Hydraulic Interpretation of Cross-Stream Variations in Bed-Load Transport

The hydraulic control of bed-load transport rates in Nahal Yatir and Nahal Eshtemoa, two coarse-grained ephemeral channels in the semiarid northern Negev, Israel, provides a rare opportunity to infer the spanwise variation in bed-shear stress from an analysis of cross-stream variations in bed-load transport rate. Automatic sediment transport monitoring stations were used to obtain synchronous measurements of bed-load discharge at a number of locations across the widths of two straight channel reaches. In both streams, channel-average bed-load fluxes demonstrated a common and well-defined response to changing channel-average shear stress and approximated the transporting capacity of the flow over much of the range of monitored discharges. However, transport rates measured at the channel margins are only half those at the channel centerline, and, at high discharges, a marked asymmetry in the pattern of bed-load transport develops across the central section of the widest channel. This variation in bed-load discharge over the two channel cross sections is thought to reflect lateral variations in shear stress induced by sidewall drag and, more tentatively, the generation and disposition of cellular secondary currents. But no systematic relation is found for the ratios of sediment fluxes at off-center sampling locations and those recorded at the channel center, even though the off-center locations are thought to move into and out of the region affected by sidewall drag as aspect ratio of the flow decreases and increases with changing water-stage. The results suggest that it is difficult to generalize about the changing influence of the sidewall on local shear and bed load as aspect ratio changes during the course of a flood.     

Flow Resistance over Mobile Bed in an Open-Channel Flow

This paper deals with the underlying mechanism of flow resistance in an alluvial channel: The effects of sidewall and bed form on flow resistance, Einstein’s divided hydraulic radius approach and Engelund’s energy slope division approach are reexamined. These two approaches assume that the shear stress on a mobile bed is the summation of shear stresses caused by skin friction and bed form. Using a different approach, this paper presents a theoretical relationship between the total bed shear stress with grain and bed-form shear stresses. The contribution of sidewall on the total bed shear stress is also discussed. The writers found that the size of bed form plays a significant role for the flow resistance, and developed relevant expressions for the length of the separation zone behind the bed forms. In addition, a systematical approach has been developed to compute the flow velocity in an alluvial channel. This approach is tested and verified against 5,989 flume and field measurements. The computed and measured discharge/velocity are in good agreement and 83.0% of all data sets fall within the ±20% error band.     

Numerical Simulation of Flow over a Rough Bed

This paper presents results of a direct numerical simulation (DNS) of turbulent flow over the rough bed of an open channel. We consider a hexagonal arrangement of spheres on the channel bed. The depth of flow has been taken as four times the diameter of the spheres and the Reynolds number has been chosen so that the roughness Reynolds number is greater than 70, thus ensuring a fully rough flow. A parallel code based on finite difference, domain decomposition, and multigrid methods has been used for the DNS. Computed results are compared with available experimental data. We report the first- and second-order statistics, variation of lift/drag and exchange coefficients. Good agreement with experimental results is seen for the mean velocity, turbulence intensities, and Reynolds stress. Further, the DNS results provide accurate quantitative statistics for rough bed flow. Detailed analysis of the DNS data confirms the streaky nature of the flow near the effective bed and the existence of a hierarchy of vortices aligned with the streamwise direction, and supports the wall similarity hypothesis. The computed exchange coefficients indicate a large degree of mixing between the fluid trapped below the midplane of the roughness elements and that above it.     

Errors in the Bed Shear Stress as Estimated from Vertical Velocity Profile

In this study, errors in determining bed shear stress caused by errors in theoretical bed surface data or roughness size selection using one-point velocity, two-point velocity, or a group of velocity measurements within the log-velocity region are systematically and quantitatively analyzed. The smaller the roughness element, the smaller the error in the bed shear stress estimate. For a fixed roughness size and absolute error in selecting the theoretical bed data, the closer to the bed the velocity measurement is taken, the larger the error in the friction velocity estimate. The velocity profile near the bed is very sensitive to the selection of the theoretical bed surface data. The velocity profile near the bed will deviate significantly from the true log profile if the theoretical bed surface data is over- or underestimated by 5?mm or more. This study shows conclusively that using the upper measurement data points, instead of the near-bed measurement, in the regression analysis yields better roughness size and bed shear stress estimates.     

