Probability Density Function of Instantaneous Drag Forces and Shear Stresses on a Bed

A probability density function (PDF) of the instantaneous bed shear stress in a turbulent flow is derived in this paper. It is argued that the shape of the PDF is similar to the PDF of the instantaneous drag forces on bed roughness elements. The influence of the near-bed relative turbulence intensity is included in the PDF. The shape of the distribution compares well with our measurements of the instantaneous drag force on a protruding bed element for a range of turbulence intensities. However, deviations are apparent at high turbulence intensities. The PDF also compares well with measurements of shear stresses on a smooth wall.     

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.     

Comparisons of Sidewall Correction of Bed Shear Stress in Open-Channel Flows

When investigating sediment transport in laboratory open-channel flows, it is often necessary to remove sidewall effects for computing effective bed shear stress. Previous sidewall correction methods are subject to some assumptions that have not been completely verified, and different values of the bed shear stress may be obtained depending on the approach used in making sidewall corrections. This study provides a quantitative assessment of the existing correction procedures by comparing them to a new sidewall correction model proposed in this study. The latter was derived based on the shear stress function and equivalent roughness size for both rigid and mobile bed conditions, which were obtained directly from experimental measurements. The comparisons show that the Einstein correction formula and the Vanoni and Brooks method generally predict relatively lower and higher bed shear stresses, respectively, while the Williams’ empirical function leads to more scatter. This study also demonstrates that the widely used Vanoni and Brooks approach can be well approximated by a simple formula derived based on the Blasius resistance function. The sidewall effects, when removed in the different ways, would consequently affect the presentation of the bedload function. Experimental results of bedload transport, when plotted as the dimensionless transport rate against the dimensionless shear stress with the latter being corrected using the present model, exhibit less scatter than those associated with the previous procedures.     

Shear Stress in Smooth Rectangular Open-Channel Flows

The average bed and sidewall shear stresses in smooth rectangular open-channel flows are determined after solving the continuity and momentum equations. The analysis shows that the shear stresses are function of three components: (1) gravitational; (2) secondary flows; and (3) interfacial shear stress. An analytical solution in terms of series expansion is obtained for the case of constant eddy viscosity without secondary currents. In comparison with laboratory measurements, it slightly overestimates the average bed shear stress measurements but underestimates the average sidewall shear stress by 17% when the width–depth ratio becomes large. A second approximation is formulated after introducing two empirical correction factors. The second approximation agrees very well (R2>0.99 and average relative error less than 6%) with experimental measurements over a wide range of width–depth ratios.     

Numerical Investigation into Secondary Currents and Wall Shear in Trapezoidal Channels

This paper presents the use of computational fluid dynamics (CFD) to determine the distribution of the bed and sidewall shear stresses in trapezoidal channels. The impact of the variation of the slant angle of the side walls, aspect ratio, and composite roughness on the shear stress distribution is analyzed. The shear stress data can be directly output from the CFD models at the boundaries, but they can also be derived using the Guo and Julien equations for the average bed and side wall shear stresses. These equations compute the shear stress as a function of three components; gravitational, secondary flows, and interfacial shear stress, and are hence used to gauge the respective merits of the different components of wall shear. The results show a significant contribution from the secondary currents and internal shear stresses on the overall shear stress at the boundaries. This work also extends previous work of the authors on rectangular channels.     

Bed-Shear Stress Characteristics of a Simple, Prismatic, Rectangular Channel

This paper examines the use of data obtained from large eddy simulations in exploring the instantaneous characteristics of boundary shear stress. The probability density functions (PDF) of the shear stress are shown to be non-Gaussian, whereas the PDF of the shear force per unit length is found to be approximately Gaussian. Conditional sampling is used in order to provide an insight into the average behavior of an extreme shear stress event. Finally, proper orthogonal decomposition illustrates that the behavior of the shear stress is complex and cannot necessarily be attributed to a single major flow event.     

Distribution of Bed Shear Stress in Rotating Circular Flume

Distributions of bed shear stress across the width of a rotating circular flume with smooth and rough bed surfaces were obtained by measurement and model prediction. Results with flows over smooth beds showed that the flow in the central part may be considered to be two-dimensional and that effects of flow depth over the operating range of the flume are minor for flow depths not exceeding 0.14 m. For rough beds, the bed shear stress distributions were found to be skewed toward the inner wall. This can be corrected if a compensating roughness is added to the bottom of the ring. Such measures are also effective for flumes with smooth beds. Measured bed shear stress distributions agreed well with the predicted distributions for smooth beds and reasonably well for rough beds. The modified Preston tube, for measurement of bed shear stress in flows over rough beds, was found to give promising results. Further tests are required to completely define the uncertainty in bed shear stress measurements made with this instrument.     

