Hydraulic Structures 978-3-030-34085-8, 978-3-030-34086-5

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C S James

Hydraulic Structures

Hydraulic Structures

C S James

Hydraulic Structures

123

C S James School of Civil and Environmental Engineering University of the Witwatersrand Johannesburg, South Africa

ISBN 978-3-030-34085-8 ISBN 978-3-030-34086-5 https://doi.org/10.1007/978-3-030-34086-5

(eBook)

© Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This textbook is intended to provide for the needs of senior undergraduate and postgraduate students in civil engineering programmes and graduate engineers in civil engineering practice. It aims to provide a solid grounding in the theory underlying the design and analysis of hydraulic structures, including underflow gates, transitions, spillways, culverts, energy dissipators, flow measuring structures and river intakes, as well as the prediction and prevention of scour. Well-established theory and procedures are presented, as well as recent developments gleaned from the research literature with a design-oriented perspective. As a learning resource, the book is not intended to provide complete design details, but rather to develop understanding and competence in applying basic theoretical concepts; the reader is directed to many freely available design guides for more detailed treatment. There is, however, sufficient detail for preliminary design of many structures and complete design of some standard structures such as measuring weirs and flumes and stilling basins. Worked examples are presented in each chapter and exercise problems are provided. The content of the book is based on a postgraduate course in hydraulic structures presented at the University of the Witwatersrand, Johannesburg over many years. At this level, mastery of basic hydraulic theory is assumed. For revision, the introductory chapter provides an overview of the basic concepts relevant to the subsequent content, particularly covering steady rapidly and gradually varied flow in open channels, and flow resistance. At postgraduate level, students and practitioners should be aware of the incompleteness of current knowledge, and be adept at keeping abreast of new developments. Knowledge of some topics presented in the book is clearly not yet definitive, but their inclusion exemplifies the evolving state of the art and the references cited provide first recourse for pursuing advancements. I am indebted to the authors of the many books and articles from which I have gained the knowledge that is presented in this work. I have acknowledged all direct sources as far as possible, but there is much accumulated influence that is implicit

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Preface

and difficult to source and attribute. I am especially indebted to Dr. Heinz Weiss for inspiring my interest in open channel hydraulics as an undergraduate student so many years ago, and to my many students since who have challenged my understanding and explanations. Johannesburg, South Africa

C S James

Contents

1 Basic Hydraulic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Flow Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Conservation Laws in Hydraulics . . . . . . . . . . . . . . . . . 1.3.1 Conservation of Mass—The Continuity Equation . . . 1.3.2 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Conservation of Momentum . . . . . . . . . . . . . . . . . . . 1.4 Steady Uniform Flow and Flow Resistance . . . . . . . . . . . . . 1.4.1 The General Resistance Equation . . . . . . . . . . . . . . . 1.4.2 The Chézy Equation . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 The Darcy–Weisbach Equation . . . . . . . . . . . . . . . . . 1.4.4 The Manning Equation . . . . . . . . . . . . . . . . . . . . . . 1.5 Steady Rapidly Varied Flow . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Application of Energy and Momentum Conservation . 1.5.2 The Control Concept . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Steady Gradually Varied Flow . . . . . . . . . . . . . . . . . . . . . . 1.6.1 The Gradually Varied Flow Equation . . . . . . . . . . . . 1.6.2 Classification of Gradually Varied Profiles . . . . . . . . 1.6.3 Gradually Varied Flow Computation . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 2 7 7 8 12 14 15 17 18 25 28 28 35 49 50 51 52 59

2 Underflow Gates . . . . . . . . 2.1 Introduction . . . . . . . . 2.2 Unsubmerged Analysis 2.3 Submerged Analysis . . 2.4 Hysteretic Behaviour . References . . . . . . . . . . . . .

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3 Open Channel Transitions . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Subcritical Flow Transitions . . . . . . . . . . . . . . . . . 3.3 Supercritical Flow Transitions . . . . . . . . . . . . . . . . 3.3.1 Straight Transitions . . . . . . . . . . . . . . . . . . 3.3.2 Curvilinear Transitions . . . . . . . . . . . . . . . . 3.3.3 Suppression of Standing Wave Propagation . 3.4 Dual Stable States and Hysteresis . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Spillways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction to Conveyance Structures . . . . . . . . . 4.2 Spillway Structures . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 The Overflow Spillway . . . . . . . . . . . . . . 4.2.2 Labyrinth and Piano Key Weirs . . . . . . . . 4.2.3 The Side-Channel Spillway . . . . . . . . . . . 4.2.4 The Side Weir . . . . . . . . . . . . . . . . . . . . . 4.2.5 Shaft (Morning Glory) Spillways . . . . . . . 4.2.6 Siphon Spillways . . . . . . . . . . . . . . . . . . . 4.2.7 Chutes . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.8 Stepped Chutes and Spillways . . . . . . . . . 4.3 Cavitation and Aeration on Spillways and Chutes . 4.3.1 Cavitation . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Aeration . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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105 105 108 108 117 119 126 130 134 144 146 150 150 153 166

5 Culverts . . . . . . . . 5.1 Introduction . . 5.2 Inlet Control . . 5.3 Outlet Control References . . . . . . .

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169 169 170 174 181

6 Energy Dissipation Structures . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . 6.2 The Hydraulic Jump . . . . . . . . . . . . . . 6.2.1 Hydraulic Jump Characteristics . 6.2.2 Controlled Hydraulic Jumps . . . 6.3 Standard Stilling Basins . . . . . . . . . . . 6.4 Other Energy Dissipators . . . . . . . . . . 6.4.1 Bucket-Type Dissipators . . . . . 6.4.2 Impact-Type Dissipators . . . . . . 6.4.3 Baffled Spillways . . . . . . . . . . .

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6.4.4 Stepped Chutes and Spillways . . . . . . . . . . . . . . . . . . . . . 231 6.4.5 Spillway Splitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 7 Flow-Measuring Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Weirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Sharp-Crested Weirs . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Broad-Crested Weirs . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Advantages and Disadvantages of Weirs for Flow Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Flumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Throated (Venturi) Flume . . . . . . . . . . . . . . . . . . . 7.3.2 The Parshall Flume . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 The Cutthroat Flume . . . . . . . . . . . . . . . . . . . . . . 7.4 Long-Throated Structures . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Errors and Measuring Ranges . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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243 243 244 244 253

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8 Intake Structures . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Reservoir Intakes . . . . . . . . . . . . . . . 8.1.2 River Intakes . . . . . . . . . . . . . . . . . . 8.2 River Intake Design for Sediment Control . . 8.2.1 Vertical Sediment Distribution . . . . . 8.2.2 Bed Load Movement Around Bends . 8.2.3 Sediment Exclusion Structures . . . . . 8.3 Pump Sumps and Intakes . . . . . . . . . . . . . . 8.3.1 Desirable Flow Conditions . . . . . . . . 8.3.2 Intake and Sump Design . . . . . . . . . 8.3.3 Model Testing for Intakes . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9 Scour and Scour Protection . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 9.2 Theoretical Analysis . . . . . . . . . . . . . . . . 9.3 Empirical Approach . . . . . . . . . . . . . . . . 9.4 Design Applications . . . . . . . . . . . . . . . . 9.4.1 Critical Shear Stress Design . . . . . 9.4.2 Permissible Velocity Design . . . . . 9.4.3 Protection of Underlying Material .

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Contents

9.5 Scour Around Bridge Piers . . . . . . . . . 9.5.1 Scour Depth Estimation . . . . . . 9.5.2 Bridge Scour Countermeasures . References . . . . . . . . . . . . . . . . . . . . . . . . .

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341 342 351 361

Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

Symbols

a a a a a a ap a* A A A Ar b b b b b B B B B B c c c c ci co C C

Acceleration (m/s2) Bridge pier width (m) Empirical coefficient Reference height for sediment concentration profile (m) Sediment particle weight lever arm (m) Sluice gate opening (m) Projected width of bridge pier (m) Effective bridge pier diameter (m) Cross-sectional area (m2) Half horizontal ellipse axis for spillway profile (m) Plan area of settling basin (m2) Channel expansion area ratio Bridge abutment width (m) Channel width (m) Sediment particle drag force lever arm (m) Siphon barrel width (m) Weir crest width (m) Average width of trapezoidal channel (m) Box culvert/cutthroat flume width (m) Channel width (m) Flow surface width (m) Half vertical ellipse axis for spillway profile (m) Centrifugal pressure adjustment (m) Empirical aeration-bulked flow coefficient Sediment particle lift force lever arm (m) Wave celerity (m/s) Contraction loss coefficient Expansion loss coefficient Air concentration Chézy resistance coefficient (m1/2/s)

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C C Ca Cc Cd Ce Ce∞ Cf Cf Ch CL Co Co Cp Csg C1 d d D D D D D D D Dr Ds Dv D15 upper D15 lower D16 D50 D84 D85lower E E Eo f f fa fe fs f 1, f 2, f 3 F F

Symbols

Empirical spillway discharge coefficient (m1/2/s) Sediment concentration Reference sediment concentration Contraction coefficient Weir/spillway/culvert/flume discharge coefficient Concentration of entrained air Equilibrium concentration of entrained air Energy loss coefficient Stability factor adjustment for riprap stability Contraction coefficient Channel expansion loss coefficient Shaft spillway discharge coefficient (m1/2/s) Air concentration at chute surface Pressure coefficient Density correction for riprap stability Coefficient for submerged cutthroat flume equation Outlet flow depth (m) Siphon barrel depth (m) Circular bridge pier diameter (m) Culvert barrel height (m) Flow depth (m) Inlet bellmouth diameter (m) Offset height for cavitation (mm) Pipe diameter (m) Sediment particle size (m) Riprap stone size (m) Grain size of material underlying filter layer (mm) Depth of Crump weir vee (m) 15% grain size of upper filter layer (mm) 15% grain size of lower filter layer (mm) 16% sediment grain size (mm) Median grain size (mm) 84% sediment grain size (mm) 85% grain size of lower filter layer Sediment settling efficiency Specific energy (m) Limiting sediment settling efficiency Darcy–Weisbach friction factor Submergence correction for Crump weir Aerated flow friction factor Effective friction factor for stepped chutes Riprap factor of safety Factors in bridge scour equation Force (N) Force on bank particle (N)

Symbols

FB Fd FD FD Ff FH FL FR Fr Fr0 FW F1, F2 F2 F*2 g G Go GV h h h h h h hb hd hdam he hexit hf hi hL ho hs hv H H H/ Ha Ha Hc Hc Hd Hdam Hmax

xiii

Force on baffle blocks per unit width (N/m) Densimetric particle Froude number Drag force (N) Pump inlet Froude number Friction force Hydrostatic pressure force (N) Lift force (N) Shear resistance force (N) Froude number Approach flow Froude number Weight component (N) Variables in submergence correction for Crump weir Downstream hydrostatic force for hydraulic jump Downstream hydrostatic force for simple hydraulic jump Gravitational acceleration (m/s2) Modified Froude number in equation for hydraulic jump on slope Radial gate opening (m) Gradually varied flow Baffle block height (m) Height of energy level above weir crest (m) Height of stepped spillway step (m) Height of water level above base of spillway (m) Height of water surface above shaft spillway throat (m) Pressure head (m) Bend head loss (m) Height of downstream water level above weir crest (m) Dam height (m) Entrance head loss (m) Exit head loss (m) Friction loss (m) USBR stilling basin feature heights (m) Local head loss (m) Outlet head loss (m) Hydrostatic pressure head (m) Velocity head (m) Reservoir water level (m) Total energy (m) Energy level above weir crest (m) Head on shaft spillway throat (m) Minimum head for spillway splitter operation (m) Energy at siphon crest (m) Maximum head for spillway splitter operation (m) Design head on spillway (m) Height of dam structure (m) Total energy upstream of dam structure

xiv

Ho Hs Ht H1, H2 HGL HW I k k ke kn ks K K K, K1 K Ka Kp K1 l l L L L L L L L/ Lb LB Leff Lj Li LII Lr L*r Ls Ls m m m M M 1, M 2, M 3 n n

Symbols

Design head for shaft spillway (m) Potential flow velocity head over spillway (m) Total head across shaft spillway (m) Horizontal bed gradually varied profiles Hydraulic grade line Headwater, reservoir water level (m) Intensity of sediment motion (t−1) Exponent in settling efficiency equation Roughness height (m) Entrance loss coefficient Proportionality constant for Manning n related to sediment size Nikuradse roughness (m) Coefficient in sediment incipient motion equation Empirical spillway profile coefficient Discharge coefficients for Parshall/cutthroat flume Laminar flow friction coefficient Spillway abutment contraction coefficient Spillway pier contraction coefficient Transverse slope factor for riprap stability Distance of hydraulic jump from slope junction (m) Length of stepped spillway step (m) Aeration inception distance (m) Cutthroat flume/spillway/weir crest length (m) Length (m) Length of settling basin (m) Length of spillway splitter (m) Wavelength (m) Effective spillway crest length (m) Distance from toe of hydraulic jump to baffle block (m) Stilling basin length (m) Effective weir crest length (m) Hydraulic jump length (m) Distance from chute blocks to sill in stilling basin (m) Length of USBR Basin II Hydraulic jump roller length (m) Simple hydraulic jump roller length (m) Distance from hydraulic jump toe to sill (m) Width of spillway splitter shelf (m) Mass (kg) Number of sediment particles displaced in time interval Shape factor for throated flume Momentum function (m3 or m2 for unit width) Mild slope gradually varied profiles Crump weir side slope (1:n) Empirical spillway profile exponent

Symbols

n n n1, n 2 N N N N N p pa pc po pv P P P P q Q Qa Qf Qi Qp Q* r r ri ro R R R R0 Re Re* Rmax Rmin Rs RV s s si S S S S

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Manning resistance coefficient (s/m1/3) Number of concentric rows of riprap stones Cutthroat flume equation exponents Length factor in hydraulic jump equation Number of piers on spillway Number of sediment particles in observed area Particle weight component normal to bank (N) Shape factor for bridge pier Pressure (Pa) Atmospheric pressure (Pa) Pressure at siphon crest (Pa) Reference pressure (Pa) Vapour pressure (Pa) Location of spillway splitters below crest (m) Net force (N) Wetted perimeter (m) Spillway/weir approach depth (m) Unit width discharge (m3/s/m) Discharge (m3/s) Air discharge (m3/s) Free flow discharge for gated spillway (m3/s) Pump sump inflow discharge (m3/s) Sump pumping rate (m3/s) Weir discharge corrected for submergence (m3/s) Channel expansion width ratio Radius of curvature (m) Inner radius of siphon spillway barrel (m) Outer radius of siphon spillway barrel (m) Force resisting motion of bank particle (N) Hydraulic radius (m) Reaction force on sediment particle (N) Approach flow hydraulic radius (m) Reynolds number Shear Reynolds number Distance of outer spray surface from spillway with splitters (m) Distance of inner spray surface from spillway with splitters (m) Shaft spillway crest radius (m) Rapidly varied flow Expansion flow depth ratio Sill or step height (m) USBR stilling basin feature spacings (m) Pump intake submergence (m) Slope Spacing between spillway splitters (m) Submergence ratio

xvi

S Sf So Ss Ss St Sw SF S1, S2, S3 t t te tr t90 T T T T TW u u* u*c v vc V V Va Vc Vo Vo Vth V0 V0p V0p1 V0p2 V w w wi W W W W W

Symbols

s/y1 Energy gradient Bed slope Channel side slope Sediment specific gravity Transition submergence ratio Water surface slope Stability factor for riprap Steep slope gradually varied profiles Riprap layer thickness (m) Time or time interval (s) Time to equilibrium bridge pier scour (days) Normalizing reference time for scour evolution (s) Time of scour to 90% of equilibrium depth (days) Dimensionless time variable Distance between top of spillway splitters and shelf (m) Particle weight component down bank slope (N) Sump pump cycle time (s) Tailwater height (m) Discharge exponent for Parshall flume Shear velocity (m/s) Critical entrainment shear velocity (m/s) Local velocity (m/s) Velocity at siphon crest (m/s) Average velocity (m/s) Pump sump volume (m3) Actual velocity at foot of spillway (m/s) Critical average velocity for sediment incipient motion (m/s) Approach velocity to weir (m/s) Reference velocity (m/s) Theoretical velocity at foot of spillway (m/s) Average approach velocity (m/s) Approach velocity for maximum live-bed scour (m/s) Approach velocity for maximum live-bed scour (m/s) Approach velocity for maximum live-bed scour (m/s) Average velocity over transverse distance (m/s) Baffle block width (m) Sediment settling velocity (m/s) USBR stilling basin feature widths (m) Cutthroat flume throat width (m) Weight or submerged weight of sediment particle (N) Weight (N) Weight of sediment leaving settling basin (N) Weir height (m)

Symbols

W W Wb Wo x x x x x x* X y y y yb yc yo yR ys yse yt y0 y1 y2 y*2 Y Y* z z z z z/ a a a a a a a a/ b b b b b

xvii

Width (m) Width of spillway splitter (m) Radial gate width (m) Weight of sediment entering settling basin (N) Coordinate direction Depth ratio for throated flume Distance (m) Distance to critical section in side channel (m) Location of spillway splitters below reservoir level (m) Distance from aeration inception point (m) Distance between chute blocks and baffle piers in stilling basin (m) Coordinate direction Flow depth (m) Height of spillway splitters above tailwater (m) Aeration-bulked flow depth (m) Critical flow depth (m) Uniform flow depth (m) Normalizing reference length for scour depth (m) Scour depth (m) Equilibrium scour depth (m) Tailwater depth (m) Approach flow depth (m) Flow depth upstream of hydraulic jump Flow depth downstream of hydraulic jump Downstream flow depth for simple hydraulic jump (m) Sequent depth ratio for hydraulic jump Sequent depth ratio for simple hydraulic jump Baffle block spacing (m) Coordinate direction Depth to flow section centroid (m) Elevation (m) Height of bed above datum (m) Bed or bank slope (degrees) Coefficient in aeration inception equation Coriolis coefficient; kinetic energy correction factor Flow approach angle for bridge pier Labyrinth weir side wall angle (degrees) Triangular weir angle (degrees) Water surface slope in USBR Basin II Pressure coefficient for curvilinear flow Bed shear correction factor in hydraulic jump equation Exponent in aeration inception equation Momentum correction factor Radial gate angle (degrees) Ratio of pier diameter to channel width

xviii

Symbols

b b/ b1 c cs C1 d d DYf DYs Dz Dh e eh eQ es η η0 h h h h1 j K l m q q1 r r so sc scb s* s*c u u w1 w2

Wave disturbance angle (degrees) Pressure force correction for curvilinear flow Angle component in curved transition solution (degrees) Fluid specific weight (N/m3) Sediment specific weight (N/m3) Variable in equation for hydraulic jump on slope Boundary layer thickness (m) Ratio of riprap size to underlying material size Hydraulic jump sequent depth correction to account for wall friction Hydraulic jump sequent depth correction to account for sill Change in bed level (m) Side wall deflection angle (degrees) Multiple of riprap stone size for distance specification Indication error Discharge error Sediment diffusivity (m2/s) =w/(w + z) Riprap stone stability number Angle of underflow gate lip (multiples of 90o) Bed slope angle (degrees) Varying side wall angle (degrees) Integration constant for curved transition (degrees) Von Karman constant =Ls/L*r Absolute fluid viscosity (kg/m/s) Kinematic fluid viscosity (m2/s) Fluid density (kg/m3) Ratio of water to water + air discharge Cavitation index Sediment geometric standard deviation Boundary shear stress (N/m2) Critical boundary shear stress at incipient sediment motion (N/m2) Critical shear stress on bank particle (N/m2) Dimensionless shear stress (Shields parameter) Critical dimensionless shear stress (critical Shields parameter) Angle of repose of sediment material (degrees) Sediment particle pivot angle (degrees) Size factor in hydraulic jump calculation with baffles Blockage factor in hydraulic jump calculation with baffles

Chapter 1

Basic Hydraulic Concepts

1.1

Introduction

Hydraulic structures can be broadly classified into four groups: conveyance structures are designed to pass a specified discharge safely from one location to another; regulation structures control flow by inducing required water levels or discharges; measuring structures enable discharges to be determined from measured water levels and energy dissipation structures cause excess flow energy to be expended safely. The design and analysis of such structures require establishing relationships between flow characteristics such as discharge, pressure, flow depth and boundary shear stress, and the bounding geometry of the structure. Most commonly, it is required to predict the flow depth (y) around and through a structure for a particular design discharge (Q). (For measuring structures, it is required to predict the discharge corresponding to a measured flow depth or depths.) As an example, a simple structure installed in a channel in the form of a smooth rise and fall (Fig. 1.1) could affect the flow depths over and around it in a number of ways; possible water surface profiles are shown in Fig. 1.1. There may be only a local effect, with the water surface rising or dipping over the structure (Figs. 1.1a, c), or the effect could extend considerable distances upstream and downstream, involving a hydraulic jump (Figs. 1.1b, d). Only one of these profiles can actually occur for a particular discharge and structure size, and which it is depends on some characteristics of the flow. Before developing the relationships required for describing flow responses to boundary geometry, it is therefore necessary to characterize the different flow ‘types’ that influence its behaviour.

© Springer Nature Switzerland AG 2020 C S James, Hydraulic Structures, https://doi.org/10.1007/978-3-030-34086-5_1

1

2

1 Basic Hydraulic Concepts

(a)

(b)

Q

Q

(c)

(d)

Q

Q

Fig. 1.1 Possible water surface responses to a simple structure

1.2

Flow Classification

Both free surface and closed conduit flow are encountered in hydraulic structures. The following classification is presented for the more general case of free surface flow (Fig. 1.2). Some characteristics and definitions are the same for closed conduit flow, while some are unique to free surface flows. The first distinction between flow types is their classification as steady or unsteady according to the variation with time. If flow conditions (such as velocity and depth) do not change with time, then the flow is steady. If they do change with Free Surface Flow

Steady

Uniform

Unsteady

Nonuniform

Gradually Varied Fig. 1.2 Basic types of free surface flow

(Uniform)

Rapidly Varied

1.2 Flow Classification

3

time, then the flow is unsteady. The small-scale variations of flow associated with turbulent fluctuations and slow variations over long periods are not here considered to be violations of steadiness. Flow can also be classified as uniform or nonuniform according to its variation with position. If the flow conditions are the same everywhere in the flow, then the flow is uniform. If the flow conditions are different at different locations, then the flow is nonuniform. In most practical applications, flow is assumed to be uniform if average flow conditions do not change along a conduit, so that changes of direction through bends and the variation of velocity through the boundary layer are not considered as violations of uniformity. Unsteady nonuniform flow is the general type, and commonly needs to be described fully, particularly in natural situations. For many problems, the flow can be considered to be steady and sometimes also uniform without introducing significant errors in practical solutions. Steady uniform flow is the simplest flow type, and its analysis provides results which have useful application to the more general types as well. Unsteady uniform flow is practically impossible; it would require a water surface to change with time but remain parallel to its original form, which is impossible for an incompressible fluid like water. Nonuniform flow may be steady or unsteady. In either case, it can be further classified as gradually varied (GV) or rapidly varied (RV). In gradually varied flow, the changes in flow condition take place over long distances, such as the backing up which occurs upstream of a dam or weir. In rapidly varied flow, the changes take place over comparatively short distances, such as those over a structure, through a constriction, or in a hydraulic jump. Rapidly and gradually varied changes are fundamentally different in nature, the former being associated primarily with local changes in boundary geometry with no significant influence of surface resistance, and the latter being controlled primarily by flow resistance. The profile shown in Fig. 1.1b can be classified for steady flow in these terms, as shown in Fig. 1.3. Unsteady gradually varied flow is the most general flow type in nature, and needs to be analysed for describing the passage of flood waves along rivers. Unsteady rapidly varied flow also occurs as surges in rivers (as, for example, the nonuniform uniform GV

RV

GV

RV

Q uniform

Fig. 1.3 Steady water surface profiles induced by a simple structure in a channel

4

1 Basic Hydraulic Concepts

famous tidal bore in the River Severn) and canals (for example, as resulting from rapid closure of gates, particularly in hydroelectric power installations). Steady rapidly varied flow is the typical situation for hydraulic structure analysis and design. Another flow classification which applies to open channel flow as well as to closed conduit flow distinguishes between laminar and turbulent flow. The distinction can be defined in terms of the Reynolds number (Re), which expresses the relative significance of inertial and viscous forces in determining the flow behaviour. Laminar flow is not common in practical applications of open channel flow and will not be given much emphasis. It could occur in physical models or in sheet flow such as drainage of paved surfaces, however, and this possibility should be recognized and its occurrence established when it occurs. This can be done by checking the value of the Reynolds number against established values defining the ranges of each flow type. The Reynolds number is defined for pipes as Re ¼

VD m

ð1:1Þ

in which V is the average flow velocity, D is the pipe diameter and m is the kinematic viscosity of the fluid. The threshold values for laminar and turbulent flow for pipes are laminar flow turbulent flow

Re \ 2000 Re [ 4000

Between these values, there is an uncertain transition range. Obviously, D is not defined for open channels and Re is usually defined in terms of a parameter known as the hydraulic radius, R. This is defined as R ¼

A P

ð1:2Þ

in which A is the cross-sectional area and P is the length of wetted perimeter. If D is replaced by R in the definition of Re, the threshold values for laminar and turbulent flows will be different because, as can easily be shown, R = D/4. The threshold values, and other important relationships, can also be applied unaltered to open channels if the Reynolds number is defined as Re

¼

VD m

¼

4VR m

ð1:3Þ

The inclusion of the factor 4 in the definition of the Reynolds number simplifies the use of results obtained from pipe data considerably, and is widely used. There are different types of turbulent flow, depending on the structure of the boundary layer. Close to the solid boundary the flow is very slow and a viscous sub-boundary layer can develop with a thickness (d) determined by the magnitude

1.2 Flow Classification

5

Fig. 1.4 Types of turbulent flow past a solid boundary

k

δ

k

δ

(a) Hydraulically smooth

(b) Transitional

k δ (c) Hydraulically rough of the boundary shear stress, the fluid viscosity and the size of the roughness elements on the boundary, k (Fig. 1.4). Hydraulically smooth flow occurs if the roughness elements are well submerged by the viscous flow layer, i.e. k < d. Transitional flow occurs when the roughness elements are about the same size as the viscous layer thickness, and starts influencing the flow further away, i.e. k  d. Hydraulically rough flow occurs when the roughness elements are more than about five times the viscous layer thickness appropriate to the prevailing flow conditions, and break it up completely. The transition between laminar and turbulent flow can be defined by the Reynolds number but the transition between the different types of turbulent flow takes place at different values of Re, depending on the surface roughness. The Reynolds number is therefore not sufficient to characterize the different types of turbulent flow. An appropriate parameter can be determined for doing this by considering quantitatively the physical reason for the differences, i.e. the relationship between the thickness of the viscous sublayer and the size of roughness elements, k/d. The thickness of the viscous sublayer can be calculated as d¼

11:6 m u

ð1:4Þ

in which u* is the shear velocity (or friction velocity) defined by rffiffiffiffiffi sffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi so cRS ¼ gRS u ¼ ¼ q q

ð1:5Þ

6

1 Basic Hydraulic Concepts

in which so is the boundary shear stress, c (=qg) is the fluid specific weight, q is the fluid density and g is gravitational acceleration. The ratio of the roughness element size to the viscous sublayer thickness is therefore k k u ¼ d 11:6 m Removing the constant 11.6, and replacing k by the Nikuradse roughness, ks, (which is related to, but not necessarily equal to the actual roughness size), means that k d

a

k s u ¼ Re m

ð1:6Þ

in which Re* is a dimensionless number, very similar in form to the Reynolds number, known as the shear Reynolds number, Re*. The shear Reynolds number can be used to define the regimes of turbulent flow as follows: Re \ 5 5\Re \ 70 Re [ 70

hydraulically smooth flow transitional flow hydraulically rough flow

Note that flow in a particular channel may be laminar or may be turbulent in any of these regimes, depending on the prevailing discharge. These are hydraulic concepts, and the flow type cannot be classified according to the channel characteristics only—the discharge must be known as well. A further very important flow classification for open channel flow reflects the relative importance of gravitational and inertial forces on the flow, which is significant because of the presence of the free surface and can be characterized by the Froude number. This is defined as Q Fr ¼ qffiffiffiffiffiffiffi g AB

ð1:7Þ

in which g is gravitational acceleration and B is the flow width at the surface. For rectangular channels, the surface width is equal to the channel width, so V Fr ¼ pffiffiffiffiffiffi gy

ð1:8Þ

where y is the flow depth. If the value of Fr is 1 it is supercritical, and if Fr = 1, it is critical. This is a very important classification for open channel flow because the flow behaviour of subcritical and supercritical flow is quite different. The flow in Fig. 1.1a is wholly subcritical, in Fig. 1.1c it is

1.2 Flow Classification subcritical

7 critical

Q

supercritical

subcritical

Fig. 1.5 Subcritical, critical and supercritical flow in a composite profile

wholly supercritical and in Figs. 1.1b, d all types occur, as shown in Fig. 1.5 for the profile of Fig. 1.1b. The following sections present the essential results for analysing steady uniform and nonuniform flow conditions.

1.3

The Conservation Laws in Hydraulics

Most practical problems in hydraulics are solved by application of the three basic conservation laws, viz., conservation of mass, energy and momentum. These are expressed in equations with various forms. The most complete forms are obviously the most general, but they are also the most complicated. By making certain assumptions, particularly of steady and incompressible flow, simpler forms can be derived which are adequate for many practical applications. Only the simple forms necessary for the applications considered in this text are presented here; more complete forms and their derivations are presented in standard introductory hydraulics texts. These simple forms do have limitations resulting from their underlying assumptions, which should be noted.

1.3.1

Conservation of Mass—The Continuity Equation

Under the assumptions of steady, incompressible flow, the form of the mass conservation equation is particularly simple. The incompressibility assumption enables variations of density to be ignored and the law considered in terms of volume. The simplest form of the continuity equation applies where the discharge does not vary in the flow direction, and concern is with flow in one direction only. Then Q ¼ V A ¼ constant

ð1:9Þ

This form will be sufficient for most applications in analysing hydraulic structures. In cases where the discharge varies spatially or velocity components in more

8

1 Basic Hydraulic Concepts

than one direction need to be considered, more complete forms are required. These will be introduced where necessary.

1.3.2

Conservation of Energy

The law of energy conservation as applied in hydraulics states simply that the energy of the flow at a particular location, or section, is equal to the energy at a specified section upstream minus the losses between the two sections, i.e. Hdownstream ¼ Hupstream

losses

ð1:10Þ

where H represents the total energy of the flow and the subscripts represent the spatial order of the sections. The total hydraulic energy includes both potential and kinetic components. It is transformation between these components that determines the variation of flow conditions induced by hydraulic structures, and these can be described using the Bernoulli equation. This equation can be derived by integrating Newton’s second law with respect to distance, subject to the assumptions of steady, incompressible, irrotational flow, resulting in the form p v2 þ zþ ¼ constant ¼ H c 2g

ð1:11Þ

in which p is pressure, c is the specific weight of the fluid (=qg), z is elevation and v is local velocity. The terms p/c and z represent potential energy and V2/2g represents kinetic energy. The limitation of irrotational flow means that the energy conservation Eq. (1.10) with H defined by Eq. (1.11) cannot be applied across streamlines within a (rotational) boundary layer. The common application to cross-section average flow conditions (v = V) at successive locations in the flow direction is acceptable, however, even with rotational flow. Note that energy is a scalar quantity and is therefore unaffected by changes in flow direction. The derivation of Eq. (1.11) included a division of the energy components by c, meaning that the final terms represent the energy per unit weight of fluid; the total energy therefore depends on velocity, but not on discharge. The consequent dimensions of the terms in the Bernoulli equation are therefore length, and the terms are commonly referred to as ‘heads’. Having length dimensions enable the terms to be represented graphically, as shown in Fig. 1.6. The elevation and pressure head are commonly referred to in combination as the piezometric head, indicating the location of the hydraulic grade line (HGL). Sketching energy lines is extremely useful for understanding and analysing variations of flow velocity and pressure in hydraulic systems. For example, the energy lines shown in Fig. 1.7 for flow through a pipe contraction enable the variations of velocity and pressure to be visualized easily.

1.3 The Conservation Laws in Hydraulics

9

Total Energy Line, H

V2/2g

Hydraulic Grade Line, HGL

velocity head pressure head

p/

total head

piezometric head

Pipe centre line

z Datum

Fig. 1.6 Energy terms and energy lines for flow in a pipe

Fig. 1.7 Energy lines for flow through a pipe contraction

H V12 /2g

V2

2 /2g

HGL p1 /

z1

p2 /

z2 2

1

Datum

In most problems, energy conservation is applied between sections in the direction of flow. It is then convenient to express the kinetic energy term of the Bernoulli equation in terms of the average velocity over the section, V = Q/A. Because of the nonuniformity of flow within boundary layers, this will give a result different from that obtained by integration over the section, and will therefore be inaccurate. To account for this, a correction factor called the Coriolis coefficient or kinetic energy correction factor (a) is applied to the mean velocity head, which then becomes aV2/2g. A value for this factor can easily be determined if the velocity distribution is known from a¼

R

v3 dA V 3A

ð1:12Þ

For laminar flow, a has a value of about 2.0. Laminar flow is rare in practical problems although it may occur in small-scale physical models of hydraulic structures. For turbulent flow, the value averages about 1.06 and rarely exceeds about 1.15, and is often assumed to have a value of 1.0 in practical calculations. For open channel applications, the Bernoulli equation is usually modified slightly. If the pressure distribution is hydrostatic, then the hydraulic grade line,

10

1 Basic Hydraulic Concepts

Fig. 1.8 Energy terms and energy lines for flow in an open channel

αV 2/2g

H E

p/ .A

HGL

y cos θ y

z

z/

θ

Datum which represents the piezometric head (z + p/c) at any point on a vertical (such as A in Fig. 1.8), coincides with the water surface. In this case, the Bernoulli equation can also be written H ¼ z0 þ y cos h þ a

V2 2g

ð1:13Þ

where z0 is the height of the bed level above the datum, y is the flow depth measured normal to the bed and h is the inclination of the bed to horizontal, as shown in Fig. 1.8. The specific energy, E, is defined as the total energy relative to the channel bed as datum, i.e. E ¼ y cos h þ a

V2 2g

ð1:14Þ

This concept was introduced by Bakhmeteff in 1912. It rests heavily on the assumption of a hydrostatic pressure distribution, and it is important to identify conditions where this assumption might not hold. The cos h term is usually ignored because bed slopes are small and it is then close to 1.0. Chow (1959) points out that this correction amounts to less than 1% until the slope exceeds 6°, or about 0.10. For greater slopes, cos h should be included. Another difficulty associated with steep slopes is that velocities are usually high, and flowing water entrains air when velocities exceed about 6 m/s. This can be clearly seen on spillway faces, where the flow becomes aerated quite suddenly some distance below the crest. When this happens, ‘bulking’ occurs and the flow depth increases. Predicted pressures are then higher than observed. If air entrainment is expected, the water density used in the calculations should be that of the air–water mixture.

1.3 The Conservation Laws in Hydraulics

11

hydrostatic

p

Fig. 1.9 Pressure distribution at a free overfall

(a)

(b) hydrostatic p hydrostatic p

Fig. 1.10 Pressure distributions in convex (a) and concave (b) flow

The pressure distribution will also not be hydrostatic if there is distinct curvature of the flow. An important instance of non-hydrostatic pressure is in the drawdown to critical flow at a free overfall, shown in Fig. 1.9. In addition to the strong vertical curvature, aeration of the underside of the nappe causes the pressure to be atmospheric at the brink. The geometry of a structure can also induce curvilinear flow, for example, at the base of a spillway. In this case, the flow would be concave, but convex flow can also occur. These situations are illustrated in Fig. 1.10. In concave flow, the centrifugal forces reinforce gravity and the pressures are greater than hydrostatic. In convex flow, the centrifugal forces oppose gravity and the pressures are less than hydrostatic. The actual pressure can be determined by adding a centrifugal component to the hydrostatic. The approximate centrifugal pressure can be computed as the product of the mass of a column of water with unit area (cy/g) and the centrifugal acceleration (v2/r, where r is the radius of curvature). Dividing by c gives this pressure as a head. The total pressure head is then h ¼ hs 

y v2 g r

ð1:15Þ

in which hs is the hydrostatic pressure head and the + and − signs correspond to concave and convex flow, respectively.

12

1 Basic Hydraulic Concepts

For simplicity, the pressure head for curvilinear flow is often represented by a0 ycos h, where a0 is a pressure coefficient. It can be shown that a0 is given by Z 1 c v dA ð1:16Þ a0 ¼ 1 þ Qy in which c is the centrifugal term in Eq. (1.15), i.e. c = y/g v2/r. For complicated curved profiles, the pressure distribution should be determined using ideal flow theory (flow nets), numerical modelling or model testing.

1.3.3

Conservation of Momentum

Integration of Newton’s second law with respect to time leads to a formulation of momentum conservation. For a solid object, this shows that a force acting on the object for a certain time duration will result in a change in its momentum. For fluids, the continuum of fluid and continuous application of forces must be considered. A force applied to a fluid (F) causes a change in its momentum flux or the rate at which momentum passes a section. For steady, incompressible flow, this can be expressed as X

 Fx ¼ q Q Vfx

Vix



ð1:17Þ

which is known as the force–momentum flux equation. The subscript x refers to the direction considered, Vxf and Vxi are the final and initial velocities in this direction. The final and initial locations are the inflow and outflow sections of an identified free body or control volume of fluid. The forces in the prescribed direction include pressure forces at the initial and final locations, boundary forces between these locations, and the weight of the free body if the flow is not horizontal. The force–momentum flux equation applies to the whole flow under consideration, not to a unit weight of fluid as does the Bernoulli equation, because it involves discharge as well as velocity. It has the advantage that it deals with overall influences on the free body of fluid defined by the bounding geometry and the inflow and outflow sections—the distribution of forces between the sections is not of concern, only the net effect. Note also that because the equation is in terms of vector quantities, direction is an important consideration—energy does not change as flow goes around a bend, but momentum flux does. Evaluating the total force between the inflow and outflow sections may require application of the equation in mutually perpendicular directions to establish the force components before their vector addition. As in the energy equation, the use of an average velocity (V = Q/A) will introduce an error if the flow is not uniformly distributed over the section. A correction factor (b) to be applied to the velocities can be determined from the velocity distribution as

1.3 The Conservation Laws in Hydraulics

b¼

13

R

v2 dA V2 A

ð1:18Þ

For laminar flow, b has a value of about 1.33. For parallel turbulent flow in open channels, the value is about 1.05. As this can be considered close to 1.0 in most practical applications, it is often assumed to be equal to 1.0 and may not appear in the equation. For open channels, the sum of forces will include pressure forces at the sections at the beginning and end of the free body to which Eq. (1.17) is applied. If these sections are both located in flow regions where streamlines are straight and parallel, then these forces may be calculated from the hydrostatic pressure distribution. If the flow is curvilinear at either or both sections, then the pressure forces must be corrected. Chow (1959) recommends multiplying the hydrostatic pressure force by a force coefficient b0 , given by Z 1 c dA ð1:19Þ b0 ¼ 1 þ Az in which z is the depth of the centroid of the flow section below the water surface and c = y/g v2/r, as used for the correction factor a0 in Eq. (1.16). In open channel applications, the net force in the force–momentum flux equation will always include pressure forces at the end sections. If these are hydrostatic, Eq. (1.17) can be expressed as P ¼ M2 c

M1

ð1:20Þ

in which section 2 is downstream of section 1, P is the net force applied to the free body between the sections in the direction of flow and M is the momentum function, defined by M¼

Q2 þAz gA

ð1:21Þ

in which z is the depth of the section centroid below the water surface. Note that M includes the hydrostatic forces at the ends of the free body. For flow over unit width M is defined as M¼

q2 y2 þ gy 2

ð1:22Þ

in which q is the discharge per unit width. If the unit width definition is used in Eq. (1.20), then P is the force per unit width. The form of the momentum equation as expressed by Eqs. (1.20)–(1.22) has no great practical advantage. It just means that the pressure forces at the sections do not

14

1 Basic Hydraulic Concepts

need to be accounted for separately, which makes some applications more efficient. For nonprismatic channels, it is often easier to use the form of Eq. (1.17) directly. Applications of the conservation laws for analysing the flow situations identified in Fig. 1.2 are discussed in the following sections. Attention is limited to steady flow, as most hydraulic structures can be analysed under this assumption.

1.4

Steady Uniform Flow and Flow Resistance

Steady uniform flow is the most fundamental flow condition in channels and, at least conceptually, the easiest to analyse. It also provides a good opportunity for developing resistance relationships that can also be used in nonuniform situations— under steady uniform flow, the rate of energy dissipation through resistance matches the rate at which it is made available by the channel gradient, both of which are constant. The energy loss term in Eq. (1.10) is then defined by the channel slope, and this is related to the flow velocity by an empirically based resistance equation. Steady uniform flow is the type that occurs for a steady discharge in a long channel with constant geometry and roughness characteristics. The engineering problem is to establish the relationship between discharge and flow depth. This is necessary for designing a canal to convey a certain discharge, where the flow depth would determine the size of channel required. The same type of relationship can be used inversely to determine the flow capacity of a channel with known characteristics. For this condition, the physical characteristics that determine the flow depth are those that are associated with the surface resistance at the boundaries. The relationships to be developed describe the influence of boundary surface resistance, and will be used wherever flow resistance must be accounted for, not just for steady uniform flow. The relationship between discharge and flow depth is an expression of the continuity equation (1.9), in which the cross-sectional flow area (A) and the depth-averaged velocity (V) depend on the flow depth (y) and the channel characteristics, i.e. Q ¼ AðyÞ Vðy; channel characteristicsÞ

ð1:23Þ

where A(y) is defined by the cross-sectional geometry of the channel and the relationship V(y, channel characteristics) is determined by the resistance of the channel to flow. As well as matching energy dissipation and channel gradient, uniform flow can be interpreted as an equilibrium situation in which the downstream component of the weight of a flow element driving its motion is exactly balanced by the shear stress on the boundary resisting its motion (Fig. 1.11). The driving force depends on the weight of the water and hence the flow depth (yo, with the subscript o indicating uniform flow) and the resisting force depends on the velocity—otherwise, the element would not be in equilibrium at a particular velocity. The required relationship V(y, channel characteristics) can be determined

1.4 Steady Uniform Flow and Flow Resistance

15

A Sf

V2/2g L

Sw

Q yo

FW

FH1 FR

FH2 θ

1

W

P H HGL x

So

2

Fig. 1.11 Motion of an element in uniform flow

by first evaluating the resisting boundary shear stress (so) in terms of the channel characteristics and flow depth (so = f(yo, channel characteristics)), formulating an independent relationship between the shear stress and velocity (so = f(V)), and then combining these relationships by eliminating so.

1.4.1

The General Resistance Equation

The relationship between the resisting stress and channel characteristics, i.e. so = f (yo, channel characteristics), can be obtained by analysing the motion of an element isolated in uniform flow (Fig. 1.11). Because the flow is uniform, the flow depth and velocity do not change in the flow direction. Therefore, the water surface is parallel to the bed, i.e. Sw = So and, because the energy line is a distance V2/2g above the water surface (at the hydraulic grade line), the energy line is also parallel to the water surface, so Sf = Sw = So. The motion of the free body can be analysed by applying Newton’s second law, F ¼ ma where F is the net force acting on the free body, m is its mass and a is its acceleration in the direction of F. There are four forces comprising F acting on this free body: – The component of its weight (W) in the flow direction, given by FW ¼ W sin h W is the product of density (q), gravitational acceleration (g) and volume AL. Because channel slopes are generally very small, sin h can be approximated by tan h, which is the bed slope So. Therefore

16

1 Basic Hydraulic Concepts

FW ¼ W sin h ¼ q g A L So – The resisting force arising from a shear stress (with average value so) over the contact surface between the water and the channel is given by FR ¼ so P L – Hydrostatic pressure forces on the upstream and downstream faces of the free body (FH1 and FH2). Because the flow is uniform and the flow depths are equal, these forces act in opposite directions and are equal in magnitude and therefore cancel. Uniform flow implies that the flow depth and velocity are constant in the flow direction, and there is therefore no acceleration. The net force on the element is therefore zero, and so FR þ FW ¼ 0 i.e. s o P L þ q g A L So ¼ 0 and so so P L ¼ q g A L S o The average shear stress on the boundary can therefore be expressed as so ¼ q g

A So P

The quantity A/P is the hydraulic radius, R. Therefore s o ¼ q g R So or, using c ¼ qg; s o ¼ c R So

ð1:24Þ

This is a particular solution for uniform flow. A similar relationship can be derived for steady flow generally, i.e. including nonuniform flow where So 6¼ Sw 6¼ Sf, where the hydrostatic forces on the upstream and downstream faces of the free body do not cancel and the acceleration is not zero. This gives the general result

1.4 Steady Uniform Flow and Flow Resistance

17

s o ¼ c R Sf

ð1:25Þ

This general equation always applies. If the flow is known to be uniform, then it is known that Sf is numerically equal to So and Eq. (1.24) applies. The resistance equations to follow include slope, which is conventionally designated as S, without subscript. This is always Sf and is also So if the flow is uniform. The same equations are used to evaluate Sf in gradually varied nonuniform flow analysis. As well as providing the basis for resistance equations, Eqs. (1.24) and (1.25) are useful for evaluating shear stresses for assessing sediment erosion in rivers and for determining u* as required for describing vertical velocity profiles or classifying turbulent flow through Re*. Formulating the required relationship V(y, channel characteristics), also requires an independent relationship between the shear stress and velocity (so = f(V)). This was first done by Antoine Chézy in 1768.

1.4.2

The Chézy Equation

Chézy used dimensional analysis to relate the bed shear stress and velocity as so ¼ a q V 2

ð1:26Þ

in which a is a dimensionless coefficient which could depend on channel characteristics. Equating the right-hand sides of Eqs. (1.25) and (1.26) gives so ¼ a q V 2 ¼ c R S from which V¼

rffiffiffiffiffiffiffiffiffiffiffi g RS a

ð1:27Þ

pffiffiffiffiffiffiffi RS

ð1:28Þ

This is usually written as V ¼C

and is known as the Chézy equation, with C known as the Chézy resistance coefficient. This enables calculation of the flow velocity from the channel slope, cross-section dimensions and flow depth, provided a value for C is known. With both the flow depth and velocity known, the discharge can be simply calculated from the continuity equation.

18

1.4.3

1 Basic Hydraulic Concepts

The Darcy–Weisbach Equation

Through empirical work and dimensional considerations by Henri Darcy and Julius Weisbach in the mid-nineteenth century, Eq. (1.29) was developed for relating the friction loss (hf) in a pipe to its length (L), diameter (D) and roughness. hf ¼

f L V2 2gD

ð1:29Þ

The friction factor, f, is related to the physical roughness of the boundary and the turbulence characteristics of the flow. The friction factor is dimensionless, making the equation more theoretically appealing than the Chézy equation. Noting that hf /L = Sf and D = 4R, Eq. (1.29) can be reformulated to provide a resistance equation for open channel applications, i.e. sffiffiffiffiffiffi 8 g pffiffiffiffiffiffiffi RS V¼ f

ð1:30Þ

Experimental work in the first half of the twentieth century produced ways for estimating values for the friction factor, in particular, the Colebrook–White equation and the Moody diagram. The Moody diagram (Fig. 1.12) shows the relationship between the friction factor, Reynolds number and the relative roughness, ks/D. Although the diagram and the corresponding equations were developed for pipe data, the effect of cross-section geometry has been found to be small and they apply for channels as well, as long as the Reynolds number is defined as 4VR/m and the relative roughness as ks/4R. Comparing Eqs. (1.28) and (1.30) shows that the Chézy and Darcy–Weisbach equations are equivalent, with sffiffiffiffiffiffi 8g C¼ f

ð1:31Þ

so the results for f can easily be used to calculate values of C. The Moody diagram shows the variation of f in two major zones, for laminar flow (Re < 2000) and turbulent flow (Re > 4000). The turbulent zone can be further subdivided into zones of hydraulically smooth (Re* < 5), transitional (5 < Re* < 70), and hydraulically rough (Re* > 70) flow. The variations in each of the zones have been represented by equations, which are easier and more accurate to apply in practice than the diagram. In the laminar flow region, f is independent of relative roughness and inversely proportional to Reynolds number and can be represented by

1.4 Steady Uniform Flow and Flow Resistance laminar

19

turbulent

0.10 0.08 5·10-2 0.06

rough 0.04

10-2

transitional 0.03

f

ks/4R

laminar 10-3

0.02

10-4 0.01 10-5

103

104

105

106

107

108

Re = 4VR/ν Fig. 1.12 Moody diagram for friction factor

f ¼

K Re

ð1:32Þ

For pipes K = 64, but different values have been reported for free surface flow. Free surface laminar flow is not common in engineering practice, and further details are not provided here. In the turbulent flow region, different equations describe the variation of f with Re and relative roughness in the different zones. For hydraulically smooth flow (Re* < 5), two different equations apply over different ranges of Re. The Blasius equation describes the variation for Re < 105, i.e. f ¼

0:316 Re0:25

ð1:33Þ

For Re > 105, the variation is described by pffiffiffi 1 Re f pffiffiffi ¼ 2 log 2:51 f

ð1:34Þ

Having these two equations means that if flow is hydraulically smooth (Re* < 5), Re must still be determined to enable the better equation to be selected and used.

20

1 Basic Hydraulic Concepts

Note that for hydraulically smooth flow f depends on Re but not on the surface roughness. This means that provided the roughness elements are completely submerged within the viscous sublayer, their size does not influence the resistance. This does not apply for transitional flow. The shapes of the curves on the Moody diagram indicate that f depends on both Re and ks. The corresponding equations must reflect this. For transitional flow (5 < Re* < 70), the Colebrook–White equation applies. For pipes, this is 1 pffiffiffi ¼ f

2 log



ks 2:51 pffiffiffi þ 3:7 D Re f



ð1:35Þ

Substituting D = 4R to make the equation applicable for channels gives 1 pffiffiffi ¼ f

  ks 2:51 pffiffiffi þ 2 log 14:8 R Re f

ð1:36Þ

Note that if Re is defined as VR/m, then the number in the last term will be different from 2.51; only for Re = 4VR/m it is 2.51. It has been found empirically that for channels, the coefficients should be slightly modified, giving 1 pffiffiffi ¼ f



ks 2:5 pffiffiffi þ 2 log 12 R Re f



ð1:37Þ

In this case, f depends on both Re and ks. Note that the second term in the transitional equation corresponds to hydraulically smooth flow and the first term to hydraulically rough flow (to follow below). For hydraulically rough flow (Re* > 70), f depends only on the relative roughness, and not at all on Re (as can be seen on the Moody diagram). The Re term in the modified Colebrook–White equation therefore falls away, giving 1 12 R pffiffiffi ¼ 2 log ks f

ð1:38Þ

The transitional equation can be used in this zone, but the Re term will become negligibly small as flow becomes fully turbulent. Equations (1.33) to (1.38), and the Moody diagram, show that f and hence C depends on the shape and size of the conduit (through R) and on the flow condition (through Re) as well as the surface roughness. Even for hydraulically rough flow, f and C are not constant for the same channel as the discharge and hence the relative roughness varies. ks, however, is a representation of the channel roughness that is independent of flow conditions and is therefore a more reliable characterization than direct estimates of f or C. Some representative values of ks are given in Table 1.1.

1.4 Steady Uniform Flow and Flow Resistance Table 1.1 Values of ks for common surfaces

21

Surface

ks (mm)

PVC (plastic) Steel pipe Very smooth concrete Smooth cement-plastered surfaces with flush joints Steel-trowelled concrete Unfinished concrete Shotcrete

0.01–0.02 0.045 0.15 0.30 1.5 7 14

Calculating f for smooth (using Eq. (1.34) and transitional flow (Eq. (1.37) requires iterative solution because it appears on both sides of the equations as they stand. For many problems, it is useful to make the equations explicit by substituting Eq. (1.30) for V in the Reynolds number. The f terms then cancel, and the smooth term becomes 2:5 2:5 m pffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi Re f 4 R 8 g R S

ð1:39Þ

allowing an explicit solution. Iterative solution cannot be avoided when using Eq. (1.33), but its applicable range of Reynolds number is unlikely to occur in practical engineering applications. Solutions for problems where the flow depth is known and the discharge is required are straightforward. The flow regime can be determined directly by calculating Re*, enabling the applicable equation for f to be identified. If the flow regime is smooth, the Reynolds number should also be calculated to enable selection between Eqs. (1.33) and (1.34). The flow velocity can then be calculated using Eq. (1.30), and the discharge from the continuity equation. Example 1.1 A long steel flume on a slope of 0.0030 has a rectangular cross section with a width of 0.50 m. Determine the discharge when the uniform flow depth is 0.30 m. Solution The discharge is given by the continuity equation

22

1 Basic Hydraulic Concepts Q ¼ AV

A ¼ W yo ¼ 0:50
0:30 ¼ 0:15 m2 sffiffiffiffiffiffi 8 g pffiffiffiffiffiffiffiffiffi R So V¼ f

A A 0:15 ¼ ¼ 0:136 m ¼ P W þ 2 yo 0:50 þ 2
0:30 The equation to use for f depends on Re: u ks Re ¼ m pffiffiffiffiffiffiffiffiffiffiffiffi u ¼ g R So pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 9:8
0:136
0:0030 ¼ 0:063 m/s ks for steel ¼ 0:045 mm ðTable 1:1Þ

R¼

m for water  1
10 6 m2 =s

Therefore Re ¼

0:063
0:045
10 1
10 6

3

¼ 2:8

Re* < 5, so flow is smooth. f is given by either Eqs. (1.33) or (1.34), depending on Re, which is unknown as yet. Assume Eq. (1.34) applies, and check Re when V is known, i.e. pffiffiffi 1 Re f pffiffiffi ¼ 2 log 2:5 f

and making the substitution from Eq. (1.39), i.e. 2:5 2:5 m pffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi gives Re f 4 R 8 g R S pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4R 8 g R S pffiffiffi ¼ 2 log 2:5 v f pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4
0:136 8
9:8
0:136
0:0030 ¼ 9:18 ¼ 2 log 2:5
1
10 6 Therefore

Check Re:

pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ¼ 9:18 8
9:8 0:136
0:0030 ¼ 1:64 m/s Re ¼

4 V R 4
1:64
0:136 ¼ ¼ 8:9
105 m 1
10 6

Re > 105, so Eq. (1.34) is correct.

1.4 Steady Uniform Flow and Flow Resistance

23

Therefore Q ¼ 0:15
1:64 ¼ 0:25 m3 =s (Note that the Chézy C and Manning n can be calculated from f using Eqs. (1.31) and (1.41.) So

and

pffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffi C ¼ pffiffiffi 8 g ¼ 9:18
8
9:8 ¼ 81 f n¼

R1=6 0:1361=6 ¼ ¼ 0:0089Þ C 81

In problems where discharge is known and flow depth is required, it is not possible to establish the flow regime until the flow depth is known. The equations for f also require a value for R, which depends on the flow depth. An iterative solution is therefore required, as follows: – assume a value for yo, – use this yo to calculate R and A from the channel cross-sectional geometry and V from continuity, – calculate Re = 4VR/m and Re* = (gRS)1/2ks/m, – calculate f using the appropriate equation, – calculate V using the Darcy–Weisbach equation and this f, – calculate Q = AV and – compare this Q with the known value, and adjust yo until they are equal. Most practical engineering work involves channels which are large enough that flow is invariably in the hydraulically rough turbulent flow regime. It is therefore convenient to assume this condition, as its equation for f is the easiest to use, and then to check the regime and revise the calculations if necessary. (For hydraulically rough flow, the Manning equation (to follow) is even simpler and usually used.) Example 1.2 A long drainage channel on a slope of 0.0010 has a 0.60 m wide rectangular cross section lined with steel-trowelled concrete. Determine the flow depth when the discharge is 0.12 m3/s. Solution The flow depth must be found by iteration. Only calculations for the final (correct) trial value are shown. Assume yo = 0.30 m. The discharge for this depth is calculated and compared with the specified value. The discharge is given by the continuity equation

24

1 Basic Hydraulic Concepts Q ¼ AV

A ¼ W yo ¼ 0:60
0:30 ¼ 0:18 m2 sffiffiffiffiffiffi 8 g pffiffiffiffiffiffiffiffiffi R So V¼ f

A A 0:18 ¼ ¼ 0:15 m ¼ P W þ 2 yo 0:60 þ 2
0:30 The equation to use for f depends on Re : u  ks Re ¼ m pffiffiffiffiffiffiffiffiffiffiffiffi u ¼ g R So pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 9:8
0:15
0:0010 ¼ 0:038 m/s ks for steel-trowelled concrete ¼ 1:5 mm ðTable 1:1Þ

R¼

m for water  1
10

6

m2 =s

Therefore Re ¼

0:038
1:5
10 1
10 6

3

¼ 57

Re* is between 5 and 70 and so the flow is transitional. f is given by Eq. (1.37), i.e. 1 pffiffiffi ¼ f

2 log



 ks 2:5 pffiffiffi þ 12 R Re f

and making the substitution from Eq. (1.39), i.e.

2:5 2:5 v pffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi gives Re f 4R 8 g R S   1 ks 2:5 m pffiffiffi ¼ 2 log pffiffiffiffiffiffiffiffiffiffiffiffiffi þ 12 R 4 R 8 g R S f   0:0015 2:5
1
10 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ¼ 2 log 12
0:15 4
0:15 8
9:8
0:0010 ¼ 6:14 Therefore

and

pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V ¼ 6:14 8
9:8 0:15
0:0010 ¼ 0:67 m/s Q ¼ 0:18
0:67 ¼ 0:12 m3 =s

The calculated discharge is correct. If it were not then a new value for yo would be selected and the calculations repeated. (Again the Chézy C and Manning n can be calculated from f using Eqs. (1.31) and (1.41.) Then

1.4 Steady Uniform Flow and Flow Resistance

and

25

pffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffi C ¼ pffiffiffi 8 g ¼ 6:14
8
9:8 ¼ 54 f n¼

R1=6 0:151=6 ¼ ¼ 0:0135 C 54

which is slightly higher than the tabulated value for steel-trowelled concrete because the flow is transitional and not hydraulically rough.

Use of the Moody diagram, or the corresponding equations, to determine C makes the Chézy and Darcy–Weisbach equations exactly equivalent. The Chézy equation is still widely used for historical reasons and because the range of flow conditions encountered in channels allows some simplifications to be made, allowing C to be defined more simply than through these equations. For the most common condition of hydraulically rough turbulent flow pffiffiffiffiffiffi 1 C ¼ 8 g pffiffiffi f   pffiffiffiffiffiffi 12 R ð1:40Þ ¼ 8 g 2 log ks 12 R ¼ 18 log ks It is important to remember when using this equation (and others) how it was developed, so that it is used appropriately. Equation (1.40) should be used only if the flow is turbulent and hydraulically rough, and if the flow resistance arises only from shear at the boundary.

1.4.4

The Manning Equation

Before the development of the Darcy–Weisbach friction factor relationships, various attempts were made to replace the Chézy resistance coefficient, C, with one that is determined by the surface roughness only, independent of the flow condition. The Manning equation was developed to provide this, based on the recognition of the relationship between C and the size and shape of the flow cross section, expressed as C¼

R1=6 n

ð1:41Þ

in which n is characteristic of the surface roughness only. This can be introduced into the Chézy equation, which becomes V¼

R1=6 pffiffiffiffiffiffiffi RS n

26 Table 1.2 Values of Manning n for common surfaces

1 Basic Hydraulic Concepts Surface

n

Glass, plastic, machined metal Corrugated metal Cement plaster Smooth concrete Steel-trowelled concrete Float-finished concrete Unfinished concrete Brick in cement mortar Gunite Rough asphalt

0.010 0025 0.011 0.012 0.013 0.015 0.017 0.015 0.019 0.016

leading to V¼

1 2=3 1=2 R S n

ð1:42Þ

in SI units. This is the Manning equation, with n being the Manning resistance coefficient. The Manning equation is not dimensionally homogeneous and the 1 should be replaced by the conversion factor 1.49 if used with foot-second units. For lined conduits, n can be related to C or f through Eqs. (1.31) and (1.41), i.e. R1=6 n¼ ¼ C

sffiffiffiffiffiffiffiffiffiffiffiffi R1=3 f 8g

ð1:43Þ

so the Moody diagram and associated equations can be used indirectly to quantify n. Strickler in 1923 proposed a direct relationship between n and ks, n¼

ks1=6 6:7 g1=2

ð1:44Þ

Because of the widespread use of the Manning equation, tables of values of n for different common surfaces are found in most hydraulics textbooks. Table 1.2 provides some representative values. Values of n are approximately constant with depth, unlike f and C. All the tables of n values implicitly assume that resistance depends only on the physical roughness. The known dependence on R is accounted for in the form of the equation, but the dependence on Re is not. The implication of this is that the Manning equation, using standard tabulated values of n, is only valid for hydraulically rough turbulent flow, which is the usual case for engineering-scale channels. If this is not the case, then n should be adjusted or the Darcy–Weisbach equation used with the friction factor defined by the equations given before.

1.4 Steady Uniform Flow and Flow Resistance Example 1.3 A long channel lined with steel-trowelled concrete has a slope of 0.00080 and a trapezoidal cross section with a base width of 1.0 m and side slopes (Ss) of 1V:1H. Determine the discharge when the uniform flow depth is 0.40 m. Solution The discharge is given by the continuity equation Q ¼ AV

  1 yo yo 2 Ss   1 0:40 0:40 ¼ 0:56 m2 ¼ 1:0
0:40 þ 2 2 1:0 1 V ¼ R2=3 So1=2 n A R¼ P ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s  yo 2 2 P ¼ W þ2 þ yo Ss ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   0:40 2 ¼ 1:0 þ 2 þ 0:402 ¼ 2:13 m 1:0 0:56 R¼ ¼ 0:26 m 2:13 n ¼ 0:013 for steel-trowelled concrete ðTable 1:2Þ A ¼ W yo þ 2

So V¼

1 0:262=3 0:000801=2 ¼ 0:89 m/s 0:013

and Q ¼ 0:56
0:89 ¼ 0:50 m3 =s This is a fairly small channel and the flow may not be hydraulically rough turbulent, which is a condition for use of the Manning equation with a fixed value of n. The flow regime should therefore be checked by calculating Re*. Re ¼

u ks m pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ¼ g R So ¼ 9:8
0:26
0:00080 ¼ 0:045 m/s

ks can be estimated from Eq. (1.38), assuming that the flow is hydraulically rough, i.e. 1 12 R pffiffiffi ¼ 2 log ks f

27

28

1 Basic Hydraulic Concepts So ks ¼

12 R pffi 101=2 f

with f calculated from n using Eq. (1.43), i.e. sffiffiffiffiffiffiffiffiffiffiffiffi R1=3 f n¼ 8g or f ¼

8 g n2 8
9:8
0:0132 ¼ ¼ 0:021 R1=3 0:261=3

So ks ¼ Therefore Re ¼

12
0:26 pffiffiffiffiffiffiffiffi ¼ 0:0011 m

101=2

0:021

0:045
0:0011 ¼ 50 1
10 6

Re* is less than 70, so the flow is transitional and Manning’s equation with a constant value of n is strictly not applicable. The flow condition can be located on the Moody diagram with Re ¼ and

4 V R 4
0:89
0:26 ¼ ¼ 9:3
105 m 1
10 6

ks 0:0011 ¼ 1:1
10 ¼ 4 R 4
0:26

3

This point plots on Fig. 1.12 just within the transitional zone. However, the increase in f (and hence n) within the transitional zone is gradual towards the rough zone and the error in using the fixed value of n will be small and within the uncertainty in the estimate of its value.

The Manning equation is the most widely used for natural channels, where boundary shear is not the only source of flow resistance. Resources are available for estimating Manning’s n for natural channels, but are beyond the scope of this text.

