FLUVIAL HYDRAULICS
by
S. Lawrence Dingman
FIGURES
1.1
Stocks and annual fluxes in the global hydrological cycle…………………....5
2.1
2.2
2.3
2.4
2.5
A watershed is topographically defined as the area that contributes……….22
Drainage-network patterns…………………………………………………….24
Plots illustrating the laws of drainage-network composition………………...26
Examples of longitudinal profiles of large rivers……………………………..29
Downstream decrease in sediment size in the River Noe, England………….32
2.6
Sinuosity of a reach of the South Fork Payette River, Idaho………………………..33
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
2.21
2.22
2.23
2.24
2.25
2.26
2.27
2.28
2.29
2.30
2.31
2.32
2.33
2.34
2.35
2.36
2.37
An intensely meandering stream in central Alaska…………………………..34
A braided glacial stream in interior Alaska…………………………………..35
Schumm’s (1985) classification of channel patterns………………………….36
Braiding/meandering discriminant-function lines……………………………37
Braiding/meandering discriminant-function lines of van den Berg (1995)…38
Planform of a meandering river showing definitions………………………...39
Local-scale plan and longitudinal profile of channel bed……………………41
Planforms and local longitudinal profile of mountain-stream types…….….44
Classification of channel boundaries…………………………………………..45
Sediment-particle shape idealized as a tri-axial ellipsoid…………………….46
Particle-size designations and typical sediment grainsize distribution……...47
Particle shape, sphericity, and roundness….…………………………………49
Angle of repose as a function of particle size and roundness………………..50
Surveyed cross sections of the Cardrona River………………………………51
Definitions of terms used to describe channel geometry……………………..54
Measurements used to characterize the bankfull channel cross section…….57
Ratios of wetted perimeter to width and hydraulic radius to depth………...58
Cumulative frequency of bankfull width/depth ratios of natural channels...59
Definitions of terms for equations 2.19 and 2.20……………………………...60
The Cardrona River cross section……………………………………………..65
Definitions of terms defining discharge and stage……………………………66
Groundwater–stream relations………………………………………………..70
Idealized groundwater–stream relations……………………………………...71
An underflow-dominated stream and a baseflow-dominated stream……….72
Bank storage in a gaining stream……………………………………………...73
Flow paths in a small upland watershed and watershed response………….74
Streamflow in response to a rainfall or snowmelt event……………………..75
Evolution of a hydrograph in response to a rainfall event…………………..76
Examples of intra-annual flow-variability patterns………………………….77
Flow-duration curve for the Boise River……………………………………...79
Flood-frequency curve for the Boise River……………………………………80
2.38
2.39
2.40
2.41
2.42
2.43
Time series of annual peak discharges of the Boise River…………………...81
Relation of length scale of channel form to time scale of adjustment……….84
Interrelations among variables in the fluvial system…………………………85
Diagram showing values of exponents in hydraulic geometry relations…….88
Plots of width, average depth, and velocity versus discharge………………..89
Construction of duration curves for width, depth, or velocity………………92
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
Diagram of hydrogen atom, oxygen atom, and water molecule……………..95
Diagram of a water molecule…………………………………………………..96
Cagelike arrangement of water molecules in liquid water…………………..96
Melting and boiling temperatures of group VIa hydrides…………………..99
Surface temperatures and pressures of the planets………………………….99
Model of the crystal lattice of ice……………………………………………..100
Freezing at the edge of an ice sheet or a frazil disk…………………………101
Processes involved in river ice-cover formation……………………………..104
The stages of river-ice breakup………………………………………………105
Water-vapor flux near a water surface………………………………………106
Effects of sediment concentration on density of water–sediment mixtures..110
Intermolecular forces acting on surface and nonsurface molecules……….111
Thought experiment for surface tension……………………………………..112
Definition sketch for computation of the height of capillary rise…………..113
Thought experiment for viscosity…………………………………………….116
Results of viscosity thought experiment……………………………………...117
Viscous flow can be thought of as the sliding of layers……………………...117
Effects of concentrations of clay minerals on kinematic viscosity………….120
Velocity gradients near a boundary create quasi-circular eddies………….121
3.20
Paths of fluid elements as flow changes laminar to turbulent……………………...122
3.21
3.22
3.23
3.24
3.25
3.26
3.27
3.28
3.29
Dye injected into flows shows laminar and turbulent flow………………....123
Turbulence generated by boundary friction in flows of air………………...124
Turbulent eddies in natural river flows……………………………………...125
Richardson’s (1926) 3/2-power law of turbulent diffusion………………....127
Vertical and horizontal turbulent eddy fluctuations………………………..129
