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.     

Friction Factors for Spatially Varied Flow with Increasing Discharge

This paper describes an experimental investigation of how friction factors change for spatially varied flow in sloping channels receiving lateral inflow. The results are compared with those of Beij in 1934, and it is concluded that uniform flow resistance coefficients are not always appropriate for spatially varied flow. Moreover, the common technique of assuming a constant friction factor over the entire length of the channel has been found to have little theoretical justification. The method of Keulegan in 1952 for calculating friction factors in spatially varied flow gives a better estimate, but does not explicitly take account of the lateral inflow rate or velocity. Beij’s 1934 experimental data, which was used by Keulegan does not show a systematic variation of friction factor with lateral inflow rate for a constant Reynolds number although this may be due to the low flowrates used. The results of the present study indicate that the friction factor increases with lateral inflow rate for a constant Reynolds number in the experiments that included subcritical and supercritical flow conditions. A method for calculating friction factors which allows for lateral inflow is presented as a precursor to the development of a general method of evaluating friction factors for spatially varied flow with increasing discharge.     

Maximum Velocity and Regularities in Open-Channel Flow

Maximum velocity in a channel section often occurs below the water surface. Its location is linked to the ratio of the mean and maximum velocities, velocity distribution parameter, location of mean velocity, energy and momentum coefficients, and probability density function underpinning a velocity distribution equation derived by applying the probability and entropy concepts. The mean value of the ratio of the mean and maximum velocities at a given channel section is stable and constant, and invariant with time and discharge. Its relationship with the others in turn leads to formation of a network of related constants that represent regularities in open-channel flows and can be used to ease discharge measurements and other tasks in hydraulic engineering. Under the probability concept, the ratio of mean and maximum velocities being constant means that the probability distribution underpinning the velocity distribution and other related variables is resilient, and that the same probability distribution is governing various phenomena observable at a channel section and explains the regularities in open-channel flows.     

Flume for Teaching Spatially Varied Open-Channel Flow

Spatially varied flow in open channels is a topic that is often included in undergraduate open channel hydraulics courses. Physical and computational models are developed to enhance the presentation of spatially varied flow to engineering students at the late undergraduate or early graduate level. The physical model is inexpensive and easy to build and the computational model is easily developed using commercially available spreadsheet software. The physical model consists of a 30.48 cm nominal-diameter PVC pipe that is 6.1 m in length and has circular orifices approximately 1.40 cm in diameter drilled on 15.24 cm centers along the pipe invert. A relationship between the orifice discharge coefficient and a modification of the Froude number, as measured in the flume upstream of the orifice in question, was developed in repeated trials having varying flume slope, volumetric inflow rate, and end conditions. With this relationship, a stepwise solution to the energy equation is used to predict the water surface profile. Differences between the water surface profiles observed and predicted in repeated trials averaged approximately 2 mm.     

Velocity Distribution of Turbulent Open-Channel Flow with Bed Suction

This study investigates theoretically and experimentally the velocity distributions of turbulent open channel flow with bed suction. A velocity profile with a slip velocity at the bed surface and an origin displacement under the bed surface is proposed and discussed. Based on this assumption, a modified logarithmic law is derived. The measured experimental velocity distribution verifies the accuracy of the theoretically derived profile. The data show a significant increase in the near bed velocity and a velocity reduction near the water surface, resulting in the formation of a more uniform velocity distribution. The values of the origin displacement, slip velocity and shear velocity are found to increase with increasing relative suction. The measured data show the occurrence of two flow regions in the suction zone: a transitional region in which the velocity readjusts rapidly; and an “equilibrium” region.     

Modeling Depth-Averaged Velocity and Boundary Shear in Trapezoidal Channels with Secondary Flows

The Shiono and Knight method (SKM) offers a new approach to calculating the lateral distributions of depth-averaged velocity and boundary shear stress for flows in straight prismatic channels. It accounts for bed shear, lateral shear, and secondary flow effects via 3 coefficients—f,λ, and Γ—thus incorporating some key 3D flow feature into a lateral distribution model for streamwise motion. The SKM incorporates the effects of secondary flows by specifying an appropriate value for the Γ parameter depending on the sense of direction of the secondary flows, commensurate with the derivative of the term Hρ(UV)d. The values of the transverse velocities, V, have been shown to be consistent with observation. A wide range of boundary shear stress data for trapezoidal channels from different sources has been used to validate the model. The accuracy of the predictions is good, despite the simplicity of the model, although some calibration problems remain. The SKM thus offers an alternative methodology to the more traditional computational fluid dynamics (CFD) approach, giving velocities and boundary shear stress for practical problems, but at much less computational effort than CFD.     

