fig.6.12. AE/Eo versus a ; 45-CL-BL mode1 ..................

Fig.6.13. AE/EO versus x/I& ; 45-CL-BS mode1 ..................

Fig.6.14. MEO versus x/H, ; 45-CS-BL mode1 . . . . . . . . . . . . . . . . . .

Fig.6.15. AE& versus x/H, ; 45-CS-BS modei ..................

Fig.6.16. AE/E, versus x/H, ; 50-CL-BL mode1 . . . . . . . . . . . . . . . . . . Fig.6.17. AE/EO vernis x/H, ; 50-CL-BS mode1 . . . . . . . . . . . . . . . . . .

Fig.6.18. AEE, versus xM, ; 50-CS-BL mode1 . . . . . . . . . . . . . . . . . . Fig.6.19. AJX,, versus xi& ; 50-CS-BS mode1 . . . . . . . . . . . . . . . . . . .

Fig.620. AE& versus x/H, ; 60-CL-BL mode1 . . . . . . . . . . . . . . . . . . Fig.6.2 1 . Aë/Eo versus xi- ; 60-CL-BS mode1 ..................

Fig.6.22. MEo versus x/B, ; 60-CS-BL mode1 . . . . . . . . . . . . . . . . . . Fig.6.23. AEE, versus x& ; 60-CS-BS mode1 . . . . . . . . . . . . . . . . . . Fig.6.24. AE/EO versus XE& ; 45 " smwîh mode1 . . . . . . . . . . . . . . . . . Fig.625. AE& versus XE& ; 50' srnooth mode1 . . . . . . . . . . . . . . . . . Fig.6.26. AER0 versus x/H, ; 60' srnooth mode1 ................. Fig.6.27. AEE0 versus y& ; 45" stepped models . . . . . . . . . . . . . . . .

................ Fig.6.28. Mo versus y& ; 50' stepped models

. . . . . . . . . . . . . . . . Fig.6.29. AE&, versus y& ; 60' stepped models

Fig.6.30. Cornparison of eq(6.14) with experimentai data 45" stepped

models ......................................... Fig.6.3 1 . Cornparison of eq(6.14) with expetimental data 50 ' stepped

rnodels .........................................

Fig.6.32. Cornparison of eq46.14) with experimentai data 60 ' stepped

models .........................................

Fig.6.33. Obsemed AEE, versus cornputeci AE/E, . . . . . . . . . . . . . . . .

Fig.7.1. Fiow over a spillway ...............................

............................... Fig.72. Finite element mesh

Fig.7.3. Construction of the interpolation functions for a typical quadratural

................................. îrianguiar elements

Fig.7.4. Cornparison of depth of water between the numericai simulation

output and experimentai data for 45' smooth mode1 . . . . . . . . .

Fig.7.5. Comparison of depth of water between the numericai simulation

output and experimentai data for 50' srnooth mode1 .........

Fig.7.6. Comparison of depth of water between the numericd simulation

......... output and experimentai data for 60' smooth model

Fig.7.7. Comparison of depth of water between the numerid simulation

. . . . . . . . . . output and experimentai data for 45-CL-BL model

Fig.7.8. Comparison of depth of water between the numerical simulation

output and experimental data for 45-CS-BS mode1 ..........

Fig.7.9. Comparison of depth of water between the numerid simulation

output and experimentai data for 50-CL-BL mode1 . . . . . . . . . .

Fig.7.10. Cornpariscm of depth of water between the numerical simulation

output and experimentai data for 50-CS-BS mode1 . . . . . . . . . . Fig.7.11. Comparison of depth of water between the numerical simulation

.......... output and experïmentai data for 60-CL-BL model

Fig.7.12. Comparison of depth of water between the numericai simulation

.......... output and experimentai daîa for 60-CS-BS model

.................... Fig.7.13. Observed y versus Simulatedd d

Plate 4.1. Mode1 45-CL-BL ................................. 70

Plate 42 . Mode1 45-CS-BL ................................. 71

Plate 4.3. Mode1 50-CLBL ................................. 72

Plate 4.4. Mode1 50-CS-BL ................................. 73

Plate 4.5. Mode1 60-CL-BL ................................. 74

List of Tables

Table 2.1. K- and N- values deduced by different researchers (Senturk,

1994) ........................................... 21

Table 2.2. USBR- recornmended vdues of K and N . . . . . . . . . . . . 21

Table 3.1. Examples of Gabion stepped spillways ................. 42

Table 3 3 . RCC dams containing stepped spillways ................ 43

Table 3.3. Embanhnent dam with stepped spillway protection in the LX,

............................. and Russia (Baker, 1992) 44

Table 3.4. Stepped spIIiways for protection of the downstream dope of

............... embankment dams in the USA Qrizeil. 1992) 45

Table 5.1. Statistical parameters of the developed equations for a single

verticaldrop ...................................... 97

Table6.1.VduesofHandqforallmodels .................... 123

Table 7.1. Samplhg points and weights in Gauss quadrature (Bathe.

1996) ........................................... Table Al . Location of water d a c e pronle for 45' smooth mode1 . . . .

Table A 2 . Location of watet d a c e pronle for 45-CL-BL model .....

Table A.3. Location of water d a c e profile for 45-CL-BS mode1 . . . . .

Table A.4. Location of water surface profile for 45-CS-BL mode1 . . . . .

Table A S . Location of water d a c e pmnle for 45-CS-BS mode1 . . . . .

Table A.6. Location of water suface profile for 50' smooth mode1 . . . .

Table A.7. Location of water d a c e profile for 50-CL-BL mode1 . . . .

Table k 8 . Location of m e r surface profile for 50-CL-BS model . . . .

Table A.9. Location of m e r d a c e profile for 504s-BL mode1 . . . . . Table A.10. Location of water d a c e profile for 50-CS-BS mode1 . . . .

... . . Table A 1 1 Location of water d a c e prome for 60' smooth mode1

.... Table A.12. Location of water d a c e pronle for 60-CL-BL model

. . . . Table A.13. Location of water d a c e pronle for 60-CL-BS model

. . . . Table A.14. Location of water d a c e profile for 60-CS-BL model

Table AlS. Location of water surface profle for 60-CS-BS mode1 . . . . 230

Chapter 1

A spillway is usuaily the most important appurtenant facility to a dam. The

fimction of the spillway is to provide an efficient and safe means of conveying

flood discharges to the downstream channel. The spiliway design primarily

depends on the design flood, dam type and location, and reservoir size and

operation. SpilIways can be classifïed based on (i) their function (main,

emergency, and a d a r y ) , @) their hydrauiic type (fme o v e ~ d l , overjbw, chute,

siphon, etc.), and ( X i ) their mode of control (ungured, and gated).

While the number of new dams under constniction in the world is declining, the

number of existing dams that have had to be upgraded to meet current hydradic

and seismic criteria is increasing. This is especially tme for smaiier embankment

dams that have been judged to have inabquate spillway capacity and are unable

to store or pas safely fioods in accordance with the cwrent criteria of at least one

haif the theoreticai hbrible Mmchtimi Hood (PMF). Many failures of dams have

been caused by improperly designed spillways or by spillways of insufficient

capacity. Severai embankment dams have been identified as unable to p a s their

design flows without failure due to overtopping (Frizell, 1991).

In addition to providing sufficiait capacity, the spillway must be hydrauIicaiIy and

structuraüy adequaîe. For a typical ovefflow spiilway the flow accelerates dong

the downstrearn face and high veIocities are attained at the spiIlway's toe. The

spillway's outiet structure must be designed to ensure that the spillway discharges

will not erode the downstream chamel bed, or undermine the toe of the dam itseK

Therefore, the spiiiway's surface must be erosion cesistant to withstand the high

scouring velocities created by the &op fiom the reservoir d a c e to the tailwater.

In dam design the traditional approach to hancüing the excessively high kinetic

energies associated with large flows is through some fom of stinnig busin structure

located at the foot of the spillway. Dependkg on the expected Froude Number

range of die incoming Bow, the form of the stilling basin can range fiom a simple

wncrete apron to a cornplex structure that may indude rows of chute blocls, baftle

piers and a plain or dentated end SU (Peterka 1958, USBR 1977)- If the basin

requires all k e e feanires this can add substantidy to the overd costs,

consequendy alternative solutions to the problem merit investigation. One possible

solution is tu consider a ~ ~ s p k t h v a y instead of the traditionai miooth ogee-

profile spillway.

The stepped spilhuay is a spillway whose face is provided with a series of steps

fiom near the crest to the toe. The steps signincantiy increase the rate of energy

dissipation Ealong place on the spillway face. They aiso reduce the size of the

required downstream energy dissipation basin A stepped spiihuuy offers a number

of advantages over an equivaient smooth spiilway:

1. compatibility with Roller Compvrced Concr~te (WC) and @ion hydraulic

structures;

2. reduction of dissipation basin ske;

3. reductioa of the risk of cavitation;

4. in the case of an auxiliary spiliway the consüuction programme will be

much shortened;

5. an inaease in the discharge capacity; and

6, low c o s

For diese reasons, there has b a n renewed mterest in this fonn of spïliway (Young*

1982, Sorensen, 1985, FnEeil ,1991, Diez-Cascon, et al., 1991).

1.2. Statement of the proMem

To the author's knowledge, only one praaicai guideline for the design of stepped

spilhvays has been produced (CIRIA, 1978). In this guideline, however, there is

no indication of the degree to which the spillway steps may impact on energy

dissipatioa In practice, therefore, one has to rely on a physical model study of a

particuiar spillway design to estimate the energy dissipated. This kind of model

has been investigated for the six following cases:

1. The Upper Stillwater Dam, (Young, 1982);

2. Monksville Dam, (Sorensen, 1985);

3. The Upper StiUwater Dam, (Huston, 1987);

4. Mode1 built in the Hydtaulic Lab. of Iberduero, SA, (Diez-Cascon, et al.

1991);

5. m a l i dam, (Bindo, et al., 1993); and

6. Salado Creek Site 10 ( Rice, and Kadavy, 1996).

In general, the resuits obtained for each of the above cases are vaiid only for bat

case, i.e. they are site-specifiic. In Iight of diese limitations, Sorensen (1985) points

the need for M e r research on srepped spilhuqy structures. Degoutîe et al. (1992)

state diat "... until now, the sire mqtn'temenu for stilhg bmhs downstmam of

stepped spiIhuqys wete not well known". Frize11 (1992) identifies the lack of

general design criteria diat quantifies the enetgy dissipation on stepped spillways,

and Christodoulou (1993) states that "... the pefomace of stepped spiIhuays is not

well known". Aithough several researchers have studied sfepped spilhuays in tbe

past, there remain considerable gaps in knowledge in the design of these structures.

For example, we are still unable to predict the energy dissipated on the structure,

or the spillway water d a c e profile without building a physical model. The water

surface profile is an important consideration when establishing the height of the

spiilway side-walIs. Given lbtst ~ e d aspecls of stkppd spilhvqy

design d opedou, tbch is d w i y a nad h r fintber meawô do a d d m s these

important issnes.

The analysis of water fiow over a spillway is an important engineering problem,

however the d y s i s preseflts several difficuities in view of the foilowing

considerations:

1. it is a non-linear problem, therefore, it might have non-unique solutions in

certain cases;

2. the fiee-water surface is unknown beforehand;

3. the flow is srrbcnticd upstrearn of the spiilway crest and is superrcriticd

downstream of the spiIlway crest; and

4. there is a control section near the crest.

Furthermore, any numerical simulation of the spiliway flow must account for

possible siimificant depatture fiom the hydrostalic of fluid-induced pressures on the

spillway face, produced by centripetal fluid acceleration at the crest and toe

sections.

1.3. S t u d y Objecîives

This study examines the hydrauiïcs of flow on various stepped ogee-profile

spiliway configurations, both fiom heomtical and physicd-modelling viewpoints.

The main objectives of the research were:

(i) to investigate the hydtaulic performance of several physicd models having

dBerent downstream dopes and Werent step coniigurations, and

(5) to develop a numericd modei for predicting the stepped spillway water

surface profile co~esponding to the design-discchorrge condition.

The study requirernents included:

1. a review of the hydrauiics of fiow on stepped spillways and the crfzeria for

&w clmst~cdion;

2. in the physical-modeiling portion of the study, mesure energy dissiwon

for different spillway downstrem s h p s and dz'fle~nt srep sires;

3. Uivestigate thepe@onnmces of different stepped spiliway models for a wide

range of discharge; and

4. deveiop a two-dimensional Finite Element (FE) mode1 to predict the

spillway water surface profile and relaîed energy dissipdon for the design-

dischqe condition.

1.4. Scope of the Researth

From the Literature review it was found that, so far, stepped spiZZwuys studies have

been largely site-specifc and that the results of these studies have been used purely

as a guide for other similar cases. This study examined and compared the

hydraulic performance of various Ogee-pmfiie stepped spiZhvays, having different

downsmmn slopes and step configu-ons. The effectiveness of the various

designs was judged by comparing the hydraulic performance of each stepped pronle

with its equivalent smooth profile over a wide range of discharges. Although

Sorensen (1985) was, the first to initiate a combination of s d and large steps on

the spillway pronle, this application was not related to considerations of energy

dissipation. The present study is the fbst to examine the use of a combination of

step sizes ta optimize energy dissipation occurring on stepped spiUways. Based on

the results of this study and previous studies several new criteria for the design of

stepped spiI1ways were developed

A 2-D Finite Element (FE) mode1 was developed which determines: (Q the free

water surEace of the spiiiwq flow, and (ii) estimates the amount of energy

dissipaîed by the stepped profile, based on the designdischmge condition. The

daîa obtained h m the Iaboratory investigations were employed to validate the

numerical modeL Tnis sûuiy is mique in two mspecb:

(1) dik d r M e s conceming st- spiiiways (which am Iargely si*

spccific), lbis s î u d y inwdgates auï compaws the hydiulic perf 'ome of

diffe~nt stuppd s-ay confîgutaîioas, consequcntiy it is more general

inname,&

(2) the numeficd mode1 devdopcd in du's shidy mphscnts a usefd design aid

in mgad to dmaîing: (i) tbe spiiiway water s d m e pmfde, and (Ü) the

eneiey dissi- by îhe sfepprd spiiiwag configuxaîion.

Chapter 2 briefly reviews the hydraulics of Ogee-PmfiZe spillways and their outlet

energV dissipaor structures. Chapter 3 is a literature review on the subject of

stepped spilhuqys- In Chapter 4 the experimental programme is descnbed and in

Chapter S the hydrauiics of stepped spillwqys is explained In Chapter 6 the

experimentai r d t s are auaiyted and several new design criteria are proposed

Chapter 7 deaIs with the development of the numencal model. Finally, Chapter

8 combines the summary, conclusions and recommendations for future research

Chapter 2

The "0gee"-hîïie Spiilway

Spillways are hydraulic stnictures provided at storage and diversion dams. At storage

dams, they release surplus or flood water which can not be contained in the allotâed

reservoir storage space. At diversion dams, they are designed to bypass ffows

exceeding those which are directed into the diversion system.

To propose a composite design for a spillway, it is necessary to consider the various

factors influenciug spiliway size and type, and to corrdate alternatively selected

components. The components of a spillway are: (i) the control structure; (ii) the

discharge charnel; (iii) the terminal structure; and (fi) the entrauce and/ or outlet

channels.

Afier the hydraulic size and ouifiow characteristics of the spillway are detennined by

routing of the design flood, the generai dimensions of the control structure can be

selected A specfic spiliway layout c m be developed by considering (i) the type of

dam; (ii) the mpography; ( i the foundaiion conditions; (iv) the hydraulic condition

of the control structure; and (v) the overd1 economy. Site conditions greatly innuence

the selection of Iocation, type and components of a spillway. Ia the design of a

spillway the following consideraiions are also necessary:

1. îhe steepness of îhe terrain traversed by the spillway control and discharge

c h e l ;

2. the class and amount of excavation matenal and the possibility for its use as

embanhent material;

3. the potential for scour of the spillway and downstream channe1 surfaces and the

need for Luiing; and

4. the stability of the excavated slopes.

2.2. Sj~Uway Types

A spiilway can be constnicted in several locations: within the body of the dam, at the

one end of it, or independently in a saddle. Spillways are ordinarily classifieci

according to their most prominent feanues, either as they relate to the control, to the

discharge channel, or to some other component. The foiiowing are common types:

1. A fme-overjüll (smght h p ) spillway is one in which the flow drops freely fiom

the crest (Fig.2- La).

2- An Ogee (oveflow) spiiiway has an ogee- or S-shaped profile. The upper cuve of

the ogee is generaliy made to conform closely to the profile of the Iower nappe of a

ventilated sheet f a h g fkom an equivdent sharp-crested weir (Fig-2.l.b).

3. A side-chopinel spillway is one in which the control weir is placed dong the side

of and approximately paraMe1 to the upper portion of the spillway discharge channel

(Fig.2.L.c).

4. A n open-chrainel (tmugh or chute) spiiiway is a spillway whose discharge is

conveyed h m the reservoir to the downstrearn river level through an open channeI,

placeci either dong a dam abutment or througb a saddle (Fig.2.2.a).

5. A conduit or mmel spiiiway is a closed channel used to convey the discharge

around or under a dam.

6. A swt (monring glory or dmp inler) spiiiway is one in which the water enters over

a horizontal-positioned lip, drops through a vertical or sloping shaft, and then flows

to the downsaeam river channel through a horkmnîai or near horizontal conduit or

tunnel (Fig.2.2.b)-

7. A siphon qiIhuay is a closed conduit system formed in the shape of a U, and

positioned so that the inside of the bend of the upper passageway is at normal reservoir

storage level. Siphonic action takes place after the air in the bend over the crest has

been expelled fiom the conduit (Fig.2.2.c).

2.3. The Ogee Spillway

An ogee spillway can be used on concrete-gravity, arch, and buttress dams, where part

of the dam's length may be used as a spillway. ft is probably the most

extensively-used spillway to safeIy pass the flood flaw out of a reservoir. An ogee

spillway has tbree main parts: (i) the cresr, (hj the sloping downstream face, and

(iii) the energy dissipator at the toe.

The crest profile is so chosen as to provide a high discharge coefficient without

causing dangerous cavitation conditions and vibrations. The profile is usually made

to conform to the lower nappe flow fiom a well-ventilated sharp-crested equivalent

rectangular weir with an approach velocity V, = O. Therefore, for the design discharge

condition, theoreticaily the pressures dong the crest section are atmospheric and a high

discharge coefficient is assured This idea is believed to have been proposed by

Muller (1908) (Subramanya, 1989)- Operating heads srnaller than the design head

@&) cause smaller nappe trajectories and hence resuit in positive pressures (above

atmospheric) on the spillway face and consequentiy lower discharge coefficients.

Similady, for operathg heads greater than E&, the lower nappe trajectory tends to puil

away fiom the spiilway Nface thereby producing negative pressures (ab-aîmosphenc)

and higher discharge coefficients.

The shape of the ciest is based on H, which is seiected for a given site such that the

minimum pressure head at the crest is higher than -6 m in order to prevent cavitation.

Fig.2.3 rnay be used to select H, so that this condition for the minimum pressure is

satisfied For operating heads (H) greater than H, the nappe is projected beyond the

control structure profile and hence a negative pressure is developed in the space

between the structure and the water column. Usually, the design head is selected such

that 1.3 < H/% (1.5. In diis range, acceptable levels of ab-atmospheric pressures are

produced on the spillway face. This results in an increase in die discharge capacity

of the spilIway and at the same time does not resdt in cavitation damage of the

spillway surface.