1.5 1.5.1

Steady Rapidly Varied Flow Application of Energy and Momentum Conservation

Rapidly varied flow is the class of nonuniform flow in which changes in flow conditions take place over short distances. In this context, ‘short’ is taken to mean

1.5 Steady Rapidly Varied Flow

29

distances over which friction losses are negligibly small. In the absence of abrupt features presenting expansion losses or drag forces, the loss term in the energy conservation equation is then usually ignored. In most applications, it can also be assumed that the bed slope is small, so cos h  1.0, the flow is approximately parallel, so the pressure distribution is hydrostatic, and the effect of the velocity distribution through the depth is negligible, so a  1.0. The specific energy relationship (Eq. (1.14) can then be written as E ¼ yþ

V2 2g

ð1:45Þ

If there are departures from these assumptions, appropriate corrections should be made. It is usually convenient to incorporate the continuity equation into the specific energy, to express it in terms of discharge, rather than velocity, i.e. E ¼ yþ

Q2 2 g A2

ð1:46Þ

For rectangular channels, this is often expressed in terms of the unit width discharge, q (=Q/b, where b is the channel width), i.e. E ¼ yþ

q2 2 g y2

ð1:47Þ

Equation (1.47) leads to some useful results for rectangular channels and is also a convenient form for explaining some general principles in free surface flow. The equation is cubic in y, implying that three values may exist for a given discharge and specific energy pair. One of these is negative and clearly of no practical interest. The relationship between the two positive flow depths and specific energy for a given value of q is shown in Fig. 1.13. It is clear from Fig. 1.13 that for a given value of q, there is a wide range of conditions for which two distinctly different flow depths (known as alternate depths) would have the same value of specific energy. The upper limb of the curve represents relatively deep, slow subcritical flow while the lower limb represents relatively shallow, fast supercritical flow. The energy of subcritical is mainly potential, while that of supercritical flow is mainly kinetic. Figure 1.13 can be used to explain the response of a flowing water surface to a change in bed elevation, such as shown in Fig. 1.14. Surface resistance energy loss is negligible over the short distance between the sections, and if the step is smooth, there will also be negligible expansion loss, so H1 ¼ H2 ¼ H3

30

1 Basic Hydraulic Concepts

y + V2/2g

y

q

alternate depths

y + V2/2g

E Fig. 1.13 Specific energy relationship

From the definition of specific energy, this can be expressed as E1 ¼ E2 þ Dz ¼ E3 If the flow conditions at one of the sections (say subcritical flow at section 3) is known, then it is a straightforward matter to determine the value of E at that section (from Eq. 1.47) and, from the above relationship, at sections 1 and 2. For small Dz, solution of the energy relationship at section 2 will yield two possible values of y. Considering that all flow conditions between sections 1 and 2 must lie on the given curve (unless q is changed by a change in width), and that the decrease in E is limited by the magnitude of Dz, the flow at the two sections must be represented by points on the same limb of the E-y curve, i.e. flow cannot be changed from subcritical to supercritical (or vice versa) by a simple upward step. For a small value of Dz, the flow depth will decrease from section 1 to section 2, and then increase back to the original value from section 2 to section 3, following the curve (Fig. 1.14a). A flow regime change can only be induced by an upward step large enough to induce flow at the turning point of the curve (critical flow), followed by a downward step. If the vertical step is large enough, the subtraction of Dz from E1 will produce an infeasible value of E2, i.e. a value for which there is no value of y for the specified discharge. In this case, the flow will ‘choke’, and the flow at section 1 will back up until E1 is large enough for Dz to be subtracted to give a feasible value of E2. The flow at section 2 will then be critical, i.e. at the turning point of the E-y curve, and the subsequent downward step will enable supercritical flow to occur at section 3 (provided that this is permitted by conditions further downstream) (Fig. 1.14b). If the undisturbed flow in the channel were supercritical and Dz relatively small, the flow depth would increase from section 1 to section 2 and reduce again at

1.5 Steady Rapidly Varied Flow

31

(a)

q

y H E1

E2

E3

Δz

Δz 1

2

3

2

E

1 3

backup

q

y

(b) H E2 = Ec

Δz

E3

E1 Δz 1

2

3

2

q

y

(c)

E

1 3

H E1

E2

E3 Δz Δz

1

2

3

2

1 3

E

Fig. 1.14 Flow response to changes in bed level

section 3, following the E-y curve on the supercritical limb (Fig. 1.14c). If Dz were large enough for its subtraction from E1 to produce an infeasible value, the flow would again choke, and in this case, the flow at section 1 would change to subcritical and the profile would again be as shown in Fig. 1.14b. The reason for this regime change at section 1 is clarified later. The definition of specific energy implies that curves for higher values of q would plot to the right of that shown in Fig. 1.13, and that curves for lower values would plot to the left. The E-y diagram can therefore also be used to explain the response of the water surface to a lateral channel contraction (Fig. 1.15). If the bed is level then E1 = E2 = E3. For subcritical flow, a moderate contraction will cause a reduction in flow depth as the flow condition moves from a lower to a higher value

32

1 Basic Hydraulic Concepts

1

2

3

(a) Plan

y H

q2 q1

E1

1,3 2

E3

E2

E 1

2

3

(b) y

q2

H

E2 = Ec E3

E1

backup 1 q1

2 3 E

1

2

3

(c) Fig. 1.15 Flow response to a lateral contraction

of q, with the same value of E; an expansion to the original width will cause the flow depth to increase to its original level (Fig. 1.15b). If the width at section 2 is small enough, the curve for q2 will not extend to the original E value, so no flow depth is feasible at section 2 for this value of E. The flow will choke, and the flow depth at section 1 will back up, thereby increasing E until a feasible flow condition at section 2 is attained; this will be for the minimum value of E on the q2 curve, i.e. critical flow (Fig. 1.15c). As for the upward step, a lateral contraction cannot induce a change from subcritical to supercritical flow unless the contraction is sufficient to induce critical flow, and is followed by an expansion. Again, the diagram can be used to trace the solution for supercritical flow and again choking would produce critical flow at section 2 and subcritical flow at section 1 (Fig. 1.15c). The critical flow condition clearly has great significance. Any feature which requires greater energy to pass a given discharge then would otherwise be available, and will induce backing up. If the flow maintains a free surface through the feature, backing up will be just sufficient to induce critical flow, as this is the condition for which the specific energy is the least possible for the given discharge. The critical condition is represented by the turning point of the E-y curve, and the relationship between discharge and flow depth at this point can be obtained by equating to zero

1.5 Steady Rapidly Varied Flow

33

the derivative of E with respect to y. This gives the following useful relationships for rectangular channels: sffiffiffiffiffi 2 3 q yc ¼ g

ð1:48Þ

2 Ec 3

ð1:49Þ

yc ¼

in which the subscript c denotes the critical condition. For nonrectangular sections, q is not defined and Eqs. (1.48) and (1.49) cannot be used. In general, the relationship between critical flow depth and discharge should be obtained from the definition of the Froude number applied at the critical condition, i.e. Fr2 ¼ 1 ¼

Q2 B g A3

ð1:50Þ

in which Fr is the Froude number, B is the surface width of the flow and A is the cross-sectional area of the flow. The relationship between critical depth and specific energy should be obtained from the more general form of Eq. (1.49), derived from Eq. (1.46), giving Ec ¼ yc þ

A 2B

ð1:51Þ

Conservation of energy cannot be applied if there are significant energy losses unless relationships (usually empirical) are available for quantifying them. Energy considerations are also not useful for quantifying forces associated with changing flow conditions. In these cases, application of momentum conservation is useful. This can be done using the force–momentum flux equation (Eq. (1.17) or the momentum function relationship (Eqs. (1.20) and (1.21)). As for energy applications, it is usually assumed that the flow is approximately parallel at the sections considered, so the pressure distribution is hydrostatic, and the effect of the velocity distribution across the flow section is negligible, so b  1.0. Again, if there are departures from these assumptions, appropriate corrections should be made. Although the use of the momentum function has no great practical advantage over the force–momentum flux equation, its form can be used conceptually to understand distinctions between subcritical and supercritical flow behaviour. The form for rectangular channels (Eq. (1.22)) is a cubic equation in y, rather similar to the specific energy relationship (Eq. (1.47)), and again there are two positive values of y with the same value of momentum function for a given value of q, as shown in Fig. 1.16. These are known as conjugate or sequent depths. As for the specific energy diagram, the upper limb of the curve represents subcritical flow and the lower limb represents supercritical flow. The momentum

34

1 Basic Hydraulic Concepts

y

y2/2 + q2/gy

q

conjugate (sequent) depths

y2/2 + q2/gy

M Fig. 1.16 Momentum function relationship

characteristics of subcritical flow are dominated by the flow depth, representing the hydrostatic pressure contribution, and those of supercritical flow arise mainly from the dynamic component associated with the velocity. The relationship between momentum function values at successive sections is given by Eq. (1.20), i.e. P ¼ M2 c

ð1:20Þ

M1

The horizontal distance between flow conditions plotted for adjacent sections in Fig. 1.16 therefore represents the force applied to the flow between the sections divided by c. In Fig. 1.17, for example, the force exerted by the sluice gate on the y P P/

M 1

2

Fig. 1.17 Flow response to applied force

2

1

1.5 Steady Rapidly Varied Flow

35

flow (which is in the negative flow direction) can be indicated on the momentum function diagram. The force exerted by the flow on the sluice gate is equal in magnitude and opposite in direction. For the case of a simple hydraulic jump, there is no force between the upstream supercritical and downstream subcritical flows, and the flow depths are therefore a conjugate pair. (Hydraulic jumps are discussed further in Chap. 6.)

1.5.2

The Control Concept

Most open channel flow problems are concerned with evaluating relationships between the flow depth (y) and the discharge (Q). Typical problems would be to calculate the flow depth at a particular location in a channel system for a given discharge, or to calculate the discharge given the flow depth at a particular point. For a given value of Q, the flow depth at any point is caused by some physical characteristics in the channel. This cause needs to be identified its effect predicted. In the examples of energy principle applications used so far, it has been assumed that the flow conditions are completely known at one section (e.g. just upstream or downstream of a step or contraction), i.e. values are known for Q (or q), y and E. In real problems, all this information may not be known. For example, only Q or E or y may be known and the others must be determined. (Note that the specific energy relationship enables any one of these to be calculated if the other two are known.) Once the flow conditions at one section are known, the conditions at the other sections can be related to them through energy conservation. This process depends on being able to define the conditions at the first section. This first section, where the calculations begin, is the one where the flow conditions are directly related to their cause. It is called a ‘control’ section. A control is a feature that determines a unique relationship between flow depth and discharge. This means that the flow depth for a given discharge (or vice versa) can be calculated by analysing that feature in isolation—it is not necessary to relate the flow conditions to those at any other section. Two control features are already familiar: – A uniform reach of channel determines the flow depth through boundary resistance. The control effect is described by a resistance equation (such as Manning’s) which gives the unique relationship between Q and y. – A contraction induces critical flow. If it is known that flow is critical, the flow depth can be related directly to discharge (e.g. by Eq. (1.48) for rectangular channels). In a real channel system, there will be many features which could possibly act as controls. A particular feature may be a control at certain discharges and not at others. The effect of one control may override the effect of another. A major difficulty in open channel flow analysis is identifying the active controls.

36

1 Basic Hydraulic Concepts

Q

yo (> yc)

So, n, geometry Fig. 1.18 Basic channel with resistance control only

Q

yo (> yc) Δz So, n, geometry

Fig. 1.19 Basic channel with structure incorporated

As an illustrative example, consider a long channel with known discharge, slope, roughness and cross-sectional geometry (Fig. 1.18). The flow will be uniform and only one flow depth is possible—that is determined by the resistance characteristics of the channel. The effects of these characteristics are described by Manning’s equation, so the flow depth can be calculated. The channel itself is the control over the whole length. Suppose here that the uniform flow depth is subcritical. Now suppose a short, smooth hump is incorporated in the channel (Fig. 1.19). Is this feature a control? It will certainly modify the flow conditions locally, but it is a control only if it creates a location where the flow depth can be calculated independently of the conditions anywhere else in the channel. If Dz is small, the hump will just cause a local change in depth without affecting upstream or downstream conditions, as discussed before (Fig. 1.20). The feature is not a control because the depth change it induces is relative to the depth determined by the channel, i.e. y2 can only be calculated from y1 or (more correctly) y3, each of

q

y

1,3

H 2 yo

Δz

Δz 1

2

3

Fig. 1.20 Structure not acting as a control

E

1.5 Steady Rapidly Varied Flow

37

backup

q

y H Ho

1

yo

Δz Δz

Eo

yc

yo

2 3 Δz

1

2

E

Eo

3

Fig. 1.21 Structure acting as a control

yo

Q

yc Δz

yo So, n, geometry

Fig. 1.22 Extended influence of control structure

which can be calculated from Manning’s equation; y2 cannot be calculated independently. If Dz is large enough, the specific energy associated with the uniform flow will be insufficient to enable the given discharge to negotiate the hump, and no value of y will satisfy the energy equation, i.e. Eo – Dz is less than the minimum value of E for the given q. Choking will then occur: the flow at section 1 will back up and flow will go through critical at section 2 (Fig. 1.21). The hump is now acting as a control. The flow conditions at sections 1 and 3 can no longer be determined using Manning’s equation. However, knowing that flow at section 2 is critical enables this flow depth to be calculated directly. The depths immediately upstream and downstream can then be calculated relative to this condition. The flow upstream and downstream may be affected for long distances through gradually varied flow (Fig. 1.22). So, the control section is where the flow depth can be calculated directly from the discharge; if the flow depth calculation requires a flow depth somewhere else then that section is not a control. Whether the hump in this example is a control or not depends only on the magnitude of Dz, but also on the discharge and the channel characteristics. If it is a control then supercritical flow must be maintained immediately downstream. This requires that the supercritical flow induced by the feature must have sufficient momentum to keep the hydraulic jump from encroaching on the control section. To determine the flow depths around a feature, it is essential to be able to establish whether or not it is a control for a specified discharge. Here, the hump will be a control if the uniform flow has insufficient specific energy for the flow to get

38

1 Basic Hydraulic Concepts

over the hump. This can be determined by initially assuming uniform conditions at section 1; if there is no solution to E2 ¼ Eo

Dz

then the hump is a control. More quickly, by knowing that the minimum energy is the critical value, the hump is a control if Eo

Dz \ Ec

and, for rectangular channels, 3 3 Ec ¼ yc ¼ 2 2

sffiffiffiffiffi 2 3 q g

The effect of a control depends significantly on whether the uniform flow without it is subcritical or supercritical. Consider again the channel with the hump, starting off with a small Dz (Fig. 1.20), and what would happen if Dz were increased enough to become a control. As Dz increases, there is just a small change in flow depth at section 2 until Dz is large enough to reduce E to the critical point. If Dz is then increased further q will momentarily decrease, causing a slight, local increase in depth. This local disturbance to the water surface will propagate upstream as a surface wave, eventually causing a region of flow with greater depth, lower energy dissipation rate, and hence increased E. This can only occur if the disturbance can propagate upstream, which it can do only if the surface wave celerity is less than the flow velocity, i.e. flow is subcritical. If the channel flow is supercritical it must first change to subcritical before the extra energy can be accumulated. So, if the channel was steep in the above example, a value of Dz constituting a control would induce subcritical flow and a hydraulic jump on its upstream side. In supercritical flow, flow conditions cannot be affected by anything downstream —the water ‘doesn’t know what happens downstream’. A control has been defined as providing a unique relationship between discharge and flow depth. This can only hold if the flow at the control section is unaffected by the downstream flow, and this requires there to be supercritical flow immediately downstream of the control section. Therefore: Subcritical flow is controlled from downstream. Supercritical flow is controlled from upstream. It follows that if a feature is acting as a control, the flow upstream of it will be subcritical and the flow downstream of it will be supercritical. For example, in Fig. 1.23, the hump is a control only in the third case (c). In the first case (a), the flow is subcritical throughout and therefore controlled by the uniform channel

1.5 Steady Rapidly Varied Flow

(b) control direction

(a) control direction

yo 1

2

39

(c) control direction

yc

yo 1

3

2

2

1

3

3

Fig. 1.23 Occurrence and direction of control

characteristics downstream of the hump: y1 depends on y2 which depends on y3, which is equal to yo. In the second case (b), the flow is supercritical throughout, and therefore controlled by the uniform channel characteristics upstream: y3 depends on y2 which depends on y1, which is equal to yo. In the third case (c), y2 = yc and both y1 and y3 are controlled by and depend on this depth. Example 1.4 Water flows in a long, 2.50 m wide rectangular channel at a uniform flow depth of 1.07 m and a uniform velocity of 1.68 m/s. Calculate the flow depths immediately before, on top of, and immediately after a smooth 0.12 m hump on the channel bed. Solution H

yo = 1.07 m

Vo = 1.68 m/s

0.12 m 1

2

3

The effect of the hump will depend on whether the flow is subcritical or supercritical. This is determined by calculating the Froude number: V 1:68 Fr ¼ pffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:52 gy 9:8
1:07

Fr < 1, so the uniform flow is subcritical. The hump is ‘smooth’, so energy losses may be neglected. Then H1 ¼ H2 ¼ H3 and E1 ¼ E2 þ D z ¼ E3 Whether the hump is a control is determined by assuming y1 = yo and comparing the consequent value of E2 with Ec, i.e.

40

1 Basic Hydraulic Concepts E2 ¼ Eo

Dz

Eo ¼ yo þ

Vo2 1:682 ¼ 1:07 þ ¼ 1:21 m 2g 2
9:8

So E2 ¼ 1:21

0:12 ¼ 1:09 m

And Ec ¼

3 yc 2

with

yc ¼

sffiffiffiffiffi 2 3 q g

and q ¼ Vo yo ¼ 1:68
1:07 ¼ 1:80 m3 =s/m So rffiffiffiffiffiffiffiffiffiffiffi 2 3 1:80 ¼ 0:69 m yc ¼ 9:8 and Ec ¼

3 0:69 ¼ 1:04 m 2

E2 > Ec, so choking does not occur. Note that the maximum hump height for no choking is Dzmax ¼ Eo

Ec ¼ 1:21

1:04 ¼ 0:17 m

The flow depth at section 2 is given by the solution of E2 ¼ 1:09 ¼ y2 þ

q2 2 g y22

i.e. 1:09 ¼ y2 þ

1:802 2
9:8 y22

The two positive solutions to this equation are y2 = 0.88 m (subcritical) and y2 = 0.55 m (supercritical). Because the flow at section 1 is subcritical and there is only a rise in bed level between sections 1 and 2, the flow at section 2 must also be subcritical. The flow depth at section 2 is therefore 0.88 m.

1.5 Steady Rapidly Varied Flow

41

The flow depth at section 3 could be obtained from the second part of the energy relationship, i.e. E2 þ D z ¼ E3 and reversing the above calculations. However, the energy relationship also gives E3 = E1 and the flow must be subcritical throughout, so y3 = y1 = yo = 1.07 m. Example 1.5 The hump height in Example 1.4 is changed to 0.75 m. a. Calculate the flow depths immediately before, on top of, and immediately after the hump. b. Calculate the force exerted by the flow on the hump. Solution 0.69 m

1.73 m

0.34 m

P

yo = 1.07 m

0.75 m 2

1

3

a. The hump is now higher, so the possibility of it being a control must be checked. Following the same procedure as before: E2 ¼ Eo

Dz ¼ 1:21

0:75 ¼ 0:46 m

which is less than Ec = 1.04 m. The hump is therefore a control, y2 = yc and y1 is no longer equal to yo but determined by the control. It is found from E1 ¼ E2 þ D z with E2 ¼ Ec ¼ 1:04 m Therefore E1 ¼ 1:04 þ 0:75 ¼ 1:79 ¼ y1 þ

1:802 2
9:8 y21

giving the subcritical solution y1 = 1.73 m. Again, E3 = E1 and because the hump is a control, y3 will be the supercritical solution of the same equation, i.e. y3 = 0.34 m. b. The force on the hump is equal to the force by the hump on the flow, which is given by Eq. (1.20), i.e.

42

1 Basic Hydraulic Concepts P ¼ M3 c

M1

P ¼ M1 c

M3

ðP is in the

ve flow direction)

So

with q2 y2 þ 1 2 g y1 1:802 1:732 þ ¼ 1:69 m2 ¼ 9:8
1:73 2

M1 ¼

and q2 y2 þ 3 2 g y3 1:802 0:342 þ ¼ 1:03 m2 ¼ 9:8
0:34 2

M3 ¼

Therefore P ¼ 1:69 c

1:03 ¼ 0:66 m2

So P ¼ 0:66
9:8
103 ¼ 6:5
103 N/m width or 6.5 kN/m, and the total force on the hump is Ptotal ¼ P W ¼ 6:5
2:50 ¼ 16:3 kN

Calculations of rapidly varied flow conditions should strictly proceed in the direction in which the control is being exercised, i.e. upstream from the control for subcritical flow and downstream from the control for supercritical flow. For testing whether a feature is a control or not, it is best to consider the conditions of the approaching flow without the feature being there. In the illustrative examples used up to now, the starting condition has been assumed to be uniform flow, but this is not necessarily so—it could be a condition induced by some other control. For example, if a second hump is located downstream from another which acts as a control (Fig. 1.24), whether the second hump is a control or not is determined by the energy of the flow approaching it, as controlled by the first hump. So the second hump will be a control if E4 – Dz < Ec. (If the humps are far apart, a gradually varied profile will occur between them and then y4 > y3 and E4 < E3.) If the two humps have the same Dz, the second is less likely to be a control than the

1.5 Steady Rapidly Varied Flow

43

Δz 2

1

4

3

Fig. 1.24 Interaction between control features

first one, because the control effect of the first one will have increased the available energy; the second hump might be a control without the first one in place and not a control with it in place. If the second hump is larger than the first, its upstream subcritical flow may submerge the first, which would then no longer be a control. This would need to be checked using the momentum function. If the subcritical M4 with the second hump as a control is greater than the supercritical M3 with the first hump as a control, then the subcritical flow will override the supercritical flow at section 3. If y3 is then greater than yc + Dz, the first hump will not be a control. If the supercritical M3 is greater than the subcritical M4, then the supercritical flow will continue over the second hump and it will not be a control. To summarize, whether a feature is a control or not depends not only on its own potential effect, but also on the presence of other features upstream or downstream. Example 1.6 A long, concrete-lined (n = 0.013) channel on a slope of 0.010 has a rectangular cross section with a width of 4.0 m. The features shown below are installed in the channel a short distance apart. Determine the flow depths at sections 1 to 6 if the discharge is 40.0 m3/s. 0.90 m

0.80 m

1

2

3

4

5

6

Solution Check if the first hump is a control, ignoring possible influence of the second hump. It is not a control if Eo Dz1 [ Ec q2 2 g y2o Q 40:0 ¼ 10:0 m3 =s/m q¼ ¼ W 4:0

Eo ¼ yo þ

The uniform flow depth can be determined from the Manning equation together with continuity, i.e.

44

1 Basic Hydraulic Concepts

Q¼

A 2=3 1=2 R S n A W yo 4:0 yo ¼ R¼ ¼ P W þ 2 yo 4:0 þ 2 yo

i.e. 40:0 ¼

4:0 yo 0:013



2=3 4:0 yo 0:0101=2 ð4:0 þ 2 yo Þ

From which, by trial, yo = 1.46 m Therefore Eo ¼ 1:46 þ

10:02 ¼ 3:85 m 2
9:8
1:462

and 3 3 Ec ¼ yc ¼ 2 2

sffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffi 2 3 3 10:02 3 q ¼ 3:25 m ¼ 2 g 9:8

Therefore Eo

Dz1 ¼ 3:85

0:80 ¼ 3:05 m\Ec

The first hump is therefore a control unless overridden by the second. Check if the second hump is a control with the approach flow determined by the first. It is not a control if E4

Dz2 [ Ec

The distance between sections 2 and 4 is short, so E4 ¼ E3 ¼ Ec þ Dz1 ¼ 3:25 þ 0:80 ¼ 4:05 m Therefore E4

Dz2 ¼ 4:05

0:90 ¼ 3:15\Ec

The second hump is therefore a control. y6 is determined from energy conservation between sections 5 and 6, i.e. E6 ¼ E5 þ Dz2 ¼ 3:25 þ 0:90 ¼ 4:15 m Therefore E6 ¼ 4:15 ¼ y6 þ

10:02 2
9:8y26

From which, taking the supercritical solution, by trial, y6 = 1.35 m.

1.5 Steady Rapidly Varied Flow

45

Check if the control effect of the second hump overrides the control of the first. This will be the case if M4 [ M3 M4 ¼

q2 y2 þ 4 2 g y4

y4 is determined from energy conservation between sections 4 and 5, i.e. E4 ¼ E5 þ Dz2 ¼ 3:25 þ 0:90 ¼ 4:15 m Therefore E4 ¼ 4:15 ¼ y4 þ

10:02 2
9:8y24

From which, taking the subcritical solution, by trial, y4 = 4.14 m So M4 ¼

10:02 4:142 þ ¼ 11:0 m2 9:8
4:14 2

M3 ¼

q2 y2 þ 3 2 g y3

Similarly

with E3 ¼ E2 þ Dz1 ¼ 3:25 þ 0:80 ¼ 4:05 m Therefore E3 ¼ 4:05 ¼ y3 þ

10:02 2
9:8y23

From which, by trial, y3 = 1.38 m. So M3 ¼

10:02 1:382 þ ¼ 8:3 m2 9:8
1:38 2

Therefore M4 [ M3 and y3 is controlled by the second hump, i.e. y3 = y4 = 4.14 m. y3 is greater than yc + Dz1 = 2.17 + 0.80 = 2.97 m, so critical flow no longer occurs at section 2. y2 is now found by energy conservation between sections 2 and 3, i.e. E2 ¼ E3

Dz1

46

1 Basic Hydraulic Concepts with

E3 ¼ y3 þ

10:02 10:02 ¼ 4:14 þ ¼ 4:44 m 2
9:8
4:142 2
9:8y23

Therefore E2 ¼ 4:44

0:80 ¼ 3:64 m

Therefore E2 ¼ 3:64 ¼ y2 þ

10:02 2
9:8y22

From which, by trial, y2 = 3.11 m And since E1 = E3 and both are subcritical, y1 = y3 = 4.14 m. The water surface profile over the humps is therefore as shown below:

4.14 m 1

3.11 m

4.14 m

2

3

4.14 m 3.25 m 4

5

1.35 m 6

In these applications, it has been assumed that Q is known and the flow depths are required. Measuring structures act as controls to enable Q to be determined from a measurement of flow depth (Chap. 7). An important problem involving the identification of control features and using control relationships is that of determining the discharge from a reservoir into a long, uniform channel (Fig. 1.25). The discharge depends on what is controlling the flow. The control is what limits the amount of water leaving the reservoir. Two cases need to be considered: 1. If the channel is steep enough for the flow in the channel to be supercritical, then the entrance is just like a very large upward–downward step (Fig. 1.26). Then flow will be critical at the entrance, i.e. the entrance is the control section, and the critical flow relationships can be used to calculate the discharge. Note that as long as critical flow occurs at this section, the discharge is determined only by the characteristics of this section. It does not depend at all on the characteristics of the channel, because supercritical flow cannot be influenced from Fig. 1.25 Flow from a reservoir into a channel

H Q? So, n, geometry

1.5 Steady Rapidly Varied Flow Fig. 1.26 Flow from a reservoir into a steep channel

47

H

yc H

Fig. 1.27 Flow from a reservoir into a mild channel

H

yc Eo

yo

H

downstream. Therefore, the discharge cannot be changed by changing the channel characteristics, such as the slope, size or roughness. 2. Under some conditions, however, the channel characteristics do determine the discharge. If the channel is mild, i.e. the uniform flow is subcritical, then yo will extend right back to the entrance, because it is controlled from downstream and its control is the channel itself (Fig. 1.27). Critical flow at the entrance section is then submerged, yo > yc and the discharge is reduced. The entrance is no longer a control section because it is not followed by supercritical flow. Because subcritical flow extends right up to the reservoir, it is the channel characteristics which determine the discharge, and a relationship other than that describing critical flow is required to calculate the discharge. This is provided by Manning’s equation (or another similar resistance equation) combined with the continuity equation, i.e. Q¼

A 2= 1= R 3S 2 n

ð1:52Þ

This equation has two unknowns (Q and yo), however, so some other relationship involving them is required to get a solution. The discharge must also be influenced by the levels of the water and channel bed at the entrance—this defines the amount of energy available, and the control relationship (Eq. (1.52)) relates this to discharge. At the entrance, the specific energy is equal to the height of the water surface above the channel bed (Fig. 1.27). Because the subcritical flow extends right back to the reservoir, this height must also be the uniform specific energy, so H ¼ Eo ¼ yo þ

Q2 2 g A2o

ð1:53Þ

48

1 Basic Hydraulic Concepts

Simultaneous solution of Eqs. (1.52) and (1.53) gives the discharge and the uniform flow depth. Note that the solution approaches for these two cases cannot be interchanged. It would be wrong to use the critical flow equations if the channel has subcritical uniform flow because critical flow does not occur. It would also be wrong to use Manning’s equation and the specific energy relationship if the uniform flow in the channel is supercritical because the flow is nonuniform in the channel for some distance and so the specific energy at the entrance is not the same as for uniform flow, i.e. H 6¼ Eo. It is usually not obvious if the uniform channel flow is subcritical or supercritical, and an assumption must be made and subsequently checked. It is usually easier to assume a supercritical channel first. Example 1.7 Water flows from a lake into a 2.0 m wide rectangular channel with a Manning n of 0.013 and a slope of 0.0025. Calculate the discharge in the channel when the water level in the lake is 1.8 m above the channel bed at the outfall. Assume zero entrance loss. Solution It is not obvious whether the channel is mild or steep. Assume it is steep and then check and correct if necessary. For a rectangular steep channel Q ¼ qW with q¼

qffiffiffiffiffiffiffiffi g y3c

and 2 2 yc ¼ H ¼ 1:80 ¼ 1:20 m 3 3 So q¼ and

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9:8
1:203 ¼ 4:12 m3 =s/m

Q ¼ 4:12
2:00 ¼ 8:24 m3 =s Check if yo with this discharge is subcritical or supercritical. yo can be determined from the Manning equation together with continuity (Eq. (1.52)), i.e.

1.5 Steady Rapidly Varied Flow

Q¼

49

A 2=3 1=2 R S n

A W yo 2:0 yo ¼ ¼ P W þ 2 yo 2:0 þ 2 yo  2=3 2:0 yo 2:0yo 0:00251=2 8:24 ¼ 0:013 ð2:0 þ 2 yo Þ R¼

i.e.

From which, by trial, yo = 1.50 m. yo > yc so the channel is mild. The discharge is therefore incorrect and must be recalculated by simultaneous solution of Eqs. (1.52) and (1.53). This is done by trial, assuming a value for yo, calculating Q from Eq. (1.52), using this to calculate H from Eq. (1.53) and then comparing this with the specified value. Only the final, correct, trial is shown here. Try yo = 1.42 m. Then from Eq. (1.52)

A 2=3 1=2 R S n   2:0
1:42 2:0
1:42 2=3 ¼ 0:00251=2 ¼ 7:71 m3 =s 0:013 2:0 þ 1:42

Q¼

Inserting yo = 1.42 m and Q = 7.71 m3/s in Eq. (1.53) gives

H ¼ E o ¼ yo þ

Q2 2 g A2o

¼ 1:42 þ

7:712 2
9 : 8 ð2:0
1:42Þ2

¼ 1:80 m

which is the specified value. Therefore Q = 7.71 m3/s.

1.6

Steady Gradually Varied Flow

Flow in a channel will always tend to be uniform, and will depart from this condition if it is disrupted by structural features or changes in slope, roughness or cross-sectional geometry. Such disruptions can induce gradually varied nonuniform flow conditions both upstream and downstream. If a disrupting feature acts as a control, the induced conditions will be subcritical upstream and supercritical downstream. Analysis and design of many hydraulic structures must include determination of the associated gradually varied profiles. Gradually varied profiles (unlike rapidly varied profiles) depend on flow resistance, and result from the existence of a difference between the energy gradient (Sf) and the slope of the bed (So). Upstream of a control, the flow depth is greater than uniform, and therefore Sf < So and specific energy increases in the downstream direction. Downstream of the control, the excess energy is dissipated by bed shear,

50

1 Basic Hydraulic Concepts

and through the gradually varied profile Sf > So. The bed shear is described by the general resistance equation in terms of Sf, s o ¼ c R Sf

ð1:25Þ

For uniform flow Sf = Sw (the water surface slope) = So, and the value of So can be used for Sf in the resistance equations (Manning, Chézy and Darcy–Weisbach). This assumption cannot be made for nonuniform flow, and Sf must be used, which varies along the profile. It is assumed, however, that the value of Sf at a section in gradually varied flow is the same as it would be for uniform flow with the same velocity and hydraulic radius. This enables Sf to be evaluated at sections in gradually varied flow through application of the conventional resistance equations, using the same resistance coefficients as used for uniform flow.

1.6.1

The Gradually Varied Flow Equation

The problem in gradually varied flow analysis is to describe the form of the water surface profile, i.e. to quantify the variation of flow depth (y) with distance along the channel (x). An equation for this can be derived by considering the change in energy in the direction of flow. It is assumed that the slopes to be considered are small enough that cos h  1.0 (the error would be less than 1% for a slope of 8°). It is also assumed that a = 1.0, implying a uniform velocity distribution across the section. (The error in the velocity head would be around 6% for turbulent flow, and so the error in total energy would be less, considerably so for subcritical flow. If more accurate calculations are required, a value of 1.06 for a may be used for turbulent flow.) Equation (1.13) for the total energy at a section can then be written as H ¼ z0 þ y þ

V2 2g

ð1:54Þ

The variation of the different components in the flow direction is obtained by differentiating this equation with respect to the flow direction, x, resulting in dE ¼ So dx

Sf

ð1:55Þ

in which So is the downstream change in bed level, dz′/dx, and Sf is the downstream change in the total energy level through friction, dH/dx, both assumed positive when decreasing. This indicates that specific energy will change along a channel if the energy and bed gradients are different. If Sf = So, then dE/dx = 0 and flow is uniform. Backing up behind a control causes Sf < So, so that dE/dx > 0 and specific energy increases to provide the energy required by the control.

1.6 Steady Gradually Varied Flow

51

Rather than the change in energy along the channel, it is the change in flow depth that is usually important. Equation (1.55) can be modified accordingly by applying the chain rule of differentiation and noting that dE/dy can be shown to be represented as dE ¼1 dy

Fr2

ð1:56Þ

dy So Sf ¼ dx 1 Fr2

ð1:57Þ

Equation (1.55) then becomes

Equation (1.57), to be referred to as the gradually varied flow equation, shows that the rate of change of flow depth with distance depends on how far from uniform the flow is (S0 – Sf) and how far from critical it is (1 – Fr2). The closer the flow is to critical, the more rapid will be the change in flow depth; the closer to uniform it is the more gradual the change. If Sf = So, then dy/dx = 0 and flow is uniform; otherwise, it is nonuniform.

1.6.2

Classification of Gradually Varied Profiles

The computation of gradually varied profiles requires prior knowledge of the range of water levels expected. The general forms of gradually varied water surface profiles under different conditions can be described by qualitative application of Eq. (1.57). The forms depend significantly on whether uniform flow in the channel is subcritical or supercritical, and slopes are defined in these terms. A mild slope is one on which uniform flow is subcritical. This can be defined by the relationship between the uniform and critical flow depths, i.e.

yo [ yc A steep slope is one on which uniform flow is supercritical, i.e.

yo \yc A critical slope is one on which uniform flow is critical, i.e.

yo ¼ yc Note that these definitions are hydraulic rather than geometric—it is generally not possible to specify a value of channel gradient as steep or mild, and a particular channel may be mild for some discharges and steep for others.

52

1 Basic Hydraulic Concepts

Whether the flow depth increases or decreases in the flow direction is determined by the signs of the numerator and denominators of Eq. (1.57). The sign of the numerator is determined by the value of the flow depth, y, in relation to the uniform flow depth, yo. The Manning equation can be arranged in terms of Sf as Sf ¼

V 2 n2 R4=3

ð1:58Þ

Clearly, if y = yo then Sf = So. It follows from Eq. (1.58) that if y < yo, then R is less than the uniform value and V is greater than the uniform value, and hence Sf > So and the numerator is negative. Similarly, if y > yo, then Sf < So and the numerator is positive. The sign of the denominator is determined by the value of the flow depth in relation to the critical flow depth, yc. If y < yc, then the flow is supercritical, Fr > 1 and the denominator is negative. If y > yc, then the flow is subcritical, Fr < 1 and the denominator is positive. Consideration of these inequalities in the regions of flow depth defined by the channel bed, the uniform flow depth and the critical flow depth allows the general forms of the water surface profiles to be described and classified. The possible profiles are shown in Figs. 1.28, 1.29 and 1.30 for mild, steep and horizontal slopes. Notice in these profiles that when the water surface approaches the uniform flow depth it does so asymptotically. When it approaches the bed and the critical flow depth, it does so steeply.

1.6.3

Gradually Varied Flow Computation

Quantitative description of gradually varied profiles requires solution of the gradually varied flow equation in one of its forms, i.e. dH ¼ dx

ð1:59Þ

Sf

Fig. 1.28 Gradually varied flow profiles on a mild slope

M1 M2

yo

M3

yc

Mild

1.6 Steady Gradually Varied Flow

53

Fig. 1.29 Gradually varied flow profiles on a steep slope

S1

yc

S2

yo

S3

Steep

Fig. 1.30 Gradually varied flow profiles on a horizontal slope

H1

yc H2 Horizontal

dE ¼ So dx

Sf

ð1:55Þ

dy So Sf ¼ dx 1 Fr2

ð1:57Þ

or

Three approaches to solution of these equations have been used, viz. direct integration, graphical integration and numerical integration. Different circumstances may dictate the use of different methods. Graphical integration is rarely used because of the ready availability of efficient numerical solution software. Direct integration techniques have been developed which are accurate, easy and quick to apply, but are limited to simple situations with prismatic channels with regular geometric cross sections and require tables of integration values. Numerical integration is now the most widely used approach and several different methods have been developed. Some methods are simpler to apply than others, but these are

54

1 Basic Hydraulic Concepts

generally more restrictive in terms of applicable conditions, e.g. the simplest methods can only be used for prismatic channels. Whatever method is used, the computation must begin at a control section—or at least a position where the depth is known—and proceed in the direction in which the control is exercised. The computation should proceed in the downstream direction for supercritical flow and in the upstream direction for subcritical flow. This requires that before the computations can be done, the control feature must be analysed to determine the control depth on the profile, and the type of profile must be identified so that the range of possible flow depths is known. Computation of Distance from Flow Depth The Direct Step Method is a simple method for computing profiles in prismatic channels, using Eq. (1.55) (or alternatively Eq. (1.57)) in finite difference form. A series of flow depths known to occur along the profile are specified (which is why the profile type must be known), and the distances between them are calculated, starting at the control depth (determined by the control analysis). For example, for an M1 profile upstream of a sluice gate (Fig. 1.31), the sluice gate is first analysed to determine the control depth at the downstream end of the profile. Then a number of flow depths (yi) between this depth and the uniform flow depth (yo) are nominated. The distance from the control depth to the next smaller depth (Dx1) is then calculated, and then the distance from this to the next smaller (Dx2), and so on until a sufficient length of the profile has been defined. An M1 profile approaches uniform flow asymptotically, so its extent is infinite, and a flow depth satisfactorily close to uniform must be chosen as an approximation for calculation purposes. The distances (Dxi) between adjacent nominated depths are calculated from Eq. (1.55) in finite difference form and expressed in terms of Dx, i.e. Dx ¼

DE So

ð1:60Þ

Sf

DE is the difference between the specific energy values at the adjacent sections under consideration. This can be calculated directly from the nominated flow

M1

Q y7

Δx7

y6

Δx6

control depth y5

Δx5

Fig. 1.31 Step computation procedure

y4

Δx4

y3

Δx3

y2

Δx2

y1

Δx1

yo

1.6 Steady Gradually Varied Flow

55

depths, the discharge and the channel cross-sectional geometry. Sf is the energy gradient between the sections, and is approximated as the average of the values at the sections. The values of Sf at a section can be calculated from the flow depth using one of the resistance equations, for example, using Manning’s equation, Sf ¼

V 2 n2 R4=3

ð1:58Þ

Computation is direct, and no iteration is required. This approach is generally satisfactory for regular, prismatic channels where an extended water surface profile must be defined. It is less satisfactory in cases where the flow depth at a particular location must be determined, for example, at the bottom of a chute or drop structure so that the Froude number can be determined for stilling basin design, or at the inlet into a channel from a lake for calculating the discharge if the flow is subcritical and nonuniform. In these cases, the calculations would have to extend past the location required and the flow depth found by interpolation, or the last calculation step could be repeated iteratively until the required distance results. Such problems can still be solved by the Direct Step Method, but the procedure is cumbersome. The following method allows the nomination of cross-section locations at which the flow depths are calculated, i.e. x is nominated and y is calculated. Example 1.8 The channel in Example 1.5 has a slope of 0.0010 and a Manning’s n of 0.013. Determine the distance from the hump to the location of a hydraulic jump. Solution As a control, the hump induces supercritical flow on its downstream side. The channel is mild, so the flow assumes an M3 profile which ends in a hydraulic jump to the subcritical uniform flow controlled from downstream. M3 0.75 m

0.34 m

yo = 1.07 m 0.42 m 2

1 x

The distance required, x, is from y1 as controlled by the hump to y2 which is the conjugate of the uniform flow depth. This is given by q2 y2 þ o 2 g yo 1:802 1:072 þ ¼ 0:88 m2 ¼ 9:8
1:07 2

M2 ¼ Mo ¼

56

1 Basic Hydraulic Concepts Therefore 0:88 ¼

1:802 y2 þ 2 2 9:8 y2

From which, by trial, y2 = 0.42 m. The problem is to calculate a distance from flow depths, so the Direct Step Method is applicable. Calculations are shown below, using equal depth increments of 0.010 m. In the table flow depth, flow area = W
y, wetted perimeter = W + 2
y, hydraulic radius = A/P, velocity = Q/A, specific energy = y + V 2/2g, energy gradient = V 2 n2/R4/3, average energy gradient over Dx.

y A P R V E Sf Sf

y (m)

A (m2)

P (m)

R (m)

V (m/s)

V2/ 2 g (m)

E (m)

Sf

0.34

0.850

3.180

0.267

5.294

1.430

1.770

0.02751

0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42

0.875 0.900 0.925 0.950 0.975 1.000 1.025 1.050

3.200 3.220 3.240 3.260 3.280 3.300 3.320 3.340

0.273 0.280 0.285 0.291 0.297 0.303 0.309 0.314

5.143 5.000 4.865 4.737 4.615 4.500 4.390 4.286

1.349 1.276 1.207 1.145 1.087 1.033 0.983 0.937

1.699 1.636 1.577 1.525 1.477 1.433 1.393 1.357

Sfm

So − Sf

∆E

∆x (m)

0.026348

−0.02535

−0.071

2.78

0.024153

−0.02315

−0.064

2.76

0.022198

−0.0212

−0.058

2.74

0.020452

−0.01945

−0.053

2.71

0.018887

−0.01789

−0.048

2.68

0.01748

−0.01648

−0.044

2.65

0.016212

−0.01521

−0.040

2.62

0.015066

−0.01407

−0.036

2.58

x (m) 0

0.025186

2.78

0.023119

5.54

0.021276

8.28

0.019627

10.99

0.018147

13.67

0.016814

16.32

0.01561

18.94

0.014521

21.52

The distance to the hydraulic jump is therefore approximately 22 m Computation of Flow Depth from Distance In situations where it is required to calculate flow depths at specified locations, Eq. (1.57) should be used in finite difference form, i.e. Dy ¼



 So Sf Dx 1 Fr2

ð1:61Þ

1.6 Steady Gradually Varied Flow

57

As before, calculations begin at a control section where the flow depth is known. A series of distance increments (Dx) along the channel are specified and changes of flow depth through them (Dy) are calculated sequentially (Fig. 1.31) using Eq. (1.61). Because the average values of Sf and Fr through Dx depend on the unknown depth at the end section, conventional practice is to assume a value for this depth and recalculate it iteratively. For regular channels, such as in most hydraulic structure applications, it is acceptable to use the characteristics of the known section to represent the entire reach up to the unknown section, especially if Dx is kept small. (This just assumes the values of Sf and Fr to represent the reaches between the midpoints of adjacent sections rather than the reaches between the sections themselves.) Iteration is then unnecessary and the change in flow depth can be calculated directly from Eq. (1.61). For irregular channels, and especially natural rivers, iterative solution is necessary. The Standard Step Method, which applies Eq. (1.61) iteratively and provides for checking assumed flow depths, is described in basic open channel hydraulics texts. For analysis of some hydraulic structures, it is necessary to extend this theory to allow for a varying discharge along the profile, such as for side weirs where water leaves the channel or side-channel spillways where water enters the channel. The necessary adjustments and extensions are made where these structures are presented in Chap. 4.

Problems 1:1 Water flows at a uniform depth of 0.50 m in a 1.0 m wide rectangular channel constructed on a slope of 0.0010. Determine the discharge a. if the channel is lined with concrete with ks = 0.50 mm and b. if the channel is made of steel with ks = 0.080 mm. 1:2 A rectangular channel is 2.50 m wide and has a slope of 0.0020 and n = 0.015. a. b. c. d.

What is the discharge when the flow depth is 1.20 m? What is the flow depth when the discharge is 2.50 m3/s? What are the values of C and f for this channel? Suggest a representative value of ks for this channel.

1:3 A rectangular concrete-lined channel is 3.0 m wide and conveys a discharge of 7.5 m3/s. What are the alternate depths if the specific energy is 2.4 m? 1:4 The floor elevation of a 3.6 m wide rectangular channel rises 0.22 m. If the mean velocity upstream of the rise is 1.2 m/s and the flow depth is 1.2 m, what is the change in water surface elevation through the transition? 1:5 Water flows at 2.0 m3/s in a 1.0 m wide rectangular channel. There is a smooth downward step of 0.20 m in the channel and the flow depth downstream of the step is 1.2 m. What is the flow depth upstream of the step?

58

1 Basic Hydraulic Concepts

1:6 Find the critical flow depth, critical specific energy and critical slope for the following concrete-lined channels when the discharge is 1.0 m3/s. a. Rectangular, 1.0 m wide. b. Trapezoidal with base width = 0.50 m and side slopes of 45°. c. Triangular with side slopes of 2H:1V. 1:7 Show that the general definitions or expressions for Fr, yc, Vc and E reduce to the simpler, particular expressions applicable for rectangular channels. 1:8 Water flows at a discharge of 9.0 m3/s and a depth of 1.5 m in a long, rectangular, concrete-lined channel with a width of 3.0 m. A short, smooth hump is installed in the channel. Determine the flow depths on top of the hump and immediately upstream and downstream if the height of the hump is (a) 0.20 m and (b) 0.30 m. 1:9 A long channel has a rectangular section 3.0 m wide and incorporates a short, smooth contraction to a width of 2.60 m, followed by an expansion to the original width. If the uniform flow has a velocity of 3.0 m/s and a depth of 3.0 m, determine a. the flow depth in the contraction, b. the minimum width in the contraction for the upstream flow to be uniform and c. the flow depths in the contraction and immediately upstream and downstream if the width in the contraction is 2.20 m. 1:10 Water is flowing at 2.0 m3/s in a 1.0 m wide rectangular channel. There is an abrupt downward step of 0.25 m in the channel and the flow depth downstream of the step is 1.20 m. a. Using the specific energy and momentum function principles, calculate the energy loss caused by the step. Sketch the energy line and water surface profile, indicating values upstream and downstream of the step. b. What minimum step height would cause the upstream flow depth to be critical? 1:11 A long, wide, rectangular channel has a Manning’s n of 0.014 and incorporates a short, smooth, upward step of 0.20 m. If the discharge in the channel is 2.0 m3/s/m wide, determine the flow depths immediately upstream and downstream of the step a. if the channel slope is 0.00010, b. if the channel slope is 0.0010 and c. if the channel slope is 0.010. 1:12 A long, concrete-lined (n = 0.013) channel on a slope of 0.0010 has a rectangular cross section with a width of 4.00 m. The channel incorporates a smooth contraction to a width of 2.80 m followed immediately by a smooth expansion back to the original width. A short distance downstream is a

1.6 Steady Gradually Varied Flow

59

similar contraction to a width of 2.60 m. Determine the flow depths before, after and within each contraction for a discharge of 20.0 m3/s. 1:13 A long, concrete-lined (n = 0.013) channel with a rectangular section 4.0 m wide and a slope of 0.0045 incorporates the features shown below. Both humps are 0.10 m high and the contraction reduces the channel width locally to 3.0 m. All the features are ‘smooth’ and do not induce local energy loss, and are close enough to each other that gradually varied flow changes between them are insignificant. If the discharge in the channel is 40 m3/s, what are the flow depths immediately before and after each feature? Longitudinal section

flow hump

contraction

flow

hump

Plan

1:14 Water flows from a lake into a long channel with a Manning’s n of 0.014. The water level in the lake is 2.7 m above the channel bed at the outfall. a. Calculate the discharge in the channel if it has a 3.0 m wide rectangular cross section and a slope of (i) 0.0070 and (ii) 0.0020. b. Calculate the discharge if the slope is 0.0070 and the channel is trapezoidal in section with a bottom width of 4.5 m and side slopes of 2H:1V. Assume zero entrance loss.

Reference Chow, V. T. (1959). Open-channel hydraulics. McGraw-Hill.

Further Reading Chadwick, A., Morfett, J., & Borthwick, M. (2013). Hydraulics in civil and environmental engineering (5th ed.). CRC Press. Chanson, H. (1999). The hydraulics of open channel flow: An introduction. Elsevier. Chaudhry, M. H. (1993). Open channel flow. Prentice-Hall. French, R. H. (1985). Open-channel hydraulics. McGraw-Hill. Henderson, F. M. (1966). Open channel flow. Macmillan.

Chapter 2

Underflow Gates

2.1

Introduction

Gates are frequently used to control discharge in canals or on spillway crests. The vertical sluice gate is particularly common, but there are also a variety of other types as well, such as the radial (or Tainter) gate and the drum gate (Fig. 2.1). The different types have relative advantages in different situations. The vertical gate may require costly roller and track assemblies, while the radial gate may require more structural expense. Radial gates are common on dam crests and outlets (Fig. 2.2) and have the advantage of being able to be lifted clear of the water surface to allow debris to pass through. They are not often used in small canals. (The use of gates on spillway crests is discussed further in Chap. 4.) As for other flow regulation structures, gates can operate as true controls, in which case the outflow is unsubmerged and flow downstream is supercritical. If the flow downstream is subcritical and sufficiently deep, the outflow may be submerged and the structure will not be a true control. In this case, it is still necessary to be able to analyse the flow to determine discharge (such as from a reservoir into a mild channel) or the change in flow depth from downstream to upstream.

2.2

Unsubmerged Analysis

The unsubmerged analysis is straightforward, and based on energy conservation. The gate is a control; the flow upstream is subcritical and the flow downstream is supercritical (Fig. 2.3). Energy losses need not be considered because there is no flow expansion and friction is negligible over short distances, so H1 ¼ H2

© Springer Nature Switzerland AG 2020 C S James, Hydraulic Structures, https://doi.org/10.1007/978-3-030-34086-5_2

ð2:1Þ

61

62

2 Underflow Gates

(b) Radial

(a) Vertical

(c) Drum

Fig. 2.1 Types of underflow gate

Fig. 2.2 Radial gates at a reservoir outlet

and, if the bed is horizontal between the sections E1 ¼ E2

ð2:2Þ

i.e. y1 þ

q2 q2 ¼ y þ 2 2gy21 2gy22

ð2:3Þ

2.2 Unsubmerged Analysis

63

H

y1

vena contracta

a y2

q

2

1 Fig. 2.3 Unsubmerged flow through vertical sluice gate

Under free flow conditions, the flow depth at the vena contracta is defined by the geometry of the opening, and can be estimated as y2 ¼ Cc a

ð2:4Þ

For a vertical gate, the value of the contraction coefficient Cc is typically 0.61 for a/y1 less than about 0.7. The value changes with the inclination of the lip of the gate. It is therefore greater than 0.61 for a radial gate, and varies with the gate opening (Fig. 2.4); for a drum gate, it is close to 1.0. Equations are available for relating Cc to the angle h, such as that proposed by Henderson (1966) for preliminary estimates, i.e. Cc ¼ 1

0:75 h þ 0:36 h2

Fig. 2.4 Contraction at an inclined gate

θ

ð2:5Þ

64

2 Underflow Gates

vena contracta yo conjugate

yo

Fig. 2.5 Flow profile for unsubmerged flow through vertical sluice gate

in which h is in units of 90°. This equation gives values of Cc to within ±5% for h  1. If required, the specific energy relationship can be developed to give an equation for Q as a function of the upstream water level. However, most problems can be solved from first principles. For the unsubmerged case, on a long mild slope, the downstream flow follows a supercritical gradually varied profile before a hydraulic jump to the subcritical uniform flow (Fig. 2.5). The position of the jump is where the flow depth on the supercritical profile is the conjugate of the uniform flow depth, i.e. where the M values of the supercritical and subcritical flows are equal. With deeper uniform flow (greater M) the hydraulic jump will be nearer to the gate, and if it is deep enough (Mo > Mvena contracta), subcritical flow will submerge the exiting flow (Fig. 2.6). The gate will then not be a control. (Note that the downstream subcritical flow might be determined by another control feature rather than being uniform.)

2.3

Submerged Analysis

If the gate opening is submerged, the upstream water level cannot be determined by analysing the gate alone, and depends on the flow conditions downstream as well as the gate opening. The above analysis no longer applies, and requires some extension. Flow will still issue as a jet under the gate, but there will be a turbulent mass of water above it. This mass has no net motion in the downstream direction and therefore does not contribute to the discharge but, by virtue of its weight, it has a strong influence on the pressure in the jet. The common analysis problem is to determine the upstream flow depth, y1, for a known discharge, q. The flow is subcritical throughout, and control is therefore from downstream. The analysis must therefore begin by establishing the downstream flow depth, y3, and working upstream. The depth y3 will be the uniform flow depth if the channel is regular over a long distance, or it will depend on some other downstream control which would have to be analysed first. It is not possible to relate the flow conditions between sections 1 and 3 directly. Energy conservation cannot be applied because the flow expansion and associated turbulent mixing

2.3 Submerged Analysis

65

H

y1 y3

y y2

1

3

2

Fig. 2.6 Submerged flow through vertical sluice gate

between sections 2 and 3 cause significant, but unquantifiable energy loss. Momentum conservation can also not be applied because the force imposed on the flow by the sluice gate cannot be easily quantified (it is not hydrostatic because the water levels on either side are different). The problem can be solved by including section 2 and relating the conditions here to those at sections 1 and 3. Section 2 is located at the vena contracta of the jet issuing under the sluice gate. The major energy loss occurs in the expansion after the vena contracta so energy conservation can be applied between sections 1 and 2, with energy loss considered negligible. No forces are applied to the flow between sections 2 and 3 (apart from the boundary shear force, which is negligible over short distances) and so momentum conservation can be applied between these two sections. This approach of using energy and momentum conservation together is also useful in analysing submerged flow conditions for many other structures, such as transitions and culverts. Beginning with a known flow condition at section 3, the conditions at section 2 are determined through momentum conservation, i.e. M2 ¼ M3

ð2:6Þ

The expression for M3 is straightforward, but the flow at section 2 is concentrated within the jet (Fig. 2.7) and the formulation must be modified to account for this. The first term of the momentum function (q2/gy) represents the dynamic contribution and so the correct depth to use at section 2 is that associated with the velocity, through continuity, i.e. y2 = Cca. The second (y2/2) term represents the hydrostatic pressure force contribution, and so the appropriate flow depth is that associated with the hydrostatic pressure, i.e. y as shown in Fig. 2.7. Knowing the gate opening, a, therefore enables the actual flow depth at section 2 (y) to be determined by evaluating Eq. (2.6) as

66

2 Underflow Gates

y p/

v

Fig. 2.7 Pressure and velocity distributions downstream of submerged gate

q2 y2 q2 y2 þ ¼ þ 3 2 gð C c aÞ 2 gy3

ð2:7Þ

Once the conditions at section 2 are known, these can be related to those at section 1 through energy conservation, i.e. E1 ¼ E2

ð2:8Þ

E1 can be expressed in terms of y1 and q in the usual way but, again, the jet-flow conditions at section 2 require adjustment of the usual specific energy formulation, taking cognizance of the meanings of the individual terms. The first term in the specific energy function (y) represents the piezometric contribution to energy, i.e. the sum of the elevation and pressure energy terms. Because the pressure at section 2 is determined by the distance below the water surface, the full flow depth (y as shown in Figs. 2.6 and 2.7) must be used for this term. The second term in the specific energy function (q2/2gy2) is the velocity head, and the depth here represents the flow area (through continuity). It must therefore be the depth associated with the jet, i.e. Cc a. The equation to solve for y1 is therefore y1 þ

q2 q2 ¼ yþ 2 2gy1 2gðCc aÞ2

ð2:9Þ

If q is known and y1 is required, the solution described above is straightforward. If q is to be determined from known values of y1 and y3 (a typical discharge measurement problem) then an iterative solution is required because the momentum function and specific energy equations both contain two unknowns (q and y). The two equations must be set up and solved simultaneously.

The simplifications made in this analysis are not entirely realistic. There is an energy loss between sections 1 and 2; the velocity distribution in the jet at the vena contracta is not uniform; the roller above the vena contracta does contribute momentum flux to the jet; the kinetic energy correction factor (a) and momentum correction factor (b) should not be assumed to equal 1.0 at all sections; the contraction coefficient value is not always equal to 0.61. Castro-Orgaz et al. (2013) proposed refinements to the analysis to account for the various assumptions but observed that, owing to compensating inaccuracies, the standard approach (as outlined above) would provide results with similar accuracy. The approach becomes questionable, however, if a > 1.64yc when the gate acts as a subcritical flow transition. It may not be known beforehand whether the gate is submerged or not. The best way to find out is first to assume that it is not submerged, and then to check if M2 > M3. If it is, then there will be a hydraulic jump some distance downstream and the unsubmerged analysis will be correct. If M3 > M2, then the outflow will be submerged and the analysis must be redone as described above. Example 2.1 Water is released from a large reservoir through a vertical sluice gate into a long, rectangular, 4.0 m wide, concrete (n = 0.013) channel on a slope of 0.00050. Plot the relationship between the water level in the reservoir and the discharge when the gate is at its maximum opening of 0.80 m. Solution

68

2 Underflow Gates Check submergence: M1 ¼

q2 y2 þ 1 2 gy1 q¼

Q 2:0 ¼ ¼ 0:50 m3 =s=m W 4:0

¼

0:502 0:4882 þ ¼ 0:224 m2 9:8
0:488 2

M2 ¼

0:502 0:5232 þ ¼ 0:185 m2 9:8
0:523 2

And

M1 > M2, so gate is unsubmerged. Assuming negligible velocity in the reservoir and no energy loss through the gate, H ¼ E1 ¼ y1 þ

q2 2gy21

¼ 0:488 þ

0:502 ¼ 0:542 m 2
9:8
0:4882

For Q = 3.5 m3/s: y1 = Cc
a = 0.61
0.80 = 0.488 m, y2 = yo, from Manning equation A 2=3 1=2 R So n  2=3 4:0yo 4:0yo ¼ 0:000501=2 0:013 ð4:0 þ 2yo Þ

Q ¼ 3:5 ¼

From which, by trial, y2 = yo = 0.758 m. Check submergence: M1 ¼

q2 y2 þ 1 2 gy1 q¼

Q 3:5 ¼ ¼ 0:875 m3 =s=m W 4:0

¼

0:8752 0:4882 þ ¼ 0:302 m2 9:8
0:488 2

M2 ¼

0:8752 0:7582 þ ¼ 0:390 m2 9:8
0:758 2

And

M1 < M2, so gate is submerged Calculate y1 by momentum conservation:

2.3 Submerged Analysis

69

M1 ¼

q2 y2 þ 1 ¼ M2 2 gðCc aÞ

y1 ¼

  2 M2

y1 ¼

  2 0:390

Therefore q2 gCc a

1=2

i.e. 0:3752 9:8
0:61
0:8

1=2

¼ 0:679 m

Assuming negligible velocity in the reservoir and no energy loss through the gate, q2

H ¼ E1 ¼ y1 þ

2gðCc aÞ2 0:3752 ¼ 0:843 m ¼ 0:679 þ 2
9:8
0:4882

Similar calculations for other discharges produce the rating curve below. Note the distinct change in the rate of increase of H with Q once the gate is submerged. 1.2 1.0

H (m)

0.8 0.6 0.4 0.2 0.0

2.4

0

1

2

3 Q (m3/s)

4

5

6

Hysteretic Behaviour

Whether a sluice gate in a steep channel acts as a control or simply allows undisturbed supercritical flow to pass through it depends on both the gate opening and the preceding flow conditions. Figure 2.8 shows the possible water surface profiles, assuming a long channel and hence uniform undisturbed flow. Figure 2.8a shows a gate opening greater than the undisturbed water depth. If the gate opening is reduced until it just touches the surface, the gate will immediately become a control—the water will back up as subcritical flow on the upstream side and a hydraulic jump will occur a long distance upstream before an S1 profile (Fig. 2.8b). If the gate is closed further, the upstream water level will increase and the hydraulic jump will advance upstream. If the gate is then opened progressively, the upstream level will decrease and the hydraulic jump will move downstream. The gate will remain a control, with the associated S1 profile and hydraulic jump, with openings greater than the original undisturbed water depth until the hydraulic jump has

70

2 Underflow Gates

Fig. 2.8 Hysteretic behaviour of a vertical sluice gate

yo (< yc ) (a) Undisturbed flow S1

S3

yo (b) Gate-controlled flow

yo conj.

yo (c) Limit of gate-controlled flow

yo (d) Undisturbed flow

2.4 Hysteretic Behaviour

71

reached the gate (Fig. 2.8c). Any further opening will restore the undisturbed uniform flow (Fig. 2.8d). There is therefore a range of gate openings greater than the undisturbed depth for which the flow depth upstream is controlled either by the gate, for which the unsubmerged analysis above applies, or by the channel characteristics through a resistance equation. This behaviour is described in more detail by Defina and Susin (2003). They also observe that similar hysteretic behaviour can occur in a mild channel, provided that the Froude number of the approach flow exceeds about 0.8. Example 2.2 A vertical sluice gate is located in a long, rectangular, concrete channel with a width of 3.0 m and a slope of 0.015. Plot the relationship between the flow depth immediately upstream of the gate and the gate opening if the gate opening reduces from 1.70 m to 0.60 m and then increases back to 1.70 m when the discharge is 15.0 m3/s. Solution

a

yo 1

2

So = 0.0150 n = 0.013

The flow depth will not be affected if the gate opening is initially greater than the uniform flow depth. Calculate this using the Manning and continuity equations: A 2=3 1=2 R So n  2=3 3:0yo 3:0yo ¼ 0:0151=2 0:013 3:0 þ 2yo

Q ¼ 15:0 ¼

From which, by trial, yo = 0.813 m. Therefore, as a reduces, y1 will be constant at yo = 0.813 m until a = 0.813 m (dashed profile above). As soon as the gate touches the water surface, backing up will occur (solid profile above). Then E1 ¼ y1 þ

q2 q2 ¼ E2 ¼ y2 þ 2gy21 2gy22 q¼

Q 15:0 ¼ ¼ 5:0 m3 =s=m W 3:0

72

2 Underflow Gates y2 ¼ Cc a ¼ 0:61
0:813 ¼ 0:496 m Therefore 5:02 5:02 ¼ 5:68 m ¼ 0:496 þ 2 2
9:8
0:4962 2
9:8y1

y1 þ

From which, by trial, y1 = 5.64 m. Similar calculations give y1 values for a reducing to 0.60 m, and then increasing until y1 is the conjugate of yo. This is found from Mo conj ¼

y2o conj q2 q2 y2 ¼ Mo ¼ þ þ o 2 2 g yo conj g yo

Therefore y2o conj 5:02 5:02 0:8132 ¼ þ ¼ 3:47 m2 þ 2 9:8yo conj 9:8
0:813 2 From which, by trial, yo

conj

= 2.13 m.

From energy calculations for increasing a values, y1 is found to equal 2.13 m for a = 1.52 m. Any further increase in a would result in undisturbed flow through the gate once more, i.e. y1 = yo. Therefore, y1 = yo = 0.813 m for a increasing from 1.52 m to 1.70 m, as for the initial condition. The relationship between y1 and a is therefore as shown below: 12

a decreasing 10

a increasing

y1 (m)

8 6 4 2 0 0.4

0.6

0.8

1.0

1.2

a (m)

1.4

1.6

1.8

2.4 Hysteretic Behaviour

73

Problems 2:1 A concrete-lined channel has a width of 5.0 m, a slope of 0.0060 and Manning’s n of 0.013. A vertical sluice gate in the channel is set at an opening of 0.80 m. If the discharge is 20.0 m3/s, locate the resulting hydraulic jump relative to the sluice gate using a numerical step procedure with three approximately equal steps. 2:2 A vertical sluice gate is installed in a long, 2.0 m wide, concrete-lined, rectangular channel on a slope of 0.00050. a. If the water depth immediately upstream of the gate is 3.0 m and the gate opening is 0.50 m, what is the discharge in the channel? b. For the flow conditions defined in (a), determine the average boundary shear stress immediately downstream of the sluice gate. c. What is the water depth immediately upstream of the gate if the gate opening is increased to 1.00 m?

References Castro-Orgaz, O., Mateos, L., & Dey, S. (2013). Revisiting the energy-momentum method for rating vertical sluice gates under submerged flow conditions. Journal of Irrigation and Drainage Engineering, ASCE, 139(4), 325–335. Defina, A., & Susin, F. M. (2003). Hysteretic behavior of the flow under a vertical sluice gate. Physics of Fluids, 15(9), 2541–2548. Henderson, F. M. (1966). Open Channel Flow, Macmillan.

Chapter 3

Open Channel Transitions

3.1

Introduction

A transition is a structure associated with a change in flow conditions. The most common transitions in practice provide either expansions or contractions in conveyance structures, often incorporating changes in cross-sectional shape. The following cases are common: • A change in the size and/or shape of an artificial canal in association with a change in gradient or required capacity. • Connection of a structure such as a diversion, reservoir outlet or a siphon to a canal (Figs. 3.1 and 3.2). This could be a canal–canal or a pipe–canal connection. • A change from a lined to an unlined canal. • Entry to a chute or drop structure in a canal or from a spillway. • Entry to and exit from a flume through a tunnel or over a crossing. Transitions may be controls, in which case they can affect the flow for great distances both upstream and downstream. If they do not act as controls they may still affect the upstream water level under subcritical flow conditions to accommodate the energy changes and losses they induce; these can influence conditions for a long distance upstream, but obviously have no effect downstream. Such extensive influences require analysis by gradually varied flow methods and will not be considered here, although the analysis would obviously have to include establishment of control depths which are determined by the local effects of the structure. The design of a transition structure requires the specification of its geometry to ensure satisfactory performance in terms of flow characteristics. Depending on particular requirements, it may be necessary to predict the variation of flow depth through the structure, as this will determine the height required for side walls and define initial depths for analysing gradually varied flow upstream and downstream. If a transition is not a control it may still be necessary to evaluate the energy loss it induces; it is commonly required to produce a design that minimizes energy loss. It © Springer Nature Switzerland AG 2020 C S James, Hydraulic Structures, https://doi.org/10.1007/978-3-030-34086-5_3

75

76

Fig. 3.1 A transition from a reservoir outlet to a canal

Fig. 3.2 A transition from a tunnel exit to a canal

3 Open Channel Transitions

3.1 Introduction

77

may also be necessary to predict the distribution of flow velocity through the structure to check for distortions and possible separation zones and, especially for supercritical flow, the occurrence and pattern of induced surface waves. The design is usually done by trial, i.e. a geometry is configured and then analysed to establish its performance, and modified if necessary. All transitions considered here are relatively short and the flow through them will be rapidly varied, associated with changes in direction, slope or cross-section size and/or shape. Simple expansions and contractions have already been discussed in the development of specific energy and momentum function concepts in Chap. 1. The situation in practical structures requires application of the same principles, but with some additional considerations. The effect of a transition structure on flow characteristics depends greatly on whether the flow is subcritical or supercritical, with the latter usually presenting greater design difficulties. These situations are treated separately.

3.2

Subcritical Flow Transitions

If flow is subcritical through a transition the analysis is fairly straightforward, using energy and momentum principles. Problems which cannot be adequately addressed theoretically are the estimation of energy loss for some geometries and undesirable flow conditions, such as waves and distorted velocity profiles, which require some empirical input. The main purpose of the analysis is to determine the flow depths upstream and downstream for a specified discharge, Q. The approach is similar for expansions and contractions. For illustration purposes, an abrupt flat bed expansion is shown in Fig. 3.3. Variations of this geometry include straight or curved gradually tapered transitions, and warped transitions where changes in cross-sectional shape occur. Because flow is subcritical, y3 must be known from an analysis of its control further downstream; it is commonly the uniform flow depth. The upstream flow depth, y1, can then be related to this using energy conservation, i.e. E1 ¼ E3 þ loss

ð3:1Þ

Chow (1959) expanded this relationship to give the change in water surface as y1

y3 ¼ ð 1 þ ci Þ

V32 2g




V12 2g



ð3:2Þ

for a contraction, where the water level drops through the transition, and y3

y1 ¼ ð 1

co Þ

V12 2g




V32 2g



ð3:3Þ

78

3 Open Channel Transitions

Fig. 3.3 An abrupt channel expansion, shown in plan

b1

b3

1

Table 3.1 Energy loss coefficients for flat-bedded contractions and expansions (Chow 1959)

2

3

Transition type

ci

co

Warped Cylinder quadrant Simplified straight line Straight line Square-ended

0.10 0.15 0.20 0.30 0.30+

0.20 0.25 0.30 0.50 0.75

for an expansion, where the water level rises through the transition. For most geometries, the loss coefficients ci and co must be determined empirically. Chow (1959) provides values for different transition geometries, as listed in Table 3.1. Note that energy losses through expansions are considerably greater than through contractions. Formica (1955) expressed losses in terms of the downstream velocity head and found values of up to 0.23 V23/2 g for square-edged contractions in rectangular channels and 0.11 V23/2 g for rounded contractions; he also found the coefficient values to vary with the ratio y3/b2. Yarnell’s (1934) results for bridge piers would also give indications of suitable coefficient values. The upstream and downstream depths can also be related using momentum conservation in terms of the momentum function, i.e. M1 ¼ M3

P c

ð3:4Þ

in which P is the total force exerted in the flow direction by the transition surfaces, or using the force–momentum flux equation, i.e. X

F ¼ q QðV3

V1 Þ

ð3:5Þ

3.2 Subcritical Flow Transitions

79

P in which F includes the forces from the transition surfaces as well as the hydrostatic forces at sections 1 and 3. Application of momentum conservation avoids having to estimate energy losses, but requires estimating the forces acting on the free body between the upstream and downstream sections. This presents the difficulty that the forces from the transition surfaces depend on the flow depth at section 2 (equal to the depth at section 1), which is the unknown that is sought, and the solutions to Eqs. (3.4) and (3.5) therefore require iteration. The calculation is nevertheless fairly straightforward for abrupt expansions where the transition forces depend only on the depth at the upstream section. It is more complicated if the transition is not abrupt, where the hydrostatic pressure varies through the transition under an unknown water surface profile. Najafi-Nejad-Nasser and Li (2015) presented an analytical solution for a tapered expansion incorporating a hump, but this depends on an empirical description of the water surface variation. For an abrupt flat bed expansion, as shown in Fig. 3.3, Henderson (1966) derived an expression for the energy loss between sections 1 and 3 by inserting section 2 and using M2 = M3 and E1 = E2. This yields the expression DE ¼ E1

E3 ¼


V12 2 1þ 2 2g Fr1

1 r 2 s2

2s Fr21



ð3:6Þ

in which r = b2/b1 and s = y3/y1. Henderson (1966) showed that if Fr1 is small enough for Fr41 to be considered negligible, then this can be expressed as DE ¼ E1

V2 E3 ¼ 1 2g


1

1 r

2

Fr2 ðr 1Þ2 þ 1 4 r

!

