Sequent Depth Ratio of a B-Jump

A B-jump is defined as the jump having the toe section located on a positively sloping upstream channel and the roller end on a downstream horizontal channel. This jump often occurs in the stilling basins with a horizontal bottom and located downstream of a steep channel. For a B-jump, a completely theoretical approach is not sufficient to solve the momentum equation and to establish the sequent depth ratio. In this paper, by using the laboratory measurements carried out in this investigation, some available empirical relationships useful for estimating the sequent depth ratio are tested. Then, by using the Π theorem of the dimensional analysis and the incomplete self-similarity theory, a generalized functional relationship for estimating the sequent depth ratio for different types of jumps is deduced. The estimate of the coefficient appearing in this relationship is dependent on the particular type of jump. In conclusion, the analysis established that the sequent depth ratio for a B-jump depends on a parameter E accounting for the toe section position, the upstream Froude number F1, and the channel slope.     

Flow Conditions of Undular Hydraulic Jumps in Horizontal Rectangular Channels

The flow conditions of undular jumps for fully developed inflow condition have been investigated systematically. If the inflow Froude number is larger than 1.2, an undular jump has lateral shock waves near the toe of the jump. For a narrow channel, the shock waves cross upstream of the first wave crest, and the flow conditions of undular jumps depend on the aspect ratio and the inflow Froude number. In contrast, for a wide channel the shock waves do not cross upstream of the first wave crest, and the flow conditions of undular jumps are independent of the aspect ratio. The flow conditions of undular jumps are classified by considering the cross position of the lateral shock waves and the inflow Froude number. Also, the hydraulic conditions for the formation of nonbreaking and breaking undular jumps are determined. The effect of the Reynolds number on undular-jump formations is discussed, and changes of the flow conditions with the Reynolds number are described.     

Limiting and Sill-Controlled Adverse-Slope Hydraulic Jump

This note analyzes, from a theoretical and experimental point of view, hydraulic jumps on adverse slopes in rectangular prismatic channels. The analysis is carried out for the classical adverse-slope hydraulic jump and the jump forced by the presence of a sill. Data collected in two series of experiments involving different equipment were added to available results to obtain a general relationship for the sequent depth ratio as a function of the upstream Froude number and adverse-slope angle. The presence of a sill stabilizes the jump.     

Hydraulic Jumps in Sloping Channels: Sequent Depth Ratio

A large variety of hydraulic jumps on horizontal and sloping inverts at the end of an ogee standard weir is investigated. An ogee standard weir was used to create supercritical flow and slopes of 0.0, ?0.025, ?0.05, ?0.075, and ?0.10 were built downstream of the weir. Based on the momentum equation in the horizontal direction, a method to predict the sequent depth ratio is presented. The theory agrees well with the results of the writers and previous investigators. A correlation was developed to predict the minimum Froude number needed to establish jumps on negative slopes. Observations showed that in those cases where the gravity force component in the jump was opposite to the flow direction, the water surface of the surface roller became undular and unstable. The hydraulic jump on an entirely adverse slope was almost impossible to control. The analysis of experimental data showed that the negative slope of the basin reduces the sequent depth ratio, while a positive slope increases the sequent depth ratio.     

Screen-Type Energy Dissipator for Hydraulic Structures

Laboratory experiments have shown that screens or porous baffles with a porosity of about 40% could be used as effective energy dissipators below small hydraulic structures, either in a single wall or a double wall mode. The experiments were carried out for a range of supercritical Froude numbers F1 from about 4 to 13, and the relative energy dissipation was appreciably larger than that produced by the corresponding classical hydraulic jumps. These screens or porous baffles produced free hydraulic jumps, forced hydraulic jumps, and in some cases submerged jumps. The flow leaving these screens was found to be supercritical with a Froude number approximately equal to 1.65 and a tailwater depth equal to 0.28 times the subcritical sequent depth y2* of the classical hydraulic jump with the same F1. To produce a secondary jump downstream of the screens, the tailwater depth needed was found to be about one half of y2*.     