Effects of Velocity Gradients and Secondary Flow on the Dispersion of Solutes in a Meandering Channel

Two contrasting mechanisms, created by channel curvature which strongly affect longitudinal dispersion of solutes in rivers are examined. In natural channels the large cross-sectional variability of the primary velocity component tends to increase longitudinal dispersion by providing a large difference between adjacent fast and slow moving zones of fluid. By contrast secondary circulation tends to decrease longitudinal dispersion by enhancing transverse mixing. A series of tests have been carried out in a very large flume containing a meandering water-formed sand bed channel to measure the longitudinal dispersion coefficient at various locations around a meander. These experimental observations are compared with experimental data obtained from meandering channels with smooth, fixed sides and regular cross-sectional shapes. All the data has been compared against predictions from two current modeling approaches. Finally, the significance of the two competing mechanisms in curved channels is discussed with regard to their relative influence on longitudinal mixing.     

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.     

Higher-Order Moments of Velocity Fluctuations in an Open-Channel Flow with Large Bottom Roughness

The present study was undertaken to examine the effect of surface roughness on the higher-order velocity moments in a turbulent open-channel flow. The wall roughness is large and the ratio of the roughness size to the depth of flow ranges from 0.1 to 0.15. Flow over two types of roughness conditions (ribs and dunes) are examined and compared with that over a smooth open channel. In the case of the rib roughness, three different spacings are used: p/k = 4.5, 9, and 18; here, p=distance between the leading edges of two consecutive ribs and k=height of a rib with square cross section. The shape of the smooth wall dunes was geometrically similar to that commonly used in previous studies. In an effort to understand the influence of local wall roughness, rough-wall dunes are also used in the present study. The variables of interest include higher-order moments, conditional statistics based on a quadrant analysis, relationship of higher-order moments to the Reynolds stress and energy production. Although there are some similarities between open-channel flows and turbulent boundary layers, there are important differences as indicated by the third-order moments. The triple products and the second quadrant events are sensitive to the wall condition. Ejection events are found to be prevalent throughout the depth. It is also clear from the study that one needs to explicitly account for the surface condition in any model development to be able to predict transport characteristics.     

Flow and Velocity Distribution in Meandering Compound Channels

An investigation of flow and velocity distribution in meandering compound channels with over bank flow is described. Equations concerning the three-dimensional variation of longitudinal, transverse, and vertical velocity in the main channel and floodplain of compound section in terms of channel parameters are presented. The flow and velocity distributions in meandering compound channels are strongly governed by interaction between flow in the main channel and that in the floodplain. The proposed equations take adequate care of the interaction affect. Results from the formulations, simulating the three-dimensional velocity field in the main channel and in the floodplain of meandering compound channels are compared with their respective experimental channel data obtained from a series of symmetrical and unsymmetrical test channels with smooth and rough sections. The aspect ratio of the test channels varies from two to five. The equations are found to be in good agreement with the experimental data. The formulations are verified against the natural river and other meandering compound channel data. The power laws used for simulating the three-dimensional velocity structure are found to be quite adequate.     

Laboratory Measurements of Flow and Turbulence in Discontinuous Distributions of Ligulate Seagrass

Turbulent flow characteristics were investigated in laboratory flume studies of a ligulate plant canopy interrupted by a gap representing discontinuities observed in seagrass prairies. The reliability of velocity measurements obtained using an acoustic Doppler velocimeter within the canopy was shown using specifically designed experiments. In relatively fast flow (mean velocity 5.5?cm?s?1), the mean flow profile was logarithmic above the canopy, had an inflection point near its top, and uniformly low values within it. Within the gap, a recirculation cell formed. Reynolds stress maxima were approximately coincident with the mean flow inflection point. Quadrant analysis revealed an ejection-dominated upper layer, a sweep-dominated region around the top of the canopy and within the gap, and no dominant quadrant within the canopy. In slower flow (mean velocity 1.7?cm?s?1) the plants were quasiemergent and the flow fields more uniform. Sweeps similarly dominated the region near the top of the canopy and within the gap. In both flows, autocorrelation of longitudinal velocity fluctuations showed a Lagrangian time scale maximum at the downstream end of the gap.     

Velocity Profile and Flow Resistance Models for Developing Chute Flow

A developing boundary layer starts at the spillway crest until it reaches the free surface at the so-called inception point, where the natural air entrainment is initiated. A detailed reanalysis of the turbulent velocity profiles on steep chutes is made herein, including mean values for the parameters of the complete turbulent velocity profile in the turbulent rough flow regime, given by the log-wake law. Accounting both for the laws of the wall and the wake, a new rational approach is proposed for a power-law velocity profile within the boundary layer of turbulent rough chute flow. This novel approach directly includes the power-law parameters and does not require for a profile matching, as is currently required. The results obtained for the turbulent velocity profiles were applied to analytically determine the resistance characteristics for chute flows. The results apply to the developing flow zone upstream of air inception in chute spillways.     

