Figures

							                        FLUVIAL HYDRAULICS
                                       by
                              S. Lawrence Dingman

                                    FIGURES

1.1    Stocks and annual fluxes in the global hydrological cycle…………………....5

2.1    A watershed is topographically defined as the area that contributes……….22
2.2    Drainage-network patterns…………………………………………………….24
2.3    Plots illustrating the laws of drainage-network composition………………...26
2.4    Examples of longitudinal profiles of large rivers……………………………..29
2.5    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    An intensely meandering stream in central Alaska…………………………..34
2.8    A braided glacial stream in interior Alaska…………………………………..35
2.9    Schumm’s (1985) classification of channel patterns………………………….36
2.10   Braiding/meandering discriminant-function lines……………………………37
2.11   Braiding/meandering discriminant-function lines of van den Berg (1995)…38
2.12   Planform of a meandering river showing definitions………………………...39
2.13   Local-scale plan and longitudinal profile of channel bed……………………41
2.14   Planforms and local longitudinal profile of mountain-stream types…….….44
2.15   Classification of channel boundaries…………………………………………..45
2.16   Sediment-particle shape idealized as a tri-axial ellipsoid…………………….46
2.17   Particle-size designations and typical sediment grainsize distribution……...47
2.18   Particle shape, sphericity, and roundness….…………………………………49
2.19   Angle of repose as a function of particle size and roundness………………..50
2.20   Surveyed cross sections of the Cardrona River………………………………51
2.21   Definitions of terms used to describe channel geometry……………………..54
2.22   Measurements used to characterize the bankfull channel cross section…….57
2.23   Ratios of wetted perimeter to width and hydraulic radius to depth………...58
2.24   Cumulative frequency of bankfull width/depth ratios of natural channels...59
2.25   Definitions of terms for equations 2.19 and 2.20……………………………...60
2.26   The Cardrona River cross section……………………………………………..65
2.27   Definitions of terms defining discharge and stage……………………………66
2.28   Groundwater–stream relations………………………………………………..70
2.29   Idealized groundwater–stream relations……………………………………...71
2.30   An underflow-dominated stream and a baseflow-dominated stream……….72
2.31   Bank storage in a gaining stream……………………………………………...73
2.32   Flow paths in a small upland watershed and watershed response………….74
2.33   Streamflow in response to a rainfall or snowmelt event……………………..75
2.34   Evolution of a hydrograph in response to a rainfall event…………………..76
2.35   Examples of intra-annual flow-variability patterns………………………….77
2.36   Flow-duration curve for the Boise River……………………………………...79
2.37   Flood-frequency curve for the Boise River……………………………………80
2.38   Time series of annual peak discharges of the Boise River…………………...81
2.39   Relation of length scale of channel form to time scale of adjustment……….84
2.40   Interrelations among variables in the fluvial system…………………………85
2.41   Diagram showing values of exponents in hydraulic geometry relations…….88
2.42   Plots of width, average depth, and velocity versus discharge………………..89
2.43   Construction of duration curves for width, depth, or velocity………………92

3.1    Diagram of hydrogen atom, oxygen atom, and water molecule……………..95
3.2    Diagram of a water molecule…………………………………………………..96
3.3    Cagelike arrangement of water molecules in liquid water…………………..96
3.4    Melting and boiling temperatures of group VIa hydrides…………………..99
3.5    Surface temperatures and pressures of the planets………………………….99
3.6    Model of the crystal lattice of ice……………………………………………..100
3.7    Freezing at the edge of an ice sheet or a frazil disk…………………………101
3.8    Processes involved in river ice-cover formation……………………………..104
3.9    The stages of river-ice breakup………………………………………………105
3.10   Water-vapor flux near a water surface………………………………………106
3.11   Effects of sediment concentration on density of water–sediment mixtures..110
3.12   Intermolecular forces acting on surface and nonsurface molecules……….111
3.13   Thought experiment for surface tension……………………………………..112
3.14   Definition sketch for computation of the height of capillary rise…………..113
3.15   Thought experiment for viscosity…………………………………………….116
3.16   Results of viscosity thought experiment……………………………………...117
3.17   Viscous flow can be thought of as the sliding of layers……………………...117
3.18   Effects of concentrations of clay minerals on kinematic viscosity………….120
3.19   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   Dye injected into flows shows laminar and turbulent flow………………....123
3.22   Turbulence generated by boundary friction in flows of air………………...124
3.23   Turbulent eddies in natural river flows……………………………………...125
3.24   Richardson’s (1926) 3/2-power law of turbulent diffusion………………....127
3.25   Vertical and horizontal turbulent eddy fluctuations………………………..129
3.26   Prandtl’s mixing-length hypothesis…………………………………………..130
3.27   Mixing length as a function of distance from the bottom…………………...132
3.28   Growth of boundary layer thickness…………………………………………134
3.29   Flow states as a function of flow depth and average velocity……………....136