ð3:7Þ

(Note that the result given as Eq. (7.1) in Henderson (1966) is incorrect, and should read as Eq. (3.7) above.) Using one of these expressions for the loss term in Eq. (3.1) will enable the flow conditions at section 1 to be determined from the conditions at section 3. The loss term depends on the conditions at section 1, however, so an iterative solution is again required. According to Henderson, the last term in the brackets in Eq. (3.7) is not very significant unless the Fr1 exceeds 0.5 or the ratio b2/b1 is less than 1.5 (this assumption would allow easier solution for preliminary design). The first condition is not common, however, and the second would imply a very small head loss anyway. If this term is neglected and it is assumed that y1= y2= y3, then a much simpler equation for head loss results, i.e. DE ¼

ðV 1

V3 Þ2 2g

ð3:8Þ

Comparison of the predictions of this equation with experimental results obtained by Formica (1955) indicates that it overpredicts energy losses by about

80

3 Open Channel Transitions

10%. None of these expressions for DE enable Eq. (3.1) to be solved without iteration, and direct use of either Eq. (3.4) or (3.5) would provide the solution anyway. They do, however, provide a rational form for developing empirical quantification of the loss. For example, Eqs. (3.6) and (3.7) show that the loss can be expressed as the upstream velocity head multiplied by a factor that depends on flow conditions and the transition geometry. Equation (3.8) suggests a dependence on both upstream and downstream velocities, while Eqs. (3.2) and (3.3) give the loss in terms of the difference between upstream and downstream velocity heads. Formica (1955) expressed the loss in terms of the downstream velocity head. All forms have been used in the definition of empirical loss coefficients. For some transition designs, it is advantageous to minimize the energy loss (although whether this is a real advantage in all cases should be carefully considered). The energy loss for an expansion can be reduced by tapering the side walls and curving them smoothly to eliminate sharp corners. The normally recommended taper is 1:4 (1 unit width increment for every 4 units of length) for subcritical flows; a more gradual taper is considered not to make further reductions in losses commensurate with the associated increase in cost. A tapered expansion is more difficult to analyse theoretically through momentum conservation, because the variation of flow depth through the expansion makes the boundary force difficult to estimate. The most practical approach is to apply energy conservation, through Eq. (3.1) with the loss term accounted for empirically. It is reasonable to assume that the form of the expansion loss equation would be similar to Eq. (3.8), and that the effect of tapering can then accounted for by introducing an empirical coefficient. The energy loss for a 1:4 straight taper, for example, is given by Henderson (1966) as DE ¼ 0:3

ðV1

V 3 Þ2 2g

ð3:9Þ

Smith and Yu (1966) investigated the energy losses in abrupt and tapered transitions from a rectangular lined channel to a trapezoidal unlined channel. They expressed the loss in terms of the upstream velocity head only (cf. Eqs. (3.6) and (3.7)), with a term to represent the geometry and an empirical coefficient (CL) to account for other effects, i.e. DEc ¼ CL ð1

Ar Þ2

V12 2g

ð3:10Þ

in which Ar = by1/By3, b is the upstream channel width and B is the average width of the downstream channel. They found that for gradual transitions (a taper of about 1:10) a straight wall flare produces smaller losses than a curved one (CL = 0.25 compared with 0.40). For a rapid transition with a taper of 1:4, CL is 0.50. Skogerboe et al. (1971) investigated this type of expansion further. They found that CL is not constant, but varies with the Froude number at the inlet and the expansion ratio B/b. This is not surprising considering appearance of these terms in the complete rational energy loss Eqs. (3.6) and (3.7) for the abrupt expansion.

3.2 Subcritical Flow Transitions

81

Short transitions can also include changes in bed level, with or without lateral expansions or contractions. Empirical loss formulations are not widely documented for these, but abrupt steps can be easily treated using momentum conservation, similarly to lateral expansions and contractions. The shape of a transition between the inlet and outlet sections has been identified as important for reducing energy losses and ensuring satisfactory flow behaviour, especially for avoiding flow separation and distorted velocity profiles. Vittal and Chiranjeevi (1983) have reviewed and improved methods for shaping rectangular– trapezoidal transitions and describing the flow conditions through them. Smith and Yu (1966) found for their rectangular–trapezoidal transitions that separation from one side occurred unless the taper is long (about 1:10). With their shorter transition, separation occurred and the flow concentrated along one side, resulting in scour in the downstream, unlined channel. They proposed evening out the velocity profile by installing baffles extending within the transition and providing riprap protection at the beginning of the unlined channel. These increased the value of CL to 0.8, so some energy loss is incurred in order to even out the velocity profile. The structure is much shorter, however, than it would have to have been without baffles to prevent separation. Najafi-Nejad-Nasser and Li (2015) showed that flow separation and energy loss in tapered expansions can be reduced by placing a low, gradual hump in the transition. They investigated a 1:4 expansion with the bed rising from the entrance to the expansion up to a crest with a height of 5–9% of the approach flow depth at the end of the expansion and then dropping back over the same distance in the wider channel. The hump has the effect of accelerating the flow and thereby neutralizing the deceleration associated with the lateral expansion; this can reduce the head loss by up to a factor of 4. The wider setting of a transition should be considered in its design. A transition will induce departure from uniform flow that can extend for a considerable distance upstream. A contraction causes an afflux, which is exacerbated by energy loss, requiring raised side walls for some distance. The afflux can be reduced or eliminated by including a downward step in the transition, but this requires lowering of the downstream channel which may be difficult to accommodate. It is therefore advantageous to minimize the energy loss in a contraction. For an expansion, it may not be desirable to reduce the energy loss. An expansion causes drawdown in the upstream channel, resulting in the specific energy at the entrance being lower than for the uniform flow upstream. The difference in specific energies between the upstream and downstream uniform conditions is all dissipated along the gradually varied profile and through the transition. Reducing the loss through the transition will increase the water level drawdown immediately upstream and this will extend the length of the gradually varied profile to increase the energy dissipated along it. This will also increase the boundary shear stress upstream, which may cause scour problems in unlined channels. Allowing a

82

3 Open Channel Transitions

loss through the transition will reduce the drawdown and hence the extent of the gradually varied flow profile upstream. The drawdown can also be reduced or eliminated by incorporating an upward step in the transition. For expansion transitions, subcritical flow will occur through the transition only under certain conditions, even if the flow downstream would be expected to be subcritical. If b3/b1 is too large, the drawdown of the upstream flow depth will approach critical and flow after the expansion will be supercritical, with a hydraulic jump to the downstream subcritical flow. Whether this happens depends on b3/b1, the discharge and the downstream flow depth, and can be determined using specific energy principles. This situation can be avoided by incorporating an upward step in the transition. The formation of waves can affect the flow through transitions, even for subcritical flow, especially if the flow is close to critical, but this is of more concern for transitions with supercritical flow. Example 3.1 A long, concrete-lined (n = 0.013) channel with a rectangular cross section and a gradient of 0.00050 conveys a discharge of 35 m3/s. The channel width increases abruptly in the direction of flow from 4.0 to 6.0 m. Determine the flow depths immediately upstream and downstream of the expansion and the energy loss through the expansion a. using energy conservation, with losses determined using Henderson’s (1966) equation and b. using momentum conservation. Sketch the water surface profile between the uniform conditions upstream and downstream. Solution F 6.0 m

4.0 m

Plan ΔE H

F 2

1 Long section

Q = 35 m3/s So = 0.00050 n = 0.013

3.2 Subcritical Flow Transitions

83

Flow is uniform far upstream and downstream from the transition. Calculate flow depths using the Manning equation with continuity, i.e. Q¼

A 2=3 1=2 R So n

35 ¼


2=3 6:0 yo 6:0 yo 0:000501=2 0:013 6:0 þ 2
yo

Downstream:

from which, by trial, yo = 2.69 m. Upstream 35 ¼


2=3 4:0 yo 4:0 yo 0:000501=2 0:013 4:0 þ 2
yo from which, by trial, yo = 4.16 m

a. y2 = yo = 2.69 m. E1 ¼ E2 þ DE E2 ¼ y2 þ

q2 2 g y22 ð35=6:0Þ2 ¼ 2:93 m 2
9:8
2:692

¼ 2:69 þ Energy loss by Henderson (1966): DE ¼

V12 2g




!

2

þ

r¼

b2 6:0 ¼ 1:5 ¼ b1 4:0

1 r

1

Fr21 ðr 1Þ2 r4

with

V1 ¼

V1 35=4:0 y1 Fr1 ¼ pffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffi g y1 9:8 y1

Therefore E1 ¼ y1 þ

Q 35 ¼ A 4:0 y1

 35 2 4:0

2
9:8 y21

¼ 2:93 þ



35 4:0 y1

2

2
9:8




1

! 
 1 2 35=4:0 y1 2 ð1:5 1Þ2 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1:5 1:54 9:8
y1

from which, by trial, y1 = 2.3 m.

84

3 Open Channel Transitions b. y2 = yo = 2.69 m M1 þ

P

c

F

¼ M2 Q2 þAy gA 352 2:69 þ 6:0
2:69
¼ 2 9:8
6:0
2:69 ¼ 29:45 m3 1 F ¼ c y21 ð6:0 4:0Þ 2 F ¼ y21 c M2 ¼

Therefore Q2 þ A y þ y21 ¼ 29:45 gA

i.e. 352 y1 þ 4:0
y1 þ y21 ¼ 29:45 2 9:8
4:0
y1 from which, by trial, y1 = 2.3 m. The water surface will approach the upstream uniform flow through an M2 profile, as shown below. Energy is lost by friction through the gradually varied profile and by expansion through the transition (DE). ΔE yo = 4.16 m

y1 = 2.30 m

y2 = yo

H yo = 2.69 m

1 2

3.3

Supercritical Flow Transitions

Supercritical flow transitions are common in chutes for spillways, diversions and drop structures. The major problem experienced with them is the formation of surface waves which raise the water level locally and require higher confining walls. If the transition is not carefully designed a pattern of standing waves may be set up which extends a long distance downstream. This section (drawn largely from Henderson (1966)) contributes to understanding and predicting the locations and heights of waves in transitions.

3.3 Supercritical Flow Transitions

85

Surface waves occur because any obstacle in a flow path generates a disturbance which moves across the water surface. In supercritical flow, the disturbance cannot be propagated upstream, and an oblique standing wave is formed. The formation of such waves can be visualized by considering a series of instantaneous disturbances on the surface of a body of still water. As shown in Fig. 3.4a, an instantaneous disturbance will cause a circular wavefront centred at the point of disturbance (A1) to travel outwards at a celerity, c, which depends on the water depth and the wavelength. If the disturbance is repeated at regular time intervals, a series of concentric circular wavefronts will result (Fig. 3.4b). If the point of disturbance moves as well (to points A2, A3…, etc.) (or the water moves relative to a stationary disturbance point), then the pattern of circular wavefronts will no longer be concentric. If the velocity of the point of disturbance is less than c, then the pattern shown in Fig. 3.4c will result, with the wavefronts closer together on the upstream side of the point than on the downstream side, but still moving away from the point. If the velocity of the point of disturbance is just equal to c then the waves generated by the disturbance accumulate ahead of the disturbance point, forming a shock front which advances at V = c (see Fig. 3.4d). If the velocity of the point of disturbance is greater than c then each new disturbance starts beyond the influence of the previous one (Fig. 3.4e). By the time the point of disturbance has reached point An there will be a series of wavefronts which can be enveloped by the common tangent AnP1. If the disturbances are continuous rather than discrete, then a continuous wavefront will form along this tangent.

c

(a)

c

(b)

c

(c)

c

c

c . A1

c

. A1,2,...

(d)

c

... A3 A2 A1

V

c

P1

(e)

c

V>c

c V

. . . A3 A2 A1

V

V2 sin ( -Δθ)

V1 V1

> V1 cos >

V2 cos ( -Δθ) V1 sin

V2

V2 ( -Δθ)

shock front Fig. 3.6 Standing wave set up by wall deflection (plan view showing streamlines)

88

3 Open Channel Transitions

and so
q2 1 g y1

1 y2



¼

1 2 y 2 2

y21



Substituting q = V1 sin b y1 and rearranging gives
 V12 sin2 b 1 y2 y2 þ1 ¼ g y1 2 y1 y1 which is the same as for an ordinary hydraulic jump, except that V1 is replaced by V1sinb. On the left-hand side of this equation, V21/gy1 can be replaced by Fr21. The equation can then be rearranged to give an equation for b which is applicable for waves with finite amplitude, i.e.

1=2 1 1 y2 y2 sin b ¼ þ1 Fr1 2 y1 y1

ð3:15Þ

Note that if the amplitude is small then y2/y1 tends to unity and Eq. (3.15) reduces to Eq. (3.14), which applies to small amplitude long waves. To solve Eq. (3.15) for finite amplitude waves, it is also necessary to know something about the ratio of the flow depths, y2/y1. A simple relationship between y1 and y2 can be derived by applying continuity across the wavefront and relating the velocities parallel to it. Applying continuity normal to the wavefront gives V1 y1 sin b ¼ V2 y2 sinðb

DhÞ

ð3:16Þ

The velocity along the front must be the same on both sides because there is no force in this direction associated with the jump. Therefore V1 cos b ¼ V2 cosðb

DhÞ

ð3:17Þ

Eliminating V1/V2 from Eqs. (3.16) and (3.17) leads to y2 tan b ¼ y1 tanðb DhÞ

ð3:18Þ

The height of the disturbance wave and the direction of the shock front can be determined by simultaneous solution of Eqs. (3.15) and (3.18). This can be done by substituting Eq. (3.18) for y2/y1 in Eq. (3.15), solving for b, and then using this value to determine y2/y1 from Eq. (3.18).

3.3 Supercritical Flow Transitions

89

Example 3.2 The width of a rectangular, concrete-lined channel is reduced in width from 8.0 to 5.0 m through an 18.0 m long straight wall transition. At the design discharge of 50 m3/s, the flow depth in the wider channel approaching the transition is 0.55 m. Determine a. the minimum height required for the side walls through the transition and b. the maximum flow depth within the transition and its location. Solution 18.0 m Δθ l

8.0 m

5.0 m

2 (downstream of shock front)

1

a.

The minimum side wall height is the flow depth after the shock front (y2), given by simultaneous solution of Eqs. (3.15) and (3.18), i.e.

sin b ¼



1=2 1 1 y2 y2 þ1 Fr1 2 y1 y1

ð3:15Þ

and y2 tan b ¼ y1 tanðb DhÞ

ð3:18Þ

Substituting (3.18) in (3.15): sin b ¼



1=2 1 1 tan b tan b þ1 Fr1 2 tan ðb DhÞ tan ðb DhÞ Dh ¼ arctan

1:5 ¼ 4:76 18

V1 Fr1 ¼ pffiffiffiffiffiffiffiffi g y1

Q 50 ¼ 11:4 m/s ¼ A1 8:0
0:55 11:4 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 4:91 9:8
0:55

1=2 1 1 tan b tan b þ 1 ¼ 4:91 2 tanðb 4:76 Þ tan ðb 4:76 Þ V1 ¼

from which, by trial, b = 15.7°

90

3 Open Channel Transitions Substitute b = 15.7° in Eq. (3.18) to get y2: tan b tanðb DhÞ tan 15:7 ¼ 0:80 m ¼ 0:55 tanð15:7 4:76 Þ

y2 ¼ y1

Therefore, the minimum side wall height is 0.80 m. b.

The maximum flow depth within the transition occurs where the shock front s from the two sides meet on the centre line. Height of combined shock wave ¼ y1 þ 2ðy2 y1 Þ ¼ 0:55 þ 2ð0:80 0:55Þ ¼ 1:05 m Location from beginning of transition, 8:0=2 tan b 4:0 ¼ ¼ 14:2 m tan 15:7

l¼

3.3.2

Curvilinear Transitions

The previous analysis applies to straight wall transitions where Dh is finite and constant. For curvilinear transitions, Dh varies continuously through the transition and there is no abrupt shock front, but rather a gradual variation through a smooth wave. The change in flow depth can be analysed by applying Eq. (3.18) to very small changes. According to Henderson (1966), setting y2 = y1 + Dy and letting Dh tend to zero leads to dy y ¼ dh sin b cos b

ð3:19Þ

Because very small increments of y are now being considered, Eq. (3.11) is valid and sinb can be replaced by c/V. Also, for small amplitudes c = (gy)1/2. Therefore dy y sin b ¼ dh sin b cos b sin b y ¼ 2 tan b sin b y V2 ¼ 2 tan b c y V2 tan b ¼ gy

3.3 Supercritical Flow Transitions

91

and so dy V 2 ¼ tan b dh g

ð3:20Þ

This equation can be used to describe the variation of flow depth along the wall of a curved transition. At any angle h, the flow depth y can be calculated and this will also be the depth along a line radiating from the wall at a tangential angle b, as defined by Eq. (3.14). This is illustrated in Fig. 3.7. Ippen (1950) integrated Eq. (3.20) under the assumption that there is no energy loss along such a continuous change in depth and that the specific energy is therefore constant. The resulting relationship between y and h (in degrees) along the wall is pffiffiffi h ¼ 3 tan

1

pffiffiffi 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Fr 1

tan

1

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fr2 1

h1

ð3:21Þ

The constant of integration h1 can be evaluated by inserting the boundary condition h = 0° at y = y1. If the specific energy, E, is calculated for the initial flow depth and this is assumed to remain constant, then Fr can be calculated for any specified value of y from the specific energy relationship expressed as
 Fr2 E ¼ y 1þ 2

ð3:22Þ

The angular position of the specified flow depth can then be calculated from Eq. (3.21). The procedure for defining the water surface profile along a side wall is then first to determine the flow depth at the entrance to the transition (where h = 0) from upstream conditions, calculate the corresponding value of Fr and hence the value of

θ

θmax ymax

Flow θ increasing y increasing converging +ve waves

θ decreasing y decreasing diverging -ve waves

Fig. 3.7 Wave pattern along a curved boundary (plan view; the oblique lines are water surface contours)

92

3 Open Channel Transitions

h1 from Eq. (3.21). The value of E through the contraction is assumed to be constant, and can therefore also be calculated from the entry condition. A range of flow depths anticipated to occur through the transition is then specified, and for each the corresponding value of Fr can be calculated from Eq. (3.22) and then its angular position h from Eq. (3.21). Alternatively, values of h can be specified and the corresponding values of Fr obtained (by trial) from Eq. (3.21) and hence y from Eq. (3.22). The maximum flow depth at the maximum wall angle, hmax, is particularly important for design. The transition geometry will define the relationship between angular and linear position. Ippen (1950) also produced a graphical solution to Eq. (3.20) (which is reproduced in Henderson (1966)). Henderson (1966) reports good agreement of Eq. (3.21) with experimental results, except that the actual point of maximum depth occurred slightly beyond the inflection point rather than right at it as predicted. Ippen (1950) also integrated Eq. (3.20) under the assumption that the velocity (rather than E) would be constant through the deflection. The resulting equation is simpler and gives slightly lower values than Eq. (3.21). This equation is
 y2 h ¼ Fr21 sin2 b1 þ 2 y1

ð3:23Þ

with b1 = sin−1 (1/Fr1). A contraction of supercritical flow presents the possibility of choking if the downstream channel is too narrow for the available upstream energy to accommodate. This can be checked for using the specific energy concepts presented in Chap. 1. Henderson (1966) identifies a further range of conditions where choking is not assured, but is possible. Although the theory for supercritical flow through transitions has been presented for the contraction case, it applies for expansions as well. Example 3.3 The width of a rectangular, concrete-lined channel is reduced in width from 8.0 to 5.0 m through an 18.0 m long curvilinear transition. In plan, the side walls follow a simple reverse curve formed by two equal circular arcs. At the design discharge of 50 m3/s, the flow depth in the wider channel approaching the transition is 0.55 m. Determine a. the minimum height required for the side walls through the transition and b. the maximum flow depth within the transition and its location. Solution a. The minimum side wall height is the maximum flow depth, which occurs at the location of hmax. (The curvature shown in the plan sketch is exaggerated for clarity.)

3.3 Supercritical Flow Transitions

93

18.0 m θmax

90o-α

α

= tan −1

θ max

1.5 m

90o-α

α

1.5 18.0

= 4.76o

= α +α = 4.76 + 4.76 = 9.52

o

Assuming no energy loss through the transition, the variation of y is described by Ippen’s (1950) relationship (Eq. 3.21): pffiffiffi pffiffiffi 1 3 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p tan 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h1 h ¼ 3 tan Fr2 1 Fr2 1

The constant h1 is obtained by defining h = 0 at the beginning of the transition, where Fr1 = 4.91 (from Example 3.2). Therefore pffiffiffi pffiffiffi 3 1 h1 ¼ 3 tan 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 22:6 4:912 1 4:912 1 ymax occurs where h = hmax, i.e.

hmax ¼ 9:52 ¼

pffiffiffi 3 tan

1

pffiffiffi 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Fr 1

tan

1

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fr2 1

22:6

from which, by trial, Fr = 3.32. For no energy loss through the transition, E = constant = E1, and (with V1 = 11.4 m/s from Example 3.2) V12 2g 11:42 ¼ 7:18 m ¼ 0:55 þ 2
9:8

E1 ¼ y1 þ

Therefore, at hmax
 Fr2 E ¼ 7:18 ¼ ymax 1 þ 2 i.e. ymax ¼

7:18 2 ¼ 1:10 m 1 þ 3:32 2

ð3:22Þ

94

3 Open Channel Transitions Therefore, the minimum side wall height is 1.10 m. b. The maximum flow depth occurs where the wave height corresponding to hmax from either side meets on the centre line.

θmax

4.0 m 3.25 m

2.5 m 9.0 m

l

Combined wave height ¼ y1 þ 2ðymax y1 Þ ¼ 0:55 þ 2ð1:10 0:55Þ ¼ 1:65 m Location: 1 Fr 1 ¼ ¼ 0:301 ) b ¼ 17:5 3:32

sin b ¼

Then l ¼ 3:25 tanð90 ¼ 3:25 tanð90

3.3.3

ðhmax þ bÞÞ ð9:5 þ 17:5 ÞÞ ¼ 6:38 m

Suppression of Standing Wave Propagation

At points where disturbance lines from opposite sides of the channel intersect, the resulting change in depth is the algebraic sum of the depth changes on the intersecting lines. The point of intersection acts as a new disturbance, producing disturbance lines directed back towards the channel sides. These are, in turn, reflected once more to produce disturbance lines which again intersect in the centre of the channel. This process of alternate disturbance reflection against the walls and at the channel centre produces the commonly observed diamond-shaped pattern of oblique standing waves, which can persist for a considerable distance downstream. This is an undesirable flow condition and can be minimized by using a contraction with straight side walls and careful choice of the flare angle. The idea behind this design is shown in Fig. 3.8. The points of contraction at the entrance to the transition create wavefronts from each side of the channel at angles b1 to the original wall directions. These

3.3 Supercritical Flow Transitions

95

wavefronts intersect and reflect at the channel centre, directing wavefronts back to the side walls. If they were to reach the side walls before or after the end of the transition, they would be reflected back towards the channel centre. At the end of the transition, the change in direction of the walls again creates a disturbance, but because the change represents an expansion rather than a contraction, the wavefronts emanating from these points are negative. The idea for suppressing the wave propagation is simply to select the transition length, L, (and hence the angle h) so that the wavefronts reflected at the channel centre meet the side walls exactly at the end of the transition, so that the positive wave is counteracted by the negative wave. If the discharge varies, it is difficult to produce a design that will be correct for the whole range. However, if the approach is controlled by friction, as for a long chute, then the variation of velocity with discharge is not great and the solution will not be very sensitive to discharge variations. Sturm (1985) has provided a solution method for designing a straight wall contraction to minimize standing wave propagation. He shows that the correct solution will be obtained by applying Eqs. (3.15)–(3.18) across the two wavefronts simultaneously with an equation for continuity through the transition. This continuity equation can easily be expressed in terms of the transition contraction ratio (r) as
 1 b1 Fr3 y3 3=2 ¼ ¼ r b3 Fr1 y1

ð3:24Þ

The solution process is laborious because of the simultaneous application over two wavefronts and Sturm provided a graphical solution, which is reproduced in Fig. 3.9. In using this diagram, the upstream flow conditions (Fr1) and the ratio of channel widths to be connected (r) will be known. The lower part of the diagram provides the best value of h, from which the transition length, L, can be determined, and the upper part provides the ratio of upstream and downstream flow depths for

θ 1

2

b1

b3 V3 V1

V2

2-θ

L Fig. 3.8 Standing wave patterns in a straight-walled contraction designed to suppress wave propagation (plan view)

96

3 Open Channel Transitions

this design. This part shows clearly how the wave height increases with Fr1, and just how large the waves can be. Supercritical contractions will choke if the contraction ratio is too small for the design flow conditions. The critical contraction ratio can be determined by applying specific energy principles, as discussed in Chap. 1. This analysis implies the occurrence of a hydraulic jump upstream of the contraction and critical flow at the end; the associated choking conditions are represented by line A in Fig. 3.9. A second possible choking mechanism has been recognized (Henderson 1966; Sturm 1985) where the gradually rising water level through the contraction reaches the critical depth. This condition can be predicted through Sturm’s (1985) analysis for Fr3 = 1, with allowance for energy loss, and is indicated by line B in Fig. 3.9. Sturm suggests that choking is possible for design conditions between lines A and B, but certain for designs to the right of line B. The near vertical orientation of line A suggests a limit of about 4o (1:14) for h to ensure no choking. More detailed

12 10 20

8

y3/y1

6

10

Fr1

5 4 3 0

A B

r = b3/b1

2 Fr1

2.5 0.5

3 4 5

6

8 10

0 10

20

θ (degrees) Fig. 3.9 Design graph for straight wall transitions (adapted from Sturm 1985)

30

3.3 Supercritical Flow Transitions

97

analysis of the choking condition in a linear contraction is presented by Defina and Viero (2010). In addition to being designed to eliminate the reflection of wavefronts, straight wall transitions cause smaller increase in flow depth than curved ones. Equation (3.21) shows that the maximum depth is determined by the maximum deflection angle. For curved transitions, whatever their form, the maximum angle is greater than the constant angle in a straight transition, and hence the increase in depth must be greater. It is also worth noting that the curve radius does not feature in the equation and therefore has no effect on the maximum depth; it will only affect the relative positions of different depths through the profile. More advanced methods have moved away from analytical and experimental results and towards computational modelling of flow (e.g. Abdo et al. 2019; Bhallamudi and Chaudhry 1992; Jiminez and Chaudhry 1988; Mazumder and Hager 1993). Example 3.4 Determine the best length of the straight wall transition in Example 3.2 for suppression of standing wave propagation. Solution The length of the transition can be expressed in terms of the difference in upstream and downstream channel widths, and the angle Dh, i.e. L¼

ð8:0 5:0Þ=2 tan Dh

For suppression of standing wave propagation, Dh is obtained from the graphical solution of Sturm (1985) (Fig. 3.9) in terms of the width ratio b2/b1 and Fr1. From Example 3.2, Fr1 = 4.91 and b3 b1 5:0 ¼ 0:63 ¼ 8:0

r¼

Hence, from Fig. 3.9, Dh = 3.5o and so L¼

3.4

ð8:0 5:0Þ=2 ¼ 24:5 m tan 3:5

Dual Stable States and Hysteresis

For transitions involving changes from supercritical flow upstream to subcritical flow downstream, situations occur where different stable states of flow are possible, and the one that actually occurs depends on prior conditions (similar to the hysteretic behaviour described for underflow gates).

98

3 Open Channel Transitions

The change from supercritical to subcritical flow takes place through a hydraulic jump, which occurs either upstream or downstream of the transition structure, depending on the relative magnitudes of the momentum function immediately before or after the structure. If the approaching supercritical flow has sufficient energy to pass the structure and the resulting supercritical flow after the structure has greater momentum than the downstream subcritical flow, then the jump will occur further downstream. Similarly, if the downstream subcritical flow can persist through the structure and the resulting subcritical flow before the structure has greater momentum than the approaching supercritical flow, then the jump will occur further upstream. Analysis of the transition can be related to the control conditions either upstream or downstream, i.e. beginning either with the supercritical flow upstream or the subcritical flow downstream. However, for certain flow conditions and transition geometries, contradictory conclusions arise from the two approaches, with the jump location being predicted as either downstream or upstream (see Example 3.5). Both are stable states and the actual condition depends on prior flow conditions, i.e. the behaviour is hysteretic. For an upward step in a channel (Fig. 3.10), for example, the conditions for the hydraulic jump to occur further downstream (solid water surface profile) are therefore E1

Dz

DE1 [ Ec

ð3:25Þ

and M2 [ Md

ð3:26Þ

where DE1 is the energy loss associated with supercritical flow over the step. Equation (3.26) means that y2 must be less than the conjugate of yd. The conditions for the hydraulic jump to occur further upstream (dashed water surface profile) are E1 ¼ Ed þ Dz þ DE2

ð3:27Þ

M1 [ Mu

ð3:28Þ

and

where DE2 is the energy loss associated with subcritical flow over the step. For a smooth step, the energy losses are negligible and the hydraulic jump will be far upstream or downstream from the step after a gradually varied profile. If the step is steep, the energy loss becomes appreciable, and the hydraulic jumps move closer to the step and the range of conditions for dual states decreases. For an abrupt step (such as in a stilling basin), the range becomes negligibly small, and only one state is likely; this has been confirmed by data obtained by Muskatirovic and Batinic (1977), as presented by Viero and Defina (2019).

3.4 Dual Stable States and Hysteresis

99

Fig. 3.10 Supercritical– subcritical transitions across an upward step

yd yu

Δz 1

2

Viero and Defina (2017) and Defina and Susin (2006) have generalized these concepts for application to different transition types. Example 3.5 The slope of a long, wide, concrete-lined channel changes from 0.020 to 0.00020 over a short distance. Determine if the resulting hydraulic jump is upstream or downstream of the transition when the discharge is 12 m3/s/m a. if the two slopes meet at the same level and b. if a short, smooth, upward step of 1.0 m is included in the transition. Solution First, establish the uniform flow conditions on the two slopes. For So = 0.020, Calculate yo from Manning equation with continuity (for a wide channel R  y) (subscript u indicates upstream value): q¼

y 2=3 1=2 y S n  

Therefore qn

yo u ¼ ¼

1=2

S




!3=5

 12
0:013 3=5 ¼ 1:06 m 0:0201=2

sffiffiffiffiffi rffiffiffiffiffiffiffi 2 2 3 12 3 q ¼ 2:45 m Critical flow depth ¼ yc ¼ ¼ g 9:8 you < yc, so first slope is Steep Also y2 q2 þ ou g yo u 2 122 1:062 þ ¼ 14:4 m2 ¼ 9:8
1:06 2

Mo u ¼

100

3 Open Channel Transitions For So = 0.00020, yod (subscript d indicates downstream): qn

yo d ¼ ¼

1=2

So


!3=5

12
0:013 0:000201=2

3=5

¼ 4:22 m

yod > yc, so second slope is Mild Also y2 q2 þ od 2 g yo d 122 4:222 þ ¼ 12:4 m2 ¼ 9:8
4:22 2

Mo d ¼

a. Compare the M values for each slope at the slope transition: Mo u [ Mo d ð14:4 m2 [ 12:4 m2 Þ and therefore the supercritical flow advances on to the Mild slope and the hydraulic jump is downstream of the transition. b. Assume upstream control, i.e. you persists up to the step, and so y1 = you.

you

Δz = 1.0 m 1

2

The step is smooth, so there are no significant energy losses between 1 and 2. y2 can then be calculated by energy conservation, i.e. E2 ¼ E1

Dz

or y2 þ y2 þ

q2 q2 ¼ y1 þ 2 g y22 2 g y21

Dz

122 122 ¼ 1:06 þ 2 2
9:81
1:062 2
9:8 y2

1:00 ¼ 6:60 m

from which, by trial, y2 = 1.16 m and the supercritical flow has sufficient energy to pass the step.

3.4 Dual Stable States and Hysteresis

101

Then y2 q2 þ 2 g y2 2 122 1:162 þ ¼ 13:3 m2 ¼ 9:8
1:16 2

M2 ¼

and M2 [ Mo d so the hydraulic jump is further downstream. Now assume downstream control, i.e. yod persists up to the step, and so y2 = yud.

yod Δz = 1.0 m 1

2

Then E1 ¼ E2 þ Dz or y1 þ y1 þ

q2 q2 ¼ y2 þ þ Dz 2 g y21 2 g y22

122 122 þ 1:00 ¼ 5:63 m ¼ 4:22 þ 2 2
9:81
4:222 2
9:8 y1 from which, by trial, y1 = 5.38 m.

Then y2 q2 þ 1 g y1 2 122 5:382 þ ¼ 17:2 m2 ¼ 9:8
5:38 2

M1 ¼

and

M1 [ Mo u so hydraulic jump is further upstream.

The two assumptions regarding the direction of control both yield valid, but contradictory solutions, indicating hysteretic behaviour. Similar calculations for a range of discharges show that the two assumptions will both predict the hydraulic jump upstream for q < * 9 m3/s/m and downstream for q > * 22 m3/s/m. Contradictory predictions occur for discharges between these values.

Problems 3:1. A long, rectangular, concrete (n = 0.013) channel has a gradient of 0.00050 and is required to change in width from 3.5 to 5.0 m. The design discharge is 25 m3/s. It is required to determine the flow depths immediately upstream and downstream of the transition and the energy loss through the transition.

102

3 Open Channel Transitions

a. The first design to be considered is for a flat bed abrupt transition. Analyse the flow through the transition using the following approaches: i. Energy conservation, assuming no losses. ii. Energy conservation, assuming losses as described by Henderson’s (1966) equation (Eq. (3.7)). iii. Energy conservation, assuming losses according to Eq. (3.8). iv. Momentum conservation, using the momentum function. v. Momentum conservation, using the force–momentum flux equation. Compare the results obtained from the different analyses, and comment on the differences. vi. Sketch the water surface profile upstream, through and downstream of the transition. vii. Explain how the transition affects the bed shear stress in the upstream approach flow. viii. How would the upstream bed shear stress be changed if an abrupt upward step of 0.50 m was included in the transition? What would be the energy loss through the transition? ix. What step height would maintain uniform flow from upstream right to the transition? What would be the energy loss? By how much would the water level change over the transition? b. The second design to be considered is a flat bed transition with a taper of 1:4. Analyse the flow through this transition using the following approaches: i. Energy conservation, accounting for losses appropriately. ii. Momentum conservation, assuming the flow depth to vary linearly through the transition. Compare these results and comment. How does the taper influence the upstream flow depth and energy loss in comparison with the abrupt expansion? iii. How does this transition affect the bed shear stress upstream? Recommend a design for this transition. 3:2. Repeat problem 3.1.a.iii but for a channel on a slope of 0.0010. Explain the flow condition and sketch the water surface profile. What would be a practical design configuration for this situation? 3:3. A long, rectangular, concrete (n = 0.013) channel has a gradient of 0.00050 and is required to change in width from 5.0 to 3.5 m. The design discharge is 25 m3/ s. Determine the flow depths immediately upstream and downstream of the transition using Chow’s (1959) method and a. assuming no losses, b. assuming a square-ended transition and c. assuming a cylinder quadrant transition.

3.4 Dual Stable States and Hysteresis

103

3:4. Water is to be released from a reservoir into a 5.0 m wide rectangular channel at a flow depth of 1.0 m and a velocity of 12.0 m/s. The channel is to be connected immediately to a 4.0 m wide chute through a 10.0 m long transition. Two transition shapes are being considered. The first is curvilinear, with each side wall following the shape of a simple reverse curve consisting of two equal circular arcs. The second has straight walls. For each transition shape, estimate a. the maximum flow depth along the side walls (using both of Ippen’s (1950) solutions for the curvilinear transition) and b. the maximum flow depth along the centre line of the transition and its location. 3:5. Water is to be released from a reservoir into a 6.0 m wide rectangular channel at a discharge of 98.8 m3/s with a specific energy of 10.8 m. A short distance downstream the channel must be contracted to a width of 3.6 m. Determine the length of straight wall transition to minimize surface wave transmission using the graphical solution proposed by Sturm (1985).

References Abdo, K., Riahi-Nezhad, C. K., & Imran, J. (2019). Steady supercritical flow in a straight-wall open-channel transition. Journal of Hydraulic Research, 57(5), 647–661. Bhallamudi, S. M., & Chaudhry, M. H. (1992). Computation of flows in open channel transitions. Journal of Hydraulic Research, 30(1), 77–93. Chow, V. T. (1959). Open-channel hydraulics. McGraw-Hill. Defina, A., & Susin, F. M. (2006). Multiple states in open channel flow. In M. Brocchini & F Trivellato (Eds.), Vorticity and turbulence effects in fluid structure interactions—Advances in fluid mechanics (pp. 105–130). Wessex Institute of Technology Press. Defina, A., & Viero, D. P. (2010). Open channel flow through a linear contraction. Physics of Fluids, 22(5), 056601. Formica, G. (1955). Esperienze preliminari sulle perdite di carico nei canali dovute a cambiamenti di sezione (Preliminary tests on head losses in channels due to cross-sectional changes), L’Energia elletrica, Milan, 32(7) (cited in Henderson (1966)). Henderson, F. M. (1966). Open channel flow. Macmillan. Ippen, A. T. (1950). Channel transitions and controls. In H. Rouse H (Ed.), Engineering hydraulics (Ch. VIII). Wiley. Jiminez, O. F., & Chaudhry, M. H. (1988). Computation of supercritical free-surface flows. Journal of Hydraulic Engineering, 114(4), 377–395. Mazumder, S. K., & Hager, W. H. (1993). Supercritical expansion flow in Rouse modified and reversed transitions. Journal of Hydraulic Engineering, 119(2), 201–219. Muskatirovic, D., & Batinic, D. (1977). The influence of abrupt changes of channel geometry on hydraulic regime characteristics. In Proceedings 17th IAHR Congress (pp. 397–404). Baden Baden. Najafi-Nejad-Nasser, A., & Li, S. S. (2015). Reduction of flow separation and energy head losses in expansions using a hump. Journal of Irrigation and Drainage Engineering, 141(3), 04014057.

104

3 Open Channel Transitions

Skogerboe, G. V., Austin, L. H., & Bennett, R. S. (1971). Energy loss analysis for open channel expansions. Journal of the Hydraulics Division, ASCE, 97(HY10), 1719–1736. Smith, C. D., & Yu, J. N. G. (1966). Use of baffles in open channel expansions. Journal of the Hydraulics Division, ASCE, 92(HY2), 1–7. Sturm, T. W. (1985). Simplified design of contractions in supercritical flow. Journal of Hydraulic Engineering, 111(5), 871–875. Viero, D. P., & Defina, A. (2019). Multiple states in flow through a sluice gate. Journal of Hydraulic Research, 57(1), 39–50. Vittal, N., & Chiranjeevi, V. V. (1983). Open channel transitions: Rational method of design. Journal of Hydraulic Engineering, 109(1), 99–115. Yarnell, D. L. (1934). Bridge piers as channel obstructions. Technical Bulletin, 442 (US Department of Agriculture). Viero, D. P., & Defina, A. (2017). Extended theory of hydraulic hysteresis in open-channel flow. Journal of Hydraulic Engineering, 143(9), 06017014.

Chapter 4

Spillways

4.1

Introduction to Conveyance Structures

Conveyance structures include spillways, canals, transitions, culverts, diversions, drops, intakes and siphons. The fundamental design problem for such structures is the specification of the size and geometry of the structure to accommodate the required discharge. This is commonly achieved through a process of iterative analysis, i.e. a certain configuration is assumed and then analysed to determine its discharge capacity. The initial configuration is then modified and reanalysed until the required discharge capacity is obtained. While the discharge through a structure is influenced by various geometric characteristics, it is controlled at one position only. Estimation of the discharge through the whole structure requires identification of the effective control feature, determination of the energy level at its location and application of its control relationship. A hydraulic structure may incorporate a number of features which could act as controls. For any particular condition, one of these will determine the discharge through the structure. At other conditions, however, different features may be the effective controls. It is important, therefore, to be able to predict which feature will be the effective control for the condition under consideration. These concepts can be illustrated by considering a closed conduit leading from a reservoir and incorporating three adjustable gates, as shown in Fig. 4.1. Associated with each gate is a control relationship between the energy level immediately upstream (H) and the discharge (Q), i.e. Q = fi(Hi), in which the subscript i denotes the particular control. Suppose gate 1 is set to be able to pass a very much smaller discharge than either gate 2 or gate 3 (Fig. 4.2a). The discharge through the whole structure will then be determined by the control relationship for gate 1, with the energy level defined by the water level in the reservoir (HW), i.e. Q = f1(HW). Any reasonably small adjustments to the settings of gates 2 and 3 will have no influence on the discharge. Clearly, gate 1 is the effective control. Flow downstream of it is

© Springer Nature Switzerland AG 2020 C S James, Hydraulic Structures, https://doi.org/10.1007/978-3-030-34086-5_4

105

106

4 Spillways

Q1 = f1(HW) 1

Q3 = f3(H3) 3

Q2 = f2(H2) 2

HW Q

Fig. 4.1 Hypothetical conveyance structure

supercritical and no features or characteristics of the structure further downstream can have any influence on the discharge. If gate 2 is set to be able to pass a very much smaller discharge than the other two (Fig. 4.2b), then it will be the effective control and the discharge will be given by Q = f2(H2). The local energy level, H2, must be calculated by subtracting from HW all the losses up to section 2, i.e. H2 ¼ HW

hL1

hf 1

ð4:1Þ

2

in which hL1 is the loss associated with flow through gate 1 and hf1-2 is the friction loss between gates 1 and 2. The conduit will flow full upstream of the control and flow downstream will be supercritical. Small adjustments to control 3 will have no effect on the discharge through the structure. Adjustments to gate 1 will affect the discharge, however, because hL1 depends on the gate setting. Similarly, if gate 3 is set to be able to pass a very much smaller discharge than the others (Fig. 4.2c), then it will be the effective control and the discharge will be given by Q = f3(H3), with H3 given by H3 ¼ HW

hL1

hf 1

2

hL2

hf 2

3

ð4:2Þ

in which hL2 is the loss through gate 2 and hf2-3 is the friction loss between gates 2 and 3. Determining the discharge through the structure therefore requires identifying the effective control and applying the appropriate control relationship using the local energy level, which accounts for all losses upstream of the control section. Nothing downstream of the control affects the discharge, so no downstream characteristics need to be accounted for. The major difficulty is identifying the effective control. In the example above, this was made quite obvious, but it is not always so (it would not be, for example, if all the gates in the hypothetical structure were set with equal openings). In many real structures, the potential control features are often quite different, such as overflow sections, contractions or lengths of closed conduit. The effective control

4.1 Introduction to Conveyance Structures

(a)

107

Q = f1(HW)

HW Q

(b)

Q = f2(H2) hL1

hf 1-2

HW H2 Q

(c)

Q = f3(H3) hL1

hf 1-2 hL2

hf 2-3

HW H3 Q

Fig. 4.2 Different positions of effective control in hypothetical structure

should be identified by applying the principle that it is the feature that has the highest energy requirement to pass a given discharge. For a given HW, therefore, the discharge can be calculated assuming each potential control to be the effective one. The lowest discharge will be the correct one, and indicates the effective control for that value of HW. Sometimes the discharge is specified and HW is required. By the same principle, this can be determined by calculating it assuming each potential control to be the effective one and selecting the highest value.

108

4.2

4 Spillways

Spillway Structures

The most common use of spillways is in dam structures, where they are installed to discharge excess or flood water which cannot be absorbed into storage. They are also used at diversion structures to bypass flows in excess of diversion requirements, and as control structures to induce required flow depths. The spillway is often the most expensive component of a dam and, from a safety point of view, one of the most critical design aspects. Many dam failures in the past can be attributed to inadequate spillway design. An essential preliminary to the hydraulic design of a spillway is a hydrological analysis to determine the design flood. This will begin by the adoption of an acceptable level of risk of failure. Usually expressed as a flood frequency or return period, this may be defined by legislation or policy, or may be based on economic considerations if there is no danger to life or infrastructure downstream. A design flood hydrograph for the site is then produced by an appropriate hydrological model. If the reservoir is large, the attenuation effects should also be accounted for by reservoir routing. If there are no constraints on flood discharge downstream, the size of the spillway may be optimized together with the flood storage capacity to obtain the most economical design. Various types and configurations of spillways are commonly used. Selection depends on factors such as the type of dam, the capacity required, and the geology and topography of the site. A spillway structure will generally comprise the following components: – A control structure, which defines the relationship between the water level in the reservoir and the discharge, such as a simple overflow crest, for example. – A discharge channel, to convey the flow away from the control structure, such as a spillway face, chute or closed conduit. – A terminal structure, usually in the form of an energy dissipator (to be discussed in Chap. 6). – Entrance or outlet channels may be required to convey the flow to the control structure, as in a col spillway, or from the terminal structure to the river downstream. This chapter concentrates on the hydraulics of the different elements that may be incorporated in a design and some of the problems commonly encountered in their design or operation.

4.2.1

The Overflow Spillway

Also known as an “ogee” spillway because of the reverse-curved shape of the entire structure from the crest to the terminal structure, this is the most common type of spillway and is widely used in gravity, arch and buttress dams, as well as for small

4.2 Spillway Structures

109

Fig. 4.3 A low-head overflow spillway

weirs and control structures in conveyance channels (Fig. 4.3). Embankment dams often incorporate concrete gravity sections to accommodate this type of spillway. Hydraulically, it is a very simple structure, having only one possible control location, which is at the crest. Design requires (i) specification of the crest shape, (ii) determination of the crest level and length and (iii) determination of the flow conditions over the crest, on the spillway face, and at the entry into a terminal structure or stilling basin. The shape of the crest is usually assumed to be ideally the same as the profile of the underside of the nappe of a sharp-crested weir, as shown in Fig. 4.4. This ensures approximately zero (atmospheric) pressure on the surface. The shape of the profile depends on the upstream head, the inclination of the upstream face and the height of the crest above the bed of the approach channel. The desired shape for a particular set of conditions could be determined by constructing a flow net or performing a numerical potential flow analysis, but for practical purposes various procedures for defining an appropriate shape have been published. The curves downstream and upstream of the crest are usually described separately. Both the United States Bureau of Reclamation (1973) and the U S Army Corps of Engineers (1995) describe the downstream profile using the power function xn ¼ KHdn 1 y

ð4:3Þ

in which x and y are the coordinates (in feet) as indicated in Fig. 4.5, Hd is the design head (in feet), n is a variable usually assumed to be 1.85 and K is a

110

4 Spillways

Fig. 4.4 Overflow spillway crest profile

energy H/

H

nappe 0.11H/

Fig. 4.5 Definition sketch for overflow spillway geometry

reservoir level Hd

Eqn (4.4) x Eqn (4.3)

P y

coefficient which depends on the approach depth below the crest, P and Hd. For P/Hd > about 1.5, K has a constant value of 2.0; values for low-head structures with smaller P/Hd may be obtained from the U S Army Corps of Engineers (1995). For the upstream segment of the profile, the United States Bureau of Reclamation (1973) proposes a single or multiple circular arcs to connect tangentially to the downstream profile at the crest. The Bureau also provides a standard profile comprising multiple circular arcs with different radii for the entire profile including the upstream and downstream segments; this is recommended for small spillways where P exceeds one half of the maximum anticipated head—otherwise, Eq. (4.3) should be used for the downstream segment. The U S Army Corps of Engineers (1995) have adopted a procedure proposed by Murphy (1973) for the upstream segment. This uses an elliptical shape described by the dimensionless relationship x2 ðB yÞ2 þ ¼1 A2 B2

ð4:4Þ

for −x < −A and y < B, in which A and B are the half ellipse axes in the horizontal and vertical directions, respectively. For P/Hd > about 2.0, A = 0.28Hd and B = 0.164Hd; values for smaller P/Hd can be obtained from U S Army Corps of Engineers (1995). Both the upstream and downstream segments are joined

4.2 Spillway Structures

111

tangentially to the respective face surfaces to conform with the dam wall geometry. Although the hydraulic behaviour is quite sensitive to the upstream segment geometry, Murphy (1973) found that little accuracy is lost with any slope on the upstream face, so long as it joins the curved segment tangentially; the upstream face is commonly vertical for concrete gravity dams. For slender dam sections, such as in arch dams, the required crest profile may not fit within the section as designed structurally. The profile can then be extended to form a corbel on the upstream face, as shown in Fig. 4.6a, or the profile can be made to bulge on the downstream face, as shown in Fig. 4.6b, although this increases the possibility of cavitation. Another alternative would be to incorporate the profile in the design of a splitter-type energy dissipator, as shown in Fig. 4.6c. As indicated by Eqs. (4.3) and (4.4) in Fig. 4.5, the spillway profile depends on the design head. The profile designed will therefore result in atmospheric surface pressures only for the design head specified. Lesser heads will result in positive pressures while greater heads will cause negative pressures. Determination of the level and length of the spillway crest requires application of a relationship between the discharge over the crest and the upstream water level. The design discharge will be a known value, having been previously established through a hydrological analysis, and the spillway size needs to be designed to accommodate this. At the design head, the nappe will be the same shape as for flow over the corresponding sharp-crested weir and the pressures on the spillway surface will be close to atmospheric. The equation for discharge (Q) is therefore similar to that for a sharp-crested weir (a derivation is provided in Chap. 7) and usually applied in the form Q ¼ CLH 3=2

ð4:5Þ

in which C is an empirical discharge coefficient, L is the length of the crest and H is the total head above the crest which can usually be assumed to coincide with the reservoir water level. Application of Eq. (4.5) requires a value for C, which depends on – the approach flow depth, – the difference between the actual head and the value for which the crest shape was designed,

(a)

Fig. 4.6 Modified crest profiles

(b)

(c)

112

4 Spillways 2.2 2.1 2.0 C 1.9 (m1/2/s) 1.8 1.7 1.6 0.0

0.5

1.0

1.5 P/H

2.0

2.5

3.0

Fig. 4.7 Discharge coefficients for vertical-faced ogee spillways [adapted from United States Bureau of Reclamation (1973)]

– the slope of the upstream face of the structure and – the downstream flow conditions if the spillway is not high. Values of C and their variations with the above influences have been determined empirically and presented by the agencies that have provided specifications for the crest profile. The value used should be consistent with the profile shape for which they were determined, but as the different profile formulations all approximate the same shape of the underside of the nappe over a sharp-crested weir, values from any of the different sources can be accepted for practical purposes. The effect of approach depth on coefficient values for vertical-faced ogee crests is shown in Fig. 4.7. The value of C depends on the head on the spillway. If the head is less than the design head, then positive pressures develop on the spillway and the value is reduced, decreasing the efficiency of the structure. If the head is greater than the design head, then pressures are negative and C is greater. It is therefore sensible to design the shape of the spillway for a head which is less than anticipated, to take advantage of the enhanced discharge coefficient. Information provided by the United States Bureau of Reclamation (1973) indicates that C is increased by about 6% if the actual head exceeds the design head by 50% and decreased by about 4% if the actual head is 50% less than the design head. There is a danger in operating a spillway at higher than design heads, however, as separation may occur if the pressure reduction is too great. This can happen intermittently leading to vibrations which can cause noise and structural damage. In extreme cases, cavitation can occur which can result in severe pitting and erosion of the surface. The occurrence and prevention of cavitation are discussed further in Sect. 4.3. Chadwick and Morfett (1986) report research results indicating that separation will not occur until the actual head is about 3 times the design head, although Novak et al. (2001) recommend a maximum of about 1.65 times the

4.2 Spillway Structures

113

design head. The U S Army Corps of Engineers (1995) suggest designing the spillway to keep the average pressure on the crest above −4.6 m, and provide cavitation safety curves relating actual and design heads to crest pressures for crests with and without piers. Cassidy (1970) presented a rational method for ‘underdesigning’ a spillway crest, which considers the minimum pressure occurring on the crest during possible flow at heads higher than the design value. The United States Bureau of Reclamation (1973) also presents additional information to account for the effects on C of upstream face slopes and downstream influences. Reese and Maynord (1987) tested the profile defined by Eqs. (4.3) and (4.4) and provided further quantifications of C for varying relative heads, upstream slopes and P/H  2.0. Further details are provided by U S Corps of Engineers (1995), including values for milder upstream face slopes and low approach depths. It should be noted that C is not dimensionless and the values given by the United States Bureau of Reclamation and U S Army Corps of Engineers publications are generally for use with foot-second units. They must be converted when using SI units. It is common to regulate the flow over a spillway by installing gates on the crest (Fig. 4.8). Gates are also installed to increase reservoir capacity. Various types of gates are used, including radial (tainter), drum, vertical lift and flap types. Radial and vertical gates (Fig. 4.9) are the most commonly used, and are described in some detail by U S Army Corps of Engineers (1995). Crest gates change the flow conditions over the spillway, which becomes more an orifice type of flow than a free surface flow. This affects the trajectory of the issuing jet, and hence the ideal spillway surface profile, and also the discharge–head relationship. The trajectory for a vertical orifice can be expressed (in foot-second or SI units) by the parabolic equation (United States Bureau of Reclamation 1973)

Fig. 4.8 Radial gates on a side-channel overflow crest

114

4 Spillways

H1 H

H

H2

Go

Go

x/Hd

x/Hd

(a) Radial Gate

(b) Vertical Lift Gate

Fig. 4.9 Common gate types for controlled spillway crests

y¼

x2 4H

ð4:6Þ

where H is the head on the centre of the opening, and for an orifice inclined at h to the vertical can be described by y ¼ x tan h þ

x2 4H cos2 h

ð4:7Þ

A profile designed according to Eq. (4.7) will be wider than the usual one, such as described by Eq. (4.3). Gates are often installed on existing structures and the negative pressures that would occur on the downstream segment of the usual profile can be reduced by locating the sills of the gates downstream of the crest (by x/ Hd = 0.2, as recommended by Novak et al. (2001) and shown in Fig. 4.9). Different gate configurations obviously define different control relationships. For the radial gate, the U S Army Corps of Engineers (1995) suggest using the high head orifice equation Q ¼ CWb Go ð2gH Þ1=2

ð4:8Þ

in which Wb is the gate width, Go is the gate opening (the minimum distance from the gate lip to the spillway surface) and H is the distance from the water surface to the centre of the gate opening (see Fig. 4.9). The discharge coefficient, C, depends on the angle, b, between the tangents to the gate lip and the spillway surface at the point defining Go. The U S Army Corps of Engineers (1995) provide a graph for C, giving values for x/Hd between 0.1 and 0.3 varying from about 0.67 for b = 50° to about 0.73 for b = 100°.

4.2 Spillway Structures

115

For a vertical lift gate, the U S Army Corps of Engineers (1995) propose the relationship 3=2

Q ¼ Qf

H2

3=2

H1

H 3=2

ð4:9Þ

in which Qf is the free flow (ungated) discharge [according to Eq. (4.5) and H1 and H2 are the depths below the water surface of the bottom of the gate and its seating level on the spillway surface (see Fig. 4.9)]. The length, L, in Eq. (4.5) does not account for the presence of piers (for gate mountings, for example) and contractions of flow at the abutments. These may be accounted for by reducing the net crest length according to (United States Bureau of Reclamation 1973) =

L ¼L


 2 NKp þ Ka H

ð4:10Þ

in which L/ is the effective length of the crest, N is the number of piers, Kp is a pier contraction coefficient and Ka is an abutment contraction coefficient. The contraction coefficients depend on the geometry and H. Kp varies from 0 for pointed nose piers to 0.020 for square-nosed piers with rounded corners. Ka varies from 0 for angled, rounded abutments to 0.20 for square abutments. Further details are given by the United States Bureau of Reclamation (1973), the U S Army Corps of Engineers (1995) and Hager (1988). Application of the control relationships described above enables an appropriate spillway length and crest level to be determined. Detailed design also requires prediction of flow characteristics over the crest, down the spillway face and at the entry to the terminal structure at the end of the spillway face. The U S Army Corps of Engineers (1995) provide guidance on the conditions favourable for cavitation on the crest surface and for defining the upper nappe profile of the flow over the crest. Knowledge of this profile is valuable for designing the geometry of the side walls adjacent to the crest and gate mounting structures. The flow conditions at the entry to the terminal structure are determined by the flow down the spillway face. This surface is usually very steep (about 60°), making the flow too complex for the usual open channel theory and gradually varied flow techniques to be used reliably. Over the crest, the flow is rapidly varied and friction has only a small influence until the boundary layer has developed fully (Fig. 4.10). Once the boundary layer is fully developed, air entrainment is likely and the flow cannot be analysed in the usual way. (Some aspects of aerated flow are discussed in Sect. 4.3.) Bradley and Peterka (1957) presented the results of investigations of this problem undertaken by the United States Bureau of Reclamation. Figure 4.11 (also presented in Henderson 1966) was developed from their prototype tests on the Shasta and Grand Coulee Dams. Figure 4.11 gives the correction to be applied to the “theoretical” velocity (Vth) to obtain the actual velocity (Va) at the foot of the spillway as a function of the

116

4 Spillways

Fig. 4.10 Boundary layer development on spillway

H limit of boundary layer

h

Fig. 4.11 Velocity at the foot of a spillway (adapted from Bradley and Peterka 1957)

180

H (m) 0.76

160

1.5 2.3 3.0 4.6 6.1 7.6 9.1 12.2

140 120 100 h (m)

80 60 40 20 0

H h

0.2

0.4

0.6 Va / Vth

0.8

1.0

spillway height and the head on the crest. This theoretical value is calculated by equating the energy values at the crest and toe, assuming no losses, to give sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   H Vth ¼ 2g h 2

ð4:11Þ

in which g is gravitational acceleration and h is the height of the reservoir level above the base of the spillway. The correction given by Fig. 4.11 is considered by the authors to be sufficiently accurate for preliminary design for spillway slopes in the range 0.8H–0.6H:1V. Note that for a high head on a low spillway, the

4.2 Spillway Structures

117

theoretical velocity is quite accurate. In this case, the boundary layer is not fully developed and friction is not controlling the flow. For a low head on a high spillway, friction will control the flow and substantial correction is required. Model testing is not reliable for this situation because the effects of aeration are not reliably reproduced.

4.2.2

Labyrinth and Piano Key Weirs

The discharge capacity of an overflow spillway is limited by the allowable upstream water level and the length of the crest. The labyrinth design provides an increased crest length by folding the crest in plan to form a series of cells (Fig. 4.12). This form provides similar discharge to a straight crest, but at a lower head, allowing greater reservoir storage capacity without compromising flood safety. Labyrinth weirs are often used for increasing reservoir storage of existing dams, in preference to installing control gates, and simultaneously increasing flood discharge capacity. Geometries vary widely, with the plan shape of cells commonly triangular, trapezoidal (a > 0) or rectangular (a = 0). The discharge is usually related to the upstream energy level by the standard weir equation (see Chap. 7), Q ¼ Cd L

2 pffiffiffiffiffi 3=2 2gH 3

ð4:12Þ

in which Cd is the (dimensionless) discharge coefficient and L is the total crest length. The value of Cd depends on H and various factors describing the weir geometry, including the crest height (P), the width and shape of the crest, and the plan geometry defined by the side wall angle (a) and the apex and exit widths. Various formulations for Cd in terms of these geometric parameters have been

Flow A

flow path 2

H

α

P flow path 1 A 1

2

1 2 (a) Plan

Fig. 4.12 Labyrinth weir geometry

(b) Section A-A

118

4 Spillways

proposed (e.g. by Khode et al. 2012 and Crookston and Tullis 2013a). The value peaks at a low head and then declines rapidly as interference of the nappes from opposite sides and submergence increases, as shown, for example, (not for design) in Fig. 4.13. The value of Cd for a straight weir peaks at around 0.80 for H/P = 0.30–0.50 and declines to around 0.70 for H/P = 1.0. Cd for a labyrinth weir is therefore always less than for a straight weir, but the reduction in efficiency is usually more than compensated for by the increased crest length. The complex flow patterns produced in labyrinth weirs can cause undesirable conditions, such as nappe vibration with associated noise, surging flow and fluctuating pressure forces on the walls. Crookston and Tullis (2013b) have described these phenomena, identified the hydraulic conditions under which they occur and proposed some measures for countering their effects. The piano key weir is a modification of a rectangular labyrinth weir, with the cells having sloping floors and extending beyond the base on which they are founded (Figs. 4.14 and 4.15). This geometry allows an even longer crest than a labyrinth layout with the same ‘footprint’; this makes it particularly useful for raising the storage level on existing dam structures. The overhanging structure reduces the construction material considerably, and improves the hydraulic efficiency slightly. The discharge can be described by the same equation as for labyrinth weirs (Eq. (4.12)), with Cd now depending also on the overhang geometry and bed slope. As for labyrinth weirs, the hydraulic efficiency is greatest at low heads, and declines significantly as the head increases (Fig. 4.13). The hydraulic performance of piano key weirs has been described by Anderson and Tullis (2012), Leite Ribeiro et al. (2012), Machiels et al. (2014), amongst others.

Fig. 4.13 Typical Cd variation for labyrinth and piano key weirs (adapted from Anderson and Tullis 2012)

0.55 0.50 Piano Key 0.45 Labyrinth 0.40

Cd 0.35 0.30 0.25 0.20 0.0

0.2

0.4

0.6

H/P

0.8

1.0

4.2 Spillway Structures

119

Fig. 4.14 Piano key weir (photograph by F. Denys)

Flow

flow path 2 A

H P flow path 1

A 1

1 2 (a) Plan

2

(b) Section A-A

Fig. 4.15 Piano key weir geometry

4.2.3

The Side-Channel Spillway

A side channel spillway is commonly used where there is insufficient space to accommodate the required spillway length in line with the dam axis, such as where the abutments are steep. The spillway crest is usually placed along one abutment and perpendicular to the dam axis, as shown in Fig. 4.16. The crest is usually a conventional overflow spillway, and in some cases, inflow is from both sides of the channel. An L-shaped configuration, with water entering the top end of the side channel in the main flow direction is also commonly used

120

4 Spillways

(b) Side-channel cross section

(a) Plan

(c) Side-channel long section

Fig. 4.16 Side-channel spillway

(Fig. 4.17); this assists in changing the direction of the lateral flow and reducing surface waves. Water flows over the control section into a channel or trough at right angles to the main inflow, usually leading to a chute. The side channel will often have a control structure (such as a weir) at its downstream end to ensure subcritical flow along its length. This enhances energy dissipation through plunging of the

Fig. 4.17 An L-shaped side-channel spillway

4.2 Spillway Structures

121

entry flow and prevents transverse wave action. It also facilitates entry into a chute or tunnel by allowing sharper changes in flow direction. The channel should be designed so that the inflow is unsubmerged at the design discharge, or the spillway discharge coefficient will be reduced and the analysis will be much more complex. The main design problem is therefore to select the depth, width and slope of the side channel to ensure this condition. The size of the side channel can be kept small by making it steep enough to induce supercritical flow and not including a control structure at the downstream end. Supercritical flow is undesirable, however, because it has insufficient depth to absorb and damp the incoming flow and the side-channel flow will be swept towards the opposite wall resulting in violent wave action. It is good practice, therefore, to include a control structure to force deep, subcritical flow along the length of the side channel. Sizing the channel requires an analysis of the water surface profile. The flow along the side channel is gradually varied, but the conventional analysis techniques are inadequate because they do not allow for a varying discharge along the profile. Also, the standard gradually varied flow equation is based on the energy equation and assumes that the friction gradient accounts for all energy losses. In the side channel, there is an appreciable energy loss associated with the turbulent mixing of the water plunging in from the side. This additional loss cannot easily be estimated, and without its inclusion, the energy equation cannot be used. The incoming flow has no momentum in the flow direction, however, and therefore does not affect the total momentum in the side channel. The flow can therefore be analysed on the basis of momentum conservation. If the bed is horizontal and the effect of friction can be considered small compared with the effect of the varying discharge, then no forces act on the water in the flow direction and the momentum function is constant. The water profile can then be computed simply by establishing the momentum function at the downstream control and solving for the flow depth at any position with the appropriate discharge. If the channel has a slope (which introduces a weight component in the flow direction) and friction is significant, then a more complete solution is required. This can be obtained by considering the momentum of an element in the profile with a length Dx, as shown in Fig. 4.18. The forces acting on this element in the flow direction are – the component of its weight in the x direction, cADx sin h in which A is the cross-sectional area and h is the angle of the bed to horizontal, – the shear force on the boundary, so PDx

122

4 Spillways

ΔQ Δx

F1 x 1

F2

θ 2

A Δx

Fig. 4.18 Side channel with lateral inflow

in which so is the boundary shear stress = cRSf, R is the hydraulic radius (=A/P), P is the wetted perimeter and Sf is the energy gradient and – pressure forces (F) at sections 1 and 2. The change in momentum between sections 1 and 2 is equal to the sum of forces acting on the element in the flow direction. In terms of the momentum function (which incorporates the pressure forces), therefore DM ¼

cADx sin h c

cRSf PDx c

For small slopes, sin h  tan h ¼ So , and R = A/P. Dividing through by Dx and replacing finite differences by differentials give
dM ¼ A So dx

Sf



ð4:13Þ

From the definition of the momentum function M¼

Q2 þ Az gA

in which z is the depth of the cross-section centroid below the water surface. M can be differentiated with respect to x to obtain dM 2Q dQ ¼ dx gA dx

Q2 dA dy þA dx gA2 dx

ð4:14Þ

4.2 Spillway Structures

123

The right-hand sides of Eqs. (4.13) and (4.14) can be equated to eliminate dM/ dx. The term dA/dx can be expressed as (dA/dy)(dy/dx) and dA/dy = B, the surface width, because incremental changes at the surface are being considered. Making this substitution and rearranging terms give 2Q dQ dy So Sf gA2 dx ¼ dx 1 Fr2

ð4:15Þ

in which Fr is the Froude number. Note that if there is no lateral inflow, then dQ/ dx = 0 and Eq. (4.15) reduces to the familiar ordinary gradually varied flow equation. For a side-channel spillway, the inflow over the spillway section is usually uniformly distributed and so the lateral inflow term dQ/dx has a constant value. The total discharge in the side channel, Q, however, varies with the distance, x. (Some sources include a in the third term of the numerator and in Fr.). Equation (4.15) can be solved in finite difference form by specifying sections along the side channel, separated by known Dx distances. Starting at the control section, Dy can then be calculated progressively from section to section. The flow depth is required for calculating A, Sf and Fr, which should strictly be calculated for each Dx as the mean of the values at the bounding sections. This would require an iterative solution using an initial estimate for the flow depth at the new section. However, if Dx is kept small, the known flow depth can be assumed to apply through Dx, enabling direct solution. The procedure is then to calculate dy/dx from Eq. (4.15) using the current section values yi and Qi. At the next section, yi+1 = yi + dy/dx
Dx and Qi+1 = Qi + dQ/dx
Dx, and these then become the current values for the next calculation. If the side channel has a smooth transition to a chute, with no control structure, at its end critical flow must occur at some position somewhere along its length. The full water surface profile must then be computed from the critical section in the upstream direction for the subcritical portion and downstream for the supercritical portion. Henderson (1966) provides the following formula for the distance, x, of the critical section from the head of the side channel: x¼

8Q2x


gB2 So

 gP 3 C2 B

ð4:16Þ

in which Qx is the inflowing discharge per unit length of channel (i.e. dQ/dx) and C is the Chézy resistance coefficient (which can be substituted by the Manning coefficient, n, if desired, with C = R1/6/n). Equation (4.16) must be solved iteratively, i.e. an initial value of x is assumed, which is used to calculate the discharge at that section (=Qx
x); the critical depth can then be calculated from Fr2 = 1 and used to calculate P, B and R; x is then calculated from Eq. (4.16) and compared with the initially assumed value; the initial estimate can then be updated and the procedure repeated until satisfactory convergence is obtained. If the calculated

124

4 Spillways

value of x is greater than the length of the channel, then subcritical flow will persist up to the entrance to the chute. If there is a control structure at the end of the channel, it may override the critical section and produce subcritical flow over the whole channel length. As mentioned before, flow conditions in the side channel can be improved by introducing a longitudinal flow component to the upstream end of the side channel by realigning the crest to an L-shape, with a portion normal to the channel flow direction and a portion parallel to it. The water surface profile analyses outlined above need modification for this situation and an appropriate design procedure has been presented by Knight (1989). Example 4.1 A 75 m long ogee spillway discharges to an 8.0 m wide, rectangular side channel with a slope of 0.15 and Manning n of 0.013. The channel leads through a smooth transition to a steep chute at the end of the spillway crest. Compute the water surface profile along the channel when the discharge over the spillway is 3.5 m3/s/m. Solution Determine if a critical flow section occurs along the channel using Eq. (4.16), i.e. x¼

8Q2x g B2 So

with the substitution C = R1/6/n, x¼




 gP 3 C2 B

8Q2x  3 2 g B2 So RgPn 1=3 B

Q = Qx
x = 3.5
48.7 = 170.4 m3/s

Solving by trial, for x = 48.7 m: for which

1=3

yc ¼



Q2 g B2

¼



170:42 9:8
8:02

1=3

¼ 3:59 m

A ¼ Byc ¼ 8:0
3:59 ¼ 28:7 m2 P ¼ B þ 2yc ¼ 8:0 þ 2
3:59 ¼ 15:2 m A 28:7 ¼ 1:89 m R¼ ¼ P 15:2 Therefore x¼

8
3:52  3 ¼ 48:7 m 2 9:8
8:02 0:15 9:8
15:2
0:013 1:891=3 8:0

4.2 Spillway Structures

125

x < L = 75 m, so a critical flow section occurs within the side-channel length. Profile computation must therefore begin at this section and work upstream for the subcritical portion and downstream for the supercritical portion. The profiles are computed by solving Eq. (4.15) in finite difference form. Because dy/dx cannot be calculated at the critical section, the first step is done by iteration of Dy, using variables averaged over the first length increment and subsequent calculations are done directly using variable values at the nominated sections. Calculations for the subcritical and supercritical profiles are shown below. In the tables

x Dx Dy y A Q B Fr2 P R V Sf Dy/Dx

distance along the channel, distance between nominated sections, change in flow depth between sections, iterated for first step, then = Dy/Dx
Dx, flow depth, = y at previous section +Dy, flow area, = B
y, discharge at section, = 3.5
x, channel width, Froude number2, = Q2B/gA3, wetted perimeter, = B + 2
y, hydraulic radius, = A/P, velocity, = Q/A, energy gradient, = V2n2/R4/3, water surface gradient, by Eq. (4.15).

