FLUVIAL HYDRAULICS
by
S. Lawrence Dingman
Chapter 1:
1.1
1.2
1.3
1.4
Chapter 2:
2.0
2.1
2.2
2.3
2.4
2.5
INTRODUCTION TO FLUVIAL HYDRAULICS……………………3
Rivers in the Global Context ……………………………………………3
1.1.1 Natural Cycles……………………………………………………3
1.1.2 Human Significance……………………………………………...5
The Role of Fluvial Hydraulics………………………………………….8
A Brief History of Fluvial Hydraulics…………………………………..9
Scope and Approach of This Book…………………………………….16
NATURAL STREAMS: MORPHOLOGY, MATERIALS, AND
FLOWS………………………………………………………………….20
Introduction and Overview…………………………………………….20
The Watershed and the Stream Network……………………………..21
2.1.1 The Watershed…...……………………………………………….21
2.1.2Stream Networks…………………………………………………..21
2.1.2.1 Network Patterns……………………………………….23
2.1.2.2 Quantitative Description……………………………….23
2.1.3 Watershed-Scale Longitudinal Profile…………………………..27
2.1.4 Downstream Decrease of Sediment Size………………………...31
Channel Planform: Major Stream Types..……………………………31
2.2.1 Classification……………………..………………………………31
2.2.2 Relation to Environmental and Hydraulic Variables…………35
2.2.3 Meandering Reaches……………………………………………..39
2.2.4 Braided Reaches………………………………………………….40
2.2.5 “Straight” Reaches……………………………………………….42
2.2.6 Anabranching Reaches…………………………………………..42
Channel Boundaries……………………………………………………42
2.3.1 Boundary Characteristics………………………………………..42
2.3.2 Sediment Size and Shape…………………………………………45
2.3.2.1 Particle Size……………………………………………..45
2.3.2.2 Particle Shape…………………………………………..46
2.3.2.3 Particle Weight…………………………………………48
2.3.3 Angle of Repose…………………………………………………..48
The Channel Cross Section…………………………………………….50
2.4.1 General Characteristics and Definitions………………………..50
2.4.2 The Width/Depth Ratio and “Wide” Channels………………...52
2.4.3 Models of Cross-Section Shape………………………………….57
2.4.3.1 Lane’s Stable Channel Cross-Section Model…………58
2.4.3.2 General Cross-Section Model………………………….60
Streamflow (Discharge)………………………………………………...61
2.5.1 Definition………………………………………………………….61
2.5.2 Relation to Channel Dimensions and Slope…………………….64
2.6
Chapter 3:
3.0
3.1
3.2
3.3
2.5.3 Measurement……………………………………………………..66
2.5.3.1 Contact Methods……………………………………….66
2.5.3.2 Remote-Sensing Methods………………………………69
2.5.4 Sources………………………………………………………….…69
2.5.5 Stream Response to Rainfall and Snowmelt Events……………70
2.5.6 Timing……………………………………………………………..74
2.5.6.1 Hydroclimatic Regimes………………………...………74
2.5.6.2 Flow-Duration Curves…………………….……………75
2.5.6.3 Flood-Frequency Curves……………………………….78
Summary: Variables and Their Spatial and Temporal Variability....81
2.6.1 Principal Variables and Time and Space Scales………………..81
2.6.2 Channel Adjustment, Equilibrium, and the Graded Stream….85
2.6.3 Hydraulic Geometry……………………………………...………86
2.6.3.1 Temporal Changes: At-a-Station Hydraulic
Geometry………………………………………………..87
2.6.3.2 Spatial Changes: Downstream Hydraulic Geometry...93
STRUCTURE AND PROPERTIES OF WATER……………………94
Introduction and Overview…………………………………………….94
Structure of Water……………………………………………………...94
3.1.1 Atomic, Molecular, and Intermolecular Structures…………....94
3.1.2 Dissociation………………………………………………………..97
3.1.3 Isotopes……………………………………………………………97
Phase Changes………………………………………………………….98
3.2.1 Freezing/Melting and Condensation/Boiling Temperatures…..98
3.2.2 Freezing and Melting……………………………………………100
3.2.2.1 Physics of Freezing and Melting…………………...…100
