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Failure of bridges due to local scour has motivated many investigators to explore the causes of scouring and to predict the maximum scour depth at abutments. In this paper, a detailed review of the up-to-date work on scour at abutments is presented including all possible aspects, such as flow field, scouring process, parameters affecting scour depth, time-variation of scour and scour depth estimation formulae.

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Department of Civil & Infrastructure Engineering

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Sadhana - Academy Proceedings in Engineering Sciences

Local scour at abutments: A review

Experimental investigation of the 3D turbulent flow field around a 45° wing-wall abutment, resting on a rough rigid bed, is reported. The experiment was conducted in a laboratory flume using the Acoustic Doppler Velocimeter (ADV). Profiles of time-averaged velocity components, turbulent intensity components, turbulent kinetic energy and Reynolds stresses at different azimuthal planes are presented. Vector plots of flow fields at azimuthal and horizontal planes show the presence of a primary vortex associated with the downflow in the upstream side of the abutment and a wake vortex on the downstream side. The shear stresses acting on the bed around the abutment are estimated from the Reynolds stresses and velocity gradients. The data presented in this study would be useful to researchers for future development and comparison of theoretical models of flow fields around bridge abutments.

Velocity and turbulence at a wing-wall abutment

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. © ASCE/SEPTEMBER 2004.

Journal of Engineering Mechanics

Sediment threshold with upward seepage

This paper presents the findings of an experimental study on the three-dimensional turbulent flow field in the vicinity of a rectangular cross-section cylinder placed at the sidewall of a rectangular open channel. Uniform sand of median diameter 0.52 mm was glued over the channel bottom to make it rough. Detailed measurements of the time-averaged velocity components, turbulent intensity components, turbulent kinetic energy and Reynolds stresses were taken by the acoustic doppler velocimeter (ADV) at different vertical sections around the cylinder. In the upstream, presentation of flow fields through vectors at vertical and horizontal sections show a primary vortex associated with the downflow near the cylinder. On the other hand, in the downstream, the flow field shows a three-dimensional separated flow with vortex shedding. Using the Reynolds stresses and velocity gradients, the bed shear stresses are determined. In front of the cylinder, the velocity and the bed shear stress are found to be approximately 1.6 and 3.3 times of their magnitudes at free flow upstream zone, respectively. The data presented in this paper would be useful to the investigators for the development and verification of mathematical models of flow field in the vicinity of vertical-wall abutments. © 2004 Elsevier Ltd. All rights reserved.

Flow Measurement and Instrumentation

Turbulent flow measurement by the ADV in the vicinity of a rectangular cross-section cylinder placed at a channel sidewall

The paper presents the application of artificial neural network (ANN) to determine the end-depth-ratio (EDR) for a smooth inverted semicircular channel in all flow regimes (subcritical and supercritical). The experimental data were used to train and validate the network. In subcritical flow, the end depth is related to the critical depth, and the value of EDR is found to be 0.705 for a critical depth-diameter ratio up to 0.40, which agrees closely with the value of 0.695 given by Dey [Flow Meas. Instrum. 12 (4) (2001) 253]. On the other hand, in supercritical flow, the empirical relationships for EDR and non-dimensional discharge with the non-dimensional streamwise slope of the channel are established. © 2004 Elsevier Ltd. All rights reserved.

End depth computation in inverted semicircular channels using ANNs

The flow upstream of a free overfall from smooth inverted semicircular channels is theoretically analysed to compute the end-depth ratio (EDR), applying an energy equation based on the Boussinesq assumption. This approach eliminates the need for an experimentally determined pressure coefficient. Experiments were conducted with horizontal channel conditions. The EDR related to the critical depth, which occurs upstream from the end section, is found to be around 0.695 for a critical depth-diameter ratio up to 0.40. A simple method is presented to estimate the discharge from a known end-depth. The theoretical model corresponds closely with the experimental data.

Nordic Hydrology

End-depth in inverted semicircular channels: Experimental and theoretical studies

Theoretical and experimental investigations are presented on the threshold of noncohesive sediment motion under a steady-uniform stream flow on a combined transverse (across the flow direction) and longitudinal (streamwise direction) sloping bed. Theoretical analysis of the equilibrium of a sediment particle lying on a combined transverse and longitudinal sloping bed shows that the critical shear stress ratio (ratio of critical shear stress for a sloping bed to that for a horizontal bed) is a function of the transverse bed slope, longitudinal bed slope, angle-of-repose of sediment particles and lift-drag ratio. As in laboratory flume study, the uniform flow is difficult to establish in a steeply sloping channel and is impossible to obtain in an adversely sloping channel, experiments were conducted in two inverted semicircular ducts (closed-conduit flow) for three sizes of uniform sediments. The critical shear stresses for experimental runs were determined considering side-wall correction. The experimental results correspond closely with the results obtained using the present model. The model can be applicable to both the channel beds and banks.