Calculations for the subcritical profile, upstream from critical section: x

∆x

(m)

(m)

∆y (m)

48.7

y (m)

3.591 −3.7

−0.120

45

3.471

A (m2)

Q

B

(m3/s)

(m)

Fr2

P

R (m)

V (m/s)

Sf

(m)

28.72

170.45

8.0

1.000

28.24

163.98

8.0

0.973

27.76

157.50

8.0

0.945

15.18

14.94

∆y/

∆y (m)

∆x 1.89

5.93

0.002543

1.88

5.80

0.002462

1.86

5.67

0.00238

−0.120 0.033

40

−5.0

−0.167

3.304

26.43

140.00

8.0

0.866

14.61

1.81

5.30

0.002151

0.036

35

−5.0

−0.180

3.124

24.99

122.50

8.0

0.784

14.25

1.75

4.90

0.00192

0.038

30

−5.0

−0.188

2.936

23.49

105.00

8.0

0.694

13.87

1.69

4.47

0.001674

0.041

25

−5.0

−0.204

2.732

21.85

87.50

8.0

0.598

13.46

1.62

4.00

0.00142

0.044

20

−5.0

−0.222

2.510

20.08

70.00

8.0

0.494

13.02

1.54

3.49

0.001153

0.049

15

−5.0

−0.246

2.263

18.11

52.50

8.0

0.379

12.53

1.45

2.90

0.000869

0.056

10

−5.0

−0.281

1.983

15.86

35.00

8.0

0.250

11.97

1.33

2.21

0.000565

0.067

5

−5.0

−0.335

1.648

13.19

17.50

8.0

0.109

11.30

1.17

1.33

0.000242

0.087

0

−5.0

−0.437

1.211

126

4 Spillways

Calculations for the subcritical profile, downstream from the critical section: x (m)

∆x (m)

∆y (m)

1.3

0.040

48.7

50

y (m)

A (m2)

Q (m3/s)

B (m)

Fr2

P (m)

R (m)

V (m/s)

Sf

3.591

28.72

170.39

8.0

0.999

15.18

1.89

5.93

0.002541

28.88

172.70

8.0

1.009

1.90

5.98

0.002571

3.630

29.04

175.00

8.0

1.020

15.26

1.90

6.03

0.002602

0.033

∆y/ ∆x

∆y

0.040

55

5.0

0.166

3.797

30.37

192.50

8.0

1.079

15.59

1.95

6.34

0.002791

0.022

60

5.0

0.108

3.904

31.23

210.00

8.0

1.180

15.81

1.98

6.72

0.003082

0.037

65

5.0

0.186

4.090

32.72

227.50

8.0

1.205

16.18

2.02

6.95

0.003196

0.024

70

5.0

0.118

4.208

33.66

245.00

8.0

1.283

16.42

2.05

7.28

0.003436

0.027

75

5.0

0.136

4.344

Repeating the calculations using iteration at each step resulted in a difference of 0.12 m at x = 0 and no difference at x = 75 m for the nominated x increments. The calculated water surface profile is shown below:

Elevation (m)

14 12

Channel bed

10

Water surface

8 6 4 2

critical section

0 0

20

40

60

80

Distance (m)

4.2.4

The Side Weir

A side weir is a weir along one side of a channel with its crest parallel to the main flow direction in the channel. Side weirs are used to divert flow from the channel into a lateral supply canal or wasteway. The discharge in the main channel will decrease along the length of the side weir by the amount diverted. The design requirements include the crest level (to determine the discharge at which diversion begins), the crest length (to determine the magnitude of discharge diverted) and the water surface profile along the length of the side weir (to determine the channel side wall heights). Determining the diverted discharge and the side wall heights both require computation of the water surface profile over the crest length. The flow over the length of the crest is gradually varied, but cannot be described by the normal gradually varied flow equation [Eq. (1.29)] because this implicitly

4.2 Spillway Structures

127

assumes a constant discharge along the profile. The momentum approach followed for gradually varied flow with lateral inflow, as used for the side-channel spillway [Eq. (4.15)], is also not applicable because it is based on the assumption that the incoming discharge does not change the momentum of the flow in the side channel because its initial direction is normal to the side-channel flow. In the lateral outflow case, the diverted flow does carry momentum with it, so the longitudinal change in momentum cannot be assumed to result only from the forces applied to the flow and the varying Q. In the lateral inflow case, an energy analysis was not possible because of the considerable and unquantifiable energy loss associated with the turbulent mixing of the incoming flow. For the lateral outflow case, however, there is no energy loss associated with the diverted flow in Bernoulli terms, because this represents the energy per unit weight of water and is unaffected by changing discharge. The gradually varied flow with lateral outflow can therefore be described using an energy analysis similar to that for normal gradually varied flow, but also accounting for the variation of discharge with distance. As for the constant discharge case, the change in specific energy along the channel results from the change in bed elevation and the loss through friction, dE ¼ So dx

ð1:55Þ

Sf

in which E is specific energy, x is the longitudinal direction, So is the bed slope and Sf is the energy gradient. Therefore   d Q2 yþ ¼ So dx 2gA2

Sf

in which y is the flow depth and A is the cross-sectional flow area. Because both Q and A vary with x, dy Q dQ þ dx gA2 dx

Q2 dA dy ¼ So gA3 dy dx

Sf

and because dA/dy = B, the surface width, and Q2B/gA3 = Fr2, Q dQ dy So Sf gA2 dx ¼ dx 1 Fr2

ð4:17Þ

which is similar to the equation for lateral inflow, except for the coefficient of the third term in the numerator. According to Eq. (4.17), the water surface will rise in the flow direction for subcritical flow and drop in the flow direction for supercritical flow. It is also possible to have a change from supercritical to subcritical flow through a hydraulic jump within the weir crest length.

128

4 Spillways

Equation (4.17) can be solved numerically to define the water surface profile. The solution requires evaluation of the term dQ/dx at each computational step. This can be evaluated from the standard weir equation 2 pffiffiffiffiffi q ¼ Cd 2gh3=2 3

ð7:2Þ

in which q is the discharge per unit crest length (=dQ/dx), Cd is an empirical discharge coefficient and h is the height of the energy line above the crest. The discharge coefficient is different from that for a normal weir because of the more complicated geometry and flow pattern. The value has been found to depend on the angle of the flow over the crest and the Froude number of the flow in the main channel. Various formulations have been derived from laboratory measurements, Borghei et al. (1999), for example, used laboratory data for Fr < 0.80 to propose for sharp-crested side weirs with subcritical flow (using h = y – w) Cd ¼ 0:7

0:48Fr

0:3

w L þ 0:06 y B

ð4:18Þ

in which w is the crest height, y is the flow depth, L is the crest length and B is the surface width in the main channel. Both Fr and y vary along the profile, and so Cd needs to be calculated at each computational step. Example 4.2 A canal with a 3.5 m wide rectangular cross section, a slope of 0.00080 and a Manning n of 0.013 incorporates a 12.0 m long sharp-crested side weir with its crest 1.00 m above the bed. Compute the water surface profile along the length of the weir if the discharge in the channel downstream 10.0 m3/s. Solution The flow depth at the downstream end of the weir is the uniform flow depth in the canal. Calculate this from the Manning and continuity equations, A Q ¼ R2=3 So1=2 n A ¼ Wyo n ¼ 0:013 for concrete A Wyo R¼ ¼ P W þ 2yo So ¼ 0:00080 Therefore Q¼

 2=3 Wyo Wyo S1=2 o n W þ 2yo

4.2 Spillway Structures

129 i.e. 10 ¼

 2=3 3:5yo 3:5yo 0:000801=2 0:013 3:5 þ 2yo

from which yo ¼ 1:51 m Compute the profile from this depth in an upstream direction over the length of the side weir using Eq. (4.17) in finite difference form. At each step, the lateral outflow (=dQ/dx) is calculated using Eq. (7.2) with Cd given by Eq. (4.18). Calculations are shown below. In the table x y A Q V R Sf Fr2 Cd dQ/dx Dy/Dx z h Crest

x (m)

y (m)

distance along the channel, flow depth, = y at previous section + Dy/Dx(x – x at previous section), flow area, = W
y, discharge at section, = Q at previous section + dQ/dx
Dx, velocity = Q/A, hydraulic radius = A/(W + 2y), energy gradient = V2n2/R4/3, Froude number = Q2B/gA3, discharge coefficient, Eq. (4.18) weir discharge per unit length, Eq. (7.2), water surface gradient, by Eq. (4.15), elevation of bed relative to level at downstream end of weir, water surface elevation = z + y, elevation of weir crest.

A (m2)

Q (m3/s)

V (m/s)

R (m)

Sf

Fr2

Cd

dQ/dx

∆y/∆x

z (m)

h (m)

Crest (m)

0

1.510

5.29

10.00

1.89

0.81

0.0008

0.242

0.471

−0.507

0.0244

0

1.510

1

−1

1.486

5.20

10.51

2.02

0.80

0.000923

0.280

0.450

−0.450

0.0246

0.0008

1.487

1.0008

−2

1.461

5.11

10.96

2.14

0.80

0.001051

0.320

0.429

−0.397

0.0246

0.0016

1.463

1.0016

−3

1.437

5.03

11.35

2.26

0.79

0.001181

0.362

0.408

−0.348

0.0244

0.0024

1.439

1.0024

−4

1.412

4.94

11.70

2.37

0.78

0.001315

0.405

0.388

−0.303

0.0240

0.0032

1.416

1.0032

−5

1.388

4.86

12.00

2.47

0.77

0.001451

0.448

0.368

−0.263

0.0236

0.004

1.392

1.004

−6

1.365

4.78

12.27

2.57

0.77

0.001588

0.493

0.349

−0.227

0.0230

0.0048

1.370

1.0048

−7

1.342

4.70

12.49

2.66

0.76

0.001726

0.538

0.330

−0.195

0.0224

0.0056

1.347

1.0056

−8

1.319

4.62

12.69

2.75

0.75

0.001865

0.584

0.312

−0.166

0.0217

0.0064

1.326

1.0064

−9

1.298

4.54

12.86

2.83

0.75

0.002003

0.630

0.294

−0.141

0.0209

0.0072

1.305

1.0072

−10

1.277

4.47

13.00

2.91

0.74

0.002142

0.676

0.276

−0.119

0.0202

0.008

1.285

1.008

−11

1.257

4.40

13.11

2.98

0.73

0.00228

0.722

0.259

−0.099

0.0194

0.0088

1.266

1.0088

−12

1.237

4.33

13.21

3.05

0.72

0.002417

0.768

0.243

−0.083

0.0187

0.0096

1.247

1.0096

The calculated water surface profile is shown below:

130

4 Spillways

1.6

Elevation (m)

1.4 1.2 1.0 0.8

Water surface

0.6

Channel bed

0.4

Weir crest

0.2 0.0

-12

-10

-8

-6

-4

-2

0

Distance (m)

4.2.5

Shaft (Morning Glory) Spillways

In a shaft spillway, water flows over a circular control crest (Fig. 4.19), through a transition, into a vertical shaft which changes to a horizontal conduit leading to an exit downstream of the dam. A typical arrangement is shown in Fig. 4.20. Shaft spillways are frequently used where there is insufficient space for a conventional spillway and the flood discharges are not extreme. They are particularly common for embankment dams with small catchment areas. One advantage of this type is that the horizontal conduit may be used for river diversion during construction.

Fig. 4.19 A shaft spillway crest

4.2 Spillway Structures

131

Fig. 4.20 Typical shaft spillway arrangement

The control relationship for a shaft spillway is complex (as shown in Fig. 4.21d) because the position of control can change as the water level changes. A different control relationship is associated with each possible position of control. In accordance with control principles, the actual discharge at any reservoir water level is the lowest predicted value for any possible control and the resulting composite control relationship is indicated by the solid curve in Fig. 4.21d. At low heads, the crest is unsubmerged, as shown in Fig. 4.21a. Control is then at the crest and the discharge is related to head by the usual weir equation, i.e. Q / H 3/2, represented by curve ab in Fig. 4.21d. At higher heads, the throat, or transition between the crest profile and the conduit, may control. In this case, the crest is submerged and the structure flows full up to the end of the throat section, as shown in Fig. 4.21b. The discharge is now described by an orifice-type equation, i.e. Q / H1/2 a , represented by curve cd in Fig. 4.21d. At still higher heads, the entire structure may flow full (Fig. 4.21c) and the discharge may be controlled by the resistance of the conduit or the level of the tailwater, i.e. Q / H1/2 t , represented by curve ef in Fig. 4.21d. If the control is by the conduit resistance, the water level at the exit will most likely be at the top of the conduit, although flow may go through the critical condition at this point if the critical flow depth is less than the height of the conduit; if the tailwater controls, then the water level will be above the top of the conduit, as determined by conditions further downstream. For conduit control, a certain increase in discharge will cause a much greater increase in head than for crest control, and shaft spillways are generally designed to operate under crest control at the design flood discharge. The relative positions of the individual curves in Fig. 4.21d, and hence the form of the composite curve, are determined by the control feature characteristics such as the diameters of the crest, shaft and horizontal conduit. The effects of these design parameters will be apparent in the following discussion of the individual control relationships.

Crest Discharge The crest, as for the straight overfall spillway, is best designed to the shape of the underside of the nappe of the flow over a corresponding sharp-crested weir. Because of the convergence of the flow, this shape is slightly different from that for a straight crest. The United States Bureau of Reclamation (1973) presents tables of coordinates of the underside nappe profiles for different ratios of total crest height to radius, based on experimental results.

132

4 Spillways H

Ha

(a) Crest control

(b) Throat control

Ht

(c) Conduit or tailwater control

f

Reservoir water level

conduit / tailwater control Q α Ht 1/2

throat control Q α Ha 1/2

d

b

crest control Q α H 3/2

a

e

c Discharge (d) Rating curve

Fig. 4.21 Rating curve for a shaft spillway

For low heads, the discharge will be determined purely by the crest characteristics. The vertical transition beyond the crest will flow partly full, with the flow following the sides of the shaft. As the head increases, the annular nappe will become thicker and will eventually converge into a solid, vertical jet. When this happens the boil of water above the crotch where the nappe converges will begin to

4.2 Spillway Structures

133

affect the flow over the weir. The discharge coefficient will therefore be different from that for a straight spillway. The United States Bureau of Reclamation (1973) assumes that the orifice flow which occurs once the nappe has converged and the crest is submerged can still be described by a weir formula, provided the discharge coefficient is specified appropriately. In terms of the head measured from the apex of the crest and the length defined by its outer periphery (i.e. the position of the equivalent sharp-crested weir), the discharge relationship is given as Q ¼ Co ð2pRs ÞHo3=2

ð4:19Þ

in which Rs is the radius of the outer periphery of the crest and the subscripts o indicate the values at the condition for which the crest profile is designed. Because of the effects of submergence and back pressure in converging flows, Co depends on both Ho and Rs. For the design head with the corresponding crest profile, the United States Bureau of Reclamation (1973) provides a graphical relationship between Co and Ho/ Rs determined by model testing. Free flow occurs for values of Ho/Rs up to about 0.45, and the value of Co is similar to that for a conventional overflow weir in this range. Partial submergence occurs as Ho/Rs increases above 0.45, becoming complete at Ho/ Rs about 1.0, and Co reduces very considerably. The United States Bureau of Reclamation (1973) also presents a diagram for adjusting the discharge coefficient for heads different from that for which the crest profile was designed. Many shaft spillway designs incorporate radial guide vanes on the crest to inhibit vortex action, as recommended by the United States Bureau of Reclamation (1973), for example. Novak et al. (2001), however, maintain that vortex formation is actually advantageous because the spiral flow induced along the shaft walls substantially reduces vibration and pressure fluctuations associated with a free-falling jet at low (and hence frequent) discharges. They recommend installing guide vanes below the crest to induce vortex formation.

Throat Discharge The shape of the throat or transition between the overflow crest and the outflow conduit should be designed to follow the shape of a vertical jet falling under gravity. The jet will have an area A = pR2 and a velocity V = (2 gh)1/2, where R is the jet radius and h is the height to the water surface in the reservoir. By setting Q = AV and expressing this in terms of R, an equation for the shape of the throat can be obtained, i.e. sffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q pffiffiffiffiffiffiffiffi R¼ p 2gh

ð4:20Þ

The shape defined by Eq. (4.20) will give the minimum size to accommodate the flow without restriction and without developing pressures on the side of the shaft.

134

4 Spillways

The actual profile will be wider than this for at least some distance to accommodate the crest profile, and flow will be under pressure in this region. Free flow will occur if the radius of the jet, as defined by Eq. (4.20), becomes less than the radius of the shaft, and the location where this happens is the point of control for the throat. By substituting the corresponding value for h, Eq. (4.20) can be used as the discharge equation under conditions of throat control.

Conduit Discharge If the entire structure is flowing full, the position of control will be at the outlet of the conduit. The rating curve for this condition can be derived simply by applying energy conservation between the crest and the outlet and accounting for friction losses using a resistance equation, such as that of Manning or Darcy–Weisbach. Entrance losses at the inlet, bend losses and expansion losses will usually be negligible in comparison with the friction losses; if they are not considered to be negligible, they can be accounted for using the approaches and results described for siphon spillways and culverts. Conduit control is actually an undesirable flow condition and should be avoided because the transition to full flow may be accompanied by intermittent aeration and sealing, causing erratic discharge, vibrations and surges. To avoid this, the United States Bureau of Reclamation (1973) recommends designing the conduit so that it will not flow full beyond the throat section. To allow for air bulking and surging, it is recommended that the conduit should not flow more than 75% full (in area) at the downstream end at the maximum discharge.

4.2.6

Siphon Spillways

A siphon spillway is a closed conduit which rises above the water level in the reservoir and discharges at a lower level downstream (Figs. 4.22 and 4.23). This enables a greater head to be applied to the flow than if a free surface existed, and the discharge for a given upstream water level is consequently much greater. Two control mechanisms apply over different ranges of water level, and the rating curve is therefore a combination of the relationships describing these. The operation of a siphon spillway can be examined by considering a gradually rising water level. Discharge will begin when the water level rises above the crest level and will be related to the water level by the usual free flow relationship, i.e. Q / H 3/2. As the water level rises, the velocity will increase and some air will be entrained by the flow and removed from the cavity formed by the conduit or barrel (Fig. 4.24). Provided air cannot enter the barrel, all the air will ultimately be removed, and the siphon will be primed and running full. Once this has happened, the discharge will be related to the total head across the structure by Q / h1/2.

4.2 Spillway Structures

135

Fig. 4.22 A three-barrel siphon spillway

H ro rc

h

Energy Line

Fig. 4.23 A simple siphon spillway arrangement

Fig. 4.24 Initiation of priming in a model siphon spillway

136

4 Spillways

Reservoir water level

fully primed Q α h 1/2

air-partialized flow

crest control Q α H 3/2

Discharge

Fig. 4.25 Typical siphon spillway rating curve

Priming usually occurs when the water level is about a third of the throat height and causes a sudden and considerable increase in discharge, as shown in Fig. 4.25. Although superficially similar, the composite rating curve for a siphon differs from that for a shaft spillway because the crest control relationship does not persist to its intersection with the fully primed conduit relationship; flow switches suddenly between the two types as priming and de-priming occur. Priming is accompanied by air entrainment and the flow is a mixture of air and water for a range of water levels. Some siphons are designed to operate under this air-partialized condition to prevent hunting and give a more gradually varying rating curve. In these designs, the inlet lip is carefully located and shaped to admit the right amount of air. For siphons operating under fully primed conditions, full conduit flow will persist until the water level drops sufficiently for air to enter the barrel once more. Because the lip of the inlet is usually located below the crest level to prevent air entry by gulping or vortices, an air vent is installed at about the crest level to control priming and de-priming, as shown in Fig. 4.26. If the outlet is not submerged, air is prevented from entering the barrel from the downstream side by installing a deflector in the barrel or giving it an adverse slope, as shown in Fig. 4.26, to create a water curtain. Siphon spillways are frequently used where it is required to pass a high discharge within a small range of water levels, or where there is a severe limitation on space. A siphon can be formed by installing a hood over an overflow or a shaft type spillway crest, and the outlet can be set to the elevation required to provide the appropriate head, irrespective of the height of the crest. A siphon can respond rapidly and operate at full capacity if the upstream water level rises suddenly. This is useful for forebays of turbine plants or water treatment works to discharge incoming water to waste in the event of sudden closure. The abrupt priming characteristic can be a disadvantage, however, if sudden spates downstream are

4.2 Spillway Structures

137 water curtain minimum level

minimum level

vent

vent water curtain

(a) Deflector type

(b) S-barrel type

Fig. 4.26 Siphon sealing and air venting arrangements

undesirable, but this can be remedied by using multiple siphon barrels set to prime at different levels. As indicated by the control relationship, once a siphon is fully primed, a significant increase in water level accompanies any appreciable increase in discharge, effectively limiting the capacity. Although compact, siphons are difficult and expensive to construct and need to be structurally stronger than conventional spillways to withstand vibration. A further disadvantage is their inability to pass debris and their tendency to clog (also with ice in cold climates). The hydraulic design problems for siphon spillways are to define the geometry, establish the rating curve (particularly for the fully primed condition), ensure that priming takes place at the required water level and ensure that throat pressures are high enough to avoid cavitation. Siphon spillways are not as common as the other types, and there appear to be no general guidelines for geometric design. Because the design discharge will occur in the primed condition, there is no need to shape the crest profile after a free jet,

he hb1 + hf hf

h hb2 + hf h o + hf hexp

Fig. 4.27 Energy loss through a fully primed siphon spillway

Energy Line

138

4 Spillways

which allows greater freedom in the design. Each design must be carried out for its particular conditions, and physical models are often used for testing and refining initial designs. For unprimed flow, the control relationship is similar to that for a conventional overflow spillway. The discharge coefficient would not be the same as for a conventional spillway unless the standard crest profile was used, and would probably also be slightly lower because of energy loss in the approach to the crest. However, the design discharge is not intended to occur in the unprimed condition and great accuracy is generally not required for the lower range of flows. For primed flow, the spillway can be analysed as a closed conduit controlled at the outlet, and a discharge equation is derived from energy considerations. This requires accounting for the friction and other energy losses between the upstream and downstream flows (Fig. 4.27). Conceptually this is easy, as the entrance, exit and bend losses can be expressed as appropriate coefficients multiplied by the throat velocity head, and friction loss can be evaluated using conventional resistance equations. In practice, the calculation is not easy because the loss coefficients depend on particular geometric characteristics and must be determined empirically, and the usual friction equations are not reliable because the barrel is usually too short for flow to be fully developed. If the coefficients and friction factor are known, the energy loss can be estimated by h ¼ he þ hb1 þ hb2 þ ho þ hexp þ hf

ð4:21Þ

in which h is the total head difference across the structure, he is the entrance and converging transition loss, hb1 is the upper bend loss, hb2 is the lower bend loss, ho is the outlet transition loss, hexp is the exit loss and hf is the friction loss. The United States Bureau of Reclamation (1973) gives the following estimates for these loss components for typical rectangular siphon structures, which can be used for preliminary design purposes. he ¼ 0:2

V2 2g

ð4:22Þ

hb1 ¼ hb2 ¼ 0:42

V2 2g

ð4:23Þ

for bends with a centre-line radius equal to 2.5 times the barrel height. ho ¼ 0:2 for diverging outlets and



V2 2g

Vo2 2g



ð4:24Þ

4.2 Spillway Structures

139

V2 ho ¼ 0:1 o 2g

V2 2g





ð4:25Þ

for converging outlets. hexp ¼



A Ao

2

V2 2g

ð4:26Þ

(which is actually just the velocity head at the outlet, V2o/2g). hf ¼ 0:25

V2 2g

ð4:27Þ

In these expressions, V is the barrel velocity, Vo is the outlet velocity, A is the barrel area and Ao is the outlet area. These relationships give indications of absolute and relative magnitudes, and should be used with caution and judgement. The two bend losses are unlikely to be equal, for example, as the upper bend is usually longer and tighter than the lower one. Assuming the exit loss to be the exit velocity head ignores the downstream flow conditions, and could be more rigorously accounted for in the way described for culverts in Chap. 5, i.e. by assuming h to represent the difference between upstream energy and the hydraulic grade line (water surface) at the outlet. The water level at the outlet can be determined by applying momentum conservation between this section and the tailwater as controlled from downstream. The friction loss, hf, could also be better estimated for particular structures using a resistance equation such as Manning or Darcy–Weisbach, although the boundary layer will not be fully developed for some distance downstream from the inlet (the equations relating boundary layer thickness to distance for a spillway (Sect. 4.3.2) would give indications of the length of barrel required for the boundary layer from the top and bottom surfaces to reach the centre line). With all the loss components expressed in terms of the barrel velocity, Eq. (4.21) provides a relationship between V (and hence Q = VA) and h, the head difference across the structure. If the upstream and downstream flow velocities are small, h can be assumed to be the difference between the corresponding water levels. Tadayon and Ramamurthy (2013) used computational fluid dynamics modelling to simulate the flow through a fully primed siphon spillway and used the results to characterize the discharge capacity in terms of a discharge coefficient (Cd) in the equation q ¼ Cd d

pffiffiffiffiffiffiffiffi 2gh

ð4:28Þ

in which q is the unit width discharge and d is the siphon barrel depth. They found good agreement between their predictions and Cd values measured in two physical models with different geometries. They make the point that numerical modelling is less expensive and time-consuming than physical modelling.

140

4 Spillways

It is not possible to predict analytically or computationally the water levels at which priming and de-priming will occur, and hence the ranges of levels for which the unprimed and primed control relationships apply. Final designs of siphon spillways usually require physical model testing to establish the priming conditions and to refine the control relationships. Model tests are necessary for determining the rating curves for air-regulated (whitewater) siphons. Siphon spillways are vulnerable to cavitation, especially in the region of the internal crest where the pressure is lowered considerably by the flow curvature. It is widely accepted that the local absolute pressure head should not be less than 3.0 m, i.e. about 6.71 m below atmospheric at sea level. The variation of atmospheric pressure with altitude must be accounted for when specifying the allowable pressure head, pc /c above sea level. The United States Bureau of Reclamation (1973) specifies allowable subatmospheric pressure heads for conduits flowing full, which can be represented by   pc ¼ 6:71 c allowable

0:0010z

ð4:29Þ

where z is the elevation (in m) above sea level. The pressure in the throat, pc, can be estimated using energy considerations. Taking the datum at the crest level, the total energy at the crest, Hc, is Hc ¼

pc v2 þ c c 2g

in which vc is the local velocity. The pressure head at the crest is therefore pc ¼ Hc c

v2c 2g

ð4:30Þ

Under the assumption of irrotational flow, the total energy over the crest section is constant and the Bernoulli equation can be applied across the section. Hc can therefore be found by subtracting the intervening losses from the upstream energy level, i.e. Hc ¼ H

losses

ð4:31Þ

in which H is the height of the upstream energy line above the crest level and the losses would include the entrance loss, some bend loss and some friction loss. The velocity at the crest, vc, can be determined from ideal flow theory. The region of flow beyond the developing boundary layer in the upper portion of the siphon is irrotational and the flow through the upper bend is approximately the same as for an irrotational vortex, for which the local velocity (v) is inversely proportional to the radius of curvature (r), and so

4.2 Spillway Structures

141

vr ¼ vc rc

ð4:32Þ

in which v is the local velocity at radius r and rc is the radius of the crest segment. (The velocity is thus highest, and the pressure consequently lowest, at the crest.) The velocity at the crest, vc, can be related to the discharge, Q, through the siphon by determining the discharge as the integral of the local velocity across the section and describing the local velocity by Eq. (4.32), i.e. Q¼

Z

Z

ro

vbdr rc ro

rc vc bdr r rc ro ¼ vc rc b ln rc

¼

in which ro is the outer radius of the barrel and b is the width of the barrel. For a given discharge, the velocity at the crest is therefore given by vc ¼

Q rc b ln rroc

ð4:33Þ

Substituting Hc from Eq. (4.31) and vc from Eq. (4.33) into Eq. (4.30) enables the local pressure head at the crest to be determined, which can then be compared with the allowable value according to Eq. (4.29). Because of the assumptions required in analysing siphon and shaft spillways, the results should be regarded as preliminary. Physical model studies are often used to confirm and/or refine the analysis and design details. Example 4.3 An S-shaped siphon spillway has a rectangular barrel which is 16.0 m long, 2.00 m wide and 0.80 m deep, with a diverging outlet reaching a depth of 1.20 m at the exit. The centre-line radius of the upper and lower bends is 2.0 m. The concrete inner surface has a Manning n of 0.012. The spillway serves a reservoir at an altitude of 550 masl. a. Using the United States Bureau of Reclamation (1973) typical energy loss estimates, determine the upstream water level above the inner crest when the discharge is 15.2 m3/s and the tailwater level is 8.0 m below the crest. Assume fully primed conditions, the outlet to be submerged and the velocity heads upstream and downstream to be negligible. b. Assess the likelihood of cavitation in the structure at these conditions. Solution a. The height of the water level above the crest, Hu, is given by Hu ¼ h

8:0

where h is the total energy difference across the structure which is dissipated by friction, entrance, outlet, exit and bend losses (as shown in Fig. 4.27), i.e.

142

4 Spillways h ¼ he þ hb1 þ hb2 þ ho þ hexp þ hf where he ¼ 0:20

V2 2g

V2 hb1 ¼ hb2 ¼ 0:42 2g  2  V Vo2 ho ¼ 0:20 2g 2g with Vo ¼ V

A D ¼V Ao Do 0:80 ¼ 0:67V ¼V 1:20

so  V2
V2 1 0:672 ¼ 0:11 2g 2g  2 2  2 2 A V D V ¼ ¼ Ao 2g Do 2 g  2 2 0:8 V V2 ¼ ¼ 0:44 1:2 2 g 2g

ho ¼ 0:2 hexp

hf ¼ 0:25

V2 2g

(Note that calculating hf by the Manning equation would give a slightly smaller loss.) and V¼

Therefore

and

Q 15:2 ¼ ¼ 9:50 m/s A 2:0
0:80

h ¼ ð0:20 þ 0:42 þ 0:42 þ 0:11 þ 0:44 þ 0:25Þ Hu ¼ 8:47

9:502 ¼ 8:47 m 2
9:8

8:00 ¼ 0:47 m

b. The likelihood of cavitation can be assessed by comparing the pressure head at the crest with the permissible value. The pressure head at the crest is given by pc ¼ Hc c

v2c 2g

4.2 Spillway Structures

143

in which Hc and v-c are the energy and local velocity at the crest. Hc can be estimated by subtracting losses from the upstream energy up to the crest. It is assumed that the losses comprise the entrance loss, half of the first bend loss and friction over the length of half the first bend (a quarter circle). The friction loss this calculated is an overestimate, and therefore conservative, because the boundary layer will not be fully established over this short distance. Then Hc ¼ Hu

he

1 hb1 2

Lbend Sf 2

where Lbend is the length of the first bend and Sf is the friction gradient. Using the USBR loss coefficients and Manning’s equation for Sf, Hc ¼ Hu ¼ 0:47

V2 1 V 2 2pr V 2 n2 0:42 4 R4=3 2g 2 2g 2 9:50 1 9:502 2p2:0 9:502 0:0122 0:42
 0:20 4=3 4 2
9:8 2 2
9:8 2:0
0:80

0:20

2
ð2:0 þ 0:8Þ

¼ 0:47 0:92 ¼ 1:64 m

0:97

0:22

The velocity at the crest is given by Eq. (4.33), i.e. Q rc b ln rroc

vc ¼

D 2 0:80 ¼ 1:60 m ¼ 2:00 2 D ro ¼ rcentre line þ 2 0:80 ¼ 2:00 þ ¼ 2:40 m 2

rc ¼ rcentre line

Therefore vc ¼

15:2 ¼ 11:7 m/s 1:60
2:0
ln 2:40 1:60

and pc ¼ c

1:64

11:72 ¼ 2
9:8

8:62 m

The permissible pressure head for no cavitation is given by Eq. (4.29), i.e.   pc ¼ 6:71 c allowable where z is the altitude.

0:0010 z

144

4 Spillways So   pc ¼ c allowable

6:71 þ 0:0010
550 ¼

6:16 m

The actual pressure head at the crest is lower than the permissible value, so cavitation is very likely for this design and appropriate modifications to the geometry should be made.

4.2.7

Chutes

A chute is a steep channel which is often used in a spillway configuration to convey the discharge from a control section to the river downstream (Fig. 4.28). Chute spillways are the most common type for embankment dams because of their simplicity, adaptability and overall economy. They may be used in conjunction with straight (Fig. 4.29) or side-channel (Fig. 4.16) spillways or even through a col separate from the embankment. This arrangement significantly reduces the size of concrete structure required and avoids disruption of the embankment construction and the problems of sealing between the fill and the concrete. Often the spillway structure is cut into an abutment or col and the excavation material used for embankment fill. The chute itself is light and can be constructed on a wide range of foundation materials. Flow in a chute is invariably supercritical and fast. The major hydraulic problems experienced in designing chutes are interference waves, translatory waves,

Fig. 4.28 A chute leading from a side-channel spillway

4.2 Spillway Structures

145

Fig. 4.29 Alternative overflow spillways for an embankment dam

(a) Looking upstream

(b) Plan

cavitation and aeration. Some of these problems are also experienced in other structures and spillway components, and this discussion is not restricted in application to chutes only. Interference Waves Interference waves are shock waves set up by any disruption of supercritical flow. The patterns and depths of standing waves and some control measures have been discussed in the chapter on transitions. Application to chutes would be similar. Translatory Waves Supercritical flow with very high velocity or on very steep slopes will become unstable and a series of roll waves will form. These present the problems that higher freeboards than otherwise are necessary, and that they can cause unsteady flow impulses in terminal structures. Roll waves can occur in channels with a wide range of slopes (0.02 < S0 < 0.35). They can be prevented by artificially roughening the chute surface, but this may promote cavitation. They can also be prevented by designing the geometry so that the ratio of flow depth to wetted perimeter is large enough to inhibit their formation. Novak et al. (2001) recommend ensuring a ratio of at least 0.1 for the maximum discharge, and accepting the presence of roll waves at lower discharges when extra freeboard is not required and the effect on the terminal structure is less crucial. More rigorous criteria for the onset of unstable flow and methods for predicting some characteristics of roll waves have been developed. Some of these are presented by French (1985).

146

4.2.8

4 Spillways

Stepped Chutes and Spillways

Flow at the downstream end of a spillway or chute is fast and high in kinetic energy, and its release into downstream channels can cause considerable damage. The excess energy is usually dissipated by deflecting the flow jet away from the structure or in a stilling basin at the end of the slope (see Chap. 6). An alternative is to provide a series of steps along the spillway face or chute, dissipating much of the energy before the end of the slope and hence requiring a smaller terminal energy dissipator (Fig. 4.30). This design is ideally suited to roller-compacted concrete dam spillways, where steps are created in the construction process. The characteristics of stepped chutes and spillways have been comprehensively presented by Chanson (2002) and research into different aspects continues. Flow down a series of steps can occur as either ‘nappe’ or ‘skimming’ flow (Fig. 4.31), depending on the discharge and the step geometry. Nappe flow occurs as a succession of freely falling nappes, with a fully or partially developed hydraulic jump or decelerating supercritical flow on each step, and voids in the step spaces (Fig. 4.31a–c). Skimming flow occurs as a uniform stream over the step edges, with recirculating vortices within the step spaces (Fig. 4.31d). A gradual but rather chaotic transition occurs between the two main regimes. Nappe flow is usually associated with relatively low discharges or relatively large steps on relatively mild slopes, and skimming flow with higher discharges, smaller steps and steep slopes. Both types can occur in the same structure, but Chanson (2002) recommends that the transition condition be avoided if possible because of the load fluctuations on the structure associated with oscillations between the flow types. It is important to be able to predict the flow type for design purposes because the associated energy dissipation and flow depths are different.

Fig. 4.30 A laboratory model of a stepped chute

4.2 Spillway Structures

147

yc

h

(a) Nappe flow, fully developed hydraulic jump

l

(b) Nappe flow, partially developed hydraulic jump

(c) Nappe flow, no hydraulic jump

(d) Skimming flow

Fig. 4.31 Flow types in a stepped channel

Many empirical formulations have been presented for distinguishing between the flow types, the differences between them arising from the different data used for their determination as well as different interpretations or definitions of the types (Fig. 4.32). Chanson (2002) defines the upper limit of nappe flow as yc ¼ 0:89 h

0:4

h l

ð4:34Þ

where yc is the critical flow depth related to the unit width discharge by q = (gy3c )0.5, h is the step height and l is the step length, and the lower limit of skimming flow is given by

148

4 Spillways 3.5 3.0

Eqn (4.38)

2.5

Eqn (4.37)

skimming flow

2.0

yc /h 1.5 Eqn (4.36)

Eqn (4.35)

1.0 Eqn (4.34)

0.5

nappe flow 0 0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6 1.7

h/l Fig. 4.32 Criteria for distinguishing flow types in a stepped channel

yc ¼ 1:2 h

0:325

h l

ð4:35Þ

Conditions between these limits correspond to the transition zone. Boes and Hager (2003) define the onset of skimming flow, with no reference to a transition, which is given by yc ¼ 0:91 h

0:14

h l

ð4:36Þ

James et al. (1999) and Chamani and Rajaratnam (1999) found skimming flow to be more and more difficult to obtain with decreasing chute slope, leading to nonlinear criteria. James et al. (1999) proposed Eq. (4.37) for the upper limit of nappe flow, which agrees well with Chanson’s criterion [Eq. (4.34)] for steep slopes.   yc h ¼ 0:541 l h

1:07

ð4:37Þ

Chamani and Rajaratnam (1999) proposed Eq. (4.38) for the onset of skimming flow. h ¼ l

ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  

yc 1 yc  0:34 0:89 þ 1:5 1 h h

ð4:38Þ

4.2 Spillway Structures

149

Equations (4.37) and (4.38) are virtually coincidental, suggesting negligible transition, although James et al. observed a small transition zone for low channel slopes and a large transition for steep slopes (>*h/l = 0.7). Chanson (1996) presented an analysis to predict the onset of skimming flow, assuming the transition to occur when the air cavity beneath the falling nappe disappears. The result predicts a trend similar to Eqs. (4.37) and (4.38), although less steep at low values of h/l. Energy dissipation mechanisms are different for the two types of flow. In nappe flow, energy dissipation occurs through jet impact on the step surface and through turbulence within the hydraulic jump, while in skimming flow it occurs through mixing between the primary flow and the recirculating water in the step spaces. Various formulations for energy loss have been presented, some of which are presented below. For nappe flow, Chanson (1993, 1994) used a combination of theoretical and empirical analysis to quantify the energy loss along the chute as DH ¼1 Hmax

0:54


yc 0:275 h

3 2

þ 1:72 þ Hydam c


yc  h

0:55

!

ð4:39Þ

in which DH is the total head loss along the chute, Hmax is the energy head upstream of the structure and Hdam is the height of the dam structure. James et al. (1999) found Eq. (4.39) to overestimate the energy loss for high values of yc/h (>*1.3). It is usually useful for designing an energy-dissipating terminal structure to calculate the residual energy, as Hres = Hmax − DH. For skimming flow, Chanson (1993, 1994) proposed that the energy dissipation is given by DH ¼1 Hmax

1=3
fe  2=3 ! fe cos a þ 12 8 sin 8 sin a a Hdam 3 þ yc 2




ð4:40Þ

in which a is the channel slope and fe is the effective friction factor. The effective friction factor is influenced significantly by aeration and is difficult to predict. Chanson (2002) suggests fe = 0.20 as an average, first approximation, value for stepped chutes. Equations (4.39) and (4.40) apply for long chutes where (quasi-) uniform flow conditions are attained. Slightly modified forms of these equations were also presented for chutes controlled by underflow crest gates. Energy loss on low chutes may be considerably less than predicted by these equations. As a general rule, the rate of energy dissipation increases with dam height and decreases with increasing discharge, so the steps become less effective during flood flows. Various modifications to step geometry have been proposed in attempts to increase energy dissipation. Chinnarasri and Wongwises (2004, 2006) compared the performance of upwardly sloping steps and end sills at the ends of the steps, and found modest improvement. James et al. (2001) tested a wedge-notch step geometry with the

150

4 Spillways

step edge deflecting flow towards the centre of the chute and towards the sides on alternate steps. This geometry had little effect on energy dissipation in the nappe flow regime, but increased it considerably in the transitional and skimming regimes. Most results for stepped chutes and spillways are empirical. The limited prototype data available makes for considerable uncertainty in design, suggesting that physical modelling would be advisable. The high degree of aeration, especially with skimming flow, limits the reliability of physical modelling, however. Chanson (2002) suggests that to ensure proper scaling of air entrainment and flow resistance with Froude similitude, the scale should not be less than 1:10 to 1:12.

4.3 4.3.1

Cavitation and Aeration on Spillways and Chutes Cavitation

Cavitation occurs when cavities or ‘bubbles’ of water vapour and other liberated gases form in flowing water. These may be carried by the flow to regions of higher pressure where they implode. Severe damage can result if implosion of cavities is repeated in quick succession against a concrete or steel surface where pressures as high as 1500 MPa have been recorded (Lesleighter 1988). Dong et al. (2010) cite some impressive reports of cavitation damage. At Hoover Dam, cavitation was triggered by a bulge in the concrete tunnel lining and led to erosion through the lining and 13.7 m into the underlying rock over a length of 35 m (Warnock 1947). Similar damage occurred at the Liujiaxia Hydropower Station in China, where three 8 mm steel bars protruding through the concrete surface precipitated cavitation and led to erosion of a hole which is 100 m long and 4.8 m deep (Liu 1983). Cassidy and Elder (1984) report other cases: On a chute in the Reza Shah Kebir Project in Iran, cavitation caused a hole which eroded through the 1.5 m concrete floor of the chute and a further 1 m into the underlying rock. On the overflow spillway of the Bratsk Dam in Russia, damage included a hole of 7.5 m by 10.5 m in plan and 1.2 m deep. Kells and Smith (1991) give details of further cases. The consequences of cavitation are therefore not trivial, and the conditions under which it occurs are not extraordinary. Assessing its likelihood is essential in the design of any hydraulic structure conveying water at high velocities, and this requires understanding its origins, recognizing the structural features associated with it and being able to prevent or control it. Falvey (1990) and Kells and Smith (1991) have provided comprehensive and detailed treatment of the phenomenon and recommended design methods for its prevention. Cavities form in water if the local pressure is the same as the water vapour pressure, which depends on temperature. Cavities can therefore be induced by increasing the temperature of the water to where the vapour pressure is equal to the ambient pressure; this is ‘boiling’, which therefore takes place at different temperatures at different altitudes. More relevant here is cavitation induced by the reduction of local pressures

4.3 Cavitation and Aeration on Spillways and Chutes

151

to the vapour pressure corresponding to the ambient temperature. The reduction in pressure necessary to initiate cavitation can result from local flow velocities being sufficiently increased by small changes in boundary geometry. Cavitation problems can be anticipated where velocities exceed about 20 m/s and irregularities exist on the flow surface, such as associated with joints or poor concrete finishes. Convex curvatures, such as on spillway crests or in chute transitions, also contribute to creating conditions favourable for cavitation by reducing pressure. Predicting the conditions at which cavitation is likely to occur is not practically possible through theoretical analysis, but many empirical relationships have been developed from experiments with different boundary geometries by (for example) Ball (1963, 1975, 1976) and Johnson (1963) and presented by Cassidy and Elder (1984). These authors present graphs defining the critical velocity at which incipient cavitation can be expected for different geometries of discrete roughness features such as positive and negative vertical offsets. A commonly used indicator of cavitation potential is the ‘cavitation index’, r, which accounts for both flow velocity and pressure. This index is derived by equating the energy at two points in the flow in terms of the Bernoulli equation, i.e. Vo2 po V2 p þ þ zo ¼ þ þz c 2g c 2g

ð4:41Þ

in which V is velocity, p is pressure, g is gravitational acceleration, c is the specific weight of water and z is elevation. The subscript o represents a reference location and the unsubscripted terms apply at a position on the boundary. Assuming the elevation terms to be relatively insignificant, Eq. (4.41) can be rearranged to 

V Vo

Cp ¼ 1

2

¼

p

po qVo2 2

ð4:42Þ

where q is the fluid density and Cp is the pressure coefficient or Euler number. For cavitation conditions, the boundary pressure p becomes the vapour pressure pv and the pressure coefficient takes its minimum value. The resulting parameter is the cavitation index expressed as r¼

po

pv qVo2 2

ð4:43Þ

which can also be expressed as r¼ in which H is the pressure head (p/c).

Ho

Hv Vo2 2g

ð4:44Þ

152

4 Spillways

Different reference locations have been used by different researchers, such as the undisturbed approach condition, the top of the turbulent boundary layer, the top of roughness elements or protrusions potentially causing cavitation and the level of an offset potentially causing cavitation. Using the undisturbed approach conditions will usually be conservative. It is important to note that both the reference and vapour pressures are absolute values, i.e. they should include atmospheric pressure and the gauge pressure. (Atmospheric pressure at sea level is 101.3 kPa (or 10.3 m of water), decreasing to about 81.0 kPa at 2000 m above sea level and about 61.5 kPa at 4000 m above sea level. The vapour pressure of water is 1.23 kPa at 10 °C, 2.33 kPa at 20 °C and 4.23 kPa at 30 °C.) The pressure should also include any centrifugal components present and the slope term for steep chutes, i.e.   y V2 p ¼ pa þ c y cos h  g r

ð4:45Þ

in which pa is the atmospheric pressure, y is the flow depth, h is the bed slope, r is the radius of curvature of the bed and the + and − signs correspond to concave and convex surfaces, respectively. The U S Army Corps of Engineers (1995) has consolidated much of the experimental work and presented curves defining incipient cavitation for different situations; those for vertical offsets into and away from the flow are shown in Fig. 4.33. Other curves presented by the U S Army Corps of Engineers indicate 2.5

D

Cavitation Index

2.0

1.5

D 1.0

0.5

0 0.1

1

10

100

D (mm) Fig. 4.33 Cavitation characteristics for abrupt offsets into and away from the flow (adapted from U S Army Corps of Engineers 1995)

4.3 Cavitation and Aeration on Spillways and Chutes

153

Table 4.1 Criteria for prevention of cavitation damage (adapted from Falvey 1983) Cavitation index

Design measures

>1.80 0.25 − 1.80

No cavitation expected Protection by surface treatment, including grinding of roughness elements to specified chamfers Protection by modifying chute profile (e.g. reducing curvature) or including aeration devices Protection by including aeration devices Protection probably not possible

0.17 − 0.25 0.12 − 0.17 9.0, a strong jump forms. Energy dissipation is effective, up to 85%, but the behaviour is unsatisfactory. The incoming velocity is very high and the difference between upstream and downstream water depths is large. The fast jet intermittently entrains slugs of water at the face of the jump, causing waves and a

6.2 The Hydraulic Jump

185

rough water surface downstream. The water surface becomes increasingly rough as Fr1 increases and is usually unacceptable for values greater than about 13. All of these jump types are encountered in energy dissipation design, and some require the use of structural features to ensure satisfactory performance. The weak jump requires no such features and the design must just ensure an appropriate downstream water level and provide for the length of the jump, which is relatively short. The oscillating jump presents wave problems. Baffle blocks and other commonly used appurtenances have little effect on waves. The problem is generally not serious for wide flows, such as on aprons below dams, but can be serious in canals, where the width is small and waves can travel long distances, causing damage to unlined banks and riprap and surging at measuring devices. Variations in channel geometry and alignment cause reflections which can dampen or intensify waves. Special types of stilling basin (such as the United States Bureau of Reclamation (USBR) Basin IV described in Sect. 6.3) are required for this condition or wave suppressors can be used. Peterka (1978) recommends two wave suppressor designs. The raft-type wave suppressor, shown in Fig. 6.1, is the most effective type when additional submergence is not possible. The rafts are porous panels set at fixed heights and spacing. Surges over the rafts are dissipated by the flow through the holes. In experiments, the wave height was found to be reduced by about 50% by each raft. The rafts are set with their top surfaces at the mean water surface and are made deep enough so that the wave troughs do not separate from the underside. The ratio of open area to total area of raft is in the range of 1:6–1:8. The spacing between rafts must be at least three times the raft width. The underpass-type wave suppressor, shown in Fig. 6.2, is the most effective type, but is more expensive than the raft type. It consists of a horizontal roof with a vertical headwall high enough to force all the flow to pass beneath the roof. The height of the roof may be set to accommodate a wide range of water levels and

w

3w min

Fig. 6.1 Raft-type wave suppressor (adapted from Peterka 1978)

186

6 Energy Dissipation Structures

Fig. 6.2 Underpass-type wave suppressor

wave heights. The degree of wave suppression for any particular roof height is determined by the length of the roof. Peterka (1978) provides some design recommendations for the underpass-type wave suppressor, but these are based on limited experimental results. Practical design of either the raft or underpass structures usually requires specific model studies. The steady jump is the most desirable and wherever possible the incoming flow should be induced within the appropriate range of Froude number. Baffles and sills are sometimes incorporated in stilling basins for this condition to shorten the length required and to stabilize the jump position. This is particularly important where the discharge is not precisely known, or is expected to vary. The effects of these features are discussed below and their incorporation in standard designs is presented in Sect. 6.3. For Fr1 greater than about 10, the jump becomes very sensitive to tailwater depth and a tailwater greater than the conjugate of the inflowing depth is advisable to ensure that the jump is kept within the stilling basin. For such high Froude numbers, the difference between the conjugate depths is large and high training walls are required. For high spillways, stilling basins are therefore not usually economically competitive with other types of energy dissipators, such as buckets or splitters. Analysis of the hydraulic jump for design purposes is necessary to enable prediction of the sequent depth ratio, the energy loss, the location and length of the jump, and the water surface profile through the jump. These issues have been the subject of many experimental investigations over many years. Comprehensive reviews of progress have been provided by Carollo et al. (2007, 2009), Hager et al. (1990) and Schulz et al. (2015). The sequent depth ratio can be predicted by applying conservation of momentum and continuity. The upstream (1) and downstream (2) flow conditions are related by M1

M2 ¼

P c

ð6:1Þ

in which M is the momentum function, P is the force applied to the flow between the sections in the upstream direction and c is the unit weight of water. For a simple hydraulic jump (with no features to apply a force to the flow) on a horizontal bed, and assuming the effect of boundary shear through the jump to be

6.2 The Hydraulic Jump

187

Fig. 6.3 Simple hydraulic jump on horizontal bed

Lj Lr

y

y2

x

y1 1

2

negligible, Eq. (6.1) can be developed to obtain a relationship between the upstream and downstream (sequent) flow depths (see Fig. 6.3). In this case M1 ¼ M2 which, for rectangular sections, can be written as q2 y2 q2 y2 þ 1¼ þ 2 2 gy2 2 gy1 By substituting q = V1 y1, this can be developed to give y2 1 ¼ y1 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 8Fr21

1



ð6:2Þ

1



ð6:3Þ

and by substituting q = V2 y2, y1 1 ¼ y2 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 8Fr22

Hager and Bremen (1989) found that the sequent depth ratio is also influenced by the inflow Reynolds number (Re1 = V1y1/m, where V1 is the inflow velocity and m is the kinematic viscosity). They proposed correction to Eq. (6.3) in terms of Fr1, Re1 and y1/b where b is the channel width, but the error in ignoring this effect is unlikely to exceed about 3%. The neglect of boundary shear in the development of Eqs. (6.2) and (6.3) results in exaggerated estimates of y2/y1 (e.g. Hughes and Flack 1984; Corollo et al. 2007, 2009; Schulz et al. 2015). The ratio y2/y1 reduces consistently with increasing bed roughness, the effect becoming more pronounced with increasing Fr1. Various formulations have been proposed for prediction, all based on laboratory

188

6 Energy Dissipation Structures

information. For example, Corollo et al. (2007) proposed that Eq. (6.2) be adjusted according to y2 1 ¼ y1 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 8ð1 bÞFr21

1



ð6:4Þ

with  0:75 ! 2 ks b ¼ arctan 0:8 p y1

ð6:5Þ

in which ks is d50 size of the bed roughness particles. Equation 6.5 was formulated and tested with data for ks up to 32 mm, ks/y1 in the range 0 to 2.025 and Fr1 up to about 10. Once the flow conditions on either side of the jump are known, the energy loss can easily be found as the difference between the specific energy values. By solving the energy and momentum relationships simultaneously, the following general expression for energy loss (DE) can be derived: DE ¼

ð y2 y1 Þ 3 4y1 y2

ð6:6Þ

Peterka (1978) presents experimental results which agree very closely with the predictions of Eq. (6.6). The energy loss varies from a negligible amount for Fr1 less than about 2 up to around 85% for Fr1 = 20. The length of a hydraulic jump is an important design parameter because it determines the size of stilling basin required or, in the absence of a basin, the location beyond which bed protection is not required. The length cannot be predicted theoretically, and many experimental studies have been carried out to formulate empirical prediction relationships (e.g. Peterka 1978; Mehrotra 1976; Hager et al. 1990; Carollo et al. 2007, 2012; Schulz et al. 2015). The length of a hydraulic jump (Lj) is difficult to measure because the end is not distinct and specifying its location involves some degree of subjectivity; the end of the jump may be assumed to be where the flow depth reaches its maximum or where the flow depth becomes approximately uniform. The length of the roller (Lr) may be more objectively defined and measured, but the difference is not great for practical purposes. The length of the jump, as well as the sequent depth, is reduced by bed roughness (Hughes and Flack 1984; Carollo et al. 2007). In most prediction formulations, the length is normalized in terms of the upstream flow depth and expressed as a function of the upstream Froude number (Fr1) or the ratio of sequent depths (y2/y1). Peterka (1978) used a large data set to produce a graphical relationship for Lj/y1(Fr1) for a wide range of Fr1 (Fig. 6.4); this can be approximated by

6.2 The Hydraulic Jump

189

180 160

Peterka (1978)

Lr /y1 or Lj /y1

140 Equation (6.8)

120 100

Equation (6.9)

80 Equation (6.10)

60 40

Equations (6.11) & (6.13)

20 0

Equation (6.12) 0

5

10 Fr1

15

20

Fig. 6.4 Comparison of predictions of jump or roller length on a smooth bed

Lj ¼ y1

0:13Fr21 þ 11Fr1

12

ð6:7Þ

Hager et al. (1990) proposed different relationships for Lr/y1 for different ranges of Fr1. For 2.5 < Fr1 < 8 Lr ¼ 8ðFr1 y1

1:5Þ

ð6:8Þ

For Fr1 > 8, they found the jump length to depend also on the aspect ratio, y1/b. For y1/b < 0.10 Lr ¼ y1

  Fr1 12 þ 160 tanh 20

Lr ¼ y1

12 þ 100 tanh

ð6:9Þ

and for y1/b > 0.10 

Fr1 12:5



ð6:10Þ

Using their own data together with those of Hughes and Flack (1984) and Hager et al. (1990), Carollo et al. (2007) proposed three alternative formulations, i.e.  Lr y2 ¼ 4:616 y1 y1 Lr 2:244 ¼  1:272 y1 y1 y2

1



ð6:11Þ ð6:12Þ

190

6 Energy Dissipation Structures

both of which require prior adjustment for bed roughness if required and explicitly accounting for bed roughness, Lr ¼ y1

   ks ðFr1 6:525 exp 0:60 y1

1Þ

ð6:13Þ

Equations (6.8) to (6.13) for smooth beds (ks = 0) are plotted together with Peterka’s (1978) curve in Fig. 6.4. Peterka’s curve indicates longer jumps than the others, mainly because of his different definition of length (Hager and Li (1992) have also suggested that Lj/Lr is approximately equal to 4/3). The Hager et al. and Carollo et al. (6.11) and (6.12) relationships give similar predictions for Fr1 up to about 16, with Carollo et al. (6.12) departing from these for Fr1 greater than about 10. The difference between Eqs. (6.9) and (6.10) of Hager et al. shows the effect of channel width to be considerable and increasing with Fr1 above 8. Relationships for Lj/y2 = f(Fr1) have also been proposed (e.g. Peterka 1978; Mehrotra 1976; Hughes and Flack 1984). These show Lj/y2 to be approximately constant for Fr1 in the desirable range of 4.5–9. The data of Peterka (1978) (see also Fig. 6.14) show Lj/y2  6.1 in this range, while the relationships of Hager et al. (1990) (Eqs. (6.8) and (6.9)) and Carollo et al. (2007) (Eqs. (6.11) and (6.13)) suggest Lr/y2  4.1. Knowledge of the water surface profile through the jump is also important in stilling basin design because it will determine the height of the retaining walls. Valiani (1997) produced an analytical solution describing the surface profile, i.e. y ¼ y1

 3 ! !1=3 y2 x 1 1þ Lj y1

ð6:14Þ

in which y is the water depth within the roller, x is the distance from the entrance to the roller and Lj is the length of the jump (the distance between the sequent depths), assumed to be the same as Lr (Fig. 6.1). This relationship was shown by Castro-Orgaz and Hager (2009) to be supported by data for Fr1 in the range of about 2.5 to about 9, and considered by them to give reasonable results for practical purposes. Chow (1959) presented a dimensionless graph for describing the profile based on the experimental results obtained by Bakhmeteff and Matzke (1936). This shows a similar trend with Fr1 as Eq. (6.14), but a more gradual initial rise of the surface profile for all values of Fr1. The jump occurs at the location where the sequent depth relationship is satisfied (Eqs. (6.2) or (6.3)). However, the length of the jump between the sequent depths may need to be accounted for if accurate location is required. The flow depth on at least one side of a hydraulic jump will be nonuniform, and its profile must be computed to enable the jump to be located. If the downstream (subcritical) flow is uniform, the jump will begin at the location on the upstream (supercritical) profile where the sequent depth of the uniform flow occurs. The jump length can then be calculated from this depth according to one of the above relationships to define the

6.2 The Hydraulic Jump

191

M3 sequent A

Lj M2

y2 M3

B

y1

Fig. 6.5 Location of a hydraulic jump between nonuniform profiles

location of the end of the jump. If the upstream (supercritical) flow is uniform, then the jump will end at the location on the downstream (subcritical) profile where the sequent depth of the uniform flow occurs. The jump length is then calculated using the upstream, uniform, flow depth to locate the beginning of the jump. If the flow is nonuniform both upstream and downstream, then both nonuniform profiles must be computed, as well as a profile of the sequent depths of one of them. An initial estimate of the jump location is then given by the intersection of one of the water surface profiles with the sequent profile of the other. The upstream flow depth at this location enables a jump length to be calculated, which is then fitted between the water surface profile and the sequent profile. This will define a location different from the initial estimate, so the jump length calculation may need to be refined. Figure 6.5 shows an example for a jump between an M3 profile and an M2 profile (as discussed by Chow 1959). The intersection of the M2 profile with the sequent M3 profile (point A) defines a supercritical flow depth on the M3 profile (point B) for calculating the jump length (Lj). The jump length is then fitted between the M3 sequent curve and the M2 profile to locate the sequent depth pair at the beginning and end of the jump. Example 6.1 Water is released from a storage facility into a long, 4.5 m wide, rectangular, concrete-lined channel on a slope of 0.00040. The design discharge is 15.0 m3/s, resulting in a velocity at the entrance to the channel of 12.0 m/s. Determine a. the flow depths immediately upstream and downstream of the hydraulic jump in the channel, b. the energy dissipated in the jump, c. the location of the jump in the channel, d. the length of the jump and roller and e. the water surface profile through the jump.

192

6 Energy Dissipation Structures Solution

W = 4.5 m

Q = 15 m3/s V = 12 m/s So = 0.00040

1

Determine the flow depth at section 1 from the continuity equation, Q ¼ AV

Therefore so

Q ¼ 15 m3 =s A ¼ Wy1 ¼ 4:5y1 V ¼ 12 m/s

15 ¼ 4.5y1
12 y1 ¼

15 ¼ 0:278 m 4:5
12

Determine the uniform flow depth in the channel from the Manning and continuity equations, A 2=3 1=2 R So n A ¼ Wyo n ¼ 0:013 for concrete A Wyo R¼ ¼ P W þ 2yo So ¼ 0:00040

Q¼

Therefore Q¼

 2=3 Wyo Wyo 1=2 3So n W þ 2yo

i.e. 15 ¼

 2= 3 1 4:5 yo 4:5 yo 0:00040 =2 0:013 4:5 þ 2 yo

from which yo ¼ 2:06 m Determine the sequent depth of y1 from Eq. (6.2),

6.2 The Hydraulic Jump

193 y2 1 ¼ y1 2

Therefore

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 8Fr21

1



V 12 Fr1 ¼ pffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 7:27 gy1 9:8
0:278 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1 þ 8
7:27 2 ¼ 2:72 m

y2 ¼ 0:278

 1

a. y*2 > yo, so the hydraulic jump is some distance along the channel, and at the jump, the subcritical flow will be yo and the supercritical flow will be its sequent depth, given by Eq. (6.3), i.e. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  yo sequent 1 1 þ 8Fr2o 1 ¼ 2 yo Vo 15=ð4:5
2:06Þ Fro ¼ pffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:360 gyo 9:8
2:06 Therefore

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi 1 þ 8
0:360 2 ¼ 0:440 m

yo sequent ¼ 2:06

1



b. The energy dissipated by the jump can be calculated as the difference between the specific energy (E) values before and after it, or from Eq. (6.6) which was derived from this, i.e.  3 yo yo sequent DE ¼ 4
yo sequent
yo ð2:060 0:440Þ3 4
0:440
2:06 ¼ 1:173 m

¼

c. The jump will be located where the sequent of the uniform flow depth occurs on the M3 profile from the entrance to the channel,

M3 y1 = 0.278 m

yo sequent = 0.440 m

yo = 2.060 m

x The distance x can be found using the Direct Step Method, i.e. by solving

194

6 Energy Dissipation Structures

Dx ¼

DE So Sf

sequentially for increments of y between y1 and yo sequent. Sf is the average energy gradient for the increment, as calculated for each location from the Manning equation, i.e. Sf ¼

V 2 n2 4 R =3

Using 10 flow depth increments, x is found to be 54 m. (Similar to Example 1.8.) d. The length of the jump or roller can be related to the supercritical Froude number by Eqs. (6.7)–(6.12). Vyo sequent Fryo sequent ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi g yo sequent

Q W yo sequent 15 ¼ 4:5
0:440 ¼ 7:58 m/s

Vyo sequent ¼

Therefore 7:58 Fryo sequent ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9:8
0:440 ¼ 3:65 The hydraulic jump length, by Eq. (6.7), is therefore Lj ¼ yo sequent

0:13 Fr2yo sequent þ 11 Fryo sequent

12

Therefore  Lj ¼ 0:440 0:13
3:652 þ 11
3:65 ¼ 11:6 m

 12

Figure 6.4 indicates that for Fr1 = 3.65, all the equations for the roller length ((6.8) to (6.12)) give similar results. Using Eq. (6.8), for example, Lr yo sequent Therefore

 ¼ 8 Fryo sequent

 1:5

6.2 The Hydraulic Jump

195 Lr ¼ 0440
8ð3:65 ¼ 7:6 m

1:5Þ

e. The water surface profile through the jump is given by Eq. (6.14), i.e.

y yo sequent

¼

1þ



yo yo sequent

3

! !1= 3 x 1 Lj

Using Lj = Lr = 7.6 m as calculated above, the water surface profile is as shown below:

Flow Depth (m)

2.5 2.0 1.5 1.0 0.5 0.0

6.2.2

-2

0

2

4 Distance (m)

6

8

10

Controlled Hydraulic Jumps

A hydraulic jump occurs if the upstream supercritical flow and downstream subcritical flow satisfy Eq. (6.1). The upstream conditions are controlled by the discharging structure and are usually not able to be manipulated for the design. Ensuring a jump therefore requires that the subcritical flow conditions, controlled from further downstream, are appropriate. For the simple hydraulic jump in a horizontal channel, the required downstream depth is given by Eq. (6.2), and is denoted y*2 hereafter. If the existing downstream conditions do not meet this requirement, then additional measures must be designed to ensure the jump takes place where required. The different possible conditions, using an ogee spillway structure, for example, are illustrated in Fig. 6.6. The ideal situation is where the tailwater (yt) is exactly equal to y*2, as shown in Fig. 6.6a. The momentum equation is satisfied and the jump forms immediately and unaided. This condition is unlikely if water is being released into a natural channel,

196

6 Energy Dissipation Structures

yt = y2* y1 (a) yt = y2*

y2*

yt

yt (b) yt < y2*

y2*

yt

yt

yt

(c) yt > y2* Fig. 6.6 Effect of tailwater on hydraulic jump formation

but could possibly be contrived for a fixed discharge in an artificial canal by designing the geometry appropriately. In practice, there may be inaccuracy in some design assumptions and some feature (such as a row of baffle blocks) is usually installed to control the jump position and ensure that it does not move downstream. If the tailwater is less than y*2, an unaided jump would occur some distance downstream, as shown in Fig. 6.6b. The supercritical flow would assume an M3 profile, in which the flow depth increases downstream until it reaches the sequent depth of yt, where the jump occurs. The tailwater level can be increased using a weir, sill or an upward step created by excavating a basin just downstream of the structure. As indicated by Eq. (6.1), any feature that imposes an upwardly directed force against the flow reduces the downstream momentum function value, and hence flow depth required for a jump to occur. If the tailwater is greater than y*2, the hydraulic jump is forced upstream and may be submerged, as shown in Fig. 6.6c. Although this is a safe condition because the position is easily fixed, the energy dissipation is less effective than for an unsubmerged jump. An unsubmerged jump can be ensured by raising the level of the apron, thereby creating a downward step to the tailwater. Another design option for the case where the tailwater is greater than the required conjugate depth is to install a sloping entrance to the stilling basin or apron. The jump will then advance up the slope to the position where momentum balance is achieved.

