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A simplified model is presented for the computation of shape and dimensions of the cross section of a self-formed straight threshold channel with noncohesive uniform sediments. The model is based on the equilibrium of the individual sediment particles lying on the channel bed under the threshold condition, due to the hydrodynamic force acting on it. The transverse momentum diffusion, caused by the Reynolds stresses, is assumed as a function of the transverse distance from the center. Channel dimensions predicted by this model are in agreement with those obtained from previous models. However, the model slightly underestimates the experimental data.

Department of Civil & Infrastructure Engineering

Journal of Irrigation and Drainage Engineering

Bank profile of threshold channels: A simplified approach

World University root node

Sediment pick-up on stream-wise sloping beds

This paper presents an experimental investigation on noncohesive sediment pickup under a unidirectional steady-uniform stream flow on streamwise steeply sloping (down slope and adverse) sedimentary beds. The characteristic parameters affecting the sediment pickup, identified based on the physical reasoning and dimensional analysis of the sediment particle movement under stream flow, are the transport-stage parameter, particle parameter, and geometric standard deviation of sediment particles. A large number of experiments (426 runs) were carried out in two long rectangular ducts (closed-conduit flow) with nine types of sediments (six uniform and three nonuniform sediments), having a variation of bed slope from- 15° (adverse slope) to 25° (down slope). In an open channel flow (laboratory flume study), the uniform flow is a difficult, if not impossible, proposition for a steepy sloping channel and is impossible to obtain in an adversely sloping channel. To avoid this problem, the tests were conducted with a closed-conduit flow. Measurements included flow discharge and sediment pickup rate. The bed shear stress for a particular run was computed considering side wall correction. The experimental data were used to determine the equation of sediment pickup function through a regression analysis. The equation is adequate to estimate sediment pickup not only on horizontal and mild slopes but also on steep and adverse slopes.

Sediment pickup on streamwise sloping beds

This paper presents a simplified approach to determine the end depth of a free overfall in horizontal or mildly sloping circular channels. The flow over a free overfall in a circular channel is simulated by that over a sharp-crested weir to calculate the end-depth ratio (EDR). The theoretical model is calibrated by the experimental data, making the coefficient of velocity as a free parameter. The EDR, related to the critical depth, varies almost linearly from 0.72 to 0.74 for a critical depth-diameter ratio up to 0.86. The method of estimation of discharge from the known end depth is also presented.

EDR in circular channels

In open channels, critical depth is an important aspect because it divides the two important flow regimes: sub-critical and super-critical flows. Critical depth is usually computed solving an implicit equation by tedious trial and error method. In this paper, a generalised numerical solution in the Chebyshev form to compute the critical depth is presented. The necessary parameters of some particular cases, namely rectangular, triangular, trapezoidal, circular and exponential channels, are furnished. The use of the Chebyshev approximation has the advantage requiring less iteration than any other iterative methods, such as Muller and Newton- Raphson approximations. © 2000 Taylor & Francis Group, LLC.

ISH Journal of Hydraulic Engineering

Chebyshev solution for critical depth in open channels

The paper presents a theoretical model to compute the end depth of a free overfall in steeply sloping rough rectangular channels. A momentum equation based on the Boussinesq approximation is applied to obtain the equation of the end depth. The effect of streamline curvature at the free surface is utilized to develop the differential equation for the flow profile upstream of the free overfall of a wide rectangular channel. As direct solutions for the end depth and flow profile cannot be obtained owing to implicit forms of the developed equations, an auto-recursive search scheme is evolved to solve these equations simultaneously. A method for estimation of discharge from the known end depth and Nikuradse equivalent sand roughness is also presented. Results from the present model correspond satisfactorily with experimental observations except for some higher roughnesses.