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.     

Assessment of Methods Used in 1D Models for Computing Bed-Load Transport in a Large River: The Danube River in Slovakia

Comprehensive measurements of bed-load sediment transport through a section of the Danube River, located approximately 70?km downstream from Bratislava, Slovakia, are used to assess the accuracy of bed-load formulas implemented in 1D modeling. Depending on water discharge and water level, significant variations in the distribution of bed load across the section were observed. It appeared that, whatever the water discharge, the bed shear stress τ is always close to the estimated critical bed shear stress for the initiation of sediment transport τcr. The discussion focuses on the methods used in 1D models for estimating bed-load transport. Though usually done, the evaluation of bed-load transport using the mean cross-sectional bed shear stress yields unsatisfactory results. It is necessary to use an additional model to distribute the bed shear stress across the section and calculate bed load locally. Bed-load predictors also need to be accurate for τ close to τcr. From that point of view, bed-load formulas based on an exponential decrease of bed-load transport close to τcr appear to be more appropriate than models based on excess bed shear stress. A discussion on the bed-load formula capability to reproduce grain sorting is also provided.     

Turbulent Flow over a Channel with Fluid-Saturated Porous Bed

The characteristics of fully developed turbulent flow in a hybrid domain channel, which consists of a clear fluid region and a porous bed, are examined numerically using a model based on the macroscopic Reynolds-averaged Navier–Stokes equations. By adopting the classical continuity interface conditions, the present model treats the hybrid domain problem with a single domain approach, and the simulated results are noted to coincide with the existing experimental data and microscopic data. The effects of porosity ? and Darcy number Da on the flow properties over and inside the porous bed are further investigated in the selected ranges of 0.6 ? ? ? 0.8, and 1.6×10?4 ? Da ? 1.6×10?2. It has been demonstrated that the presence of the porous bed causes the significant reduction of the flow velocities inside the clear fluid region relative to that of a smooth impermeable bed, and also reduces the magnitude of the integral constant B of the velocity logarithmic distributions from its traditional value 5.25. Moreover, turbulent shear stress within the upper part of the porous bed increases significantly with the porosity ? and Darcy number Da. The thickness of turbulence penetration remains proportional to the values of porosity ? and Darcy number Da.     

Sediment Threshold with Upward Seepage

An analytical model is presented to determine the threshold bed shear stress for noncohesive sediment motion subject to upward seepage on horizontal sedimentary bed under a stream flow. Hydrodynamic, seepage, and micromechanical forces acting on a solitary sediment particle, resting over a sedimentary bed under slip-spinning condition, are analyzed. The correlation coefficient between the results obtained using the present model and the experimental data of threshold bed shear stress with upward seepage on the horizontal bed is 0.767. It indicates that the model predicts satisfactorily the threshold bed shear stress with upward seepage.     

Initiation of Motion of Calcareous Sand

This paper examines the initiation of motion of four natural and five sieved calcareous sand samples in unidirectional flow. Flume experiments yield the sediment transport rate as a function of bed shear stress up to bed-form development. Reference-based criteria are supplemented by visual observations to determine the critical shear stress. The results are compared with published data for rounded and irregular particles in terms of the median sieve size and the corresponding nominal and equivalent diameters as functions of particle Reynolds number. The comparison shows that the critical shear stresses of the irregular particles are higher than the Shields curve in the hydraulically smooth flow regime and lower in the rough turbulent flow regime.     

Movable Bed Roughness in Alluvial Rivers

A method for the prediction of a movable bed roughness coefficient is proposed by relating the roughness parameter A to the nondimensional shear stress τb′/τc50 and the Froude number of flow. This relationship is tested with a large number of flume and field data from different sources, and also compared with other prediction methods. The calculated flow depth based on the bed roughness coefficient predicted by the proposed method agrees very well with the measured data.     