Applyùig the principle of projectile motion, it is found that the e q d o n of the lower

nappe profile is a second-degree parabola. The generai fonn of this equation is written

as:

in which x and y are the coordinates of the downstream curve of the spillway with the

ongin of coordinates being located on the apex. K and N are constants, whose values

depend upon the inclination of the spillway's upstrearn face and the approach veiocity

of the fluid Many attempts have been made by various researchers to determine the

values of K and N. Table 2.1. prrsents some examples of these attempts. The crest

shapes of uncontroiled ogee spillways have been extellsively used by the US Bureau

of Reciamation (USBR). For low approach veiocities, typical values of K and N can

be found in table 22 .

The crest profile upstream of the apex is usually given by a series of compound

curves. Cassidy (1970) reportai the equation for the upstream portion of a vertical

faced spïiIway as:

which is valid for the following regions:

Based on extensive Iaboratory investigations the U.S. Army Corps of Engineers (1977)

deveioped s e v d empiricai relationships for the crest shape. These relationships are

shown in Fig.2.4.

2.3.2. Downsûeam Face

The downstream face of an ogee spillway usuaily has a very steep gradient and

consequently the flow on the face quickly becornes supercritical. Because of this

steepness some difficuities axise in the analysis of the flow conditions. The

development of a bounday Iayer starts near the crest Once this boundary layer

extends to the water surface, air is entrained in the high velocity flow. Air entrainment

or insufflation is the process by which air enters the body of water as a result of

turbulence of the f i e water surface (Falvey, 1980, Wood 1991). The intersection

point of the bouaàary iayer and the water surface is called the point of inception.

Based on Keller and Rostagi's (1975) study, Wood (1991) presented the following

equation for development of the boundary Iayer in the spüiway flow field:

in which

6 = boundary layer thickness;

= equivdent sand roughness;

x, = distance dong the dope which the boundary layer grows; and

h, = static head at the inception point

This equation is appIicable in the range 5' I 8 5 70; where 0 is the dope of the

spiilway face.

The designer needs to know the amount of air entrainment in order to select the height

of the spiliway si& walls. If the walls are designed to insufncient heights,

overtopping may occur and this may Iead to dangerous erosion on the outsides of the

w a k Knowing the amount of air entrainment is also helpfid in assessing the

cmitation potentiai, sùlce air near the channel bottom reduces the possibility of

mimion. Air entrainment decreases the density of the air-water mixture by

increasing the velocity of flow. At the same tirne, the excess velocity of flow

increases the air friction, which decreases the velocity of flow.

Flow over a spillway that inchdes air entrainment may be divided into the foilowing

four vertical maes (Fig.2.5).

1. Qper zone: comprises a small mass of fîying water particles ejected fÎom the

mixing mne. This zone is not important for engineering applications.

2. Miking zone: has d a c e waves of random amplitude and fiequency. To prevent

overtopping of the side waiis. it is necessary to take the height of these waves into

consideration-

3. Undedymg zone: the nnfaçe waves do not peneaate the underlying zone* and the

air concentration at any depth in this zone is determined by the number and the size

of air bubbles-

4. AiFfme zone: exïsts only in that part of the channel where aeration is stiU

developing.

Wood (1991) divided the development of ~ e ~ a e t a t e d flows in a wide charnel into the

following four regions (Fig.2.6):

1. Nonaerated flow region: in this region the turbulent boundary layer does not reach

the water sudace and hence no air is entrained,

2. Partiaily-aeraîed flow region: air concentration profiles Vary with distance, however,

the air does not reach the chamel bottom in the partiaiiy-aerated region.

3. Fully-aerated flow region: air concentration profiles vary with distance, however, the

air reaches the channel bottom in the partiaily-aerated region.

4. Uniform (ful1y-developed) aerated flow region: the flow conditions reach an

equiiibrium -te and are constant with distance. A large amount of air is entraineci,

and bulking of the fîow occun.

In the d o m aerated region mean-flow properties, such as flow depth &) and

depth-averaged air concentration, (Cm) are functions of discharge, charme1 bottom

dope, bottom roughness, and the fluid properties. The US. A m y Corps of Engineea

recommends increasing the flow depth by 20 percent to accomt for bulking (US

Amy Corps of Engrs., 1965).

By using Straub and Anderson's data (1958) nom an artificialiy-roughened flume,

Hager (1 99 1) developed the foiiowing empirical expression in the SI system for a wide

rectangular channel in the d o m aerated region :

in &ch

y, = fiow depth wrresponding to 99 percent au concentration;

y, = depth conespondllig to a pu= water column;

n = Manning's constaut;

0 = the slope; and

g = gravitationai acceleratioa,

At the toe of the spiiiway a smooth transition is needed This smoodi transition avoids

excessive vibration that could induce structural damage and possible failure. It aln,

prevents the toe of the spillway ftom scouring the foundation due to the impact of the

fdhg water. Therefore, the surface at the spillway toe is usuaily designed as a

circuiar c m e or bucket. To be thoroughiy effective the bucket should tangent to the

foundation as well as to the spiiiway's terminal downstream slope. The radius R of

the bucket, muwred in fw may be estimated approximately by the foilowing

empirical formula (Chow, 1 959):

V = the velocity, (Wsec) of the flow at the spiiiway toe, and

= the dam height, (fi).

To find the radius of bucket, Creager et ai. (1964) suggested the dimensions shown in

Fig.2-7.

To prevent scour and erosion of the toe of a dam, as weli as the dowustream channel,

energy dissipators are provided to dissipate a sufficient amount of energy before water

enters the downstream chaud. The flow velocity at the toe of a hi&-head spiilway

is usuaily amund 32-35 d s e c - This high veloeity may cause serious scour and erosion

of the downstream channel if proper precautions are not taken. The comxnon types of

energy dissipators which are used as outlet structures in spiilway design, are:

(a) Straight drop: This kind of energy dissipator is commody used in m a I I drainage

structures, and low head spiiiways. The aerated fiee-falling nappe in the straight &op

usualiy produces supercritical flow on the apron. Therefore, a hydraulic jump may be

formed downstream.

(5) S t i h g basin: The stilling basin siructure cornes in several forms and is designed

to dissipate the energy of the flow by utilizing the development and enhancement of

the hydraulic jump within the basin. This is a solution for dams up to 60-70 m high.

The location of the hydraulic jump is important for determining the lengfh of channel

requiring protection, Energy loss in a hydraulic jump is expressed as:

in which

E, = energy at section 1;

y, = depth at section 1;

E, = energy at section 2;

y,= depth at section 2; and

AE = energy lost in the jump.

Depending upon the range of Froude number of the upstream flow, the following

devices may be incorporateci, either singularly or in combination, in stilling basin

design:

(i) Paved apron: A simple paved apron may be used in the case of low - h e d stmctures.

The length of paved apron depends on the maximum Iength of the anticipated range

of hydraulic jumps.

(ii) Chute blocks: These devices are placeci at the entrance of the stilling basin to

cclmgute the jet, and lift a portion of it h m the basin floor. This creates a greater

number of energy dissïpating eddies, &ch r d t s in a shorter length of jump than

would be possible without them. These b1ocks also d u c e the tendency of the jump

to sweep off the apron at taiI water elevations beIow conjugate depth.

(iii) Bafne blocks: These structures are located in the stilling basin a certain distance

downstream fiom the chute blocks. Their funcbon is to dissipate energy mostiy by

impact action. In certain circumstances, they must be designed to withstand impact

fiom ice or floating debris and/ or damage due to cavitation.

(iv) End d l : This device is located at the exit of the stilling basin. It cm be either

solid or dentated. Its fûnction is to reduce m e r the length of the jump and to

control scour immediately downstream fiom the stnicture. For large basins that are

designed for high incornhg velocities, the silI is u d y dentated to perfonn the

additional fimction of diffusing the residuai portion of îhe hi&-velocity jets that may

reach the end of the basin.

Fig.2.8 shows the basic types of stilling basins commoniy used as outlet structures.

(c) Flip buckets: When the downstream charnel bed is relatively stable, theflp-bucket

or ski-jump spiIlway can be considerd Fig.2.9. This f o m of dissipator structure

projects the high velocity jet away h m the base of the dam into the downstream

chamel. This is achieved via a concave section at the bottom of the chute sectioa

The spiilway overflow is discharged h o aîmosphere completely above the tailwater

level. Under ideal conditions, these kinds of energy dissipators have been found to be

very efficient in dissipating the excess energy.

(d) Rouer buckets: When the discharge channel has an appreciable natural erosion

resistance, the d e r bucket can be d This kind of dissipator requires substantiaiiy

higher tailwater levels than conventional hydradic jump basins. Two types of roller

buckets have been developed, Fig.2. IO. Scour will occur in the streambed at the point

of impingement when the jet dives, but will be filled in by the ground rolier when the

jet rides.

(e) Stepped spillways: Another possible energy dissipator option is to dissipate part of

the kinetic energy of the spillway flow on the spillway face itself- An ogee stepped

spillway is an ogee spiilway containhg a series of drops or steps in the invert At a

point just downstream of the spiilway crest, steps are introduced into the profile so that

the envelope of their tips foUows the standard pronle d o m to the me of the spillway,

Fig.2.11. Therefore, the total fd is divided intD a ntunber of =der fds. At each

fdl, retarciing forces are derived nom the reaction of the steps to the descendhg flow.

The steps signinci~tly increase the rate of energy dissipation taking place on the

spiUway fafe and eIiminate or greatiy reduce the need for a large energy dissipation

basin at the toe of the spilIway. Step geometries are eitber horizontal, incluied or

pooled, Fig.2.12. FnZeii (1992) examined the hydraulic performance of Merent step

dopes and showed that horizontal steps would have the greatest energy dissipation.

2.4. Discharge Coefficient

A m e relaing the upstream reservoir level and the spilIway discharge is d l e d the

spilhvuy &g curve. The relationship talres the fom:

where

C = discharge coefficient,

L = effective length of the crest,

- = velocity head component 2g

An o v e ~ o w spillway can be considered as a short-crested weir. me basic merence

between a broad-crested weir and a short-crested weir is that nowhere above the short-

crested structure can the curvimire of the streadhes be neglected; i.e. die pressure

distribution is other than hydrostatic.

From an ewnomical point of view, spiiiways must safely discharge a peak flow under

the smaiiest possible head, white on the other hand the negative pressures on the crest

must be limited to avoid the danger of cavitation. Theoreticaily, there should be

atmosptieric pressure on the crest In practice, however, fiiction between the surface

of the spilIway and the lower nappe wiil introduce some negative pressures. If the

spillway is operating under a head lower than its design head, positive pressures occur

throughout the crest region and the discharge coefficient is reduced.

An operating head (E) greater than the design head ofd) will cause negative pressure

at al1 points of the crest profile and will increase the discharge coefficient. The

avoidance of severe negative pressure on the crest, which may cause cavitation on the

crest or vibration of the structure, should be considered an important design criterion

on high-head spillways. Mode1 experiments indicate, however, that the design head

may be safely exceeded by up to 50%, but beyond this harmfd cavitation may develop

(Chow, 1959).

The discharge coefficient, C, is influenced by a number of factors, such as (0 the depth

of the approach flow, (ii) relation of the actuai crest shape to the ided lower-nappe

shape, (fi0 upstream spiliway slope, (iv) downstream apron interference, and (v)

downstream submergence (USBR, 1977). Through extensive laboratory tests, the U.S.

Bureau of Reclamation has compiled information on the coefficient of discharge for

smooth spillways. There are two conditions for which the value of C is determined

(Baban, 1995):

(0 d e n WH, > 1.33 the velocity head is negligible, then C = 2.225 and

(ii) when HA& < 1.33 the veiocity head should not be neglected The value of C

caicuiated fiom the experïmentai curves is presented in Fig. 2.13.

Researchers

Creager-Scimemi (1945)

Smetana (1948)

de Marchi (1928)

Taôle 2.1. K- and N- values deduced by different researdiers (Sentt.uk, 1994)

Table 2-2 USBR- recommended values of K and N

Section A-A -

Fig.2.2. Further examples of spillway types: (a) Open-channel. ( 6 ) Shaft, (c)

Siphon (after Novak et al., 1990)

Fig.2.4. Spillway crest r h a p

(fier US. Army Corps of Engrs., 197 - -

Fig.2.3. CaviMion prevention criterion for

design head - ogee spiliways (after Chaudhxy, 1993)

Fig2.5. Principal mes of aerated flow on rpillway fue

24

Fig.2.8. Examples of standard stilling b a h : (a) Type II, @) Type III ( a f k

Peterka, 1958)

26

Fig.2.12. Various types of step geometry

Fig. 2.13. H-Q relationship for standard ogee spillway (after B a b a 1995)

Chapter 3

Litemtum Review

Drop structures aud stepped spillway profiies have been used for more than 2500

years. The oldest stepped spiiiways are the spillways of the two Khosr dams, in Iraq,

built by the A m a n King Sennacherib in 694 B.C. These dams were designed to

supply water to the Assyrian capitai city Nmeveh (near the present city of Mosul).

Botfi dams feature a stepped downstrearn face and were intended to discharge the

entire river flow over their crests- Much later, the Romans built stepped ovedow

dams in Syria and Tunisia Mer the faii of the Roman empire, several stepped

spillways in Iraq and Spain were designed by Moslem civil engineers. Spanish

engineers, who benefitted fiom Roman and Modem precedents, were the rnost

exceptional engineers in dam expertise before 1850. Towards the end of the 14th

cenmy they built the Almansa dam, in Spain, which in 1576 was raised while

keeping the lower original part intact (Fig.3.l.a). A similar structure is the

Tibi-Aiicante Dam built between 1579 and 1594 on the river Monegre, in Spain

(Fig.3.l.b). The largest dam with a stepped spillway is the Puentes dam, in Spain,

which was built in 179 1. This dam was washed out in 1802 after a fomdation failme.

In centrai Mexico, several stepped overflow dams, like the PabeIlon Dam, 1730

(Fig3. l .c) were built by Spanish Engineers during the 1 8th- 19th centuries.

In aii of the above examples, stepped spillways were selected (if) to contribute to the

stability of the dam, ( f i ! for their simplicity of shape, or (fi) for a combination of the

two. The spillway of the New Croton dam, USA, (1906) is probably the fht stepped

spillway designed specincally to dissipate energy (Wegman, 1907).

Gabions are a common construction material for stepped spillways (Peyras, et ai.,

1992). Their simple geometric fonn and easy positioning make the construction of

gabion spillway structures easy and cheap. Stepped gabion spillways are stnicturally

stable, resistant to water loads, and efficient energy dissipators, Fig.3.l.d. The

flexibility that they provide to dams and their drainage capacity make them a highly

suitable matend for spiIlway construction. Stepped gabion spiliways are fiequently

used in dams less than 5 m hi& From a technical point of view, stepped gabion

spillways can withstand unit-width flowfates q up to 3 m3/dm without great damage,

if the setting of the gabions complies faithfully with the code of ptactice (Peyras, et

al., 1992). Table 3.1 includes two examples of gabion stepped spillways.

Peyras, et al. (1992) studied the hydraulics of gabion stepped spillways, They

recommended stepped spillways as a good option for small dams. They found about

10-30% saving on stiliing basin length by using a stepped spilIway. Theu research

focussed on describing fiows over small, stepped, homogeneous gabion spillways to

quatl* accuraîely the energy dissipation. B a s 4 on their experiments they developed

a graph (Fig.3.2) for estimaîing the amount of dissipated energy. With increasing

$/(@), the head loss per unit height dmps sharply, due to the change from q p e to

skïmming flow.

3.3. RCC and Concritte Sîcpped Spillways

Rouer Compacted Concrete @CC) is regarded as a p w e m new material by dam

builders around the w r l d In addition to its use for grady dams, RCC has found

increasing use for spïiiways as well as for foundation improvements for dams- Stepped

spiiiways are naturai extensions of RCC placement techniques. Table 3.2 lists some

examples of RCC dams with steppeà spillways.

In Norway, steps were introduced in a tunnel spillway with a free d a c e flow to

enhance the energy dissipation (Chanson, 1994). This design was seiected to d o w

detrainment upstream of a vertical shaft aud to prevent entrainment of air in the shaft

In the UK a stepped intake of a morning-glory type spiiiway was seIected for

Ladybower reservoir, following mode1 tests whïch showed that this arrangement

provided a larger discharge capacity than a smooth intake (Chansa1994).

Recently the use of RCC for existing smaii earth embanlcments has received a great

amount of interest. RCC block-steps are more fiequently being placed on the

downstream slopes of dams to aliow for overtopping during flood conditions (Hansen,

1986).

3.3.1. Block Stepped Spiiiway

Dam Safety inspections have concluded that a large number of both srnaIl and large

embankment dams are unsafe due to predicted overtopping during extreme flood events

(FrizeU, 1992). In addition, the capacity of a spiliway often needs to be increased,

either because of changes in methods of hydrological assesment, or requirements for

higher standards of safety. In some cases it is even necessary to replace the exisbng

spiIiway completely. However, o h die preferred solution is to augment capacity

with an auxiliary spdiway, which generally cornes into operation at a higher reservoir

levei than the existing spdway (Pravdivets, and Bramley, 1989).

Recexttly there has been considerable interest in the use of stepped-block spïIlways for

the construction of such auxiliary spillways, because of: (i) the ease of construction,

and (ri) the Iow wst per installecl unit discharge capacity (Pravdivets, and Bramley,

1989). Stepped block protection has considerable potential as a low-cost method for

the construction of chute spiIlways and the protection of embankments fiom erosion

by overtopping flow. By using stepped blocks, the overaii construction programme is

shorter because of the reduction in preparatory works required before dam construction-

The use of RCC to solve an ùiadequate spdiway capacity condition was deveioped by

Freese and Nicholas (Hansen, 1986) for the Brownwood Country Club dam, Texas

(Fig. 3.3). After severai possible solutions were investigated, it was decided to place

stair-stepped RCC on the dam's downstream slope. The entire project was completed

in two weeks at a cost of about one-third of the other options studied. This soIution

has already produced similar desigus for modeing two existing embankments in

Colorado and two more in North Caroiina (Hansen, 1986).

Wedge-block spdimys differ fiom conventionai stepped spüiways in a number of

respects. Apart fiom the drainage holes, an important consideration is tùat

wedge-block spiliways are considerably flatter than traditional stepped spillways. In

addition, an under-drain is an important feature with adequate water egress at the me.

The wedge block choice for the spillway at BNshes Clough, in the UK, proved to be

an economical solution and straight forward to construct. The loose blocks have not

moved or shown auy sign of distress due to hydradic forces and have been tested up

to q4.75 m2/sec (Baker, and Gardurer,I995).

Research in Russia, the UK and the USA has shown that spillways constructed h m

small, lightweight, wedge-shaped concrete blocks, can utilize fïow hydrodynarnics to

achieve a high degree of stability- Table 3.3 lia several examples of stepped block

protection in the UK and Russia, and Table 3.4 Lists similar examples in the USA.

Pravdivets, and Bramley (1989) studied the hydraulics of stepped block spillways.

They found that the optimum block stability is given by:

where

1 = Iength of steps, and

h = height of sreps.

FrizeII (2992) studied the hydraulics of block stepped spillways through a physical

mode1 test programme. The emphasis of his research was on producing a stable

stepped spillway overlay diat di provides energy dissipation on 2@9:1(V)

embanlrment dopes. Frize1.s resuits showed that the steps provide a good energy

dissipator, and thgr are ais very stabIe.