Fulltext

Sadhana - Academy Proceedings in Engineering Sciences

End depth in steeply sloping rough rectangular channels

This paper presents an experimental investigation of the noncohesive sediment threshold under a unidirectional steady-uniform stream flow on streamwise sloping (down slope and adverse) sedimentary beds. The theoretical analysis of the equilibrium of a solitary sediment particle lying on a sloping sedimentary bed shows that the threshold shear stress ratio, that is, the ratio of threshold shear stress for sloping bed to that for plane bed, is a function of the streamwise bed slope and angle of repose of sediment particles. Experiments carried out in two long rectangular ducts (closed-conduit flow) with nine types of sediments show that the equation derived from theoretical analysis corresponds closely to the experimental data. In an open channel flow (laboratory flume study), uniform flow is a difficult, if not impossible, proposition for a steeply sloping channel and is impossible to obtain in an adversely sloping channel. To avoid this problem, the tests were conducted with a closed-conduit flow. This paper presents an experimental investigation of the noncohesive sediment threshold under a unidirectional steady-uniform stream flow on streamwise sloping (down slope and adverse) sedimentary beds. The theoretical analysis of the equilibrium of a solitary sediment particle lying on a sloping sedimentary bed shows that the threshold shear stress ratio, that is, the ratio of threshold shear stress for sloping bed to that for plane bed, is a function of the streamwise bed slope and angle of repose of sediment particles. Experiments carried out in two long rectangular ducts (closed-conduit flow) with nine types of sediments show that the equation derived from theoretical analysis corresponds closely to the experimental data. In an open channel flow (laboratory flume study), uniform flow is a difficult, if not impossible, proposition for a steeply sloping channel and is impossible to obtain in an adversely sloping channel. To avoid this problem, the tests were conducted with a closed-conduit flow.

Influence of streamwise bed slope on sediment threshold under stream flow

The standard step method is commonly used to compute free surface profiles in gradually varied flow (GVF) through open channels. In this study, generalized numerical solutions in the Chebyshev form are presented for the standard step method to compute the free surface profiles in GVF without using look-up tables, interpolation procedures, or simplified assumptions concerning the cross-section geometry. The solutions are obtained using the flow resistance equations of Manning, Chezy, and Colebrook-White. The necessary parameters of some particular cases, namely rectangular, triangular, trapezoidal, circular, and exponential channels, are furnished. The use of the Chebyshev approximation has the advantage of requiring less iteration than the Newton-Raphson approximation. The standard step method is commonly used to compute free surface profiles in gradually varied flow (GVF) through open channels. In this study, generalized numerical solutions in the Chebyshev form are presented for the standard step method to compute the free surface profiles in GVF without using look-up tables, interpolation procedures, or simplified assumptions concerning the cross-section geometry. The solutions are obtained using the flow resistance equations of Manning, Chezy, and Colebrook-White. The necessary parameters of some particular cases, namely rectangular, triangular, trapezoidal, circular, and exponential channels, are furnished. The use of the Chebyshev approximation has the advantage of requiring less iteration than the Newton-Raphson approximation.

Chebyshev solution as aid in computing GVF by standard step method

A model is presented to compute the threshold shear stress for non-cohesive sediment (uniform and non-uniform) motion on a horizontal sedimentary bed, under a unidirectional steady-uniform stream flow. On the basis of hydrodynamical and micro-mechanical considerations, forces on a solitary sediment particle, resting on a compact bed formed by equal sized sediment particles, under slip-spinning condition were analyzed with the aid of a computational scheme. The experimental data of sediment threshold reported by various investigators were used to calibrate the model making the lift coefficient as a free parameter. An excellent agreement between the present model and the experimental data of uniform sediments was possible owing to the calibration of the model. The computational results are presented in the graphical form where the variation of normalized threshold shear stress with particle Reynolds number for various angles of repose is shown. It is revealed that the threshold shear stress increases with an increase in angle of repose of bed sediment. The results obtained using the present model are compared with the curves proposed by different investigators and also have an agreement with the experimental data of non-uniform sediments. Diagram is presented for the direct estimation of threshold shear stress from the information on sediment size and angle of repose. A model is presented to compute the threshold shear stress for non-cohesive sediment (uniform and non-uniform) motion on a horizontal sedimentary bed, under a unidirectional steady-uniform stream flow. On the basis of hydrodynamical and micro-mechanical considerations, forces on a solitary sediment particle, resting on a compact bed formed by equal sized sediment particles, under slip-spinning condition were analyzed with the aid of a computational scheme. The experimental data of sediment threshold reported by various investigators were used to calibrate the model making the lift coefficient as a free parameter. An excellent agreement between the present model and the experimental data of uniform sediments was possible owing to the calibration of the model. The computational results are presented in the graphical form where the variation of normalized threshold shear stress with particle Reynolds number for various angles of repose is shown. It is revealed that the threshold shear stress increases with an increase in angle of repose of bed sediment. The results obtained using the present model are compared with the curves proposed by different investigators and also have an agreement with the experimental data of non-uniform sediments. Diagram is presented for the direct estimation of threshold shear stress from the information on sediment size and angle of repose.