Ski Jump Hydraulics

Ski jumps are a major element of each dam spillway because these are the only structures able to accomplish satisfactory energy dissipation for takeoff velocities in excess of some 20?m/s. This research aims to add to several hydraulic problems with ski jumps that have not yet been systematically solved so far. Based on an experimental campaign, the following problems were addressed: (1) pressure head maximum and pressure distribution along a circular-shaped flip bucket; (2) takeoff characteristics for a certain bucket deflection and a relative bucket curvature including the jet trajectories of both the lower and the upper nappes; (3) impact characteristics in a prismatic tailwater channel with details of shock wave formation and height of recirculation depth; (4) energy dissipation across the ski jump, from the upstream channel to downstream of jet impact; and (5) choking flow conditions by the flip bucket. These results demonstrated the significant effect of the approach Froude number, the relative bucket curvature and the bucket angle. The results allow immediate application to the design of ski jumps in hydraulic engineering.     

Bubbly Two-Phase Flow in Hydraulic Jumps at Large Froude Numbers

A hydraulic jump is a sudden, rapid transition from a supercritical flow to a subcritical flow. At large inflow Froude numbers, the jump is characterized by a significant amount of entrained air. For this paper, the bubbly two-phase flow properties of steady and strong hydraulic jumps were investigated experimentally. The results demonstrate that the strong air entrainment rate and the depth-averaged void-fraction data highlight a rapid deaeration of the jump roller. The results suggest that the hydraulic jumps are effective aerators and that the rate of detrainment is comparatively smaller at the largest Froude numbers.     

Direct Equations for Hydraulic Jump Elements in Rectangular Horizontal Channel

Many problems in the design of stilling basins require a knowledge of various elements of a hydraulic jump with known values such as discharge intensity and the energy loss in the jump. Even for a simple case of a rectangular horizontal channel the solutions of equations involve tedious methods of trial and error. In this paper, direct explicit empirical equations for prejump and postjump depths and specific energies in a rectangular horizontal channel have been developed. These present forms of equations for hydraulic jump elements having very high accuracy and are applicable for a very wide range of values of discharge intensity and head loss without any limitations in comparison to other methods attempted so far.     

Undular Hydraulic Jumps in Circular Conduits

Undular hydraulic jumps in circular conduits are considered with an experimental approach. Based on previous findings in rectangular channels, this research indicates differences in terms of shape effects. All present results depend on the filling ratio of the upstream conduit flow in addition to the upstream Froude number. The results include information on the wave crests and troughs, wave lengths, and generalized axial surface profiles. The wall surface profile is shown to be similar to the axial wave profile, but with smaller wave extrema and a wave shift. The design of conduits containing undular jumps should be avoided because of unstable flow. It is also demonstrated that conduits may choke in the presence of undular jumps, with a previously established choking number relating to a design limit. For flows with choking numbers in excess of 1, choking occurs associated with a transition from the free surface to the pressurized conduit flow.     

Conservative Formulation for Natural Open Channels and Finite-Element Implementation

This investigation considers an approximate formulation of the St. Venant equations for natural channels, in which the fully conservative form is developed by revising the boundary pressure term accounting for the topographic variation in the momentum equation. As such a formulation has the potential to enhance the performance of existing models used in practice, the accuracy implications for this approximate formulation are examined using an error analysis for a simplified case. Further, an energy calculation is performed which illustrates that an earlier formulation actually results in energy gain for some cases. A more general formula for the constant water surface elevation that corrects this is introduced and tested. It is found that the refined formulation presented here is accurate for hydraulic jumps, steep surge waves, and flood wave propagation in natural channels. The shock capturing capability of the approximate formulation is illustrated for both steady- and unsteady-flow situations using the finite-element method, for which this approximate equation formulation adapts naturally. Using the characteristic-dissipative-Galerkin finite-element scheme, good results are obtained for the case of a hydraulic jump in a diverging rectangular channel, with the maximum percent error associated with the approximate formulation determined to be only 0.34%. For the case of dam break wave propagation in a converging and diverging rectangular channel, the model performs similarly well, with the maximum error only 0.0064%. Further, the approximate formulation is used to simulate the flood routing in a natural channel, the Oldman River in southern Alberta. The computational results are in good agreement with the observed data. The arrival time of peak flow is 5?h earlier and the magnitude of peak discharge is only 3.8% lower than the observed value.     