Journal of Hydraulic Research

Threshold of sediment motion on combined transverse and longitudinal sloping beds [Seuil de mise en mouvement des sédiments sur des lits de pente transversale et longitudinale]

A model is presented to determine the critical shear stress for the incipient motion of bivalve shells on a horizontal sand bed, under a unidirectional flow of water. Hydrodynamic forces on a solitary bivalve shell, resting over a sand bed, are analyzed for the condition of incipient motion including the effect of turbulent fluctuations. Hydrodynamic forces such as drag and lift are taken into consideration. Three types of bivalve shells, namely Coquina Clam, Cross-barred Chione and Ponderous Ark, were tested experimentally for the condition of incipient motion. The shape parameter of bivalve shells is defined appropriately. Experiments were conducted in a flume with a horizontal bed, and the critical shear stresses were computed using Vanoni's side-wall correction. The experimental data are used to calibrate the model making lift coefficient a free parameter. The results obtained using the present model agree satisfactorily with the experimental data.

Incipient motion of bivalve shells on sand beds under flowing water

This paper presents a theoretical and experimental study on free overfall in a smooth inverted semicircular channel. Applying the momentum equation based on the Boussinesq approximation, the flow upstream of a free overfall is theoretically analyzed to calculate the end-depth-ratio (EDR). This approach eliminates the need of an empirical pressure coefficient. In subcritical flows, the EDR related to the critical depth is found to be 0.705 for critical depth-diameter ratio up to 0.42. In supercritical flows, the Manning equation is used to express the end-depth as a function of the streamwise slope of the channel. Methods to estimate discharge from the end-depth in subcritical and supercritical flows are presented. The discharge is also related to the end-depth and a characteristic parameter of the channel. The upstream free surface profiles are computed using the streamline curvature at free surface. The computed and experimental results are in satisfactory agreement in subcritical flow and have a slight discrepancy in supercritical flow.

Journal of Hydraulic Engineering

Free overfall in inverted semicircular channels

Free overfall in U-shaped channels is theoretically analyzed, applying a momentum equation based on the Boussinesq approximation, to obtain the equation of end depth. The end-depth ratio is 0.75 up to normalized critical depth 0.50; and then it decreases gradually with an increase in normalized critical depth. A method to estimate discharge from the end depth is presented. The upstream flow profile of a free overfall is computed using the streamline curvature at the free surface. Experiments were conducted in two U-shaped channels to verify the model. The results obtained using the model agree satisfactorily with the experimental data.

Overfall in U-shaped channels

The results of an experimental investigation of the time variation of scour hole and the flow characteristics of the quasi-equilibrium state of scour of a cohesive bed downstream of an apron due to a submerged horizontal jet issuing from a sluice opening are presented. Experiments were carried out with natural cohesive sediment for various sluice openings, jet velocities, and lengths of apron. Attempts are made to explain the similarity existing either in the process of scour or in the scour profiles that the scour holes follow downstream of an apron. The scour profiles at different times follow a particular geometrical similarity and can be expressed by a polynomial using relevant parameters. The characteristics parameters affecting the time variation of scour depth are identified based on the physical reasoning and dimensional analysis. An equation for time variation of maximum scour depth is obtained empirically. The diffusion characteristics of the submerged jet, growth of boundary layer thickness, velocity distribution within the boundary layer, and shear stress at the quasi-equilibrium state of scour are also investigated. The expression of shear stress is obtained from the solution of the von Kármán momentum integral equation.

Hydraulics of submerged jet subject to change in cohesive bed geometry

Journal of Irrigation and Drainage Engineering

Closure to “edr in circular channels”

A free overfall at the end of an open channel offers a simple means of measuring flow discharge. In this paper, two methods are presented for the computation of end-depth and discharge of a free overfall from smooth circular channels with flat base. Firstly, applying the momentum equation based on the Boussinesq approximation, the flow upstream of a free overfall is theoretically analyzed to calculate the end-depth-ratio (EDR). This approach eliminates the need of an experimentally determined pressure coefficient. In subcritical flows, the EDR is related to the critical-depth that occurs far upstream. In supercritical flows, the Manning equation is used to express the end-depth as a function of streamwise slope of the channel. Methods to estimate discharge from the end-depth in subcritical and supercritical flows are presented. The upstream flow profiles of a free overfall are computed using the streamline curvature at free surface. Secondly, an alternate method for analyzing free overfall from circular channels with flat base is also presented, where a free overfall in a circular channel with flat base is simulated by the flow over a sharp-crested weir to calculate the EDR. The comparisons of the computed results with the experimental data are satisfactory for subcritical flow and acceptable for supercritical flow. © 2002 Elsevier Science Ltd. All rights reserved.