Experimental Observations of Flow and Bed Processes in Large-Amplitude Meandering Flume

Meanders of large amplitude often exhibit asymmetric planform shape or subsidiary bends. The present work is aimed at improving on understanding of the morphodynamic phenomena affecting the bed evolution of large amplitude meandering channels. Attention is focused on the development of the steady point bar-pool configuration and of the superimposed large-scale migrating bed forms; of particular interest is the role of the changing channel curvature and bed topography variation on flow pattern. A series of experiments was carried out in a sine-generated large-amplitude meandering flume, for two values of width-to-depth ratio. Maps documenting the bed topography and the flow pattern along the meandering bends are reported. Two point bars per bend were observed and seem to be part of a series of damped oscillations developing in response to the changing channel curvature. In response to the bed deformation, the maximum flow velocity moves at the outer bank at the entrance of the bend.     

Hydraulic Radius for Evaluating Resistance Induced by Simulated Emergent Vegetation in Open-Channel Flows

The resistance induced by simulated emergent vegetation in open-channel flows has been interpreted differently in the literature, largely attributable to inconsistent uses of velocity and length scales in the definition of friction factor or drag coefficient and Reynolds number. By drawing analogies between pipe flows and vegetated channel flows, this study proposes a new friction function with the Reynolds number that is redefined by using a vegetation-related hydraulic radius. The new relationship is useful for consolidating various experimental data across a wide range of vegetation density. The results clearly show a monotonic decrease of the drag coefficient with the new Reynolds number, which is qualitatively comparable to other drag coefficient relationships for nonvegetated flows. This study also proposes a procedure for correcting sidewall and bed effects in the evaluation of vegetation drag.     

Spatially Averaged Open-Channel Flow over Rough Bed

In this paper it is suggested that the double-averaged (in temporal and in spatial domains) momentum equations should be used as a natural basis for the hydraulics of rough-bed open-channel flows, especially with small relative submergence. The relationships for the vertical distribution of the total stress for the simplest case of 2D, steady, uniform, spatially averaged flow over a rough bed with flat free surface are derived. These relationships explicitly include the form-induced stresses and form drag as components of the total stress. Using this approach, we define three types of rough-bed flows: (1) Flow with high relative submergence; (2) flow with small relative submergence; and (3) flow over a partially inundated rough bed. The relationships for the double-averaged velocity distribution and hydraulic resistance for all three flow types are derived and compared with measurements where possible. The double-averaged turbulent and form-induced intensities and stresses for the case of regular spherical-segment-type roughness show the dominant role of the double-averaged turbulence stresses and form drag in momentum transfer in the near-bed region.     

Test of a Method to Calculate Near-Bank Velocity and Boundary Shear Stress

Detailed knowledge of the flow and boundary shear stress fields near the banks of natural channels is essential for making accurate calculations of rates of near-bank sediment transport and geomorphic adjustment. This paper presents a high-resolution laboratory data set of velocity and boundary shear stress measurements and uses it to test a relatively simple, fully predictive, numerical method for determining these distributions across the cross-section of a straight channel. The measurements are made in a flume with a fairly complex cross-section that includes a simulated, cobble-roughened floodplain. The method tested is that reported by Kean and Smith in Riparian Vegetation and Fluvial Geomorphology in 2004, which is modified here to include the effects of drag on clasts located in the channel. The calculated patterns of velocity and boundary shear stress are shown to be in reasonable agreement with the measurements. The principal differences between the measured and calculated fields are the result of secondary circulations, which are not included in the calculation. Better agreement with the structure of the measured streamwise velocity field is obtained by distorting the calculated flow field with the measured secondary flow. Calculations for a variety of narrower and wider configurations of the original flume geometry are used to show how the width-to-depth ratio affects the distribution of velocity and boundary shear stress across the channel.     

Experimental Study of Bed Load Transport through Emergent Vegetation

Vegetation is an important agent in fluvial geomorphology and sedimentary processes, through its influence on the local hydraulics that determine sediment transport. Within stands of emergent vegetation, bed shear is substantially reduced through the absorption of momentum by drag on the stems. This stimulates deposition of sediment and reduces capacity for bed load transport. The effect of emergent vegetation on hydraulic parameters (including equilibrium bed gradient, flow depth, and velocity) and on bed load transport rate has been investigated experimentally for one sediment size, stem diameter, and stem spacing. Bed load transport rate was found to be closely related to bed-shear stress, which must be estimated by partitioning total flow resistance between stem drag and bed shear.     