ïhe Construction Industry Research and Information Association (CIRIA) design guide

(Baker, and Gardiner, 1995) recommends a minimum block thickness of 100 mm for

a discharge intensity q = 1.525 m2/sec on a 1(H):3(V) slope. However, North West

Water Engineering (NWW) opted to increase the average block thickness to 212 mm,

giving a block weight of 120 kg and hence a greater resistance to removai or damage

(Baker, 1995).

Poggi (1 949) (Essery and Horner, 1978) performed extensive experimental research on

the flow behaviour of a scde mode1 of a stepped spiilway operating as a series of

pooled sreps. The objective of his investigations was to provide subcritical approach

flaw to every &op by creating a series of hydmIicjmps. 'ïhus, almost complete

dissipation of l e energy attained by the change in level was achieved on aii steps,

The resulting design cansists of large pools and s m d drops, inherently a costly design.

When a flow enters a spiliway having steps of constant geometq, it accelerates over

the initial section of the structure. This section is termeci the fnmsition zone. On steps

lower down the spillway, equilibrium is established and the flow geometry is the same

at each step, although the depth varies across each step. This section is refened to as

the unifom zone. The length of the transition zone is influenced by the physicai

characteristics of the spillway and by the discharge per unit width. On a prototype

spillway the length of the transition zone is generally mail in cornparison with the

Length of the spiliway. Therefore, in order to provide design information, it is only

necessary to maice energy measurements on flows in the uniform zone.

In the transition zone containing the fint series of sreps, when the flow accelerates ta

a maximum velocity (Vmd and air starts to be entraineci, the flow is smooth with slight

unddations reflecîing the steps below. Then at V,, air is drawn into the flow and

intense turbulence starts and grows toward the bottom of the spillway. The water

cushion on îhe steps gives way to bottom rollers in which air bubbIes can be seen to

gyrate. The cucrent skims over these roUers and the step iips. The air entrainment

veiocity is solety govemed by spillway features, me siep height and iength. Pqrras,

et al. (1992) recorded values of the order of 5 .56 m/s depending on spillway slope.

Once V,, is reeched, flow velocity drops slightly. In addition to a loss of h m

potential energy per step, the flow dso dissipates some of its kinetic energy.

It must be noted that caviwon damage may occur on stepped spillways, but the risk

of cavitation damage is reduced by the flow aeration phenornenon. Peterka (1953) and

Russell and Sheehan (1974) showed that 508% of air concentration next to the spillway

bottom may prevent cavitaîion damage on coacrete suffaces. Further, the hi& rate of

energy dissipation dong stepped qillways reduces the flow momentum. Essery and

Horner (1978) perfonned a physical model study of the complex flow behaviour down

stepped spillways. To the author's knowiedge their report is the only available

guideline for design of stepped spillways. Unfortmately the results obtained from this

study apply oniy to a restricted range of designs.

Young (1982) studied the feasibility of a stepped spillway for the Upper Stillwater

Dam, Utah. He estimated a 75 percent reduction of energy through the use of a

stepped spillway. He noted that "the stepped spilhuay cm lie used with dams w hem

the unit discbge is lm ad stiIIing basin construction is to be limite#. He fomd

that at s m d discharges (0.25 S q i 0.75 m2/s), the flow strikes each step and loses

nearly ai l of its velocity before accelerating to the next srep. The r d t s of his study

conf'irmed an enonnous saving in concrete and excavation dong with reducing the area

over which uplifk pressures can deveiop.

Sorenson (1985) performed a physical model investigation of stepped spillways. He

found that the best way to eiimuiaîe the defleaing jet of water was to add a few steps

further up the face of the spillway. For a modei spiliway, the scaled steps fom the

dominant surfme mghness. He stated that "the model s @ i e mughness md riesulting

turbulent boumhy Iqver gnnuth and air entminment w erie not suffiCient to adepately

simulire the pmtotypicai conditions". So, if the Reynolds numbers are in die order of

16, the scde e f k t s d be minimai and predicted prototypical toe velocities should

be closer to m e prototypical vaIues.

The spillway sections adopted for De Mist Krdl weir on the Litde Fish river, and

Zaaihoek dam on the Slang river, boîh in South Africa, are shown in Fig.3.4 and

Fig.3 -5 respectively (Hollingworth and Druyts, 1986). Choosing stepped spillways for

these dams was for two purposes:

1. to help to dissipate the energy; and

2. to facilitate the placing of rollcrete and facing concrete.

The upstream faces of these two structures are vertical and îheir downstream faces are

szepped. The De Mist Kraal spillway was hydraulically tested on a smail-scaie model

(1:75). A srep size 1 m high by 0.6 m wide was found to be optimum, for the

required discharge range. The same general configuration was adopted for the

Zaaihoek dam.

Houston (1987) studied the hydradic mode1 performance of the Upper Stiilwater

stepped spillway. She stated that "the c ~ s t section must be appmpriately designed ro

pmduce the desired flow conditions down the m i n d e r of the stepped fixe." In her

studies, she fowd a reduction of 85 percent for the required stilling basin length. She

emphasized the ne& for M e r research, regardhg the design data that relates the

fiow depths to energy dissipation on stepped spillways.

fijaramam (1990), in his work on stepped spiUway flow, defined a coefficient of

fiuid fiction, Cf, as:

where

y, = normal depth ,and

In his experiments Cf was found to Vary between 0.1 1 S Cf I 0.2, with an average of

0.18 and for s m d fiow rates the range was found to be 0.25 I Cf I 0.28.

Stephenson (1991) found that the energy loss over stepped spillways is proportiond

to the length of the face once uniform depth is reached It means the incrementai

energy ioss equals the incrementai height of the dam, Therefore, the energy los ratio

increases with dam height and is not only a funcbon of step configuration. He also

noted that "... a riecessed top step kept the jet quinst the dm fme, giving a mon

unijionn bss gmdient down the ctrlm face." His results show that energy dissipation

increases up to a certain limit as the step sizes are increased, beyond which there is

limited advantage in increasing the sep height

Diez-Cascon, et al. (1991) studied the hydrauiic behaviour of a prototype stepped

spiiiway, based on tests on a scale modd, The 1/10 scaie mode1 represented a stepped

spiilway containing steps of 30 cm length and 60 cm height. They observed that the

stepped spillway behaved in the same way as a smooth one, with some unknown

roughness, and they identsed the trapped vortices in the steps as the main reason for

energy loss. They presented a theoretical-experimental approach to the h y d r d c

behaviour of skimmmg flow on the maximum air entrainment and sweliing of flow.

They dso observed a 91% reduction in upstrearn fluid energy as a resdt of using a

stepped spiiiway.

Kensen and Reinhardt's (1991) resuits are reproduced in Fig.3.6. It shows that with

increase in discharge the rate of energy dissipation decreases. At Monksville dam,

New Jersey, no terminal energy dissipator structure was required because of the dense

foundation rock at the end of the spillway. A 0.61 m high step appears to produce a

good combination of hydraulic efficiençy and simple constructibility. Also, for

Monksville Dam the designers determined that the training wall height sbouid be twice

that rquired by mode1 similitude at maximum discharge to accoimt for fkeeboard,

accommodate buIlring of the fiow, and reduce spray. For Upper Stillwater Dam,

laboratory studies indicated that, for the design discharge condition, the stepped

spiliway produced a 70 % energy reduction over a conventional smooth spillway.

This, in tuni, led to an 85 percent reduction in the necessary length of stiIliag basin.

Frizell(1992) mentioned that many embanlnnent dams have been identified as unable

to p a s their design 40w without failure due to overtopping. She discussed the use of

stepped spiliways, for providing erosion protection as weil as dissipating energy.

Energy dissipation characteristics of severai site-specific mode1 tests are shown in

Fig.3.7. These data are plotted in prototype values and show the ratio of the kinetic

energy at the dam toe to the total available head versus the unit discharge. The kinetic

energy was calculated using measured velocities on the s t e p near the dam me. The

ratio reduces to the average stepped spillway veiocity over the theoretical maximum

velocity for a given dam height. Most of the information is for dams with 0.6 cm step

height.

Peyras, et al. (1992) studied the hydraulics of gabion stepped spiUways, based on 1 5

scaie mode1 tests. They examineci gabion deformations and factors affecting resistance

to floods. The tests revealed some defonnation of gabions, due to movement of the

Etone f i g . They recommended, for expected floods exceeding q = 1.5 m2/s, that

wire mesh and lachg must be strengthmed It was also advised to seiffen the

zinc-coiited box with a thlld row of extra binding wire and increase the number of

diaphragms in the gabions. In generai, fiom an economic point of view, they fomd

a saving of 540% on project cost, using stepped spiuways.

Chanson (1993) stated that up to 99% of the total head developed at a dam site cau

be dissipated by using a stepped spillway. In his study he consideml the effects of

air entrainment in stepped spiliways. He noticed better eniciency for a stepped

spillway with dopes steeper than 309

Cbristododou (1993) found that energy loss due to the steps depend primarily on the

ratio of yJh, where y, = ait icai depdi, as weU as on the number of steps N, He foimd

that for higher vaiues of y&, the eff- of N, becomes appreciable at a cerrain y&,

and that the relative energy loss inmeases with N,. In his experiments 10 and 13 step

models were w d and discharges rangeci between 10 and 45 Vs. He indicaed that the

most important parameters governing energy dissipation are yJh, and the number of

steps N,. Using Sorensen's (1985) data, he developed a graph to find the amont of

energy dissipaîed (see Fig.3.8). Lt has been established that energy dissipation

decreases with an increase in yJNb .

Bindo, et aL's (1993) tests for the case of the M'Bali dam showed that

1. the discharge coefficient was not affected by the presence of steps;

2. the s&p heights progressively increase toward the toe of the spillway, so that

the flow would not jump for low discharges;

3. the energy dissipation dong the chute varied from 90 % (for low discharges)

to 50 % (design discharge);

4. the hydraulic jump was located within the basin; an4

5. downstream scouring was iimited

'fhey also recommended that "the height of the steps 6e chosen in such u way M, for

the maximum flow , air enten the* some 5 to 10 ni higher t h the elevdon of the

sn1Iing bCLSnI-" They developed Fig.3.9 for detemiining the flow velocity.

Chanson (1994) uidicated bat energy dissipation in a stepped spillway increases widi

dam height Also, for a given dam height and using a stepped spillway, the rate of

energy dissipaiion demeases when the discharge increases. Charnani and Rajaratmm

(1994) presented a method to estimate the energy loss on a stepped spillway with

nappe flow. By andyzing the data of Homer (1969), and Moore (1943), they found

that the relative energy loss couid be up to 97% for y& < 0.8, which is a very low

discharge.

Rice and K a d a . (1996) studied die physicai models for the Saido Creek Site 10, San

Antonio, Texas. It is a RCC stepped spillway with slope 2.5(H) : 1 (V). This saidy

was iimited to ody three discharges. They found the energy dissipation for stepped

models was 71 %, whereas for a smooth model it was 25 %. Figs. 3.10.a and 3.10.b

present typical velocity distributions for stepped and mooth models respectively.

From these figures it can be concluded that with the use of steps die momentum

transfer caused by the rougbness of steps penetrates vertically into the fiow and results

in lower velocities compareci to the smooth model.

RietspnUt outfall,

Souîh Africa

San Paolo weirs,

Brazil 1986

Slope (')

Step

height

(ml

No. of

steps

9,Il and

8

- -- -

3 successive sîepped

weirs

Solid transport and

flood flow controls.

3 successive weirs . - -

Table 3.1. ExampIes of gabion steppd spillways

No. of Steps

Ns

Type of Steps

Slope

( 3 Dam

Height H (m) I Dam Name

De Mist Kraal weir,

South Afnca 1986

Zaaihoek dam, South Afnca

1986

Monksville, USA 1987

Olivettes dam, France 1987

Horizontal

Horizontal

Upper Stillwater, 1 USA 1987

M'Bali dam, CAR 1990

Horizontal

- .

Horizontal New Victoria dam, Ausfralia

1993

Petit-Saut dam Guyana 1994

Table 3.2. RCC dams containing stepped spillways

Dam Name

Bnishes Clough

dam, UK, 18591

1991

Dneiper hycïro plant,

Ukraine, 1976

1 Russia, 1978 l

Slope ('1

LukhoVitsLy dams,

Russia 1978, 1980,

and 1981 1

Transbaikal region 14

1 dam, Russia, 1986 1

Dam

height

(ml

Mpt, Dis.

(m2/s)

Type of Steps

Wedge concrete

blocks inched

downward 8=-5-6

Concrete block

system. Horizontai

steps -

Stepped block systern

Stepped block system

-

Stepped block system

Table 3.3 Embanicment dams with stepped spillway protection in the UK, and Russia;

(Baker, 1989)

Dam and location

Lahontan, NV

-

Brownwood Country

Club, TX

K e d e , IX

McClure, NM

Upper Las Vegas

Unit Dis-

m2/s

6.32

2.29

31.12

Spcing Creek, CO

Goose Lake, CO

Table 3.4. Stepped spiliways for protection of the dow~l~tream dope of embankment

dams in the USA (Frizeii, 1992)

9.04

Hydrau-

tic

height

(m)

33.53

5.79

6.40

2.62

0.85

36.27

Head

(ml

1.83

1.68

7.32

1524

10.67

3-13

Downst-

ream

Slope

@ I V )

2: 1

2: 1

0.8: 1

1.36

0.73

Downstream

concrete facing I

and Placement

technique

Conventional

formed

RCC Mformed

RCC dormed

2.18:l RCC unformed

2.3: 1

1:l

RCC imfomed

RCC unformed

Fig.3.1. Examples of stepped spîiiway shuctures: (a) Aimansa dam (Spain,

1576). @) Tibi-Alicante d.m (Spain, 1594), (c) PabeUon dam (Mexico, 1730)

(cl) Gabion stepped spillway profile

Fig 3.2. Unit h d loss over plain gabion steps (afier Peyras, et ai., 1992)

1 i

T -' - exirting embankmem -

- . . . . ~ * ground Iine

15-24 cm diaineter toc drain

Fig.33. Brownaiwd Coktry Club dam ( a f k Hansen, 1986) (USA)

Fig.3.6. Srepped spluway energy dissipation for Upper Stillwater, MonLnille,

and Stagecoach Dam (after Hensen aud RcuibnrdS 1991)

Fig3.7. Energy dissipation ch&stics at the toe of ncpped rpülways (ifter

F M , 1992)

Fig.3.8. iWE,, w y m rdationship (after Christododou, 1993)

Hbl h

Fig3.9. Row vdocity at the toe of a stepped spillway ( a h Bindo, et ai., 1993)

50

1 .s b O MUN

MIN .W.S. V M M

1 3 a

Fig.3.10. Velocity profile on a spillway: (a) smwth, @) Stepped (fier Rice, and

Kadavy, 1996)

CI

E - t.0

8 I

t SMoOï'H w STATION 134 m O 5 5 t

q-14.5 rn /s/m 0 0.s - J LL

0.0

b

O M U N . MIN

0 MAX

- W.S. .

m

Chapter 4

Expeiimenîal Studies

Physical rnodels of hydrauiic structures are probably the most common type of

hydrauiic model to be studied They are cheap, generaily easy to operate, d easy to

interpret Physical mode1 audies are used to investigate the anticipaîed performauce

of hyeiraulic structures. The model is usually a reduced-size representation of the

proposed hydrdc structure and is designed to investigate specific areas of concem

related to the hydrodynamic performance of the waterways. Modeiling laws based to

the theoretical analysis and laboratory experience have been developed which permit

correct simulation of the pdcular hydrodynamk phenornena under investigation.

Once the model laws ara understood and proper consideration is given to heir lirniting

range of application, successful laboratory investigations can be completed which will

in fact simufate the perfomance of die Ml scaie structure.

The hydraulic model plays a key role in the development of finai design concepts for

hydrauiic structures. Although its most important role is to ver* the adequacy of the

design concept, it is also used to f i e iune details such as length of a h i h g basin, or

the height of a spiI1way's training waüs, etc., wtiich normaiiy produce construction

cost saving that more than offset îhe cost of the hydraulic mode1 investigation,

The principal goai of this study was to experimentaiiy investigate the energy

dissipation and the location of the fme surJace w-r on ogee Jteppcd spiUwqy

models installed in a rectangular channel. This chapter provides details on the

experimentd setup and apparatus used as well as the procedures foiiowed during the

course of the laboratory investigations.

4.2. Experimentsl Setup and Meastuemen&

The experimental programme was performed in the large adjustable-dope 5ume

located in the Hydraulics Laboratory of the Department of Civil Engineering,

University of Ottawa- The glass-wall flurne is 12 m long and has a cross section

0.387 m wide x 0.61 m deep. A head tank with a sluice gate control section is located

at the upstream end of the flume. Water supply to the tank is via a 203 mm diameter

pipe with aperfomted-T end section that ensure as even fiow distribution to the tank.

Screens are instaiied in the tank to eliminate any residual large-scale turbulence in the

flows exiting the tank Discharge to the 5ume is adjusted by a butteffly control valve

in the 203 mm diameter supply line. A variable-height weir is located at the flume's

downstream end and flows exiting the flurne discharge to a collection tank situated

directly under îhe flume's outiet Various types of sharp-crested weirs can be installed

in the collection tank to mesure fiow rates in the flurne.

The flume is supported on two (wide flange) steel beams, which are pivoted close to

the upstream end of the flume and supported on two manually-driven screw-type jacks

close to the downstream end of the flume. Longitudinal dope adjustment of the flume

is achieved through vertical movement of the jacks. A drawing of the general

arrangement of the flume is shown in Fig.4.I.

Water d b c e pronles within the test section of the flume were recorded as foUows:

(0 a series of piezorneter tapping points, located at intervals along the centreline of

the flume floor and connected to a manometer board by polyethylene tubing, provided

depth readings upstream and downstrearn of the various model spillways tested.

a vernier point-gauge, having on accuracy of 0.1 mm, were used to record water

surface profiles along the downstream faces of the model spillways. po t e : due to the

inherenîiy unstable water surface associated with a model spillway flow, the actual

measurement accuracy when recording our mode1 spillway flow profiles was closer to

* 2-3 mm].

Flow over hydraulic structures involves significant vertical velocity components and

it is therefore necessary to constnict the model to a naturd (Le. undistorted) scale.

Any exaggeration of the vertical scale would cause an unacceptable exaggeration of

vertical flow cornponents and for this reason must be avoided .

Because the flow is dorninated by gravitational effects, rnodels of free surface flow

around hyâraulic structures mut be scaled according to the Froudian criterion. To

ensure freedorn from scale according to the fairly large and scaies ranging fiom about

1 : 1 5 to 1:60 are common (Sharp, 198 1). U.S. Bureau of Reclamation (1 980) spillway

models for large dams have typically been constnacîed to scale ratios between 1:30 and

1: 100. The Bureau recommends that medium-size spillway models not be smaller than

The scaie chosai will generally depend on the f d t i e s available. Facilities Wre flurne

skes, discharge and s6 on The model scde rnust saîisfL two requiremmts:

(i) it must allow the whole model to be incorporated in the diannel available, and

(ri) it must be such that the model discharge is within the capabiliw of the existing

P-P-

Because of cmitatrkm considerations in spiliway design, stepped spiIlways are practical

ody for SA)&- t~ modemte- height dams. Accordingiy, based on the dimensions of

the University of Onawa tilting fiurne and also on the capaaty of the laboratory pump,

a hypothetical spillway (height EI, = 19.0 m and design head, H, = 2.5 m) was

selected as the prototype for this model study. The model to prototype scaie ratio

adopted for the study was 150 which, given the dimension of the flume, meant that

the model spÏllways were 38.0 cm high and the design head was 5 cm. The model

spilIways, which were constmcted nom piexiglass blocks. included (ij smooth ogee-

profiIe spillway models. aud (R) various combinaîions of stepped ogee-pronle

spillway models.