Deflector Ski Jump Hydraulics

Ski jumps are a standard element of dam spillways for an efficient energy dissipation if takeoff velocities are large, and stilling basins cannot be applied. This laboratory study investigates the hydraulic performance of a triangular-shaped, rather than the conventional circular-shaped, bucket placed at the takeoff of ski jumps. The following items were addressed: (1) pressure head maximum and pressure distribution along the triangular-shaped bucket; (2) takeoff characteristics as a function of the bucket deflector angle and the relative bucket height including the lower and the upper jet trajectories; (3) jet impact characteristics in a prismatic tailwater channel including the shock wave formation and the height of recirculation depth below the jet cavity; (4) energy dissipation across the ski jump, from the approach flow channel to downstream of jet impact; and (5) choking flow conditions of the flip bucket. A significant effect of the approach flow Froude number, the relative bucket height, and the deflector angle is found. A comparison with previous results for the circular-shaped bucket geometry indicates a favorable behavior of the novel bucket design.     

Discharge Coefficient for Sharp-Crested Side Weir in Subcritical Flow

To estimate the outflow over a rectangular sharp-crested side weir, the discharge coefficient in the weir equation needs to be known. Although this type of structure has been designed and used extensively by hydraulic engineers, a universally acceptable discharge coefficient does not exist. In this study over 250 laboratory tests were conducted, and the results were analyzed to find the influence of the flow hydraulics and the geometric, channel, and weir shapes on the coefficient. The results show that for subcritical flow the De-Marchi assumption of constant energy is acceptable, and the weir discharge can therefore be used. Furthermore, it was discovered that the De-Marchi coefficient of discharge is a function of the upstream Froude number and the ratios of weir height to upstream depth and weir length to channel width, whereas the channel slope in subcritical flow can be ignored. Hence, an accurate equation for the coefficient of discharge is introduced.     

Simple Flume for Flow Measurement in Sloping Open Channel

First, this paper presents a new flume for measuring flow discharge in sloping channels, originally proposed by Samani and Magallanez for use in a horizontal channel. The flume is obtained by inserting two semicylinders in a rectangular cross section. Then, using dimensional analysis and the self-similarity theory, the stage-discharge relationship of the flume is theoretically deduced. For determining the two coefficients of the power stage-discharge equation, some experimental runs are carried out using flumes characterized by different values of the contraction ratio (ranging from 0.17 to 0.81) and of the flume slope (ranging from 0.5 to 3.5%). Finally, for a given range of the contraction ratio, the relationships for estimating the two coefficients of the stage-discharge equation are obtained.     

Aeration Characteristics of Ski Jump Jets

Scour downstream of ski jumps may be avoided by jet deflection to an area where the energy dissipation is accomplished. The main purpose of this experimental study was the analysis of the jet air entrainment downstream of a ski jump, both for pure water and preaerated approach flow conditions. A systematic variation of the Froude number and the flow depth in the approach flow channel resulted in a range of discharge characteristics, whereas the geometry of the ski jump was maintained for all tests. The air concentration profile was measured at different locations downstream from the ski jump to evaluate the: (1) jet air concentration distribution; (2) location of minimum air concentration along the mixture flow jet and development of the minimum and the cross-sectional average air concentrations; (3) jet trajectories; and (4) process of air entrainment characteristics and jet disintegration. The results demonstrate the significant effect of the approach flow Froude number, the approach flow depth, and of preaeration on jet disintegration.     

Turbulence Structure of Hydraulic Jumps of Low Froude Numbers

Turbulence characteristics of hydraulic jumps with Froude numbers of 2.0, 2.5, and 3.32 are presented. A Micro Acoustic Doppler velocimeter was used to obtain measurements of the velocities, turbulence intensities, Reynolds stresses, and power spectra. The maximum turbulence intensities and Reynolds stress at any section were found to decrease rapidly from the toe of the jump towards downstream within the jump and then gradually level off in the transition region from the end of the jump to the friction dominated open channel flow downstream. The maximum turbulence kinetic energy at each section decreases linearly with the longitudinal distance within the jump and gradually levels off in the transition region. The Reynolds stress and turbulence intensities within the jump show some degree of similarity. The dissipative eddy size was estimated to vary from 0.04 mm within the jump to 0.15 mm at the end of the transition region. The dominant frequency is in the range from 0 to 4 Hz for both horizontal and vertical velocity components.     