Velocity Distribution in the Roughness Layer of Rough-Bed Flows

Several models for the vertical distribution of the double-averaged (in time and in the plane parallel to the mean bed) longitudinal velocity in the flow region between roughness troughs and roughness tops are suggested. We found that the same model for velocity distribution may be applicable to a range of flow conditions and roughness types, which share some common features. The suggested models for velocity distribution in the near-bed region are: (1) Constant velocity; (2) exponential velocity distribution; and (3) linear velocity distribution. The measured velocity distributions may be approximated by a single model or by a combination of models depending on roughness geometry and flow conditions. The validity of these models for velocity distribution is supported by laboratory data.     

Response of Velocity and Turbulence to Sudden Change of Bed Roughness in Open-Channel Flow

The experimental study shows how an open-channel flow would respond to a sudden change (from smooth to rough) in bed roughness. Using a two-dimensional acoustic Doppler velocimeter and a laser Doppler velocimeter, the velocity, turbulent intensities, and Reynolds stress profiles at different locations along a laboratory flume were measured. Additionally, the water surface profile was also measured using a capacitance-type wave height meter. The experimental data show the formation of an internal boundary layer as a result of the step change in bed roughness. The data show that this boundary layer grows much more rapidly than that formed in close-conduit flows. The results also show that the equivalent bed roughness, bed-shear stress, turbulent intensities, and Reynolds stress change gradually over a transitional region, although the bed roughness changes abruptly. The behavior is different from that observed in close-conduit flows, where an overshooting property—which describes the ability of the bed-shear stress to attain a high-peak value over the section with the larger roughness, was reported. A possible reason for the difference is the variation of the water surface profile when an open-channel flow is subjected to a sudden change in bed roughness.     

Reduction of Bend Scour by an Outer Bank Footing: Flow Field and Turbulence

River bank protection is a costly but essential component in river management. Outer banks in river bends are most vulnerable to scour and erosion. Previous laboratory experiments illustrated that a well-designed horizontal foundation of a vertical outer bank protruding into the cross section, called a footing, can reduce the scour depth and thereby protect the bank. This paper provides detailed experimental data in a reference experiment without footing and an experiment with footing carried out under similar hydraulic conditions, which suggest a delicate interaction between bed topography, downstream and cross-stream velocity, and to lesser extent turbulence. The presence of the outer bank footing modifies this delicate interaction and results in a more favorable configuration with respect to bank stability including: reduced maximum scour depth, more uniformly distributed downstream velocity, and weaker cross-stream circulation cells.     

Upstream Discharge Distribution in Compound-Channel Flumes

Common inlet design for compound-channel flumes does not ensure a proper upstream discharge distribution. As the total head in the upstream tank is the same for both main-channel and floodplain subsections, the velocity in the upstream section is also the same in both subsections. The floodplain discharge is therefore too large and a mass transfer towards the main channel occurs along the flume. This Technical Note investigates how long a compound-channel flume must be to ensure that equilibrium between subsection discharges is achieved. The required length is found to be significant compared to the actual length of experimental flumes reported in the literature.     

Depth-Averaged Velocity Distribution in Straight Trapezoidal Channels

In trapezoidal channels that are not “wide,” the banks exert form drag on the fluid and thereby control the depth-averaged velocity distribution. As such, commonly used equations for predicting depth-averaged velocities in wide channels are not well suited for predicting depth-averaged velocities in trapezoidal channels. Using data from three previous studies, we developed two models for predicting depth-averaged velocity distributions in straight trapezoidal channels. The data used to develop the models had a range of discharges (8.05–4,248?L/s), velocities (0.16–1.03?m/s), bottom widths (0.305–3.62?m), flow depths (0.0518–0.805?m), and bank slopes (1.0–3.0, horizontal/vertical). The first model requires measured velocity data for calibrating the model coefficients, whereas the second model uses prescribed coefficients. The first model yielded velocity distributions with coefficients of determination (r2) from 0.84 to 0.90 and we recommend its use when possible because it yields predictions that are more accurate. The second model also yielded good results (r2 = 0.86 and 94% of the predicted velocities were within 20% of the observed values).     

Velocity Distribution and Energy Dissipation along Stepped Chutes Lined with Wedge-Shaped Concrete Blocks

Stepped channels lined with wedge-shaped concrete blocks may constitute a low-cost alternative to provide overtopping protection of embankment dams if the discharge capacity of existing spillways is not adequate or even to be used as the main spillway of newly built embankment dams. This paper addresses the velocity distribution and the energy dissipation, downstream of the inception point, on stepped chutes lined with wedge-shaped concrete blocks. An experimental setup was developed with two flumes designed with a relative scale of 1∶2.5. Air concentration was measured with an optical probe in several cross sections of both flumes. The velocity profiles along chutes lined with wedge-shaped blocks with the upper face sloping downstream were analyzed. The measurements’ accuracy was checked by comparing discharges indicated by a facility flowmeter and obtained by the integration of velocity and air concentration profiles. The effect of the steps-slope in the energy dissipation is studied. Values of the Darcy-Weissbach friction factor are proposed for this type of chute lining, for transition flows, and for skimming flows.