Prandtl’s mixing-length hypothesis…………………………………………..130
Mixing length as a function of distance from the bottom…………………...132
Growth of boundary layer thickness…………………………………………134
Flow states as a function of flow depth and average velocity……………....136
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
Coordinate systems used in this book………………………………………..140
Distance traveled by a fluid element in in time dt…………………………..142
Streamlines in steady flows…………………………………………………...145
Definitions of terms for deriving the expression for pressure……………...146
Pressure as a function of depth in open-channel flows……………………...148
Thought experiment showing that pressure is equal in all directions……..149
Diagram for derivation of the “microscopic” continuity equation………...150
Diagram for derivation of macroscopic continuity equation……………….151
Definitions of terms for determining magnitude of potential energy………155
Movement of a fluid element along a streamline…………………………....157
4.11
4.12
4.13
4.14
4.15
Conceptual diagram of the diffusion process………………………………..159
Diffusion of momentum in an open-channel flow…………………………...160
Force-balance diagram for a fluid element in a steady uniform flow……...161
Combined plot of U/(g∙Y∙sin SS)1/2 versus Y/yr…………..…………………..169
U/(g∙Y∙sin SS)1/2 versus Y/yr for 29 stream reaches………………………….173
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
Relation between shear stress and distance above the bottom……………..177
Linear relation between shear stress and distance above the bottom……...178
Relative velocity as a function of relative distance above the bottom……...180
Maximum depth at which laminar flow occurs as a function of slope……..182
Relative velocity as a function of relative distance above the bottom……...183
Velocity structure in a turbulent boundary-layer flow……………………..185
Diagram of hydraulically smooth and rough turbulent flow……………….188
The zero-plane-displacement adjustment………………………………..…..190
Relative velocity as a function of relative distance above the bottom……...191
P-vK velocity profile for a turbulent flow with Yw = 1 m…………………...193
Ratio of local mean velocity to local surface velocity………………………..194
5.12
Velocity profiles given by the P-vK law and the velocity-defect law………………197
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
5.21
5.22
Velocity profiles as given by the P-vK law and the power-law……………..199
Velocity profile given hyperbolic-tangent profile…………………………...200
Velocity profile in central portion of the Columbia River, Washington…..201
Two profiles measured in Casper Kill, New York…………………………..202
Velocity components in a rectangular flume………………………………...203
Velocities in one half of a parabolic channel………………………………...205
Measured and simulated velocities and velocity profiles in two flows……..207
Isovels in a meander bend of the River Klarälven, Sweden………………...208
Diagram of a meander bend…………………………………………………..208
Isovels in two natural channels……………………………………………….209
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
6.17
6.18
Scales of local, cross-section-averaged, and reach-averaged quantities…...212
Idealized development of uniform flow………………………………………214
Celerity of shallow-water gravity waves as a function of flow depth………216
Flow states and flow regimes as a function of average velocity and depth...217
Ratio of wave amplitude to mean depth as a function of Froude number...217
Roll waves on a steep driveway during a rainstorm………………………...218
Definitions of terms for development of the Chézy relation………………..219
The Moody diagram…………………………………………………………..224
Baseline resistance as a function of relative smoothness……………………225
Ratio of “excess” resistance to baseline resistance 664 flows……………….227
Three categories of channel irregularity that increase flow resistance…....228
Isovels in the near-bank portion of an idealized flow……………………….229
Ratio of excess to baseline resistance for gravel and boulder-bed streams..230
Effects of plan-view curvature on flow resistance…………………………..233
Excess resistance due to slope variations…………………………………….234
Plot showing the effect of surface instability on flow resistance……………235
Ripples………………………………………………………………………….238
Dunes……………..….…………………………………………………………239
6.19
6.20
6.21
6.22
6.23
6.24
6.25
6.26
Side view of antidunes in a laboratory flume………………………………..240
Sequence of bedforms and flow resistance in sand-bed streams…………...240
Plan view and cross sections of the Deep River at Ramseur, NC…………..242
U.S. river reaches covering a range of values of Manning’s nM……………247
Variation of Manning’s nM with hydraulic radius and bed grain diameter.251
The Hutt River at Kaitoke, New Zealand……………………………………260
Surveyed cross section in the center of the Hutt River reach………………261
Comparison of estimated and actual hydraulic relations, Hutt River….….263
7.1
7.2
7.3
7.4
7.5
7.6
7.6
7.7
7.8
7.9
7.10
7.11
7.12
7.13
7.14
7.15
7.16
7.17
Diagram for deriving expressions to calculate force magnitudes………….270
