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.     

Reynolds Stress and Bed Shear in Nonuniform Unsteady Open-Channel Flow

Expressions for the Reynolds stress and bed shear stress are developed for nonuniform unsteady flow in open channels with streamwise sloping beds, assuming universal (logarithmic) velocity distribution law and using the Reynolds and continuity equations of two-dimensional open-channel flow. The computed Reynolds stress distributions are in agreement with experimental data.     

Turbulent Bed Shear Stress under Symmetric and Asymmetric Waves: Experimental and Numerical Results

A new instrument built to directly measure the bottom shear stress under different conditions of periodic waves is presented. Similar to classical Taylor’s cylindrical viscosimeter, the instrument is formed by two concentric cylinders, and the space between them is filled with water. The internal cylinder is fixed whereas the external cylinder, with a specific roughness, rotates following a given unsteady velocity time history. The shear stress at the outer cylinder wall is measured by using a torque meter. Both monochromatic and sawtooth-shaped waves have been tested, and the experimental shear stress time histories were compared with the results obtained from a numerical model.     

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.     

Critical Bed Shear Stress for Unisize Sediment

The overall, spatially averaged, mean magnitude of local, spatially averaged (over a small area enclosing the particles’ projected area), instantaneous, critical Shields shear-stress parameters required for incipient motion of uniform-sized sand grains, independent of the bed shear-velocity particle Reynolds number, equal to 0.16, is obtained from calibration of a theory for bed load sediment transport, by minimizing the sum of the squares of the deviations between theoretical and experimental bed load rates. Additionally, optimized expressions for a proposed probability density distribution of the bed shear stresses, for its standard deviation, for finite, maximum, and minimum bed shear stresses, and a bed load rate are obtained. In terms of the mean fluid shear stress, a dimensionless, critical, shear-stress parameter equal to 0.0513 is obtained. Investigation of the probability density distribution of the spatially varying, critical shear stresses would allow a more accurate formulation for the case of low transport rates.     

Wall Shear Stress in the Early Stage of Unsteady Turbulent Pipe Flow

Conventionally, wall shear stress in an unsteady turbulent pipe flow is decomposed into a quasi-steady component and an “unsteady wall shear stress” component. Whereas the former is evaluated by using “standard” steady flow correlations, extensive research has been carried out to develop methods to predict the latter leading to various unsteady friction models. A different approach of decomposition is used in the present paper whereby the wall shear in an unsteady flow is split into the initial steady value and perturbations from it. It is shown that in the early stages of an unsteady turbulent pipe flow, these perturbations are well described by a laminar-flow formulation. This allows simple expressions to be derived for unsteady friction predictions, which are in good agreement with experimental and computational results.     

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.     

Depth-Averaged Shear Stress and Velocity in Open-Channel Flows

Turbulent momentum and velocity always have the greatest gradient along wall-normal direction in straight channel flows; this has led to the hypothesis that surplus energy within any control volume in a three-dimensional flow will be transferred toward its nearest boundary to dissipate. Starting from this, the boundary shear stress, the Reynolds shear stress, and the velocity profiles along normal lines of smooth boundary may be determined. This paper is a continuous effort to investigate depth-average shear stress and velocity in rough channels. Equations of the depth-averaged shear stress in typical open channels have been derived based on a theoretical relation between the depth-averaged shear stress and boundary shear stress. Equation of depth mean velocity in a rough channel is also obtained and the effects of water surface (or dip phenomenon) and roughness are included. Experimental data available in the literature have been used for verification that shows that the model reasonably agrees with the measured data.     

Wall Shear Stress in Transient Turbulent Pipe Flow by Local Velocity Measurement

The modeling of unsteady wall shear stress plays a crucial role in the analysis of fast transients in pressurized pipe systems, since it allows to evaluate transient energy dissipation properly. The main aim of this paper is to give a contribution to the understanding of transient pressurized flow dynamics in turbulent regime by measuring not only pressure but also the instantaneous axial velocity profile at two sections of the laboratory pipe. Specifically, by means of ultrasonic Doppler velocimetry—a completely nonintrusive technique—instantaneous velocity gradients at pipe wall are measured allowing to evaluate the time history of the actual wall shear stress by coupling velocity measurements to a two-zone stress model. As a result, the behavior of accelerating and decelerating flows with respect to the corresponding steady ones, i.e., with the same value of the discharge, is pointed out. Due to the characteristics of the laboratory pipe—a 352-m long high density polyethylene pipe—transients phenomena are investigated both at short and long time scales.     

Bed Shear Stress Boundary Condition for Storage Tank Sedimentation

Computational fluid dynamics-based (CFD) software tools enable engineers to simulate flow patterns and sediment transport in ancillary structures of sewer systems. Lagrangian particle tracking represents a computationally efficient technique for modeling sediment transport. In order to represent the process of sedimentation in storage tanks, careful consideration must be given to the boundary condition at the bottom of the tanks. None of the boundary conditions currently available in the FLUENT CFD software appears to represent the observed behavior of sediment particles, which may become resuspended after first contact with the bed if the local flow velocity is sufficiently high. In this study, a boundary condition based on bed shear stress has been implemented in FLUENT and evaluated against laboratory data. A particle is trapped if the local bed shear stress is below the critical bed shear stress; otherwise, the particle is resuspended. The approach gives satisfactory agreement with measured sedimentation efficiency data, and the simulated spatial distribution is very similar to the sediment distribution observed in a laboratory tank.     