3.2.2.2 Freezing and Melting of Lakes and Ponds…………..101
3.2.2.3 Freezing and Melting of Streams……………………..103
3.2.3 Evaporation, Condensation, and Sublimation……………...…105
Properties of Liquid Water…………………………………………...107
3.3.1 Density………………………………………………………..…..108
3.4
Chapter 4:
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
3.3.1.1 Definitions………………………………………….…..108
3.3.1.2 Magnitude……………………………………….……..109
3.3.2 Surface Tension and Capillarity………………………….…….110
3.3.2.1 Surface Tension………………………………….…….110
3.3.2.2 Magnitude of Surface Tension…………………….….111
3.3.2.3 Capillarity………………………………………….…..112
3.3.3 Viscosity……………………………………………………….…115
3.3.3.1 Viscosity, Shear Stress, and Velocity Gradients….…115
3.3.3.2 Magnitude of Dynamic Viscosity………………….….118
3.3.3.3 Viscosity and Momentum Flux…………………….…118
3.3.3.4 Magnitude of Kinematic Viscosity……….…………..119
3.3.3.5 Summary……………………………….………………119
3.3.4 Turbulence…………………………………….…………………120
3.3.4.1 Qualitative Description………….…………………….121
3.3.4.2 Statistical Description…………………………………123
3.3.4.3 Eddy Viscosity…………………………………………125
3.3.4.4 Prandtl’s Mixing-Length Hypothesis………………..128
3.3.4.5 Summary……………………………………………….132
Flow States, Boundary Layers, and the Reynolds Number……...…133
3.4.1 Flow States and Boundary Layers….…...……………………..133
3.4.2 The Reynolds Number………….……………………………….135
BASIC CONCEPTS AND EQUATIONS……………………………137
Introduction and Overview……………………………………...……137
Basic Mathematical Concepts………………………………….……..138
4.1.1 Fluid Continuum………………………………………….……..138
4.1.2 Fluid Element….………………………………………….……..138
4.1.3 Coordinate Systems……………………………………….…….139
4.1.4 The Lagrangian and Eulerian Viewpoints……………….……141
Kinematics and Dynamics……………………………………………141
4.2.1 Kinematics……………………………………………………….142
4.2.1.1 Velocity…………………………………………………142
4.2.1.2 Acceleration……………………………………………143
Equations Based on Conservation of Mass (Continuity)………...…149
4.3.1 “Microscopic” Continuity Relation…………………………….149
4.3.2 Macroscopic Continuity Relations……………………………..150
Equations Based on Conservation of Momentum…………………..152
Equations Based on Conservation of Energy………………………..154
4.5.1 Mechanical Potential Energy…………………………………...154
4.5.2 Mechanical Kinetic Energy……………………………………..156
4.5.3 Total Mechanical Energy and the Laws of Thermodynamics..157
Equations Based on Diffusion……………………………………...…159
Force-Balance and Conductance Equations…………………………161
Other Bases for Equations……………………………………………162
4.8.1 Equations of Definition………………………………………….162
4.8.2 Equations Based on Dimensional Analysis…………………….163
4.8.2.1 Theory of Dimensional Analysis…………..………….163
4.8.2.2 An Application of Dimensional Analysis to OpenChannel Flow…………………………………………..164
4.8.3 Empirical Equations………….………………………………....170
4.8.3.1 Regression Equations………………………………....170
4.8.3.2 Empirical Equations Based on Dimensional
Analysis……………………………………….…..……171
4.8.4 Heuristic Equations…………………………………….……….173
Chapter 5:
5.0
5.1
5.2
5.3
5.4
Chapter 6:
6.0
6.1
6.2
VELOCITY DISTRIBUTION………………………………………..175
Introduction and Overview…………………………………………...175
“Vertical” Force Profile in Uniform Flows……………………….....176
Velocity Profile in Laminar Flows……………………………………179
5.2.1 Derivation………………………………………………………..179
5.2.2 Average “Vertical” Velocity…………………………………….181
Velocity Profile in Turbulent Flows………………………………….181