4.1    Coordinate systems used in this book………………………………………..140
4.2    Distance traveled by a fluid element in in time dt…………………………..142
4.3    Streamlines in steady flows…………………………………………………...145
4.4    Definitions of terms for deriving the expression for pressure……………...146
4.5    Pressure as a function of depth in open-channel flows……………………...148
4.6    Thought experiment showing that pressure is equal in all directions……..149
4.7    Diagram for derivation of the “microscopic” continuity equation………...150
4.8    Diagram for derivation of macroscopic continuity equation……………….151
4.9    Definitions of terms for determining magnitude of potential energy………155
4.10   Movement of a fluid element along a streamline…………………………....157
4.11   Conceptual diagram of the diffusion process………………………………..159
4.12   Diffusion of momentum in an open-channel flow…………………………...160
4.13   Force-balance diagram for a fluid element in a steady uniform flow……...161
4.14   Combined plot of U/(g∙Y∙sin SS)1/2 versus Y/yr…………..…………………..169
4.15   U/(g∙Y∙sin SS)1/2 versus Y/yr for 29 stream reaches………………………….173

5.1    Relation between shear stress and distance above the bottom……………..177
5.2    Linear relation between shear stress and distance above the bottom……...178
5.3    Relative velocity as a function of relative distance above the bottom……...180
5.4    Maximum depth at which laminar flow occurs as a function of slope……..182
5.5    Relative velocity as a function of relative distance above the bottom……...183
5.6    Velocity structure in a turbulent boundary-layer flow……………………..185
5.7    Diagram of hydraulically smooth and rough turbulent flow……………….188
5.8    The zero-plane-displacement adjustment………………………………..…..190
5.9    Relative velocity as a function of relative distance above the bottom……...191
5.10   P-vK velocity profile for a turbulent flow with Yw = 1 m…………………...193
5.11   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   Velocity profiles as given by the P-vK law and the power-law……………..199
5.14   Velocity profile given hyperbolic-tangent profile…………………………...200
5.15   Velocity profile in central portion of the Columbia River, Washington…..201
5.16   Two profiles measured in Casper Kill, New York…………………………..202
5.17   Velocity components in a rectangular flume………………………………...203
5.18   Velocities in one half of a parabolic channel………………………………...205
5.19   Measured and simulated velocities and velocity profiles in two flows……..207
5.20   Isovels in a meander bend of the River Klarälven, Sweden………………...208
5.21   Diagram of a meander bend…………………………………………………..208
5.22   Isovels in two natural channels……………………………………………….209

6.1    Scales of local, cross-section-averaged, and reach-averaged quantities…...212
6.2    Idealized development of uniform flow………………………………………214
6.3    Celerity of shallow-water gravity waves as a function of flow depth………216
6.4    Flow states and flow regimes as a function of average velocity and depth...217
6.5    Ratio of wave amplitude to mean depth as a function of Froude number...217
6.6    Roll waves on a steep driveway during a rainstorm………………………...218
6.7    Definitions of terms for development of the Chézy relation………………..219
6.8    The Moody diagram…………………………………………………………..224
6.9    Baseline resistance as a function of relative smoothness……………………225
6.10   Ratio of “excess” resistance to baseline resistance 664 flows……………….227
6.11   Three categories of channel irregularity that increase flow resistance…....228
6.12   Isovels in the near-bank portion of an idealized flow……………………….229
6.13   Ratio of excess to baseline resistance for gravel and boulder-bed streams..230
6.14   Effects of plan-view curvature on flow resistance…………………………..233
6.15   Excess resistance due to slope variations…………………………………….234
6.16   Plot showing the effect of surface instability on flow resistance……………235
6.17   Ripples………………………………………………………………………….238
6.18   Dunes……………..….…………………………………………………………239
6.19   Side view of antidunes in a laboratory flume………………………………..240
6.20   Sequence of bedforms and flow resistance in sand-bed streams…………...240
6.21   Plan view and cross sections of the Deep River at Ramseur, NC…………..242
6.22   U.S. river reaches covering a range of values of Manning’s nM……………247
6.23   Variation of Manning’s nM with hydraulic radius and bed grain diameter.251
6.24   The Hutt River at Kaitoke, New Zealand……………………………………260
6.25   Surveyed cross section in the center of the Hutt River reach………………261
6.26   Comparison of estimated and actual hydraulic relations, Hutt River….….263