6.2 The Hydraulic Jump

197

In practice, the relationship between the sequent depth required for jump formation and the tailwater depth determined by downstream conditions may vary with discharge. It would be highly unlikely for yt to be equal to y*2 over the whole range of expected discharges. It would be quite common, however, for yt to be either greater or less than y*2 over the entire range. In these cases, the designs discussed above would be satisfactory for a wide range of discharges, although ideal only for the particular value used for the design. It is possible, however, for the relationship between yt and y*2 to reverse within the expected range of discharges. If y*2 > yt at low discharges and y*2 < yt at high discharges, a combination of the designs described above would be effective, i.e. the provision of a sill or upward step to raise the tailwater at low discharges and a sloping apron to allow the advance of an unsubmerged jump at high discharges. If y*2 < yt at low discharges and y*2 > yt at high discharges, it is not possible to design a satisfactory configuration for all discharges. The most practical approach would be to provide an upward step to raise the tailwater at high discharges and accept a submerged jump at low discharges. Commonly used features used to reduce the tailwater depth requirement are sills, baffle blocks, abrupt upward steps and sloping approach surfaces. These features add stability to the jump and also increase the energy dissipation. Jumps associated with such features are classified according to their locations. In particular, Type A jumps are wholly located upstream or downstream of the controlling feature and Type B jumps extend across the feature. Analyses for these features are presented in the following sections. Sloping Surfaces Many designs for energy dissipators induce hydraulic jumps on sloping surfaces, such as at the ends of spillways or drop structures or sloping entrances to stilling basins. A sloping entrance or apron assists in stabilizing the jump location and ensuring its occurrence over a range of discharges. The design again requires knowledge of the sequent depth ratio, the length of the jump and its location. The momentum analysis for sequent depths is complicated considerably if the jump occurs in a channel with a slope steep enough that gravitational forces need to be considered. Evaluation of the forces acting on the free body is then difficult and empirical input is essential. The weight component of the free body is difficult to estimate because the length and shape of the jump are not well defined. A further complication is that significant aeration occurs, changing the specific weight of the water, and hence affecting estimation of both the free body weight component and the pressure forces. A number of distinct situations are recognized for hydraulic jumps associated with sloping channel beds (e.g. Chow 1959; Henderson 1966; French 1985), as shown in Fig. 6.7. These situations are characterized by the position of the jump, defined by its length, Lr, the distance of its beginning from the change in slope, l and the relationship between the sequent depths, y1 and y2. In the illustrated cases, yt is the tailwater depth, and y1 and y2 are the upstream and downstream sequent depths. The sequent depth for y1 on a horizontal bed (i.e. as given by Eq. (6.2)) is denoted as y*2. If the jump begins at or after the end of the

198

6 Energy Dissipation Structures Lr

y1

y2 = y2* = yt

θ

Type A

Lr

Lr l

y1

y1

y2 = yt θ

Type E

Type B

y2 θ

Lr Lr y1

y2 = yt

Type C y2

θ y1

Lr

y2

y1

Type F

yt

Type D

θ

Fig. 6.7 Types of hydraulic jumps in sloping channels (adapted from French 1985)

slope then y2 = y*2 = yt and the jump will be Type A. In this case, the usual analysis and results apply. If yt is greater than that needed for a jump on a horizontal bed, the jump will advance onto the slope and the jump will be Type B, C or D. Here the slope introduces a weight component which enhances the upstream value of M, and also provides a force to affect the change in flow direction. If yt is insufficient to force the entire jump onto the slope, the jump is Type B. If yt is just great enough for the end of the jump to coincide with the end of the slope, then the jump is Type C. For greater values of yt, the jump will occur completely on the slope as Type D. Ohtsu and Yasuda (1991) found that for these types if yt/y1 > 3.0 and h > about 20° a surface roller does not form and the upstream supercritical flow plunges directly into the subcritical flow. Type E jumps occur on long slopes with no break to horizontal. Type F jumps are unusual, but may occur below some drop structure designs. The analysis for Type A jumps is straight forward, and the results already presented for horizontal channels apply. No analytical solution is yet available for jumps of Types B, C and D but useful empirically based relationships have been developed. Kindsvater (1944) developed the following equation for the sequent depth ratio for jumps completely on the slope, i.e. Types C and D.

6.2 The Hydraulic Jump

199

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 3h cos 1 þ 8Fr21 1 2N tan h

y2 1 ¼ y1 2 cos h

1

!

ð6:15Þ

in which N is an empirical factor related to the jump length and h is the channel slope. Equation (6.15) can be written as y2 =

y1

¼

1 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 8G21

1



ð6:16Þ

in which y1′= y1/cos h, and with G21 ¼ C21 Fr21

ð6:17Þ

and C21 ¼

cos3 h 1 2N tan h

ð6:18Þ

N has been found to be primarily a function of h and Peterka (1978) presented a graphical relationship between N and tanh for 5° < h < 16°, which can be represented by N¼

0:938 lnðtan hÞ þ 0:258

ð6:19Þ

Rajaratnam (1967) related C1 directly to h as C1 ¼ 100:027h

ð6:20Þ

with h in degrees. For the same situation, Ohtsu and Yasuda (1991) proposed  y2  ¼ 0:077 h1:27 þ 1:41 ðFr1 y1

1Þ þ 1

ð6:21Þ

which agrees well with Eqs. (6.16)–(6.20) over practically realistic ranges of Fr1 and h. Various empirical formulations have been presented for the sequent depth ratio for Type B jumps and have been reviewed and tested by Carollo et al. (2011). Solution of all the equations requires knowledge of the location of the beginning of the jump, in terms of its distance from the junction between the sloping and horizontal beds (l) or the height of this point above the horizontal bed level. For most design applications, this calculation is not necessary, however. The sequent depths are determined by the upstream and downstream controls, and the position of the

200

6 Energy Dissipation Structures

jump adjusts to these. If the tailwater level necessary to hold a jump against a specified inflow depth is required, a value can be found by trial that ensures a suitable jump type and location. The jump location is defined by its length (Lj) and the distance of its beginning from the slope junction (l). Ohtsu and Yasuda (1991) proposed equations for the length of the jump for 4 < Fr1  14 and for two ranges of slope. For h < 19° Lj ¼ 5:75 tan h þ 5:70 y2

ð6:22Þ

and for 19° < h < 60°  Lj yt ¼ 4:6   y2 y2



1 þ 5:7

ð6:23Þ

Equation (6.22) is in accordance with the graphical relationship of Peterka (1978) over the specified range of Fr1, but Peterka’s graph indicates considerable dependence on Fr1 beyond this range. He presented no data for slopes greater than 17°. Ohtsu and Yasuda (1991) showed Eq. (6.23) to represent roller length (Lr) and jump length (Lj) data equally well. Ohtsu and Yasuda (1991) proposed Eq. (6.24) for the distance l, applicable for the full range of h up to 60°, 6  Fr1  14 and 1.1  yt/y*2  3.0. l ¼ y2

2:3 ðtan hÞ0:73

0:8

!

yt y2

1

0:75

ð6:24Þ

This agrees well with the graphical relationship of Peterka (1978). In these analyses, it is usually assumed that y1 will not vary along the slope over the distance within which the hydraulic jump may be expected. This is a reasonable assumption for relatively high head structures. A suggested application of the relationships for describing the characteristics of a hydraulic jump associated with a transition from a sloping to a horizontal channel proceeds as follows: – Calculate y1, yt and y*2. y1 is the flow depth upstream of the jump, and is determined by the upstream control feature. yt is the tailwater depth, determined from the downstream control; if this is a required quantity for the design, a trial value is assumed and adjusted until a value resulting in a satisfactory condition is produced. y*2 is the sequent depth of y1 as would occur on a horizontal surface (i.e. for a Type A jump), given by Eq. (6.2). – Compare yt with y*2. If yt < y*2, then a Type A jump will occur at some location further downstream. If yt > y*2, then the jump will occur at least partially on the slope. – If yt > 3y*2, then there will be no distinct roller and the upstream supercritical flow will plunge as a jet directly into the tailwater.

6.2 The Hydraulic Jump

201

– If yt < 3y*2, then an unsubmerged jump will occur. Calculate the length of the jump (Lj) from Eq. (6.22) if h < 19° or Eq. (6.23) if 19° < h < 60°. Calculate the location of the beginning of the jump (l) from Eq. (6.24). – Compare Lj with l. If Lj > l, then the jump is Type B, and is now fully described. If Lj = l, then the jump is Type C and is also now fully described. If Lj < l, then the jump is Type D. – For jump Type D, if h > 19°, then there will be no distinct roller and the upstream supercritical flow will plunge as a jet directly into the tailwater. If h < 19°, then the sequent depth (y2) is calculated from Eq. (6.21) or from Eqs. (6.16) to (6.19). The end of the jump will be located where y2 occurs on the subcritical S1 profile. The energy loss is best determined by calculating the energy values upstream and downstream of the jump relative to the same datum and taking the difference. For Type A jumps, Eq. (6.6) may be used. Example 6.2 The channel in Example 6.1 is modified to have an initial steep slope leading to a horizontal reach. A control structure further downstream induces a flow depth of 4.30 m in the vicinity of the slope transition. Locate the hydraulic jump in the channel and determine the associated energy loss a. if the initial slope is 0.25 (14°) and b. if the initial slope is 0.10 (5.7°). Solution a. Slope = 0.25 Assume the jump begins close enough to the exit that the supercritical flow at the toe of the jump is approximately the same as at the exit. The analysis could be refined if necessary by computing the gradually varied profile before the jump. Determine if the jump occurs on the horizontal (Type A) or sloping (Type B or C or D) surface: yt = 4.30 m (given), y*2 = 2.72 m (from Example 6.1), yt > y*2, so the hydraulic jump occurs on the slope. Determine if the jump is submerged or unsubmerged: yt < 3y*2 (4.30 < 3
2 .72 = 8.16 m), so an unsubmerged jump will occur on the slope. Calculate the length of the jump: The slope of 0.25 (14°) is less than 19° so the jump length is given by Eq. (6.22), i.e. Lj ¼ 5:75 tan h þ 5:70 y2

202

6 Energy Dissipation Structures So

Lj ¼ 2:72ð5:75 tan 14 þ 5:70Þ ¼ 19:4 m

Calculate the location of the beginning of the jump, from Eq. (6.24), i.e. !

2:3

l ¼ y2

0:8

ðtan hÞ0:73

0:75

yt y2

1

!

4:30 2:72

So l ¼ 2:72

2:3

0:8

ð0:25Þ0:73

1

0:75

¼ 10:0 m

Lj > l, so the jump is Type B and the water surface profile is

19.4 m 10.0 m 0.28 m

4.30 m 14o

1

2

The energy loss can be determined by relating the total energy values at sections 1 and 2, with the datum selected at the horizontal bed level, Loss ¼ H1 H2     V2 V2 y2 þ 2 ¼ y1 cos h þ 1 þ l tan h 2g 2g   12:02 ¼ 0:28
cos 14o þ þ 10:0 tan 14o 2
9:8 ¼ 10:11 4:33 ¼ 5:78 m

0

@4:30 þ



15=ð4:5
4:30Þ2 2
9:8

b. Slope = 0.10 As for the slope of 0.25, an unsubmerged jump will occur on the slope. Calculate the length of the jump: The slope of 0.10 (5.7°) is less than 19° so the jump length is given by Eq. (6.22), i.e.

1 A

6.2 The Hydraulic Jump

203

Lj ¼ 5:75 tan h þ 5:70 y2 So Lj ¼ 2:72ð5:75 tan 5:7 þ 5:70Þ ¼ 17:1 m Calculate the location of the beginning of the jump, from Eq. (6.24), i.e. 2:3

l ¼ y2

ðtan hÞ0:73

0:8

!

0:75

yt y2

1

!

4:30 2:72

So 2:3

l ¼ 2:72

0:73

¼ 20:9m

ð0:10Þ

0:8

1

0:75

Lj < l, so the jump is Type D. The sequent depth ratio is most easily obtained from Eq. (6.21), i.e.  y2  ¼ 0:077h1:27 þ 1:41 ðFr1 y1

1Þ þ 1

So   y2 ¼ 0:278 0:0775:71:27 þ 1:41 ð7:27 ¼ 3:96 m

1Þ þ 1

Alternatively, using Eq. (6.16), y2 =

y1

1 2

¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 8G21

 1

with =

y1 ¼

y1 0:278 ¼ ¼ 0:279 cos h cos 5:7

with G21 given by Eqs. (6.17) and (6.20), i.e. G21 ¼ C21 Fr21

C1 ¼ 100:027h ¼ 100:027
5:7 ¼ 1:43

So G21 ¼ 1:432 7:272 ¼ 108 Therefore

204

6 Energy Dissipation Structures  1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 8
108 y2 ¼ 0:279 2

1

¼ 3:96 m





Alternatively, using Eq. (6.16) with G21 given by Eqs. (6.17), (6.18) and (6.19), i.e. G21 ¼ C21 Fr21 C21 ¼

1

cos3 h 2N tan h

N ¼ 0:938 lnðtan hÞ þ 0:258 ¼ 0:938 lnðtan 5:7 Þ þ 0:258 ¼ 2:42 So C21 ¼

1 ¼ 1:91

cos3 5:7 2
2:42 tan 5:7

and G21 ¼ 1:91
7:272 ¼ 101 Therefore  1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y2 ¼ 0:279 1 þ 8
101 2 ¼ 3:83 m

1





The three results for y2 are similar; accept y2 = 3.96 m. The water surface profile is 20.9 m 17.1 m

3.96 m 0.28 m

4.30 m

5.7o 1

2

The energy loss can again be determined by relating the total energy values at Sects. 6.1 and 6.2, with the datum selected at the horizontal bed level,

6.2 The Hydraulic Jump

205

Loss ¼ H1 H2     V2 V2 y2 þ 2 ¼ y1 cos h þ 1 þ l tan h 2g 2g   2 12:0 þ 20:9
tan 5:7 ¼ 0:278
cos 5:7 þ 2
9:8 ! ð15=ð4:5
3:96ÞÞ2 3:96 þ 2
9:8 ¼ 9:71 4:00 ¼ 5:71 m

Sills A sill is a transverse, thin, vertical obstruction extending over the full flow width. The effect of sills on hydraulic jump performance has been investigated by (amongst others) Rajaratnam and Murahari (1971), Ranga Raju et al. (1980), Ohtsu (1981), Ohtsu et al. (1996) and Hager and Li (1992). Following from previous work, Hager and Li (1992) defined different types of jumps associated with sills, depending on the height of the sill (s), its distance from the toe of the jump (Ls) and the tailwater depth (y2) (Fig. 6.8). The Type A jump is similar to the simple jump, with the sill located at the end of the surface roller. The sill is relatively small and has little effect on the jump location or sequent depth ratio. A Type B jump occurs with a slightly lower tailwater, allowing the jump to move downstream and the deflection of the bottom current by the sill causing a small standing wave just downstream of the sill. A further reduction in tailwater depth allows a second surface roller to form. This is regarded as the limiting condition for practical energy dissipation design, and is designated the Type Bm (B minimum) jump. A Type C jump with a pulsating standing wave is also recognized, but should be avoided in design. An extreme situation with a large disturbance wave over the sill and flow returning to supercritical downstream can also occur. Hager and Li (1992) (with some reliance on data provided by Bretz (1987)) proposed accounting for the effect of the sill on the sequent depth ratio by applying an empirical correction to the ratio for the simple hydraulic jump (Eq. (6.2)), i.e. Y¼

y2 ¼ Y y1

DYf

DYs

ð6:25Þ

in which Y* is the sequent depth ratio for a simple jump, as calculated by Eq. (6.2), DYf is a correction to account for wall friction (if applicable) and DYs is a correction to account for the effect of the sill. The correction DYs is given by DYs ¼ 0:7 S0:7 þ 3Sð1

KÞ2 for K [ 0:5

ð6:26Þ

in which S = s/y1, K ¼ Ls =Lr . Lr is the roller length of a simple hydraulic jump given by Eqs. (6.8) or (6.9) or (6.10). This equation shows that the sequent depth

206

6 Energy Dissipation Structures

y1

y2

Type A

y2

Type B

y2

Type Bm

s Ls

y1

s Ls

y1

s Ls

Fig. 6.8 Hydraulic jumps controlled by sills

ratio decreases with increasing sill height and also depends on the distance between the sill and the toe of the jump, the effect increasing with decreasing distance. A typical design problem would be to determine the sill height and location to force a jump to occur between known entry (y1) and tailwater (y2) depths. Because Eq. (6.26) includes both the sill height and its location (Ls), one of these needs to be specified for a design, and the other calculated from Eqs. (6.25) and (6.26) (DYf would be negligible in most cases.) The sill height should be constrained by the upper limit for appearance of a Type Bm jump, given by Smax ¼  9ð 1

1 KÞ3



ð6:27Þ

The design should be checked for the full range of expected discharges. To maximize energy dissipation, the sill should be high and close to the toe of the jump. Hager and Li (1992) suggest, however, that the potential for scour in the tailwater region should also be considered. They recommend the Type A jump if the tailwater channel is easily erodible and the Type Bm only if the tailwater channel is highly resistant to scour.

6.2 The Hydraulic Jump

207

A bottom roller occurs downstream of a sill and Hager and Li (1992) suggest including its length within the stilling basin. Using Bretz’s (1987) data for the extent of the bottom roller, they proposed the length of the basin (LB) to be given by  LB 4 1 ¼ 3 Lr

1 0:6 S =3 ð1

KÞ



ð6:28Þ

Example 6.3 For the situation described in Example 6.1, determine the height and location of a sill to induce a hydraulic jump near the outlet. Solution A hydraulic jump near the outlet without a sill would require a tailwater depth to give a sequent depth ratio according to Eq. (6.2), i.e. Y ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  y2 1 1 þ 8 Fr21 1 ¼ y1 2  1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 ¼ 1 þ 8
7:27 1 ¼ 9:79 2

The actual tailwater (y2) is the uniform flow depth (=2.06 m, from Example 6.1), which is less than y*2. The amount by which the sill must reduce the required tailwater, DYs (=Dy/y1), is given by Eq. (6.25), i.e. DYs ¼ Y 

Y

with Y = y2/y1, and assuming the effect of friction within the jump to be negligible. Therefore DYs ¼ 9:79

2:06 ¼ 2:38 0:278

Equation (6.26) relates DYs to the step height and distance, DYs ¼ 0:7 S0:7 þ 3 Sð1 with S¼

s y1

K¼

Ls LR

and

From Eq. (6.8)

KÞ2

for K [ 0:5

208

6 Energy Dissipation Structures LR ¼ y1 8ðFr1 1:5Þ ¼ 0:278
8ð7:27 1:5Þ ¼ 12:8 m An appropriate design can be obtained by choosing a sill height (s), solving Eq. (6.26) for K and checking the limit for K and the maximum sill height for the onset of a Bm jump, according to Eq. (6.27), 1 Smax ¼ 9 ð1 KÞ3 The smallest sill height that satisfies both limiting conditions is s = 0.78 m, for which S¼

s 0:78 ¼ 2:81 ¼ y1 0:278

which gives, from Eq. (6.26), K¼

Ls ¼ 0:79 LR

So Ls ¼ 0:79
12:8 ¼ 10:1 m and Smax ¼

1 9ð1

0:79Þ3

¼ 12

which is greater than the 2.81 m selected. The length of basin required is given by Eq. (6.28), i.e.

So

 LB 4 1 ¼ LR 3

1 0:6 S =3 ð1

LB ¼ 12:28


 4 1 3

¼ 12:3 m

 KÞ

1 0:6
2:81 =3 ð1

0:79Þ



The water surface through the jump is then

y2 = 2.06 m s = 0.78 m

y1 = 0.278 m Ls = 10.1 m LB = 12.3 m

6.2 The Hydraulic Jump

209

Baffle Blocks Ranga Raju et al. (1980) considered the effect of a row of baffle blocks in combination with a sill. Their results for baffle blocks can be extracted to describe their effect independently. From dimensional consideration, they showed that   FB Lb h ¼ f ; ; Fr ; g ð6:29Þ 1 1 F2 y1 y1 in which FB is the force on the blocks per unit width, F*2 is the hydrostatic force associated with the sequent depth for a simple jump (i.e. 0.5 cy*2 2 ), Lb is the distance between the toe of the jump and the front of the block, h is the block height and η is the blockage ratio (=w/(w + z), where w is the block width and z is the spacing between blocks). Equation (6.29) was expressed as   W1 W2 FB Lb ¼ f2 ð6:30Þ F2 y1 with W1 accounting for h/y1 and W2 accounting for η. The force on the blocks is related to the upstream and downstream conditions by Eq. (6.1), which can be expressed as  2    FB y1 y1 1 þ 2Fr21 2Fr21 ¼ 1 ð6:31Þ F2 y2 y2 in which F2 is the hydrostatic force per unit width at the downstream section (i.e. 0.5 cy22). Using data from Basco and Adams (1971) and Murahari (pers. comm. to Ranga Raju), Ranga Raju et al. (1980) developed relationships for Eq. (6.30) and for W1 and W2. These were presented graphically, but can be approximated by the following equations: W1 W2 FB ¼ 0:56 exp F2



  Lb 0:045 y1

ð6:32Þ

for trapezoidal blocks (such as the standard shape in USBR stilling basins), and W1 W2 F B ¼ 0:52 exp F2



  Lb 0:054 y1

ð6:33Þ

for rectangular blocks (as used by Murahari), with   h W1 ¼ 3:08 y1

0:7

ð6:34Þ

210

6 Energy Dissipation Structures

and W2 ¼ 0:33 g

0:99

ð6:35Þ

For a design application, the location and geometry of baffles would be required to hold a jump between the upstream flow depth (y1) and the tailwater depth (y2) which is less than the sequent depth for a simple jump (y*2). The force on the baffles can first be calculated from Eq. (6.31) and the simple hydraulic jump sequent depth from Eq. (6.2). Then two of the three design parameters (h, η and Lb) must be specified and the third determined through Eqs. (6.32) to (6.35); Lb should be well within the undisturbed jump length. (Note that for standard stilling basins, USBR uses η = 0.5, i.e. w = z.) Ranga Raju et al. (1980) observed that a second row of baffle blocks had little effect on the drag force, and the relationship represented by Eq. (6.32) applied equally well to this situation. An additional row of baffles may assist to even out the velocity distribution within the jump and reduce downstream scour, but does not further reduce the sequent depth ratio. Example 6.4 For the situation described in Example 6.1, determine the dimensions and location of a row of baffle blocks to induce a hydraulic jump near the outlet, Solution The force on the baffles per unit width (FB) can be calculated from the upstream and tailwater conditions by Eq. (6.31). FB is then related to the block height (h), blockage ratio (η) and distance between the toe of the jump and the baffles (Lb) by Eqs. (6.32), (6.34) and (6.35). Two of these variables must be selected and the third calculated from the equations. Calculate FB by Eq. (6.31):  2  FB y1 ¼ 1 þ 2Fr21 F2 y2

2 Fr21

F2 ¼ 0:5 c y22

  y1 y2

1

¼ 0:5  9:8  103  2:062 ¼ 20:8  103 N/m width Fr1 ¼ 7:27 from Example 6:1

Therefore FB ¼ 20:8
10

3



  0:278 2 1 þ 2
7:272 2:06

2

2
7:27



 0:278 2:06

¼ 14:2
103 N/m width Lb and η will be chosen and h calculated. Ranga Raju et al. (1980) recommend that Lb should be well within the undisturbed jump length (L*r ); assume Lb/L*r = 0.30. (Note that h decreases with Lb, so a small value is preferable.)

!

1

6.2 The Hydraulic Jump

211

For their stilling basins, USBR recommends η = 0.50; assume this value. Calculate the left-hand side of Eq. (6.32) for USBR standard shape blocks,    w1 w2 FB Lb ¼ 0:56 exp 0:045 F2 y1 Lb ¼ 0:30 ðchosenÞ LR LR ¼ 12:8 ðfrom Example 6:3Þ So Lb ¼ 0:30
12:8 ¼ 3:84 m and    w1 w2 FB 3:84 ¼ 0:300 ¼ 0:56 exp 0:045 0:278 F2 Calculate W2 for η = 0.50 from Eq. (6.35), i.e. w2 ¼ 0:33 g

0:99

¼ 0:33
0:50

0:99

¼ 0:655

and F2 ¼ 0:5c y2 2 y2 ¼ 2:72 m ðfrom Example 6:1Þ So F2 ¼ 0:5
9:8
103
2:722 ¼ 36:3
103 N/m width Now calculate W1 from

w1 w2 FB ¼ 0:300 as calculated above, i:e: F2

w1 ¼

0:300 F2 0:300
36:3
103 ¼ ¼ 1:17 w2 FB 0:655
14:2
103

Calculate h from Eq. (6.34), i.e. w1 ¼ 3:08

  h y1

0:7

So

h¼



w1 3:08

 1= 0:7

y1 ¼



 1= 0:7 1:17 0:278 ¼ 1:11 m 3:08

212

6 Energy Dissipation Structures For their stilling basin baffles, USBR recommends the spacing and width to be 0.75 times the baffle height; accept this recommendation, so width and spacing will be 0.75
1.11 = 0.83 m. The water surface through the jump is

y2 = 2.06 m h = 1.11 m

y1 = 0.278 m Lb = 3.8 m

Upward and Downward Steps An upward (positive) step can be used to stabilize a hydraulic jump if the tailwater depth is lower than that required for an unaided jump to form. In practice, this is usually obtained by excavating the stilling basin to lower its bed relative to the fixed level downstream. Hager and Bretz (1986) classify jumps with positive steps as Type A if the step coincides with the end of the roller, and Type B (with a lower tailwater) if the step is within the length of the roller (Fig. 6.9). The relationship between the sequent depths and the height of the step can be described by Eq. (6.1) provided the force on the step can be quantified. Forster and Skrinde (1950) tested Type A jumps and assumed a hydrostatic pressure distribution on the step, related to the flow depth immediately upstream of it. This assumption led to overestimation of Y and they used their results to produce a diagram for relating Y, Fr1 and S. Hager and Sinniger (1985) and Hager and Bretz (1986) assumed that the water level before and after the step would be the same, and the hydrostatic pressure force per unit width would then be equal to qg(y2 + s/2)s. Equation (6.1) can then be written as q2 y2 þ 1 2 gy1







q2 y2 þ 2 2 gy2



 s ¼ y2 þ s 2

ð6:36Þ

This was found to underestimate the force, which is not actually hydrostatic and includes a dynamic contribution associated with the jet beneath the jump. Better agreement with Forster and Skrinde’s (1950) and Hager and Bretz’s (1986) data was found by assuming a uniform pressure distribution over the step height, calculated from the total depth; the unit width force is then equal to qg(y2+s)s, and Eq. (6.1) becomes 

q2 y2 þ 1 gy1 2



q2 y2 þ 2 gy2 2





¼ ðy2 þ sÞs

ð6:37Þ

6.2 The Hydraulic Jump

213 Lr

y2 y1

Type A

s

Lr

Type B y2 y1

s

Fig. 6.9 Hydraulic jumps with upward step

This assumption produces better agreement with Hager and Bretz’s (1986) data for both Type A and Type B jumps, suggesting that a greater than hydrostatic force on the step persists for jumps at least up to the Type A position. The force would tend to hydrostatic for jumps located further upstream than Type A. Other more complicated adjustments to the pressure distribution have been proposed by Fiuzat (1986) and Karki and Mishra (1986) but provide inconsequential improvement. Hager and Sinniger (1985) and Hager and Bretz (1986) expressed Eq. (6.37) as a relationship between Fr1, s and the sequent depths as Fr21 ¼

 Y ð Y þ SÞ 2 þ S2 2ð Y

1ÞÞ

1



ð6:38Þ

Application of Eq. (6.37) to Hager and Bretz’s (1986) data shows it to overestimate the required step height for given values of y1 and y2 by about 25%; its application would therefore lead to a conservative design. As for sills, the relationship expressed by Eqs. (6.37) and (6.38) would be expected also to depend on the position of the jump, as the force on the step must depart further from hydrostatic as the toe of the jump becomes closer to the step (Fiuzat 1986). This effect is yet to be satisfactorily quantified, as the documented experiments were performed with fixed toe locations. Hager and Bretz (1986) positioned their Type B jumps with the step about halfway along the length of the roller, which they showed to be about 4.25(y2 + s) for Type B jumps and about 4.75(y2 + s) for Type A jumps. Forster and Skrinde (1950) fixed the distance from the toe to the step at 5(y2 + s) for all their experiments. As the results from these different sources are quite consistent, the effect of the jump location on the sequent depth ratio is probably not great. Knowing the location of the jump is important for

214

6 Energy Dissipation Structures

designing the basin length, however, and there is evidence (Fowler 2019) that the distance of the toe of the jump from the step varies consistently with the step height. It is important to note that the tailwater depth determined from downstream conditions cannot always be assumed to occur immediately after the step, and the possibility of sweep out by the incoming supercritical flow must be considered. If the tailwater depth for a given step size or the step size for a given tailwater depth is less than indicated by Eq. (6.37), the supercritical flow may sweep out past the step and push the hydraulic jump a considerable distance downstream. To prevent sweep-out, the subcritical depth just after the step must be greater than the sequent depth of the sweeping out supercritical flow at that location. Hager and Sinniger (1985) maintain that these conditions are similar for abrupt steps, so sweep-out is unlikely. This is not so for smooth steps, however, and sweep-out is highly likely. Moreover, with a smooth step, preventing sweep-out will induce a subcritical flow depth before the step considerably greater than the sequent depth of the approaching supercritical flow, pushing the hydraulic jump far upstream and most likely submerging the upstream control. However, as the objective of a stilling basin is to effect energy dissipation, a smooth step would be less appropriate anyway. Example 6.5 For the situation described in Example 6.1, determine the height of an upward step to induce a hydraulic jump near the outlet. Solution The step height can be calculated from the momentum equation with an assumption of the pressure distribution on the step. Equation (6.37) applies for both Types A and B jumps, i.e. M1

M2 ¼ ðy2 þ sÞs M1 ¼

q2 y2 þ 1 gy1 2 Q 15 ¼ 3:33 m3 =s/m q¼ ¼ W 4:5

So M1 ¼

3:332 0:2782 þ ¼ 4:11 m2 9:8
0:278 2

M2 ¼

3:332 2:062 þ ¼ 2:67 m2 9:8
2:06 2

Similarly

Therefore 4:11 i.e.

2:67 ¼ 1:44 ¼ ðy2 þ sÞs

6.2 The Hydraulic Jump

215

s2 þ y2 s

1:4 ¼ 0

Therefore

s¼

ð2:06Þ 

¼ 0:55 m

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2:06Þ2 4
1
ð 1:44Þ

OR

2
1 2:61 m

So s ¼ 0:55 m It is not possible to determine the position of the jump, so the basin length should accommodate the length of at least a Type A roller. The minimum length is therefore Lb ¼ Lr ¼ 4:75 ðy2 þ sÞ ¼ 4:75 ð2:06 þ 0:55Þ ¼ 12:4 m; say 15 m

A downward (negative) step can be used to stabilize the position of a hydraulic jump if the tailwater depth is greater than required for an unaided jump to form. This can be incorporated in a design by setting the apron or channel bed level at the exit from a structure above the bed level of the downstream channel. As for an upward step, the jump can occur at different locations relative to the step, and different types are recognized (e.g. Hsu 1950; Rajaratnam and Ortiz 1977; Hager and Bretz 1986) (Fig. 6.10). Similar to the situations with an upward step, a jump is Type A if it occurs wholly upstream of the step and Type B if it extends over the step. The transition from Type A to Type B as the tailwater level reduces passes through an unstable sequence of standing wave and then upward and downward oriented jets. Hager and Bretz (1986) define the minimum Type B jump as the limiting type, where the jump is wholly downstream of the step with the supercritical depth at the toe equal to that just before the step. For lower tailwater depths, the jump would occur as a simple type further downstream. Design requires analysis of Types A and minimum B as well as quantification of the standing wave height to establish side wall heights. For the Type A jump, the force on the step is equal to qg(y2 − s/2)s and Eq. (6.1) is written as q2 y2 þ 1 2 gy1







q2 y2 þ 2 2 gy2



 y2

¼

s s 2

ð6:39Þ

or, in terms of Fr1, Fr21 ¼

 Y ðY

2ð Y

SÞ 2 1 1Þ



ð6:40Þ

216

6 Energy Dissipation Structures

y1

y2

Type A

y2

Transition profiles

s

y1 s

y1

y2

Type B

s

y1

y2 = y2*

s

minimum Type B

y1

Fig. 6.10 Hydraulic jumps with downward step

For the Type B jump, Hsu (1950) assumed the force on the step to be hydrostatic as determined by the upstream flow depth, i.e. equal to qg(y1 + s/2)s, i.e. q2 y2 þ 1 2 gy1







q2 y2 þ 2 2 gy2



¼



y1 þ

s s 2

ð6:41Þ

leading to

Fr21 ¼

 Y Y2

2ð Y

ð 1 þ SÞ 2 1Þ



ð6:42Þ

Hager (1985) found better agreement with data by assuming for Type B jumps that the incoming flow is similar to a free jet. The pressure is then atmospheric at the top of the step and hydrostatic over the step height; the force on the step is then equal to qgs2/2 and Eq. (6.1) is written as

6.2 The Hydraulic Jump

217



q2 y2 þ 1 gy1 2

q2 y2 þ 2 gy2 2







¼

s2 2

ð6:43Þ

or, in terms of Fr1, Fr21 ¼

Y ðY 2 S2 1Þ 2ð Y 1Þ

ð6:44Þ

For the minimum B jump, the jet reaches the channel bed just after the step, with the same flow depth as immediately upstream. The sequent depth ratio is therefore as for a simple jump on a horizontal bed. Hsu (1950) presented experimental data confirming the applicability of Eqs. (6.39) and (6.41) for the sequent depth ratios for Type A and B jumps, and particularly the conditions at the transition between them. Hager and Bretz (1986) found the length of the roller to be approximately 3.5y2 for Type A jumps (decreasing slightly with Fr1) and about 4.25y2 for Type B jumps. They accounted for variation with Fr1 in their response to the discussion by Ohtsu and Yasuda (1987), but acknowledged the limited data available. A downward step design should allow for the side walls to accommodate the height of the standing wave in the transition between Type A and B jumps. Based on data from Rajaratnam and Ortiz (1977) and their own experiments, Hager and Bretz (1986) propose the maximum wave height to be given by ðY þ SÞmax ¼ 2Fr1

6.3

ð6:45Þ

Standard Stilling Basins

The purpose of a stilling basin is to effect energy dissipation by means of a hydraulic jump within the basin, so that the high energy supercritical flow entering the basin is converted to lower energy subcritical flow before reaching the downstream channel. The aims of the design are to ensure formation of the jump, to keep it stable in one position and to make it as short as possible. Features commonly used to control hydraulic jumps were discussed in the previous section, and these are usually used in combination for stilling basin design. Just as the effects of the features acting individually were determined empirically, so must their effects in combination. Original stilling basin design therefore invariably requires physical modelling testing. Because many stilling basins need to be designed for similar conditions, various general-purpose, standard designs have been developed. In particular, the United States Bureau of Reclamation (USBR) (Bradley and Peterka 1957; Peterka 1978) has proposed a number of designs for specific applications and conditions. These are applicable for a wide range of

sloping floor baffle piers

chute blocks

end sill floor level

(a) Longitudinal section

chute blocks

end sill

baffle piers

(b) Plan Fig. 6.11 Features of USBR stilling basins

conditions, but care should be taken to comply with the specified restrictions; otherwise, specific model studies should be carried out. A typical USBR stilling basin will incorporate some or all of the features illustrated in Fig. 6.11. The sloping floor at the entrance assists in maintaining an unsubmerged jump under a range of discharges, as described in the previous section. The floor level is set to match the tailwater level to the downstream subcritical flow depth required for jump formation. This should be checked for the full range of discharges expected, with the tailwater level determined by the downstream channel characteristics. The chute blocks are sometimes used to form a serrated entrance. The blocks disperse the incoming flow by splitting it in jets and raising some of the flow off the floor, as well as enhancing aeration. This has the effect of stabilizing the jump, shortening its length and generally improving the performance. The sill is placed at the end of the basin to ensure that a jump occurs and to control its position over the whole operating range of discharges. It also protects the channel bed from direct action of the current. Sometimes it is also used to set up a reverse roller, which directs any mobilized bed material back towards the structure to prevent undermining (Fig. 6.12a).

scour hole

6.3 Standard Stilling Basins

219

For large basins with high incoming velocities, the sill is usually dentated, as shown in Fig. 6.12b. This helps to diffuse the remaining part of the high-velocity jet which may reach the end of the basin, resulting in a more uniform velocity profile, less wave action and less likelihood of the intermittent entrainment and downstream displacement of water masses in the jump. Baffle piers are blocks placed on the basin floor. These assist in energy dissipation by impact action. By exerting additional force on the control volume, they reduce the downstream depth required for a jump to form, and therefore allow a lower tailwater to be tolerated. Baffle piers are used most in small structures with low velocities. They are unsuitable for high velocities because of the likelihood of cavitation. Three standard USBR stilling basin designs incorporating these features are recommended by Bradley and Peterka (1957), Peterka (1978) and United States Bureau of Reclamation (1987). They have been developed through analysis of many successful designs as well as laboratory studies, as described in detail by Peterka (1978). The USBR Basin II (note that Basin I refers to a simple, unaided hydraulic jump) is intended for use with high dam spillways and large canal structures, with high incoming velocities and Froude numbers greater than 4.5. The recommended geometry and dimensions, in terms of the inflow and sequent tailwater depths, are shown in Fig. 6.13 and ensure safe, conservative design for spillways up to about

α y1 y2* 0.02 y2* h2 = 0.2 y2*

h1 = y1 LII

slope = 2:1 (a) Longitudinal section 0.5 y1 w1 = y1 s1 = y1

s2 = 0.15 y2* w2 = 0.15 y2*

(b) Plan Fig. 6.13 USBR Basin II geometry (adapted from Peterka 1978)

220

6 Energy Dissipation Structures

60 m high with discharges up to about 45 m3/s/m of basin width. This type does not incorporate baffle piers because of the possibility of cavitation at high velocities. The chute block spacing may be varied to avoid having fractional blocks, but the width and spacing should be kept equal. The width and spacing of the sill dentates indicated in Fig. 6.13 are recommended maximums and may be reduced in narrow basins to increase their number; again the width and spacing should be kept equal. The basin length, LII, is equal to the hydraulic jump length, which is related to the entry Froude number, as shown in Fig. 6.14. This shows the significant reduction in jump length affected by the stilling basin features. The basin floor level should be set so that the tailwater is at least at the sequent depth of y1 for all discharges anticipated. The incipient sweep-out level is about 5% less than this, so setting for the sequent depth provides this margin of safety. The basin performance is not affected appreciably by the slope of the entrance, but for slopes greater than 45o the transition to the stilling basin floor should be curved (rather than sharp as shown in Fig. 6.13a) with a radius of at least 4y1. The water surface profile, which can also be considered as the pressure profile, can be approximated by a straight line at angle a drawn from the tailwater level at the end of the basin (Fig. 6.13). The value of a depends on Fr1 according to a ¼ 5:06 ln Fr1

ð6:46Þ

1:43

The USBR Basin III incorporates baffle piers, resulting in a more compact and therefore more economical design. Baffle piers introduce vulnerability to cavitation, however, so the flow conditions are more restrictive than for Basin II. The design is appropriate for entry Froude numbers greater than 4.5 but with an upper limit on entry velocity of about 15–18 m/s and on discharge of about 18 m3/s/m of basin width. The recommended geometry and dimensions are shown in Fig. 6.15.

7 6

L/y2*

5 4 3 Basin I, IV 2

Basin II

1 0

Basin III 2

4

6

8

10 Fr1

12

14

Fig. 6.14 USBR Basin lengths (adapted from Peterka 1978)

16

6.3 Standard Stilling Basins

221

y1 0.5 y2

y2*

0.2 h3

*

slope = 1:1 h3

h1 = y1

h4

0.8 y2* slope = 2:1

LIII (a) Longitudinal section 0.5 y1

0.375 h3

w1 = y1 s1 = y1

w3 = 0.75 h3 s3 = 0.75 h3

(b) Plan Fig. 6.15 USBR Basin III geometry (adapted from Peterka 1978)

The height of the baffle piers is related to the entry Froude number according to h3 ¼ 0:59 þ 0:17 Fr1 y1

ð6:47Þ

The width and spacing may deviate from 0.75 times the height as recommended, but the sum of the widths should remain equal to the sum of the spaces. The baffle pier shape may be as shown, or they may be cubes, but the corners should not be rounded as this would reduce the effectiveness of the flow separation producing energy-dissipating eddies. The recommended distance of the baffle piers from the downstream face of the chute blocks should be adhered to. The end sill should be solid rather than dentated as for Basin II. Its height is not crucial, but the USBR recommends a slight variation with Fr1 according to h4 ¼ 1:0 þ 0:056 Fr1 y1

ð6:48Þ

222

6 Energy Dissipation Structures

The basin length (LIII) is given by the lowest curve in Fig. 6.14, showing that the appurtenances reduce the jump length to less than half of the length of a simple jump. As for Basin II, the tailwater level should correspond to the sequent of the inflow depth, although the incipient sweep-out level is about 15–18% lower, giving a greater margin of safety. The entrance transition should also be rounded for entry slopes steeper than 45%. The water surface profile (and pressure profile) shows a very steep rise from about half the tailwater depth to the full depth in the vicinity of the baffle piers (Fig. 6.15a). The USBR Basin IV is designed for low-head dam and canal structures where the entry Froude number is in the range 2.5–4.5, i.e. corresponding to the oscillating jump where wave suppression is necessary. This basin is therefore an alternative to the wave suppressors described in Sect. 6.2. The recommended geometry and dimensions are shown in Fig. 6.16. The design is intended to reduce appurtenances to a minimum. The usual chute blocks are replaced by larger deflector blocks with sloping top surfaces. Their width can be increased up to a maximum of y1, but the width to spacing ratio should be kept at 1:2.5. The sill (which is optional) can be sized as for Basin III, i.e. with a slope of 2:1 and a height according to Eq. (6.48). This geometry does not reduce the hydraulic jump length, so the recommended basin length is as indicated by the top curve in Fig. 6.13. The floor level should be set for a tailwater depth equal to 1.1 times the sequent depth of the incoming flow. Performance will be satisfactory for the full range of discharges if this is designed for the maximum discharge.

2 y1 min

5o slope 2 y1

h4 LI (a) Longitudinal section

s1 = 2.5 w1 w1 = 0.75 y1

(b) Plan Fig. 6.16 USBR Basin IV geometry (adapted from Peterka 1978)

slope = 2:1

6.3 Standard Stilling Basins

223

0.2 y1 0.2 hs y1

y1

hs = 0.2 y2

*

slope = 2:1

X L1 L = 3 y2* (a) Longitudinal section 0.375 y1 min 0.7 y1 0.7 y1 0.15 y2*

(b) Plan Fig. 6.17 Geometry of USBR Basin for low Fr1 (adapted from George 1978)

Another basin design developed by the Bureau of Reclamation (George, 1978) for low Froude numbers (2.5 to about 6.0) includes chute blocks and baffle piers (similar to Basin III), resulting in a much shorter basin. This was considered superior to the St Anthony Falls basin (see below) which causes rough flow and high scour potential beyond the basin at low Froude numbers. The geometry and dimensions are shown in Fig. 6.17. The sill is dentated, with equal dentate widths and spacings. The distance from the chute blocks to the baffle piers (X) is given by X ¼ 2:65 Fr1 0:69 y2

ð6:49Þ

and the distance from the chute blocks to the sill (L1) by L1 ¼ 3:5 y2

0:30 Fr1

ð6:50Þ

The overall length of the stilling basin is the length of the hydraulic jump, which may be assumed to be equal to 3y*2 over the applicable range of Fr1. This is significantly less than for Basin IV (intended for the same range of Fr1) and similar to that for Basin III. For very low Fr1, the distance L1 plus the length of the sill may exceed L, in which case the basin should be extended to include the sill.

224

6 Energy Dissipation Structures

Another common design for small structures is the St Anthony Falls (SAF) stilling basin (Blaisdell, 1959). It has a particularly wide range of application conditions (1.7 < Fr1 < 17) and is very effective in shortening the jump. It also incorporates baffle blocks, a sloping entrance and an end sill. Because the size of the basin depends on the entry flow depth, it is sometimes advantageous to flare the transition from the discharging structure or chute to the basin in order to increase the width and hence reduce the depth. The diverging side walls are then extended through the stilling basin, resulting in a trapezoidal plan shape. The geometry and dimensions for a straight-sided basin are shown in Fig. 6.18. The length of the basin (LB) is related to the incoming Froude number by LB ¼

4:5y2 Fr10:76

ð6:51Þ

The chute blocks may be placed with either a block or a space next to the side wall, as long as the blocks are symmetrical about the outlet centre line. The baffle

top of side wall

y2* /3

TW

y1

y1

0.07 y2*

LB /3 LB (a) Longitudinal section wingwall 0.375 y1 min 0.75 y1 0.75 y1

(b) Plan Fig. 6.18 St. Anthony Falls stilling basin geometry

45o pref.

6.3 Standard Stilling Basins

225

piers are placed directly downstream of the openings between the chute blocks at a distance equal to LB/3 from the ends of the chute blocks. They should occupy between 40% and 50% of the basin width, and no pier should be closer to the side wall than 0.375 y1. For diverging stilling basins, the widths and spacings of the baffle piers should be increased in proportion to the increase in the basin width at their location. The end sill should be extended as a cutoff wall to prevent undermining of the basin by erosion. The floor of the basin should be set for a tailwater (TW) height of TW y1

1:4 Fr10:90

ð6:52Þ

The side wall height should allow a freeboard of y*2/3 above the tailwater level to accommodate the rough surface, which may extend beyond the basin at high Froude numbers. Wingwalls should be provided for the transition from the basin to the downstream channel. They should slope downwards at 45° from the end of the basin, and preferably be flared at 45°.

6.4 6.4.1

Other Energy Dissipators Bucket-Type Dissipators

If the tailwater at the bottom of a spillway is too deep to allow a free hydraulic jump to form, energy can be dissipated effectively in a roller bucket (Fig. 6.19).

Fig. 6.19 A slotted bucket dissipator

226

6 Energy Dissipation Structures

Fig. 6.20 Bucket-type dissipator

The dissipator works on the principle of induced turbulence and mixing, which is initiated by a sharp upward deflection of the flow (Fig. 6.20). This causes a roller motion within the bucket, which induces a reverse roller immediately downstream. The motion of the rollers and mixing with incoming water dissipate energy and prevent scour. Two basic types of submerged bucket dissipator have been developed (Fig. 6.21), with many variations of the basic designs. The solid bucket is simply a continuation of the spillway profile, with a circular radius. The slotted bucket is dentated on an extended apron. The solid and slotted configurations operate on the same principle, but the hydraulic action of each has characteristics that impose certain limitations on its application. The two types are therefore suited to different conditions. For the solid bucket, the high-velocity flow is all directed upwards, creating a high boil and a violent ground roller. The ground roller continuously draws bed material back towards the lip of the bucket and keeps some material in constant agitation. In the slotted bucket, the high-velocity jet leaves the lip at a much flatter angle, with only part of it at an angle sufficient to cause a surface boil. The boil is therefore less violent and there is better flow dispersion above the ground roller. There is less high energy flow within the bucket and the downstream flow is smoother. These characteristics have relative advantages and disadvantages in practice. The solid bucket may be undesirable because of abrasion of the concrete surface by

(a) Solid bucket Fig. 6.21 Basic submerged bucket designs

(b) Slotted bucket

6.4 Other Energy Dissipators

227

the agitated bed material. Also, the higher surface turbulence generated by the violent boil can cause damage to river banks downstream. The slotted bucket provides more effective energy dissipation with less severe disturbance of the water surface and the stream bed. It is, however, susceptible to sweep out at low tailwaters, which results in high-velocity flow and potential scour problems downstream (Fig. 6.22a). Effective operation therefore requires the tailwater to be above a minimum depth (Fig. 6.22b). If the tailwater is too deep for a slotted bucket, then diving flow can occur (Fig. 6.22c). Here, the jet leaving the bucket is not deflected to the surface but is depressed to the river bed, causing scour immediately downstream of the bucket and deposition of the scoured material a short distance downstream. There is also, therefore, a maximum tailwater requirement. By virtue of its better energy dissipation characteristics, the slotted bucket is therefore a preferable design, provided its more stringent tailwater requirements can be met. Full details of the design geometries are given by Peterka (1978) and USBR (1973), including the limitations on tailwater depths for the slotted bucket. As for the hydraulic jump stilling basins, model studies are advisable if the specified limits of application are violated.

yt minimum

(a) Sweep-out condition

yt maximum yt minimum

(b) Normal operating condition

yt maximum

(c) Diving flow condition Fig. 6.22 Flow conditions for slotted buckets

228

6.4.2

6 Energy Dissipation Structures

Impact-Type Dissipators

Impact-type stilling basins have been developed for pipe or relatively narrow open channel outlets (Peterka, 1978) (Fig. 6.23). Energy dissipation is effected by the impact of the incoming jet on a vertical baffle spanning across the basin (Fig. 6.24) and turbulence in eddies formed after the impact. This basin’s capacity is generally limited by the feasibility of the structural design, which must allow for impact loads and vibration, and to an incoming velocity of about 15 m/s. Designs have been used for discharges up to 11 m3/s. It has been found to be significantly more effective in dissipating energy than a hydraulic jump on a horizontal floor. The performance is not very dependent on tailwater conditions, although is best when the tailwater is less than about halfway up the baffle. A standard design geometry with dimensions for a range of discharges is presented by Peterka (1978).

Fig. 6.23 Impact-type stilling basin at an impoundment outlet

Fig. 6.24 Impact-type stilling basin action

6.4 Other Energy Dissipators

6.4.3

229

Baffled Spillways

For low weirs, chutes and drop structures, energy can be effectively dissipated by a system of baffles over the whole slope length (e.g. Peterka, 1978; Rhone, 1977), as illustrated in Figs. 6.25 and 6.26.

Fig. 6.25 Baffles on a low-head spillway

Fig. 6.26 Baffled spillway layout

(a) Section

(b) Plan

230

6 Energy Dissipation Structures

The baffles limit the acceleration of flow down the slope and ensure a reasonable terminal velocity for any slope length. Peterka (1978) has reviewed the results of model and prototype experiments on baffled spillways and drop structures and has formulated a procedure for their design. The design is effective for unit width discharges up to about 5.6 m3/s/m and for slopes between 1 V:2H and 1 V:4H. The entrance to the slope should be set above the bottom of the approach channel to create a pool and ensure that the approach velocity does not exceed 1.5 m/s less than the critical velocity. Recommended baffle dimensions are shown in Fig. 6.27, in which the baffle height (H) is equal to 0.8yc. Apart from the baffle widths and spacings being equal and not less than H, the dimensions are not critical. In narrow structures, partial baffles with widths of 0.5H–1.0H should be placed adjacent to the training walls wherever necessary to complete rows. The bottom of the slope should be constructed below the stream bed and backfilled, with riprap protection placed at the ends of the training walls.

Fig. 6.27 Baffle dimensions

(a) Section

1.5H

1.5H (b) Plan

6.4 Other Energy Dissipators

6.4.4

231

Stepped Chutes and Spillways

Energy can be effectively dissipated over the length of a spillway if it constructed as a series of vertical steps (Chanson 1993, Christodoulou 1993, Essery and Horner 1978, Rajaratnam 1990). This arrangement is particularly attractive for roller-compacted concrete dams. The hydraulics of stepped chutes and spillways is dealt with in Chap. 4.

6.4.5

Spillway Splitters

A flow splitter system placed on the face of a dam spillway a short distance below the crest provides an effective way of dissipating energy (Fig. 6.28). The design, first conceived by Roberts (1943), comprises a row of evenly spaced discrete features (splitters) along the spillway face located above a solid, horizontal shelf, which may be cantilevered from or inset into the spillway face (Fig. 6.29). The splitters separate the flow into two streams. The upper stream is deflected away from the spillway face along the upper surface of the splitters. The lower stream passes between the splitters and is deflected by the shelf. The trajectories of the two dispersed and aerated streams intersect, further increasing the aeration and resulting in a spray with greatly reduced impact on the downstream apron or riverbed. Roberts (1980) used laboratory test results, including those of Roberts (1943), to derive geometric and performance relationships for designing a splitter arrangement

Fig. 6.28 Splitter arrangement on a gated spillway, showing operation at opened gate

232

6 Energy Dissipation Structures

H x

x

L

P

W

S

T hdam

y Ls θ

(b) Splitter details yt Rmin Rmax

(a) General profile Fig. 6.29 Spillway splitter dimensions

for spillway heads up to 3 m. These, as also reported by Calitz and Basson (2015), are presented below. The splitter arrangement is suitable for relatively high dams with x þ y [ 4HD

ð6:53Þ

where HD is the design head and x and y are defined in Fig. 6.29. Performance is satisfactory for heads between a minimum (Ha) necessary for effective dispersion of the flow streams and a maximum (Hc) above which the splitters become drowned, ineffective and vulnerable to cavitation. These limiting heads are related to the design head by Hc ¼ 1:2 HD

ð6:54Þ

and Ha ¼

Hc 4:4

ð6:55Þ

The splitter dimensions are related to the number (N) installed over the length (B) considered, which may be the total spillway length or the distance between piers if these are installed. The splitter width (W) is then given by W¼

3B 7N

ð6:56Þ

6.4 Other Energy Dissipators

233

and the splitter spacing (S), length (L) and surface height above the shelf level (T) by S¼L¼T¼

4B 4 ¼ W 7N 3

ð6:57Þ

The shelf width (Ls) should be in the range 1:25 

Ls  1:50 L

ð6:58Þ

The performance of the splitters depends on the level of the splitter surface below the spillway crest (P). The splitters should be far enough below the crest for sufficient kinetic energy to have been developed to ensure effective dispersion, but nearer the crest than the distance to the point of aeration inception. The maximum value of P can therefore be determined using the methods for locating the inception point presented in Chap. 4, such as Wood et al.’s (1983) Eq. (4.51). The optimal splitter width for effective dispersion is related to P. Roberts (1943) presented experimental data relating the optimal W/P to Hc/P, which can be represented (with R2 = 0.995) as  1:24 W Hc ¼ 0:324 P P

ð6:59Þ

The design can proceed either by choosing the splitter geometry and determining a corresponding value for P, or by choosing P and determining an appropriate geometry. The geometry can be chosen by specifying a value for either W or N, from which the other dimensions follow through Eqs. (6.56) to (6.58). P is then determined from Eq. (6.59), rearranged as P¼



0:324 Hc1:24 W

4:17

ð6:60Þ

and checked to be less than would allow the point of aeration inception to be reached before the splitter location. If P is chosen first, subject to satisfying the aeration inception point limit, W is then obtained from Eq. (6.60), and the other dimensions from Eqs. (6.56) to (6.58). Lastly, the extent of the spray jet (Rmin to Rmax) must be determined to ensure that it falls within the planned plunge pool or to dimension an apron if one is to be constructed. These distances are given by Eqs. (6.61) and (6.62). pffiffiffiffiffi Rmax ¼ 2 f xy pffiffiffiffiffi Rmin ¼ f xy

y cot h y cot h

ð6:61Þ ð6:62Þ

234

6 Energy Dissipation Structures

The distances x and y are shown in Fig. 6.29 (for practical purposes x  HD + P). Roberts (1943) presented experimental data relating f to (P + T) and HD, which can be described (with R2 = 0.991) by f ¼ 1:025 e

0:022ððP þ T Þ=HD Þ

ð6:63Þ

The design is not unique for a specified design head, as different combinations of splitter size and P can perform satisfactorily. The combination selected must ensure that the point of aeration inception does not occur before the splitters and that the aerated jet clears the toe of the dam. The smaller the splitter size chosen, the greater will be P and the smaller will be Rmin. The relationships presented above are applicable for unaerated splitters, originally considered to be effective for design heads less than 3.0 m (corresponding to a unit width discharge (q) of about 12 m3/s/m for a conventional ogee spillway). Through specific model testing, designs including aeration of the splitters have been used for considerably greater heads (Jordaan 1989). Calitz and Basson (2018) conducted model tests for splitters designed as above for HD = 6.7 m (q = *40 m3/s/m) and found satisfactory performance up to the design head, above which the splitters became drowned and cavitation became inevitable. By providing artificial aeration satisfactory performance, without cavitation risk, was extended to at least HD = 7.6 m (q = * 50 m3/s/m). Example 6.6 Dimension an effective spillway splitter arrangement for a 30.0 m high dam with a 150 m long ogee spillway sloped at 54°. The design head is 2.40 m and the tailwater depth at the corresponding discharge is 4.0 m. Determine also the extent of the spray jet at the base of the dam. Solution Refer to Fig. 6.29 for notation. Check the suitability of the splitter option for energy dissipation, according to Eq. (6.53), i.e. x þ y [ 4 HD x þ y ¼ hdam þ HD ¼ 30:0 þ 2:4

yt 4:0 ¼ 28:8 m

4 HD ¼ 4
2:4 ¼ 9:6 m

x + y = 28.8 m > 4HD = 9.6 m. Therefore, the suitability condition is satisfied. Establish head limits for satisfactory performance: Upper limit from Eq. (6.54), i.e. Hc ¼ 1:2 HD ¼ 1:2
2:40 ¼ 2:88 m Lower limit from Eq. (6.55), i.e.

6.4 Other Energy Dissipators

235

Ha ¼

Hc 2:88 ¼ 0:66 m ¼ 4:4 4:4

Determine Pmax to ensure splitters are located beyond the air entrainment inception point: Pmax  L sin h The distance to the inception point (L) is insensitive to H, so calculate Pmax for HD. Calculate L using Wood et al.’s (1983) Eq. (4.51), i.e.  0:11   d x x ¼ 0:0212 x Hs ks

0:10

ðAÞ

by iteration, with x = L and d = y, the flow depth at the inception point. Calculations with the successful iteration values are shown below: Try L = 45.3 m. With L = 45.3 m, try y = 0.296 m Then Hs ¼ H þ L sin h

y cos h

with h ¼ 54:0o and H ¼ HD ¼ 2:4 m So Hs ¼ 2:40 þ 45:3 sin 54:0

0:296 cos 54:0 ¼ 38:9 m

and V¼

pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2gHs ¼ 2
9:8
38:9 ¼ 27:6 m/s

y¼

q V

Check:

3= q ¼ CD HD 2

3 ¼ 2:2
2:4 =2 for a conventional ogee spillway

So

¼ 8:18 m3 =s/m

236

6 Energy Dissipation Structures

y¼

8:18 ¼ 0:296 m 27:6

which is equal to the trial value. Therefore y = 0.296 m. Now calculate RHS of (A) with the trial values. (The result is insensitive to ks; a value of 0.30 mm is representative.) RHS ¼

0:0212

0:296 45:30:11  38:9

45:3 0:00030



0:10

¼ 45:2 m

which is approximately equal to the trial value of L. Therefore L = 45.3 m. Therefore Pmax  45:3 sin 54 ¼ 36:6 m which is greater than the dam height, so there is no effective restriction on the location of the splitters. Select P and determine splitter dimensions: With no restriction on Pmax, the choice of P is arbitrary. A value of P = 7.5 m is used for example. The splitter dimensions then follow, according to Eqs. (6.59), (6.57) and (6.58), as follows. W is defined by Eq. (6.59), i.e.  1:24 W Hc ¼ 0:324 P P Therefore  1:24 Hc W ¼ P 0:324 P   2:88 1:24 ¼ 7:5
0:324 ¼ 0:74 m 7:5 Then, from Eq. (6.57) 4 4 S ¼ L ¼ T ¼ W ¼ 0:74 ¼ 0:99 m 3 3 And, from Eq. (6.58), Ls must be in the range 1:25 

Ls  1:50 L

i.e. Ls min ¼ 1:25 L ¼ 1:25
0:99 ¼ 1:24 m

6.4 Other Energy Dissipators

237

and Ls max ¼ 1:50 L ¼ 1:50
0:99 ¼ 1:49 m Choose Ls ¼ 1:36 m Determine the extent of the spray jet: Rmin will occur at the low end of the operating range, i.e. for H = Ha = 0.66 m. Then, from Eq. (6.62), pffiffiffiffiffi Rmin ¼ f xy

y cot h

with, from Eq. (6.63),

f ¼ 1:025 e

¼ 1:025 e

0:022ððP þ T Þ=Ha Þ 0:022ðð7:5 þ 0:99Þ=0:66Þ

¼ 0:772

and x ¼ Ha þ P ¼ 0:66 þ 7:5 ¼ 8:16 m y ¼ hdam P yt ¼ 30:0 7:5 4:0 ¼ 18:5 m Therefore pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rmin ¼ 0:772 8:16
18:5

18:5 cot 54 ¼

3:96 m

So the spray jet will fall on the spillway face. Rmax will occur at the high end of the operating range, i.e. for H = Hc = 2.88 m. Then, from Eq. (6.61), pffiffiffiffiffi Rmax ¼ 2f xy

y cot h

with, from Eq. (6.64),

f ¼ 1:025 e

¼ 1:025 e

0:022ððP þ T Þ=Hc Þ 0:022ðð7:5 þ 0:99Þ=2:88Þ

¼ 0:961

and x ¼ Hc þ P ¼ 2:88 þ 7:5 ¼ 10:38 m y ¼ hdam P yt ¼ 18:5 m as before

238

6 Energy Dissipation Structures Therefore pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rmax ¼ 2
0:961 10:38
18:5

18:5 cot 54 ¼ 13:2 m

which indicates the length of apron required for protection of the river bed.

Problems 6:1 A long concrete (n = 0.013) chute has a slope of 0.10 and is designed to convey a discharge of 25 m3/s. The chute terminates in a stilling basin with a horizontal bed followed by an abrupt upward step to a long channel with a slope of 0.0010 and n = 0.020. The width is 5.0 m throughout. a. Determine an appropriate step height and basin length to keep the hydraulic jump within the basin and on the horizontal bed. b. Determine the positions of the hydraulic jump if the step height is (i) 1.20 m and (ii) 2.50 m. For each case sketch and dimension the water surface profile and the total energy line. 6:2 Water is to be released from a reservoir at a discharge of 20.0 m3/s and a velocity of 12.0 m/s into a 6.0 m wide, rectangular, concrete-lined (n = 0.013) canal on a slope of 0.00050. Energy dissipation is required before the water enters the canal. a. Size a stilling basin with only a smooth upward step, checking for sweep-out. b. Size a stilling basin with only an abrupt upward step, to create a Type A hydraulic jump. c. Size a stilling basin with only an abrupt upward step, to create a Type B hydraulic jump. d. Size a dissipator using just a sill, with the smallest satisfactory sill height. e. Size a row of dissipating baffle blocks to be placed 4.0 m from the reservoir outlet. Assume standard USBR block shapes with a blockage ratio of 0.50. 6:3 A reservoir releases water onto an 8.0 m wide, horizontal apron at a discharge of 30.0 m3/s and a velocity of 12.0 m/s. A 1.40 m high sill is placed 10.0 m from the release point. A further 10.0 m downstream from the sill, the apron leads to an 8.0 m wide rectangular, concrete-lined channel on a slope of 0.0010 through an abrupt step. Determine the height of the step to ensure that the hydraulic jump forms with its toe close to the release point. (Assume the effects of the sill and the step to be independent, and friction between the sill and the step to be negligible.)

References

239

References Bakhmeteff, B. A., & Matzke, A. E. (1936). The hydraulic jump in terms of dynamic similarity. Transactions of the American Society of Civil Engineers, 101, 630–647. Basco, D. R., & Adams, J. R. (1971). Drag forces on baffle blocks in hydraulic jumps. Journal of the Hydraulics Division, ASCE, 97(HY12), 2023–2035. Blaisdell, F. W. (1959). The SAF stilling basin, agriculture handbook No. 156, U S Department of Agriculture. Bradley, J. N., & Peterka, A. J. (1957). The hydraulic design of stilling basins. Journal of the Hydraulics Division, ASCE, 83(HY5), 1401–1406. Bretz, N. V. (1987). Ressaut hydraulique force par seuil, Thèse No. 699, Ecole Polytechnique Fédérale de Lausanne, Switzerland. Calitz, G., & Basson, G. R. (2015). The design of Roberts splitters for energy dissipation at dam spillways. Design and Construction of Hydraulic Structures: Stellenbosch University. Calitz, G., & Basson, G. R. (2018). The effect of aeration through an internal gallery of a dam on the cavitation risk of Roberts splitters. Journal of the South African Institution of Civil Engineering, 60(1), 31–43. Carollo, F. G., Ferro, V., & Pampalone, V. (2007). Hydraulic jumps on rough beds. Journal of Hydraulic Engineering, 133(9), 989–999. Carollo, F. G., Ferro, V., & Pampalone, V. (2009). New solution of classical hydraulic jump. Journal of Hydraulic Engineering, 135(6), 527–531. Carollo, F. G., Ferro, V., & Pampalone, V. (2011). Sequent depth ratio of a B-jump. Journal of Hydraulic Engineering, 137(6), 651–658. Carollo, F. G., Ferro, V., & Pampalone, V. (2012). New expression of the hydraulic jump roller length. Journal of Hydraulic Engineering, 138(11), 995–999. Castro-Orgaz, O., & Hager, W. H. (2009). Classical hydraulic jump: basic flow features. Journal of Hydraulic Research, 47(6), 744–754. Chanson, H. (1993). Stepped spillway flows and air entrainment. Canadian Journal of Civil Engineering, 20(3), 422–435. Chow, V. T. (1959). Open-channel hydraulics, McGraw-Hill. Christodoulou, G. C. (1993). Energy dissipation on stepped spillways. Journal of Hydraulic Engineering, 119(5), 644–650. Essery, I. T. S., & Horner, M. W. (1978). The hydraulic design of stepped spillways, 2nd Edition, CIRIA Publication R33. Fiuzat, A. A. (1986). Flow characteristics of the hydraulic jump in a stilling basin with an abrupt bottom rise. Journal of Hydraulic Research, 24(3), 207–209. Forster, J. W., & Skrinde, R. A. (1950). Control of hydraulic jump by sills. Transactions of the American Society of Civil Engineers, 115, 973–987. Fowler, N. (2019). Hydraulic jumps and steps: the relationship between jump location and sequent depth ratio using a step, MSc(Eng) project report, University of the Witwatersrand, South Africa. French, R. H. (1985). Open-channel hydraulics, McGraw-Hill. George, R. L. (1978). Low Froude number stilling basin design, Report REC-ERC-78–8, Engineering and Research Center, United States Department of the Interior, Bureau of Reclamation, Denver, Colorado, USA. Hager, W. H. (1985). B-jumps at abrupt channel drops. Journal of Hydraulic Engineering, 111(5), 861–866. Hager, W. H., & Bremen, R. (1989). Classical hydraulic jump; sequent depths. Journal of Hydraulic Research, 27(5), 565–585. Hager, W. H., Bremen, R., & Kawagoshi, N. (1990). Classical hydraulic jump; length of roller. Journal of Hydraulic Research, 28(5), 591–608.