5.3.1 The Prandtl-von Kármán Velocity Profile…………………….181
5.3.1.1 Derivation……………………………………………...181
5.3.1.2 The P-vK Law and Shear Distribution………………183
5.3.1.3 Shear Velocity (Friction Velocity)……………………184
5.3.1.4 Value of von Kármán’s Constant, ………………….184
5.3.1.5 Velocity Near the Boundary…………………………..185
5.3.1.6 Smooth and Rough Flow and the Determination
of y0……………………………………………………..187
5.3.1.7 Zero-Plane Displacement Adjustment……………….188
5.3.1.8 The P-vK Law: Summary…………………………….189
5.3.1.9 Average “Vertical” Velocity…………………………..191
5.3.2 The Velocity-Defect Law………………………………………..194
5.3.3 Power-Law Profiles……………………………………………...197
5.3.4 The Hyperbolic-Tangent Profile………………………………..198
5.3.5 Other Theoretical Profiles………………………………………199
5.3.6 Observed Velocity Profiles……………………………………...200
5.3.7 Summary: Velocity Profiles in Turbulent Flow……………….201
Velocity Distributions in Cross Sections……………………………..204
5.4.1 Velocity Distribution in an Ideal Parabolic Channel………....204
5.4.2 Observed Velocity Distributions…………………………….….204
5.4.2.1 Narrow Channels……………………………………...204
5.4.2.2 Bends…………………………………………………...204
5.4.2.3 Irregular Natural Channels…………………………..206
5.4.3 Statistical Characterizations of Velocity Distribution………...209
FLOW RESISTANCE
Introduction and Overview……………………………………..……211
Boundary Characteristics…………………………………………….212
Uniform Flow in Open Channels……………………………………..213
6.2.1 Basic Definition………………………………………………….213
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
Chapter 7:
7.0
7.1
6.2.2 Qualifications……………………………………………………215
6.2.2.1 Uniform Flow as an Asymptotic Condition………….215
6.2.2.2 Water-Surface Stability……………………………….215
6.2.2.3 Secondary Currents…………………………………...216
Basic Equation of Uniform Flow: The Chézy Equation…………….218
Definition of Reach Resistance……………………………………….220
Factors Affecting Reach Resistance in Uniform Flow………………223
Factors Affecting Reach Resistance in Natural Channels …………226
6.6.1 Effects of Channel Irregularities……………………………….227
6.6.1.1 Cross-section Irregularities…………………………..227
6.6.1.1 Cross-section Irregularities…………………………..231
6.6.1.3 Longitudinal-Profile Irregularities…………………..231
6.6.2 Effects of Vegetation…………………………………………….234
6.6.3 Effects of Surface Instability……………………………………235
6.6.4 Effects of Sediment……………………………………………...236
6.6.4.1 Effects of Sediment Load……………………………...236
6.6.4.2 Effects of Bedforms……………………………………236
6.6.5 Effects of Ice……………………………………………………..239
Field Computation of Reach Resistance……………………………..241
The Manning Equation……………………………………………….243
6.8.1 Origin…………………………………………………………….243
6.8.2 Determination of Manning’s nM……………………………….245
6.8.2.1 Visual Comparison with Photographs……………….245
6.8.2.2 Tables of Typical nM Values………………………….245
6.8.2.3 Formulas That Account for Components of Reach
Resistance…………..……………….…………………245
6.8.2.4 Formulas That Relate nM to Bed-Sediment Size and
Relative Smoothness…………………………………..249
6.8.2.5 Statistically Derived Formulas That Relate nM
to Hydraulic Variables………………………………..251
6.8.2.6 Field Measurement of Discharge and Hydraulic
Variables……………………………………………….252
6.8.3 Summary…………………………………………………………252
Statistically-Derived Resistance Equations………………………….253
Application of Resistance Equations…………………………………255
6.10.1 Determining the Velocity–Discharge and Depth–Discharge