Free overall in circular channels with flat base: A method of open channel flow measurement

In this paper, two methods are presented to analyse the free overfall in Δ-shaped (equilateral triangle-shaped) channels. First, the flow upstream of a free overfall from smooth horizontal or mildly sloping Δ-shaped channels is analysed theoretically to determine the end-depth-ratio (EDR), applying the momentum equation based on the Boussinesq approximation. Second, an alternate method for analysing free overfall in Δ-shaped channels is also presented where the flow over a free overfall in a Δ-shaped channel is simulated by that over a sharp-crested weir to calculate the EDR. The method of estimation of discharge from the known end depth is also presented for both the methods. These approaches eliminate the need of an experimentally determined pressure coefficient. Experiments are conducted to verify the results obtained from the present methods. Comparisons of the computed and experimental results are satisfactory.

Hydraulics of free overfall in Δ-shaped channels

An experimental study on incipient motion of gravel and coal beds under unidirectional steady-uniform flow is presented. Experiments were carried out in a flume with various sizes of gravel and coal samples. The critical bed shear stresses for the experimental runs determined using side-wall correction show considerable disagreement with the standard curves. The characteristic parameters affecting the incipient motion of particles in rough-turbulent regime, identified based on physical reasoning and dimensional analysis, are the Shields parameter, particle Froude number, non-dimensional particle diameter and non-dimensional flow depth. Equations of critical bed shear stress for the initial movement of gravel and coal beds were obtained using experimental data. The method of application of critical bed shear stress equations is also mentioned.

Incipient motion of gravel and coal beds

The secondary motion of incompressible fluid for a fully developed turbulent flow in horizontally laid curved pipes is theoretically analysed with the aid of the turbulent boundary-layer approach, integrating the Navier-Stokes equations and satisfying the continuity equation, having mean-time-average parameters. Outside the boundary layer that is developed due to the secondary flow, the inviscid motion of fluid is treated using the Euler and continuity equations. Inside the thin boundary layer, the Pohlhausen method is applied considering a one-seventh power distribution of velocity for solving the integrals of the Navier-Stokes equations. The expression of shear given by Blasius for the turbulent flow in pipes is incorporated into the integrals of the Navier-Stokes equations as an apparent shear to derive an implicit equation that is numerically solved to determine the boundary-layer thickness and subsequently the wall shear stress. The computational results of the boundary-layer thickness and the wall shear stresses are presented in non-dimensional (graphical) form for different Reynolds numbers and curvature ratios. The model is applicable over a wide range of Dean numbers (D) 57 ≤ D ≤ 5 × 105.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Secondary boundary layer and wall shear for fully developed flow in curved pipes

Seventy years ago, Hunter Rouse observed that a free overfall in rectangular channels could be used as a simple flow measuring device that requires no calibration. Due to its practical importance, since then many investigators studied the free overfall in various channels. Although the laboratory experimental results are rather simple, the mathematical treatments of free overfall problems have been found to be deceptively complicated. This article presents a comprehensive state-of-the-art review of the important laboratory experimental and theoretical investigations on free overfall in various channels highlighting the hydraulic background and the mathematical treatment of the problem. In this review article, 120 references are included. © 2002 Elsevier Science Ltd.All rights reserved.

Free overall in open channels: State-of-the-art review

A free overfall at the end of an open channel provides a simple means for measuring flow discharge. The paper presents a simplified approach for the computation of end-depth of a free overfall in horizontal or mildly sloping inverted semicircular channels. Using a known end-depth the discharge can be estimated. The flow over a free overfall in an inverted semicircular channel is simulated by that over a sharp-crested weir to calculate the end-depth-ratio (EDR). The mathematical model is calibrated by the experimental data, making the coefficient of velocity a free parameter. The EDR, related to the critical depth, is around 0.705 for a critical depth-diameter ratio up to 0.42. The computed results agree satisfactorily with the experimental data. © 2001 Elsevier Science Ltd. All rights reserved.

Flow measurement by the end-depth method in inverted semicircular channels

Generalized solution for the location of critical flow section in collector channels is presented. Based on the concept of the singularity, the dynamic equation of spatially varied flow (SVF) is solved using the flow resistance equations of von Karman (for rough regime) and Jain (for transitional and smooth regimes). The advantage of using Jain's equations is that they provide the explicit forms of the Colebrook-White and Nikuradse equations. Computational steps for the determination of critical flow section in a collector channel, being dependent on channel geometry, roughness, longitudinal bed slope and inflow discharge, are given for different channel shapes.