Vector diagram showing effect of Coriolis force on velocity……………….277
Cumulative distribution of gravitational force per unit mass……………..282
Cumulative distribution of pressure force per unit mass………………….283
Cumulative distribution of ratio of pressure force to gravitational force...284
Cumulative distribution of turbulence force per unit mass………….…….284
Cumulative distribution of turbulence force per unit mass………….…….284
Cumulative distribution of the ratio of turbulent to viscous force….……..285
Cumulative distribution of typical centrifugal force per unit mass……….285
Cumulative distribution of ratio of centrifugal force to turbulent force….286
Cumulative distribution of the magnitude of convective acceleration…….286
Cumulative distribution of ratio of convective to gravitational force……..287
Discharge hydrograph of the Diamond River……………………………….287
Hydraulic geometry relation for the Diamond River…………………….…288
Range of values of forces per unit mass typical of natural channels…….…289
Depth, velocity, slope, Reynolds number, and resistance vs. scale…………290
Forces per unit mass as a function of flow scale…………………………….291
Cumulative distribution of Froude numbers………………………………..293
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
8.11
8.12
8.13
Derivation of the macroscopic one-dimensional energy equation………….296
Derivation and evaluation of the energy and momentum coefficients……..297
The energy coefficient and momentum coefficients…………………………304
Velocity head and pressure head for flows in natural channels……………305
Derivation of the one-dimensional energy equation……………………...…306
Specific-head diagram…………………………………………………….…..308
Specific-head relations in a channel………………………………………….309
Relations between depth and discharge for a rectangular channel………..312
Specific-head diagram for example………………………………….……….313
Derivation of expressions for stream power…………………………………314
Specific-force diagram………………………………………………………...318
Hydraulic drops and expansion eddies………………………………………320
Difference between the energy and the momentum coefficient…………….321
9.1
9.2
9.3
9.4
Flooded house………………………………………………………………….324
Relations between normal depth and critical depth…………….………..…329
Situations associated with common types of water-surface profiles……….330
Water-surface profiles associated with controls due to changes in slope….332
9.5
9.6
9.7
9.8
Diagram illustrating partial section controls………………………………..333
A high flow in a small New England stream…………………………….…..338
Division of a cross section for computation of conveyance……………..…..343
Computed water-surface profile for the example in table 9.4…….………..344
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8
10.9
10.10
10.11
10.12
10.13
10.14
10.15
10.16
10.17
10.18
10.19
10.20
10.21
10.22
10.23
10.24
10.25
10.26
10.27
10.28
10.29
10.30
10.31
10.32
10.33
10.34
Relationship between Y/Yc and Fr………….………………………………..349
A channel eroded in ice in central Alaska…………………………………...351
Local supercritical flow over a stone block with a hydraulic jump….…….352
Hydraulic jumps at engineering structures………………………….………353
Types of hydraulic jumps………………………..…………………………....354
Hydraulic jump types in a laboratory flume………………………………...355
Water-surface profiles through submerged and unsubmerged jumps…….356
Definitions of terms for analyzing hydraulic jumps…………………….…..357
Jump conditions as a function of upstream Froude number……………….359
More jump conditions as a function of upstream Froude number………...360
Specific-head diagrams for an abrupt decrease in channel elevation…...…363
Specific-head diagrams for an abrupt increase in channel elevation………365
Dimensionless specific-head diagram………………………………………...366
Idealized diagram of the form of the water surface over bedforms………..368
Definition diagram for analysis of a width contraction……………………..373
Depth ratio as a function of width ratio and Froude number……………...376
Four cases of rapidly varied flow induced by a constriction……………….379
Diagram for computing backwater effect due to a width constriction…….380
Backwater effect as a function of downstream Froude number……………381
Critical value of Froude number as a function of width constriction….…..382
Backwater effect for supercritical flow through a width constriction……..383
Definition of terms for describing flow over weirs………………………….384
Flow over a rectangular sharp-crested weir in a laboratory flume………..385
Definition diagram for flow over a sharp-crested weir……………………..386
Discharge coefficient for sharp-crested rectangular weirs…………………387
Weir coefficient of contracted rectangular sharp-crested weir…………….388
V-notch sharp-crested weirs for stream gaging in research watersheds…..389
Diagram for the equation for discharge through a V-notch weir……….....390
Weir coefficient as a function of weir head for a V-notch weir…………….391
Combination V-notch weirs…………………………………………………..392
Weir coefficient for broad-crested weirs as a function of weir height……..393