240

6 Energy Dissipation Structures

Hager, W. H., & Bretz, N. V. (1986). Hydraulic jumps at positive and negative steps. Journal of Hydraulic Research, 24(4), 237–253. Hager, W. H., & Li, D. (1992). Sill-controlled energy dissipator. Journal of Hydraulic Research, 30(2), 165–181. Hager, W. H., & Sinniger, R. (1985). Flow characteristics of the hydraulic jump in a stilling basin with an abrupt bottom rise. Journal of Hydraulic Research, 23(2), 101–113. Henderson, F. M. (1966). Open channel flow, Macmillan. Hsu, E. Y. (1950). Discussion of “Control of the hydraulic jump by sills” by J W Forster and R A Skrinde. Transactions of the American Society of Civil Engineers, 115, 988–991. Hughes, W. C., & Flack, J. E. (1984). Hydraulic jump properties over a rough bed. Journal of Hydraulic Engineering, 110(12), 1755–1771. Jordaan, J. M. (1989). The Roberts splitter: Fifty years on. The Civil Engineer in South Africa, 31 (10), 319–321. Karki, K. S., & Mishra, S. K. (1986). Flow characteristics of the hydraulic jump in a stilling basin with an abrupt bottom rise. Journal of Hydraulic Research, 24(3), 210–2015. Kindsvater, C. E. (1944). The hydraulic jump in sloping channels. Transactions of the American Society of Civil Engineers, 109, 1107–1154. Mehrotra, S. C. (1976). Length of hydraulic jump. Journal of the Hydraulics Division, ASCE, 102 (HY7), 1027–1033. Ohtsu, I. (1981). Forced hydraulic jump by a vertical sill. Transactions of the JSCE, Journal of the Hydraulic and Sanitary Engineering Division, 13, 165–168. Ohtsu, I., & Yasuda, Y. (1987). Discussion of “Hydraulic jumps at positive and negative steps”. Journal of Hydraulic Research, 25(3), 407–413. Ohtsu, I., & Yasuda, Y. (1991). Hydraulic jump in sloping channels. Journal of Hydraulic Engineering, 117(7), 905–921. Ohtsu, I., Yasuda, Y., & Hashiba, H. (1996). Incipient jump conditions for flow over a vertical sill. Journal of Hydraulic Engineering, 122(8), 465–469. Peterka, A. J. (1978). Hydraulic design of stilling basins and energy dissipators, Engineering Monograph No. 25, revised, United States Department of the Interior, Bureau of Reclamation. Rajaratnam, N. (1967). Hydraulic jumps, Advances in Hydroscience (Vol. 4, pp. 197–280). New York: Academic Press. Rajaratnam, N. (1990). Skimming flows in stepped spillways. Journal of Hydraulic Engineering, 116(4), 587–591. Rajaratnam, N., & Murahari, V. (1971). A contribution to forced hydraulic jumps. Journal of Hydraulic Research, 9(2), 217–240. Rajaratnam, N., & Ortiz, N. V. (1977). Hydraulic jumps and waves at abrupt drops. Journal of the Hydraulics Division, ASCE, 103(HY4), 381–394. Ranga Raju, K. G., Kitaal, M. K., Verma, M. S., & Ganeshan, V. R. (1980). Analysis of flow over baffle blocks and end sills. Journal of Hydraulic Research, 18(3), 227–241. Rhone, T. J. (1977). Baffled apron as spillway energy dissipator. Journal of the Hydraulics Division, ASCE, 103(HY12), 1391–1401. Roberts, C. P. R. (1980). Hydraulic Design of Dams, Report, Division of Special Tasks, Department of Water Affairs, Forestry and Environmental Conservation, Pretoria. Roberts, D. F. (1943). The dissipation of energy of a flood passing over a high dam, Transactions, South African Institution of Civil Engineers, XLI:48–92. Schulz, H. E., Nobrega, J. D., Simoes, A. L. A., Schulz, H., & Porto, R. D. M. (2015). Details of hydraulic jumps for design criteria of hydraulic structures, in Hydrodynamics—Concepts and Experiments, Schulz, H. E. (Ed.), Chapter 4: 73–116, ISBN: 978-953-51-2034-6, https://doi. org/10.5772/58963. United States Bureau of Reclamation. (1987). Design of Small Dams (3rd ed.). Bureau of Reclamation, United States Government Printing Office, Washington, D.C.: United States Department of the Interior.

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Valiani, A. (1997). Linear and angular momentum conservation in hydraulic jump. Journal of Hydraulic Research, 35(3), 323–354. Wood, I. R., Ackers, P., & Loveless, J. (1983). General method for critical point on spillways. Journal of Hydraulic Engineering, 109(2), 308–312.

Further Reading Chanson, H. (Ed.) (2015). Energy Dissipation in Hydraulic Structures, CRC Press.

Chapter 7

Flow-Measuring Structures

7.1

Introduction

The discharge in a river or canal can be determined either directly by measuring and integrating the velocity distribution or indirectly by relating measured water levels to discharge through a known rating function. The former approach is particularly useful where only one-off measurements are needed, such as for compiling a stage– discharge relationship, or where velocities are also required, such as for ecohydraulic applications. Velocities are commonly measured using propeller-type current meters but modern instruments for acoustic Doppler current profiling have made the approach considerably more efficient. For continuous monitoring, such as for hydrological purposes or water supply allocations, the indirect approach is reliable and practical and widely used. A stable rating function can be obtained by installing a permanent structure in the channel. Measuring structures are designed, as far as possible, to operate as true controls where the discharge can be related to the upstream head only—often referred to as the ‘modular’ flow condition. They can, however, still function under submerged flow conditions, but this requires measurement of the downstream as well as the upstream flow depth. Some structures which are installed for purposes other than flow measurement can also be used for measurement, including sluice gates, culverts, dam spillways, bridges and diversion structures. Many different kinds of measuring structures have been used, but most are variations of two basic types: weirs, which induce control through vertical contraction of the flow, and flumes, which induce control through horizontal contraction of the flow. This chapter will identify the more common types, provide their rating functions and enable the influence of measurement error on their accuracy to be made.

© Springer Nature Switzerland AG 2020 C S James, Hydraulic Structures, https://doi.org/10.1007/978-3-030-34086-5_7

243

244

7.2

7 Flow-Measuring Structures

Weirs

A weir is a structure which obstructs the flow, causing the water level to rise and resulting in flow over a well defined crest. The water level in the pool upstream of the crest is functionally related to the discharge and can, therefore, be used as a measure of the discharge. Many different types of weir are available for use under different conditions. They are classified according to the shape of the crest or the opening through which the water must pass. The broadest classification distinguishes between broad-crested and sharp-crested (also known as thin-plate) weirs.

7.2.1

Sharp-Crested Weirs

The crest of a sharp-crested weir is formed from steel, with a 90° sharp front edge, a flat horizontal top surface 1–2 mm wide, and a 45° chamfer on the downstream side. A variety of crest alignment shapes have been used, the most common being rectangular and triangular. Rectangular Weirs A rectangular weir has a horizontal crest and vertical sides so that a rectangular opening is formed. There are two types of rectangular weir: 1. A contracted weir has a crest length which is less than the width of the channel, and the vertical sides form part of the sharp crest. The flow must, therefore, contract horizontally as well as vertically through the opening (Fig. 7.1). 2. A suppressed weir has a crest that extends across the full width of the channel, with the channel walls forming the vertical sides (Fig. 7.2). In this case there is no horizontal contraction.

(a) Side view Fig. 7.1 Contracted weir

(b) Front view

7.2 Weirs

245

(a) Side view

(b) Front view

Fig. 7.2 Suppressed weir

The relationship between water level and discharge cannot be determined completely by theoretical analysis. However, by making certain simplifying assumptions, the general form of the relationship can be derived. This can be calibrated or refined for particular geometries by empirical correction. The analysis can be done most simply for the suppressed rectangular weir, where the flow can be considered to be two-dimensional (Fig. 7.3). It is required to relate the discharge over the weir to the upstream energy (H) or preferably to the upstream flow depth (h), which is directly measurable. (Note that critical flow cannot be assumed over the crest because the non-hydrostatic pressure distribution violates the assumptions under which critical flow results were derived.) The discharge over the weir shown in Fig. 7.3 can be determined by integrating the velocity over the flow depth above the crest (this could be done at any other location, but the velocity distribution is best known at the crest). This requires knowledge of the thickness of the nappe above the crest and the distribution of the

total energy Vo2/2g h

H p

W

Fig. 7.3 Two-dimensional flow over a sharp-crested weir

z

246

7 Flow-Measuring Structures

velocity through the nappe. These are not easy to determine, and for the purpose of this general analysis, the following simplifying assumptions are made, with the understanding that empirical corrections will have to be made to account for them later. • There is no vertical contraction over the crest. • The pressure is atmospheric right across the nappe. • There is no energy loss in the approach flow to the crest. Although unrealistic, the first assumption defines the nappe thickness and hence the limits for integration along the vertical direction of the velocity distribution, i.e. z from V2o/2 g to H, where Vo is the approach velocity, g is gravitational acceleration and H is the total energy relative to the weir crest. (The kinetic energy correction factor is neglected for simplicity, and will also be accounted for in the empirical correction to follow.) The second assumption is also unrealistic: the pressure is indeed atmospheric at the upper and lower surfaces, but it is greater than atmospheric in between, the actual distribution being difficult to determine. But this assumption implies that the distance from any point within the nappe to the energy line is the velocity head only, and this enables the velocity distribution to be determined as v¼

pffiffiffiffiffiffiffi 2gz

in which v is the local velocity and z is the distance measured downwards from the energy line. (Note that this application of the Bernoulli equation across streamlines is valid because the flow across the nappe can be assumed to be irrotational.) The third assumption is reasonable because the crest causes flow contraction only, and does not introduce any energy dissipating mechanisms. This enables the position of the energy line to be defined above the crest, which is necessary to quantify z. Making these assumptions, the unit width discharge over the weir crest is given by

and so

H

q¼

Z

¼

Z

¼

2 pffiffiffiffiffi 3=2 2gz 3

Vo2 2g

vdz

H

Vo2 2g

pffiffiffiffiffiffiffi 2gzdz

H

Vo2 2g

7.2 Weirs

247

2 pffiffiffiffiffi 2g q¼ 3

H 3=2

 2 3=2 ! Vo 2g

ð7:1Þ

It is generally more convenient in practice to relate discharge to upstream flow depth, rather than the upstream energy level because it is the water level that is measured. Therefore, substituting H ¼ hþ

Vo2 2g

into Eq. (7.1) gives  3=2  2 32 ! Vo2 Vo hþ 2g 2g  3=2  2 32 ! 2 pffiffiffiffiffi 3=2 Vo2 Vo 2gh 1þ ¼ 3 2gh 2gh

2 pffiffiffiffiffi q¼ 2g 3

A contraction coefficient, Cc, is introduced to account for the assumption of no contraction over the crest, i.e. 2 pffiffiffiffiffi q ¼ Cc 2gh3=2 3

 3=2  2 32 ! Vo2 Vo 1þ 2gh 2gh

The contraction coefficient and the term in brackets both depend on the approach flow conditions, and the two terms are generally combined and denoted the discharge coefficient, Cd. Then 2 pffiffiffiffiffi q ¼ Cd 2gh3=2 3

ð7:2Þ

Discharge coefficients are often grouped with constant terms in the equation, and variants of Eq. (7.2) may not explicitly include the 2/3 term or both this and the (2 g)0.5 term. These will then be incorporated in Cd, and care should always be taken when using published empirical coefficient values to ensure that they are appropriate for the equation forms actually used. The discharge coefficient accounts for the assumptions in the derivation and must be determined empirically for particular weir geometries. It is clear from the above derivation that Cd in Eq. (7.2) depends on the approach velocity and the upstream water level. It is, therefore, not a constant for a given weir, and depends on both the water level and the approach geometry. Various formulations have been proposed for Cd in terms of the geometric characteristics of the weir and the approach flow depth (see, for example, Herschy (1995), Ackers et al. (1980) and

248

7 Flow-Measuring Structures

Wessels and Rooseboom (2009b)). Henderson (1966) presents a relationship derived from experimental work by Rehbock in 1929 for Cd in terms of h and W (the height of the crest above the bed), i.e. Cd ¼ 0:611 þ 0:08

h W

for

h \5 W

ð7:3Þ

Notice that for large W, Cd tends to 0.611. A large W implies a small approach velocity, in which case Cc = Cd (according to the substitution made above). This value of Cc (0.611) is the same as the theoretical Cc for a two-dimensional jet from a large tank. Equation (7.3) does not give good results for large values of h/W. As W tends to zero, the condition approaches that for a free overfall, where critical flow occurs a short distance upstream from the brink. For h/W > 20, Cd can be approximated by Henderson (1966) 

W Cd ¼ 1:06 1 þ h

3=2

for

h [ 20 W

ð7:4Þ

For 5 < h/W < 10 Eq. (7.3) can be used, but accuracy decreases (Cd = 1.135 for h/W = 10). No results are available for 10 < h/W < 20. The following factors also affect Cd. – The thickness of the crest edge. – Lateral contraction, if present. – The pressure below the nappe. It was assumed in the derivation of Eq. (7.2) that the pressure was atmospheric right across the nappe. For suppressed weirs the edges are sealed and, unless ventilation is ensured, negative pressures will develop and increase Cd. – The distribution of the approach velocity. For a contracted weir with a relatively short length, the velocity of approach of the water going directly over the crest will be greater than the average over the channel width. – The roughness of the upstream face of the weir. A rougher surface will reduce the contraction effect and increase Cd. – Adhesion and surface tension at very low heads. For suppressed weirs the total discharge can be calculated by multiplying the unit width discharge by the crest length, L, i.e. pffiffiffiffiffi 2 Q ¼ qL ¼ Cd L 2gh3=2 3

ð7:5Þ

For contracted weirs there is a lateral contraction and the effective length of the weir is less than the total length and, therefore,

7.2 Weirs

249

Q ¼ q Leff in which Leff is the effective length. The Francis formula (Henderson 1966) approximates the contraction as h/10 on each side, if L > 3 h, and therefore 2 Q ¼ Cd ðL 3

pffiffiffiffiffi 3=2 2gh

ð7:6Þ

 h 0:1 L

ð7:7Þ

0:2hÞ

The effect of the end contractions can also be accounted for by adjusting the discharge coefficient. Webber (1971) cites the Hamilton Smith formula in the British Standard for the discharge coefficient for a weir with end contractions, i.e.  Cd ¼ 0:616 1

The analysis and equations presented above apply only to modular flow, i.e. where the weir acts as a true control and the discharge is uniquely related to the upstream water level. This requires that the downstream water level is too low to affect the flow over the crest. It is, therefore, advisable to ensure this free flow condition when designing a measuring weir. Sometimes free flow cannot be ensured over the whole range of discharges to be encountered. Measurement is still possible when flow is submerged (Fig. 7.4), but the conventional rating equations are obviously not applicable because the discharge depends on both the downstream and upstream water levels. When submerged, the weir induces an energy loss which manifests as an afflux of the upstream water level that can be related to the discharge. For the submerged condition, both water levels must be measured and accounted for in the rating equation. Webber (1971) presents the Villemonte formula for these conditions. This provides a correction to be applied to the discharge calculated assuming free flow, i.e.

Fig. 7.4 Submerged rectangular weir

h hd

250

7 Flow-Measuring Structures

Q ¼ Q 1

 3=2 !0:385 hd h

ð7:8Þ

in which Q* is the actual discharge, Q is the free discharge associated with h, and hd is the height of the downstream water level above the weir crest. Equation (7.8) suggests that the downstream water level will begin to affect the flow over the weir as soon as it exceeds the height of the weir crest. Because of the empirical nature of this equation, it should not be applied directly to other types of weir. Generally, the accuracy of flow measurement is decreased by submergence, so it is advisable to ensure free flow as far as possible. This can be done in practice by setting the crest sufficiently high or including a downward step in the canal bed to lower the downstream water level. Example 7.1 A sharp-crested weir is installed in a 4.0 m wide rectangular canal. The weir extends across the full width of the channel and its crest is 1.20 m above the bed. When the discharge is 3.20 m3/s the uniform flow depth in the channel is 1.50 m. What is the flow depth immediately upstream of the weir at this discharge? Solution The uniform flow depth downstream is above the weir crest, so the weir is submerged and the discharge is related to the upstream flow depth by Eq. (7.8), i.e.  hd 3=2 0:385 (see Fig. 7.4) Q ¼ Q 1 h with

pffiffiffiffiffi 2 Q ¼ Cd L 2gh3=2 3

h W h ¼ 0:611 þ 0:08 1:2

Cd ¼ 0:611 þ 0:08

L ¼ 4:0 m hd ¼ y o

W ¼ 1:50

1:20 ¼ 0:30 m

Therefore     pffiffiffiffiffiffiffiffiffiffiffiffi 2 h 0:611 þ 0:08 4:0 2
9:8h3=2 1 3:20 ¼ 3 1:20



 !0:385 0:30 3=2 h

from which, by trial, h = 0.62 m and so the flow depth upstream = h + W = 0.62 + 1.20 = 1.82 m.

7.2 Weirs

251

Triangular Weirs The triangular weir or V-notch (Fig. 7.5) is used when accuracy is important at low discharges, but a range of higher discharges must also be measured. The effective flow width clearly varies with h (Fig. 7.6). Assuming a small approach velocity and applying the same principles as for the rectangular weir, the discharge can be shown to be given by Q¼

8 a pffiffiffiffiffi 5=2 2gh Cd tan 15 2

ð7:9Þ

and Cd = Cc if the small approach velocity condition holds. The most commonly used triangular weirs have a = 90o, for which Cd = 0.585 and then Q ¼ 1:382h5=2

ð7:10Þ

in SI units.

Fig. 7.5 A laboratory triangular weir Fig. 7.6 Triangular weir geometry

α

h

252

7 Flow-Measuring Structures

General Many equations for sharp-crested weirs have been presented over a long period of time. These show considerable discrepancies because of their inherent empiricism and the different conditions for which they were developed. Unless these conditions are reproduced exactly in design, the equations will be inaccurate. It is, therefore, meaningless to provide equations without details of the weir geometry for which they are applicable. For this reason various authorities have developed standard weir shapes with calibrated rating formulas. Henderson (1966) presents the British Standard Specifications, for example. If a weir is constructed which does not conform to a standard geometry, the rating equation can be expected to be inaccurate and it should be calibrated. This should be done using an independent measuring technique with significantly greater accuracy than that of the weir itself, e.g. volumetric measurement. Further details concerning discharge coefficient values and design requirements are provided by Herschy (1995) and Ackers et al. (1980). Other Weir Geometries Many other different weir geometries have been proposed for particular purposes, and standard geometries and design details with associated rating equations have been proposed by various authorities. A design commonly used in the United States of America is the Cipolletti weir, which has a trapezoidal shape with the sides sloping at 1:4 (horizontal to vertical). This shape is intended to compensate for the reducing effect of the vertical sides of contracted weirs, as accounted for by Eqs. (7.6) and (7.7). Design details and discharge coefficient values are provided by Herschy (1995). Other geometries have been proposed for contriving particular discharge characteristics for controlling water levels, matching simple recording devices or establishing certain error characteristics. Some of these unusual weirs, with their discharge relationships, are described by Troskolanski (1960). It is sometimes useful to extend the length of a weir crest if upstream water levels are restricted, or if water levels are required to remain within a small range over a wide range of discharges. This can be done by orienting the crest at an angle across the channel or using a curved or labyrinth planform. The general weir equations are applicable to such extended weirs, but the discharge coefficients may need calibration, especially if the flow is made to converge significantly. For flow measurement, it is desirable for the water level to vary significantly over the range of measurement, and this can be enhanced by using compound weir geometries where low flows are confined to narrow portions of the structure. It is common in river measuring weirs to use a short rectangular or triangular section for very low flows, and increasing lengths of rectangular weir as discharge increases (Fig. 7.7). The rating curves for such weirs are complicated and usually require individual calibration.

7.2 Weirs

253

Fig. 7.7 A compound river measuring weir

7.2.2

Broad-Crested Weirs

A broad-crested weir is simply an extended horizontal surface with sufficient length and height above the channel bed to induce critical flow with parallel streamlines (Fig. 7.8). The discharge, Q, over the weir is given, by continuity, as Q ¼ AV A is the flow area at the critical section, given by A ¼ byc

Energy H

Fig. 7.8 Broad-crested weir

yc

254

7 Flow-Measuring Structures

where b is the width of the crest and yc is the critical flow depth. The velocity at the critical condition is related to the flow depth by V¼

pffiffiffiffiffiffiffi gyc

Assuming negligible energy loss over the crest, the critical flow depth can be related to the upstream energy level above the crest, H, by 2 yc ¼ H 3 The discharge is, therefore, given by Q ¼ AV  sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 2 2 g H ¼b H 3 3 or Q ¼ Cd b

pffiffiffiffiffiffiffiffiffi gH 3

ð7:11Þ

with Cd = (2/3)3/2 = 0.544. Unfortunately, the assumptions underlying this derivation are not sufficiently realistic for the result to have the accuracy required for flow measurement. The flow separates at the sharp front edge of the crest, causing energy loss and sloping and curvature of the streamlines where critical flow occurs at the point where the separation zone is thickest. Various attempts have been made to describe this phenomenon and calibrate Cd. Separation at the leading edge can be avoided by rounding the corner and thereby increasing Cd. Herschy (1995) provides discharge coefficients for both sharp and rounded broad-crested weirs. The Crump Weir The Crump weir overcomes the difficulties associated with normal broad-crested weirs. It has a triangular longitudinal section (Fig. 7.9) and either a horizontal or V-shaped crest, with lateral crest slopes of 1:10, 1:20 or 1:40. This weir is commonly used for river gauging and is one of the main types used in South Africa (Wessels and Rooseboom 2009a). It is simple to construct, has a stable and relatively constant discharge coefficient in the modular range and is not very sensitive to submerged conditions. A typical installation is shown in Fig. 7.10. The discharge relationship for horizontal Crump weirs under modular conditions is given by BSI 3680 (1986) as Q ¼ Cd

2 3

rffiffiffiffiffiffi 2 gbH 3=2 3

ð7:12Þ

7.2 Weirs

255

total energy V H

2/2g

h

Dv

P

upstream measurement

1

1

1

n

5

2

≥2Hmax

≥Hmax 2Hmax

(b) (alternative) V-shaped crest

(a) Longitudinal section Fig. 7.9 Crump weir geometry

Fig. 7.10 A Crump weir showing structure (a) and in operation (b)

with  Cd ¼ 1:163 1

0:0003 h

3=2

ð7:13Þ

The equation is applicable for h  0.060 m, P  0.060 m, b  0.30 m, h/ P  3.5 and b/h  2.0. Wessels and Rooseboom (2009b) point out that within these limitations Cd varies by less than 1%, and they suggest using a constant value of 1.163 for practical applications. For submerged conditions, a second flow depth measurement is required. The standard design provides for measurement through tappings just downstream of the crest, but Wessels and Rooseboom (2009b) suggest measurement downstream of the hydraulic jump in rivers because the standard tappings tend to block with sediment in river structures. The discharge equation for submerged conditions is given by Ackers et al. (1980) as

256

7 Flow-Measuring Structures

2 Q ¼ Cd f 3

rffiffiffiffiffiffi 2 gbH 3=2 3

ð7:14Þ

with Cd given by Eq. (7.13). The factor f accounts for submergence according to f ¼ 1:00 f ¼ 1:035 0:817 f ¼ 8:686



H2 H

for

H2  0:75 H

4 !0:0647 

H2 8:403 H



ð7:15Þ

for 0:75\

for 0:93\

H2  0:93 H

H2  0:985 H

ð7:16Þ ð7:17Þ

In Eqs. (7.15)–(7.17) H2 is the downstream energy level above the crest. The discharge relationship for V-Crump weirs under modular conditions is Q ¼ Cd if H is within the V of the crest, and

4 pffiffiffi 5=2 gnH 5

4 pffiffiffi 5=2 gnH 1 Q ¼ Cd 5

 1

ð7:18Þ

DV H

5=2 !

ð7:19Þ

if H is above the top of the triangular part of the crest profile, i.e. H > DV, where DV is the depth of the V. The discharge coefficient in both cases is CD = 0.633 for the specified longitudinal profile, and n is the lateral crest slope as 1:n (Wessels and Rooseboom (2009a, b) suggest n = 10 for normal situations). This calibration holds for h  0.06 m, DV/P  2.5, and H/P  2.5. For submerged conditions (H2/H < 0.78), both Eqs. (7.18) and (7.19) are corrected by the factor f (as for the horizontal crest case), with f given by f ¼ F1 f ¼ F2 þ ðF1

F2 Þð1:5 f ¼ F2

for Hd \0:5 Hd Þ

for 0:5  Hd  1:5

for Hd [ 1:5

ð7:20Þ ð7:21Þ ð7:22Þ

with Hd = H/DV and F1 ¼ 1:2 1

 4 !0:40147 H2 H

ð7:23Þ

7.2 Weirs

257

and F2 ¼ 1:1019 0:914

7.2.3

 4 !0:17353 H2 H

ð7:24Þ

Advantages and Disadvantages of Weirs for Flow Measurement

Weirs have a number of advantages over other means of measuring flows. These include the following. – They can measure a relatively wide range of flows with reasonable accuracy. – They are simple in form and easy to construct. – They can be used in combination with other structures, such as division structures, turnouts, drops, etc. – To a certain degree, they can be made adjustable and portable for small canal applications. They do, however, have some disadvantages, including the following. – They require a greater operating head than other devices, and, therefore, introduce significant head losses to a system. – Weir pools can clog or fill with sediment and must be kept clean to ensure consistency of the rating formula. – Weirs disrupt the passage of migrating fish.

7.3 7.3.1

Flumes Throated (Venturi) Flume

A flume operates in a way similar to the Venturi meter in a pipe, i.e. the flow is contracted over a short distance and variations in the head (or water surface in this case) over the contraction can be related to discharge. Such a contraction can be incorporated in a channel with any cross-sectional geometry. A typical contraction and associated water surface profiles are shown in Fig. 7.11. An expression for discharge in terms of flow depths can be obtained by equating specific energy upstream (section 1) and at the throat (section 2), i.e.

258

7 Flow-Measuring Structures

throat

b3 = b1

b2

b1

(a) Plan geometry energy line V12/2g

V32/2g

V22/2g

y1

y3

y2

(b) Submerged flow condition energy line V1

2/2g

y1

V22/2g

V32/2g y3

y2 = yc

(c) Unsubmerged flow condition Fig. 7.11 Throated flume

E1 ¼ E2 i.e. y1 þ

V12 V2 ¼ y2 þ 2 2g 2g

y1

y2 ¼

from which V22

V12 2g

7.3 Flumes

259

Also, from continuity, Q ¼ V1 b1 y1 ¼ V2 b2 y2 from which V1 ¼

V2 b2 y2 b1 y 1

which can be substituted into the energy relationship to give y1

V2 y2 ¼ 2 2g

1

  ! b2 y 2 2 b1 y 1

This can be rearranged to obtain the following expression for the velocity at section 2: pffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2g y1 y2 V2 ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1

b 2 y2 b 1 y1

2

ð7:25Þ

A coefficient of discharge, Cd, is introduced to account for the effect of nonuniform velocity distribution and a small energy loss. The discharge is then given by Q ¼ V2 Cd b2 y2 and, substituting for V2 from Eq. (7.28) gives

in which

pffiffiffiffiffi 2gCd b2 y2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi y1 y2 1 m2 m¼

ð7:26Þ

b2 y 2 b1 y 1

Cd has a value in the range of 0.96–0.99. Equation (7.26) is similar to the equation for discharge through a pipe contraction but, because of the free surface, the flow area and m are variable. Equation (7.26) is general and applies to both flow conditions shown in Fig. 7.11. The submerged condition is, however, unsatisfactory for practical flow measurement, although it is sometimes unavoidable. In this case, the use of the

260

7 Flow-Measuring Structures

equation is complicated and two water level measurements are required. Also, y2 is not very stable, making its measurement and the small difference (y1 – y2) inaccurate. For these reasons, throated flumes are designed to operate unsubmerged or ‘free’, as far as possible. This is achieved by selecting a throat width that will induce critical flow in the throat, backing up upstream, and a short reach of supercritical flow downstream, followed by a hydraulic jump. Under these conditions flow is critical in the throat and so

which implies

V2 Fr ¼ 1 ¼ pffiffiffiffiffiffiffi gy2 V2 ¼

pffiffiffiffiffiffiffi gy2

The discharge can, therefore, be expressed as Q ¼ b2 y2 V2 pffiffiffi 3=2 ¼ b2 gy 2

It is difficult to locate the critical section exactly, so it is usually the upstream depth (y1) that is measured. It can be shown (e.g. Linford 1961) that the ratio y2/y1 is constant for all discharges, although it varies with the ratio b1/b2. The discharge can, therefore, be expressed in terms of y1. Therefore, pffiffiffi 3=2 Q ¼ b2 gy1 x3=2

in which x = y2/y1. As before, to account for nonuniform velocity and energy losses, a discharge coefficient (Cd) is introduced, and pffiffiffi 3=2 Q ¼ Cd b2 gy1 x3=2

ð7:27Þ

In this equation, Cd has a value of about 0.95. The application requires an evaluation of the x3/2 factor, which depends on b1/b2. If the ratio b1/b2 is large, the approach velocity will be small and the upstream water level (y1) will be approximately equal to the upstream energy level which can be equated to the (critical) energy level in the throat, i.e. y1  y2 þ

V22 gy2 ¼ y2 þ 2g 2g 3 ¼ y2 2

7.3 Flumes

261

from which x¼

y2 2 ¼ y1 3

and therefore,  3=2 pffiffiffi 3=2 2 Q ¼ C d b2 gy 1 3 pffiffiffi 3=2 ¼ 0:544Cd b2 gy1

ð7:28Þ

The value of x3/2 in Eq. (7.28) (i.e. (2/3)3/2 = 0.544) applies for small values of b2/b1. Linford (1961) provided a graph of x3/2 for a range of values of b2/b1, which is reproduced in Fig. 7.12. In practice, b2/b1 will not be less than about 0.70. For Eq. (7.27) to be used in practice, it must be ensured that the flume will operate freely and that the downstream flow will not encroach into the throat. If this happens, then the ratio y2/y1 will not be constant and Fig. 7.12 cannot be used. Two flow depths have to be measured and the general Eq. (7.26) applied. For rectangular channels, it may be assumed that free flow will result if y3/y1  0.8, where y3 is the flow depth in the channel downstream. The design of a free-flowing Venturi flume, therefore, requires an initial calculation of the depth–discharge relationship for the canal. The throat width can then be calculated to ensure that the upstream flow depth is within the range of the measuring instrument and greater than the downstream depth at all discharges. In many cases free flow can be maintained at high discharges but not at low discharges. Rather than narrowing the throat and thereby raising the upstream depth excessively for high discharges, the base of the flume can be raised to induce critical flow or the channel bed level dropped on the downstream side.

0.70

x3/2

0.65 0.60 0.55 0.50 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

b2 /b1 Fig. 7.12 Relationship between depth and width ratios for Venturi flumes under unsubmerged conditions (adapted from Linford 1961)

262

7 Flow-Measuring Structures

As for weirs, compound arrangements may be used to measure a wide range of discharges. For example, a number of flumes may be installed side by side with different base levels.

7.3.2

The Parshall Flume

The most widely used standard measuring flume was developed by Ralph Parshall in 1920 (Parshall 1926). It has the advantage of having a standard shape that can be used over a wide range of flows, with empirically determined discharge equations valid over the whole range. Parshall flumes have been installed with throat widths ranging from 75 mm to 15 m, to measure discharges from 0.25 l/s up to 15 m3/s. They can be used to measure flows using one depth measurement when operating unsubmerged, or using two depth measurements when operating submerged. The geometry of the Parshall flume is rather complicated (Fig. 7.13), making it expensive and difficult to construct. The flume consists of an entrance transition, a converging section, a throat section, a diverging section and an exit section in plan. In the longitudinal section there is a gradual rise of the base through the entrance transition, a level base in the converging section, a downward slope through the throat, followed by a gradual rise through the diverging section. This geometry induces critical conditions at the beginning of the throat section, yielding a relationship between upstream depth and discharge as for throated flumes generally. Dimensions for the different flume sizes are tabulated in Table 7.1 and associated measuring ranges in Table 7.2. For unsubmerged flow, the discharge equation is Q ¼ Khu

ð7:29Þ

in which h is the water level measured at the upstream measuring point, K is a dimensional coefficient that depends on the throat width, and u is an exponent between 1.522 and 1.607, depending on the flume size. According to Herschy (1995), Eq. (7.29) applies for 0.015 m < h < 1.83 m, and with h measured at the location specified in Fig. 7.13 and Table 7.1, i.e. for the dimension c measured along the wall, equal to (b/3 + 0.813) m or 2/3A, upstream from the crest. The values of K and u for use in Eq. (7.29) for the different standard sizes are given in Table 7.2. The following steps provide an approach for sizing a Parshall flume for unsubmerged flow conditions: • Identify all throat width sizes (b) that can measure the required discharge range. • Check the suitability of sizes, starting with the smallest identified.

7.3 Flumes

263

M

B

curved entrance

L

G

h1

R

(h2) c

A

A b

D

C

A alternative straight entrance

diverging section

converging section throat section

entrance transition

exit section

(a) Plan

submerged flow E

h

h2

N

free flow K

Y X

(b) Section A-A Fig. 7.13 Parshall flume geometry

– Check encroachment on the freeboard by calculating h for the maximum discharge and applying energy conservation to a section immediately upstream of the flume entrance. – Check for submergence by ensuring that h2/h is less than the limit given in Table 7.2. h2 is the height of the downstream water surface above the flume bed at the location of h, given by h2 = yo – M/4, where yo is the uniform flow depth downstream (this assumes a horizontal water level at the downstream height).

D (m)

0.167 0.213 0.259 0.396 0.573 0.844 1.02 1.21 1.57 1.93 2.3 2.67 3.03 3.4 4.75 5.61 7.62 9.14 10.67 12.31 15.48 18.53

b (m)

0.025 0.051 0.076 0.152 0.229 0.305 0.457 0.610 0.914 1.22 1.52 1.83 2.13 2.44 3.05 3.66 4.57 6.10 7.62 9.14 12.19 15.24

0.305 0.305 0.305 0.305 0.305 0.381 0.381 0.381 0.381 0.457 0.457 0.457 0.457 0.457 0.457 0.457 0.457 0.457 0.457 0.457 0.457 0.457

M (m)

0.093 0.135 0.178 0.393 0.381 0.610 0.762 0.914 1.22 1.52 1.83 2.13 2.44 2.74 3.66 4.47 5.59 7.31 8.94 10.57 13.82 17.27

C (m)

0.357 0.405 0.457 0.610 0.862 1.34 1.42 1.50 1.64 1.79 1.94 2.09 2.24 2.39 4.27 4.88 7.62 7.62 7.62 7.92 8.23 8.23

B (m) 0.076 0.114 0.152 0.30 0.30 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.61 0.91 0.91 1.22 1.83 1.83 1.83 1.83 1.83

L (m) 0.204 0.253 0.30 0.61 0.46 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 1.83 2.44 3.05 3.66 3.96 4.27 4.88 6.10

G (m) 0.153−0.229 0.153-0.253 0.305−0.610 0.61 0.76 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 0.91 1.22 1.52 1.83 2.13 2.13 2.13 2.13 2.13

E (m) 0.029 0.043 0.057 0.114 0.114 0.228 0.228 0.228 0.228 0.228 0.228 0.228 0.228 0.228 0.34 0.34 0.46 0.68 0.68 0.68 0.68 0.68

N (m) 0.019 0.022 0.025 0.076 0.076 0.076 0.076 0.076 0.076 0.076 0.076 0.076 0.076 0.076 0.152 0.152 0.229 0.31 0.31 0.31 0.31 0.31

K (m) 0.363 0.415 0.466 0.719 0.878 1.37 1.45 1.52 1.68 1.83 1.98 2.13 2.29 2.44 2.74 3.05 3.50 4.27 5.03 5.79 7.31 8.84

A (m) 0.241 0.277 0.311 0.415 0.588 0.914 0.966 1.01 1.12 1.22 1.32 1.42 1.52 1.62 1.83 2.03 2.34 2.84 3.35 3.86 4.88 5.89

c (m) 0.008 0.016 0.025 0.051 0.051 0.051 0.051 0.051 0.051 0.051 0.051 0.051 0.051 0.051

X (m)

0.013 0.025 0.038 0.076 0.076 0.076 0.076 0.076 0.076 0.076 0.076 0.076 0.076 0.076

Y (m)

Table 7.1 Dimensions of standard Parshall flumes (adapted from Herschy 1995). Note: flumes with throat sizes 0.076–2.44 m have approach aprons with 1:4 slopes and rounded entrances with R = 0.4 m for 0.076–0.228 m throats, R = 0.51 m for 0.30–0.90 m throats and R = 0.60 m for 1.2–2.4 m throats

264 7 Flow-Measuring Structures

7.3 Flumes

265

Table 7.2 Discharge characteristics of Parshall flumes (adapted from Herschy 1995) Throat width b (m)

Discharge range Minimum Maximum (l/s) (l/s)

K

u

Head range Minimum Maximum (m) (m)

Modular limit h2/h

0.025 0.051 0.076 0.152 0.229 0.305 0.457 0.610 0.914 1.219 1.524 1.829 2.134 2.438

0.09 0.18 0.77 1.50 2.50 3.32 4.8 12.1 17.6 35.8 44.1 74.1 85.8 97.2 (m3/s) 0.16 0.19 0.23 0.31 0.38 0.46 0.60 0.75

0.0604 0.1207 0.1771 0.3812 0.5354 0.6909 1.056 1.428 2.184 2.953 3.732 4.519 5.312 6.112

1.55 1.55 1.55 1.58 1.53 1.522 1.538 1.550 1.566 1.578 1.587 1.595 1.601 1.607

0.015 0.015 0.03 0.03 0.03 0.03 0.03 0.046 0.046 0.06 0.06 0.076 0.076 0.076

0.21 0.24 0.33 0.45 0.61 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76 0.76

0.50 0.50 0.50 0.60 0.60 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70

7.463 8.859 10.96 14.45 17.94 21.44 28.43 35.41

1.60 1.60 1.60 1.60 1.60 1.60 1.60 1.60

0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09

1.07 1.37 1.67 1.83 1.83 1.83 1.83 1.83

0.80 0.80 0.80 0.80 0.80 0.80 0.80 0.80

3.048 3.658 4.572 6.096 7.620 9.144 12.192 15.240

5.4 13.2 32.1 111 251 457 695 937 1427 1923 2424 2929 3438 3949 (m3/s) 8.28 14.68 25.04 37.97 47.14 56.33 74.70 93.04

Parshall flumes can be used to measure discharges under submerged conditions, which requires the additional measurement of the flow depth (h2) at the downstream measuring point. As with all measuring structures this is not a desirable flow condition, but may be unavoidable considering the relatively low modular limits of Parshall flumes, especially for the smaller sizes, (see Table 7.2). For submerged flow, corrections to the free flow calculated discharge can be made using the second flow depth measurement, as presented by Aisenbrey et al. (1974), ISO 9826 (1992) and the United States Bureau of Reclamation (1997). Any flume which is to operate under free flow conditions only may be modified downstream of the throat section without affecting the discharge relationship. It is therefore quite common to install a Parshall flume at the head of a drop structure by continuing the throat slope into the slope of the drop. The complicated geometry and low modular limits of Parshall flumes have led to their declining popularity in favour of simpler types, such as the cutthroat flume and long-throated structures.

266

7 Flow-Measuring Structures Example 7.2 A concrete-lined (n = 0.013) canal on a slope of 0.00060 has a trapezoidal cross section with bottom width (B) of 2.0 m and side slopes (S) of 1.2H:1V. Select a Parshall flume size to be installed in the canal at the construction stage to measure discharges ranging from 0.22 to 2.40 m3/s. Modular flow should be ensured at all discharges and the maximum allowable afflux is 0.20 m above the uniform flow depth. (The uniform flow depths for the maximum and minimum discharges are 0.72 m and 0.18 m respectively.) Solution From Table 7.2, the minimum throat width for measuring the maximum discharge is 1.524 m and the maximum throat width for measuring the minimum discharge is 3.658 m. The selected size should be the smallest within this range that doesn’t cause unacceptable afflux or submergence of the throat. Try throat width (b) = 1.524 m. From Table 7.2, the discharge equation is Q ¼ 3:732h1:587 and the modular limit is 0.70. From Table 7.1, the dimensions necessary for calculating afflux and submergence conditions are (refer to Fig. 7.13). A ¼ 1:98 m; c ¼ 1:32 m; D ¼ 2:30 m; and M ¼ 0:457 m Check for allowable afflux: H afflux huniform

Eu

1

u

Eu ¼ E1 þ M

E1

h1

hu M

1 4

E1 ¼ h1 þ

V12 2g

1=1:587 Q 3:732   2:40 1=1:587 ¼ ¼ 0:757 m 3:732 Q V1 ¼ A1

h1 ¼ h ¼



  c ð D bÞ A 1 ¼ h1 b þ 2 A 2   1:32 ð2:30 1:524Þ ¼ 0:757 1:524 þ 2 1:98 2 ¼ 1:545 m2

7.3 Flumes

267

Therefore V1 ¼

2:40 ¼ 1:553 m/s 1:545

and E1 ¼ 0:757 þ

1:5532 ¼ 0:880 m 2
9:8

So E u ¼ hu þ

Vu2 1 ¼ 0:880 þ 0:457 4 2g Q Vu ¼ Au Au ¼ Bhu þ Sh2u

¼ 2:00
hu þ 1:20
h2u

So Vu ¼

2:40 2:00
hu þ 1:20
h2u

Therefore hu þ

 2 1 2:40 ¼ 0:994 2
9:8 2:00
hu þ 1:20
h2u

from which, by trial, hu = 0.963 m Therefore the afflux = hu – huniform = 0.96 – 0.72 = 0.24 m, which exceeds the allowable value of 0.20 m. The throat width of 1.524 m is therefore unacceptable and the next larger size is tested. Try throat width (b) = 1.829 m. From Table 7.2, the discharge equation is Q ¼ 4:519h1:595 and the modular limit is 0.70. From Table 7.1, the dimensions necessary for calculating encroachment and submergence conditions are (refer to Fig. 7.13) A ¼ 2:13 m; c ¼ 1:42 m; D ¼ 2:67 m; and M ¼ 0:457 m Check for allowable afflux: Following the same procedure as for the previous trial throat width, the afflux is 0.14 m, which is acceptable. Check for submergence by ensuring the modular limit is not exceeded:

h

M

h2

hd

268

7 Flow-Measuring Structures If the channel bed downstream is at the same level as upstream, and the water level is assumed to be horizontal on the downstream side, h2 ¼ hd

M

1 4

where hd is the uniform flow depth. At the maximum discharge, therefore, h2 ¼ 0:72

1 0:457 ¼ 0:606 m 4

The throat flow depth is h¼



 2:40 1=1:595 ¼ 0:673 m 4:519

and the submergence ratio is h2 0:606 ¼ 0:900 ¼ 0:673 h which is greater than the modular limit (0.70) and, therefore, unacceptable. At the minimum discharge, h2 ¼ 0:18

1 0:457 ¼ 0:067 m 4

The throat flow depth is h¼



 0:22 1=1:595 ¼ 0:150 m 4:519

and the submergence ratio is h2 0:067 ¼ 0:447 ¼ 0:150 h which is less than the modular limit (0.7) and, therefore, acceptable. A flume with a throat width of 1.829 m and the same bed level upstream and downstream is unacceptable because submergence will occur at the higher range discharges. A larger throat width will increase the submergence ratio, and so to ensure modular flow the channel bed downstream of the flume should be lowered (which is feasible at the construction stage). To ensure modular flow, h2 at the maximum discharge should be no greater than h2 ¼ h
modular limit ¼ 0:673
0:70 ¼ 0:471 m Again assuming a horizontal water surface on the downstream side, the level of the downstream channel bed should be below the flume crest by at least the uniform flow depth less h2, i.e. 0.720 – 0.471 = 0.249 m, i.e. 0.249 – 0.457/4 = 0.135 m below the original level.

7.3 Flumes

7.3.3

269

The Cutthroat Flume

The cutthroat flume was originally developed by Skogerboe et al. (1967) and has been widely used in irrigation canals. The geometry is very simple (Fig. 7.14): it has a flat base and straight and vertical converging and diverging side walls. This simple geometry enables the same forms to be used for installations with different throat widths. The flat base enables it to be placed in a lined canal with no excavation, making it suitable for portable construction. Flume lengths ranging from 0.45 to 2.7 m have been used to measure discharges up to 1.4 m3/s. The flume can operate under free flow conditions, or with submergence up to 95%. For free flow the discharge equation is Q ¼ Chna1

ð7:30Þ

in which Q is the discharge in m3/s, ha is the water level measured at the upstream piezometer tapping point, C is the free flow discharge coefficient, and n1 is an empirical exponent. C is given by C ¼ KW 1:025

ð7:31Þ

in which W is the throat width in metres, and K is an empirical coefficient. Both n1 and K depend on the flume length (L). Skogerboe et al. (1972) presented graphical relationships for these, which can be expressed as n1 ¼ 1:828L

0:172

for 0:45\L\2:7

ð7:32Þ

K ¼ 3:414L

0:492

for 0:84\L\2:7

ð7:33Þ

and

B = W + L/4.5

3

6 1

1

ha piezometer tap

2L/9

5L/9

L/3

2L/3 L

Fig. 7.14 Cutthroat flume plan geometry

hb piezometer tap

W

270

7 Flow-Measuring Structures

For submerged flow, two flow depth measurements are required. The discharge is given by Q¼

C1 ðha hb Þn1 ð log SÞn2

ð7:34Þ

in which S is the submergence, hb/ha, where hb is the water level measured at the downstream piezometer tapping point, and C1 is the submerged flow coefficient, given by C1 ¼ K1 W 1:025

ð7:35Þ

K1 and n2 also depend on the flume length according to graphical relationships provided by Skogerboe et al. (1972), which can be expressed as K1 ¼ 2:065L

0:733

for 0:45\L\2:7

ð7:36Þ

n2 ¼ 1:452L

0:055

for 0:84\L\2:7

ð7:37Þ

and

It is necessary to determine whether or not the flume is submerged at a particular discharge so that the correct discharge equation can be selected. The transition submergence, St, can be found by equating the right hand sides of Eqs. (7.30) and (7.34) (which must give the same result at this condition), and solving for S. The solution for St (%) is given graphically by Skogerboe et al. (1972) and can be expressed for 0.45 < L < 2.7 as St ¼ 67:54L0:158

ð7:38Þ

This value can be compared with the actual submergence to select the appropriate equation. Growing acceptance of the cutthroat flume has led to more detailed examination of the discharge equations and further developments and refinements in calibration have been presented by Keller (1984), Skogerboe et al. (1993), Weber et al. (2007), Manekar et al. (2007) and Torres and Merkley (2008). The equation for free flow presented by Manekar et al. (2007) is based on experiments with a wide range of sizes and is demonstrably more accurate than Eqs. (7.30) and (7.31) with coefficients given by Eqs. (7.32) and (7.33). They propose the following single equation in non-dimensional terms for all flume sizes.  1:7053 Q ha pffiffiffi 1:5 ¼ 0:9169 L gWL

ð7:39Þ

7.3 Flumes

271

ha

hb Δs

yo

Fig. 7.15 Cutthroat flume with raised bed

Weber et al. (2007) found the discharge relationship for submerged conditions to be improved if it was expressed in terms of specific energy rather than flow depth at the two measurement points, but their results are based on experiments with just one flume size. Torres and Merkley (2008) found that a single equation could be used for both submerged and free flow conditions but, again, their results are for just one flume size and are yet to be generalized. As for the Parshall flume, it is highly desirable for the cutthroat flume to operate freely over the whole range of discharges to be measured. It has been suggested that this can be ensured by raising the flume base above the channel bottom by Ds (Fig. 7.15) so that hb  haSt at all discharges. However, this will result in an increased afflux of approximately the same amount, even if the throat width is also increased. If the modular flow is not possible within the afflux constraint, then the flume should be designed for submerged flow with two water level measurements. Equation (7.34) should then be used for afflux and submergence calculations. Submerged flow causes increased afflux, so a wider throat would be necessary to meet the same afflux limit as for modular flow. As with most hydraulic structures, the design is an iterative process of analysis and adjustment of an initially assumed geometry. A suitable procedure would be as follows: • Choose a value for B to be accommodated within the canal width. • Assume a value for W which, together with B implies a value for L (see Fig. 7.14). • Use this value of L to obtain values for n1, K and St from Eqs. (7.32), (7.33) and (7.38) if Eqs. (7.30) and (7.31) are to be used or just for St if Eq. (7.39) is to be used. • Calculate ha for the design discharge from Eqs. (7.30) or (7.39) • Check possible encroachment on the canal freeboard by calculating the afflux, as follows: – Calculate the specific energy at the position of ha, i.e. E a ¼ ha þ

Va2 2g

272

7 Flow-Measuring Structures

– where Va = Q/Aa with Aa calculated from the geometry shown in Fig. 7.14. – Calculate the specific energy upstream of the structure as Eu ¼ Ea • Calculate the flow depth upstream from the specific energy and compare it with the allowable value (yo + afflux allowed). • Check for submergence, i.e. whether hb < ha St. If this condition is not satisfied, and increased afflux can be accommodated, the bed should be raised by Ds = yo – ha St, where yo is the uniform flow depth downstream (this assumes a level water surface downstream from the position of hb and uniform flow). If the bed is raised the afflux should be recalculated, with Ds added to Ea and compared with the allowed value. • If hb < ha St cannot be obtained within the allowable afflux, then modular flow cannot be assured and the flume will have to be operated under submerged conditions, with both ha and hb measured. Example 7.3 A concrete-lined (n = 0.013) canal on a slope of 0.00050 has a trapezoidal cross section with a bottom width of 1.40 m and side slopes (S) of 1H:1 V. Select a cutthroat flume size to be installed in the canal after construction to measure discharges ranging from 0.30 m3/s to 1.20 m3/s. The maximum allowable afflux is 0.20 m above the uniform flow depth. (The uniform flow depths for the maximum and minimum discharges are 0.63 m and 0.29 m, respectively.) Solution The size selection is determined by trial to ensure acceptable afflux and modular flow if possible. Calculations are presented only for the successful trials. A sensible base width (B) is the bottom width of the channel, i.e. 1.40 m. The throat width (W) determines the afflux and the submergence condition: the smaller the throat, the greater the afflux, the larger the throat the more likely it is to be submerged. By trial, the smallest throat width with acceptable afflux is W = 0.81 m. Calculations are shown for this width. For B = 1.40 m and W = 0.81 m, L ¼ 4:5ðB

W Þ ¼ 4:5ð1:40

0:81Þ ¼ 2:655 m

Using Eq. (7.39),

i.e.

 1:7053 Q ha pffiffiffi 1:5 ¼ 0:9169 gWL L   Q ha 1:7053 pffiffiffiffiffiffiffi ¼ 0:9169 2:475 9:80:81
2:4751:5

7.3 Flumes

273

Therefore Q ¼ 1:903h1:7053 a Check for allowable afflux: At the maximum discharge ha ¼



   Qmax 1=1:7053 1:20 1=1:7053 ¼ 0:763 m ¼ 1:903 1:903

Then the upstream flow depth can be calculate from Eu ¼ Ea where E a ¼ ha þ

Va2 2g

with Va ¼

Qmax Aa Aa ¼ ha Wa

and, referring to Fig. 7.14 WÞ 3 2ð1:40 0:81Þ ¼ 0:81 þ 3 ¼ 1:203 m

Wa ¼ W þ

2ðB

So Aa ¼ 0:763
1:203 ¼ 0:918 m2 Va ¼

1:20 ¼ 1:307 m/s 0:918

Ea ¼ 0:763 þ

1:3072 ¼ 0:850 m 2
9:8

Therefore Eu ¼ hu þ

Vu2 ¼ 0:850 2g

274

7 Flow-Measuring Structures

with Vu ¼

Qmax Au

and 1 Au ¼ Bhu þ 2
Sh2u ¼ 1:40hu þ 1
h2u 2 Therefore, hu þ

 2 1 1:20 ¼ 0:850 2
9:8 1:40hu þ h2u

from which, by trial, hu = 0.829 m. Therefore, Afflux ¼ hu huniform ¼ 0:83 0:63 ¼ 0:20; allowable: Afflux is satisfactory for the maximum discharge, and will be less for lower discharges; it is, therefore, satisfactory for the whole range. Check for submergence: Transition submergence is calculated from Eq. (7.38), i.e. St ¼ 67:54L0:158 ¼ 67:54
2:6550:158 ¼ 78:8% At maximum discharge submergence is S ¼

huniform 0:63 ¼ 83% ¼ 0:763 ha

At minimum discharge submergence is S ¼

huniform 0:29 ¼ 85% ¼ 0:339 ha

with  Qmin 1=1:7053 1:903   0:30 1=1:7053 ¼ 1:903

ha ¼



¼ 0:338 m Therefore, S¼

huniform 0:29 ¼ 86% ¼ 0:338 ha

Submergence exceeds the transition submergence over the full discharge range and so modular flow is not possible for W = 0.81 m on a level bed. Raising the bed can ensure modular flow only by allowing greater afflux. The flume must, therefore, be designed for submerged flow, with two water level measurements. The afflux is greater for submerged flow than for modular flow, and, therefore, a larger throat width is required. By trial, the smallest throat width for acceptable afflux is W = 0.85 m. Calculations are shown for this width. The results of Skogerboe et al. (1972) are used for submerged flow. For B = 1.40 m and W = 0.85 m,

7.3 Flumes

275 L ¼ 4:5ðB

W Þ ¼ 4:5ð1:40

0:85Þ ¼ 2:475 m

The discharge is calculated by Eqs. (7.34)–(7.37), i.e. Q¼

C1 ðha hb Þn1 ð log SÞn2

with C1 ¼ K1 W 1:025 K1 ¼ 2:065L

0:733

¼ 2:065
2:475

0:733

¼ 1:063

Therefore, C1 ¼ 1:063
0:851:025 ¼ 0:900 and n1 ¼ 1:828L

0:172

n2 ¼ 1:452L

0:055

¼ 1:828
2:475

0:172

¼ 1:564

0:055

¼ 1:381

and

¼ 1:452
2:475

and S¼

hb huniform 0:63 ¼ ¼ ha ha ha

Check for allowable afflux: At Qmax

1:20 ¼

0:900ðha 0:63Þ1:564  1:381 log 0:63 ha

from which, by trial, ha = 0.767 m. Then the upstream flow depth can be calculated from Eu ¼ Ea where

276

7 Flow-Measuring Structures

Ea ¼ ha þ

Va2 2g

with Va ¼

Qmax Aa Aa ¼ ha Wa

and, referring to Fig. 7.14 WÞ 3 2ð1:40 0:85Þ ¼ 0:85 þ 3 ¼ 1:217 m

Wa ¼ W þ

2ðB

So Aa ¼ 0:766
1:217 ¼ 0:932 m2 Va ¼

1:20 ¼ 1:288 m/s 0:932

Ea ¼ 0:767 þ

1:2882 ¼ 0:852 m 2
9:8

Therefore, Eu ¼ hu þ

Vu2 ¼ 0:852 2g

with Vu ¼

Qmax Au

and 1 Au ¼ Bhu þ 2
Sh2u ¼ 1:40hu þ 1
h2u 2 Therefore, hu þ

 2 1 1:20 ¼ 0:852 2 2
9:8 1:40hu þ hu

from which, by trial, hu = 0.830 m Therefore, Afflux ¼ hu huniform ¼ 0:83 0:63 ¼ 0:20; allowable: Afflux is satisfactory for the maximum discharge, and will be less for lower discharges; it is, therefore, satisfactory for whole range.

7.4 Long-Throated Structures

7.4

277

Long-Throated Structures

A flow measuring structure that is gaining popularity is one with a horizontal crest (similar to a broad-crested weir) but with an upward approach slope at between 1:2 and 1:3 (Fig. 7.16). The longitudinal profile can be used unchanged for any cross-sectional shape, making it a very versatile design. The geometry may include a lateral contraction as well as the raised sill, so these structures may be similar to both weirs and flumes. The design and establishment of a discharge relationship are described by Bos et al. (1986). The United States Bureau of Reclamation provides computer software for design and calibration.

7.5

Errors and Measuring Ranges

When using weirs—or other measuring structures—it is necessary to know the error involved, which varies with discharge, and the range of discharge over which errors are acceptable. The discharge error, eQ = eQ/Q where eQ is the difference between the estimated and true discharge values, is mainly the result of the indication error in the measurement of the water level, eh = eh/h. The error in reading the scale on the water level gauge, eh is about ±1.0 mm to ±0.5 mm. The discharge error, eQ, can be related to the water level indication error by analysing the rating equation. For a rectangular sharp-crested weir

or

pffiffiffiffiffi 2 Q ¼ Cd L 2gh3=2 3 Q ¼ Kh3=2

Fig. 7.16 Long-throated measuring structure profile

ð7:5Þ

278

7 Flow-Measuring Structures

assuming Cd is constant. Differentiating this equation with respect to h gives dQ 3 1=2 ¼ Kh dh 2 from which 3 dQ ¼ Kh1=2 dh 2 The error in discharge can be related to the indication error as follows: eQ ¼

dQ 32 Kh1=2 dh ¼ Q Kh3=2 3 dh ¼ 2 h

i.e. eQ ¼ 1:5eh

ð7:40Þ

This means that the discharge error is 1.5 times the indication error obtained in measurement of the water level. It is preferable to express the discharge error in terms of the error in the scale reading; this can be done by substituting for h in Eq. (7.40) from the rating equation, i.e. h¼

Q2=3 K 2=3

gives dhK 2=3 Q2=3 1:5K 2=3 eh ¼ Q2=3

eQ ¼ 1:5

ð7:41Þ

in which eh is about ±1 mm. Equation (7.41) shows that the discharge error is greater at lower discharges than at higher discharges. This analysis can be used to determine the minimum head required on a weir to ensure a specified accuracy for the discharge measurement. The maximum discharge is defined by the maximum head that can be accommodated, and the measuring range is defined by

7.5 Errors and Measuring Ranges

279

Qmin ¼ Qmax



hmin hmax

3=2

ð7:42Þ

Example 7.4 If the maximum acceptable error in discharge is ±2% and the indication error is ±1 mm, determine the minimum head and discharge to be measured a. by a 2.0 m long, 1.20 m high suppressed sharp-crested weir, and b. a 90o triangular weir. Solution a. The relationship between discharge error and indication error for a rectangular weir is given by Eq. (7.40), i.e.

eQ ¼ 1:5eh Therefore, eh max ¼

eQ max 2% ¼ 1:33% ¼ 1:5 1:5

The minimum head is therefore

hmin ¼

eh 1:0 mm ¼ 75 mm ¼ 1:33% eh max

and the minimum discharge is given by Eq. (7.5), i.e. pffiffiffiffiffi 3=2 2 Qmin ¼ Cd L 2ghmin 3 with Cd ¼ 0:611 þ 0:08

h 0:075 ¼ 0:611 þ 0:08 ¼ 0:616 W 1:20

Therefore pffiffiffiffiffiffiffiffiffiffiffiffi 2 Qmin ¼ 0:616
2:0 2
9:8
0:0753=2 ¼ 0:075 m3 =s 3 b. Following the same procedure as for the rectangular weir, it can be easily shown that for a triangular weir eQ ¼ 2:5eh Therefore,

280

7 Flow-Measuring Structures

eh max ¼

eQ max 2% ¼ 0:80% ¼ 2:5 2:5

The minimum head is therefore

hmin ¼

eh 1:0 mm ¼ 125 mm ¼ 0:80% eh max

and the minimum discharge is given by Eq. (7.10), i.e. 5=2

Qmin ¼ 1:382hmin ¼ 1:382
0:1255=2 ¼ 0:0076 m3 =s Note that although the triangular weir appear less accurate, a rectangular weir would have to be quite narrow and would not be able to accommodate the same range of discharges with comparable accuracy.

Problems 7:1 The discharge through a triangular weir is described in SI units by Q ¼ 1:382h5=2 If the scale on the water level gauge can be read with an accuracy of ±1.0 mm, determine the lowest discharge that can be measured by this weir with an accuracy better than ±2.0%. 7:2 A long, rectangular, concrete-lined (n = 0.014) canal has a width of 3.0 m and a slope of 0.00030. A sharp-crested weir is installed in the canal and a vertical sluice gate is installed a short distance downstream of the weir. Both the weir and the gate extend across the full width of the canal. Determine the flow depths upstream and downstream of the two structures when the discharge is 2.5 m3/s with the following structure characteristics: a. The weir crest is 2.6 m above the bed and the gate opening is 0.20 m. b. The weir crest is 2.0 m above the bed and the gate opening is 0.20 m. c. The weir crest is 1.0 m above the bed and the gate opening is 0.30 m. 7:3 A concrete (n = 0.013) trapezoidal canal has a bottom width of 1.5 m, side slopes of 1.5H:1 V and a gradient of 0.00050. The canal is designed to convey 1.10 m3/s with a freeboard of 0.25 m. a. Select a Parshall flume to measure flows ranging from 0.11 m3/s to 1.10 m3/s under free flow conditions. Assume the bottom of the canal to be at the same level upstream and downstream of the flume. b. Design a cutthroat flume for measuring the same range of discharges under free flow conditions. Compare conditions using the calibrations originally proposed by Skogerboe et al. (1972) and by Manekar et al. (2007). c. If the same cutthroat flume design is used in another channel with the same geometry but on a gradient of 0.00020, calculate the upstream flow depth

7.5 Errors and Measuring Ranges

281

for a discharge of 0.11 m3/s. Modify the design to ensure modular flow for this discharge. 7:4 A cutthroat flume in a 2.0 m wide, rectangular, concrete channel has a throat width of 1.5 m. a. What is the discharge if ha = 0.949 m and hb = 0.713 m? b. What is the accuracy of the discharge measurement for the conditions in above if the scale on the piezometers can be read with an accuracy of ±1.0 mm? c. What is the discharge if ha = 1.618 m and hb = 1.322 m?

References Ackers, P., White, W. R., Perkins, J. A., & Harrison, A. J. M. (1980). Weirs and flumes for flow measurement (3rd ed.). New York: Wiley. Aisenbrey, A. J., Hayes, R. B., Warren, H. J., Winsett, D. L., & Young, R. B. (1974). Design of small canal structures. United States Department of the Interior, Bureau of Reclamation. Bos, M. G., Clemmens, A. J., & Replogle, J. A. (1986). Design of long-throated structures for flow measurement. Irrigation and Drainage Systems, 1, 75–92. BSI 3280. (1986). Part 4B: Measurement of liquid flow in open channels—triangular profile weirs. London: BSI. Henderson, F. M. (1966). Open channel flow. Macmillan. Herschy, R. W. (1995). Streamflow measurement (2nd ed.). E & F N Spon. ISO 9826 (1992) Measurement of liquid flow in open channels—Parshall and SANIIRI flumes. Keller, R. J. (1984). Cut-throat flume characteristics. Journal of Hydraulic Engineering, 110(9), 1248–1263. Linford, A. (1961). Flow measurement and meters (2nd ed.). E & F N Spon. Manekar, V. L., Porey, P. D., & Ingle, R. N. (2007). Discharge relation for cutthroat flume under free-flow condition. Journal of Irrigation and Drainage Engineering, 133(5), 495–499. Parshall, R. L. (1926). The improved Venturi flume, transactions. American Society of Civil Engineers, 89, 841–880. Skogerboe, G. V., Hyatt, M. L., Anderson, R. K., & Eggleston, K. O. (1967). Design and calibration of submerged open channel measurement structures. Part 3—Cutthroat flumes (pp. 1–37). Utah Water Research Laboratory, College of Engineering, Utah State University. Skogerboe, G. V., Bennett, R. S., & Walker, W. R. (1972). Installation and field use of cutthroat flumes for water measurement. Water Management Technical Report No 19, Colorado State University, Colorado, USA. Skogerboe, G. V., Ren, L., & Yang, D, (1993). Cutthroat flume discharge ratings, size selection and installation. International Irrigation Center, Department of Biological and Irrigation Engineering, Utah State University. Torres, A., & Merkley, G. P. (2008). Cutthroat measurement flume calibration for free and submerged flow using a single equation. Journal of Irrigation and Drainage Engineering, 134 (4), 521–526. Troskolanski, A. T. (1960). Hydrometry. Oxford: Pergamon Press. United States Bureau of Reclamation. (1997). Water measurement manual. Webber, N. B. (1971). Fluid mechanics for civil engineers. London: Chapman and Hall. Weber, R. C., Merkley, G. P., Skogerboe, G. V., & Torres, A. F. (2007). Improved calibration of cutthroat flumes. Irrigation Science, 25, 361–373.

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Wessels, P., & Rooseboom, A. (2009a). Flow gauging in South African rivers, Part 1: An overview. Water SA, 35(1), 1–9. Wessels, P., & Rooseboom, A. (2009b). Flow gauging in South African rivers, Part 2: Calibration. Water SA, 35(1), 11–19.

Further Reading Abt, S. R., Cook, C., Staker, K. J., & Johns, D. D. (1992). Small Parshall flume rating correction. Journal of Hydraulic Engineering, 118(5), 798–803. Charlton, F. G. (1978). Measuring flows in open channels: A review of methods. CIRIA Report 75. Clemmens, A. J., Wahl, T. L., Bos, M. G., & Replogle J. A. (2010). Water measurement with flumes and weirs. Water Resources Publications. Gill, M. A. (1985, August). Flow measurement by triangular broad-crested weir. Water power and dam construction (pp. 47–49). Keller, R. J., & Mabbett, G. O. (1987). Model calibration of a prototype cutthroat flume. Journal of Hydraulic Research, 25(3), 329–340. Ramamurthy, A. S., Rao, M. V. J., & Auckle, D. (1985). Free flow discharge characteristics of throatless flumes. Journal of Irrigation and Drainage Engineering, 111(1), 65–75. Rao, N. S. L. (1975). Theory of weirs. Advances in Hydroscience, 10, 310–406. Samani, Z., & Magallanez, H. (1993). Measuring water in trapezoidal channels. Journal of Irrigation and Drainage Engineering, 119(1), 181–186.

Chapter 8

Intake Structures

8.1

Introduction

An intake structure is a transition device through which flow is diverted from a source, such as a river, reservoir or the ocean, into a conduit, which may be a canal or a pipe. Its primary functions are to admit and regulate (and possibly also to measure) water from the source, and to ensure satisfactory quality of the admitted water (in terms of sediment content, temperature, chemical and biological characteristics). An intake may be an integral part of an impoundment or diversion structure, such as a dam or weir, or it may be entirely independent. In either case, it will generally incorporate a conduit with protective works and screens at the entrance section and gates or valves for regulating the flow. The location and design of intake works must ensure reliability of operation and satisfactory water quality. The factors affecting these considerations depend on the nature of the water body from which the supply is taken. For example, designs would be very different for extraction of water from shallow or deep lakes and reservoirs, and large and small rivers.

8.1.1

Reservoir Intakes

Intakes in large lakes or reservoirs should be located in as deep water as possible to avoid the mixed zone resulting from wind and convection. In impounded reservoirs, intakes are, therefore, usually at or near the dam wall or embankment. This also ensures operation at low water levels. Intakes for deep reservoirs should be designed to enable water to be drawn off at different depths. It is sometimes desirable to avoid drawing surface water into the supply because of excess turbidity during high winds, high temperatures during hot © Springer Nature Switzerland AG 2020 C S James, Hydraulic Structures, https://doi.org/10.1007/978-3-030-34086-5_8

283

284

8 Intake Structures

summer months and the occurrence of plankton near the surface. Many microorganisms that can produce objectionable odours also live near the surface because of their requirement for light. Water at great depths can be rich in carbon dioxide, iron and manganese, and may have excessive colour and hardness. These undesirable characteristics can be avoided by drawing water from near the surface, although it may sometimes be advantageous to draw iron-rich bottom water in order to reduce the quantity of coagulant used for treatment. These considerations are important not only for water supply systems but also for releases from large reservoirs into downstream river reaches where environmental constraints may dictate water quality and temperature ranges; these can often be satisfied by mixing water from different depths through multilevel intakes. Deep reservoir intakes are commonly constructed as towers some distance from the shore, often with connecting bridges. Intake ports are provided at two or three levels, depending on the maximum depth. Gates and screens should be installed inside the tower to facilitate maintenance. The tower may also include two parts with duplicate screens and outlets to enable uninterrupted service during maintenance and improve reliability.

