Solution of Specific Energy and Specific Force Equations

Alternate depths and sequent depths are calculated by solving the specific energy and specific force equations, respectively. The specific energy and specific force equations result in fifth degree polynomial equations for trapezoidal cross sections. A methodology is developed for simultaneous determination of the alternate depths and/or sequent depths. The methodology uses solutions of quadratic and third-order polynomial equations to identify the two flow depths.     

Forced Hydraulic Jumps below Abrupt Expansions

A new approach for studying forced hydraulic jumps below abrupt symmetric and asymmetric expansions is introduced. Intensive experiments for two expansion ratios, one asymmetry ratio, and different solid sill heights and locations were conducted specifically for the spatial jump case as a critical design case. A new parameter correlating sill height and sill location is presented and found to play an important role in the basin’s design, scour, and flow patterns. Using this parameter gives distinguished relationships for predicting sequent depth, energy dissipation, and basin length for the preliminary or operating design in expected or unexpected flow conditions. Empirical equations and experimental regression curves are introduced to predict the sequent depth and the basin length at any sill height and sill location. Scour and flow patterns are intensively studied experimentally and qualitatively described for the symmetric and asymmetric basins with or without sills for the spatial jump. A numerical example is presented to illustrate the applicability of the study in preliminary and operating designs.     

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.     

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.     

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.     

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.     

Hydraulic Jumps in an Inclined Rectangular Chute Contraction

Hydraulic jumps in an inclined rectangular chute contraction were studied in this paper. Theoretical equations for the sequent-area and sequent-depth ratios for hydraulic jumps in the contraction were developed considering the effects of contracting width and sloping bottom. The equation of the sequent-area ratio (instead of the sequent-depth ratio) has the same form as the traditional Belanger equation for the sequent-depth ratio for hydraulic jumps in a horizontal rectangular constant-width channel (HRCW channel). A modified approach Froude number M, including the effects of contracting width and sloping bottom, was introduced to replace the approach Froude number F1 in analyzing hydraulic jumps in the contraction. Laboratory experiments of hydraulic jumps in inclined contractions were also conducted to verify the theoretical sequent-area ratio and also to develop the empirical equations of the location of the start of the jump, flow depth of the start of the jump, length of the jump, and energy loss of the jump. The calculation procedure for the application of the presented equations is also provided in this paper.     

Local Scour Downstream of Positive-Step Stilling Basins

This note focuses on the temporal and spatial evolution of local scour below low-head spillways. Steady-flow experiments were carried out in a 1-m wide and 20-m long rectangular straight channel. The jet was generated by an ogee-crest spillway followed by a positive-step stilling basin. Nearly uniform sandy beds were generally tested, but additional tests were also performed with a special bed of lead spheres. To circumvent the combination of local and general scour phenomena, tailwater depths were set such that tailwater flow intensities were below the threshold of sediment motion. As a consequence, for each run a submerged hydraulic jump formed. Tests were of long durations (of order of days) mainly to achieve conditions of quasi-equilibrium. Based on the data collected, literature approaches are discussed. Then, empirical models are proposed to estimate: (1) the maximum scour depth at the quasi-equilibrium stage and its horizontal distance from edge of stilling basin; (2) the time variation of scour depth; and (3) the axial scour profiles. The proposed equations agree well with experimental data. Findings also highlight that affinity rather than similarity may be the typical property of low-angle eroding jets.     

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.     

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.     

Errors in Surface Irrigation Evaluation from Incorrect Model Assumptions

Some two-dozen methods have been proposed in the literature for estimating an infiltration function from field measurements. These methods vary in their data requirements and analytical rigor, however most assume some functional form of the infiltration equations. In this paper, if is shown that the form of infiltration and roughness equations can cause errors in the estimation of actual conditions. For example, assumptions regarding the influence of wetted perimeter on furrow infiltration can result in inappropriate infiltration equations and parameters. Also, the Manning n has been shown to vary with time during an irrigation event as the soil is smoothed by the flowing water. Thus estimates of Manning n based on the advance curve may vary substantially from those based on measured water depths. Inappropriate selection of equations or parameter values for infiltration or roughness can lead to unrealistic parameter values for the other. The estimated parameters from evaluation of a measured irrigation event usually give reasonable estimates of actual performance. However, extrapolation to future irrigation events, particularly with a different application depth or flow rate, can lead to inappropriate recommendations.     

New Solution of Classical Hydraulic Jump

This technical note, applying dimensional analysis and incomplete self-similarity, proposes a new functional relationship for the sequent depth ratio for hydraulic jumps over both smooth and rough horizontal beds. For the smooth bed condition, experimental measurements in the literature were used to calibrate the new relationship. For the rough bed condition the data of a previous investigation were used with new measurements carried out in a rectangular horizontal flume having a gravel bed. Finally, a generalized solution of the sequent depth ratio is proposed.     

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.     

Finite Volume Model for Two-Dimensional Shallow Water Flows on Unstructured Grids

A numerical model based upon a second-order upwind finite volume method on unstructured triangular grids is developed for solving shallow water equations. The HLL approximate Riemann solver is used for the computation of inviscid flux functions, which makes it possible to handle discontinuous solutions. A multidimensional slope-limiting technique is employed to achieve second-order spatial accuracy and to prevent spurious oscillations. To alleviate the problems associated with numerical instabilities due to small water depths near a wet/dry boundary, the friction source terms are treated in a fully implicit way. A third-order total variation diminishing Runge–Kutta method is used for the time integration of semidiscrete equations. The developed numerical model has been applied to several test cases as well as to real flows. Numerical tests prove the robustness and accuracy of the model.     

One-Dimensional Model for Transient Flows Involving Bed-Load Sediment Transport and Changes in Flow Regimes

This study presents a novel, simple, but rather accurate approximation of the eigenvalues of the system formed by the Saint-Venant–Exner equations, based on the comparison between eigenvalues for the complete system and eigenvalues for the water phase only. Moreover, a strategy is proposed to compute efficiently the intercell fluxes by properly adapting a Harten, Lax, and van Leer scheme for each equation. Two examples of transient transcritical flows are developed: the erosive migration of a knickpoint induced by an increase in the bed slope, and the evolution of a hydraulic jump over a mobile bed.     

Backward Flow Velocities of Submerged Hydraulic Jumps

This paper describes an experimental investigation of submerged hydraulic jumps forming downstream of overflow structures. Submergence happens when the hydrological tailwater depth in a channel exceeds the jump's subcritical sequent depth. It is a common occurrence, particularly with low overflow structures. The jump may produce a vortex having significant countercurrent free-surface velocities. This phenomenon is held responsible for frequent personal injury accidents of unwary recreationists, hence its appellation of “drowning machine.” Experimental results, supported by analytical reasoning, are presented that quantify these dangerously high velocities for all hydraulic situations.     

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.     

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.     

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*.     

Energy Dissipation on Submerged Block Ramps

Block ramps represent structures that produce high energy dissipation and have a unique characteristic of preserving the ecological balance in a river restoration project. The energy dissipation of a block ramp changes with the tailwater level. In this technical note the relative energy dissipation in submerged flow conditions has been investigated. The experiments were conducted at the Hydraulic Laboratory of the University of Pisa, Pisa, Italy on ramps characterized by different block materials, submergence conditions, and ramp slopes. The study shows that the relative energy loss, varying the hydraulic jump location on the ramp, is essentially a function of the scale roughness, the ramp slope, the ratio between the critical water depth, and the ramp height and the ratio between the ramp length and the reduced length in submerged conditions. The differences in energy dissipation for a submerged hydraulic jump in different bed conditions are also investigated.