Applied Mathematical Modelling

Sediment threshold

A model is presented to determine the threshold shear stress for noncohesive sediment (uniform and nonuniform) motion on horizontal and stream-wise sloping sedimentary beds, under a unidirectional steady-uniform streamflow. Hydrodynamic and particle-mechanic forces on a solitary sediment particle, resting over a sedimentary bed under the slip-spinning condition, are analyzed including the effect of turbulent fluctuations. Hydrodynamic forces such as drag, shear lift, and Magnus lift are taken into consideration. The drag coefficient is determined using an empirical formula. The inclusion of Magnus lift is significant because spherical particles spin just before dislodging downstream from their original position due to the differential hydrodynamic force along the vertical. The experimental data of sediment threshold are used to calibrate the model making the lift coefficient as a free parameter. The dependency of normalized threshold shear stress on particle parameter for various angles of repose and stream-wise bed slopes is presented graphically. The results obtained using the present model are compared with the curves proposed by different investigators and the experimental data of sediment (uniform and nonuniform) threshold for horizontal and stream-wise sloping beds.

Journal of Engineering Mechanics

Sediment threshold under stream flow on horizontal and sloping beds

A theoretical model is presented to estimate the time-variation of scour depth in an evolving scour hole at circular piers under clear-water and live-bed scour conditions with uniform and non-uniform sediments. The procedure developed for computing the time-variation of scour depth was based on the concept of the conservation of the mass of sediment, considering the horseshoe vortex system to be the main agent of scouring and assuming a layer-by-layer scouring process. The present model corresponds closely with the experimental data of time-variation of scour depth under clear-water and live-bed scour conditions with uniform and non-uniform sediments reported by various investigators. Computation is extended up to the equilibrium scour condition and the corresponding design formula for estimation of maximum scour depth is also proposed for given flow, sediment and pier conditions. The computed values of maximum scour depth showed adequate and safe estimation for bridge pier foundations. A theoretical model is presented to estimate the time-variation of scour depth in an evolving scour hole at circular piers under clear-water and live-bed scour conditions with uniform and non-uniform sediments. The procedure developed for computing the time-variation of scour depth was based on the concept of the conservation of the mass of sediment, considering the horseshoe vortex system to be the main agent of scouring and assuming a layer-by-layer scouring process. The present model corresponds closely with the experimental data of time-variation of scour depth under clear-water and live-bed scour conditions with uniform and non-uniform sediments reported by various investigators. Computation is extended up to the equilibrium scour condition and the corresponding design formula for estimation of maximum scour depth is also proposed for given flow, sediment and pier conditions. The computed values of maximum scour depth showed adequate and safe estimation for bridge pier foundations.

Proceedings of the Institution of Civil Engineers: Water, Maritime and Energy

Time-variation of scour in the vicinity of circular piers

The generalized equations for geometric elements of commonly used artificial irrigation channels are presented. The use of the proposed equations eliminates the need of separate computations for different channel sections. © 1998 Taylor & Francis Group, LLC.

Generalized geometric elements of artificial channels: A note

This paper presents a theoretical model to determine the end depth of a free overfall in horizontally laid rough rectangular channels. A momentum equation based on the Boussinesq approximation was applied to obtain the equation for the end depth. The effect of streamline curvature at the free surface was utilized to develop the differential equation for the flow profile upstream of the free overfall. As direct solutions for the end depth and flow profile cannot be obtained due to implicit forms of the developed equations, an auto-recursive search scheme was evolved to solve these equations simultaneously. The method for estimation of discharge from the known end depth and Nikuradse equivalent sand roughness is also presented. The present model corresponds closely with the experimental observations.

Proceedings of the Institution of Civil Engineers: Water and Maritime Engineering

Free overfall in rough rectangular channels: A computational approach

Solutions for choke-free flow in a trapezoidal channel, with rise in bed elevation that may occur with partial filling up of the channel bottom, are discussed. In this analysis, the side slopes of the channel are assumed to be the same before and after the transition. Considering smooth and gradual transition zones, equations of energy and continuity are solved for subcritical and supercritical upstream flow conditions to determine the maximum limiting rise in bed elevation. The ranges of the upstream flow depths are also obtained for the choke-free condition, using the continuity equation of the upstream flow.

Choke-free flow in trapezoidal channels with change in bed elevation

Physical solutions for the flow through circular channels due to increase in bed elevations are obtained, satisfying the condition of choke-free flow. In this analysis, three different cases of channel transitions, commonly found in practice, are considered. They are (1) when the channel bed is raised with the formation of a flat base; (2) when the channel bed is raised due to change in position of the centerline; and (3) when the channel bed is raised due to decrease in channel diameter. Considering a smooth and gradual transition zone, the equations of energy and continuity are solved for the subcritical approaching flow regime to determine the maximum limiting rise in bed elevation. Using the continuity equation of the upstream flow, the ranges of the upstream flow depths are also obtained for the choke-free condition.

