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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.

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Sadhana - Academy Proceedings in Engineering Sciences

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.

Flow Measurement and Instrumentation

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.

A note on critical flow section in collector channels

Journal of Irrigation and Drainage Engineering

Influence of streamwise bed slope on sediment threshold under stream flow

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.

Bank profile of threshold channels: A simplified approach

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.