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This note concerns variations of the friction factor in the two transitional regimes, one between laminar and turbulent flows and the other between fully smooth and fully rough turbulent flows. An interpolation approach is developed to derive a single explicit formula for computing the friction factor in all flow regimes. The results obtained for pipe flows give a better representation of Nikuradse's experimental data, in comparison with other implicit formulas available in the literature. Certain modifications are also made for applying the obtained friction formula to open-channel flows. DOI: 10.1061/(ASCE)0733-9429(2008) 134:9(1357) CE Database subject headings: Friction; Pipe flow; Open channel flow; Laminar flow: Turbulent flow; Reynolds number.

The critical condition for incipient sediment motion is formulated in this note based on the settling velocity. The formula obtained is simple, relating the ratio of critical shear velocity to settling velocity to the dimensionless sediment diameter. Comparisons are then made with other settling-velocity based formulas available in the literature. To facilitate the computation of the effective near-bed velocity at the threshold condition, a generalized law-of-the-wall function is proposed for predicting the velocity distribution under various boundary conditions. This study demonstrates that the settling velocity is equivalent to the critical near-bed velocity, which is experienced by a typical bed sediment particle under the threshold condition, but only for large sediment sizes such as sand and gravel. Comparison results show that Yang's formula is suitable for flows with small flow depth relative to sediment size while Le Roux's formula may overestimate the threshold condition for fine particles by up to 30%. DOI: 10.1061/(ASCE)0733-9429(2008)134:8(1136) CE Database subject headings: Velocity; Sediment transport; Comparative studies.

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. CE Database subject headings: Shear stress; Open channel flow: Channel beds: Laboratory tests.

Nonuniform-unsteady flow in open channels with streamwise sloping beds having uniform upward seepage is theoretically analyzed. Expressions for the Reynolds stress and bed shear stress are developed, assuming a modified logarithmic law of velocity profile due to upward seepage, and using the Reynolds and continuity equations of two-dimensional open channel flow. The computed Reynolds stress profiles are in agreement with the experimental data. DOI: 10.1061/(ASCE)0733-9399(2005)131:4(451) CE Database subject headings: Reynolds stress: Shear stress: Open channel flow: Streamflow; Nonuniform flow: Unsteady flow; Seepage; Hydraulics.

In this study, we investigated a vertical dilute sediment-laden jet both experimentally and theoretically. First, an instantaneous whole-field velocimetry tool, particle image velocimetry, was applied to measure the sediment and fluid mean and fluctuating velocities of a downward sediment-laden jet at the same time. Subsequently, an analysis was performed based on two-phase conservation equations for both downward and upward jets. The analysis shows that the mean sediment velocity can be taken as the sum of fluid velocity and the settling velocity in both cases. For the downward jets, the decay rate of the centerline sediment concentration increases with the sediment settling velocity while decreases with the initial discharge velocity. The zone of flow establishment for the sediment velocity is found to be longer than that of the fluid. For the upward jets, the maximum rise of the sediment particles and their deposition distribution on the ground were derived theoretically. The predicted results compare well to the experimental data in the literature. CE Database subject headings: Sediment transport; Submerged jets; Velocity; Two-phase flow; Pipe flow.

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. CE Database subject headings: Suspended sediments; Velocity profile; Two phase flow; Open channels; Interactions.

This study is concerned with two hydrodynamic parameters that are essential for the construction of submerged dikes in rivers with sand-filled geosynthetic bags. First, laboratory study and field observation are conducted to investigate the horizontal settling distance of sandbags in open channel flows. Sandbag models used for laboratory tests are prepared according to similarity requirements. An analytical formula for evaluating the settling distance is then derived based on the experimental data. The predictions are then compared to the field measurements. The observed settling distance varies significantly and its probability density distribution is positively skewed. This study also presents a formula for computing the critical flow velocity at incipient sandbag motion for two extreme conditions. DOI: 10.1061/(ASCE)0733-950X(2004)130:2(98) CE Database subject headings: Geosynthetics; Settling velocity; Incipient motion; Open channels; Hydraulic models.

The Ergun equation that is widely used in chemical engineering is generalized for describing seepage through the sediment boundaries in this study. Comparisons with the measurements of the hydraulic gradient for flows through packed beds show that the Ergun equation may not be suitable for the transitional zone where both viscous and inertial effects are of the same order of magnitude. The generalized Ergun equation is then used to evaluate the critical shear velocity for incipient sediment motion subject to seepage. The computed results are finally compared to the experimental data and those predicted using the previously proposed approach. CE Database subject headings: Velocity; Seepage; Equation of motion; Sediment; Computation.