7.1    Diagram for deriving expressions to calculate force magnitudes………….270
7.2    Vector diagram showing effect of Coriolis force on velocity……………….277
7.3    Cumulative distribution of gravitational force per unit mass……………..282
7.4    Cumulative distribution of pressure force per unit mass………………….283
7.5    Cumulative distribution of ratio of pressure force to gravitational force...284
7.6    Cumulative distribution of turbulence force per unit mass………….…….284
7.6    Cumulative distribution of turbulence force per unit mass………….…….284
7.7    Cumulative distribution of the ratio of turbulent to viscous force….……..285
7.8    Cumulative distribution of typical centrifugal force per unit mass……….285
7.9    Cumulative distribution of ratio of centrifugal force to turbulent force….286
7.10   Cumulative distribution of the magnitude of convective acceleration…….286
7.11   Cumulative distribution of ratio of convective to gravitational force……..287
7.12   Discharge hydrograph of the Diamond River……………………………….287
7.13   Hydraulic geometry relation for the Diamond River…………………….…288
7.14   Range of values of forces per unit mass typical of natural channels…….…289
7.15   Depth, velocity, slope, Reynolds number, and resistance vs. scale…………290
7.16   Forces per unit mass as a function of flow scale…………………………….291
7.17   Cumulative distribution of Froude numbers………………………………..293

8.1    Derivation of the macroscopic one-dimensional energy equation………….296
8.2    Derivation and evaluation of the energy and momentum coefficients……..297
8.3    The energy coefficient and momentum coefficients…………………………304
8.4    Velocity head and pressure head for flows in natural channels……………305
8.5    Derivation of the one-dimensional energy equation……………………...…306
8.6    Specific-head diagram…………………………………………………….…..308
8.7    Specific-head relations in a channel………………………………………….309
8.8    Relations between depth and discharge for a rectangular channel………..312
8.9    Specific-head diagram for example………………………………….……….313
8.10   Derivation of expressions for stream power…………………………………314
8.11   Specific-force diagram………………………………………………………...318
8.12   Hydraulic drops and expansion eddies………………………………………320
8.13   Difference between the energy and the momentum coefficient…………….321

9.1    Flooded house………………………………………………………………….324
9.2    Relations between normal depth and critical depth…………….………..…329
9.3    Situations associated with common types of water-surface profiles……….330
9.4    Water-surface profiles associated with controls due to changes in slope….332
9.5     Diagram illustrating partial section controls………………………………..333
9.6     A high flow in a small New England stream…………………………….…..338
9.7     Division of a cross section for computation of conveyance……………..…..343
9.8     Computed water-surface profile for the example in table 9.4…….………..344

10.1    Relationship between Y/Yc and Fr………….………………………………..349
10.2    A channel eroded in ice in central Alaska…………………………………...351
10.3    Local supercritical flow over a stone block with a hydraulic jump….…….352
10.4    Hydraulic jumps at engineering structures………………………….………353
10.5    Types of hydraulic jumps………………………..…………………………....354
10.6    Hydraulic jump types in a laboratory flume………………………………...355
10.7    Water-surface profiles through submerged and unsubmerged jumps…….356
10.8    Definitions of terms for analyzing hydraulic jumps…………………….…..357
10.9    Jump conditions as a function of upstream Froude number……………….359
10.10   More jump conditions as a function of upstream Froude number………...360
10.11   Specific-head diagrams for an abrupt decrease in channel elevation…...…363
10.12   Specific-head diagrams for an abrupt increase in channel elevation………365
10.13   Dimensionless specific-head diagram………………………………………...366
10.14   Idealized diagram of the form of the water surface over bedforms………..368
10.15   Definition diagram for analysis of a width contraction……………………..373
10.16   Depth ratio as a function of width ratio and Froude number……………...376
10.17   Four cases of rapidly varied flow induced by a constriction……………….379
10.18   Diagram for computing backwater effect due to a width constriction…….380
10.19   Backwater effect as a function of downstream Froude number……………381
10.20   Critical value of Froude number as a function of width constriction….…..382
10.21   Backwater effect for supercritical flow through a width constriction……..383
10.22   Definition of terms for describing flow over weirs………………………….384
10.23   Flow over a rectangular sharp-crested weir in a laboratory flume………..385
10.24   Definition diagram for flow over a sharp-crested weir……………………..386
10.25   Discharge coefficient for sharp-crested rectangular weirs…………………387
10.26   Weir coefficient of contracted rectangular sharp-crested weir…………….388
10.27   V-notch sharp-crested weirs for stream gaging in research watersheds…..389
10.28   Diagram for the equation for discharge through a V-notch weir……….....390
10.29   Weir coefficient as a function of weir head for a V-notch weir…………….391
10.30   Combination V-notch weirs…………………………………………………..392
10.31   Weir coefficient for broad-crested weirs as a function of weir height……..393
10.32   Flows over a rectangular broad-crested weir in a laboratory flume……....394
10.33   Plan and elevation of a Parshall flume……………………………………….396
10.34   Definition diagram for derivation of equations 10.59 and 10.64.……..……398