8.1.2

River Intakes

In large rivers intakes should be located so that the ports are always submerged to avoid floating debris and air entrainment. The ports should also be well clear of the bed to avoid intake of coarse sediment. River intakes are, therefore, usually located in the deepest part of the stream, particularly if large variations of river stage occur, and water is conveyed to the shore through a pipe along the bed or a tunnel below the bed. For small supplies, the intake may be simply a screened bell mouth at the end of a pipe. For large supplies, tunnels are often required and intake towers are constructed which incorporate vertical shafts to connect to the tunnel. The intake ports in the tower walls are fitted with gates and often with screens. If the river stage is relatively constant the intake may be constructed on the shore, obviating the need for a tunnel and resulting in a much simpler structure. Most large river intakes require pumping and pumping stations are usually constructed on the river bank adjacent to the intake. The intake and pump station can often be combined if the intake can be built on the shore. The design of pump sumps and intakes present special problems and is discussed in greater detail in Sect. 8.3. Intakes in small streams often requires the construction of a diversion barrage or weir. This ensures a sufficient depth of water for abstraction and also allows some settling of sediment, although the pools often fill quite rapidly. Intakes of this type are often in remote locations and maintenance is intermittent, so measures must be taken to avoid clogging by debris and sediment. Screening is, therefore, important.

8.1 Introduction

285

Bar racks and mesh screens are frequently used with small stream intakes, the former to provide protection for the latter from heavy debris. Various types of flat and drum screens are available.

8.2

River Intake Design for Sediment Control

Many problems associated with river intake structures are related to the sediment being transported by the river. The structures should be designed so that they will not clog with deposited sediment, initiate local erosion, or affect the morphological equilibrium of the natural channel. For water diversions, they should also divert the required quantity of water to the supply aqueduct with minimum sediment content (diversions intended to extract sediment from a river are not considered here). This section describes measures for minimizing the intake of sediment and removing the bulk of diverted sediment from the water at or close to the abstraction point. Complete removal of sediment for domestic or industrial water supply must be performed by fine screening or filtration, which is not usually done at the intake location. For irrigation supply it is rarely justifiable to remove all sediment from the water, in fact, it is frequently desirable to retain fine sediment as this is beneficial in sealing unlined canal laterals and may improve the soil texture and fertility of croplands. Obviously, aqueducts conveying sediment-laden water must be carefully designed to ensure that excessive deposition does not take place. This is difficult as it is usually impossible to maintain sufficiently high velocities at all times and at all points in the system. It must also be borne in mind that excluding sediment at an intake and returning it to the stream can create instability problems in the natural channel downstream, particularly if a large proportion of the stream flow is diverted. The sediment concentration below the diversion point would be increased and could disturb the equilibrium state of water discharge, sediment load and channel morphology. Expensive maintenance or corrective action in the stream, or both, may then be required. The nature and extent of sediment control problems at intakes depend on the type and quantity of sediment transported by the river, which in turn depends on the rate of supply of sediment to the river and its transport capability. For rivers transporting relatively coarse sediments, the supply rate generally exceeds the transport capability; some deposition, therefore, occurs and the actual rate of transport is defined by the transport capability, which can be related to local flow conditions through sediment transport equations. For rivers transporting relatively fine sediments the supply rate is usually less than the transport capability. The actual transport rate is, therefore, determined by the supply rate which depends on catchment characteristics and conditions as well as the history of sediment-producing flows. Under these circumstances it is not possible to correlate sediment concentration with discharge reliably, and it is extremely difficult to establish sediment discharge rates. Site-specific measurement is advisable in all cases.

286

8 Intake Structures

Water in aqueducts can be kept relatively sediment-free by diverting sediment at the headworks, removing it from the aqueduct by ejectors, or inducing deposition in settling basins from where it can be removed mechanically or by sluicing. The structures required for diversion or ejection are very difficult to analyse, and most designs are based on experience gained from model studies or existing prototype performance. A basic understanding of sediment and relevant flow behaviour is, however, essential when designing a general configuration. Of particular practical importance are the vertical distribution of suspended sediment through the flow and the response of near-bed sediment to flow curvature induced by bends.

8.2.1

Vertical Sediment Distribution

The total sediment load in a river can be considered to consist of two main components, namely suspended load and bedload. (Very fine sediment, not represented in the river bed material, is known as wash load and cannot be excluded at intakes.) The bed load consists of the coarser size fractions which are moved along the bed or by saltation under the influence of traction forces. The suspended load consists of fine particles that can be maintained within the flow by the action of turbulence. The vertical distribution of suspended material depends on both particle size and flow characteristics. The vertical concentration distribution of suspended material can be described by applying the ‘diffusion analogy’. According to this analogy, if sediment concentration varies through the flow, then particles will tend to move from regions of high-concentration to regions of low concentration at a rate proportional to the concentration gradient. The proportionality between transfer rate and concentration gradient is related to the momentum diffusivity (or eddy viscosity), which is determined by the turbulence structure of the flow. The concentration is highest near the bed, so particles will tend to be diffused upwards. Particles also tend to settle under the influence of gravity at a rate equal to the concentration multiplied by the particle settling velocity. An equilibrium distribution of suspended sediment concentration can be derived by considering a mass balance of material in an element of fluid under the influence of these two transfer mechanisms. Under equilibrium conditions, the net transfer is zero and, therefore, 0 ¼ wC þ es

@C @y

ð8:1Þ

in which C is the concentration, w is the particle settling velocity (positive downwards), es is the diffusivity for sediment, and y is the height above the bed. The diffusivity for sediment is assumed to be similar to the momentum diffusivity for small particles. Its magnitude and variation with height above the bed can be determined through Prandtl’s mixing length theory and described by

es ¼ ju

y ðD D

yÞ

ð8:2Þ

in which j is the von Karman constant, which has a value of 0.40 for clear water, D is the total flow depth and u* is the shear velocity, defined by u ¼

rffiffiffiffiffi so q

ð8:3Þ

in which so is the bed shear stress, given by so ¼ qgRS

ð8:4Þ

in which q is the water density, g is gravitational acceleration, R is the hydraulic radius (which can be assumed to equal the flow depth for wide channels) and S is the energy gradient (which is equal to the bed gradient for uniform flow). Substituting Eq. (8.2) into Eq. (8.1) and integrating yields C ¼ Ca



D

y y

a D

a

w=ju

ð8:5Þ

commonly known as the ‘Rouse equation’. This equation gives the sediment concentration, C, at any height, y, in terms of the concentration Ca at a specified height a. Ca is usually estimated very close to the bed, using an appropriate bed load equation and assuming a thickness for the bed layer for a. The equation shows that very fine particles are distributed fairly uniformly over the flow depth and that as the particle size increases more sediment is transported close to the bed (Fig. 8.1). In a natural river, where the total sediment load consists of various sizes of sediment moving as bedload and suspended load, the transported sediment will be concentrated in the lower part of the flow. This can be used to advantage when designing sediment removal structures or devices.

288

8 Intake Structures Example 8.1 A long, wide channel has a slope of 0.0020. When the discharge is 2.25 m3/s/m the flow depth is 1.0 m. Compare the suspended sediment concentration profiles over the flow depth for sediment grain sizes of 0.25, 0.125 and 0.062 mm with corresponding settling velocities of 0.0343, 0.0118 and 0.0031 m/s. Assume the sediment concentration to be 0.25 at 1.0 cm above the bed for all sizes. Solution The concentration profile is given by Eq. (8.5), i.e.  w=ju C D y a ¼ Ca y D a with Ca ¼ 0:25 a ¼ 0:010 m

D ¼ 1:0 m pffiffiffiffiffiffiffiffiffiffiffi gDSo for uniform flow in a wide channel pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 9:8
1:0
0:0020 ¼ 0:140 m/s

u ¼

Then C can be calculated for 0.010 m < y < 1.0 m. For example, for y = 0.50 m w 0:0343 ¼ 0:613 ¼ ju 0:40
0:140

For d ¼ 0:25 mm; w ¼ 0:0343 m=s; C ¼ Ca



D

y

a

w=ju

y D a  0:613 1:00 0:50 0:010 ¼ 0:015 ¼ 0:25 0:50 1:00 0:010

For d ¼ 0:125 mm; w ¼ 0:0031 m=s; C ¼ Ca



D

w 0:0118 ¼ 0:211 ¼ ju 0:40
0:140

y

a

w=ju

y D a  0:211 1:00 0:50 0:010 ¼ 0:095 ¼ 0:25 0:50 1:00 0:010

For d ¼ 0:062 mm; w ¼ 0:0118 m=s; C ¼ Ca



D

w 0:0031 ¼ 0:055 ¼ jux 0:40
0:140

y

a

w=ju

y D a  0:055 1:00 0:50 0:010 ¼ 0:25 ¼ 0:194 0:50 1:00 0:010

8.2 River Intake Design for Sediment Control

289

Concentrations for other heights are calculated similarly and plotted and tabulated below. 1.0

y (m)

0.8 0.6 0.4 0.2 0.0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

C d = 0.25 mm

d = 0.125 mm

d = 0.062 mm

y (m)

d = 0.25 mm

d = 0.125 mm

d = 0.062 mm

0.01

0.250

0.250

0.250

0.05

0.091

0.177

0.228

0.10

0.058

0.151

0.219

0.15

0.043

0.137

0.213

0.20

0.035

0.127

0.209

0.25

0.029

0.120

0.206

0.30

0.025

0.114

0.203

0.35

0.022

0.108

0.201

0.40

0.019

0.104

0.198

0.45

0.017

0.099

0.196

0.50

0.015

0.095

0.194

0.55

0.013

0.091

0.192

0.60

0.012

0.087

0.189

0.65

0.010

0.084

0.187

0.70

0.009

0.080

0.185

0.75

0.008

0.076

0.182

0.80

0.006

0.071

0.179

0.85

0.005

0.066

0.176

0.90

0.004

0.060

0.172

0.95

0.002

0.051

0.165

0.98

0.001

0.042

0.156

1.00

0.000

0.000

0.000

290

8.2.2

8 Intake Structures

Bed Load Movement Around Bends

The behaviour of flow around bends is a very important consideration when designing artificial canals or analysing river behaviour. In artificial canals, bends must be designed with radii sufficiently large to avoid excessive energy losses and to ensure satisfactory approach conditions for measuring structures and control structures. The development of tight meander bends in natural rivers may cause increased flood levels upstream and channel stability problems in addition to deleterious effects on structures. Bends also induce local variations in flow velocity which influence the location and movement of transported sediment and are, therefore, important features to be considered in designs for sediment exclusion. An analysis of energy component variations across a bend section shows that the water surface rises towards the outside of the bend, and that the depth-averaged velocity increases towards the inside of the bend. The actual flow distribution is influenced by resistance, and the maximum velocity is near the inner bank at the entrance to the bend but shifts towards the outer bank through the bend. Also, because of the increasing depth (and hence hydrostatic pressure) towards the outer bank and the low velocity near the bed, an inwardly directed pressure gradient exists close to the bed, causing inwardly directed flow. Balancing this is an outwardly directed flow near the water surface. The net result is a secondary, circulatory flow pattern over the cross section and a helical flow through the bend (Fig. 8.2). Raudkivi (1993) provides a more detailed and quantitative description, with results for predicting water surface superelevation, velocity direction and variation across the section, the elevation of the maximum velocity location (which is below the water surface) and the angular location of the completion of secondary flow development. This flow pattern has a very important effect in channel forming processes in natural rivers: sediment tends to be scoured at concave banks and deposited as bars on convex banks. The effect is extremely important for sediment exclusion design. As has already been demonstrated, most sediment tends to be transported close to the bed and at a bend will, therefore, be directed towards the inner bank and away from the outer bank. This phenomenon sometimes referred to as ‘bed load sweep’,

flow near bed flow near surface outer bank, erosion

inner bank, deposition

(a) Cross section Fig. 8.2 Flow pattern through a bend

(b) Plan

8.2 River Intake Design for Sediment Control

291

has been exploited for centuries. The remains of irrigation systems built by ancient Mediterranean civilizations show that branch canals were always connected to the main channel at straight sections or on the outsides of curves. The approach is no less useful today and ensures that sediment concentrations in a branch channel connected to the outside of a curve are much less than in the main channel (Hernandez 1969). In addition to choosing a suitable location for an intake, the angle of diversion between the river and the diversion channel must be selected as this has a significant effect on the amount of sediment diverted. Any diversion at an angle to the river flow creates an effective curvature in the flow (Fig. 8.3). The water near the surface has a higher velocity and hence more momentum than water near the bed and will, therefore, be less easily diverted and will tend to continue with the main river flow. The most easily diverted water will be the slower moving water near the bed, which also has a higher sediment concentration. If the quantity of diversion is significantly relative to the river flow, the take-off effectively creates a curve, resulting in bedload sweep into the diversion channel. As well as causing undesirable sediment content in the deviated water, deposition could occur in the diversion channel as an effective point bar and this could eventually cause blocking. The amount of sediment diverted depends on the head difference between the main and diversion channels and the diversion angle. The flow will always enter the diversion channel at an angle less than 90° so the ideal angle is something smaller than 90°. It is difficult to define an optimal angle because it depends on the diversion ratio, i.e. the ratio of discharges in the diversion and the river, and also on the position of the diversion in a bend. The optimum angle increases as the diversion ratio decreases. The best design approach is to conduct model tests and choose the best angle for the dominant diversion ratio or for the condition producing maximum bedload discharge. If a model study is not undertaken and no sediment deviator is to be installed, an angle of diversion between 10° and 45° is recommended (Avery 1989).

flow near surface flow near bed

diversion angle sediment deposition

Fig. 8.3 Sedimentation at diversion

diversion channel

292

8 Intake Structures

Results of model studies show that, for example, for the main and diversion channel beds at the same level and with an angle of diversion of 30° and a diversion ratio of about l:4, about 50% of the bedload would enter the branch. For a discharge ratio greater than about 50% practically all the bedload would be diverted (Hernandez 1969).

8.2.3

Sediment Exclusion Structures

Three types of structures are commonly used for restricting the amount of sediment in a supply aqueduct: Sediment diverters are intended to prevent most of the sediment from entering the aqueduct; sediment ejectors remove sediment already in the aqueduct; settling basins allow deposition of very fine sediment in the aqueduct for subsequent removal. The appropriate structure for a particular intake depends on the particular conditions. Sediment diversions are intended mainly for the exclusion of bedload. Rivers carrying most sediment in suspension require settling basins. Sediment ejectors would be appropriate where the sediment is transported in suspension in the river but becomes mainly bedload in the aqueduct because of the reduced transporting capacity. The more common approaches to sediment exclusion are described in the following paragraphs; further details are provided by Vanoni (1975), Avery (1989) and Raudkivi (1993). Sediment Diverters A sediment diverter is a device or structure incorporated in the intake headworks for preventing most of the sediment, especially bedload and coarser suspended fractions, from entering the aqueduct. Many different arrangements have been developed either from theoretical analyses, model tests or observations of actual systems over many years. Most take advantage of the vertical distribution of suspended

Fig. 8.4 A river diversion structure looking downstream (a) and upstream (b)

8.2 River Intake Design for Sediment Control

293

sediment and the bedload sweep phenomenon described above. The most basic application is simply to locate the intake on the outside of a bend and as high above the bed as practically possible. In the diversion structures shown in Fig. 8.4 the intake (through trashracks) is on the outside of the bend with a gated sluiceway immediately downstream and in line with the weir to direct the flow and keep the intake clear by sluicing local bed load. More sophisticated designs applying these principles have been developed, such as those described in concept below. Detailed design is made difficult by the uncertainty and variability of future flows and sediment concentrations, as well as the limitations of analytical and computational approaches. Physical model studies provide the most reliable basis for the design. Although they are not usually able to reproduce movement and distribution of sediment quantitatively, they are useful for providing qualitative evaluations and comparisons between different design options. Training Walls and Guide Banks: create an artificial curve in the flow in the vicinity of the intake in order to take advantage of the bedload sweep phenomenon. This approach is often used when the intake is constructed together with a weir-type structure and where river flow is limited, with most of the excess used for sluicing, such as illustrated in Fig. 8.5. The training walls induce curvature in the flow, effectively placing the canal intake on the outside of a bend so that relatively clear water is taken into the canal and high-concentration bedload proceeds to the sluiceway. The canal intake sill is set above the elevation of the sluiceway and has a projecting lip to further ensure that water with higher sediment concentration is washed through the sluiceway. The spacing of the training walls must be designed to ensure sufficient velocity to flush sediment through at low discharges. It should be noted that diversion weirs generally fill with sediment after some time, but a low flow channel to the diversion works can be maintained by regular sluicing.

sluiceway canal training walls weir crest A to canal A

river flow training walls (a) Plan

Fig. 8.5 Training wall configuration

to sluiceway

(b) Section A-A

294

8 Intake Structures

river banks

guide bank

induced deposition

sluiceway guide bank canal Fig. 8.6 Guide bank configuration

Guide banks (Fig. 8.6) produce the same effect as training walls by forcing a realignment of the river channel to create a convex bank at the intake location. Where intakes are required on both ends of a weir the required curvature for both intakes can be created by forming a suitably shaped central island upstream of the weir. In all applications of training walls and banks, the point of diversion must be positioned far enough downstream along the curve for the helicoidal flow pattern producing bedload sweep to have become established. This position depends on size and quantity of bedload, flow velocity and channel geometry and can only be determined reliably by model studies. The Pocket and Divider Wall: arrangement in association with a barrage structure induces deposition ahead of the intake and diverts flow above the high-concentration region near the bed. A divider wall upstream of the barrage structure creates a ‘pocket’ in front of the intake (Fig. 8.7). The divider wall directs part of the river flows into the sluiceway (with convex curvature approaching the intake) and the remainder over the barrage structure. The

Fig. 8.7 Pocket and divider wall configuration

sluiceway

canal barrage crest divider wall pocket

river flow

8.2 River Intake Design for Sediment Control

295

pocket acts as a ponding area for deposition of sediment which can be sluiced away either intermittently or continuously, depending on available stream flow. Whether sluicing is continuous or intermittent, the headgate invert must be set well above the pocket floor—at least one-third of the flow depth is recommended. This provides temporary storage for bedload if intermittent sluicing is used and a bedload exclusion device if continuous sluicing is used. The length of the divider wall is determined largely by the ratio of flow in the stream to that being diverted. If the stream flow is limited, the divider is usually extended as a training wall to create a curved approach and the length and geometry should be determined by modelling. If the stream flow is more than sufficient to provide for the abstraction and sluicing, the divider wall acts merely as a splitter and to isolate the pocket area. In such cases, the divider wall need not extend upstream beyond the headgate. The width of the pocket is best determined by physical modelling. The cross section should be large enough to convey the abstraction and sluicing water, but small enough to maintain sluicing velocities for dominant or design flow conditions. A slight convergence (up to 1:10) improves scouring action, but the pocket must not be too wide at the entrance or sluicing will be ineffective. The diversion barrage is often a gated structure, in which case the careful operation of the gates can greatly enhance the performance of the sediment division works. By opening the gates furthest from the headworks the most and those nearest the headworks the least, an effective curvature of the flow will be created, drawing sediment towards the inside of the curve and away from the headgate. If there are intakes on both sides of the barrage it would be advantageous to open the gates in the centre of the barrage the most. The relative openings of the sluice gates and barrage gates near the divider wall also affect the amount of sediment excluded. The path of the bedload depends on the ratio of the average flow velocities on the riverside and the pocket side of the divider wall. If this ratio is greater than one the bedload tends to be directed to the riverside of the wall, and vice versa. A divider wall can be effective without the provision of a pocket for relatively small diversion proportions (Fig. 8.8). The upstream orientation of the approach to the intake induces flow curvature and bedload sweep towards the main river. This effect can be enhanced by a suitably positioned island or guide walls. A sluicing facility can also be included to reduce sediment build-up ahead of the intake, if necessary. Sand Screens or Skimming Weirs: are simply low barricade walls which allow the relatively clear water near the surface to flow over to the intake, and direct the bed load back into the flow of the stream, i.e. they provide a rudimentary exploitation of the vertical sediment concentration distribution. The tops of the walls are set at a level that will effectively divert the sediment for the dominant or selected design conditions. Provision for flashboards on the tops of the walls is often made to adapt the configuration to changes in stream flow.

296

8 Intake Structures

canal

sluiceway divider wall island Fig. 8.8 Simple divider wall configuration

Sand screens are most effective where the sediment load is predominantly bedload. To be effective they must allow sufficient flow past the point of diversion to transport the bed load so that deposited ‘ramps’ of material cannot form, which would make the screen ineffective. In some cases, the efficiency of a sand screen has been increased by installing a thin horizontal projection at the top of the wall for a short distance upstream. Guide Vanes: are used to induce local helicoidal flow patterns in straight river reaches in order to generate the bedload sweep phenomenon associated with the natural helicoidal flow in river bends. Guide vanes are used to direct the flow either close to the bed or close to the water surface. Bottom guide vanes (Fig. 8.9a) are used to deflect the water near the bed, with its associated bedload or highly concentrated suspended load, away from the intake. Surface guide vanes (Fig. 8.9b) direct the relatively clear surface water towards the intake, which induces a counter flow near the bed away from the intake. Surface vanes are generally supported by a raft arrangement from which the vanes project downwards into the water far enough to influence the flow direction of surface water. Model tests conducted by USBR have shown that the two configurations are about equally effective in reducing bed load intake. Surface vanes have the drawback that they tend to trap and accumulate floating debris, causing maintenance problems. For a particular model study reported by Vanoni (1975) the use of bottom vanes reduced the ratio of sediment concentration in the diverted flow to that in the river from 2.38 to 0.10. Model tests are recommended for specific structures. Stream Inlets: are deep inlet structures built in the bed of the stream. They tend to trap a lot of sediment and require much maintenance unless some sort of sluiceway is incorporated. They are, however, much less expensive than diversion barrages with other types of sediment diverters and are, therefore, used quite commonly for small diversions. A typical layout incorporating a sluiceway is shown in Fig. 8.10. A steep channel gradient makes sluicing relatively easy as a short high head sluice pipe can be used.

8.2 River Intake Design for Sediment Control

297

to canal

to canal

near surface flow near bed flow (a) Bottom guide vanes

(b) Surface guide vanes

Fig. 8.9 Guide vanes for sediment diversion

Stream inlets are usually used for relatively small diversions, which are often on mountain streams and relatively inaccessible. They should, therefore, be designed to ensure minimal blocking by debris. The trashrack, for example, should always have a slope in the downstream direction about 1:12 greater than the stream gradient. Some more sophisticated stream inlet designs have been used for automatic intermittent sluicing, such as that developed by Raynaud (1951). In this system, water and sediment pass through trashracks into a triangular desilting basin where the sediment accumulates and clear water flows over a weir crest into a collecting channel. The desilting basin is sluiced periodically under the control of an automatic gate that operates by buoyancy created when sediment has accumulated in the desilting basin. Tunnel Type Sediment Diverters: exploit the vertical distribution of transported sediment by separating the high-concentration flow near the bed and directing it through a number of tunnels back to the river downstream (Fig. 8.11). The relatively clear water above the tunnels is directed to the canal. The tunnels are parallel to the axis of the canal intake control structure. The tunnel closest to the canal intake begins some distance upstream of the intake and the others decrease in length

298

8 Intake Structures

A canal sluice pipe sluiceway C

C

A (a) Plan

canal

sluiceway

sluice pipe (b) Section A-A

trashrack

(c) Section C-C Fig. 8.10 Stream inlet structure configuration

away from the intake. Tunnel diverters have proved to be very efficient for large and small diversions. It must be ensured that the flow separation at the tunnel entrances takes place smoothly to reduce energy losses and to prevent induced turbulence from suspending material moving as bedload into the tunnels. The tunnel dimensions must be designed to ensure that the sediment can be easily transported and clogging does not occur. This requires a velocity of about 3 m/s and it has been shown that the discharge should be at least 15–20% of the canal diversion for efficient operation (Vanoni 1975). It is extremely important to ensure that the discharge point of the lower channel is kept clear at all times, even if this requires a channel separate from the stream. A recommended design procedure is presented by Kothyari et al. (1994).

8.2 River Intake Design for Sediment Control

299

canal A

sluiceway tunnels

river flow

river flow

A barrage (a) Plan

tunnels canal

(b) Section A-A Fig. 8.11 Tunnel type sediment diverter

Sediment Ejectors Sediment ejectors are designed to remove from a canal any sediment that was not diverted at the headgates. Even if a diverter is incorporated in an intake some sediment will enter the aqueduct, and some material transported in suspension in the river may settle out and become bedload in the canal. A number of different types of ejector devices have been developed for different situations. The most common are described below (further details are given by Vanoni (1975) and Raudkivi (1993)). Tunnel Type Ejectors: operate on the same principle as tunnel type diverters, i.e. by separating the relatively clear water nearer the surface from the heavily sediment-laden water near the bed. An example configuration is shown in Fig. 8.12. A number of tunnels extending across the width of the channel are set below the floor level with their openings directed upstream. The tunnels converge and curve to one side, passing through the side of the channel and a gated control structure to an outfall. The tunnels may be divided into smaller tunnels before the convergence in order to improve the flow pattern. At the opening, the tunnels usually have a height equal to about 20–25% of the canal flow depth. The roof usually extends

300

8 Intake Structures

A

A

canal

sediment outfall

(a) Plan

sediment

clear water

(b) Section A-A Fig. 8.12 Tunnel type sediment ejector

upstream of the diversion wall to prevent resuspension of material by turbulence associated with the transition. The tunnels should be set with a differential head large enough to induce a velocity sufficient to move the sediment through; a velocity of 2.5–3.0 m/s is generally sufficient for sand-sized sediments. Efficiency increases with the differential head but so does the discharged flow rate which must be allowed for in the intake and aqueduct design. Normally the aqueduct discharge must be increased by 20–25% to operate an ejector of this type. These ejectors can be combined with canal wasteway structures that regulate the flow rate in the aqueduct. Vortex Tube Ejectors: extract sediment moving along the canal bed by means of an open tube set in the floor at an angle of about 45° to the flow direction, as shown in Fig. 8.13. Water flowing over the opening sets up a spiralling motion in the tube which helps to move the bedload along the tube to an outlet. Tests have been conducted on these structures by various researchers. Parshall (1950), for example, used a tube of about 100 mm and found that velocity of about 0.75 m/s over the lip induced rotation of about 200 rpm in the tube, which was sufficient to move gravel the size of hens’ eggs.

8.2 River Intake Design for Sediment Control

301

Fig. 8.13 Vortex tube type sediment ejector

A A

sluiceway (a) Plan

(b) Section A-A Based on theoretical analyses, confirmed by comparison with model and field data, Atkinson (1994a, b) proposed methods for predicting the trapping efficiency of a vortex tube extractor and for designing a tube. His results suggested that the tube should preferably be set perpendicular to the flow direction, and the ratio of slit width to tube diameter (he used a circular tube) should be in the range of 0.15–0.30. Settling Basins The diverter and ejector devices already discussed are intended to remove mainly coarse sediment material which is transported as bedload or as suspended load close to the bed. Finer material which is transported in suspension in the river and the aqueduct can be removed by using a settling basin downstream of the headworks. Underlying concepts are well described by Raudkivi (1993) and guidelines for the design are presented by Avery (1989). A settling basin is an expanded section of the canal, or some other feature, in which the velocity is reduced to permit settling of the suspended particles. Clear water only can then be directed into the aqueduct and the sediment can be removed from the basin by sluicing or mechanical means. The basin comprises an inlet zone, a settling zone and an outlet zone (Fig. 8.14). The inlet zone is designed to distribute the flow and suspended sediment uniformly into the settling zone. The expansion should, therefore, be gradual to avoid flow separation, and the inlet may incorporate a submerged weir, guide vanes or baffle walls. The outlet zone provides a transition back to the canal and may include measures for controlling the water level in the settling zone such as a submerged weir or gate. As for the inlet zone, the transition should ensure evenly distributed flow, but its length may be shorter for the contracting rather than expanding flow. The settling zone must be designed to allow most of the suspended sediment to settle out and to prevent settled material from being remobilized. These objectives

302

8 Intake Structures

sluiceway

inlet zone

settling zone

outlet zone

(a) Plan

settled sediment (b) Long Section Fig. 8.14 Settling basin configuration

are achieved by appropriate sizing of the length and width of the basin, as governed by the flow velocity and sediment settling velocity. Settling can be described reasonably well analytically and many relationships and mathematical models have been proposed for predicting the removal efficiency of settling basins. For the ideal situation of particles settling in the absence of turbulence and re-entrainment, the rate of sediment entering the basin is the concentration multiplied by the discharge, Q, and the rate of settling is the concentration (C) multiplied by the plan area of the basin, A, and the settling velocity, w. The settling efficiency (also referred to as the removal ratio or trap efficiency) is, therefore, given by E¼

CwA wL ¼ CQ q

ð8:6Þ

in which L is the length of the settling basin and q is the unit width discharge. For real situations, Vetter (1940) proposed the relationship W ¼ Wo e

wL=q

ð8:7Þ

in which W is the weight of sediment leaving the basin and Wo is the weight of sediment entering the basin. The settling efficiency is then given by

Wo W ¼ E¼ Wo

 1

W Wo



¼1

e

wL=q

ð8:8Þ

Equation (8.8) assumes a high degree of turbulence, with less turbulence leading to greater settling efficiency. Camp (1946) produced a graph showing the increase in efficiency with decreasing turbulence (Fig. 8.15). The effect of turbulence is accounted for by the turbulent mixing coefficient, which can be expressed in terms of the shear velocity, u*. The efficiency, therefore, depends on both the particle settling velocity and the turbulence and is related to the parameter w/u*. The shear velocity can be evaluated through a resistance equation, such as Manning’s, giving the form given in Fig. 8.15 with the hydraulic radius represented by the flow depth, D. The settling efficiencies in Fig. 8.15 agree closely with Eq. (8.8) for high turbulence (w/u* = 0.01) and increase with decreasing turbulence to agree closely with Eq. (8.6) for low turbulence (w/u* approaching 0.1 for low wA/Q and 10.0 for higher values).

100

80

60

40

20

0 0.01

0.10

1.0

10.0

304

8 Intake Structures

Results of laboratory experiments by Garde et al. (1990) contradicted predictions using the methods of Camp (1946) and Vetter (1940), especially for fine sediments. They proposed a new equation,  E ¼ Eo 1

e

kL=D



ð8:9Þ

in which Eo is the limiting efficiency, i.e. the maximum that can be achieved for a given sediment size and flow condition. Eo and k were both found to depend on w/u*. Their experimental values were presented graphically and can be approximated as k ¼ 0:042

 2:17 w u

ð8:10Þ

and   w þ 0:023 Eo ¼ 0:434 u

ð8:11Þ

Eo has a maximum value of 1.00 and Garde et al. (1990) recommend the maximum value for k to be that corresponding to w/u* = 2.20, i.e. kmax = 0.23. It was pointed out by Schrimpf (1991) that Camp’s (1946) graph is only valid for a vertically uniform concentration at the stilling basin entrance, which is not always a reasonable assumption and could account for the discrepancy found by Garde et al. (1990). Raudkivi (1993) describes further refinements for estimating settling efficiency. He also introduces the more rigorous computational solutions of the sediment diffusion equations, which are recommended for final design purposes. Example 8.2 A canal conveying 8.0 m3/s leads into a 16.0 m wide, rectangular section settling basin with Manning’s n = 0.013. The flow depth in the settling basin is maintained at approximately 2.0 m. Determine the length of settling basin required to settle 75% of the inflowing suspended sediment if its representative size is 0.10 mm with a settling velocity of 7.82 mm/s. Solution The removal ratio equations include the basin length as a variable and can be rearranged to give the length required to achieve a specified removal ratio. Assuming quiescent flow without turbulence, the removal ratio is given by Eq. (8.6), i.e. E¼ Therefore

wA wL ¼ Q q

8.2 River Intake Design for Sediment Control

L¼

305

Eq w

with q¼

Q 8:0 ¼ ¼ 0:50 m3 =s/m W 16:0

So L¼

0:75
0:50 ¼ 48 m 0:00782

Assuming a high degree of turbulence, the removal ratio is given by Eq. (8.8), i.e. E¼1

e

wL=q

Therefore L¼ ¼

q lnð1 E Þ w 0:50 lnð1 0:00782

0:75Þ ¼ 89 m

Assuming the degree of turbulence to be determined by the flow conditions, the removal ratio is given by Fig. 8.15. The ordinate is E = 75% and the abscissa is given by wD1=6 pffiffiffi nV g with D ¼ 2:0 m n ¼ 0:013 Q 8:0 V¼ ¼ ¼ 0:25 m/s A 16:0
2:0 So

From Fig. 8.15

wD1=6 0:00782
2:01=6 pffiffiffiffiffiffiffi ¼ 0:86 pffiffiffi ¼ nV g 0:013
0:25
9:8 wA wL ¼ ¼ 0:80 Q q

Therefore

306

8 Intake Structures

L ¼ 0:80

q 0:50 ¼ 0:80 ¼ 51 m w 0:00782

Figure 8.15 allows for intermediate roughness and gives a result between the extremes given by Eqs. (8.6) and (8.8), which appears to be realistic. The position of w/u* in Fig. 8.15 indicates only a small departure from the quiescent condition, consistent with the solution being closer to that given by Eq. (8.6) than by Eq. (8.8). Equations (8.9)–(8.11) also account for varying degrees of turbulence, i.e.  E ¼ Eo 1



e

kL=D

 D ln 1 k

E Eo

Therefore, L¼



with  2:17 w k ¼ 0:042 u and pffiffiffi nV g D1=6 pffiffiffiffiffiffiffi 0:013
0:25
9:8 ¼ 0:0091 m/s ¼ 2:01=6

u ¼

So   0:00782 2:17 k ¼ 0:042 ¼ 0:030 0:0091 and   w Eo ¼ 0:434 þ 0:023 u   0:00782 þ 0:023 ¼ 0:396 ¼ 0:434 0:0091 Therefore, L¼

 2:0 ln 1 0:030

 0:75 0:396

for which there is no solution. The value of Eo = 0.396 gives the maximum possible removal rate, so the required removal rate of 0.75 cannot be achieved with any basin length. Using the length determined from Fig. 8.15, the removal rate according to Eq. (8.9) is

8.2 River Intake Design for Sediment Control  E ¼ Eo 1

 ¼ 0:396 1

e

307

kL=D

e



0:030
51=2:0



¼ 0:212

which is considerably less than the required. This method suggests that the basin should be widened to reduce V and hence u*.

The dimensions of the stilling basin must also ensure flow conditions that are not competent to mobilize the settled sediment. The Shields criterion for scouring (see Chap. 9) can be used for assessing the likelihood of remobilization. The settled sediment can be removed by sluicing if there is sufficient river flow for the additional sluicing water to be diverted. The discharge, velocity and channel conditions must be such that sediment is transported downstream and does not deposit and clog the river channel near the intake. Intermittent sluicing can be done by operation of the gates controlling the sluiceway. Designs have also been developed for continuous sluicing without temporary sediment accumulation (Raudkivi 1993). If there is insufficient flow available for sluicing the accumulated sediment must be removed from settling basins by mechanical means, such as dredging or dragling, which can be very expensive.

8.3

Pump Sumps and Intakes

In many cases water cannot be extracted directly from a river and provision for pumping must be made. The intake must then incorporate a sump to provide sufficient volume and depth for the pumps to operate. The efficiency of a pump intake structure is governed to a large degree by the geometry of the configuration. As for other intake structures, design problems cannot readily be solved analytically and most design procedures and recommendations have been developed from model experiments and observations of existing structures. In practice, particular situations often require unusual or complicated solutions that necessarily depart from the general recommendations. The design solution for these problems should be obtained—or at least confirmed—by model studies.

8.3.1

Desirable Flow Conditions

Poor flow conditions resulting from careless design can lead to unanticipated operating restrictions and extra costs through delays in commissioning, increased maintenance, structural alterations and retrospective model tests. Useful guidelines for design have been produced by Hydraulic Institute (1998) and Prosser (1977), from where much of the following content is derived and adapted. For effective and efficient pumping the flow to the pump should be single-phase (i.e. containing no entrained air), uniform (i.e. the flow velocity should be constant in magnitude and direction across the approach section) and steady (i.e. not fluctuating with time in magnitude and direction).

308

8 Intake Structures

Well designed configurations are usually able to achieve the single-phase ideal but some minor departures from uniform and steady conditions are to be expected. There will always be some nonuniformity due to boundary layer formation at the walls and a certain amount of unsteadiness will be produced by small-scale turbulence. Significant departures from these conditions, however, can lead to poor performances or increased costs. Departures from the single-phase requirement can have serious effects on the pump, apart from causing possible undesirable effects of air in the water supply. High air content can cause the pump to deprime and cease to deliver. Lesser amounts of air can cause significant reductions in discharge and efficiency, the severity of which will depend on the type of pump. For example, for a centrifugal pump 3% of free air has been found to decrease efficiency by 15% and axial pumps are even more sensitive. Further, entrained air will cause uneven loading on the pump impeller which can result in vibration, rough running and damage to bearings. Approach conditions should, therefore, be designed to ensure that no free air enters the intake. There are several ways, related to intake or sump geometry, in which air can be entrained. If the water level in the sump is very close to the top of the intake air may be drawn through the intake continuously or by ‘gulping’. This tendency is increased with high intake velocities, particularly where approach velocities are relatively low and the draw-down of the water surface near the intake is accentuated. A minimum submergence of the intake is, therefore, necessary. Air can also be entrained by a falling jet of water, which is common where water enters the sump over a weir or through a culvert at a higher level than the water in the sump. Air bubbles may be carried over the length of the sump and drawn into the intake. If it is not possible to avoid high-level entry into the sump then the sump should be made long enough to allow the air bubbles to rise to the surface before reaching the intake. A minimum sump length could be calculated from the approach velocity and water depth in the sump and the rising velocity of air bubbles (about 0.2 m/s for 2–5 mm diameter bubbles). Intense vortex action can lead to air entrainment (Fig. 8.16). A vortex may be stable with a diffuse or solid air core or it may be unstable and unsteady with intermittent air entrainment only. The formation of vortices is dependent on submergence and on intake and sump geometry. High intake velocities help to initiate air entrainment by drawing bubbles off the bottom of the surface depression associated with the vortex. Departures from uniform flow, such as intense swirling can cause rapid changes in local pressure on pump impellers, which can lead to cavitation and severe vibration with associated bearing damage. Axial type pumps are generally most susceptible to this type of damage. Sometimes larger scale, less intensive swirling flow may be centered on the pump axis. Depending on the direction of swirl relative to the movement of the impeller, this could be beneficial or detrimental to pump performance. Intense swirling flow is associated with vortex action. In addition to the surface vortices which also cause air entrainment, there may be submerged vortices that

8.3 Pump Sumps and Intakes

309

Fig. 8.16 A solid air-entraining vortex in a model pump sump

originate at the floor or walls of the sump rather than the water surface. Larger scale swirl is generally caused by general circulation in the sump which is amplified as the flow converges towards the intake. The general circulation may be initiated by a distorted velocity profile in the approach flow and the sump geometry. This type of nonuniformity may be steady but can help to initiate surface or submerged vortices. Departures from a steady flow can cause variations with time of pump blade loading, resulting in vibrations and bearing wear. The effects are worst if the pump is very close to the intake: some of the unsteadiness becomes damped out if there is a length of conduit, bends or changes of cross section between the intake and pump. In addition to the small-scale turbulence which can normally be expected, large-scale turbulence where the eddy size is of the same order of magnitude as the intake cross section sometimes occurs. The major causes of this are unsteady flow patterns arising from obstacles in the sump or poor inlet conditions and vortex shedding from pillars or other pumps. Another common cause of unsteadiness is the presence of stagnant regions of water above or behind the intake. Boundaries between stagnant regions and the flow tend to be unstable and the changing position causes unsteadiness in the main flow. The presence of stagnant regions also favours vortex formation and can, therefore, also lead to nonuniformity and air entrainment.

8.3.2

Intake and Sump Design

General Layout Four functional zones can be identified in a typical sump-intake arrangement (Fig. 8.17). The first zone is the inlet to the pump station from the supply source. The entry to the pump station may be by canal or pipeline or directly from a river or lake, and often includes a control structure such as a weir. The second zone is an

310

8 Intake Structures

screen penstocks

weir A

A

inlet

sump

approach

(a) Plan motor

pump wet well sump

bellmouth

(b) Section A-A Fig. 8.17 General sump-intake arrangement

approach section which may contain screens for removing solid matter and gates or division walls for directing the flow to the appropriate sump. The sump itself constitutes the third zone. It is generally rectangular in plan with a flat floor. The purpose of the sump is to provide storage and to damp out distorted flow patterns which may result from the configuration of the distribution zone. The sump is separated from the final zone by the intake, a section where the water enters a closed conduit, i.e. a division between the free surface and internal flow. The final zone is the short section of conduit between the intake and the pump, which should be kept as short and straight as possible. This is a very general description and the actual configuration may vary considerably to meet specific requirements. For example, the sump may be left out completely when water is drawn directly from a reservoir. The pump may be installed within the sump (wet well arrangement), or in a dry well at the end of the sump (Fig. 8.18). The wet well has the advantage of the simplicity of design and hence relatively low-cost. It has the disadvantage that

motor

pump

(a) Wet well

(b) Dry well

maintenance requires drainage of the sump or removal of the pump. The wet well arrangement is ideal for conditions where intermittent pumping only is required, such as storm-water pumping, and the sump will be dry for most of the time. If reliability is important a dry well arrangement is preferable, making the pump accessible for maintenance at all times. The intake for a dry well configuration may be horizontal or with a turned down bellmouth in the sump, depending on minimum water levels required in the sump. A turned down bellmouth allows a lower water surface and is less susceptible to vortex formation. Flow towards the intake should be uniform across the width of the sump. This ensures a uniform approach and lessens the likelihood of swirl developing near the intake, which could lead to the occurrence of vortices with associated air entrainment and nonuniformity problems. If more than one pump is installed the approach flow conditions are considerably improved by placing dividing walls between them, especially if the pumps will not necessarily operate together (Fig. 8.19). Prosser (1977) recommends a sump width of twice the bellmouth diameter and a length of ten times the diameter. Possible stagnant flow regions should be filled, as they can be unstable and cause fluctuations and unsteadiness in the flow. The flow velocities should be kept low, about 0.6 m/s into the pumping station and 0.3 m/s approaching the bellmouth. Any obstructions, such as structural elements, should be streamlined to avoid flow separation with associated nonuniformity and unsteadiness near the intake. Excess kinetic energy associated with changes in level down slopes, over weirs or through control structures should be dissipated well away from the intake. If the approach section is sloped, Prosser (1977) recommends that the slope be less than 10° and if it expands the flare angle should not exceed 20°. Hydraulic Institute (1998) presents examples of various configurations and their associated flow patterns. Bellmouth Intakes The bellmouth intake is provided to prevent flow separation which would occur with a sharp-edged inlet. The turbulence associated with separation is thus avoided and uniform flow through the intake with a minimum head loss is ensured. For intake in a vertical wall, the minimum bellmouth radius required is about 0.1 of the pipe diameter; for a suspended pipe the radius would have to be larger to allow for

10

flow from behind the bellmouth. In practice the shape is normally specified by pump manufacturers and usually forms a quarter ellipse in section. The bellmouth diameter is usually 1.5–1.8 times the pipe diameter. Prosser (1977) recommends a clearance below a vertical pipe bellmouth of 0.50 of its diameter. Hydraulic Institute (1998) gives recommendations for bellmouth diameters and entry velocities for different pumping rates. Sump Volume The sump volume is an important design decision. In addition to ensuring satisfactory approach flow conditions to the pump intake, the sump provides storage to allow for constant speed pumps to switch on and off as the sump fills and empties. Factors affecting this decision include the number and capacities of pumps, their operating levels and minimum cycle times. The minimum cycle time is determined by the heat generated in an electric motor during start-up, which imposes a restriction on the frequency of starts. This is not significant if variable-speed pumps are used but constitutes an important constraint on sump volume for fixed-speed pumps. For a fixed-speed pump, the cycle time (T) for filling and emptying the active sump volume (V) is T¼

V V þ Qi Qp Qi

ð8:12Þ

in which Qi is the inflow rate and Qp is the pumping rate. This can be rearranged to give the active volume required for a specified cycle time, i.e.

8.3 Pump Sumps and Intakes

313

V ¼T

Qi  Qp Qp

Qi



ð8:13Þ

By differentiating Eq. (8.12) it can be shown that the minimum cycle time, or the maximum frequency of pump starts, occurs if the inflow rate is half the pumping rate. If this condition is applied, then (from Eq. (8.13)) V¼

TQp 4

ð8:14Þ

Hydraulic Institute (1998) presents examples of the application of these relationships to multiple fixed-speed pumps with sequential operation. Prevention of Vortices The formation of vortices is a major source of flow nonuniformity and air entrainment. Vortices may occur completely below the surface, attached to the floor or a side wall, and cause considerable swirl into the intake. Surface vortices also produce swirl and, if sufficiently intense, can entrain large quantities of air which can lead to major pump damage. Subsurface vortices at a vertical bellmouth can be eliminated or at least reduced by the careful setting of the clearance below the intake and/or the installation of swirl-preventing devices (Fig. 8.20). Surface vortices can be prevented by ensuring uniform approach flow, providing sufficient submergence of the intake and/or providing vortex-suppressing devices. Uniform flow can be ensured by adhering to the general layout recommendations above. A high intake submergence reduces the likelihood of surface vortex formation. However, the submergence defines the lowest point of the pumping station and, therefore, has a significant influence on civil costs. The minimum water level is generally defined by external conditions, so the sump floor level is determined by the submergence and the clearance below the bellmouth. The submergence should, therefore, be kept as small as possible to minimize costs.

(a) Cone

(b) Short splitter or cross

Fig. 8.20 Sub-surface vortex prevention devices

(c) Long splitter

314

8 Intake Structures

The minimum submergence to ensure vortex-free conditions depends on the size of the bellmouth and the intake velocity; generally, the minimum submergence increases with increasing intake velocity. Prosser (1977) recommends a minimum of 1.5 times the bellmouth diameter for turned down intakes and 1.0 times the diameter from the top of the bellmouth for horizontal intakes. Hydraulic Institute (1998) recommends Eq. (8.15) for specifying the submergence required for preventing strong air-entraining vortices. S ¼ 1:0 þ 2:3 FD D

ð8:15Þ

in which S is the submergence (see Fig. 8.18), D is the diameter of the inlet (the bellmouth if used) and FD is a ‘Froude number’ based on the inlet velocity and diameter, i.e. V FD ¼ pffiffiffiffiffiffi gD

ð8:16Þ

S 4 2=3 ¼ 2:1 þ FD D 3

ð8:17Þ

Werth and Frizzel (2009) carried out experiments with vertical intakes to determine the submergence required to prevent vortices weaker than able to entrain air, but strong enough to cause swirl (as indicated by a dye core) right to the intake. For this more restrictive condition, they proposed the minimum submergence to be given by

These are general recommendations and any unusual configuration should be examined carefully—there are cases where vortex action is worst at relatively high submergence. Submergence on its own cannot guarantee vortex-free conditions; strong air-entraining vortices can occur at high submergence if the approach geometry allows swirl to develop. If surface vortices are found to occur in existing structures and modifications for improving approach flow conditions or increasing submergence are not possible or ineffective, then vortex-suppressing devices can be installed. These take a variety of forms, all intended to break circulation in the vicinity of the intake. For vertical, upward-directed intakes a perforated screen wall (Fig. 8.21a) or a vertical splitter (Fig. 8.21b) may be effective. For a vertical downward-directed intake (a very vulnerable arrangement) anti-rotation baffles or guide vanes have been found to be effective (Fig. 8.21c). For horizontal intakes, the most common vortex-suppressing devices are horizontal plates, which may be fixed or floating and solid or perforated (Fig. 8.21d). Amiri et al. (2011) investigated different plate types and positions and found that most satisfactory vortex suppression was obtained with a rectangular, perforated plate positioned at the top of the intake. The optimal dimensions were 1D from the intake face in the intake flow direction and 1.5D normal to this

8.3 Pump Sumps and Intakes

315

(a) Screen wall

(b) Long splitter

A A

A

A-A

(c) Baffles

A A-A

(d) Horizontal plate

Fig. 8.21 Surface vortex prevention devices

direction, where D here is the pipe diameter (excluding bellmouth). The optimal area of perforations was 50% of the plate area. They did not investigate the effect of perforation size, but Chigura et al. (2016) found the effectiveness to increase with decreasing perforation size.

8.3.3

Model Testing for Intakes

Most guidelines for deigning pump sumps and intakes are based on limited information derived from prototype observations and model studies. Any proposed departures from the guidelines should, therefore, be tested by a model study of the particular configuration. The significance of circulation and air entrainment in the functioning of sumps and intakes presents some special challenges for their physical modelling. Free surface flow occurs up to the intake and model similarity can be ensured by maintaining equality of Froude numbers in the model and prototype, provided the scale is sufficiently large to preclude viscosity and surface tension effects. Hydraulic Institute (1998) recommends that to safely exclude these effects the Reynolds number (Re = VD/m, where m is the kinematic viscosity) for the model intake should be above about 6  104 and the Weber number (We = V2D/(r/q), where r is the

316

8 Intake Structures

interface surface tension and q is the fluid density) should be above 240. It is also recommended that to observe flow patterns reliably and to obtain sufficiently accurate measurements, the approach bay should be at least 300 mm wide, the flow should be at least 150 mm deep and the pump throat should be at least 80 mm in diameter. Anwar (1968) recommends that the scale should not be less than 1:20 for modelling vortices. Prosser (1977) suggests a minimum of 1:25, but down to 1:50 for large intakes from reservoirs. Avoidance of surface and subsurface vortices is a crucial design objective, but there is uncertainty regarding the validity of Froude similarity for vortex formation. It has been observed that similarity of air-entraining vortices occurs at model velocities greater than indicated by Froude similarity. Denny (1956) found that similarity appeared to occur at a velocity scale equal to the linear scale rather than the square root of the linear scale as implied by the Froude law. He suggests that the velocity required to form a surface dimple is scaled according to the Froude number but the local velocity required to drag air off the bottom of the dimple is unaffected by scale. There are, therefore, good arguments for using larger velocities than required for Froude similarity when checking for vortices, even though this might cause the water profile to be incorrectly represented. Prosser (1977) suggests operating the model at Froude velocities and if these tests show little or no vortex action, increasing the flow velocity to 2 or 3 times the Froude scale velocity to get an idea of the margin of safety with respect to air-entraining vortices and to observe any effects which may have been missed. Hydraulic Institute (1998) recommends that both free surface and subsurface vortices should be less severe than able to maintain coherent dye cores for more than 10% of the time. Air entrainment by gulping or a falling water jet will also not be accurately represented by the model. More air will be entrained in the prototype than as indicated by the model and air bubbles will take longer to reach the surface. Because surface vortex formation is influenced by approach flow patterns the whole of the sump-intake structure including the approach and inlet zones should be modelled. If the structure is divided into similar sections it is possible to model one section alone only if no flow between the sections could occur in the prototype. A plane of symmetry cannot be replaced by a solid boundary in the model unless there is actually a corresponding physical separation of flow in the prototype structure. If the intake is to draw water from a large body of water such as a reservoir, the model must include a sufficiently large area of the surface to enable full flow patterns to become established. It is not necessary to include the entire structure on the downstream side of the intake section—a length of a few pipe diameters is sufficient. All details such as stop-log grooves and screens should be modelled. Modelling of screens is important if the solidity is such that the flow patterns will be affected. Screen material need not be modelled as long as the model screen has the same solidity as the prototype.

8.3 Pump Sumps and Intakes

317

Problems 8:1 Water is to be extracted from a wide alluvial river directly into a canal. The river has a slope of 0.00080 and a bed composed of sediment with a median grain size of 0.12 mm (which has a settling velocity of 0.011 m/s). When the flow depth in the river is 1.50 m the suspended sediment concentration is 0.050 at a height of 0.10 m from the bed. At what height above the river bed should the bottom of the offtake canal be set to ensure that the average suspended sediment concentration of the diverted water is not more than 0.020?

References Amiri, S. M., Zarrati, A. R., Roshan, R., & Sarkardeh, H. (2011). Surface vortex prevention at power intakes by horizontal plates. Water Management, 164(WM4), 193–200. Anwar, H. O. (1968, October). Prevention of vortices at intakes. Water Power, 393–401. Atkinson, E. (1994a). Vortex-tube sediment extractors. I: Trapping efficiency. Journal of Hydraulic Engineering, 120(10), 1110–1125. Atkinson, E (1994b) Vortex-tube sediment extractors. II: Design. Journal of Hydraulic Engineering, 120(10), 1125–1138. Avery, P. (Ed.). (1989). Sediment control at intakes—A design guide. Bedford, England: BHRA, The Fluid Engineering Centre. Camp, T. R. (1946). Sedimentation and the design of settling tanks. Transactions, ASCE, 111, 895–958. Chigura, T., Mashika, B. K., & Mpofu, S. T. (2016). The use of perforated horizontal plates as vortex suppressors at horizontal intakes. Final Year Investigational Project, School of Civil & Environmental Engineering, University of the Witwatersrand, Johannesburg, South Africa. Denny, D. F. (1956). An experimental study of air-entrainment vortices in pump sumps. In Proceedings of the Institution of Mechanical Engineers (170 pp.). Garde, R. J., Ranga Raju, K. G., & Sujudi, A. W. R. (1990). Design of settling basins. Journal of Hydraulic Research, 28(1), 81–91. Hernandez, N. M. (1969). Irrigation structures. In C. V. Davis & K. E. Sorensen (Eds.), Handbook of applied hydraulics (3rd ed.). McGraw-Hill. Hydraulic Institute. (1998). American national standard for pump intake design. Report ANS/HI 9.8-1998, Parsippany, New Jersey, USA. Kothyari, U. C., Pande, P. K., & Gahlot, A. K. (1994). Design of tunnel-type sediment excluders. Journal of Irrigation and Drainage Engineering, 120(1), 36–47. Parshall, R. L. (1950). Experiments in cooperation with Colorado Agricultural Experiment Station. Fort Collins, Colorado. Prosser, M. J. (1977). The hydraulic design of pump sumps and intakes. British Hydromechanics Research Association and Construction Industry Research and Information Association. Raudkivi, A. J. (1993). Sedimentation: Exclusion and removal of sediment from diverted water. Hydraulic Structures Design Manual 6. Rotterdam: International Association for Hydraulic Research, A A Balkema. Raynaud, A. (1951). Water intakes on mountain streams, example of application to the Torrent Du Longon. International Association for Hydraulic Research, Fourth Meeting, Bombay, India (pp. 1–9). Schrimpf, W. (1991). Discussion of “design of settling basins”. Journal of Hydraulic Research, 29(1), 137–143.

318

8 Intake Structures

Vanoni, V. A. (Ed.). (1975). Sedimentation engineering. Prepared by the ASCE Task Committee for the Preparation of the Manual on Sedimentation of the Sedimentation Committee of the Hydraulics Division. New York: American Society of Civil Engineers. Vetter, C. P. (1940). Technical aspects of the silt problem on the Colorado River. Civil Engineering, 10(11), 698–701. Werth, D., & Frizzell, C. (2009). Minimum pump submergence to prevent surface vortex formation. Journal of Hydraulic Research, 47(1), 142–144.

Further Reading Anwar, H. O. (1966). Formation of a weak vortex. Journal of Hydraulic Research, 4(1), 1. Anwar, H. O. (1967, November). Flow in a free vortex. Water Power, 455. Anwar, H. O., Weller, J. A., & Amphlett, M. (1978). Similarity of free-vortex at horizontal intake. Journal of Hydraulic Research, 16(2), 95–105. Blaisdell, F. W. (1960). Hood inlet for closed conduit spillways. Journal of the Hydraulics Division, ASCE, 86(HY5), 7. Camp, T. R., & Lawler, J. C. (1969). Water supplies. In C. V. Davis & K. E. Sorensen (Eds.), Handbook of applied hydraulics (3rd ed.). McGraw-Hill. Dagget, L. L., & Keulegan, G. H. (1974). Similitude in free surface vortex formations. Journal of the Hydraulics Division, ASCE, 100(HY11), 1565–1582. Goldschmidt. (1974). Wet-well volumes for multipump systems. Journal of the Irrigation and Drainage Division, ASCE, 100(IR3), 371–385. Goldschmidt. (1978). Mixing fixed-speed pumps to variable flows. Journal of the Water Pollution Control Federation, 50(7), 1733–1741. Gordon, J. L. (1970). Vortices at intakes. Water Power, 22, 137–138. Graf, W. H. (1971). Hydraulics of sediment transport. McGraw Hill. Hattersley, R. T. (1965). Hydraulic design of pump intakes. Journal of the Hydraulics Division, ASCE, 91(HY2), 223–249. Hecker, G. E. (1981). Model-prototype comparisons of free surface vortices. Journal of the Hydraulics Division, ASCE, 107(HY10), 1243–1259. Henderson, F. M. (1966). Open channel flow. Macmillan. Razvan, E. (1989). River intakes and diversion dams. In Developments in civil engineering (Vol. 25). Elsevier Science Publishers. Vanoni, V. A. (Chmn). (1972). Chapter V: Sediment control methods: C control of sediment in canals. Journal of the Hydraulics Division, ASCE, 98 (HY9), 1647–1689.

Chapter 9

Scour and Scour Protection

9.1

Introduction

Many hydraulic structures present the potential for scour at the interface between flowing water and an erodible boundary. The boundary in question may be an integral part of the structure, such as an embankment or the bed and banks of an unlined canal, or it may be adjacent to it, such as the riverbed downstream from a culvert or dam spillway. Scour refers to the removal or redistribution of the sediment material forming the boundary. This can be deleterious, or even catastrophic to the functioning of the structure, and must therefore be considered in its design. Scour may be local or general. Scour occurs at any location where the shear stress exerted by the flow on the bed is sufficient to move the bed particles. Local scour is often associated with the rapidly varied flow patterns induced by structures, and frequently occurs around bridge piers and downstream of culverts, spillways and other river structures. General scour, although a manifestation of local scour phenomena, refers to a net bed degradation over some distance, and is associated with an excess of sediment transport capacity over the sediment supply rate. Such situations are common downstream of impoundments, which trap sediment and hence reduce the supply further downstream. Similar problems occur in urban rivers, where increased flood flows increase the sediment transport capacity. This chapter addresses the conditions under which cohesionless sediment particles (including all sizes from fine sand to large rocks) can be moved by the flow. It presents criteria for predicting whether or not scour can be expected, and for designing loose protective linings (riprap) capable of resisting scour. In many practical applications, it is required that no movement of the bed material is acceptable, while in others (such as around bridge piers) local scour occurs but stabilizes at some depth, with or without sediment movement, and the design problem is to establish that the stabilized scour depth is acceptable. Some designs for addressing general scour problems require ensuring a balance between sediment

© Springer Nature Switzerland AG 2020 C S James, Hydraulic Structures, https://doi.org/10.1007/978-3-030-34086-5_9

319

320

9 Scour and Scour Protection

supply and transport capacity, rather than ensuring no movement of material. The information in this chapter cannot be used directly to analyse such problems. Much work has been done on establishing scour conditions and predicting scour magnitudes, both theoretically and empirically, for single particles and bulk sediments. At present, no completely theoretical description of scour is possible, but a theoretical analysis does help to gain an understanding of the phenomenon.

9.2

Theoretical Analysis

The critical flow condition at which scour can just take place can be established by analysing the stability of a single particle in the bed (e.g. James 1990). The stability of a particle is determined by the forces acting on it (Fig. 9.1). These are its submerged weight (W), the reaction forces (R) and friction forces (Ff) between it and adjacent particles, and the hydrodynamic lift (FL) and drag (FD) forces imposed by the flow. At the condition of incipient motion, the reaction and friction forces through all but one or two contact points will reduce to zero, and the particle will tend to move by sliding or pivoting over the remaining contact point or points. Pivoting is probably more common in nature, but both mechanisms do occur and at about the same hydraulic conditions. Either mechanism can be assumed and will lead to similar formulations. Pivoting is the more common assumption and is followed here. The relationship between the relevant forces at incipient motion can be described by considering the equilibrium of their moments about the pivot axis, i.e. W cos a a sinðu þ aÞ ¼ W sin a a cosðu þ aÞ þ FD b þ FL c

FL F Ff

R

FD

φ

Ff

pivot axis

Ff R R W Fig. 9.1 Forces acting on a particle

α

ð9:1Þ

9.2 Theoretical Analysis

321

In Eq. (9.1), a, b and c are the distances from the pivot axis to the particle centroid, the line of action of the drag force and the line of action of the lift force, respectively, u is the pivot angle and a is the bed slope. The variables in Eq. (9.1) can be quantified in terms of sediment, geometric and flow parameters, and the equation rearranged to obtain a scour criterion with the form sc cDðSs

1Þ

¼ K tanðu

aÞ

ð9:2Þ

in which sc is the boundary shear stress at incipient motion, c is the specific weight of water, D is the sediment particle size and Ss is the specific gravity of the sediment. The coefficient K accounts for a variety of geometric and hydraulic effects, which are more explicitly accounted for in the more complete analysis of James (1990). Equation (9.2) is sufficiently rigorous for many engineering applications, but requires empirical determination of K.

9.3

Empirical Approach

It is not possible at present to develop a complete theoretical solution to the incipient motion problem—some empirical content is necessary. It is possible to address the problem completely empirically and many results have been presented which are useful for engineering applications. An empirical study begins with an identification of the variables which affect the phenomenon. These are the boundary shear stress (so), the fluid density (q), the fluid viscosity (l), the particle size (D) and the submerged specific weight of the grain (cs − c) where cs is the sediment specific weight. The functional relationship between these variables can be expressed as f ðso ; q; l; D; ðcs

cÞÞ ¼ 0

ð9:3Þ

Using dimensional analysis, two dimensionless groupings can be formed from the identified variables. Hence f




ðcs

so

u D cÞD m ;



¼0

ð9:4Þ

The first term, often referred to as the Shields parameter (s*), will be recognized as the same dimensionless shear stress which was developed theoretically. The second is the shear Reynolds number, Re*, in which u* = (so/q)1/2 and m = l/q. Both of these parameters have physical interpretations. The shear Reynolds number represents the ratio of the particle size to the thickness of the viscous sublayer. It therefore represents the degree to which grains project through this layer and are

322

9 Scour and Scour Protection

subjected to turbulent flow. The dimensionless shear stress can be interpreted as the ratio of the shear stress (or average total drag force acting on a unit area of bed) to the submerged weight, per unit area of bed, of a single layer of particles. It therefore represents the ratio of disturbing to stabilizing forces on a bed of particles. The functional relationship between these parameters requires experimental investigation. The classic work in this field was by Shields (1936) who produced the first version of what has come to be known as the Shields diagram (Fig. 9.2). It is a graphical representation of the relationship implied by Eq. (9.4), at the condition of incipient motion (i.e. so = sc), as determined from flume experiments. The curve in Fig. 9.2 represents the threshold of movement. If the combination of sediment and flow characteristics represented by the two variables plots below the curve then no movement of sediment is expected. If it plots above the curve then movement, or erosion, is expected to occur. Figure 9.3 shows a similar curve compiled by Yalin and Karahan (1979) including additional data; this shows a more gradual slope for small values of Re* and a lower constant value of sc* at high values. (A similar curve was presented by Miller et al. (1977)). The following three zones can be identified on the Shields diagram: Hydraulically smooth turbulent zone: Re* < *2. The particle diameter is smaller than the thickness of the viscous sublayer and the particle is therefore enclosed by laminar flow and particles move mainly under the influence of viscous forces. The critical entrainment shear stress depends strongly on Re*. Transition zone: 2  Re*  *400 (or *70 on more modern diagrams). The particle size is about the same as the thickness of the viscous sublayer. The particle is partly enclosed by the viscous sublayer and subjected to periodic exposure to 1

τ* =

τo γ D ( S s − 1)

D

ν 2

4

6

10

0.1( S s − 1) g D 20

40

60

100

200

400 600 1000

0.1

0.01 0.2

1.0

100

10 u D Re* = *

ν

Fig. 9.2 The Shields diagram (adapted from Vanoni 1975)

1000

9.3 Empirical Approach

323

1

τ* =

τo γ D ( S s − 1)

laminar turbulent

0.1

0.01 0.1

1.0

10

100

1000

u D Re* = *

ν

Fig. 9.3 The Shields diagram (adapted from Yalin and Karahan 1979)

turbulent flow. The dimensionless critical shear stress reaches its minimum of about 0.030 in this zone, at a value of Re* of about 10. Fully developed turbulent zone: Re* > *400 (or *70 on more modern diagrams). In this zone, particles protrude through the viscous sublayer and are fully exposed to turbulence bursts, which cause instantaneous shear stresses significantly greater than the time-averaged values. The dimensionless critical shear stress is therefore considerably less than unity, about 0.056 on old diagrams and 0.047 on more modern versions (e.g. Miller et al. 1977; Yalin and Karahan 1979), and independent of Re*. There are various shortcomings of Shields’s results and many extensions and modifications have been made since. Some of these are described below. Because u* appears in both variables, a solution is not explicit in the hydraulically smooth and transition zones. An iterative solution is therefore required. In the version presented by Vanoni (1975) (Fig. 9.2), an auxiliary scale is included. If the value of the parameter D m

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
 cs 0:1 1 gD c

is located on the scale and projected along the direction of the parallel lines, the intercept with the threshold curve indicates the required values of Re* and s*. Yalin (1977) overcomes this problem by replacing Re* by the parameter

324

9 Scour and Scour Protection

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðSs 1ÞgD3 m2 and presenting a revised form of the diagram. The constancy of the Shields parameter at about 0.047 in the rough turbulent zone (Re* > *70) enables a simplified relationship to be derived. If quartz density (Ss = 2.65) and a water viscosity of 10−6 m2/s are assumed, it can be shown that Re* = 70 corresponds to a particle size of about 2 mm. For D > 2 mm, s*c is therefore constant at 0.047, i.e. so ¼ 0:047 cDðSs 1Þ and using the general shear relationship s0 ¼ cRS in which R is the hydraulic radius and S is the gradient, yields D  13RS

ð9:5Þ

as an erosion criterion. It must be remembered that S is the energy gradient, which is equal to the bed gradient only if the flow is uniform. The energy gradient can be calculated by a resistance equation, such as the Manning equation. Shields had very little data for low values of Re* (i.e. fine sediments) and his original diagram exaggerated the threshold for Re* < *1.0. The curve has been refined to account for this by White (1970), Grass (1970), Mantz (1973, 1977) who also investigated some shape effects. Other ways of defining erosion criteria have been proposed, using the particle settling velocity as a characterizing variable. Collins and Rigler (1982), Komar and Clemens (1986) proposed relationships for the critical shear stress or shear velocity in terms of the settling velocity. The average flow velocity is an appealing indicator of bed stability, although it does not uniquely represent flow conditions at the bed; the flow depth should also be accounted for in some way. One of the earliest known criteria for erosion was proposed in 1753 by Brahms, in terms of average flow velocity, i.e. Vc ¼ kW 1=6

ð9:6Þ

where Vc is the critical average velocity, k is an empirical constant and W is the weight of the particle. This relationship shows that the critical velocity is only weakly related to particle size. For example, doubling the weight of a particle (i.e. increasing its diameter by about a quarter) implies an increase in critical velocity of only about an eighth.