Relations………………………………………………………..256
6.10.2 Determining Past Flood Discharge (Slope-Area
Measurements)………………………………………………...259
Summary……………………………………………………………….267
FORCES AND FLOW CLASSIFICATION………………………...269
Introduction and Overview…………………………………………...269
Force Classification and the Overall Force Balance………………...270
7.1.1 Classification of Forces………………………………………….271
7.1.2 Steady Uniform Flow……………………………………………272
7.2
7.3
7.1.3 Steady Nonuniform Flow……………………………………….272
7.1.4 Unsteady Nonuniform Flow…………………………………….272
Basic Geometric Relations……………………………………………273
Magnitudes of Forces per Unit Mass………………………………..274
7.3.1 Driving Forces…………………………………………………..274
7.3.1.1 Gravitational Force…………………………………...274
7.3.1.2 Pressure Force………………………………………………274
7.3.1.3 Total Driving Force…………………………………………275
7.4
7.5
7.6
7.7
Chapter 8:
8.0
8.1
7.3.2 Frictional Resisting Forces……………………………………..275
7.3.2.1 Viscous Force………………………………………….275
7.3.2.2 Turbulent Force……………………………………….276
7.3.2.3 Total Frictional Resisting Force……………………...277
7.3.3 Perpendicular Forces……………………………………………277
7.3.3.1 Coriolis Force………………………………………….277
7.3.3.2 Centrifugal Force……………………………………...278
7.3.4 Accelerations…………………………………………………….278
7.3.4.1 Convective Acceleration………………………………279
7.3.4.2 Local Acceleration…………………………………….279
Overall Force Balance and Velocity Relations………………………279
Magnitudes of Forces in Natural Streams…………………………...280
7.5.1 Database………………………………………………………….280
7.5.2 Driving Forces…………………………………………………...281
7.5.3 Resisting Forces………………………………………………….281
7.5.4 Perpendicular Forces……………………………………………281
7.5.4.1 Coriolis Force………………………………………….281
7.5.4.2 Centrifugal Force……………………………………...282
7.5.5 Accelerations…………………………………………………….283
7.5.5.1 Convective Acceleration………………………………283
7.5.5.2 Local Acceleration…………………………………….285
7.5.6 Summary of Force Magnitudes………………………………...288
7.5.7 Forces as a Function of Scale…………………………………...288
Force Ratios and the Reynolds and Froude Numbers………………292
7.6.1 The Reynolds Number………………………………………......292
7.6.2 The Froude Number…………………………………………….292
Summary……………………………………………………………….293
ENERGY AND MOMENTUM PRINCIPLES……………………...295
Introduction andOverview……………………………………………295
The Energy Principle in One-Dimensional Flows…………………...295
8.1.1 The Energy Equation……………………………………………296
8.1.1.1 Total Mechanical Energy at a Cross Section………...296
8.1.1.2 The Energy Equation………………………………….303
8.1.2 Specific Energy……………………………………………….....307
8.1.2.1 Definition………………………………………………307
8.1.2.2 Alternate Depths, Critical Depth, and the Froude
Number…….………………………………..…………308
8.1.2.3 Which Alternate Depth?...............................................310
8.2
8.3
Chapter 9:
9.0
9.1
9.2
9.3
9.4
8.1.3 Stream Power……………………………………………………313
8.1.3.1 Definitions……………………………………………..313
8.1.3.2 Applications……………………………………………315
The Momentum Principle in One-Dimensional Flows……………...315
8.2.1 The Momentum Equation………………………………………316
8.2.2 Specific Force……………………………………………………317
Comparison of the Energy and Momentum Principles…………….319
GRADUALLY-VARIED FLOW AND WATER-SURFACE
PROFILES…………………………………………………………….323
Introduction and Overview…………………………………………..323
The Basic Equations…………………………………………………..325
9.1.1 Continuity (Conservation-of-Mass) Equation…………………325
9.1.2 Energy Equation………………………………………………...326
9.1.3 Resistance Relations……………………………………………..327