Flows over a rectangular broad-crested weir in a laboratory flume……....394
Plan and elevation of a Parshall flume……………………………………….396
Definition diagram for derivation of equations 10.59 and 10.64.……..……398
11.1
11.2
11.3
11.4
11.5
11.6
Diagram for derivation of macroscopic continuity and energy equations...402
Definition diagram for discretization of the Saint-Venant equations…...…406
Flume arrangement for tests of the Saint-Venant equations………….……408
Comparisons of measured and simulated hydrographs…………….………409
Comparison of measured and simulated hydrographs…….……………….410
A sinusoidal wave (equation 11.41)………………………….……………….413
11.7
11.8
11.9
11.10
11.11
11.12
11.13
11.14
11.15
11.16
11.17
11.18
11.19
11.20
11.21
11.22
11.23
11.24
11.25
11.26
11.27
The hyperbolic-tangent function (equation 11.47)………………………….415
Wave celerity as a function of wavelength for deep-water waves………….416
Dimensionless wave celerity as a function of /Y……………………………417
Schematic showing orbital paths of water parcels beneath…………….…..417
Propagation of gravity waves created by dropping a stone into water…….418
The solitary wave generated by displacement of a gate…………….……....419
Effect of relative wave amplitude on the celerity of a solitary wave……….420
Profile of a solitary wave……………………………………………………...421
Time-space relations for a typical flood wave……………………………….422
Hydrographs of the Diamond River near Wentworth Location, NH…..….424
Hydrographs showing sudden release from hydroelectric dam……………425
Definition diagram for uniformly progressive flow…………………………427
Ratio of kinematic-wave velocity to water velocity…………………….……431
Schematic diagram illustrating steepening of kinematic wave……….…….432
Terms for estimating the effects of overbank flow on floodwave velocity....432
Diagram illustrating effect of Froude number on flood-wave diffusivity.…436
Definition diagram for the Muskingum routing procedure……………..….439
Hydrographs illustrating the Muskingum routing procedure………….….441
Graphical determination of Muskingum routing parameters……………..445
Comparison of measured and predicted output hydrographs……………..447
Effects of routing parameter on hydrograph attenuation………………….448
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
12.10
12.11
12.12
12.13
12.14
12.15
12.16
12.17
12.18
12.19
12.20
12.21
12.22
12.23
12.24
Classification of solid loads and sediment-texture terms……….…………..453
Grain saltation…………………………………………………………………456
Helley-Smith bed-load sampler……………………………………………....457
DH-48 type rod-suspended depth-integrating sediment sampler…………..458
Deployment of sediment samplers……………………………………………459
Suspended-sediment–discharge relations for the Boise River...……………461
Bed-load–discharge relation for the Boise River……………………………463
Total particulate load as a function of discharge for the Boise River……...464
The particulate-load duration curve for the Boise River…………………...468
Cumulative loads contributed by five flow ranges for the Boise River……470
Forces on a spherical particle undergoing “slow” relative motion………...473
Drag coefficient as a function of particle Reynolds number…….………….475
Flow around spheres at increasing particle Reynolds number…………….476
Fall velocity as a function of sieve diameter…………………………………479
Forces on a particle on the stream bed………………………………………479
Shields diagrams………………………………………………………………482
Depth-slope product required for initiation of motion………….…………..484
Hjulström curves………………………………………………………………486
Bedrock erosion: plucking…………………………………….…………..….488
Bedrock erosion: abrasion…………………………………….…………..….489
Conditions for cavitation……………………….……………………………..491
Diffusion-theory approach to distribution of suspended-sediment………...493
Vertical distribution of sediment diffusivity in flume………………………495
Effect of Rouse number on suspended-sediment concentration profile…...496
12.25
12.26
12.27
12.28
12.29
12.30
12.31
12.32
12.33
12.34
12.35
12.36
12.37
12.38
Calculated and measured values of Rouse number…………………………497
Sediment-concentration profiles measured in flume experiments…………498
The Rouse number and depth-slope product for 641 flows………………...499
Predicted vs. compared concentration of sand-sized particles……………..500
Measured vertical velocity profiles for water and sediment………………..501
Predictive ability of 14 methods for estimating sediment concentration…..504
Bedforms as a function of bed-material size and stream power……………505
Forces on a sediment particle on the side of a trapezoidal channel………..506
Ratio of critical bank shear stress to critical bed shear stress……………...508
Distribution of boundary shear stress in a trapezoidal channel………....…508
Definition diagram for derivation of the Lane stable-channel relation……509
Geometry of the Lane stable channel as a function of angle of repose…….511
The Lane stable channel, types A and B………….………………………….512
Comparison of the Lane stable channel with the power-law cross section..512