Stepped Spillway with Hydraulic Jumps: Application of a Numerical Model to a Scale Model of a Conceptual Prototype

Hydraulic jumps on the steps of a stepped spillway were investigated analytically, physically, and numerically. Using classic hydraulic formulae, a conceptual prototype was designed. A large scale model was adapted and an experimental study was conducted to examine similarity of hydraulic jumps on each step, minimizing hydraulic jump length and maximizing discharge per unit width. A numerical model based on the 2D Reynolds averaged equations, where the free-surface is represented using a refined volume-of-fluid algorithm, the internal obstacles are described by means of the fractional area-volume obstacle representation method, and the turbulence is represented by a specially developed RNG (Renormalization Group) k-ε closure model was used for evaluating velocities and pressures, and for characterizing hydrodynamic forces on the baffles and sills. Preliminary design criteria are proposed for stepped spillways with hydraulic jump formation of simple shapes and adequate relations of critical depth/step height and the application of computational fluid dynamics to such problems is studied.     

Hydraulic Jumps on Rough Beds

This paper presents the results of an experimental investigation on the hydraulic jump on horizontal rough beds. Experiments were carried out to study the effect of bed roughness on both the sequent depth ratio and the roller length. The investigation allowed the writers to positively test the reliability of a new solution of the momentum equation for the sequent depth ratio as a function of the Froude number and the ratio between the roughness height and the upstream supercritical flow depth. The applicability of some empirical relationships for estimating the roller length was also tested.     

Further Results to Time-Dependent Local Scour at Bridge Elements

This research intends to clarify the limitations of a local scour equation recently proposed, based on extended laboratory data collected at VAW, Zurich, Switzerland. The present project is concerned with four items: (1) clarification of the minimum laboratory dimensions required to apply Froude similitude; (2) effect of sloping abutments on scour advance; (3) extension of scour formula to spur dikes; and (4) effect of unsteady flow on scour development. These items were investigated mainly from an experimental point of view based on some 150 laboratory experiments and accounted for by a hydraulic approach. It was found that the basic scour equation mentioned may be applied provided additional limitations are specified. These are discussed in the light of the densimetric particle Froude number, the threshold Froude number, and other important parameters that influence the progress of local scour. The results of this study may be applied to practice, provided the limitations of the computational approach are respected.     

Energy Concept for Predicting Hydraulic Jump Aeration Efficiency

Hydraulic jumps, plunging jets and stepped channels are generally used as energy dissipators and self-aerators. Accordingly, it is expected to find a positive correlation between the aeration efficiency and energy dissipation. For this purpose, hydraulic jump self-aeration efficiency has been investigated with the function of energy dissipation rate per unit width. The hydraulic jump data revealed a positive linear relationship between the aeration efficiency and energy dissipation rate. This new procedure could have practical implications for predicting hydraulic jump aeration efficiency.     

Numerical Simulation of Equal and Opposing Subcritical Flow Junctions

This study presents measured and computational results of a flow pattern at a junction with equal and opposing flows in the upstream channel that collide and turn 90° into the branch channel. The computational results are obtained using a two-dimensional, depth-averaged model with the k-ε turbulent closure scheme. The aim is to predict the recirculation zones that form as the flow turns into the branch channel. The simulated depth and velocity profiles in the upstream main and the downstream branch channels are found to compare well with the measurements made in the physical model for various inlet Froude numbers and width ratios of the main channel to the branch channel. The absolute relative error between the measured and computed contraction coefficient, a measure of the recirculation zone size, is less than 4.7%. The computational model is then used to develop curves for the contraction coefficient for various inlet Froude numbers and ratios of main channel width to the branch channel width for design purposes.