9.3 Empirical Approach

325

Yang (1973) succeeded in formulating an incipient motion criterion in terms of the average flow velocity and particle fall velocity. He did this by expressing the drag and lift coefficients in terms of the particle fall velocity and analysing the stability of the particle, assuming that movement would take place by sliding. He assumed the appropriate velocity for drag and lift was at a distance equal to the particle diameter above the bed, and related this to the average velocity by integrating the velocity profile. This analysis produced the general form of a relationship between Vc /w and Re* which is hyperbolic for Re* < 5 and constant for Re* > 70 (Fig. 9.4). By calibrating with available data, he produced the following relationships: Vc 2:5 ¼ þ 0:66 w log Re 0:06

for

Re \70

ð9:7Þ

and Vc ¼ 2:05 w

for

Re [ 70

ð9:8Þ

(Yang’s 1973 comments on the use of shear stress or velocity to define incipient motion make interesting reading.)

Fig. 9.4 A riprap-protected channel bank

326

9 Scour and Scour Protection

One of the major problems in the empirical approach to establishing incipient motion conditions is the decision as to what constitutes sediment movement. Most theoretical and empirical relationships are in terms of time-averaged conditions. In fact, it is not a time-averaged force that moves a particle, but an instantaneous occurrence of a fluctuating force. The fluctuations are associated with turbulence and are randomly distributed in time. Therefore, if a single particle is observed, the definition of whether a particular flow condition is able to move the particle depends on the period of observation. Lavelle and Mofjeld (1987) query whether it is reasonable to define a critical condition at all; entrainment should properly be considered probabilistically, such as by Einstein (1950) in his bedload model. (A comparison of Einstein’s (1950) entrainment function with the Shields criterion (James 1988) suggests that the original Shields diagram represents a probability of movement of about 50% in the fully turbulent region, i.e. for grain sizes greater than about 2 mm.) Neill and Yalin (1969) and Yalin (1977) proposed criteria for defining observation conditions to ensure consistent results. To assist in quantifying incipient motion criteria, Shvidchenko and Pender (2000) defined the intensity of motion as I¼

m Nt

ð9:9Þ

in which m is the number of particle displacements during time interval t and N is the total number of particles over the sample area. The incipient motion condition then represents a specified, very small, intensity of motion rather than no motion at all. Armitage and Rooseboom (2010) adopted this measure and related it to the ‘movability number’ = u*/w (where w is the grain settling velocity). As a criterion for incipient motion, they selected a value of I = 2  10−5 s−1, corresponding to visual observation of the movement of the first few grains on a bed. Using previously published and their own data, they established the following relationships between the critical value of u*/w and Re* for different ranges of Re*: u 2:2 ¼ w Re 1:4

for

Re \6:2

ð9:10Þ

Re [ 6:2

ð9:11Þ

and u ¼ 0:17 w

for

For practical engineering applications, it is sufficient to bear in mind that the Shields and other similar criteria do not represent absolute thresholds, but a certain probability of movement, and to exercise appropriate caution. Any of the criteria presented can be used for predicting erosion conditions, but specifying settling velocities for rocks makes the criteria expressed in terms of

9.3 Empirical Approach

327

particle settling velocity inappropriate for designing loose rock protection. The critical shear stress approach, as exemplified by the Shields criterion, is therefore recommended for design.

9.4

Design Applications

Knowledge of the flow conditions at which sediment will begin to move is of great value in designing unlined channels through cohesionless material and loose stone linings (riprap) for channels subject to erosion (Fig. 9.4). Both the critical shear stress and permissible velocity approaches are commonly used, and both have been developed further for practical application. Guidelines have also been developed for ensuring that the material below a riprap lining is stable.

9.4.1

Critical Shear Stress Design

Channel Beds Predicting the likelihood of erosion and designing loose rock protection on plane surfaces are easily done through application of the critical shear stress relationships presented in the previous section. Direct application of the Shields criterion is particularly useful and widely accepted. Example 9.1 A wide rectangular channel is laid on a slope of 0.00050 and has a pebble bed with a representative size of 12 mm. The channel conveys a discharge of 1.0 m3/s/m towards a vertical drop structure. A layer of loose stones is to be used to prevent erosion near the drop structure. a. Show that the pebble bed is stable under uniform flow conditions. b. Determine a size for the protecting stones. c. Determine the distance over which the protecting layer should extend. Manning’s n for large bed particles can be calculated using the Strickler formula n¼

D1=6 pffiffiffi 6:7 g

where D is the particle size in metres. Solution For a wide channel, the hydraulic radius, R, can be approximated by the flow depth, y. a. For stability so \soc where so is the bed shear stress and soc is the critical shear stress.

328

9 Scour and Scour Protection so is given by so ¼ c yo Sf yo from Manning (for a wide channel, R  y): y q ¼ y2=3 So1=2 n Therefore y¼

qn 1=2

So

!3=5

with n¼ So y¼

qn 1=2 So

D1=6 0:0121=6 pffiffiffiffiffiffiffi ¼ 0:023 pffiffiffi ¼ 6:7 g 6:7 9:8

!3=5

¼




1:0
0:023 0:000501=2

3=5

¼ 1:017 m

and so ¼ 9:8
103
1:017
0:00050 ¼ 4:98 N/m2 soc is given by soc ¼ sc c DðSs

1Þ

with sc ¼ 0:047 ðShields for large DÞ and Ss ¼ 2:65 Therefore soc ¼ 0:047
9:8
103
0:012
ð2:65

1Þ ¼ 9:12 N/m2

so \soc and so the bed is stable. b. The bed is most vulnerable near the brink, where the flow depth approaches critical, i.e. sffiffiffiffiffi rffiffiffiffiffiffiffiffiffi 2 2 3 1:0 3 q y ¼ yc ¼ ¼ 0:467 m ¼ g 9:8 The protective stone size is given by the Shields critical shear relationship

9.4 Design Applications

329

sc ¼

c y Sf cDðSs 1Þ

Therefore D¼

y Sf sc ðSs 1Þ

with Sf ¼

So Sf ¼ Therefore

D¼




¼




V 2 n2 (Manning) y4=3 q V¼ y D1=6 n¼ pffiffiffi 6:7 g ðq=yÞ2 D1=6 pffiffiffi y4=3 6:72 g

sc ðSs

q2 1Þy7=3 6:72 g

0:047
ð2:65

3=2

ðAÞ

3=2 1:02 ¼ 0:072 m 1Þ
0:4677=3
6:72
9:8

A slightly larger size should be chosen, say 0.080 m. c. The protection should extend until the flow depth is greater than the depth below which erosion of the 12 mm material would occur. The water surface over the protected reach follows an M2 profile from the critical depth at the drop structure.

M2 M2 ymin yc

Equation (A) above can be rearranged to calculate the critical flow depth for a given bed material size, i.e.

330

9 Scour and Scour Protection

ymin ¼




¼




sc ðSs

3=7 q2 1ÞD2=3 6:72 g 1:02 1Þ
0:0122=3
6:72
9:8

0:047
ð2:65

3=7

¼ 0:78 m

The length of protection required is therefore the distance along the M2 profile from yc = 0.47 m to ymin = 0.78 m. This can be computed using the Direct Step Method, as performed in the table below: y (m)

V (m/s)

E (m)

0.47

2.128

0.701

DE (m)

Sf

0.003 0.50

2.000

0.704

1.818

0.719

1.667

0.742

1.429

0.804

1.282

0.864

−0.01030

−0.30

0.00837

−0.00787

−1.85

0.00616

−0.00566

−4.07

0.00422

−0.00372

−16.80

0.00268

−0.00218

−27.43

x (m) 0 0.30 2.16 6.23

0.00316 0.060

0.78

0.01080

0.00527 0.062

0.70

x (m)

0.00705 0.023

0.60

So − Sf

0.00969 0.015

0.55

Sf (ave)

0.01191

23.02

0.00220

50.46

The length of protection should therefore be greater than 50 m, say 55 m

Channel Banks The previous results must be extended for application to channel banks. Bank particles are inherently less stable than bed particles because of their downslope weight components. They are also exposed to boundary shear stresses different from the cross-section average, as given by s = cRS, and the distribution of local values over the section should be considered for efficient design. Both the applied shear stress and the resistance to movement vary over the cross section. A sediment particle on a channel bank may be expected to be less stable than one on the bed because it has a downslope weight component. The shear stress required to displace it is therefore less than for a bed particle. The critical shear stress for a sediment particle on the bank can be expressed in terms of the critical shear stress for a similar particle on the bed by including the downslope weight component in an analysis of its stability. The forces acting on a particle on the plane bank of a channel inclined at an angle a to the horizontal include its submerged weight (W), the force applied by the flow (F) and a force resisting movement (R) (Fig. 9.5).

9.4 Design Applications

331

Fig. 9.5 Forces on a bank particle (solid vectors are in the plane of the bank, dashed is vertical and dotted is normal to the bank)

R

F N

W

T

α

The submerged weight has a component normal to the plane of the bank equal to N ¼ W cos a and a component down the slope of the bank equal to T ¼ W sin a The submerged weight, and hence N and T, can be expressed in terms of the critical shear stress on a horizontal bed. Considering the particle to move by sliding sc / W tan u where / is the angle of repose of the material and tan / is the coefficient of friction. Therefore W/

sc tan u

The downslope weight component can therefore be expressed as T/

sc sin a tan u

and the component normal to the bank can be expressed as N/

sc cos a tan u

The shear force associated with the flow is assumed to act parallel to the longitudinal slope of the channel, but may actually be inclined as a result of secondary flow in the channel. As the incipient motion condition is being considered, this

332

9 Scour and Scour Protection

force is defined as being proportional to the critical value of shear stress on the bank, scb, i.e. F / scb The force resisting motion can be expressed in terms of limiting friction, i.e. R ¼ N tan u ¼ W cos a tan u and, substituting the above proportionality for W, R/

sc cos a tan u tan u

and so R / sc cos a At incipient motion, the forces on the particle in the plane of the bank are therefore as shown in Fig. 9.6. The proportionality constant relates force on a particle to boundary shear and can be assumed to be the same for all forces. At incipient motion, the forces are in equilibrium, and are therefore related by R2 ¼ F 2 þ T 2 Making the above substitutions and cancelling the proportionality constant gives s2c cos2 a ¼ s2cb þ

s2c sin2 a tan2 u

whence sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan2 a scb ¼ sc cos a 1 tan2 u

ð9:12Þ

Fig. 9.6 Forces on bank particle at incipient motion

R ∝ τ c cos α

F ∝ τ cb

T∝

τc tan ϕ

sin α

9.4 Design Applications

333

which can also be expressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin2 a scb ¼ sc 1 sin2 u

ð9:13Þ

The above analysis was suggested in principle by Forchheimer (1924) and introduced for design applications by Lane (1953). Equations (9.12) and (9.13) give the critical shear stress capable of moving a particle on a bank with inclination a in terms of the critical shear stress which would move it on the bed of the channel (as defined by the Shields or some other criterion) and the angle of repose of the material. The critical shear stress on the bank is always less than on the bed, but this does not necessarily mean that bank particles are less stable than those on the bed because the applied shear stress is less on the banks than on the bed. In order to apply the above criterion for stability on the banks, it is necessary to know first a value for the friction angle for the sediment, and second the actual distribution of shear stress in terms of flow conditions. The friction angle can be assumed to be equal to the angle of repose of the material. This is quite easily determined experimentally and Lane (1953) has presented a graphical, empirical relationship between angle of repose, particle size and particle shape (Fig. 9.7). Shear stress is not uniformly distributed over the boundary (so = cRS gives the average over the cross section). Lane (1953) determined the shear stress distribution

Fig. 9.7 Angles of repose of non-cohesive sediment (adapted from Lane 1953)

42 40

Angle of repose (o)

38 36 34 32 30 28 26 24 22 20 5

10

20

Particle size (mm)

50

100

334

9 Scour and Scour Protection

Table 9.1 Permissible shear stress adjustment for channel sinuosity (Lane 1955)

Degree of sinuosity

Relative limiting shear stress

Straight Slightly sinuous Moderately sinuous Very sinuous

1.00 0.90 0.75 0.60

experimentally for various cross-sectional shapes. He showed that for rectangular and trapezoidal channels, the local shear stress does not exceed cyS on the bed and about 0.76 cyS on the banks for any width–depth ratio and side slope (y is the maximum flow depth). It has also been found that channel sinuosity increases the likelihood of scour. Lane (1955) proposed adjusting the permissible shear stress by the factors listed in Table 9.1 to account for this effect. Using data from various sources, Lane (1953) also showed that the critical shear stress is increased by a sediment load, this effect becoming more pronounced and significant as the particle size decreases below about 5 mm. Example 9.2 A trapezoidal channel with a bottom width of 4.0 m and side slopes of 2.5H:1V is excavated on a slope of 0.00080 in alluvial material with a representative particle size of 18 mm and an angle of repose of 33°. Determine the maximum discharge for which there will be no erosion in the channel. Manning’s n for large bed particles can be calculated using the Strickler formula n¼ where D is the particle size in metres.

D1=6 pffiffiffi 6:7 g

Solution The maximum discharge can be found from the Manning equation for the maximum flow depth that would induce erosion which, in turn, can be found through the Shields criterion. The applied and resisting shear stresses are different for the bed and banks, and must be found separately. For the bed: The critical shear stress is found from the Shields criterion for large particles, i.e. sc c DðSs

1Þ

¼ 0:047

Therefore sc ¼ c DðSs

1Þ0:047 3

¼ 9:8
10
0:018
ð2:65

1Þ
0:047 ¼ 13:7 N/m2

9.4 Design Applications

335

The maximum applied shear stress is so ¼ c ymax S Therefore, at incipient erosion so ¼ sc i.e. c ymax S ¼ 13:7 so ymax ¼

13:7 ¼ 1:75 m 9:8
103
0:00080

For the banks: The critical shear stress is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan2 h scb ¼ sc cos h 1 tan2 / where h ¼ arctan

1 ¼ 21:8 2:5

Therefore rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tan2 21:8 ¼ 10:0 N/m2 scb ¼ 13:7
cos 21:8
1 tan2 33 The maximum applied shear stress is so ¼ 0:76 c ymax S Therefore, at incipient erosion so ¼ sc i.e. 0:76 k ymax S ¼ 10:0 So ymax ¼

10:0 ¼ 1:68 m 0:76
9:8
103
0:00080

The critical flow depth is lower for erosion of the banks than for the bed, so ymax = 1.68 m. The discharge is given by the Manning equation combined with continuity, i.e.

336

9 Scour and Scour Protection

Q¼

A 2=3 1=2 R S n

where 1 A ¼ Bymax þ 2
ðsymax Þymax 2 ¼ 4:0
1:68 þ 2:5
1:682 ¼ 13:8m2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P ¼ B þ 2 ðsymax Þ2 þ y2max qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 4:0 þ 2
ð2:5
1:68Þ2 þ 1:682 ¼ 13:0m A 13:8 ¼ ¼ 1:06m P 13:0 1=6 D 0:0181=6 pffiffiffiffiffiffiffi ¼ 0:024 n¼ pffiffiffi ¼ 6:7 g 6:7
9:8

R¼

Therefore

13:8 2=3 1=2 R S n 13:8 1:062=3 0:000801=2 ¼ 16:9 m3 =s ¼ 0:024

Q¼

9.4.2

Permissible Velocity Design

Various methods for defining bed stability in terms of a critical velocity have been proposed, but are difficult to formulate because particle movement is not related to the cross-section average velocity, but rather to the boundary shear stress or a velocity near the bed. Criteria in terms of the average velocity usually require specification of the flow depth as well. Criteria in terms of a local velocity near the bed require estimation of this velocity through a velocity distribution equation. However, the commonly used logarithmic distribution equation includes u*, which depends on the flow depth. (Yang (1973) followed this approach in developing his critical velocity criterion (Eqs. (9.7) and (9.8)), by relating the near-bed velocity to the average velocity through integration of the velocity profile, also requiring knowledge of u* and hence the flow depth.)

The HEC-11 Procedure for Riprap Rock Sizing Recognizing a preference in engineering practice for using flow velocity rather than bed shear stress as a basis for designing riprap, the United States Federal Highway Administration reformulated the Shields criterion and expressed it as a relationship

9.4 Design Applications

337

between the minimum rock size and the average flow depth and velocity (Brown and Clyde 1988). Although it offers little advantage over direct application of the Shields criterion, its development is included here because of its common use. The stability factor (SF) is defined as the ratio of the critical shear stress at incipient motion (sc) to the applied shear stress (so), SF ¼

sc so

ð9:14Þ

with stability requiring a value greater than 1.2, based on observations of stable and unstable linings. The applied shear stress is given by so ¼ c R S

ð9:15Þ

where c is the specific weight of the rock material, R is the hydraulic radius and S is the energy gradient. The critical shear stress can be expressed in terms of the dimensionless Shields parameter (s*c) as sc ¼ K1 sc cD50 ðSs

1Þ

ð9:16Þ

in which D50 is the median rock size and Ss is the specific gravity of the rock material. The factor K1 accounts for a transverse slope of the surface (such as a sloping channel bank) and is given by (from Eq. (9.13)) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin2 a K1 ¼ 1 sin2 u

ð9:17Þ

where a is the side slope angle and / is the angle of repose of the riprap material. The required value of SF can therefore be expressed as K1 sc c D50 ðSs cRS

1Þ

¼ 1:2

ð9:18Þ

The value of s*c = 0.047, as indicated on the Shields diagram for large particles, was found to be supported by field observation for typical riprap rocks. For simplicity, it was assumed that R = d, the average flow depth in the main part of the channel (between flood plains). A typical value of Ss = 2.65 was assumed, but a correction for different values allowed for subsequently. The energy gradient can be determined from the Manning equation as S¼

V 2 n2 R4=3

ð9:19Þ

338

9 Scour and Scour Protection

where V is the average flow velocity and n is the Manning resistance coefficient, which can be related to the median rock size by 1=6

n ¼ 0:0482 D50

ð9:20Þ

Making these substitutions, Eq. (9.18) can be rearranged to give the minimum stable rock size as D50 ¼

0:0068 V 3 3=2

d 1=2 K1

ð9:21Þ

This size can be adjusted to account for a different rock material specific gravity (Ss) by multiplying D50 by the factor Csg ¼

2:12 ðSs

1Þ3=2

ð9:22Þ

and for an SF different from 1.2 by multiplying by the factor Cf ¼




SF 1:2

3=2

ð9:23Þ

The method is intended for use with uniform or moderately gradually varied flows. SF is used to allow for uncertainty associated with departures from these conditions according to Table 9.2. Note that although the method accounts for the reduced stability of rocks on side slopes (through factor K1), it uses the average applied shear stress and does not allow for the reduction of local shear on the banks (as by Lane (1953)). Because erosion is related to the boundary shear, the required rock size by this method requires specification of both flow depth and velocity (Eq. (9.21)). Within the gradually varied flow regions associated with control structures, it is therefore Table 9.2 Guidelines for selecting stability factor values adapted from Brown and Clyde (1988) Condition

SF range

Uniform flow; straight or mildly curving reach (curve radius/channel width >30); minimum impact of waves and floating debris; little or no uncertainty in design parameters Gradually varied flow; moderate bend curvature (30 > curve radius/channel width >10); moderate impact of waves and floating debris Approaching rapidly varied flow; sharp bend curvature (curve radius/channel width 1.0) and reaches a maximum at an approach velocity (V0p) at which the bed planes out and bed forms disappear (Fig. 9.10). This velocity is given by the greater of V0p ¼ 5Vc

ð9:41Þ

pffiffiffiffiffiffiffiffi V0p ¼ 0:6 g y0

ð9:42Þ

or

The live-bed equilibrium scour depth for approach velocities between the live-bed threshold and the maximum value is given by " yse ¼ f1 2:2 a

V0 Vc V0p Vc

1 1

!

þ 2:5 f3

V0p Vc V0p Vc

V0 Vc

1

!#

for 1:0 

V0 V0p  Vc Vc

ð9:43Þ

The maximum live-bed scour is given by yse ¼ 2:2 f1 a

for

V0p V0 [ Vc Vc

ð9:44Þ

346

9 Scour and Scour Protection

The HEC-18 equation for predicting the maximum scour depth (Arneson et al. 2012) is widely used and performed well in the evaluation by Sheppard et al. (2011, 2014). For clear-water and live-bed scour with relatively fine sediment (D50 < 20 mm), y 0:35 yse 0 ¼ 2:0 K1 K2 K3 Fr00:43 a a

ð9:45Þ

in which a is the pier width (the short dimension), K1 accounts for pier shape, K2 for the alignment angle of the approach flow (h) and K3 for the bed-form regime. For h > 5°, the shape influence is small compared with the alignment influence and K1 = 1.0 for all shapes; for h < 5°, K1 = 1.0 for circular and rounded piers, 1.1 for square-nosed piers and 0.9 for sharp-nosed piers. For non-circular piers, the value of K2 depends on the alignment and the geometry according to
0:65 L K2 ¼ cos a þ sin a a

ð9:46Þ

where L is the length of the pier. K3 = 1.1 for clear-water conditions and beds with plane surfaces or antidunes or dunes smaller than about 3 m, increasing to 1.3 for dunes larger than about 10 m. A ‘rule-of-thumb’ is recommended that yse < 2.4a for Fr0 < 0.8 and yse < 3.0a for Fr0 > 0.8. Arneson et al. (2012) provide a separate equation for relatively coarse (D50 > 20 mm) and graded (r  1.5) bed sediments, under clear-water (V0/ Vc < 1) conditions, i.e.
 yse F2d ¼ 1:1 K1 K2 tanh 1:97 r1:5 a0:62 y00:38

ð9:47Þ

in which Fd is the densimetric particle Froude number (=V0/(g(Ss − 1)D50)0.5) and r (=(D84/D16)0.5) is the geometric standard deviation of the sediment size distribution. This equation was revised by Shan et al. (2016) to extend the conditions of its applicability as
 yse F2d ¼ 1:32K1 K2 K3 tanh 1:97r1:5 a0:62 y00:38

ð9:48Þ

Equation (9.48) is able to predict the scour depth for clear-water scour and for live-bed scour with V0/Vc up to 5.2 and Fr0 up to 1.7, and for sediments with 0.21 mm < D50 < 127 mm and r up to 7.5.

9.5 Scour Around Bridge Piers

347

Scour Depth Evolution The actual scour depth depends on the duration of the scouring flow relative to the development of the scour hole with time. Live-bed scour during floods progresses rapidly and the equilibrium scour depth is representative (Melville and Chiew 1999). Clear-water scour develops very slowly, however, and may be intermittent for piers on flood plains. Various methods have been proposed for estimating the time for equilibrium scour to develop and for describing the evolution of the scour hole towards equilibrium. Sheppard et al. (2011) described and evaluated ten scour evolution methods. They recommended use of the clear-water method proposed by Melville and Chiew (1999) for circular piers, with revised coefficient values and the equilibrium scour depth calculated by the S/M method described above. This is referred to as the Melville/Sheppard (M/S) method and can be used for live-bed as well as clear-water conditions. The scour depth (ys) at any time t after the commencement of scour is given by ys ¼ exp yse

(


 1:6 )

Vc t

0:04 ln te V0

ð9:49Þ

in which te is the time to equilibrium, given by
a V0 te ðdaysÞ ¼ 200 V0 Vc

0:40



for

y0 [ 6; a

V0 [ 0:4 Vc

ð9:50Þ

or te ðdaysÞ ¼ 127:8


a V0 V0 Vc

0:40

  y0 0:25 a

for

y0  6; a

V0 [ 0:4 Vc

ð9:51Þ

Because scour approaches the equilibrium depth asymptotically, a time to equilibrium has little practical meaning and Sheppard et al. (2011) also suggest the time to 90% of equilibrium scour as a more useful measure. This is given by t90 ¼ exp




 V0 1:83 te V

ð9:52Þ

Oliveto and Hager (2002, 2005) proposed an equation for time-dependent clear-water scour, also applicable for nonuniform sediments, which is independent of the time to equilibrium, i.e. ys ¼ 0:068 N r yR

0:5 1:5 Fd

log T

ð9:53Þ

348

9 Scour and Scour Protection

in which yR = (y0D2)1/3 for a circular pier with diameter D or yR = (y0b2)1/3 for an abutment with width b, N = 1 for a circular cylindrical pier or 1.25 for a vertical abutment. Fd is the densimetric particle Froude number (=V0/(g(Ss − 1)D50)0.5) and T = t/tr with tr = yR/(r1/3(g(Ss−1)D50)0.5). Oliveto and Hager limit the applicability of Eq. (9.53) to clear-water scour by specifying a maximum upstream densimetric Froude number of 1.2 for the local initiation of motion, and to sediments with D50 less than 5–10 times y0. These equations for describing the time variation of the scour depth can be applied to determine the pre-equilibrium depth for a specified duration of design discharge, or to a time-varying discharge for a design hydrograph if necessary. The scour depth resulting from a single flow episode may be considerably less than the equilibrium value; Melville and Chiew (1999) estimate that the scour hole develops to between 50 and 80% of the equilibrium depth after the initial 10% of the time to equilibrium under steady flow conditions. Because of the absence of sediment supply for clear-water scour, no infilling would occur between intermittent events and scour would be cumulative. The total duration of scouring discharges through the structure’s design life should therefore be considered in relation to the scour development time. For live-bed conditions, prediction of time-dependent scour development would require simulation including sediment transport calculations to determine the rate of sediment supply to the scour hole. Relationships similar to those for piers have also been developed for abutments, for example, by Cardoso and Fael (2010), Dey and Barbhuiya (2004, 2005), Mohammedpour et al. (2017) and Yanmaz and Kose (2009). Example 9.3 A rectangular, sharp-edged bridge pier 20 m long and 4.0 m wide is founded in sediment with D50 = 1.0 mm. The design flow, lasting approximately 4 h has a depth of 2.0 m and a velocity of 1.5 m/s. Using the S/M and M/S methods, determine a. the equilibrium scour depth and b. the scour depth after the design flow duration. Solution a. Determine whether clear-water or live-bed scour will occur under the design conditions by calculating the ratio of V0 to Vc. Vc is given by Eq. (9.30) or (9.31) depending on the flow condition defined by Re*. Re ¼

u ks m

At the critical condition (from Eq. (9.32))

9.5 Scour Around Bridge Piers

349



1=2 9:09
10 6 16:2 D50 D50 ð38:76 þ 9:6 ln D50 Þ 0:0050 D50 

1=2 9:09
10 6 ¼ 16:2
0:0010 0:0010
ð38:76 þ 9:6 ln 0:0010Þ 0:0050 0:0010

u ¼ uc ¼

¼ 0:023 m/s For D50  0.6 mm

ks ¼ 2:5 D50 ¼ 2:5
0:0010 ¼ 0:0025 m Therefore Re ¼

0:023
0:0025 ¼ 57:5 1
10 6

which is greater than 70 and so, from Eq. (9.31)
 2:21 y0 Vc ¼ 2:5uc ln D50
 2:21
2:0 ¼ 0:48 m/s ¼ 2:5
0:023
ln 0:0010 Therefore V0 1:50 ¼ 3:1 ¼ Vc 0:48 and so live-bed scour will occur. The approach velocity at which maximum live-bed scour occurs is the greater of V0p ¼ 5Vc ¼ 5
0:48 ¼ 2:4 m/s or

Therefore

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi V0p ¼ 0:6 gy0 ¼ 0:6
9:8
2:0 ¼ 2:7 m/s V0p 2:7 ¼ 5:6 ¼ 0:48 Vc

V0p ¼ 2:7 m/s and

V0p > V0c, so the equilibrium scour depth is given by Eq. (9.43), i.e. " yse ¼ f1 2:2 a

V0 Vc V0p Vc

1 1

!

þ 2:5 f3

V0p Vc V0p Vc

V0 Vc

1

!#

350

9 Scour and Scour Protection with

4m

f1 ¼ tanh

 

y0 0:4 a

a ¼ Ks ap

20 m





ap ¼ 4 cos 15 þ 20 sin 15 ¼ 9:04 m  p

4

Ks ¼ 0:86 þ 0:97 a

4  p

4

¼ 0:86 þ 0:97 0:262

4 ¼ 0:933

15o (= 0.262 radians)

a ¼ 0:933
9:04 ¼ 8:43 m f1 ¼ tanh

"


f3 ¼

0:4



¼

0:4



a D50

2:0 8:43

1:2

0:4 #





   þ 10:6 Da50  8:43 

 8:43 1:2 0:0010

¼ 0:41

a D50

0:13

0:0010

þ 10:6


yse 3:1 ¼ 0:51
2:2  5:6 a ¼ 0:796

¼ 0:51





8:43 0:0010

0:13




1 5:6 3:1 þ 2:5
0:41
1 5:6 1

Therefore yse ¼ 0:796
8:43 ¼ 6:71 m b. The scour depth after the design flow duration is given by Eq. (9.49) with t = 4 h or 4/24 = 0.17 days, i.e. (


 1:6 )

Vc ys t

¼ exp 0:04 ln te yse V0

with te given by Eq. (9.50) or (9.51) depending on the value of y0/a*. Here yo/a* = 2.0/8.43 = 0.24, indicating use of Eq. (9.51), i.e.

9.5 Scour Around Bridge Piers

351

te ðdaysÞ ¼ 127:8 ¼ 127:8


a V0 V0 Vc 8:43 ð3:1 1:5

¼ 1350 days

  y0 0:25 a
 2:0 0:25 0:40Þ 8:43

0:40

Therefore (


 )

1 ys 0:17

1:6 ¼ exp 0:04

ln 3:1 1350 yse and

9.5.2

¼ 0:80

ys ¼ 0:80
6:71 ¼ 5:4 m

Bridge Scour Countermeasures

If the anticipated scour depth is unacceptable, its development needs to be prevented or restricted. This can be achieved either by protecting the bed around the pier base with a stable surface, such as riprap, or by installing features that modify the local flow pattern to reduce its scouring action.

Design of Riprap for Pier Protection Riprap protection has the advantage over fixed protection devices of being flexible and able to interact and move with the bed sediments (Chiew and Lim 2000). The design of a riprap layer requires specification of the rock size, the level of the layer relative to the original bed, the thickness of the layer and the extent of the layer. Many recommendations for riprap design have been proposed, all based on data obtained from laboratory experiments. Four of the more recent and complete methods are presented below. The results obtained by using different methods should not be expected to be entirely consistent, as different experimental conditions and definitions of failure were used in their development. For clear-water conditions, Chiew (1995) defined three modes of riprap failure. Shear failure is the movement of the stones by the local shear stress associated with the flow pattern around the pier. Winnowing failure is the removal of the underlying bed material through the voids in the riprap layer. Edge failure arises from undermining of the stones by scour of the unprotected bed at the periphery of the layer. The possibility of each of these should be considered in the design. Shear failure is prevented by ensuring that the riprap stones are large enough to resist the scouring action. Chiew (1995) showed for circular piers that the approach velocity (V0) at which pier scour commences is equal to 0.3 times the critical velocity for scour without the pier (Vc). This is assumed to define the critical condition for entrainment of riprap stones, provided it is adjusted to account for the

352

9 Scour and Scour Protection

influences of the size of the riprap relative to the pier diameter and the approach flow depth relative to the pier diameter by the introduction of adjustment factors. The relationship is then V0 0:3 ¼   Vc K D K y0  D Dr50

ð9:54Þ

in which D is the pier diameter and Dr50 is the median size of the riprap stones. The adjustment factors, derived from laboratory test data, are given by


D K Dr50 K




D Dr50





¼1

for

D  50 Dr50

ð9:55Þ


 2 D D for 1   50 ð9:56Þ 0:034 ln Dr50 Dr50 y  y0 0 K 3 ð9:57Þ ¼ 1 for D D y  y 0:322 y0 0 0 K ð9:58Þ 0:106 for 0\ \3 ¼ 0:783 D D D

¼ 0:398 ln




D Dr50



Equation (9.54) can be developed as a relationship between V0 and D50 to enable sizing of the riprap. Vc can be expressed in terms of the shear velocity through Manning’s equation with Strickler’s relationship (n = 0.042D1/6 r50) as
 Vc y0 1=6 ¼ 7:66 uc Dr50

ð9:59Þ

At the critical entrainment condition, from the Shields criterion for large particles, u2c gDr50 ðSs

1Þ

¼ 0:056

ð9:60Þ

Combining Eqs. (9.59) and (9.60) and rearranging gives Dr50 ¼

Vc3 pffiffiffiffiffi 388 y0

which can be expressed in terms of V0 through Eq. (9.54), i.e.

ð9:61Þ

9.5 Scour Around Bridge Piers

Dr50

353

  3    3 V03 K DDr50 K yD0 ¼ pffiffiffiffiffi 10:47 y0

ð9:62Þ

for the minimum riprap stone size. Note that for D/Dr50 < 50 and/or y0/D < 3, the solution must be found iteratively because Dr50 appears in the adjustment factor. Unger and Hager (2006) conducted laboratory tests on single layers of uniformly sized riprap placed in concentric rows around a circular pier. Failure was assumed to have occurred when the first stone along the pier perimeter was dislodged by rolling, undermining or sliding. Conditions at failure are described according to the Shields criterion, with adjustment for the effects of the size of the pier relative to the channel width (B), the size of the riprap (Dr) relative to the underlying material and the extent of the riprap coverage. The basic Shields criterion for rough turbulent flow was expressed by Hager and Oliveto (2002) as r

1=3


1=6 Vc R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1:65 D gD50 ðSs 1Þ 50

ð9:63Þ

The extended form for riprap around the pier, including the adjustments derived from the test results, is r

1=3


1=6 V0c Ro pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1:65 ð1 D50 gD50 ðSs 1Þ

0:40

b1=4 Þd0:20n

ð9:64Þ

in which D50 refers to the underlying material, V0c is the critical approach velocity for the riprap, R0 is the approach hydraulic radius, n is the number of rows of stones, b = D/B and d = Dr/D50. The applicability of the equation is limited to rough turbulent approach flow, 2 < d < 50, 1 < r < 3, 1  n  10, 0.05 < b < 0.25 and 0.60 < V0/Vc < 1.2 with Vc calculated for the approach flow through Eq. (9.63). Froehlich (2013) developed an equation for riprap size by defining the factor of safety against overturning of a stone in terms of the Shields criterion and relating the velocity adjacent to the pier to the approach velocity through a potential flow analysis. For a horizontal riprap cover the factor of safety, defined by the ratio of static resisting and overturning moments on a single stone, is given by fs ¼

1 g0 þ

1 Ss

ð9:65Þ

in which η0 is the stone stability number representing the ratio of applied to critical shear stress multiplied by the relative difference in the densities of the riprap and water, and is given (in terms of dimensionless shear stresses) by

354

9 Scour and Scour Protection


g0 ¼ 1

 1 s Ss sc

ð9:66Þ

The applied shear stress can be expressed through Manning’s equation, with n = (kn/g0.5)D1/6 r , as
1=3 2 Dr s ¼ q kn V2 y where s, y and V are the local shear stress, flow depth and depth-averaged velocity. Froehlich suggests a value for kn of 0.145. Substituting s/(qgDr(Ss–1)) in Eq. (9.66) gives
 1 kn2 y 2=3 2 Fr g0 ¼ Ss sc Dr

ð9:67Þ

Equating Eqs. (9.66) and (9.67) yields an expression for the riprap stone size, 0 13=2 Dr @ kn2  A Fr3 ¼ y sc Sfss 1

ð9:68Þ

The local Froude number can be expressed in terms of the approach value (Fr0) through a relationship between the corresponding velocities. Using potential flow theory, Froehlich showed that the maximum average velocity with a distance eDr from the pier surface, measured normal to the approach flow direction, is given by V ¼ V0




D=ðeDr Þ þ 1 D=ð2eDr Þ þ 1



ð9:69Þ

The parameter e is a multiplier of the order of unity to define the distance from the pier to be considered, and needs to be specified. Substituting this velocity in Fr in Eq. (9.68) and assuming the local flow depth to be the same as the approach depth give 0 13=2
 Dr @ kn2 D=ðeDr Þ þ 1 3 3 A   ¼ Fr0 ð9:70Þ D=ð2eDr Þ þ 1 y0 sc Ss 1 fs

Froehlich presented a more complete expression for the bracketed term in Eq. (9.68) for riprap heaped at an angle against the pier. He also provided adjustments to account for the effect of pier width, cross-flow shear in the approach flow, multiple equally spaced piers, different pier shapes and the alignment of the approach flow for non-circular piers. Application of Eq. (9.70) requires

9.5 Scour Around Bridge Piers

355

specification of the safety factor, which depends on the confidence of the designer in the calculation of the hydraulic conditions, particularly y0 and V0. Froehlich recommends values of 1.25 for fs and 0.060 for s*c. Winnowing failure by erosion of the underlying material can lead to the development of a scour hole beneath the riprap layer, into which the entire intact layer sinks (Chiew 1995). Winnowing can be prevented by providing more than one layer of riprap. Recommendations for layer thickness (t) include t > 2Dr50 by Neill (1973) and by Lauchlan and Melville (2001), and t > 3Dr50 by Richardson et al. (1991). Edge failure can be countered by providing sufficient thickness and extent of riprap cover. A thick layer is able to maintain an armoured surface as the adjacent fine material is eroded. The further the riprap extends beyond the pier, the less pronounced is the scouring effect produced by the pier on the adjacent bed. Neill (1973) recommended an extent of 1.5D in all directions from the face of the pier; Richardson et al. (1991) recommended a lateral extent of 2D from the face of the pier, but made no recommendation for a longitudinal extent. Lauchlan and Melville (2001) recommend a total extent of 4D around the pier. Unger and Hager’s (2006) Eq. (9.64) implies a relationship between the effective size and number of rows of stones. For live-bed conditions, an additional mode of riprap failure is recognized, whereby the layer is destabilized by the passage of bed forms past the pier (Chiew and Lim 2000; Lauchlan and Melville 2001). If the trough level of the bed forms is deeper than the bottom of the riprap layer, the stones are undercut and settle into the trough and become embedded. The destabilizing effect of the bed forms also interacts with the modes of failure recognized for clear-water scour, but remains the dominant failure mode. Riprap destabilized by bed forms settles as a layer to the level of the bed-form troughs, and remains capable of providing protection. Lauchlan and Melville (2001) found that the effectiveness of a riprap layer against live-bed scour could be increased by initially placing the layer some distance below the original bed surface. They proposed an equation for the required riprap size, which accounts for this effect, i.e. Dr50 ¼ KD KS Ka Kc 0:3 Fr01:2 y0

ð9:71Þ

Equation (9.71) implicitly assumes a riprap layer with a thickness of 2Dr50 and an extent of 4D. KD, KS and Ka account for the pier diameter relative to the bed material size, the pier shape and alignment, respectively, which were all assumed to have values of 1.0. Assuming Ks = 1.0 and Ka = 1.0 would be appropriate for circular piers and, following Chiew (1995), KD = 1.0 would be valid for D/ Dr50 > *50. The factor Kc accounts for the placement depth of the riprap layer, for which failure of the layer is assumed to correspond to a riprap scour depth not exceeding 20% of the maximum unprotected scour depth. From their experimental data, Lauchlan and Melville propose

356

9 Scour and Scour Protection

Kc ¼




Y y0

1

2:75

ð9:72Þ

in which Y is the placement level of the top of the riprap layer below the initial sediment surface. Comparison with other methods showed Eqs. (9.71) and (9.72) to be conservative, predicting considerably larger riprap sizes, especially for Fr0 less than about 0.5. Example 9.4 Determine the size of riprap required to protect the riverbed around a 2.5 m diameter circular bridge pier for an approach flow with a depth of 2.0 m and a velocity of 1.8 m/s using a. the method of Chiew (1995), b. the method of Froehlich (2013) and c. the method of Lauchlan and Melville (2001) for different riprap depths. Solution a. The riprap size is given by Eq. (9.62), i.e.

Dr50

    3    3 V03 K DDr50 K yD0 ¼ pffiffiffiffiffi 10:47 y0

(y0/D) = 2.0/2.5 = 0.80 < 3, so K(y0/D) is given by Eq. (9.58), i.e. K

y 

y 0:322 0 ¼ 0:783 0:106 D D
0:322 2:0 ¼ 0:783 0:106 ¼ 0:623 2:50 0

K(D/Dr50) = 1.0 for D/Dr50 > 50, i.e. Dr50 < 2.5/50 = 0.050 m, which is unlikely. Therefore, Dr50 must be found by trial, with K(D/Dr50) given by Eq. (9.56), i.e. K




D Dr50




 D ¼ 0:398 ln Dr50


 2 D 0:034 ln Dr50

Calculations only with the final trial value are shown. Try Dr50 = 0.084 m. Then K




D Dr50



¼ 0:398 ln ¼ 0:959




 2:50 0:084




2:50 2 0:034 ln 0:084

9.5 Scour Around Bridge Piers

357

and 1:803
0:9593
0:6233 pffiffiffiffiffiffiffi 10:47 2:0 ¼ 0:084 m

Dr50 ¼

which is equal to the trial value. b. The riprap size is given by Eq. (9.70), i.e. 0 13=2
 Dr @ kn2 D=ðeDr Þ þ 1 3 3  A Fr0 ¼ D=ð2eDr Þ þ 1 y0 sc Sfss 1 with kn ¼ 0:145 sc ¼ 0:06 Ss ¼ 2:65 fs ¼ 1:25 e ¼ 1:0 V0 1:80 Fr0 ¼ pffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:407 gy0 9:8
2:0

Solution for Dr50 is by trial; calculations only with the final trial value are shown. Try Dr50 = 0.158 m. Then Dr ¼ 2:0 giving

0:1452   0:060 2:65 1 1:25

!3=2


 2:5=ð1:0
0:158Þ þ 1 3 0:4073 2:5=ð2
1:0
0:158Þ þ 1

Dr = 0.158 m, which is equal to the trial value.

If fs = 1.0, Dr = 0.095 m. If fs = 1.25 and e = 2.0, Dr = 0.141 m,

indicating quite low sensitivity to specification of e. c. The riprap size is given by Eq. (9.71), i.e. Dr50 ¼ KD KS Ka Kc 0:3Fr1:2 0 y0 with KD ¼ KS ¼ Ka ¼ 1:0
 Y 2:75 Kc ¼ 1 ðequationð9:72ÞÞ y0

358

9 Scour and Scour Protection For Y = 0 Kc ¼ 1:0 So Dr50 ¼ 1:0
1:0
1:0
1:0
0:3
0:4071:2 2:0 giving Dr50 = 0.204 m For Y = 0.30 m Kc ¼ ¼







2:75

1

Y y0

1

 0:30 2:75 ¼ 0:64 2:0

So Dr50 ¼ 1:0
1:0
1:0
0:64
0:3
0:4071:2 2:0 giving Dr50 ¼ 0:131 m For Y = 0.60 m Kc ¼ ¼







2:75

1

Y y0

1

0:60 2:0

2:75

¼ 0:375

So Dr50 ¼ 1:0
1:0
1:0
0:375
0:3
0:4071:2 2:0

giving Dr50 ¼ 0:077 m

Scour Prevention by Flow Modification Bridge-pier scour results primarily from the action of the downflow jet and horseshoe vortices and measures to reduce their intensity are effective in reducing the extent of scour. Various flow-altering countermeasures used for modifying the flow pattern have been reviewed by Tafarojnoruz et al. (2010) and classified into four distinct categories as summarized below.

9.5 Scour Around Bridge Piers

359

Openings through the piers reduce the strength of the downflow and horseshoe vortex by allowing a portion of the approach flow to pass through openings within the pier or pier group. Internal connecting tubes have entry openings at the front of the pier and exits at locations further around the perimeter and, depending on their geometry and the approach flow conditions, have been found to reduce the scour depth up to 39%. A vertical slot in the pier can divert more of the flow than internal tubes, and therefore more effectively reduce the strength of the downflow jet and up to 88% scour reduction has been recorded. Significant flow diversion can be achieved by replacing a large pier with a group of smaller piers. Devices attached to the pier surface can also reduce the strength of the scouring flow. A particularly simple measure is the threading of cables spirally around a pier. Depending on the cable number, size and spiral angle, scour reductions of up to 46% have been achieved. A collar is a thin disc around a pier, which may be placed singly at different levels or in multiples over the anticipated flow depth. Thin horizontal plates may also be attached to the upstream face of the pier; various shapes and configurations have been used and meaningful reductions in scour depth obtained. Bed attachments of various types are used to divert or weaken the approach flow, and consequently reduce the strength of the downflow and horseshoe vortex. Different forms of attachments include sacrificial piles, vanes and sills, and surface guide panels. The fourth category of countermeasures described by Tafarojnoruz et al. (2010) includes modifications to the pier shape, extraction of water through holes in the pier by a suction pump and judicious placement of the pier footing. These and the previously mentioned measures can also be used in combination. Problems 9:1. A plane bed of: sand is laid in a long, wide channel on a slope of 0.00050. Determine the flow depth, velocity and unit width discharge at which movement of the bed material will just occur for sand grain sizes of 1.0, 5.0 and 10 mm. 9:2 Water is to be released from a small storm-water retention pond at a discharge of 0.90 m3/s/m and a velocity of 6.0 m/s into a long, wide channel on a slope of 0.015 and lined with loose 100 mm stones. a. Show that erosion is unlikely once uniform flow is attained, but likely near the reservoir outlet. It is proposed to provide a rigid lining near the reservoir outlet where scour could occur. Estimate the distance from the outlet to where loose placing could commence without scour occurring b. if the rigid lining is made by setting the 100 mm stones in concrete and c. if the rigid lining is plain concrete. Gradually varied flow calculations should be done numerically using three computational steps.

360

9 Scour and Scour Protection

Note: Manning’s n for the stones set in concrete is 0.030. For loose stone, Manning’s n can be calculated from the Strickler relationship, n¼ where D is the stone size in metres.

D1=6 pffiffiffi 6:7 g

9:3 A trapezoidal channel with a bottom width of 3.0 m and side slopes of 3H:1V is excavated on a slope of 0.0020 in alluvial material with a representative size, D, of 20 mm and an angle of repose of 35o. Manning’s n can be found from the Strickler relationship, n¼

D1=6 pffiffiffi 6:7 g

in which D is the representative particle size (m). Calculate the maximum discharge for which there will be no erosion in the channel. State clearly any assumptions made in your calculations. 9:4 A canal is required to convey 20 m3/s of water on a slope of 0.0020 through well-rounded alluvial material with D50 = 25 mm. Recommend a suitable cross section after considering a. a concrete-lined canal, b. an unlined canal with stabilized banks, with no erosion of the bed and c. an unlined trapezoidal canal, with no erosion. 9:5 A 2.5 m circular bridge pier is found in river sand with D50 = 0.50 mm. For a steady approach flow depth of 3.8 m with a velocity of 2.0 m/s determine a. the equilibrium scour depth and b. the scour depth after 8 h. Analyse the sensitivity of equilibrium scour depth with respect to the pier diameter, sand size and estimation of approach flow conditions.

References Armitage, N., & Rooseboom, A. (2010). The link between movability number and incipient motion in river sediments. Water SA, 36(1), 89–96. Arneson, L. A., Zevenbergen, L. W., Lagasse, P. F., & Clopper, P. E. (2012). Evaluating Scour at Bridges (5th ed. Rep. No. FHWA-HIF-12-003, HEC-18). Washington, DC: Federal Highway Administration. Brown, S. A., & Clyde, E. S. (1988). Design of Riprap Revetment (Rep. No. FHWA-IP-89,016, HEC-11). USA: Federal Highway Administration. Cardoso, A. H., & Fael, C. M. S. (2010). Time to equilibrium scour at vertical-wall bridge abutments. Water Management, 163(10), 509–513.

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Chiew, Y.-M. (1995). Mechanics of riprap failure at bridge piers. Journal of Hydraulic Engineering, 121(9), 635–643. Chiew, Y.-M., & Lim, F.-H. (2000). Failure behaviour of riprap layer at bridge piers under live-bed conditions. Journal of Hydraulic Engineering, 126(1), 43–55. Collins, M. B., & Rigler, J. K. (1982). The use of settling velocity in defining the initiation of motion of heavy mineral grains, under unidirectional flow. Sedimentology, 29, 419–426. Dey, S., & Barbhuiya, A. K. (2004). Clear-water scour at abutments. Water Management, 157, 77–97. Dey, S., & Barbhuiya, A. K. (2005). Time variation of scour at abutments. Journal of Hydraulic Engineering, 131(1), 11–23. Dey, S., & Raikar, R. V. (2005). Scour in long contractions. Journal of Hydraulic Engineering, 131(2), 1036–1049. Einstein, H. A. (1950). The Bed-Load Function for Sediment Transportation in Open Channel Flows (Technical Bulletin No 1026, U S Department of Agriculture). Washington, D C: Soil Conservation Service. Ettema, R., Constantinescu, G., & Melville, B. W. (2017). Flow-field complexity and design estimation of pier-scour depth: Sixty years since Laursen and Toch. Journal of Hydraulic Engineering, 143(9), 03117006. Forcheimer, P. (1924). Hydraulik. Leipzig and Berlin: Teubner, Verlagsgesellschaft. Froehlich, D. C. (2013). Protecting bridge piers with loose rock riprap. Journal of Applied Water Engineering and Research, 1(1), 39–57. Grass, A. J. (1970). Limited instability of fine bed sand. Journal of the Hydraulics Division, ASCE, 96(HY3), 619–632. Hager, W. H., & Oliveto, G. (2002). Shields’ entrainment criterion in bridge hydraulics. Journal of Hydraulic Engineering, 128(5), 538–542. James, C. S. (1988). Use of the Einstein entrainment function for predicting selective entrainment and deposition of heavy minerals. Water S A, 14(4), 219–228. James, C. S. (1990). Prediction of entrainment conditions for nonuniform, noncohesive sediments. Journal of Hydraulic Research, 28(1), 25–41. Komar, P. D., & Clemens, K. E. (1986). The relationship between a grain’s settling velocity and threshold of motion under unidirectional currents. Journal of Sedimentary Petrology, 56(2), 258–266. Lambe, T. W., & Whitman, R. V. (1969). Soil Mechanics. Wiley. Lane, E. W. (1953). Progress report on studies on the design of stable channels by the Bureau of Reclamation. In Proceedings of the American Society of Civil Engineers (No 280). Lane, E. W. (1955). Design of stable channels. Transactions. American Society of Civil Engineers, 120, 1234–1279. Lauchlan, C. S., & Melville, B. W. (2001). Riprap protection at bridge piers. Journal of Hydraulic Engineering, 127(5), 412–418. Lavelle, J. W., & Mofjeld, H. V. (1987). Do critical stresses for incipient motion and erosion really exist? Journal of the Hydraulic Engineering, 113(3), 370–385. Leps, T. M. (1973). Flow through rockfill. In R. C. Hirschfeld & S. J. Poulos (Eds.), Embankment Dam Engineering (pp. 87–108). New York: Wiley. Mantz, P. A. (1973). Cohesionless, fine graded, flaked sediment discharge by water. Nature, Physical Science, 246, 14–16. Mantz, P. A. (1977). Incipient transport of fine grains and flakes by fluids—extended Shields diagram. Journal of the Hydraulics Division, ASCE, 103(HY6), 601–615. Melville, B. W. (2014). Scour at various hydraulic structures: sluice gates, submerged bridges, low weirs. In H. Chanson & L. Toombes (Eds.), Hydraulic Structures and Society—Engineering Challenges and Extremes 5th IAHR International Symposium on Hydraulic Structures, Brisbane, Australia. Melville, B. W., & Chiew, Y.-M. (1999). Time scale for local scour at bridge piers. Journal of Hydraulic Engineering, 125(1), 56–65.

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Melville, B. W., & Lim, S.-Y. (2013). Scour caused by 2D horizontal jets. Journal of Hydraulic Engineering, 140(2), 149–155. Miller, M. C., McCave, I. N., & Komar, P. D. (1977). Threshold of sediment motion under unidirectional currents. Sedimentology, 24, 507–527. Mohammedpour, R., Ghani, A. A., Zakaria, N. A., & Ali, T. A. M. (2017). Predicting scour at bridge abutments over time. Water Management, 170(1), 15–30. Neill, C. R. (1973). Guide to Bridge Hydraulics, Roads and Transportation Association of Canada. Toronto, ON, Canada: University of Toronto Press. Neill, C. R., & Yalin, M. S. (1969). Quantitative definition of beginning of movement of coarse granular bed materials. Journal of the Hydraulics Division, ASCE, 95(HY1), 585–587. Oliveto, G., & Hager, W. H. (2002). Temporal evolution of clear-water pier and abutment scour. Journal of Hydraulic Engineering, 128(9), 811–820. Oliveto, G., & Hager, W. H. (2005). Further results to time-dependent local scour at bridge elements. Journal of Hydraulic Engineering, 131(2), 97–105. Richardson, E. V., Harrison, L. J., & Davis, S. R. (1991). Evaluating Scour at Bridges (Rep. No. FHWA-IP-90-017 HEC-18). Washington, DC: Federal Highway Administration. Shan, H., Kilgore, R., Shen, J., & Kerenji, K. (2016). Updating HEC-18 Pier Scour Equations for Noncohesive Soils (Rep. No. FHWA-HRT-16-045). McLeann, VA: Federal Highway Administration. Sheppard, M., Demir, H., & Melville, B. W. (2011). Scour at Wide Piers and Long Skewed Piers. NCHRP Rep. 682, Washington, DC: National Cooperative Highway Program. Sheppard, M., Melville, B., & Demir, H. (2014). Evaluation of existing equations for local scour at bridge piers. Journal of Hydraulic Engineers, 140(1), 14–23. Shields, A. (1936). Application of similarity principles and turbulence research to bed load movement, Mitteilungen der Preussischen Versuchsanstalt fürWasserbau und Schiffbau, Heft 26, Berlin [English translation by W. P. Ott and J. C. Uchelon, California Institute of Technology, Pasadena, California, Report No 167]. Shvidchenko, A. B., & Pender, G. (2000). Flume study of the effect of relative depth on the incipient motion of coarse uniform sediments. Water Resources Research, 36(2), 619–628. Tafarojnoruz, A., Gaudio, R. & Dey, S. (2010). Flow-altering countermeasures against scour at bridge piers: a review. Journal of Hydraulic Research, 48(4), 441–452. Unger, J., & Hager, W. H. (2006). Riprap failure at circular bridge piers. Journal of Hydraulic Engineering, 132(4), 354–362. Vanoni, V. A. (Ed.) (1975). Sedimentation Engineering, American Society of Civil Engineers. White, S. J. (1970). Plane bed thresholds of fine grained sediments. Nature, 228, 152–153. Yalin, M. S. (1977). Mechanics of Sediment Transport (2nd ed.). Pergamon. Yalin, M. S., & Karahan, E. (1979). Inception of sediment transport. Journal of the Hydraulics Division, ASCE, 105(HY11), 1433–1443. Yang, C. T. (1973). Incipient motion and sediment transport. Journal of the Hydraulics Division, ASCE, 99(HY10), 1679–1704. Yanmaz, A. M., & Kose, O. (2009). A semi-empirical model for clear-water scour evolution at bridge abutments. Journal of Hydraulic Research, 47(1), 110–118.

Further Reading Kothyari, U. C. (2008). Bridge scour: Status and research challenges. ISH Journal of Hydraulic Engineering, 14(1), 1–27. Kothyari, U. C., & Kumar, A. (2010). Temporal variation of scour around circular bridge piers. ISH Journal of Hydraulic Engineering, 16(Sup1), 35–48. Lim, F.-H., & Chiew, Y.-M. (2001). Parametric study of riprap failure around bridge piers. Journal of Hydraulic Research, 39(1), 61–72.

Further Reading

363

May, R., Ackers, J., & Kirby, A. (2002). Manual on Scour at Bridges and Other Hydraulic Structures. London: Construction Industry Research and Information Association (CIRIA). Melville, B. W., & Coleman, S. E. (2000). Bridge Scour. Highlands Ranch, Colorado: Water Resources Publications. Yanmaz, A. M. (2006). Temporal variation of clear water scour at cylindrical bridge piers. Canadian Journal of Civil Engineering, 33(8), 1098–1102.

Postscript

Going Wider and Deeper For the aspirant expert this book is just a start. Hydraulic structures is a wide field and the content presented is necessarily selective in terms of both scope and detail. The emphasis has been on presenting an exposure to some of the more common hydraulic structures and developing an understanding of the basic concepts underlying their performance. For practical applications it may be necessary to obtain further details from the primary references cited or from other sources, including those recommended for further reading at the end of each chapter. Some general suggestions for sourcing further information are given here. A deeper understanding of the fundamental theory can be obtained from any of the fine books available on open channel hydraulics. These include Castro-Orgaz, O & Hager WH (2019) Shallow Water Hydraulics, Springer International Publishing. Chanson, H (1999) The Hydraulics of Open Channel Flow: An Introduction, Elsevier. Chaudhry, MH (1993) Open Channel Flow, Prentice-Hall. Chow, VT (1959) Open-Channel Hydraulics, McGraw-Hill. French, RH (1985) Open-Channel Hydraulics, McGraw-Hill. Henderson, FM (1966) Open Channel Flow, Macmillan. Many books provide further details for practical applications. In addition to the topic-specific books listed for further reading, the following more general books are useful. Davis, C V and Sorensen, K E (1969) Handbook of Applied Hydraulics, 3rd Edition, McGraw-Hill. Novak, P, Moffat, A I B, Nalluri, C and Narayanan, R (2001) Hydraulic Structures, 3rd Edition, Spon. United States Bureau of Reclamation (1987) Design of Small Dams, 3rd Edition. © Springer Nature Switzerland AG 2020 C S James, Hydraulic Structures, https://doi.org/10.1007/978-3-030-34086-5

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366

Postscript

United States Bureau of Reclamation (1978) Design of Small Canal Structures. Design standards, manuals and guidelines covering a wider range of topics as well as greater detail are available from various government agencies and research organizations, many providing free downloads. Some particularly valuable sources are CIRIA (Construction Industry Research and Information Association). United States Bureau of Reclamation. United States Army Corps of Engineers. Federal Highway Administration. The field of hydraulic structures is continually evolving, in terms of both its scope and its technology. The growing demand for sustainable and environmentally acceptable solutions in infrastructure developments requires innovative design of structures. Examples include the inclusion of fish passes in weirs and culverts, detention and retention storage facilities in sustainable urban drainage schemes, and low and high discharge dam outlets for environmental and channel maintenance flow releases. Where standard procedures are unavailable it is necessary to resort to fundamental principles in order to produce novel designs, and engagement with experts in other disciplines such as ecology, biology, planning and landscape architecture may be required. Continually improving instrumentation enables more and better data to be acquired for confirming and developing design concepts and details. Advances in analysis techniques, especially in numerical modelling capabilities, allow greater reliability and application to wider ranges of conditions in design. And, of course, there will be unexpected breakthroughs that could have major impacts on practice. The expert practitioner must therefore keep abreast of developments, in hydraulics as well as in allied fields. New contributions are generally published initially in conference proceedings and academic journals; some of the more relevant and valuable journals include the following. Ecological Engineering. ISH Journal of Hydraulic Engineering (Indian Society for Hydraulics). International Journal of River Basin Management. Journal of Hydraulic Engineering (American Society of Civil Engineers). Journal of Hydraulic Research (International Association for HydroEnvironment Engineering and Research). Journal of Irrigation and Drainage Engineering (American Society of Civil Engineers). Urban Water Journal. Water Management (Proceedings of the Institution of Civil Engineering). A wealth of knowledge is there for discovering!

Index

A Aeration, 117, 134, 145, 149, 150, 153, 154, 156, 157, 159, 162–164 Air concentration, 153, 157–161, 163, 166 Alternate depth, 29, 57 Angle of repose, 331, 333, 334, 337, 360 B Baffle blocks, 185, 196, 197, 209, 210, 224, 238 Baffled spillways, 229, 230 Bank stability, 333 Bed roughness, 187, 188, 190 Bellmouth, 311–315 Bends, 286, 290, 291, 293, 296, 309 Bernoulli equation, 8–10, 12 Blasius equation, 19 Boundary layer, 115–117, 139, 140, 143, 152, 154–157 Boundary shear stress, 1, 5, 6, 15 Bridge piers, 319, 341, 342, 348, 356, 360 Broad-crested weirs, 253, 254, 277 Bucket-type dissipators, 225, 226 Bulked flow depth, 154, 157–161 C Cavitation, 111–113, 115, 137, 140–145, 150–154, 162–166 Cavitation index, 151, 153 Chézy equation, 17, 18, 25 Chutes, 108, 120, 121, 123, 124, 144–146, 148–157, 159, 160, 163, 164, 166

© Springer Nature Switzerland AG 2020 C S James, Hydraulic Structures, https://doi.org/10.1007/978-3-030-34086-5

Cipolletti weir, 252 Circular culvert, 173, 174 Colebrook–White equation, 18, 20 Conjugate depth, 33 Conservation of energy, 8, 33 Conservation of mass, 7 Conservation of momentum, 12 Continuity, 7, 14, 17, 21, 23, 27, 29, 43, 47, 48 Contracted weirs, 244, 248, 252 Contraction, 75, 77, 78, 81, 92, 94–97 Contraction coefficient, 63, 67 Control, 1, 12, 35–39, 41–47, 49, 50, 54, 55, 57, 105–109, 114, 115, 117, 120, 121, 123, 124, 130, 131, 134, 136–138, 140, 144, 145, 150, 154, 160, 166 Controlled hydraulic jumps, 195 Conveyance structures, 105, 106 Coriolis coefficient, 9 Crest gates, 113, 149 Critical flow, 11, 30, 32, 33, 35, 45–48, 51, 52, 58 Critical shear stress, 323, 324, 327, 330, 331, 333–335, 337, 353 Critical velocity, 324, 336, 351 Crump weir, 254, 255 Culverts, 169–179, 181 Curvilinear transitions, 87, 90, 92, 103 Cutthroat flume, 265, 269–272, 280, 281 D Darcy–Weisbach equation, 18, 23, 25, 26 Design flood, 108, 131

367

368 Diffusion analogy, 286 Direct Step Method, 54–56 Discharge coefficient, 111, 112, 114, 117, 121, 128, 129, 133, 138, 139, 165, 247, 249, 252, 254, 256, 260, 269 Dual stable states, 97 E Empirical approach, 321, 326 Energy dissipation, 183–185, 196, 197, 205, 206, 214, 217, 219, 227, 228, 234, 238 Energy loss, 184, 186, 188, 201, 202, 204 Entrance loss coefficient, 176 Errors, 243, 252, 277–279 Expansion, 75, 77–82, 84, 92, 95, 102 F Filter layers, 339, 340 Flow classification, 2, 4, 6 Flow-measuring structures, 277 Flow resistance, 3, 14, 25, 28, 49 Flumes, 243, 257, 258, 261–265, 268–272, 274, 277, 280, 281 Force–momentum flux equation, 12, 13, 33 Friction factor, 18, 19, 25, 26 Froude number, 6, 33, 39, 55 G General resistance equation, 15, 50 Gradually varied flow, 3, 37, 49, 50, 52, 53, 59 Gradually varied flow computation, 52 Gradually varied flow equation, 51, 52 Gradually varied profiles, 42, 49–52 H Hydraulically rough flow, 5, 20, 23 Hydraulically smooth flow, 5, 19, 20 Hydraulic jump, 183, 184, 186–188, 190, 191, 193–198, 200, 201, 205–207, 210, 212–217, 219, 220, 222, 223, 225, 227, 228, 238 Hydraulic jump characteristics, 184 Hydraulic jump length, 188, 194, 205, 220, 222, 223, 238 Hydraulic jump on slope, 200 Hydraulic jump roller length, 205 Hydraulic radius, 4, 16, 50 Hysteresis, 97 Hysteretic behaviour, 69–71 I Impact-type dissipators, 228 Incipient motion, 320–322, 325, 326, 331, 332, 337

Index Inlet control, 170, 173, 174, 178, 179, 181 Intakes, 283–286, 291–297, 299, 300, 307–316 Intake submergence, 313 Intensity of motion, 326 Interference waves, 144, 145 L Labyrinth, 117, 118 Laminar flow, 4, 9, 13, 18, 19 Layout, 296, 309, 313 Long-throated structures, 265, 277 M Manning equation, 23, 25–28, 43, 48, 52 Model testing, 315 Momentum coefficient, 12 Momentum function, 13, 33–35, 43, 58 Moody diagram, 18–20, 25, 26, 28 Movability number, 326 N Nappe flow, 146–150 Nikuradse roughness, 6 Nonuniform flow, 3, 7, 16, 17, 28, 49, 50 O Outlet control, 169, 174, 178–181 Overflow spillway, 108–110, 117, 119, 138, 145, 150 P Parshall flume, 262–266, 271, 280 Permissible velocity, 327, 336 Piano key, 117–119 Pump intakes, 307, 312 Pump sumps, 284, 307, 309, 315 R Rapidly varied flow, 3, 4, 28, 42 Rectangular box culvert, 170, 172 Rectangular weirs, 244, 245, 249, 251, 252, 279, 280 Reservoir intakes, 283, 284 Reynolds number, 4–6, 18, 21 Riprap, 319, 325, 327, 336, 337, 339, 340, 351–357 River intakes, 284, 285 S Scour, 319–321, 334, 341–349, 351, 355, 358, 359 Scour depth, 319, 341–351, 355, 359, 360 Scour evolution, 347 Sediment control, 285

Index Sediment distribution, 286 Sediment diversion, 292, 297 Sediment ejection, 286 Sediment exclusion, 290, 292 Self-aeration, 153–155, 160, 163, 166 Sequent depth, 33 Sequent depth ratio, 186, 187, 197–199, 203, 205–207, 210, 213, 217 Settling, 284, 286, 288, 292, 301–304, 307, 317 Settling efficiency, 302–304 Shaft spillway, 130–133, 136, 165 Sharp-crested weirs, 244, 245, 250, 252, 277, 279, 280 Shear Reynolds number, 6 Shear velocity, 5 Shield diagram, 322, 323, 326, 337, 340 Shock front, 85, 87–90 Side-channel spillway, 119, 120, 123, 127, 144 Side weir, 126, 128, 129, 165 Sills, 186, 196, 197, 205–209, 213, 218–225, 238 Sinuosity adjustment, 334 Siphon spillway, 134–137, 139–141, 165 Skimming flow, 146–150 Specific energy, 10, 29–33, 35, 37, 47–50, 54, 57, 58 Spillway crest shape, 109, 111 Spillways, 105, 108–117, 119–121, 123, 124, 127, 130–141, 144–146, 150, 151, 153–156, 158, 163–166 Spillway splitters, 231, 232, 234 Standard Step Method, 57 Standing waves, 84–87, 94, 95, 97 St Anthony Falls stilling basin, 224 Steady flow, 3, 14, 16 Stepped chutes, 146, 149, 150 Stepped spillways, 146, 150 Steps, 193, 196, 197, 207, 212–217, 231, 238 Stilling basins, 185, 186, 188, 190, 196, 197, 207, 209–212, 214, 217–220, 223–225, 227, 228, 238 Straight transitions, 87, 97 Subcritical flow, 29–35, 38, 42, 43, 47, 50, 54

369 Subcritical flow transitions, 77 Submerged analysis, 64 Submerged inlet, 174, 179 Sump volume, 312 Supercritical flow, 6, 7, 29, 30, 32–34, 37, 38, 42, 43, 46, 47, 54, 55 Supercritical flow transitions, 84 Suppressed weirs, 244, 245, 248 Surface waves, 77, 84–86, 103 T Tapered expansion, 79–81 Theoretical analysis, 320 Throated flumes, 258, 260, 262 Transition loss coefficient, 78, 80 Transitions, 75–84, 87, 89–103 Translatory waves, 144, 145 Triangular weirs, 251, 279, 280 Turbulent flow, 4–6, 9, 13, 17–19, 23, 25, 26, 50 U Underflow gates, 62 Uniform flow, 3, 14–16, 21, 27, 36–39, 42, 43, 47, 48, 50–52, 54, 55, 58 Unsteady flow, 2, 3 Unsubmerged analysis, 61, 67, 71 Unsubmerged inlet, 179 USBR stilling basins, 209, 218, 219 V Vena contracta, 63, 65, 67 Venturi flumes, 261 Vortex, 300, 301, 308, 309, 311, 313–316 W Water surface profile, 184, 186, 190, 191, 195, 202, 204, 220, 222, 238 Wave celerity, 86 Wave suppression, 94, 97 Wave suppressor, 185, 186, 222 Weirs, 243–257, 262, 277–280