Department of Civil & Infrastructure Engineering

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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.

Journal of Irrigation and Drainage Engineering

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.

World University root node

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.