4.3.1. Smooth ûgee Spühvay Models

AU ogee-profile spillways investigated in this study had vertical upstream faces. The

spillway crest geometry used was that proposed by the Waterways Experiment Station

(WES) (1977). It includes the three-arc curve upstream fiom the crest section.

Downstream nom the West the following power function applies (see sec. 2.3.1 for

more detail):

This shape conbiues dom to the point where the slope of the cunre mets the

terminal spiiiway dope. In this study 3 different terminai slopes were investigated,

namely: 4S0, 50: 60'.

Based on Creager, et ai's recommendation (1964) a 14 cm radius of curvature was

chosen for the spiilway toe transition c w e for ail models. This ensured a smooth

transition of the flow fkom the spillway ta the dowumeam channel. In this shidy die

profiie of these models were produced using the cornputer program GEMLSPS.

Figs.4.2 shows the smooth 45'. 50'. 60' models. The experimental outputs were

plotted in Figs. 4.3, 4.4. and 4.5.

4.3.2. Steppcd Ogee-hflle Spülway Models

In this study the effeaiveness of different og-pmofiic st+ spiilway configurations

were examined by considering : (9 difEerent downstream slopes; (ii) different step

sizes; and (iii) diBerat arrangement of the steps.

For each spillway dope considered, the various stepped spilhuqy pmfiles were arrangeci

so that the tips of the steps codonnecl to the correspondhg smooth pronle. The steps

themselves were introduced to the pmnle just below the crest section. Each of the 3

(buse) steppd models coc~slfllcted in two parts: (0 the c ~ s t part, which was 1/3 of the

totai spiiiway height (Hdi), and (fi) the bonom part, which had a height = 2/3 H&,.

While each parts of the models had a d o m step size diroughouî its profile, the step

sizes of the base modets were different fiom each other.

Based on the iiterature review (Sorensen, 1985, Chriçtododou, 1993, Bindo, 1993,

Rice md Kadgvy, 1996) two different step sizes were chosen:

1. 1/20 which considered as Iqe-step sizes;

2. 1/40 H, which cunsidered as smd-srep sizes.

Since the CRSZ and bonom parts of the Merent base mode1s couid be interchanged,

this pennitted testing a wide variety of srep arrangements, as foiIows:

Model 45-CLBL

This model has a (1H: 1V) downstream slope. The crest part has 10 large-size steps

and the bottom part has 16 large-size steps. Except for a few steps on the upper

portion of the crest section the height of steps are identicai- The few steps on the

upper portion of the crest have a smaIler height to provide the jet (Sorensen, 1985).

Among ail 45" srepped modeis this model has the largest stcp size and the lest

nurnber of steps. Plate 4.1 shows this model- The water s u r f . profiles are plotted

in Fig- 4.6.a

Model 45-CS-BL

This rnodel bas a (LH : 1V) dowasaearn slope. The crest part has 20 smaibsize steps

and ttie bottom part has 16 large-size sreps. Therefore, tbis mode1 provides smoother

face on the upper part of the model. Plate 4.2 shows thîs rnodel. The water surface

profiles are plotted in Fig. 4.6.b.

This mode1 has a (18: IV) downstream dope. The =est part has 10 large-size steps

and the bottom part has 32 mail-size steps. In this mode1 the upper part is rougha

than the base part Thus more energy shouid dissipate on the crest part and there di

be a smooth transition to die stilhg basin at the toe of spdiway- The water Nface

pronles are plotted in Fig. 4.7.a

'Rüs mode1 has a (1EE IV) downstream dope. The crest part has 20 small-sïze steps,

and the bottom part has 3 2 sm&-size steps. This model has the highest number of

steps, and also the d e s t step ske. The water Sunace profiles are plotted in

Fig. 4.7.b.

This model has a (1H: l.lg2V) downstream slope. On the =est part this model has

7 large-sïze steps, and on the bottom part it has 12 large-size s t e p Except the few

steps on the upper part of the crest the height of steps are identical. 'Inose few steps

on the upper part of the crest have mialler height to provide the j e t Plate 4.3 shows

this model. The water surface profles are plotted in Fig. 4.8.a

This mode1 has a (1H: 1.192V) downstream slope. On the crest part this model has

12 small-size stops, and on the bottom part it has 12 large-sïze steps. The smaller

steps on the upper part provides a smooth transient to the base part Plate 4.4 shows

this model. The water surface profile are plotted in Fig. 4.8.b.

Model 50-CGBS

This model has a (1H: L.192V) downstream slope. On the crest part this model has

7 large-size steps, and on the bottom part it has 24 small-size steps. In this model the

crest part is rougher than the base part Thus more energy should dissipate on the crest

part and there will be a relative miooth tranntion to the stilling basin at the toe of

spillway. The water &$ce profiles are plotted in Fig. 4.9.a

Model 50-CSBS

This model has a (1:H 1.I92V) downstream slope. On the crest part, this model has

12 smd-size steps, and on the bottom part it has 24 smd-size steps. Among all the

50' models, this model has the largest number of steps, and the smallest step size.

The water surface pronles are plotted in Fig. 4.9.b.

Model 60-CLBL

This mode1 has a (1H: l.732V) dowastrearn slope. On the crest part this model has

7 large-size steps, and on the bottom part it has 12 IargeQze steps. Except the few

steps on the upper part of spillway the height of steps are identical. This model has

the largest step she and the least amount of srep among ail 60' models. Plate 4.5

shows this model. nie water d a c e profiles are plotted in Fig. 4.10.a

Model 6û-CS-BL

This mode1 has a (m 1.732V) downstream slope. On the crest part it has 12 smaii-

size steps and on the bottom part it has also 12 Iarge-size steps. The steps on îhe crest

part have lower height than the bottom part It provides a smoother transition to the

rough surface at the doWLlStream- The water surface profiles are plotted in

Fig. 4.10.b.

Model 60-CGBS

This model has a (1H: 1.732V) downstream slope, On the crest part it has 7 large-size

steps, and on the bottom part it has 24 small-size steps. In this model the crest part

is rougher than the bottom part. Thus more energy shodd dissipate oa the crest part

and there will be a relative smooth msition to the stiiiing basin at the toe of spiiIway.

The water surface profiles are piotted in Fig. 4.1 1 .a.

Model 6O-CS-BS

This model has a (1H: 1.732Vj downsîream slope. This model has 12 smaü-size steps

on the crest part, and 24 smd-size steps on the bottom part. The steps are identical,

This model provides smoother model than the rest of 60' models. The water surface

profiles are plotted in Fig. 4.1 1 .b.

Fi8.4.10. Observed water surface profiles for (a) 60-CL-BL model, (b) 60-CS-BL model

Plate 4-1, Modei 45-CL-BL

Plate 4.4. Mode1 50-CS-BL

Plate 4.5- Mode1 60-CL-BL

Chapter 5

5.1. Dmp SbuehuP

Drop structures or free overfalls are usualiy used in irrigation canal systems, flood

conveyance channels, and degrading streams. These structures in the bed level are

provided at suitable places to economize the cost of the earthnll. They may be

vertical, or inched, Fig.S.1. Rectangular vertical &op structures are probably as

frequently constructed in practice as the inclineci ones.

Moore (1943) studied two physical models of drop structures having 0.15 m, and

0.45 rn height. Hïs study is very useful h m a practical viewpoint. He found that

the energy loss at the base of the faIl is a function of the relative fa11 (h/yJ. He

also noted tùaî the depth of water standing behind the fa11 is greater dian the depth

of water shooting from the base of the fall. Using the momentum equaîion he

found that the pool depth, y,, can be predicted by :

White (1943), in the discussion of Moore's paper, developed a meihod to predict

the energy loss at the base of the &op. He used the following assumptions:

1. the recirculating flow ia the pool at the bottom of the drop was the same

as the backwater flow;

2. the veIocity of the supercritical stream immediately downstream of die drop

was the same as the d o m velocity in a thicker siream at the side of the

pook

3. the angle of the f-g jet was not âffected by the presence of the pool;

4. the energy loss at the drop was due to the mkïng with the pool; and

5. the effect of pool in îhe calculating the velocity of jet, V, was negiected

These assumptions were questioned by rnany researchets, including Moore, in

pariicular the velocity is not d o m in the thicker stream (2d assunption), and the

pool would effect the flow. White (1943) found

Rand (1955) used his own sets of experimental data and Moore's &ta to deveIop

the foilowing wnelaîions:

where

L, = distance h m the waü to the position y, .

Gill(1979) modined White (1943) assumptions. He assumed that

1. the d o m velocity occurs in the thinner layer below the pool;

2. the velocity is not constant in îhe mixhg zone;

3. the effect of the pool ia the calculation of velocity is considered as

4. the effect of the pool in regards to energy loss is neglected

Gill's assumptions modif'ied some of the weakness of White's assumption. But

assuming the pool would not effect the energy l o s was incorrect GiIi's

relationships are:

where

B = the angle of the jet where it hits the bed

El-Khashab, et al. (1987) developed theoreticai mode1 for a trapezoidai weir &op

structure, ch was operating in a trapezoidal channel. A good agreement was

reported between the theoretical and observeci data

Chanson (1994), using the trajectory equations of the nappe centreLine, developed

the following expressions:

Rajaratnarn and Charnani (1995) examined experimentally the energy loss at the

base of a drop structure. They developed the foilowing relationships:

where

AE =h -El

The energy upstream of the drop E, can be written as:

E,, =h+l.Sy,

and

Fig.S.1 .a shows a vertical rectanguIar drop structure. The foiiowing assumptions

apply to Fig.5.I.a and the anaiysis performed in the present study:

1. the shear force at the bed is negligible; and

2. the angle of ihe jet where it impacts the pool is the same when it impacts

the bed

Applying the momennim principle tu sections 1-1 and a-a , gives:

where

where

y = specific weigtit of the fi4 and

p = mass density of the water.

Therefore the eq.(S.20) can be written as:

The criticai depdi, y, , occurs some distance upstream of the bnnk. Applying the

momentum equaîion for sections c-c and a-a gives:

Assuming no energy loss between sections c-c and a-a, then

where

Vc = critical velacïty.

Therefore eq. (5.23) can be written as:

Based on the data obtained by Moore (1943), Rand (1955), am

Charnani (1995), one can write:

(5.25)

1 Rajaratnam and

Therefore, soiving eqs. (5.21), (5.22), (5.25), and (5.26) gives:

and

Figs.5.2, 5.3, 5.4, and 5.5 compare the above equations with the experimental data

obtained in this study. Fig.5.3 shows that widi increasing discharge the depth of

pool fonned on the &op structure would increase. When y, = h, the depth of pool

would be y, = h, which means the whole space is filIed

The coefficient of detrminatioa (9). the stcrasticai F-Observed for eqs. (5.26).

(5.27). (5.28). and (5 -29). and F-Criticai considering a I % level of sigm!çance and

the correspondhg degrees of f~edom have are presented in Table 5.1. Since for

the above equations F-Obsewed » F-Criticai, it can be concluded that good

agreement was achieved bemeen the semi-empincal pioposed equation and Moore's

(1943), Rand's (1955), and R'aratnam and Charnani's (1995) data Figs. 5.6, 5.7,

5.8, and 5.9 show the observed parameters vernis the predicted parameters.

Fig.5.4 presents the y& vs cos B. It can be concluded that the angie of the jet

where it hits the bed is decreased when the discharge is increased The value

predicted by eq.(5.28) is fairly close to the experimental data observed Fig.5.5

presents the energy loss at the base of a drop structure. It can be seen with

increasing discharge the energy loss wili decrease. When y,= h, the energy loss

rate would be the minimum (0.1). Applying eq. (5.29) shows a reliable agreement

with the available experimental data

In general, in this research some of the weokest assumptions for the theory of

hydraulics of rectangular vertical &op structures are replaced with better

assumptions. The resuiting semi-empincal equations found show very reliable

agreement with tbe available experimental data

5.2. Flow over a Stepped Spilïway

The flow over a stepped spillway can be divided into nappe flow and skrmming

flow regimes (Fig.5.10). Peyras, et ai. (1992) described two types of nappe flow:

(i) q p e flow with a My-developed hydrauiic jump for Iow discharge aud

d flow depth, and

(i) nuppe flow with partialiy-developed hydraulic jump.

In the nappe fiow reghe, the flow from each step hits the step below as a falling

jet, with the energy dissipation occurring by jet break-up in the air, and jet mixing

on the step, with or without the formation of a partial hydraulic jump on the s&p.

in the skintmzng flow regime, the water flows d o m the stepped face as a coherent

Stream, skimming over the steps and is cushioned by the recirculating fluid trapped

between them.

5.2.1. Nappe Flow

Consider a nrpttpe flow with the unit discharge q, Fig.5.11. When the flow bits the

step beiow, part of the flow i s directed upstream (against the flow direction) and

part of the flow continues downsîrearn. When the upstream-directed flow meets the

vertical step wd, it is forced to reverse direction and a pool is f o m e d This flow

is recirculated in the pool and joins the downstream-directed flow with a lesser

velociîy .

Based on the experirnental chta of Essery and Horner (1978) and Peyras et al.

(1991) for gabion stepped spiuways, the onset flow condition for chute stepped

spiiiways may be estimated as:

Skimming flow will occur when y& > (y&)-.

Rajaratnam (1990) suggested that nappe flow occurs when y& c 0.8. Based on

his experiences with South Afncaa dams, Stephenson (1991) suggested the

foiiowing conditions for nappe ftow

and

A recent te-analysis (Chanson, 1994) showed that skimming flow regime on a chute

spiUway occurs for discharges Iarger than a criticai value dehed as:

This relation was obtained using a hear regression of the data of Essery and

Horner (1978), Peyras, et al- (1991), and Beitz and Lawless (1992) with a

regression coefficient of 0.79, which indicated a relatively poor correlation. This

was mostly due to infiltration through the gabion material of Peyras, et ai. (1991).

Theoreticaliy, to have a M y nappe flow, the length of steps shouid be larger than

the length of a fûiiy-developed hydrdic jump, Lj, plus the length of drop (Là), or

For the vaiue of L, in eq(5.34) , eq.(5.13) which was deveIoped by Chanson(l994)

cm be useci Chaudhry (1993) suggested:

One can find y, fiom the classical hydrauiic jump relationship:

where F,, =Fr& No.

From eqs(5.26) and (536) w e can &te:

By applying eq@. 13) and tq(5.3 7) in eq45.34) the following critena for the nuppe

fiow can be developed:

Fig.S.12 preseats eq.(5.38) and the cnteria deveioped for the data of Essery and

Horner (1978) and Peyras et al. (1991), Rajaratmm (1990), and Stephenson (1991).

It can be seen a nappe fiow regime for higher discharges is possible, but tbis hi&

discharge needs veq fiat dope. Also for steeper slopes, it shows that the discharge

must be very low to have a nrrppe flow regime. Because of the implied large

increase in conaete volume in the case of concrete gravity dam tbese conditions

make such a structure unattractive from an economic viewupoint The step height

can aiso be a problem if the concrete thickness is Iimited This is because the

forces caused by impingïng water can crack the concrete and cause interna!

stresses, which add to the rapid detenoration of the concrete.

Fig.S.13 verines the eq.(5.38) with the experimentd data fomd in this study . It

cm be seen diat these criteria are highly applicable to die r d t s of measured

values. This figure iadicates that a nappe flow is also possible for very hi&

d o m e a m dopes, but it needs very low disdiarges. At these discharges most of

energy can be dissipateci by using the stepped spillway.

Skimming flow occurs at modemte to high discharges No nappe is visible and the

spillway is submerged bene* a strong, relatively smwth cwent In the skiniming

flow regime, the extemai edges of the steps form a pseudo-bottom over which the

flows pas . Beneath this, horizontal-axis vortices develop, nIling the zone between

the main flow and the step. These recirculating vortices are maintained through the

transmission of shear sires from the fluid flowing past the edges of the steps.

Skimmzng flows are cheiacterized by large friction losses and the air-entrainment

process.

DomStream of the point of inception, a layer containkg a mixture of air and water

extends through the fluid Far dowumeam, the flow will become d o m and for

a given discharge any measurement of flow depth, air concentration, and velocity

dishibution will not vary dong the spillway. Chanson (1994) staîes that, when the

skinrming flow becornes My-developed, a stepped spiIIway behaves in the same

way as an unstepped one.

From approximatdy die aest &on of the spinway, a turbulent boundary Iayer

wiii deveiop on the conmete surface. A short distance downstream of the point of

inception of this boundary I r y a . where the boundary layer extends to the water

surface, the turbulence next to the fne d a c e become large enough to initiate

namal alfia-surface aeration. On a srnooth spillway, Keller and Rostagi (1977)

suggested the

and

where

& = the distance from the start of the growrh of the boundary Iayer,

yr= the depth of flow at the inception point, and

k: = rougbness of spillway surface

For a smooth conaete spiliway Cain and Wood (1981) found th&

rnechanisrns of air entrainment are similat. ûnce the flow becomes My-deweloped

then a stepped spillway can act as a miooth spillway with high roughness.

In nappe fiows, air entraInment occurs near the impact point of the falling jet with

the horizontal step and the hydraulic jump. On the other hand large de-aeration

also occurs dowastream of the falling jet and downstream of jump. Therefore, the

net aeration is s m d and it is believed that the effect of air entrainment on nappe

flows cm be neglected

In skimming fîows, the air entrainment increases the flow depth afier the point of

inception. Whai the flow becomes rmiform the flow depth stays constant. It has

been observed that at low discharges the air entrainment is higher than high

discharges. In generai to h d the energy dissipation, if one applies the bulLuig

effect the r d t is highly overeshmated The local air concentration, Ca, is defined

as the volume of air per unit voiume, whÏch can be expressed as:

It is generaliy accepted that air concentration in the flow on a wide chute (widdi-

depth ratio m e r than 5 ) spillway can be estirnated by U S Amy Corps. of Engg.

formula (1977) in fps uni& as follows:

sine Ca = mi b.(F) + 09)L

The results of Knauss (1979) indicaîe a comparable rate of air entrainment for both

smooth and rough flows. Hïs relationship takes the fonn:

Chenson (1993) States that the d o m air concentration on a stepped spillway cm

be expected to be simila- to that for a smooth spillway, where the air concentration

is a function of dope only. Sentruk (1994) suggested the foiiowing ernpincal

equation for the air entrainment on an ogeesmouth spillway:

5.5. Flow Resisbuice

In any reai fluid flow, energy is continuowly being dissipated This occurs because

the fhid has to do work aga& resistance onginaihg in the fluid's viscosity.