Numerical Modeling of Abutment Scour with the Focus on the Incipient Motion on Sloping Beds

A three-dimensional computational fluid dynamics model is applied to predict local scour around an abutment in a rectangular laboratory flume. When modeling local scour, steep bed slopes up to the angle of repose occur. To predict the depth and the shape of the local scour correctly, the reduction of the critical shear stress due to the sloping bed must be taken into account. The focus of this study is to investigate different formulas for the threshold of noncohesive sediment motion on sloping beds. Some formulas only take the transversal angle (perpendicular to the flow direction) into account, but others also consider the longitudinal angle (streamwise direction). The numerical model solves the transient Reynolds-averaged Navier-Stokes equations in all three dimensions to compute the water flow. Sediment continuity in combination with an empirical formula is used to capture the bed load transport and the resulting bed changes. When the sloping bed exceeds the angle of repose, the bed slope is corrected with a sand-slide algorithm. The results from the numerical simulations are compared with data from physical experiments. The reduction of the bed shear stress on the sloping bed improves the results of the numerical simulation distinctly. The best results are obtained with the formulas that use both the transversal and the longitudinal angle for the reduction of the critical bed shear stress.     

Coherent Structures in the Flow Field around a Circular Cylinder with Scour Hole

Large-eddy simulation (LES) and laboratory-flume visualizations were used to investigate coherent structures present in the flow field around a circular cylinder located in a scour hole. The bathymetry corresponds to equilibrium scour conditions and is fixed in LES. The flow parameters in the simulation correspond to the experimental conditions in which the approach flow is fully turbulent. Detailed consideration is given to the interaction of the horseshoe vortex (HV) system within the scour hole with the detached shear layers formed from the cylinder, and the near bed turbulence. It is found that the overall structure of the HV system varies considerably in space and time, though a large, relatively stable, primary necklace vortex is present at practically all times inside the scour hole. The simulation captures the presence of bimodal chaotic oscillations inside the HV system, as well as the sharp increase in the resolved turbulent kinetic energy levels and pressure fluctuations reported in prior experimental investigations. High levels of the mean bed shear stress are observed beneath the primary necklace vortex, especially over the region where the bimodal oscillations are strong, as well as beneath the small junction vortex at the base of the cylinder. It is also found that the detachment and advection of patches of vorticity from the downstream part of the legs of the necklace vortices can induce large instantaneous bed shear stress values. When the critical bed shear stress value for sediment entrainment on a flat surface is adjusted for bed slope effects, the LES simulation correctly predicts that the distribution of the mean bed shear stress is consistent with equilibrium scour conditions.     

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.     

Two-Dimensional Solution for Straight and Meandering Overbank Flows

This paper presents a practical method to predict depth-averaged velocity and shear stress for straight and meandering overbank flows. An analytical solution to the depth-integrated turbulent form of the Navier-Stokes equation is presented that includes lateral shear and secondary flows in addition to bed friction. The novelty of the present approach is not only its inclusion of the secondary flows in the formulation but also its applicability to straight and meandering channels. The analytical solution is applied to a number of channels, at model and field scales, and is also compared with other available methods such as that of Shiono and Knight and the lateral distribution method. The present formulation gives much better predictions of velocity and shear stress, particularly in those cases where the secondary flows are dominant.     

Scale Independent Linear Behavior of Alluvial Channel Flow

For flow in a rigid open channel with no bed sediment, the achievement of the special state of stationary equilibrium yields a linear characteristic. To examine the existence of a linear characteristic in alluvial channel flow, this study presents a direct formulation of the special equilibrium state following a variational approach. It finds that a linear relationship between shear stress and width/depth ratio of alluvial channels emerges under the commonly identified flow resistance and sediment transport conditions. Most importantly, this linear relationship yields not only the theoretical equilibrium channel geometry that is very close to a widely identified empirical counterpart but also a nondimensional number H, defined as the ratio of the relative increment of shear stress to the increment of width/depth ratio. This study suggests that H needs to be adopted as a criterion of hydraulic similitude for modeling sediment (bed-load) transport in alluvial channels.     

Turbulence Characteristics of Open-Channel Flow with Bed Suction

The influence of bed suction on the characteristics of turbulent open channel flow is studied in a laboratory flume using a two-component laser Doppler velocimeter. The experimental results show how bed suction significantly affects the mean flow properties, turbulence levels, and Reynolds stress distributions. The data reveal the presence of a more negative vertical (downward) velocity. The results also show how the horizontal and vertical turbulence intensities and Reynolds shear stresses respond to suction. All these properties are found to reduce with increasing relative suctions: decreasing more rapidly around the bed region than that near the free surface. In the downstream direction, the flow structure in the suction zone undergoes a process of rapid readjustment within a transitional region. Beyond this region, the turbulence flow structures asymptotes toward an “equilibrium” region.     

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.