11.1    Diagram for derivation of macroscopic continuity and energy equations...402
11.2    Definition diagram for discretization of the Saint-Venant equations…...…406
11.3    Flume arrangement for tests of the Saint-Venant equations………….……408
11.4    Comparisons of measured and simulated hydrographs…………….………409
11.5    Comparison of measured and simulated hydrographs…….……………….410
11.6    A sinusoidal wave (equation 11.41)………………………….……………….413
11.7    The hyperbolic-tangent function (equation 11.47)………………………….415
11.8    Wave celerity as a function of wavelength for deep-water waves………….416
11.9    Dimensionless wave celerity as a function of /Y……………………………417
11.10   Schematic showing orbital paths of water parcels beneath…………….…..417
11.11   Propagation of gravity waves created by dropping a stone into water…….418
11.12   The solitary wave generated by displacement of a gate…………….……....419
11.13   Effect of relative wave amplitude on the celerity of a solitary wave……….420
11.14   Profile of a solitary wave……………………………………………………...421
11.15   Time-space relations for a typical flood wave……………………………….422
11.16   Hydrographs of the Diamond River near Wentworth Location, NH…..….424
11.17   Hydrographs showing sudden release from hydroelectric dam……………425
11.18   Definition diagram for uniformly progressive flow…………………………427
11.19   Ratio of kinematic-wave velocity to water velocity…………………….……431
11.20   Schematic diagram illustrating steepening of kinematic wave……….…….432
11.21   Terms for estimating the effects of overbank flow on floodwave velocity....432
11.22   Diagram illustrating effect of Froude number on flood-wave diffusivity.…436
11.23   Definition diagram for the Muskingum routing procedure……………..….439
11.24   Hydrographs illustrating the Muskingum routing procedure………….….441
11.25   Graphical determination of Muskingum routing parameters……………..445
11.26   Comparison of measured and predicted output hydrographs……………..447
11.27   Effects of routing parameter on hydrograph attenuation………………….448

12.1    Classification of solid loads and sediment-texture terms……….…………..453
12.2    Grain saltation…………………………………………………………………456
12.3    Helley-Smith bed-load sampler……………………………………………....457
12.4    DH-48 type rod-suspended depth-integrating sediment sampler…………..458
12.5    Deployment of sediment samplers……………………………………………459
12.6    Suspended-sediment–discharge relations for the Boise River...……………461
12.7    Bed-load–discharge relation for the Boise River……………………………463
12.8    Total particulate load as a function of discharge for the Boise River……...464
12.9    The particulate-load duration curve for the Boise River…………………...468
12.10   Cumulative loads contributed by five flow ranges for the Boise River……470
12.11   Forces on a spherical particle undergoing “slow” relative motion………...473
12.12   Drag coefficient as a function of particle Reynolds number…….………….475
12.13   Flow around spheres at increasing particle Reynolds number…………….476
12.14   Fall velocity as a function of sieve diameter…………………………………479
12.15   Forces on a particle on the stream bed………………………………………479
12.16   Shields diagrams………………………………………………………………482
12.17   Depth-slope product required for initiation of motion………….…………..484
12.18   Hjulström curves………………………………………………………………486
12.19   Bedrock erosion: plucking…………………………………….…………..….488
12.20   Bedrock erosion: abrasion…………………………………….…………..….489
12.21   Conditions for cavitation……………………….……………………………..491
12.22   Diffusion-theory approach to distribution of suspended-sediment………...493
12.23   Vertical distribution of sediment diffusivity in flume………………………495
12.24   Effect of Rouse number on suspended-sediment concentration profile…...496
12.25   Calculated and measured values of Rouse number…………………………497
12.26   Sediment-concentration profiles measured in flume experiments…………498
12.27   The Rouse number and depth-slope product for 641 flows………………...499
12.28   Predicted vs. compared concentration of sand-sized particles……………..500
12.29   Measured vertical velocity profiles for water and sediment………………..501
12.30   Predictive ability of 14 methods for estimating sediment concentration…..504
12.31   Bedforms as a function of bed-material size and stream power……………505
12.32   Forces on a sediment particle on the side of a trapezoidal channel………..506
12.33   Ratio of critical bank shear stress to critical bed shear stress……………...508
12.34   Distribution of boundary shear stress in a trapezoidal channel………....…508
12.35   Definition diagram for derivation of the Lane stable-channel relation……509
12.36   Geometry of the Lane stable channel as a function of angle of repose…….511
12.37   The Lane stable channel, types A and B………….………………………….512
12.38   Comparison of the Lane stable channel with the power-law cross section..512