Whether the fiow is l h a r or turbulent the basic resistance rnechanism is the

shear stress by which a slow-moving Iayer of fluid exerts a retardhg force on an

adjacent layer of faster moving nuid Between the adjacent Iayers moving with

different velocity there WU be a retarding force exerted on the fmer-rnoving layer,

because it is contiauously receiving low-velocity fluid and losing high-velocity

fluid. This retardabon will be in the forrn of a shear force, since it acts parailel to

the interface between the two layers. It is well established by observation that fluid

actuaüy in contact with a solid d a c e has no motion dong the surface. The

created transverse velocity gradient enables the soiid d a c e to exert on the moving

fluid a drag force, which is transmitted outwards through successive faster-moving

fluid layea Flow resistance therefore depends on: fi) the presence of solid

d a c e s ; and @) the strength of the viscosity or twbulence.

Studies of slommmg flow over large elements in open channels and pipes indicate

that the classical flow resistance must be modifieci to taice into account the shape

of the roughness elements. hdeed, the fiow resistance is the sum of the skin

resisiaace and the fum resistance of the steps. Neglecting the effkct of the flow

aetation, Chanson (1 994) showed bat the fiiction factor was a fimctîon of the step

roughaess height and channel dope.

wùere

f = Darcy-Weisbach ection factor; and

Da = hydtaulic depth, ( =4$) ; where P = wetted perimeter.

Noori (1984) experientaüy shidied the form drag resistance of two-dimensionai

flcll-stepped channels. He expressed the drag coefficient as:

Based on Gnachuk et aL's (1977) data and Noori's (1984) data for the flat dope (

8 < 12 ' ) and 0.02 < k',/D, < 0.3 the foiIowing equation is obtained:

Fig.5.16 shows good agreement betweea eq45.53) and the data obtained by

Grinchuk et al. (1977) and Noori (1984).

Fig.5.17 shows the d t s obtained fiam several authors for steep slopes ( 8 >

279. A cornparison between Figs.5.16 and 5.17 shows substantial increase of the

fnction factor. For fiat Ehannels and 0.02 < k'/D, < 0.3 the space of recirculating

fluid beneath the pseudo-bottom formed by the step edges is relatively thin. The

recirculathg vortices do not fill the entire cavity between die edges of adjacent

steps. Therefore stable recircdating eddies cannot develop. For steep dopes, the

space between the edges of the steps was fïiied completely by ihe recirculating

vortices. The energy dissipation and the flow resistance are functions of the energy

required to maintain the circuiation of vortices. Chanson (1994) indicaîed that the

stable recircuiation occurs for slopes Iarger than 27' on a stepped chute.

Vittai et al, (1977) performed experiments in open channel with two dimensionai

îrïangular roughness elements. For a roughaess height to Iength ratio of 1/5, their

red ts satisfy:

where

L, = the rougimess spacing.

Rajaramam and Katopodis (1991) studied the hydraulics of steepps fishways.

Ushg theoretical considerations and experïmental observations, they developed an

expression that relates the flow rate, the siope of the fishways and the depth of

flow. They also obtained friction factors for sreeppuss fishways in the range of 0.4

to 4.

Stephenson (1991) by assuming the skimming flow reaches d o m depth found:

This equation has a singularity point when y,& = 0.067. This equation was

recommended for y& MI.2.

Gevorkyan and Kalantarova (1992) studied the flow past stepped teerh opposed to

the flow with a rougbness height to length ratio of 1/4. ïhey reported the

following equation:

Chanson (1994) for the exphent s of aartrrng and Scheuerlein (1970) and

Scheuedein (1973) developed the following empirical eqdon:

These data were obîained for mountain rivers and mcknll channels. This equation

is a very rough estimation of the fiction fmor. The resuits obtained shows very

hi& fiction factor- Thetefore applying tbis equation estimates very high fiction

factor.

Toai (1994) developed the following expressions:

(i) for6.69(H):l(V)

(Ùf) for 2 (H) : 1 (V)

(M) for 0.75 (H) : 1 (V), when y#, >1.80

Fig.5.18 presents these equations. It can be seen that the y&: decreases, the

fiction f w r increases. Applying eqs.(5.58),(5.59), and (5.60) obtauied by Tozzi

(1994) gives very low fnaion fiaor.

Qibei and Baker (1995) studied the resistance forces on wedge-shaped block used

for spillway protection. Thy assumeci that the fiow passing over the steps is

simiiar to that passing through an abrupt enlargement For a low-siope downstream

surface, they developed the following expression

where

Cc = the Chezy coefficient-

In the next chapter thc friction factor based on the experimental data of this

research will be presented.

5.6. Energy Dissipation

Althou& the mechanisms of eaergy loss are quite Werent between the q p e flow

and s h m i n g flow regimes, but both flows can dissipate a major proportion of the

flow energy. A stepped spiilwoy with nappe flow can be considered as a

succession of dmp stnictwes. The energy dissipation occurs by jet breakup in the

air, jet impact aad jet mixïng on the steps, with the formation of My-developed

or partially-developed hydraulic jumps on the steps. In a skimmmg flow regime,

the steps act as large roughess elemenis. Most of energy is dissipated to maintain

stable hohntal vortices ben& the pseudo-bottom formed by the extemal edges

of the steps. The vortices are maintained through the transmission of turbulent

shear stress bmmen the skimming Stream and the recircuiating fluid underneath.

5.6.1. Eneqgy Dissipation for Nappe Flow

C b n (1993) assumed that the head loss at any intermediaty step equals the step

height He expresseci the total head loss for fully-developed hydraulic jump as:

in dis equabon al1 assumptions of Rand (1955) were applied. Eq,(5.62) is in good

agreement with Moore's (1943) data, but d e n die number of steps increase it

varies significantiy.

Chamani and Rajarairiam (1994) presented a method to estimate the energy loss on

a ssepped spiilway with nappe flow. Anaiyzing the data obtained by Horner

(1 969), they introduced the concept of awertllqe rieldive energy loss per step. The

found the following equaîïon:

where

p, = the proportion of the energy loss per step;

94

a, b = coefficients described by the following ecpations:

AU these equations are valid for y,& < 0.8. They found that the relative energy

loss could be up to 0.97. Al1 these correlations were based on the Eorner's (1969)

data

The rnaxllnum energy can be stated as:

E,, =H, + 13y,

Assuming the verticd distance between the crest and the first step as the height of

one step, then

Energy ai the base of a spillway can be expressed as :

Referring to eq.(5.26), eq.(5.69) can be expressed as:

Therefore the energy dissipation can be written as:

Fig.S.19 the data obtained by Moore(L943) and Essery and Homer (1978)

superimposeci on eq. (5.71). This equation does not have the Iimitafions applied

to the Randts (1955) equationo which was used by Chanson (1993), or the

correlation of Homer(l969)'s data which was used by Chamani and Rajaraînam

(1 994).

Frorn dieir experimental data, Pqnas et ai. (1991) found diat the rate of energy

dissipation of nappe flows, with partially-developed hydraulic jumps, wrts within

10% of the values obtained for nappe fiows widi Mly-developed hydraulic jwnps

for same flow conditions.

Table 5.1. Statistical parameters of die developed equations for a single vertical

drap

Eq. No.

(5.26)

(5.27)

(5.28)

(5.29)

3

0.97

0.97

0.87

0-93

F-Observed

226.3

226.3

45

86.33

F-Critical

5.85

5.85

9.5

5.93

Fig.5.1. Simple drop structures (a) vertid, (b) inclined

Fig. 5.2 h n p i s a n of eq. (5.26) wdh experimental data nOm various sources

Fig. 5.3. Compkon of cq. (527) with d e n t a l data h m various sorpces

0.00 O. 10 0.20 0.30 0.40 0.50

y p

Fig5.4. Camparison of eq. (528) wiih R q j m and chanmi's (1995) &ta

Fig. 5.5. Camparison of eq. (5.29) with ercperimental data h m various sou~ces

Observed y / h 1

Fig 5.9. Obxned aE / Elversus cornpuMd AE / Eo

- 0.00 1 1 I I I I I 1 I I 1 I 1 I

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70

Observed A E/E O

Fig-5-10. Charaaeristic fiows over a stepped spillway: (a) nappe fiow with M y

developd hydrauiic jump, @) nappe flow with partidiy developed hydnulic

jump, (c) stimming fiow

(S kim ming Fiow Regim e)

I

I 1 I I I 1 I I 1

Fig. 5.12. CQmpariScm of eq. (5.38) with various aitaia developed by 0 t h

Fig. 5-13. Comparisan of eq. (5.38) with m e n t a data

fiom various sources

Fig-5-16 Fricfion f m r for stepped channds on flat dopes (0 e 12.)

(aftu Chansen, 1994)

- s H

Fig.S.17. Friction fMor of steppcd chauncls on Sap dopa (0 > 27' )

Fig. 5.18. Friction Factor for stepped aidaces ( a f k Tozzi, 1994)

0.16 - -

0.14 - -

0.12 - -

0.10- -

0.08 - -

0.06 - -

0-04 - -

0.02

- a7SOri (v)

- 5 %

% % - - - - -

I I I I I f I I I I l I I I i

1.00 10.00 Y* c

Fig. 5-19. Comprkm of Eq. (5.71) with Exprimental data

Chapter 6

The most common types of hydrauiic mode1 studied so f a . were physicai-type

rnodels, which are used to study the performance of hydrauiÏc structures. These

models have an important mle in the design and development stages. The Literanire

review carrieci out in Chapter 3 showed that the stepped spillways studied, so far,

were site-specific. The experimental models and setup were explained thoroughiy

in Chapter 4. In this dapter the resuits obtained fiom the experimental programme

would be discussed

6.2. Discbuge Coefficient

In Chapter 2, the ra t iod for a coefficient of discharge for a smooth s p i l i w ~ was

preseated In this section, the coeflcient of dischurge for 12 stepped spillway

models is determinecl on the basis of experimentd observations for the different

models. In order to determine the effect of the steps on the coefficient of

discharge, the upstream head above the crest, and the corresponding discharges

were measured- At Low heads it was noted that the water d a c e upstream of the

crest is essentiaiiy horizontal a short distance upstream of the crest By increasing

the discharge this distance aiso increases. To ensure that the effect of the curvature

of water d h c e upstream of the crest muid not effect the measurements, the head

measuring station was Iocated about 12 & upstream from the spillway crest

Table 6.1 lists the discharges and the corresponding upstream heads, H, for al1

models. nie H-Q relasionships are also plotted in Figs. 6.1, 6.2, and 6.3. The

rating cuve for these modeIs talres the form:

Comparing with the smwîh models, it was noted that the coefficient of discharge

increased for sfepped spiïlways (by approximately 8%). AIso it was observeci that

the dowusiream dope of the models does not effect the àiscimge coeficient.

Thedore it can be concluded that

6.3. Friction Factor for Stppcd Spülways

Since the fluid bas to do work against resistance, the enetgy is being continuousIy

dissipated. h the skanmmg-flmv mgime on stepped spiLiways, the resistance

mody depends on: i) the shape of the SUrfaee, and ü) the vortices on the steps.

In Chapter 5, differer~t methods to determine the friction factors for rough

boundaries were reviewed It was fomd that the friction factor found by these

methods couid not be appiied to stepped spillways with large downstream dope.

in this setion the maiu effort was to develop a gened equation for the friction

factor for the different stepped spüiway mo&Is studid

For n o n - d m gradually-varied flows, the friction f m r cm be found from the

energy equation:

wher e

AH = îhe btai head Loss over a distance Aq and

U â s = the friction dope.

Based on the resuits of our experimental studies and using the Grid S m & Method,

the foiIowing equatîon develops:

Figs, 6.4,6.5,6.6, and 6.7 show the cornpaxkm of the experimental data obtained

in th& research with eq(6.4). Based on a gcwidness of fit test (8 = 0.968, F-

Observed = 536 » F-Cnticd = 3.6 ), a good agreement between the experimental

data and eq. (6.4) is observed Fig. 6.8 shows the obsenred f versus the computed

f h m eq. (6.4).

Figs. 6.4,6.5,6.6, and 6-7 plotted y, / k: vs f for differeut spillway dopes. It can

be seen thai, as y, / decreases the f increases. Eq. (6.4) is a general equation

to find the f . This equaîion shows that the higher the Q the d e r the f will be.

This consequeme is mostiy due to the efféct of the trapped vortices on the main

spillway fiow. With increasing discharge îhe &ect of the trapped vortices, (Ïn the

spaces between the steps) on the main stteam will be reduced-

Figs 6.9, 6.10, and 6.11 compare y,, / k'* vs f for the different stepped models.

It can be seen that the CSBS and CLBS models provided the highestf, whereas the

CLBL modal has the d e s t fiction facbr. This r d t s mostly because of the

Iarger nmnbet ofsteps in the CSBS and CLBS modek Concerning thef, it would

appear that the number of steps is more important than the size of the steps.

6.4. Encigr Dissirnon for Skïmming Flow

In the skïmrnmg-frm m e , moa of the energy is dissi-pated in the maintenance

of stable depression vortices between the steps. Chanson (1993) proposed t h e for

an open chamel stepped spillway, if uniform flow conditions are reached at the

dow~lstrearn end of the spiliway, the total head l o s can be expresed as:

To find the en- a any point dong the spillway Bernoulli's equation is use&

z = elevrtion above datrmi, and

p = hydrostatic pressure.

A hydrostatic pressure distribution is a gwd assumption for an open-chamel flow

on mild slopes. However, as the dope increases the pressure distribution no longer

remains hyhstatic. Therefore, a pressure distribution correction must be made on

steep dopes, which is:

Near the crest aad the toe of the spillway the streamhes have a substzintiai

m a t u r e in the vertical plane- Tfie piezomemc pressure at the boundary mus be

conected by an appropriate fhctor, This factor is quai to the force produceci per

unit area of a mas of mer undergoing cenû&gal acceleration- For die charme1

invert, the factor is determined h m the foilowing relatioaship:

C, = centnfugd acceleration factor,

R, = radius of mature -

Therefore, for the c ~ s t region:

P = YY,, -Cg

and for the toe region:

Substiniting the vaiues found by eqf6.9) in eqs.(6.10) and (6.1 1), the corrected

pressure for the required locations can be found Applying eqs. (6.8), (6.10), aad

(6.11) one can find the energy at any intermediate step.

In skimming-flm mgime, d e n the discharge is low (035<H/Hd < 03), the lower

nappe hits the sep tips. The water will be projected into the air and reaîîached a

few steps d o ~ ~ ~ ~ t r e m . A h , some of the water upon reattachmenî, wii l be

. sphshed into the air. Because of this eEect, measuring the water &8ce profiIe at

For different dopes the energy dissipation rate (aE/E0) dong the spillway bottom

axis (horizontal axis) were plotted in Figs. 6-12 ta 6-23, In Figs. 6.24, 6.25, and

6.26 the energy dissipation tate aE/E, vs x/EF, are also plotted for the smooth

models. These figures graphically illustrate the energy dissipation h m the crest

to the toe of the spiliway. Cornparhg Figs. 6.12 to 6.23 with Figs. 6.24 to 6.26

it cm be concludeci that the stepped modeIs dissipate about 4 times of their

corresponding smooth models- From Figs. 6.12 to 623 it cari be seen that the

energy dissipation in the spiIlway section immediately downstream fiom the crest

beginning (a 4 . 5 ) have almost the same dissipation rate for al1 discharges. For

lower discharges, when WEI,, I 1.25, and x/H, s1.5, the energy dissipation rats is

significantly higher- Upon increasing the discharge, the rare of energy dissipation

in the middle part of the spîiiway wodd decrease. At a point near the toe of the

spillway, this raie would sharply decrease and becorne aimost zero, i.e. the energy

dissipation bewrnes constant &er this point, which we wïil hereafter refer to as

the suhni~ltl-on point. It was observeci that , upon increasing the downstream slope

the saturation point moved upstream from the toe toward the middle of the

spiiiway- Since the horizontal length of the spiliway muid shorten for steeper

downstream slope, the surudon point would stay the same for each of these

models. Therefore, the dissipation muid be constant after a certain length. The

horizontal length of spillway has to be long enough to reach to the satumtion point.

The sumaion point can be found h m the foiiowing relationship:

where

X, = the horizontal length h m the crest to the -on point.

In high dams normaily this criteria is satisfïed. This is an important criterion for

low to moderate dams. It was ais0 observed that fat very high discharges

> 2), the energy dissipation rate associated with a stepped spüiway would be

insignincant

Figs 6.12 to 6.23 show that the energy dissipation rate will decrease with

increasing discharge. It appears that, at high discharges (Hl& > 2), even if the

length of spïiiway is large enough to have a My-developed flow, the effect of the

sreps wiil be very smali- rii general, a stepped spiiIway will act the same as a

smooth spiilway after > 1.5-

Figs. 6.27, 628, and 629 show the energy dissipation at the toe of the different

steppd modeIs stadied, for 8 = 45", 50' and 605 It can be seen that overd the

CLBS models provide the highest energy dissipation rate among d the models

studied The model tests indicated thttt Iarger steps at the upper portion of the

spiiIway, (where the depth of flow was lower), muid have more effect on the

spillway flow. At the toe of the spillway the small steps will have a large number

of smder vorti- Therefore, at the toe of the spiliway the transition of flow to

the stiiiing basin would be very smooth.

In spite of hi& f associaîed in the CSBS models, the vortices created in the step

spaces were not Iarge. Therefore the energy dissipation in these models is not the

highest The CLBL modeis produced the strongest vortices but the number of

vortices is small. Both the CLBS modeis and CSBL models provided a combination

of strong vortices and a large number of steps. In the case of skimming-flow, the

water shoots down the face in a regdar Stream and steps act as a fonn of

intensive-fn'ction in retarding the flow. The total energy dissipation above the

spiiiway and the r e s i d d energy at the bottom of the spiliway are functions of

L, 97 e N, andf,

For the mennt modeis studied the &don factor c m be fotmd fiom eq(6.4).

Among the 5 principal faaon influencing energy dissipation, &, 0, q, are

determined (i) by the purpose of structure, (n) by d e topography, and (m) the

design flood Moreover, once N, is seiected f is determined Using the data

obtained for dinaent 8, q, and N, and using the Grid Search Method the foilowing

equation is obtaiuied:

Figs. 6.30,6.3 1, and 6.32 compare this quatïon widi the data obtained for each set

of models. A good agreement between the experimental data and eq.(6.14) was

achieved (3 = 0.9, F-Obscrved = 152 >> F-Criticai = 3.62). Fig. 6.33 shows the

observed AEEO versus the cornputeci A E / ' h m eq. (6.14). Accordingly, this

equation enables one to estimate the amount of energy dissipated on a stepped

spillway structure.

In most of the literaîure the energy dissipation raie reported was for low discharges.

This study shows that this m e is in the range of 50-60 percent for the design h e d

When the head over the crest (H) is more than twice the design discharge the

stepped spillway WU not be & i v e anyrnore.