Choke-free flow in circular channels with increase in bed elevations

The flow upstream of a free overfall from smooth circular channels is theoretically analyzed to calculate the end-depth ratio (EDR), applying the momentum equation based on the Boussinesq approximation. The present approach eliminates the need for an empirical pressure coefficient. In subcritical approaching flows, the EDR is related to the critical depth, which occurs upstream of the end section, and the value of EDR is found to be around 0.75 for a critical depth-diameter ratio up to 0.82. On the other hand, in supercritical approaching flows, the end depth is expressed as a function of the streamwise slope of the channel using the Manning formula. Simple methods are presented to estimate the discharge from the end depth in subcritical and supercritical approaching flows. A relationship of discharge to the end depth and the channel characteristics parameter is also proposed. Streamline curvature at the free surface is used to compute the flow profiles upstream of a free overfall. This paper also presents a theoretical model to analyze the free overfall from horizontally laid rough circular channels with the aid of an autorecursive search scheme. The flow upstream of a free overfall from smooth circular channels is theoretically analyzed to calculate the end-depth ratio (EDR), applying the momentum equation based on the Boussinesq approximation. The present approach eliminates the need for an empirical pressure coefficient. In subcritical approaching flows, the EDR is related to the critical depth, which occurs upstream of the end section, and the value of EDR is found to be around 0.75 for a critical depth-diameter ratio up to 0.82. On the other hand, in supercritical approaching flows, the end depth is expressed as a function of the streamwise slope of the channel using the Manning formula. Simple methods are presented to estimate the discharge from the end depth in subcritical and supercritical approaching flows. A relationship of discharge to the end depth and the channel characteristics parameter is also proposed. Streamline curvature at the free surface is used to compute the flow profiles upstream of a free overfall. This paper also presents a theoretical model to analyze the free overfall from horizontally laid rough circular channels with the aid of an autorecursive search scheme.

Journal of Hydraulic Engineering

End depth in circular channels

A model is presented that enables us to estimate the temporal variation of sediment pick-up in evolving scour holes around circular cylinders, for uniform sediments, under a clear water scour. The procedure developed for computing the pick-up rate of sediment is based on the fundamental equation of local scour considering the horseshoe vortex system in front of the cylinder to be the main scouring agent. The present model agrees satisfactorily with experimental flume data.

Sediment pick-up for evolving scour near circular cylinders

The three-dimensional quasi-steady vortex flow field around circular piers in a quasi-equilibrium scour hole under a clear water regime has been investigated experimentally. The main characteristic features of the flow, to be modeled, are a relatively large secondary vortex flow within the scour hole and skewed velocity distributions along the circumference of the pier. The flow field has been divided into a number of parts according to the flow characteristics. The components of quasi-steady velocity for different parts have been theoretically expressed in the form of equations satisfying the continuity equation and the requirements of fitting an individual profile of the measured velocity components. Using the suitable functions, the equations of the velocity components derived for different parts have been combined and matched at the junctions of these parts to get a single equation for each velocity component. The measured data have been utilized to determine the coefficients and exponents used in these equations through curve fittings. The proposed flow model, which corresponds closely with the observations, can be utilized to simulate the flow field in prototype. © ASCE.

Clear water scour at circular piers: A model

The three-dimensional flow field around a circular cylinder in a quasi-equilibrium scour hole under a clear water regime has been investigated experimentally. The distribution patterns of tangential, radial and vertical components of velocity upstream and longitudinal component of velocity downstream of a cylinder have been presented in this paper. The flow simulation results show the existence of a helical motion of fluid around the cylinder. © 1995 Indian Academy of Sciences.

Sadhana

Three-dimensional vortex flow field around a circular cylinder in a quasi-equilibrium scour hole

The bed shear stress for a horseshoe vortex in the equilibrium scour hole around the base of a circular cylinder, embedded in a loose bed with a uniform approach flow under a clear water regime, is computed from turbulent boundary layer analysis by integrating the Navier-Stokes equations. Inside the thin boundary layer, which is developed by the reverse flow at the bed due to swirl motion, Pohlhausen's method is applied with an arbitrary 1/7th power distribution of velocity, as was assumed by Weber in solving turbulent boundary layer. Outside the boundary layer, the expressions of 3-D flow field given by Dey are used. Finally the above technique produces two unknowns, namely bed shear stress and boundary layer thickness. The expression of wall shear stress given by Blasius for turbulent flow in pipes as well as over flat plates is assumed for solving boundary layer thickness, and subsequently bed shear stress. The computational results of bed shear stress and boundary layer thickness are presented. © 1994.