With the assumption that the bed shear stress fluctuates in a lognormal fashion, the probability density function (PDF) of the standardized bed shear stress is derived as a function of the relative shear stress intensity. The PDF is more skewed with larger relative intensities, but approaches a Gaussian function when the relative intensity is small. The computed PDF agrees well with the reported experimental data for flows over a smooth boundary. The higher-order moments of the bed shear stress, skewness, and kurtosis, are shown analytically to be also dependent on the relative intensity. The theoretical dependencies are then compared to a number of measurements available in the literature. The Reynolds number effect on the relative intensity is also discussed. DOI: 10.1061/(ASCE)0733-9399(2003) 129:1(126) CE Database keywords: Shear stress; Skewness; Reynolds number; Beds.

An exponential formula that does not involve the concept of the critical shear stress is derived in this study for computing bedload transport rates. The formula represents well various experimental data sets ranging from the weak transport to high shear conditions. Comparisons of the present study are also made with many previous bedload formulas commonly cited in the literature. CE Database keywords: Bed load; Sediment transport; Transport rate; Shear stress.

A theoretical analysis of the initiation of sediment suspension is performed based on the concept of probability of suspension. The proposed condition for the initiation of sediment suspension from the top of the bed-load layer is comparable with most existing empirical relationships for large particle Reynolds numbers. The study shows that a constant Rouse parameter can be used as the critical condition for sediment suspension only for large dimensionless particle diameters. When the probability of suspension is taken to be an infinitesimal value (P = [10.sup.-7]), the derived relationship of the Shields parameter and the particle Reynolds number is in good agreement with the updated Shields diagram.

A modified logarithmic law is derived to describe the velocity distribution in open-channel flow with an upward seepage. The derived function agrees well with the experimental measurements conducted in a laboratory flume. The roughness function included in the modified logarithmic law is found to be dependent on the boundary Reynolds number and the ratio of the seepage velocity to the shear velocity. With the integration of the modified logarithmic law, the bed-shear stress subjected to upward seepage can be computed based on the depth-averaged velocity, water depth, boundary roughness, and seepage velocity. Using this method, variations of the bed-shear stress over the seepage zone are studied. The results show that the bed-shear stress decreases rapidly from the leading section of the seepage zone and then increases gradually toward the downstream end of the seepage zone.

Based on the continuity of flow and sediment, an analysis on scouring in long contractions is conducted with a formula obtained from measured sediment discharge in long contractions and a formula for the flow resistance related to local scour. The analytical solution shows that the equilibrium scour depth in the contracted section depends only on the ratio of the uncontracted to the contracted channel width, and the approach flow depth. The equation is applicable to both live-bed and clear-water scouring conditions. Previous predictive equations by Straub or Gill can also arrive at the same conclusion, but only for some special cases. A comparison between the measured scour data and various analytical models confirms that the simplified equation is sufficiently accurate for most practical purposes.

A semi-empirical analysis on the time-averaged equilibrium live-bed scour at an abutment, placed perpendicular to the flow direction, is presented. The approach is based on the flow and sediment mass continuity equations, scour geometry, flow resistance, and a generalized bed-load transport relation in an alluvial channel. The proposed equation can be reduced to the equilibrium clear-water scour equation proposed earlier by the first writer (Lim). A total of 111 live-bed scour data from various sources were analyzed and compared with the developed equation. The agreement between the computed and measured data is acceptable.

The drag coefficient for dispersed particles settling in a fluid is related to the particle Reynolds number with the same function as that used for a single particle falling in a clear fluid, after taking into account the effects of concentration both on the effective density of particles and on the viscosity of fluid-sediment mixture. A new method is developed for evaluating the settling velocity of natural sediment particles dispersed in a fluid; it agrees well with the available experimental data. The differences among various empirical expressions are illustrated using the present relationship.

A new and simplified formula for predicting the settling velocity of natural sediment particles is developed. The formula proposes an explicit relationship between the particle Reynolds number and a dimensionless particle parameter. It is applicable to a wide range of Reynolds numbers from the Stokes flow to the turbulent regime. The proposed formula has the highest degree of prediction accuracy when compared with other published formulas. It also agrees well with the widely used diagrams and tables proposed by the U.S. Inter-Agency Committee in 1957.

Turbulence generated by a vertically oscillating grid in a water tank was investigated using the digital particle image velocimetry technique. The statistical turbulence characteristics computed based on the experimental data agree well with the two main findings reported in the literature: (1) The turbulence decays following the power law; and (2) the integral length scale increases linearly with the distance from the grid. In addition, the flow structure near the grid was observed in detail with the advantage of the planar measurements. It was found that the velocity fluctuations in the region near the grid vary depending on the grid geometry. The fluctuations immediately over the bar position are significantly different from those over the grid openings. Turbulence with the highest intensity occurs above the intersection of the square bars that constitute the grid. The results imply that shear flow clearly exists near the grid and homogeneity of the turbulence can only be achieved at a distance from the grid greater than about three mesh sizes. In addition, it was found that at least 400 vector maps should be taken to ensure the accuracy of the measured velocity fluctuations.