Subelement Form-Drag Parameterization in Rough-Bed Flows

Spatial averaging of the Reynolds-averaged Navier–Stokes equations gives the double-averaged Navier–Stokes equations, for which boundary drag appears naturally and explicitly in momentum conservation equations. Increasing use of the double-averaged equations, e.g., for relating flows to three-dimensional bed roughness, for evaluation of profiles of flow stresses and velocities in ecologically significant regions below roughness tops, and for modeling purposes, requires parameterization of boundary drag at subelement scales. Based on seven flows over repeated square-rib roughness and ten flows over repeated fixed simulated sand waves, with measured velocities and bed pressures, expressions for form-drag coefficient CD = f (elevation below roughness top, relative roughness submergence, roughness steepness) are obtained for each of the two-dimensional roughness types. Using these equations, form drag variation with elevation below roughness tops can be calculated using either the double average of the square of local velocity (preferred based on conceptual considerations, trends in coefficient prediction, and also overall drag prediction) or the squared local double-averaged velocity, the roughness area being normal to the flow in each case. Integration of subelement drag given by these expressions is shown to give form-drag coefficient magnitudes and trends for complete individual elements comparable to those obtained by other authors based on measurements or numerical simulations. The ranges of roughness steepness and relative roughness submergence upon which the present equations have been derived need to be noted in consideration of application of the equations. In addition, effective application of the expressions is limited in regions of strongly negative double-averaged velocity. Further work remains to determine drag parameterization for alternative roughness geometries.     

Turbulent Boundary Layer over Symmetric Bodies with Rigid and Flexible Surfaces

This paper presents the results of an investigation concerning the development of a turbulent boundary layer over a 2D symmetrical aerofoil and a 3D axisymmetric body with rigid and flexible surfaces. The experimental work included detailed measurements of the mean velocity profiles, pressure distribution, and drag force. The thin shear layer equations were solved numerically using a modified turbulence model to obtain the characteristics of the turbulent boundary layer. The results of this study indicate a significant difference between the characteristics of flow over rigid surfaces and those of flow over flexible surfaces of the same geometry. The mean velocity of flow in the case of flexible surfaces is smaller than the corresponding velocity of flow in the case of a rigid surface for a major part of the boundary layer. The boundary layer thicknesses are consistently higher on flexible surfaces than those on the corresponding rigid surfaces. Furthermore, in the case of flexible surfaces, drag reduction was always observed. The amount of reduction was seen to be systematically dependent on the characteristics of the flexible surface.     

Three-Dimensional Modeling of Sediment Transport in a Narrow 90° Channel Bend

A three-dimensional numerical model was used for calculating the velocity and bed level changes over time in a 90° bended channel. The numerical model solved the Reynolds-averaged Navier-Stokes equations in three dimensions to compute the water flow and used the finite-volume method as the discretization scheme. The k-ε model predicted the turbulence, and the SIMPLE method computed the pressure. The suspended sediment transport was calculated by solving the convection diffusion equation and the bed load transport quantity was determined with an empirical formula. The model was enhanced with relations for the movement of sediment particles on steep side slopes in river bends. Located on a transversally sloping bed, a sediment particle has a lower critical shear stress than on a flat bed. Also, the direction of its movement deviates from the direction of the shear stress near the bed. These phenomenona are considered to play an important role in the morphodynamic process in sharp channel bends. The calculated velocities as well as the bed changes over time were compared with data from a physical model study and good agreement was found.     