Large-Eddy Simulation of High Reynolds Number Turbulent Flow Past a Square Cylinder

Flow past a square cylinder at a Reynolds number of 21,400 has been studied numerically using the large-eddy simulation technique. A dynamic subgrid-scale stress model has been used for the small scales of turbulence. The time- and span-averaged axial and transverse velocities in the downstream of the cylinder are in good agreement with the experimental results. The distribution of turbulent normal and shear stresses is also well predicted. The coherent and incoherent components of turbulent fluctuations at some specified phases have been separated and their relative magnitudes downstream of the cylinder have been compared. The comparison shows more coherence in the near wake than the far wake, while the coherent and incoherent components are of comparable magnitude in the far wake. The far wake shows irregular phase-averaged structures.     

Fine-Scale Characterization of the Turbulent Shear Layer of an Instream Pebble Cluster

This study characterizes the shear layer and associated vortex shedding around an isolated submerged pebble cluster in a gravel-bed river. The approach combines flow visualization and high frequency three-dimensional velocity (acoustic Doppler velocimeter) measurements. Two vortex shedding modes in the wake of the cluster were identified: A small scale high frequency initial instability mode and a lower frequency mode that scales with cluster height. The lower frequency mode arose from the intermittent interaction and amalgamation of the small-scale instability vortices. Reynolds shear stresses, velocity spectra, and coherence functions indicated a dominance of longitudinal-vertical shedding vortices in the wake of the cluster. Simultaneous flow visualization was required to determine the nature and behavior of the shedding modes. Quadrant analysis revealed that Q2 and Q4 events contributed 80% of the local longitudinal-vertical component Reynolds shear stress, and demonstrated a dominance of ejection events in the wake of the cluster. Through flow visualization, the behavior of the shear layer was seen to vertically expand and contract with the passage of Q2 and Q4 events, respectively.     

Determination of Boundary Shear Stress and Reynolds Shear Stress in Smooth Rectangular Channel Flows

A method for computing three-dimensional Reynolds shear stresses and boundary shear stress distribution in smooth rectangular channels is developed by applying an order of magnitude analysis to integrate the Reynolds equations. A simplified relationship between the lateral and vertical terms is hypothesized for which the Reynolds equations become solvable. This relationship has the form of a power law with an exponent of n = 1, 2, or infinity. The semiempirical equations for the boundary shear distribution and the distribution of Reynolds shear stresses are compared with measured data in open channels. The power-law exponent of 2 gave the best overall results while n=infinity gave good results near the boundary.     

Unsteady Bed Shear Stresses Induced by Navigation: Laboratory Observations

Time-dependent bed shear stresses induced by the passage of a barge tow have been measured with hot film shear stress sensors in a 1:25 scale model. Conditions typical of those observed for Upper Mississippi River navigation traffic were simulated in the experimental facility. Two sets of experiments were carried out: the first set consisted of simultaneous shear stress measurements at different locations for a variety of flow depths and boat operating conditions, providing space-time distributions of ensemble averaged wall shear stresses. The second set included a large number of realizations gathered for one particular flow condition at a single position, allowing analysis of the time evolution of the turbulence characteristics (i.e., standard deviation) of the bed shear stresses. The results of the first set of experiments show that for all the experimental conditions the basic patterns of the shear stress are similar, with two regions of high shear stress associated with the passage of the bow and the stern of the barge tow, respectively. Analysis of the second set of experiments showed that, as a result of the passage of the barge tow, the bed-shear stress standard deviation departs from the values commonly observed under steady, uniform, open-channel flow conditions. This behavior has important implications for sediment transport.     

Approximation of Turbulent Wall Shear Stresses in Highly Transient Pipe Flows

Theoretical predictions of wall shear stresses in unsteady turbulent flows in pipes are developed for all flow conditions from fully smooth to fully rough and for Reynolds numbers from 103 to 108. A weighting function approach is used, based on a two-region viscosity distribution in the pipe cross section that is consistent with the Colebrook–White expression for steady-state wall friction. The basic model is developed in an analytical form and the resulting weighting function is then approximated as a sum of exponentials using a modified form of an approximation due to Trikha. A straightforward method is presented for the determination of appropriate values of coefficients for any particular Reynolds number and pipe roughness ratio. The end result is a method that can be used relatively easily by analysts seeking to model unsteady flows in pipes and ducts.     

Bed-Shear Stress in Turbulent Wave-Current Boundary Layers

An approach for calculating turbulent flows in a wave-current boundary layer over a slowly varying bed is presented. Waves are periodic in time with several harmonics. In this paper, we adopt a time invariant eddy viscosity model, in which the eddy viscosity is linearly proportional to the distance from the bed. The boundary-layer flow field is solved analytically in terms of Fourier components. The approach allows fast computations and can be easily included in a phase resolving wave propagation model. As a part of the results, bottom shear stress and the spatial variation of the boundary layer thickness are also obtained. Present results compare well with experimental data and can explain the asymmetries in the bottom shear stress under sawtooth shaped waves.     

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.     

Shear Stress Distribution in Partially Filled Pipes

Boundary shear stresses have been calculated for circular pipes with a flat sediment bed using computational fluid dynamics (CFD). First of all, CFD simulations were carried out for rectangular channels in order to check the software package for its ability to reproduce experimental (literature) results. The influence of the applied turbulence model (isotropic or anisotropic) was also studied for rectangular channels. The simulations for circular pipes, using an isotropic turbulence model, were done for different filling ratios, mean flow velocities, and roughness heights. For validation, the numerical results were compared with former experimental work. With the help of the detailed shear stress distribution, sediment transport can be calculated more accurately than using the global shear stress, as is traditionally done. This method was applied to a simple flume experiment, subjected to a triangular inflow hydrograph, and the comparison with the traditional approach was made.     

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.     

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.