Fig. 6.1. H-Q r e l a t i d p for 45"models

0.00 I I I I I I I I I l 0.000 0.010 0.020 0.030 0.040 0.050

HI.' (m)

Fig. 6.2. H-Q relaticmship fa 50°models

Fig. 6.3. H-Q relaticmships for 60' models

Fig. 6.4. Comparkn of eq. (6.4) with CLBL modei expaimental data

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I I I 1 I 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4-00 4.50 5.N

Fig. 6.5. Campeaison of eq. (6.4) with CLBS modei exprhentd data

Fig 6.6. Camprison of eq. (6.4) with CSBL model expimental data

Fig. 6.7. Camparison of eq. (6.4) with CSBS model expaimentd data

Fig. 6.8. Comparkon of eq. (6.4) with 45" stepped modd eqehea ta l data

Fig. 6.10. Campison of eq(6.4) wah 60" steqpd modei eqaimentai data

Fig. 6.1 1. Obsaved f versus computed f

Fig 6.13. aUE vernis x/Hd ; 45-CGBS model O

Fig. 6.16. W E o versus fid ; SOCL-BL modd

Fig. 6.18. âE/Eo versus ; 5WS-BL

0.00 2.00 4.00 6.00 8.00

xlH,

Fig 6.20. Mo versus dHd ; oOCL-BL moâel

Fig. 6 . 2 1 . ~ v a s u s ~ ; Oo-CL-BSmodel

x/Hd

Fig. 6.24. AEE vernu fid ; 45' smooth model O

Fig 6.27. aUE, vexsus y /h ; 45"stepped madel0

Fig. 6.29. versus ycîh ; 60" stepped models

Fig. 6.30. Comparison of eq. (6.14) with scperimental data 45' steppeci models

Fig 6.32 C t m p i s m ofeq. (6.14) with expimentai data 60" stcppedmodels

Fig. 6.33. Obmed ABEn vasus Comprited AEEn

0.00 I I I I 1 I I I I I 0.00 0.20 0.40 0.60 0.80 1 .O0

Observed A E 1 E O

Chapter 7

Numencal Modelling of Spillway Flow

Virtuaily every phenornenon in nature, whether biologicd, geological, or mechanical,

can be describeci with the aid of the Iaw of physics, in terms of Jgebraic, differential,

or integral eqdons relating various quantities of inîerest Most enguieers and

scientists studying phyçical phenornena are involved with two major tasks:

1. Mathematical formulation of the physicai process;

2. Numerid analysis of the mathematical model.

Development of the mathematical model of a process is achieved through assumptions

conceming how the process works. In a numerical simulaiion, one uses a numencal

method and a cornputer to evaiuate the mathematical model and estimate the

characteristics of the process.

While the deridon of the goveming equations for most problems is not excessively

diEcult, their solution by exact methods of analysis is a formidable task. In such

cases, approximate methods of andysis provide alternative means of kd ing solutions.

Among these, the finite merence method (FDM) and the variationai methods, such

as the Rayleigh-Ritz and Galerkin methods, are most fiequendy used in the Literature.

In the finite difference approximation of a differential equation, the derivatives in the

latter are replaceci by dinerence quotients (or the function is expanded in a Taylor

series) that involve the values of the solution at discsete me& points of the domain.

The resuithg dgebraic equations are solved, after imposing the boundary conditions,

for the values of the solution at the mesh points. This metfiod &ers fiom the

disadvantage that the approximation functions for problems with arbitrary domains are

difficuit to consuuct

The h i t e eIement method (FEM) overcomes the disadvantage of the traditional

metfiods, such as FDM by providing a systematic procedure for derivation of the

approximation functions over subregions of the domain, in general, in the FEM, a

given domain is divided into subdomains, called finite elements, and an approximate

solution to the problern is developed over each of these. The subdivision of a whole

into parts has two advantages:

1. It allows accurate representation of complex geometries and inclusion of dissimilar

materiais.

2. Lt enables accurate representation of the solution within each element, to b h g out

local effects (e.g. large gradients of the solution).

The anaiysis of water flow over a spiliway presents a difficdt mathematical problem

because of the following :

1. the flow is non-linear;

2. îhe free water d a c e is unknown beforehand;

3. the flow is subcritical upstream of the crest of the spiiiway;

4. the fiow is supercritical dowllstfeam of the crest of the spiliway; and

5. there is a control section near the crest.

Depending on whether the flow is supercritical or subcntical, SouîhweII and Varisey

(1946) introduced two different iterative meîhods. AIthough they apply a relaxation

meîhod to fluid strerunlines, they found difficulty in determination of the fke stream

boundary. ikegawa and Washizu (1973) introduced a new iteratian procedure by

combining the use of the variational principle (VP) and the FEM In this procedute

the stream functim and the fiee surface profile are independent ~uantities, The result

obtained by their model shows good agreement with those obtained using an empirical

formula

Chan et al, (1973) using VP and FEM, studied free d a c e ideai fluid flows. Their

study was focussed on a steady two-dimensional, invisicid fluid with a îÏee surface

boundary. To f h d the location of the fiee d a c e , they introduced an iterative

algorithm. Depending on the number of elements introduced, 10-20 iterations were

normally required for Gnding the exact location of the free surface- In their study the

entire domain had only 72 elements.

Taylor et aI. (1978) used FEM to study models of turbulent flow. Dao-Yang and

Man-Ling (1979) developed a two dimensionai mathematical model for flow over a

spiIlway. They developed a synchronous iteration to h d the discharge and location

of the free surface. Bathe and Khoshgoftar (1979) presented an iterative procedure for

nnite element d y s i s of free seepage problems. The procedue does not need

changing in the finite element mesh, but instead uses a non-Iiaear pressure dependent

permeabiIity.

Saionen et al, (1981) used FEM to analyze a two-dimensionai incompressible

irrotational inviscid fluid flow with the velocity cornponents as basic uaknowns. By

using a modifieci Ieast squares metbd, i-e. a combination of the conventional least

squares and subdomaiu coiiocation method they found a remarkable increase in

accuracy compared with the conventional method Bettes and Betîess (1983) used

variable geometry isoparametric elements and a variationai method to solve free

surface problems. S k e this method allows the fkee surface to be curved rather than

be piecewise straight, the representation is more accurate.

O'CarroU and Tom (1984) presented an algorithm for flow over a weir. In theu

algorithm they tried to solve the stream fundion and fiee surface location

simultaneousiy. This is a rapid convergent iterative procedure for computing the

correct value of criacal discharge. Rasanen and Saionen (1984) using FEM suggested

the importance of curvature of the free surface on velocity. Rasanen and Salonen

(1985) used the vetocity components (yv-formulation) as basic unknowns and FEM

to develop an algorithm for the solution of f iee-dace problems. The basis of the

mediod is the modifieci lem squares function. They also used an iteration method to

fmd the location of free surface.

Heng et ai. (1986), by usuig a different iterative approach and FEM, tned to solve the

flow under a sluice gate, i.e basically a free surface flow. Their study showed that the

results are very sensitive to the f i t e element grid use& Aitchison and Karageorghis

(1988) used the boundary element method (BEM) to study the conventional flow of

an incompressibie inviscid fluid past a plate in a chamel of f i t e width and infinite

length. The r d t s show more accuracy because of the property of BEM to ded ody

with the boundary of the region under consideration- Berger and Winant (1991)

devdoped a one-dimensional numerical model for shallow water, termed CURVElD.

This model however has a significant bed cwature Limitation, The Limit is

- 2c KY 5 0.54, where K is the bed curvature and y is the depth.

Henderson et al. (1991), by a direct mathematical approach instead of trial-and-error

methods, presented a two-dimé~~sional model for fiee Surface flow over a weir. The

model mostly was developed for studying cavitation damage. They used BEM for the

numencal approach. Ratmanan and Engelman (1993), using FEM and the Galerician

approach, presented an algorithm to simulate viscous fiee-surface flows. This

aigorithm is valid only for a continuous boundary condition, and for fluid with hi&

viscosity. Bettess et al. (1 994) used FEM to find the flow profile dong a cnunp weir.

They adopted a rectangular element with variable geometry. By using the

Newton-Raphson iteration method, the location of the Gee-surface was found. Khan

and Stefner (1996) v d e d the applicability of a 1-D model velocity average and

momentum (VAM) equations on flow over curved beds. This method can be used

even for steep (near vertical) dopes. Using Montes (1994) expenmental data, they

found the VAM equations model extremely well-

In spite of many researchers having studied fluid problems using FEM and BEhrL, there

is no specific procedure for solving al1 fiee-surface problems. This is because there

is a moving boundaq at the fie-surface. The location of the fiee-surface is very

important in the study of flows over spillways.

In order to establish a stable upstream boundary condition, a large-volume reservoir

is assumed It is also asmmed that fIow in the domain can be described by velocities,

by location of the fiee d a c e , and by discharge. It is necessary to impose certain

conditions upon the flow b detive the goveming equations. First, it can be assumeci

that the flow out of the reservoir is tirne-indepeadent This assumption means that the

water level of the resemoir (the stagnation level) does not change with time and the

potentiai energy of the resemir is therefore the-independent Furthemore, it is

assumed the spillway flow is (î) twodimensiond; (ii) steady; (iii) incompressible;

(iv) non-viscous; and (v) irrotationd on srnooth and rotationai on stepped models.

For flow downstream of the ogee-profile stepped spillway, the acceleraîion and

boundary layer development are taking place on the flow goes down the spiliway façe.

M e n the bouridary iayer meets the fiee surface, the flow becomes M y developed.

At this poin.t, the point of inception, the phenomenon of air entrainment or inmon

occurs. Due to the mixture of air and water after the point of inception, denning a

precise £ree surface is very diBcuit in practice.

The andysis of fluid flow problems, principdly two-dimensionai ones, is facilitatecl

by the concomitant consideration of a scaiar. The stream function Y is a scalar field

function. This parameter was introduced by d'Alembert (1752) and was extensively

used by Lagrange (1781). The intimate reiaîion between the stream function and the

continuity relation in two-dimensionai fluid motion was demonstrated by Rankine

(1864). The Stream fimction for two-dimensionai flow of an incompressible fluid is

defined by:

and

in which

u = velocity in the xairection; and

v = velocity in the y-direction.

The Stream functions autodcally satisfy continuity. Aiso in a two-dimensional flow

in &ch

O = vorticity.

By substitutïng eq.(7.1) and eq(7.2) in eq47.3). one gets:

This is the governing equation for flow over a stepped spillway. When the flow is

irrotationai = 0, and eq(7.4) becornes the Laplace equation. For rotationai flow,

a> t O, and eq(7.4) is t e 4 the Poisson equation. The singular vortex elements in

the form of elements of circulation, are generated on solid walls in order to destroy

slip velocities.

73.1. Bo- Conditiom

The boundary conditions are as follows:

1. At the fiee d a c e , it is necessary to satisfjr:

6) i P = O

and

) An additional boundary condition on the fiee surface must be satisfied. The

pressure is prescnbed there and is taken as atmospheric. Bernoulli's law then

gives the foiiowing boundary condition:

in which

n, = the normal outward vector on the boundary;

f(x) = the distance of the free d a c e measured fiom the stagnation IeveI

(Fig.7.1); and

2. At îhe bottom of the channel:

' 4 = q

3. It is also assumed the flow is d o m on the far downstream.

4. The final assumption is that Y varies liaearly dong the boundaries.

The goal of this numerical study is to predict the location of the free surface for a

given discharge q and flow field- The following procedure is suggested.

1. for an assumed q, estimate the corresponding water surface profile;

2. find the function f(x);

3. by using FEM fhd the value of Y at nodal points in the flow region;

4. calculate aYh, using 'Y's obtaind in step 3;

5. substitute step 4 in eq. (7.6) 6nd a new vdue for @);

6. repeat steps 3,4, and 5 d l the profile of free boundary d a c e is found to

have converged.

The SPIGFIBW programme has been developed for the above algorithm-

7.4. Finite Element Mcthod

The finite element method (FE.) is a simple two-dimemional geometric shape that

can be used to approximate a gïven two dimensional domain as weU as the solution

over it. The FEM is endowed with three basic features that account for its supenority

over ofher competing methods:

(i) a geometrically complex domain of the problem is represented as a collection of

geometrically simple subdomains, caiied finite elements.

(ri) over each finite element, the approximation fimctions are derived using the basic

idea tfiat auy conîinwus function can be represented by a hea r combination of

aigebraic polynomiais; aud

(iii) aigebraic relations among the undetennineci w&cients (Le., nodal values) are

obtained by satisfyiag the goveraing equations, often in a weighted-integral seme, over

each element

Therefore, the ability to represent domains with irregdar geometries makes the nnite

element metbod a valuable practical tool for the solution of b o u n e value problems.

In FEM the approximation fundons are often talcen to be dgebraic polynomids, and

the undetennineci parameters represent the values of the solution at a finite number of

preselected points, caiied nodes, on the boundary and in the interior of the element.

The approximaîion functions are denved using concepts from interpolation tbeory, and

therefore calleci Weipolurion fimctions. One h d s the degree of the interpolation

fimctions depends on the number of nodes in the element and the order of the

differential equation being solved

In a two-dimensional problem, not only one seeks an approximate solution to a given

problem on a domain but dso one apptoximates the domain by a suitabIe finite

element me& Consequeatly, in the nnite element analysis of two dimensional

problems, the approximation emrs are due to the approximation of the solution as

weil as disaetization errors due to approximation of the dom* The f i t e element

mesh consists of simple two dimensiond elements that d o w unique derivation of

interpolation hctions. The elements are comected to each other d nodal points on

the boundaries of the elements.

The major steps are as follows:

1. discretidon of the domain into a set of f i t e elements;

2. weighted-integrai fornulahon of the goveming düfercntid equation (7.4);

3. derivation of finite element interpolatioa fiinctions;

4. development of the nnite element mode1 usïng the weighted-residuai form;

5. assembly of nnite elements to obtain the global system of algebraic equations;

6, imposition of bormdary conditions;

7. solution of equations; and

8. post-computation of solution and qwtities of interest

These steps will be explained in detail in the following.

In a two-dimensional problem, the interpolation fundons depend on:

(0 the number of nodes in the element; and

(ii) the shape of the element

nie shape of the element must be such that its geomatry is imiquely defhed by a set

of points, which save as the element nodes in the development of the interpolation

fimctions.

The representaiion of a given region by a set of elements is an important step in f i t e

element analysis. The choice of element type, number of elements, the density of

elements depends on:

(0 the geometry of the domain;

(ri) the problem to be d y z e d ; and

(in) the degree of accuracy desireci.

The generai des of mesh generation for element formulation include the following:

1. select elements that characterize the goveming equations of die problem;

2. the number, shape, and type (i-e., hear or qunAratic) of eiernents should be

mch thai the geometry of the domain is represented as accurately as desïred;

3. the density of elements should be such that regions of large gradients of the

solution are adequateIy modeled (Le., more or highersrder elements should be

used in regions of large gradients); and

4. mesh refhements shouid be gradually nom hi&-density regions to low-density

regions.

In the development of the weighted form, one needs only consider an arbitrary typical

167

element Assuming diat P is such an element (Fip7.2), one can develop the =te

element mode1 of eq(7.4) over W.

To develop the weighted-residual form for eq(7.4) one may proceed as foilows:

i ) MultipIy eq.(7-4) with a weight function w , which is assumed to be

oncadifferentiable with respect to x and y, and then integrate the resuiting equation

over the element domain Q:

ii) Distribute the differentiation equdy between Y and W. To achieve this, one can

integrate the Grst two terms in eq.(7.8) by parts. First note the identities:

a au &am a au -(w-) =-- +w-(-) a r i b r mai: maic

and

Then, recd the component form of the gradient (or divergence) theorem, Le.,

and

where q, and 4 are the components of the unit normal vector on the boundary of the

element, r", and ds is the arclength of an infinitesimal line element dong the

Using the above equaîions one obtains:

The specification of Y constitutes the essentiai boundary condition, and hence Y is the

primary variable. The specincation of the coefficient of the weight fûnction in the

boundary expression

constitutes the naturai bormdary conditions; thus, t, is the secondary variable of the

formulation. The function t, = tJs) denotes the projection of the vector am dong the

imit normal n. By definition, t, is positive outward from the surface as one moves

counter-clockwise dong the boundary P.

iii) Use the dennition eq(7.17) in eq.(7.16) and write the weighted-residuai form of

eq(7.4) as:

The weighted-tesiduai form in eq-(7.18) forms the basis of the nnite element mode1 of

eq. (7.4).

The weighted-cesiduai form in eq(7.18) requires that the approximation chosen for Y

should be at least bear in both x and y, x> that there are no terms in eq(7.4) that are

identicaüy zero. Since the primary variable is simply the fimction itseif, the Lagrange

family of interpolation finictiom is admissible. Suppose tbltt Y is approxïmated over

a typical finite e1ment P by the expression:

where is die value of Y' at the jth node (3,yj) of the element, and Pj are the

Lagwige interpolation fiinctions, with the property

Sij = 1 if id, and

s,=oifi#j.

Substituthg the finite element approximation eq(7.19) for \y into the weighted-residual

form eq. (7.1 8), one obtains:

This equation must hold for any weight fiindon W. Since n independent algebraic

e q d o n s are required to d v e for n unknowm w,, \y, ..., y, then n independent

fimctions for w: (w=&, 5 ..- , a) must be chosen This particdar choice is a naturd

one when the weight firnction is viewed as a virtual variation of the dependent

unknown (i.e.. w = 6~ = Z'3yrihi). ûne can label the algebraic equation resuiting

fiom substitution of & for w into above eq(7.21) as the fh t aigebraic equation. The

ith algebraic equation is obtaiaed substituting w = A, into eq(7.2 1):

and

Q; = fF t, a: dr

In mark notation, eq(7.23) takes the foilowing form:

This completes the deveiopment of hi te element model.

The finite element approximation YC(x,y), of %y) over an element fZ, must satisfy

the foiIowing conditions in order for the appmximate solution to be convergent to the

true one:

1. 'Y" must be differentiable, as required in the weighted-residuai form of the

pro blem;

2. the polynomials used to represent \Ye must be complete; and

3. a11 terms in the polynomid should be lineariy independent.

The number of lïnearly independent terms in the representation of Ye dictates the shape

and number of degrees of freedom of the element In generai, by using Pascai's

triangle, which contains the tenns of polynomiaIs of various degrees in the two

coordiriates x and y, pth-order triangular element has n, nodes, with

and a complete polynomial of the p,th degree is aven by

For this study, a second-order triangular elexnent &=2) was developed, which contains

6 nodes. The positions of the six nodes in the triangle are at the vertices and at the

midpoiats of the three sides (Fig.7.3). The polynomial involves six constants which can

be expresseci in tenns of the nodal values of the variable being interpolated:

where h, are the quadraîic interpolation functions.

The quaciratic poIynomid associaîed with the Quadratic (six-node) triangular element

is :

v(qy) =4 +$x +qy +a4q + q x 2 +ad2 (7.3 O)

The derivatives of yr' are:

Since the transformation fiom the global coordinate, (%y), system to the local

coordinate, (r,s), system involves only rotation and translation, the second order

polynomial in the global coordinaîe system has st i i i the second order in the local

coordinate system. Therefore,

where ali ( i=1,2, ... -6) are constants depending on the ai and the angle of rotation,

The interpolation functions of a 6 number-node element are:

and

These fiinctions are constnicted in the usual way, namely, q must be at node i

and zero at al1 other nodes,

The transformation of a qiinAnlaterd element of a =te element mesh fiom a local

coordinate (r,s) system to a global coordinate (%y) system is for the purpose of

numericdly evduathg the integrals. Consider the element coefficients

The integrand is a hction of the global coordinate. One has ta rewrite in terms of r

and s using the foiiowing transformations:

This integrand contains not only hc t ions but also derivatives with the respect to the

global coordinates (&y). Therefore, one has to relate aY f/ax and aY@y to aYJ& and

aYi'/as using the above transformations. Hence, by the chah d e of partiai

differentidon, one can write:

an; an; a; a - - ---+--

or, in matmc notation,

which gives the dation benmen die derivatives of ,'th respect to the global and

local coordinate.

The Jacobiau rnatrix is d&ed as:

Therefore

where [JI" is the invat of the Jacobian mmix.