Theory of plastic and viscous deformation

A theory of inelastic deformation, previously applied to 304 stainless steel with good quantitative agreement,² is used to study a variety of materials which differ in the degree to which dislocation motion is resisted by viscous drag forces. In the theory, mobile dislocations are injected into the material by the rising stress, move over a mean free path to create strain, and are trapped. The velocity of motion, determined by the magnitude of an effective stress relative to the viscous drag, determines the mean lifetime of mobile dislocations and thereby, in part, the mobile density. An attractive feature of the theory is its simplicity. There are only three significant physical constants, two which characterize the dislocation velocity and one, taken in this case to be material independent, which determines the strain-hardening coefficient. The calculations have been done to simulate a variety of tests done in a soft tensile machine, in which the principal control is exerted over the rate of stress increase. The results show diversetransient strain rate behavior, determined by the magnitude of the drag forces, but a commonsteady state strain rate, controlled by strain hardening. Soft materials with low viscous drag, such as copper, exhibit brief transients on change of stress rate, whereas in hard materials with high drag, such as iron-3.5 pct silicon, the transients are very long. These transients include the onset of yielding at the start of a strain-stress test, low temperature creep, and the strain rate response to a brief pulse of high stress rate. Thus for example, hard materials show long loading transients (slow approach to steady state), extensive low temperature creep, and no evident ‘rapid’ strain during a high rate stress pulse. For soft materials the converse results obtain. These differences and others distinguish, respectively, viscous and plastic deformation behavior.     

Boundary Shear Stress in Spatially Varied Flow with Increasing Discharge

The distribution of the wall shear stress on the bed and sidewalls of an open channel receiving lateral inflow was obtained from experimental measurements of the distribution of the velocity in the viscous sublayer using a laser doppler velocimeter. The experiments were conducted in a 0.4 m wide by 7.5 m long flume. Lateral inflow was provided into the channel from above via sets of nozzles positioned toward the downstream end of the flume. Lateral inflow was provided over a length of 1.9 m. The results indicate that the local boundary shear stresses are significantly influenced by lateral inflow. The significant variation occurs near and around the region where the lateral inflow enters the channel. At various measurement positions along the lateral inflow zone, mean boundary, mean wall, and mean bed shear stresses were obtained and compared. The results indicate that the mean boundary shear stresses increase from the upstream to the downstream ends of the lateral inflow zone. The results also indicate that the mean bed shear stress is always greater than the mean wall shear stress, which are approximately 30–60% of the mean bed shear stress. The friction factor in the Darcy–Weisbach equation was obtained from both the mean boundary shear stress and from the equation describing the water surface elevation in an open channel receiving lateral inflow (equation for spatially varied flow with increasing discharge). The results indicate that the estimated friction factors from the latter approach are significantly larger. Also, the estimated friction factors from both approaches are higher than the values predicted from the Blasius equation which describes the friction factor for wide uniform open channel flows. They were also higher than values predicted from the Keulegan equation, which is an empirically derived equation for flow in roof drainage gutters. The study highlights the deficiencies in the existing equations used to predict friction factors for spatially varied flow and that further research is required to explore the distribution of boundary shear stress in an open channel receiving lateral inflow.     

Sediment Threshold under Stream Flow on Horizontal and Sloping Beds

A model is presented to determine the threshold shear stress for noncohesive sediment (uniform and nonuniform) motion on horizontal and stream-wise sloping sedimentary beds, under a unidirectional steady-uniform streamflow. Hydrodynamic and particle-mechanic forces on a solitary sediment particle, resting over a sedimentary bed under the slip-spinning condition, are analyzed including the effect of turbulent fluctuations. Hydrodynamic forces such as drag, shear lift, and Magnus lift are taken into consideration. The drag coefficient is determined using an empirical formula. The inclusion of Magnus lift is significant because spherical particles spin just before dislodging downstream from their original position due to the differential hydrodynamic force along the vertical. The experimental data of sediment threshold are used to calibrate the model making the lift coefficient as a free parameter. The dependency of normalized threshold shear stress on particle parameter for various angles of repose and stream-wise bed slopes is presented graphically. The results obtained using the present model are compared with the curves proposed by different investigators and the experimental data of sediment (uniform and nonuniform) threshold for horizontal and stream-wise sloping beds.     