Water-Surface Profiles: Classification……………………………….327
9.2.1 Normal Depth and Critical Depth……………………………...327
9.2.1.1 Normal Depth………………………………………….327
9.2.1.2 Critical Depth………………………………………….328
9.2.2 Mild and Steep Reaches………………………………………...329
9.2.3 Profile Classification…………………………………………….330
Controls………………………………………………………………...331
Water-Surface Profiles: Computation……………………………….333
9.4.1 Theoretical Basis………………………………………………...334
9.4.2 The Standard Step Method……………………………………..338
9.4.2.1 Basic Approach………………………………………..338
9.4.2.2 Detailed Steps and Example Calculation…………….340
9.4.2.3 Factors Affecting Accuracy…………………………...343
Chapter 10: RAPIDLY VARIED STEADY FLOW………………………………347
10.0 Introduction and Overview…………………………………………...347
10.1 Hydraulic Jumps………………………………………………………350
10.1.1 Classification…………………………………………………...351
10.1.2 Sequent Depths and Jump Heights…………………………...352
10.1.3 Jump Length…………………………………………………...358
10.1.4 Characteristics of Waves in Undular Jumps………………...359
10.1.5 Energy Loss…………………………………………………….360
10.2 Abrupt Channel Transitions with No Energy Loss…………………361
10.2.1 Elevation Transitions………………………………………….361
10.2.1.1 Basic Approach………………………………………361
10.2.1.2 Elevation Drops………………………………………362
10.2.1.3 Elevation Rises……………………………………….362
10.2.1.4 Dimensionless Specific-Head Curve………………..364
10.2.1.5 Implications for Flow over Bedforms………………365
10.2.2 Width Transitions…………………………….………………..369
10.3 Abrupt Transitions with Energy Loss……………….………………369
10.4
10.3.1 General Theoretical Approach………………………………..372
10.3.1.1 Momentum Equation………………………………...372
10.3.1.2 Energy Equation……………………………………..374
10.3.2 Experimental Results………………………………………….377
10.3.3 Constrictions (Bridge Openings)……………….……………..378
Artificial Controls for Flow Measurement………….……………….383
10.4.1 Weirs…………………………………………….……………...383
10.4.1.1 Sharp-Crested Weirs……………….………………..385
10.4.1.2 Broad-Crested Weirs…………………….………………...391
10.4.2 Flumes………………………………………….……………….395
10.4.3 Flows through Width Constrictions………….……………….396
10.4.3.1 Conceptual Approach………………………………..397
10.4.3.2 Approach of Matthai (1967)…………………………397
Chapter 11: UNSTEADY FLOW…………………………………………………..400
11.0 Introduction and Overview…………………………………………...400
11.1 The St.-Venant Equations: The Basic Equations of Unsteady
Gradually-Varied Flow……………………………………………….401
11.1.1 Conservation of Mass Equation (Continuity)………………..401
11.1.2 Dynamic Equation (Momentum/Energy)…………………….402
11.1.2.1 Derivation…………………………………………….402
11.1.2.2 Incorporation in Resistance Relations……………...404
11.1.3 Solution of the Saint-Venant Equations………………………405
11.1.4 Tests of the Saint-Venant Equations………………………….408
11.2 Hydraulic Geometry…………………………………………………..408
11.3
Waves………………………………………………………………….411
11.3.1 Basic Characteristics…………………………………………..411
11.3.2 Classical Theory of Oscillatory Waves……………………….414
11.4 Gravity Waves in Open-Channels……………………………………418
11.4.1 Simple Gravity Waves…………………………………………418
11.4.2 The Soliton………………………………….…………..………419
11.5 Flood Waves………………………………………….………………..421
11.5.1 Qualitative Aspects………………………….…………………421
11.5.2 Kinematic Waves………………………………………………423
11.5.2.1 Kinematic-Wave Velocity……………………………426
11.5.2.2 Effects of Overbank Flow on Kinematic-Wave
Velocity………………………………………………..430
11.5.2.3 Relations between Kinematic Waves and Gravity