A very important numericd integration procedure, in which the positions of die

sampling points and the weigtits have been optirni& is the Gauss quadrature (Bathe,

1996). It can be written as foliows:

or, when extended ïo a -ensional case:

where

q and a, = the weighting factors,

ri and rj = the locations, and

m = the number of Gauss points.

Table (7.1) lias the d u e of a, and ri for one-dimemional g(r). In this shidy a 4-point

Gauss quadrature was chosen,

The assembly of hite elment equations is based on two priaciples:

1. Continuiw of primary variables, and

2. Ew'Iibrium of secondary variables.

The cbntinuity of îhe primary variables at the interelement nodes guatantees the

continuity of the primary variable dong the entire interelement bourrdary. For the

secondary variables, at the interface between the two elements, the fluxes h m them

shouId be equd in magnitude and o p p i t e in sign,

In genetal, when severai elements are connected, the assembly of the elements is

carrieci out by putting element coefficients Yi' and Q: into proper locations of the

global coefficient matrix and the right hand vector. This is done by means of the

comectivity relations, Le. the correspondence of the local node number to the global

node number.

7.4.8. Inaposilion of Bolllybrv Conditions

There are two types of boundary conditiolls for this study:

1. Essentiai (Dirichia) bundary conditions, i.e. boundary conditions on primary

variables;

2. Naturai n tu ma mi) boundary conditions , i.e. conditions on secondary

variables

The procedure Eor implementing the boimdary conditions on the primary variables

involves mo-g tbe assembIcd coefncient matrix and the right-hand column v e r

by the following three Operaaons:

1. moving the known producis to the rigbt-hmd column of the matnx equation;

2. repiacing the columns and rows of the assembled cotfficient matrix

comosponding to the known primaty variable by zeros, and serting the

coefficient on the main diagonal to unity;

3. replacing the corrc~p~~lding cornponent of the right-hand column by the

spedied d u e of the variable.

F M y the full mrtmt form is:

Gaussim eliminatibtl and back-substitution was used to solve the e q d o n system

eq47.49). This is the most &&ive direct solution technique (Bathe, 1996, Hoffmsn,

1992). Scaiing aud pivoting u e C S S C I I ~ ~ ~ elemeats of Gauss elimination.

The snite ciment solution at rny point (&y) in an &ment P is given by:

8

The abave cquations cm k used to cornpute the solution and its derivatives at my

point (%y) in the dement

7.5 Discussion

ï h e &ect of air eatmimnent in the model was studied by applying the U.S. Anny

Corps. of Engg. formuia (eq.(5.48)). Baseci on diis quaiion it un be wnduded that,

with inaease in diochrrge for a specific downstream dope, the air cnmhment WU

decrease. Afla die point of inception q(5.48) was ppplied to the m e r sutface

p r o f i found using SPIL-FLOW. It was found haî, for WH,?125, the r q d length

to establish d o m fiow couid not be achieved. Therefore our numerid model can

not be rppiied in siniir;tions whae tmïfomi flow conditions on die spillwoy face cannot

be a c h i d

In order to qply oiir model and verify it, dapiled information regarding the

coordinates of the spillway d a c e , discharge, and location of m e r surface pronie are

necesspry. Thedore, for the purpose of compdson, die mode1 was rpplied to severai

senes of test data in which H/H, ~1.25 and where d o m flow condition on the

spillway face wete vhieved Figs. 7.4 to 7.12 show the r d t s of these verifkation

tests.

The maximum total number of elements used was 576 with 1305 nodes (the SPIL-

FLOW model col hande 1000 elemenîs). The maximum number of iterations was 30.

As i n c l i d in Figs. 7.4. to 7.12, the model predicts water d a c e profiles better for

low mage of He ( W?& e1.0 ). The maximum dinennce between die numencal

simulaîion and the experimentai tesults is about 10 % @ig. 7-13), with ? = 0.947.

ki spite of applying the ha entrainment coefficient it is found that the fiow-depth

d t s of the nimierical modelling are aiways kss tiian t&e experimental data h

regards to pndicting the spiliway water spi f i elcvaticm proflies the model

rmderpredicts, by a maximum 10°/i the observed nater smface profiles. The reason

for this may be îhaî the actual air entrammat d u e n t is higher than the one

tstimated by eq.(5.48). This will lead to underestimahg the energy dissipation as well

as the height of the necessary retaining walIs.

In this chpter, an aigorithm that applies a 2-D finite elunent method tb find the

location of the fret watu d ~ e ~ for fiow over a spüiway has beai proposed. The

main conclusions are:

B d on the nature of the govrming mon and tbe boimdary conditions, the

variational method (wiîh quaàratïc triaagular elements) was a$plied to devtlop

the FEM procedures.

in order to find the location of the frte water surface for fiow over a spiiiway,

a tnid ad cmr procedure based on Bwnouiii's equation has ban used.

The &ect of air entrainment was studied by applying the wd-hown and

generally accepteci U.S. Amy Corps. of Engg. formula (q(5.48)).

The model has been applicd to the test data conesponding to H/Hd < 1.25,

which is tae data for *ch d o m fiow conditions wue achievcd on the

model spillways. For this range of data the modet predic~r a good camparison,

Table 7.1. Smpling points and the weights in Gauss quidriiaire (afker Borhe, 1996)

Fig.7.1. Flow o v a a spillway

Fig7.3. Consbution of the inte3polation firndons

Fig. 7.5. campkm of dcph Ofwatabetwealthe ninnhd simwon output a n d ~ d d a t a f ~ M 0 ~ O O f h m o d t l

Chapter 8

S u m m a ~ ~ , Conclusions, and Recommendations

for F'uture Reseailch

The sleppd ogee-pde spiiiway is a spiUw;iy dose downstream face is provideci

with a series of seps h m neu Le =est to the ta. The steps, which are amanged so

that their tips confonn to the comsponding smooth ogee-pmfiie, significantly inuease

the rate of energy dissipation taking place on the spillway face. Accordingiy, if a

stilling basin is rlso rquüed, its s k e is sigdicantly rduced by the action of the

stops. h Ins reseuch the hydraulic chuacteristics of various combination of ogee-

profiie siepped spillway were imrestigated experimentally and numerically. The

pesomance of the various stepped models were compareci to those for the

corresponding mouth profile.

Twelve sreppd ogee-pronle modeis and three smooth ogteprofile models were tested

in this stuây. AU modds had v d d upstream faees and tbe spillway cnst gwmetry

used was thrt pmposed by the U.S. Watenways Exporinient Station (WES) (1977).

The models, which induded doumstrearn dopes of 45' , 50' , and 60' hod an

asmameci modei a, prototype sule nbio of 150. The amputer program GEMLSPS

was used to generate the geometry of the models, &dei performance was examinecf

for a wide range of operational head, H (i-e. h m 0.5 H, to 2.5 a. In the case of

the stepped models, at the Iower discharges the water jet upon impacting the kt srep

was projected into the air and became regttachd a fm seps downstream. Upon

reanachment, some water was splashed into the au, and measuring this particdar

water surf- profile was virhially impossibIe. It was also found that the &kt of

sreps was insignikmt for IYa, ~2 .5 , and stepped modds acted as the smooth models.

The fiow over a stcpped spillway can be divided inm nqpe and skimming fiow

regimes. N q p flow regime can be dwcribed as (i) n q w flow with M y developed

hydraulic jump, and (ii) nrlppe flow with pariiaüy-dcvcloped hydrauiîc jump. In the

case of q r p e fiow with M y developed hydraulic jump, each mppe would perform

the same as a drop stnictute. In this research the drop stnictures were studied

extensivcly (sa sec. 5.1)- Based on the resuiîs of this research, new criteria were

developed for the drap structure. Applying these criteria, a generd equation was

derived to distinguish between a fullydeveloped q p e flow regime and a skimmmg

flow regime (eq. (5.38))- For a fullydevcloped nappe fïow regime, a general equation

based on the momumun principle was developed ta prdict the amount of dissipated

energy (eq. (5.71)). This quaion was foimd to be more applicable to the avdable

data h m other sources thm simiiar equatîons &und in the l i t m e (Chanson 1993,

Charnani, and Rajamham, 1994). Althou@ the mppe fiow regime was studied, the

main emphasis of this research was the skimmmg fiow regime. From an econornical

and practical viewpoint, in order to have a q p e flow regime one needs flat slopes.

In this study, the models studied had relativeIy steep dowllstream slopes.

It was found that compared to an equivalent smooth spiIiway the discharge coefficient

increased for the case of a siepped spiliwsy (see eq. (6.2))- Increased discharge

cepacity is one of îhe advantages of stepped spillways over equivalent smooth

spiIlways. ï h i s iilcreasc provides more deploymmt of flood to the d o m

channe1 in a *ven time period. Another important hydtaulic characteristic of a

szepprci spi11wq is in fiction f-r. This aspect ans dso investigated in

restard rnd a grnerai nlationohip was d d o p e d (q. (6.4)). This equ~a*on was dm

verified b d on die qerhmtai btr It is sâown 164 in gencrai, the modeIs with

the lrrga nimibers of sups (CSBS- type models) ptwide bigher fiction factors.

It was fouad thaî encfgy dissipation on ~1eppc<i spilIways is a h a i o n of the height

of dam a), dope (8). discharge (q), number of s t q s (NJ, md the fiction f~ctor

(f ). B a d on the entai resuits, a g e n d quation for the m e of energy

dissipation for flow over a s t e m oga-profle spillwry was devdoped (eq. (6.14)).

Also, based on the diffcrcat modeis testeci, it was found th& a ceriain horimntal lai*

was nqui~ed for the metgy dissipaîion to r d an equiIibnum rate (q. (6. 12)).

A 2-D non-lincir Finite Element m) mode1 was developed for estimating the

spillw~y mer surface profile. Because of die problems such as non-linearity of the

governing q d o n , and uuknown botmdary conditions bâorehmd, the andysis of

ffow over a spillway is a dinicult mathematicai problem Using Bernoulli's Iaw, an

algorithm (SPIL-FLOW) was intraduced to find the location of the fiee waîer sudace

over a spillway. i h k numerid mode1 was verined using the experimentai data

8.2. Main Conc111sions

Bacd on !be d a obtdned in (bir saidy, the min conclusions ae:

1. Appiyhag tbe momcaoimi principk, sed-cmpiiid epuoiiom for y& y&

cosp, Id AE& mudy qs. (536), ( S m ) , (5.28) rd (539), for a h p

stmcûue wexe devdopd in (Lis W. As iroted in Tde 5.1. chc mt3 s b w

good apemilnt wi8h (bc data pubürkd by Moom (1943), Gill (1978), md

Chmon (î993).

v o n (5.71) for Ume eœqy dissipation for folly=dtvcloped nqpr flow

over a ogeepfïïe rtcppd spi l l i rqr wrs &veJopd Iliir -on doer not

B#d on cbc d b of bcupdmm& che di- coefiicient for a Jicppod

rpührqr wm &&M (sœ eg (64). It r a s fomd thmt for rbc smœ b#d

conditioa on de urrt chir c d i c i e n t îs bigber for d m s p ü l r r q s than for

sniodb spimi.8ys.

Bacd on &e uprimcnld dili d !k Gnd Sem& Mcdiod, a gened impücit

cqprdlon (eq. (6.4)) wa devdopd for oit friction f a r (f ). ït w u round

dmt Q nimikr of si- is nom impoiimt ihrii & size of the st-.

Among & l2 modeis Md, in g t œ d *ppcd spührgr c o a f i ~ o r r b h l g

J q e ~p œarUœ errrtfollowed s d ~ r p s nt.r&e toe, (y. UBS-( lp

Future research investigations should be carrieci out to provide information in the

1. The mechankm of flow reckcuirtion in skinming fiow regime needs to be

studied in deuil. 'Ihe flow neiradation is the major SOUTC~ of energy

dissipation in skimmmg flow ngime- To snidy this mtChanism one na& to

b d d large d e physid modcls. A ratio of 1:10 is ncommcnded.

2. One of the most impo-t wrtcr q d t y puameters in nvers and rtreams is the

diJsobd oxygen content. Deep and slow pools of water upmeam of a dam

reduce the a i r - ~ e r gas &et process. Stepped spillways can d a n c e the

oxygen transfer durfng die water release. Also s i e p d spillwpys are used in

water trea!ment plauts to enhance the air-waîer transfer of atmorpheric gares.

This process nccds to be studied in detail for both q p e flow and skimming

flow regimes*

3. On a Jtcppad spillway, the high rate of energy dissipation reduces the fîow

mommhmi in cornpuison with a miooth spillway. The pmence of air within

the flow will hause the fiow depdi, which t d t s in reduaion of flow

vdocity. The reduction of flow vdocity aiso d u c s the risk of cavitation as

4. A l h * K d s (1993) among thc athers bave made prcliminary audies of the

hydraulies of gabion jicppcd spiliways, tbae is stiii a a d for fbhcc studies

of f imclm~~~tr l nature on this important topic- The modciiing of fiow through

aaduvcrthesuppcdspiiiwayneeds~bestudiedhdetaïi. I t i s v u y

important rnd -4 for @ion siepped spîiiways. In order to investigate

the flow through and mer the stcppcd spiiiway, one needs to budd several

large d e physicai modek

Chapter 9

Akin, J.E., (1987). "Finite Eiement Formulation of Fiows with Singuldes," Proc,

of îhe Second Int. C o d on Finite Elements in Waîer Resources, Imperia1 CoIlege,

London, Jkiy 9-12 , pp. 99-1 12.

Atichison, JM, Katsgerorghis, A, (1988), "Numerid SoIution o f a Boundary

Element Metbod," Int. J- for Nmhcai Meîhods in Ruids, Vol. 8, pp. 91-96.

Baban, K. (1995), " Design of Diversion Weirsm, John Widy & Sons Ltd, UK.

Baker, R, Gsrdiner, KD., (19951, "Building Bloch," Lat. Water Power & Dam

Construction, Jan., Vol. 47, No. 11, pp. 16-18,

Baker, R, (1989), "Prc-Cast Concrete Blucks for Dam Spillways," Int. Water

Power & Dam Constnicbon, July, Vol. 41, No- 7, pp. 60-66.

Bathe, KJ., (1996), "Finite Elment Procedure," Prenîice Eh& Englewood C m ,

NJ.

Bathe, KJ., KhoshgoRarr, UR, (1979), "Finite Elunent Free Surface Seepage

Analysis without Mesh Itemtbn," Int. I. fbr Numerical rad Analytical Methods in

Geomechanics, VOL 3, pp. 13-22.

Bathe, KI-, Oden, J-T., Wunddch, W., (1977), "Formulations and ComputatÏod

Algorithms in Finite Eiement Anaiysis, lMassachusetts Institute of Technology,

Boston, MA

B d e , KI, Wilson, EL., (1976). "Numerid Methods in Finite Element Analysis,"

Rmtice HdI, Englewood CiifEs. N.J.

Bayai, HO, (1991). "Steppeci Spillway Feasibility Investigation," Proc. 17"' ICOLD

Congres, Viema, Austria, pp. 18034817.

Beitq E., Lawiess, M., (1992), Wydrauiic Model Snidy for Dam on GEEL 3791

Isaac River at Burton Gorge,"Water Resources Commimmission Report, Brisbane,

Austdia

Berger, RC., (1 W), "Strength md Wmkness of ShaUow Water Equations in Steep

Open Channd Flow," Roc. Nat Conf. on Hydmdic En-, ASCE, Buffalo, N-Y..

Aug 1-5, pp. 1257-1262 .

Berger, RC., Stockstill, RL., (1994). "Considerations in 2-D Modehg of

H y d r d i d y Stap Flow," Proc. Nat Co& on Hydraulic Engrg., ASCE, Buffalo,

N.Y., Aug 1-5, pp. 1285-1289.

Berger, RC., Winrnt, EH, (1991), "One Dimensionai Finite Element Mode1 for

Spillwiy Flow," Roc. Nat C o d on Hydrrulic Engrg., ASCE, Nashviile,

Tennessee, July 29-Aug. 2.. pp. 388-391.

Bettes, O., White, W.R. Bettes, R, (1994). "Resirculobon in Flow ovet Cnunp

Weirs," Roc. Nat C o d on HydrPulic Engrg., ASCE, Buffaio. N.Y., Aug 1-5, pp.

103-1 17,

Bettes, P., Bettess, LA, (1983), "Analysis of Free S h e Fiows using

Isoparametric Finite Elements," ïnt J. for Numerical Methods in Engineering, Vol.

19, m. 1675-1689.

Bindo, M., Gautier, I., Lacn>ix, F., (1993), "The Stepped Spillway of M'Bali Dam,"

Int Water Power & Dam Constnictïon, Jm, Vol. 45, No. 1, pp. 3 5-36.

Cain, P., Wood, IR, (198 1), "M~8mfements of Self"Aeratcd Flow on a Spillway,"

J. & h u h c Div., ASCE, Vol. 107, NO. HY11, pp. 14254444

CBSEidy, J.J., (1970), "Design Spillway Crest for EIigh-Head Operatioq" I.

Hydraulic Div., Roc. ASCE, Vol. 96, No. HY3, pp. 745-753.

Cassidy, J.J., (1%5), "Irrotational Fiow over Spillways of Finite Heighc" I. Mech.

Div., ASCE, Vol, 91, No. EM6, pp. 155-1 73.

Chamani, UR, -atatnam, N., (1994), "Jet Flow on Stepped Spiiiways," J.

Hydraulic Engrg, ASCE, Vol, 120, No. 2, pp. 254259.

Chan, S-TK, Larock BE., Hemnann, LX, (1973), "Free Surface Ideai Fluid by

Finite EIem~ts," J. Hydraulic Engrg., ASCE, Vol, 99, No. 6, pp. 959-974.

Ctunison, EL, (1996) : Discussion appearing in Rajatamam, N., Chamani, MX,

(1995).

Chanson, H, (1994), "Cornparison of Energy Dissipation between Nappe and

SInmmiiig Flow Regimes on Stepped Chutes," J. of Hydraulic Res., iAHR, Vol, 32,

No. 2, pp. 213-218.

Chanson, IL, (1993), "Self-Aerated Flows on Chutes and Spiiiways," J. IIydraulic

Engrg., ASCE, Vol. 119, No, 2, pp. 220-243.

Chanson, H, (1993), "Stepped Spillway Flows and Air Entrainment," Caa. J- Civ.

Eng. CSCE, Vol, 20, pp. 422-435.

Chamon, &,(1992), "Air Entrainment in Chutes and Spillways," Research Report

No. CE 133, Dept of Civil Enginetring, Univ. of Queensland, Austraiia

Chaudhxy, MIK, (1993), "Open-Channel Flow," Preatice-Hail, Inc., New Jersey,

N.Y.

Chow, V.T., (1959), "Open Chaune1 Hydraulics," McGraw- Hill, New York, N.Y.

Christodouloy G-C., (1993), "Energy Dissipation on Stepped Spillway," J.

Hydraulic Engrg., ASCE, Vol. 119, No. 5, pp. 473-482,

Creager, WP., Justin, JD.. Hinds, J., (1964), "Engineering for Dams," John Wiely

& Sons, Inc., New York, N.Y.

Dao-Yang, D., Man-Ling, L., (1979), "Mathematicai Made1 of Flow over a

Spiliway Dam," Proc. 8' ICOLD Congress, New Delhi, pp. 959-976.

Degoutte, G., Peyras, L., and Royet, P., (1992): Discussion appearing in

Rajaratnam, N., (1 990).