Three-Dimensional Numerical Modeling of Mixing at River Confluences

Lateral mixing of a pollutant is considered as a slow process that is usually complete within 100–300 river widths. Recent studies on flow dynamics at river confluences revealed that lateral mixing can be markedly enhanced when the tributary channel is shallower than the main channel. This study uses a three-dimensional model to examine mixing processes immediately downstream of confluences as well as further downstream in the mainstream. Simulations are presented for a concordant and discordant laboratory junction and a field confluence for a low and a high flow condition. The decrease in standard deviation at a cross section of a tracer over a distance of 5 channel widths is 30% for discordant beds but only 10% for concordant beds in the laboratory simulation. At the natural site, bed discordance is more important at the low flow than at the high flow with corresponding decreases in the standard deviation of 31 and 18% over 3.5 channel widths. Mixing is completed after a distance of 25 and 37 channel widths for the low and high flow conditions, respectively. Further downstream, mixing is mainly affected by planform curvature of the channel.     

Performances of Hydraulics and Bedload Sediment Flushing in Rigid Channel Using Surge Flows

This paper presents an investigation of the performance of the hydraulic and sediment removal of a flushing system in a detention basin. A hydraulic criterion for the design of the flushing system is proposed. An equation for the maximum height of the flushing wave front as a function of the distance from the gate, the initial water depth in the chamber, and the chamber length is proposed. The Lauber and Hager equation for the maximum velocity of a flushing wave is also verified. Effective removal of sediment particles on the bed is a direct function of the bed shear stress generated by the flushing flow. This study reveals that the bed shear stress on the channel bed induced by the flushing flow can be attributed to the hydrostatic pressure, the flow acceleration, and the convection-induced momentum. The shear stress associated with fluid distortion and the turbulent viscosity may be neglected. Significant error would occur if the hydrostatic pressure component were used as an estimate of the bed shear stress on a mild slope channel. The energy slope method may provide an overestimate of the bed shear stress. Finally, an appropriate equation to evaluate the maximum bed shear stress is proposed.     

Analysis of Velocity Lag in Sediment-Laden Open Channel Flows

Laboratory experiments have recently confirmed that the streamwise particle velocity is largely less than that of the fluid in sediment-laden flows. This velocity lag is investigated analytically in the present study based on the drag force exerting on a particle in the presence of other neighbors. The normalized drag force or the hindrance coefficient is found generally dependent on the particle concentration, particle Reynolds number, and specific gravity. The velocity lag is then derived by relating the hindrance coefficient to the shear stress distribution for uniform sediment-laden open channel flows. The analysis shows that the profile of the velocity lag, when normalized by the shear velocity, is associated with the shear Reynolds number, dimensionless particle diameter, and specific gravity. For the dilute condition, the velocity lag distribution varies only with the shear Reynolds number, and the lag can be ignored if the shear Reynolds number is less than unity. The theoretical predictions are comparable to limited experimental results.     

Flow Turbulence over Fixed and Weakly Mobile Gravel Beds

Characteristics of turbulence structure in quasi-2D flows with static and weakly mobile gravel beds are presented. Three sets of measurements with acoustic Doppler velocimeters in an irrigation canal were used: two with subcritical bed shear stress (static beds) and one with the bed shear stress τo close to critical τoc (weakly mobile bed). The analyses included vertical distributions of local mean velocities, turbulence intensities, turbulent shear stresses, velocity auto- and cross-spectra, the quadrant method, and high-order velocity moments. A number of properties of turbulence intensities, high-order moments, streamwise bursting parameters, and velocity spectra appeared to be similar for all three flows, but some properties were different. The most important one was an observed reduction in the von Kármán constant for the flow with weakly mobile bed. Comparison of these results with other studies and analogies with drag-reducing flows suggest that at τo∕τoc ≈ 1 the drag on the bed for a given granular material should be minimized.     

Lateral Depth-Averaged Velocity Distributions and Bed Shear in Rectangular Compound Channels

This paper presents a method to predict depth-averaged velocity and bed shear stress for overbank flows in straight rectangular two-stage channels. An analytical solution to the depth-integrated Navier–Stokes equation is presented that includes the effects of bed friction, lateral turbulence, and secondary flows. The Shiono and Knight method accounts for bed shear, lateral shear, and secondary flow effects via three coefficients, f, λ, and Γ, respectively. A novel boundary condition at the internal wall between the main channel and the adjoined floodplain is proposed and discussed along with other conventional boundary conditions. The analytical solution using the novel boundary condition gives good prediction of both lateral velocity distribution and bed shear stress when compared with experimental data for different aspect ratios.