Waves………………………………………………….433
11.5.2.4 Kinematic Waves: Summary………………………..434
11.5.3 Quantitative Analysis of Flood Waves………………………..434
11.6 Flood-Wave Routing…………………………………………………..438
11.6.1 Overview………………………………………………………..438
11.6.2 Hydrologic Routing: The Muskingum Method………………439
11.6.2.1 Basic Development…………………………………...439
11.6.2.2 Discretization…………………………………………442
11.6.2.3 Significance of Routing Parameters…….…………..447
11.7
Unsteady Flow: Summary…………………………………………….448
Chapter 12: SEDIMENT ENTRAINMENT AND TRANSPORT……………….451
12.0 Introduction and Overview…………………………………………..451
12.1 Definitions and Measurement………………………………………..452
12.1.1 Definitions…………………………………………….………..452
12.1.2 Measurement…………………………………………………...456
12.1.2.1 Bed Load………………………………………….…..456
12.1.2.2 Suspended Load……………………………………..457
12.2 Sediment Transport and Geomorphological Concepts…………….458
12.2.1 Empirical Concentration–Load–Discharge Relations………459
12.2.2 Sediment Yield and Denudation Rate………………………..465
12.2.3 Magnitude–Frequency Relations……………………………..469
12.3 Forces on Sediment Particles…………………………………………472
12.3.1 Relative Motion of a Sphere in a Fluid……………………….472
12.3.2 Particles Settling in a Fluid: Fall Velocity……………………475
12.4 When Does Sediment Transport Begin?.............................................480
12.4.1 Critical Boundary Shear Stress: The Shields Diagram……..480
12.4.2 Critical Velocity: The Hjulström Curves…………………….485
12.4.3 Erosion of Cohesive Sediments………………………………..485
12.4.4 Bedrock Erosion………………………………………………..487
12.4.4.1 Plucking……………………………………………....487
12.4.4.2 Abrasion………………………………………………489
12.4.4.3 Cavitation…………………………….……………….490
12.5 Sediment Load………………………………………………………....490
12.5.1 Bed Load………………………………………………………..491
12.5.2 Suspended-Sediment Concentration and Load………………492
12.5.2.1 Concentration Profile: Diffusion-Theory
Approach……………………………………………..492
12.5.2.2 Concentration Profile: Two-Phase Flow?..................500
12.5.3 Total Bed-Material Load………………………………………502
12.5.4 Sediment Transport and Bedforms…………………………...504
12.6 The Stable Cross Section……………………………………………...505
12.6.1 Stability of a Trapezoidal Channel……………………………506
12.6.2 The Lane Stable Channel……………………………………...507
12.6.2.1 Derivation…………………………………………….507
12.6.2.2 Comparison with Generalized Cross-Section
Form………………………………………………….512
Appendix A: PHYSICAL QUANTITIES: DIMENSIONS, UNITS, AND
MEASUREMENT PRECISION……………………………………..514
A.1 Dimensions………………………………………………………..514
A.2 Units……………………………………………………………….515
A.3 Precision and Significant Figures…………….………………….515
A.3.1 Absolute Precision……….………….…………………..518
A.3.2 Relative Precision……….………………………………519
A.4 Unit Conversion…………………………………………………..520
A.5 Equations: Dimensional Properties and Conversion…………...523
A.5.1 Dimensional Properties of Equations.…………………523
A.5.2 Equation Conversion.…………………………………………525
Appendix B: SYNTHETIC CHANNEL MODEL………………………………….526
Appendix C: FLOW DATA …………..……………………………………………...527
C.1 Overview…………………………………………………………..527
C.2 Basic Approach…………………………………………………...528
C.2.1 Channel Shape………………….……………………….528
C.2.2 Velocity……………………….………………………….528
C.2.3 Water Properties….…………………………………….529
C.3 Displays……………………………………………………………529
Appendix D. DESCRIPTION OF WATER-SURFACE PROFILE COMPUTATION
SPREADSHEET……………………………………………………………….530