Diersch, EJ., Schirmer, A, Busch, m., (1977), "Analysis of Flow with Initidy

Unlcnown Discharge," J. ~ d r a u i i c Engrg., ASCE, Vol. 103, No. 3, pp. 213-232.

Diez-Cascon, f., Blasco, J-L., Rivilla, J., Garcia, R, (1991), "Studies on Hydraulic

Behaviour of Stepped Spiliway," Int Water Power & Dam Construction, Sept,Vol.

43, No. 9, pp. 22-26.

Dressier, RF., (1978). "New Noniinear Shaüow-Flow Equations with Cwature,"

J. Hydraulic Res., Vol. 16, No. 3, pp. 205-222.

El-Khashab, A M , Hdweg O., Aiajaji, A., (1987), "Theoretical Flow Mode1 for

Dmp Smictur~," ROC. of Not. CO& on Hyd18UIic Engrg., ASCE, Aug. 3-7,

W i a m ~ b ~ g , Virghia, pp. 1 1 12-1 1 17.

Ellis, 1, (1985). "Numerid ModeMg of Spillway Rows," 2"" International

Confmnce on the Hydrrulic of Fi& & Flood Control, Cambridge-England, Sept

24-26, pp. 73-90.

Essery, I.T.S., Homer, MW., (1978). "nie Hydrdic Design of Stepped

Spillways," CIRU &poa No. 33, 2d ed, London , U.K

Essery, LI-S., Tebbutt, THY., Rasar- SK, (1978). "Design of Spiliway for

Re-aeration of Poiiuted Waters," CIRIA, London, U K

Fdvey, Kt. (1980). "Air-Water in Hydraulic Stnictutes," USBR, Engrg.

Monograph No. 41, Denver, Co.

Fxind, E.O., Maîanga, GB., (1985). "nie Dual Formulotion of Flow for

Con-ant Tca11sport Modeilkg" Watcr Reso\uces Research, Vol. 21, N0.2,

pp. 159-169.

Frizell, Ita, (1991). "Sapped Spiiiway Design for Flow over Embsnhnent," Roc.

Nat Con£ Hyümuiic Energ., ASCE, Nashville. Tennessee, July 29, Aug. 2, pp. 1 18-

123.

Frhll, KH., (1992), "Eydraulics of Stepped Spillways for RCC Dams and Dam

Rehabilitaîions," Roc. 3 " Specialty C o d on RolIer Compacted Concrete, ASCE,

San Diego, CA, pp.423439.

Garg, SK, (1983), "Yrrigation Engineering and Hydraulic Structures," Khana

Publishers, Delhi, India.

Gevorlryan, S.G., Kalantrova, - ZK, (1992), "Design of Tsunami-Protective

Embenkment with a Cornpiex front Surface," Hydrotechnkd Construction, pp.286-

290.

Giil. MA., (1980), " E f b of Boimdq Roughness on Hydraulic Jump," Int.

Water Power & Dam Constniction, Jan., Vol. 32, No. 1, pp. 22-24.

GilI, MA, (1979), "Hydrauiic of Rectauguiar Vertical Drop Stnictures," J.

Hydraulic Res., IAHR, Vol. 17, N0.4, pp. 289-302.

Golze, A X , (1977), "Hsidbook of Dam Engineering," Van Nostrand Reinhoid Co.,

New York, N.Y.

Grinchuck, AS., Rwdivets, YS., Shektman, N.V., (1977), "Tests of Earth Slope

Revetments Permiîting Fiow of Water at Large Specific Discharge," Hydrotechnical

Construction, pp. 367-373.

Guymon, GL, (1973), "Finite Element Solution for General Fluid Motion," J.

Hydraulic Div., ASCE, Vol- 99, No. KY6, pp. 913-919.

Hager, WH., (1992), "Energy Dissipaors and Hydraulic Jump," Kiuwer Academic

PubIishers, The Naheriands.

Hager. W H , (1991). "Experiments on Standard SpiUway Flow," Roc. Inan. Civ.

m., Part 2, Sept. pp. 399-416.

Hansen, KD.. Rcllihrrdt, W.G, (1991). "Roiia Compacted Concrete Dams,"

McGraw-EGU, New York, N.Y.

Ibnsen, KD., (1986). "Mer Compaaed Concrete Ddopments in the USA," Int.

Wam Power & Dam Construction, JIIL. Vol. 38, No. 1, pp. 9-12.

Harhiag, F., Scheudein, H, (1970). "Design of Overfiow Rodrnll Dams," Proc.

IO& ICOLD Congres, Montreai, Cauda, pp. 587-598.

Henderson, HC., Kok, M., Dekoning, WL., (1991). "Cornputer Aided SpUway

Design Using the Boundary Elemcnt Method and Non-lincar Programming." Int

J. of Numerid Methods in Fluids. VOL 13, pp. 625- 641.

Hendmn, F M , (1966). "w Channel Flow," The McMiUau Company,

New York, N.Y.

Heng, FL, Mitsoulis, E., Prinos, P., (1986). "Miodeiiïng of Flow under Sluice

Gate with FEld," VI Im Conference on Finite Elemaits in Water Resources,

Lisbon, pp. 483493.

Hofian, J.D., (1992). "Numerical Metùods for Enginais and Scientists,"

McGraw- Hill, New York, N.Y.

Hollingworth, F., F B W M , (1986). "Rollcrrte: Some Applications to

Dams in Soutb Afns" Int Water Power & Dam Construction, Jan.. Vol. 38, No. 1,

pp. 13-16.

Horner, MW., (1969), " An Analysis of Flow on Cascade of Steps," PhD. Thesis,

Univ. of Birmingham, UK

Huston, KL,, (1987), "Stepped Spiliway design with a RCC Dam," Froc. Nat.

Cod. on Hydraulic Engrg., ASCE, WiIliamsburg, Virginia, Aug. 3-7, pp. 122-

Ikegawa, MT Washizu, K. (1973), "Finite Wemmt Methad Applied to Analysis of

Flow over a Spiîiway C w " ht J. for Numerical Methods in Eng. Vol. 6, p p

179-189.

Keller, RJ., Ro-, A.K., (19n), 'Design Chart for Predicting Critical Point on

Spiiiwayq" J. Hydrauüc Div., ASCE, VOL 103, No. HY12, pp. 1417-1429.

Keiis, I.A., (1993), "Spatially Varied Flow over Rockm Embankment," Can J.

Civ. Eng., Vol 20, pp. 820-827-

Kennedy, JB., Neville, A M , (1986), "Basic Stdstical Meîhods for Engineers and

Scientists," Fhper & Row Publ., New York, N-Y-

Khan, AA, Stefner, P M , (19%), "Vertidy Averaged and Momentum Equations

Mode1 for Flow over Cwed Beds," J. Hydraulic Engrg., ASCE, Vol. 122, No. 1,

pp* 3-9.

Knauss, J., (1979), "Computation of Bhimum Discharge at Overflow R o c m

Dams," Proc- 13" ICOLS Congres, New Delhi, Iadia, pp. 143-1 59.

Knight, D.W., MacDonald, LA, (1979), "HydraJic Resistrince of Artincial Stnp

Rougbess," I. Hydrauiic Div., ASCE, Vol. 105, No- HY6, pp. 675490.

Ligget, LA., (1991). "Forty Year of Cornputational Hy<fiaulics," ROC. Nat Conf.

on Hydraulic Engrg., ASCE, Nashville, Tmessee, Idy 29-Aug. 2.. pp. 1124-1 133.

Logie, C.V., U e . J X , (1987). "Hydraulic Design Pamo Spiilway." Roc. Nat

C o d on Hydraulic Engrg*, ASCE, Williamsburg, Virginia, Aug. 3-7, pp. 408-413.

Lde, J-C., (1967), "A Variational PrincipIe for a Fiuid with a Free Sdace,," J. of

nuid Mech., UK, Vol. 27, part 2, pp. 395- 397.

Marchi, E., (1993). "On the Free Overfall," J. H y d d i c Res., Vol. 31, No. 6, pp.

777-790.

May, R W E , (1989), "Cavitgtion in Hydrauiic Structures: Occurrence and

Preventation," Hydraulic Research Report, Wailingford, U K

McCorobleJ.A. Li, C.Y., (1971). "Finite Elmient Aadysis of Sluice Gate Fiow,"

T m . Eng. lnst cm, Vol. 14, pp. 189-202.

Mohamed Ali, ES., (1994), "Bed Roughened Stilling B e , " Ploc. Instn Civ.

Engrs., Wat M d & Energy, Vol. 106, Dec., pp. 33 1-341.

Montes, LS., (1994). "Potentiril-Flow Solution to 2D Transition from Mild to Steep

Slope," J. Hydruilic Engrg., ASCE, Vol. 120, No. 5, pp. 601621.

Moore, W.L., (1943), "Energy Loss at the Base of a fnc Overfd," Trans. ASCE,

Voi.108, pp. 1343-1360, Discussions: (1) White, W., (1943), Tans. ASCE, Vol.

108, pp. 1361-1364.

Neuman, SE, Witherspoon, P.A,(1971), "Aiulysis of Non-steady Flow with a F m

Surface Usiog the Finite Element Method," Water Resources Research, Vol. 7, pp.

Noorï, B.MA., (1984), Fonn Drag Resistance of Two Dimensional Stepped Steep

Open Chei,' Roc. 1 " ht C o d on Hydruilic Dcsign in Water Resources

Eng., Southrmpton, UK, pp. 1.133-1.147.

NovJr, P.. Wffac A.I.B., Nalluri* C., (1990), "Hydraulic Structures," Unwin

Hymim Ltd, London, UK

O'CamU, MI, Tom, EX., (1984), "Numerical Computations of Criticai Flow over

a Weu." Int J. for Nmerid Methods in Fluids, Vol. 4, pp. 499- 509.

O'Carroll, MI., asrison, ET., (1976). "Vlliatîon Tecbnïques for F r a StresmIine

Problems," Proc. 2*ht Symp. Finite Elanmt in Flow Roblems, Genoa, pp.

485495.

Patsntsr* S.V.. (1980). "Numerical Heat TrPiisfer and Fiuid Flow," Hemisphere

Publisbg Co., U.S.A.

Peterka, AI.. (1 W8), 'Hydrouiic Design of Stilling Basins and Energy Dissipators,"

Engineering M;onograph No. 25, USBR. Denver, Co.

Peterka, AJ.. (1953). "ïhe Effect of Entraineci Ait on Cavitation Pitting," Joint

Meeting Paper, IAHR/ASCE, Minneapoiis, Mï~esota, Aug., pp. 507-518.

Peyras, L.. Roy* P., Degoutta, G., (1992), "Flow and Energy Dissipation over

Stepped Gabion W*" J. Hydrauiic Engrg., ASCE, Vol. 118, No. 5, pp. 707-717.

Peyras, L., Roy* P., Degoutte, G., (1991), "Flowo and Dissipation of Energy on

Gabion Wei&' J. Houille Blanche, No.1, pp. 37-47.

Pravdivets, YS., Bramiey, ME, (1989), "Stepped Protection Blocks for Dam

Spiiiways," Int Water Power & Dam Con~t~cîion, Jdy , Vol. 41, No. 7, pp.

49-56.

Qibei, L., Baker, R, (1995), "Dam Spillways: Theoretical Anaiysis of FIow

Parameters fbr WedpSharped Spillway Protection Block," J. Dam Engineering,

Vol. VI, issue I, pp. 65-76.

Rajasainam, N., ChPrnani, MX, (1995), "Energy Loss at Drop," J. of Hydraulic

Res., IAHq Val. 33, No. 3, pp.373-384. Discussions: (1) Chanson, E, (1996),

J. of Hydraulic Rcs, Vol, 34, No. 2, pp. 273-278.

Rajaratnam, N., bpadis , C. (1991), "Hydraulic of Steeppass Fishways," Can. I.

Civ. Eng., Vol 18, pp- 1024-1032.

Rajuatmm, N., (199û), "Skioiming Flow in Stepped Spiliways," 1. Hydraulic

Engrg., ASCE, Aug. 3-7, Vol. 116, No. 4, pp. 587-591. Discussions: (1) Degoutte,

G., Peyras, L., and Royet, P., (1992)- L Hydrauiic Engrg., ASCE, Vol. 118, No. 1,

pp. 111-114.

Riajaramam, N., Submumya, K, Mddhar , D., (1968), "Flow Profiles over

Sharp-Cred Weir," J. EIy&aulic Div., ASCE, Vol. 94, No. HY3, pp- 843-847.

Ramaun, N. Engdnm, M., (1993), "An Alganthm for Simulation of Steady Free

Surface Flows," ProcProc of the 8" ht C o d in Numerical Methods in M a r and

TurbJent Flow, la*-23" luly, Swaasea, pp. 435-447.

Rand, W., (1955), "Flow Geometry a! Straight h p Spillways," Ptoceedings,

ASCE, 81% Peper 791, pp. 1-1 3.

Rasanen, SM. Saionen, E-M, (1985), "A qv-fornidation for Free Sinfact Flow."

Roc. of the 4& Int C o d in Numerid Metbods in Lamïnar and Tduient Flowy

9*-12* M y , S- pp. 1393-1401.

Rasenen, SM. Salonen, E-M., (1984), "A Mode1 for Understanding Free Surface

Flow," Numerid Methods for Non-hep Roblems, Val. 2, pp. 849-860.

Rice, CE, Kdnvy, KC., (1996), "Mode1 Study of a RolIer CompPacd Concreîe

Stepped Spillway," 1. Hydruilic Ea@& ASCE, Vol. 122, No. 6, pp. 292-297.

Rice, CE., Kadavy, KC., (1994). "Energy Dissipation for a RCC Stepped

Spillway," Roc. Nat. Cod. on Hydroulic Engrg., ASCE, Briffdo, N-Y., Aug 1-5,

pp. 548552.

Russel, S.O., Sheehan, G.I. (1974). "Effècî of Entraineci Air on Cavitation

Damage," Cm. J. Civ. hg., Vol. 1, pp. 97-107.

Salonen, E-M, Lehtonen, P., Ramils, A, (198 1). "Further qv-Formulation for Ided

Fluid Flow," Roc. of the Iuninnr and Turbulent Flow, Pinendge Press, pp.

1185-1 196.

Scheuedein, H7 (1973), "The Mechania of Flow in Steep, Rough Open Channels,"

Proc. lSB IAHR Congres, Istanbul, Twkey, pp. 1-9.

Sennirk, F., (1994), "Hydraulics of Duns rad Reservoirs," Waîer Resources

Publications, USA.

Sharp, J.J., (1981). "Hydraulic Modehg," Butterworths, London, UK

Smith, LM, Gnnths, DY., (1988). "ProBrpmming the FUUte Element Methoci,"

John Wiely & Sons Ltd, UK.

Smith, N., (1971). "A Himry of Dams: The Chaucer Press, Peter D8Yiest London,

UK.

SoremeqRM, (1985). "Stepped Spillway Hydraulic Model Investigation,* J.

EydrBULic Engrg., ASCE, Vol. 111, No. 12, pp. 1461-1472.

Southwell, RV., Vabey, G-, (1943), "Relaxation Methods Applied to Engineering

Problems. Vm. Plme-Potcatial Roblems Involving S p d d N o d Gradients,"

Proc. Roy. Soc. A. pp. 129-151.

Sternberg. RL., Kalinoudi, AJ, Plpndntis* M., (1979), " Non-linear Partial

DifferaitiJ Equaticms in Engineering and Applid Sciences," Marcel Paldrer Inc.,

New York., N.Y.

Stephenson, D.. (1991). "Energy Dissipation &wn Stepped Spillways," Int Watar

Power & Dam Constn~ction, Sept,Vol. 43. No. 9, pp. 27-30.

Straub. L-G., Anderson, AG., (1958). "Experiments on Self-Aerated Flow in Open

Charnels," J. Hydruilic Div., ASCE, Vol. û4, No. HY7, pp. 1 WO-I,l 890-3 5.

Subramanya, K. (1989). "Flow in Open Channel." Tata McGnw Hïi1 Publishing

Co. Ltd, New De& hdia

Taylor. C., Hughes, T-G.. Morgan., K. (1978). "Finite Elements solution of One

Equation Md& of Tiirbulait Flow," J. of Computational Wysics, Vol. 29. pp.

163-172.

Toc, W., (1994), "Bcsidurl Energy in Stepped Spillwys," Int Water Power &

Dam Co&on, May, Vol. 46. No. 5, pp. 32-34.

U.S. Amy Corps of Enginecrs. (1995), "Hydraulic Design of Spiliway," ASCE

Press. New Yotk, N.Y.

U.S. Amy Corps of Enginers Wltcnmys Experimenta Station, (1977), "Hydraulic

Design Criteria," Vicksburg, Miss.

U.S. Anny Corps of Enginam Watmmys Expriment Station, (1%1), "Prototype

Tests of Spillwiy Crest," Vicksburg, Miss

U.SBR, (1977), "Design of Srnd Dams," Bureau of Redamation, Washington,

D.C.

Vermeyen, TB., Merritt, DH. Taylor, M. (1991). "Design of High Capacity

Spillways for Rit&ud Dam,' Roc. Na C o d on Hydrauiïc Engrg., ASCE.

Nashviile, Tennessee, Juiy 29, Aug. 2, pp.394399.

Vittai, N., Ranga mu, KG., Garde, RJ.. (1977). "Rcsistance of Two Dimaisional

Tnangulu Roughness," J. of Hydraulic Ras., UHR, Vol. 15, No. 1. pp.19-35.

W a d h , K. (1985), "Some Application of Finite Element Techniques to Nonlineu

Free Sutfiace Fluid Flow Aoblems," Roc. of the 4& Int Conf. in Numerical

Mediods in rlaminm md TUrt,ulat FLOW, 9 * - 1 ~ ~ Jdy, S-4 pp.3-15.

Wegmrm, E., (1907). "The Design of the New Croton Dam," T m - ASCE, Paper

1047, June, pp.398-457.

White. W., (1943): Discussion tppcrring in Moore, W.L., (1943).

Wmd, IR, (1991), "Air Entrainment in Free-Surfse Fiows," IAHR Structures

Design Manual No- 4, HydrPulic Design Considerations, Bdema P d . , Rotterdam,

The Nettieriands.

Wortman, RT., (1989), "Spüiway Discharge Calculabons in NWSDAMBRK," J-

Hydraulic Engrg., Vol- I 15, No. 7, pp-976-98 1.

Young, W., (1982), " F h b Z î y Study of a Stepped Spillway," Roc. Hydraulic

Div- Spec. Cbnf., ASCE, New York, N-Y-, pp.96-106-

Zienkiewicz, OC, Szmeher, X, Pcrke, J., (1990), "Compressi'ble and

Incompressi'ble Flow, An Algonthm for Al1 Seasons," Computer Metho& in

Applied Mechania and Engkerïng, Vol. 78, pp.LO5-121.

Zienkiewicz, O.C, Taylor, RL, (1989), "Finite dernent m e t h e McGraw- Hill,

New York, N.Y.

Table A.1. Location of water surface profile for 45' smooth mode1

Table A.4. Location of water surface profile for 45-CS-BL model

Table AS. Location of water surface profile for 45-CS-BS mode1

Table A.8. Location of water sdace profile for 50-CL-BS mode1

Table A.9. Location of water surface profile for 50-CSBL model

Table A.lO. Location of water surface profile for 50-CS-BS mode1

Table A.11. Location of water surface profile for 60' smooth mode1

Table A.14. Location of water surface profile for 60-CS-BL mode1