Estimating Discharge From Undular Hydraulic Jumps: Feasibility Assessment Based on Flume Experiments

Free-surface undulation and velocity turbulence in shallow undular hydraulic jumps

Undular hydraulic jumps are characterized by a smooth rise of the free surface, followed by a train of stationary waves. These jumps sometimes occur in natural waterways and rivers. Numerical difficulties are especially distinct when the flow condition is close to the critical value because of the high sensitivity of the near-critical flow field to flow and channel conditions. Furthermore, the free surface has a wavy shape, which may indicate the occurrence of several transitions from supercritical to subcritical states and vice versa (i.e., undular hydraulic jumps). In this study, a flow model is used to predict an undular hydraulic jump in a rectangular open channel. The model is based on the general two-dimensional, Reynolds-averaged, Navier-Stokes flow equations. The resulting set of partial differential equations is solved using the FLOW-3D solver. The results are compared with the experimental data to validate the model. The comparative analysis shows that the proposed model yields good results. Several types of undular hydraulic jumps occurring in different situations are then simulated to prove the potential application of the model.

Numerical simulation of undular hydraulic jump on smooth bed using volume of fluid method

A theoretical and laboratory investigation was made of antidunes and associated stationary waves. The objectives were to determine the factors involved in the formation of antidunes, the characteristics of the stationary waves, and the effects of antidunes and waves on the friction factor and sediment transport capacity of streams. In the potential flow solution for flow over a wavy bed it was hypothesized that the flow shapes the erodible sand bed by scour and deposition to conform to a streamline of the flow configuration for which the energy is a minimum. Under this hypothesis, flow over antidunes is the same as the segment of flow above an intermediate streamline of the fluid motion associated with stationary gravity waves (waves with celerity equal and opposite to the flow velocity) in a fluid of infinite depth. For a velocity V the wave length, L, is given by L = (2*pi*V^2)/g and waves break when their height reaches 0.142 L. Laboratory and field data for two-dimensional stationary waves and antidunes confirmed these relations. For the same velocity, short-crested, three-dimensional waves (rooster tails) have shorter wave lengths than two-dimensional waves. Forty-three experimental runs in laboratory flumes were made for different depths and velocities and bed sands of two different sizes (0.549 mm and 0.233 mm). No general criterion for the formation of antidunes or the occurrence of breaking waves could be formulated because of inadequate knowledge of the complex sediment transport phenomenon. Qualitatively, it was found that for a given sand, the critical Froude number for the occurrence of breaking waves decreased as the depth was increased. Over a certain range of depth and velocity it was found that the flow formed waves and antidunes or was uniform depending on whether or not the flow was disturbed to form an initial wave. Waves that did not break had no measurable effect on the transport capacity or friction factor, but breaking waves increased both of these quantities.

Hydraulic jump may form when a supercritical flow runs over a vegetated bottom and change the downstream flow hydraulics and fluid properties after air‐water mixing and energy dissipation at the flow transition. Unlike the abundant literature on hydraulic jump characteristics on engineered or natural sediment rough beds, few studies of vegetation roughness effects have been reported. New experiments were conducted by creating hydraulic jumps with identical inflow Froude numbers on a smooth bed, fully and partly grated rough beds, and partially vegetated grate‐rough beds covered with artificial plants of different densities. The free‐surface dynamics and air‐water flow properties were investigated with primary focus on the interaction between the flow and flexible vegetation barrier/roughness, in addition to the effects of Froude number and inflow roughness conditions. The characteristic jump and jump roller lengths were found shortened as the jump roller was deflected over the vegetation layer. The flow structure modification was observed with a steepening of the roller free‐surface profile and enhanced surface wave propagation, especially when the flow experienced an invert roughness discontinuity at the toe. An increasing bubble detection frequency was obtained in nonvegetated channels when the bed was roughened, but additional vegetation on the rough invert caused reduction in the bubble frequency and cross‐sectional mean air concentration by inducing roller deflection and air bubble detrainment. The results indicated that the effects of perturbing artificial flexible vegetation with relatively large plant height could be different from uniformly distributed rigid bed roughness elements and complicated by the time‐varying flow‐vegetation interaction.

A hydraulic jump is a complex, rapidly varied flow phenomenon occurring in the transition from supercritical to subcritical flows. Hydraulic jumps are characterized by instationary motions including jump toe oscillations and free-surface fluctuations. A light detection and ranging (LIDAR) instrument was used to continuously record the time-varying free-surface features of fully aerated hydraulic jumps with fully developed inflow conditions and high spatial and temporal resolution. The free-surface features were analyzed with two processing methods considering a freely moving hydraulic jump and a case where each instantaneous hydraulic jump profile was shifted to the mean jump toe position. The results provided detailed dimensionless distributions of the mean profiles, as well as free-surface fluctuations and jump toe oscillations that increased with increasing Froude numbers. The high spatial resolution data provided continuous free-surface time and length scales along the hydraulic jumps. Largest free-surface scales were observed close to the jump toe linked with longitudinal oscillations of the hydraulic jumps, and the dimensionless free-surface scales increased with the Froude number. The present study provides new insights into free-surface turbulent structures, highlighting the strong relationship between jump toe movements and free-surface fluctuations in hydraulic jumps.

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.

Hydraulic jumps have been attracted to many researchers for a long period of time because of its importance in designing hydraulic structures. Stilling basins are one of the possible solutions for hydraulic jump to dissipate kinetic energy to produce safe downstream flow, which causes no bed scour and bank erosion. Generally, the dimension of a hydraulic jump is taken as an indicator of the dimension of the stilling basin or as a design parameter. Lot of study has been made on hydraulic jump, and very limited studies have been reported in literature on the performance of hydraulic jumps formed in rectangular channel with rough walls, that way the paper is a valuable contribution to the subject. The purpose of the research work was to observe the phenomenon of the hydraulic jump,and to develop an understanding of the effect of roughness in an open horizontal channel. Also, to appreciate therelationship between the experimental results and the theoretical analysis based on the application of continuityand momentum principles. A series of experiments were carried out in a rectangular flume which consists of artificially roughenedwalls formed by circular balls with homogeneous and uniform shapes. The hydraulic jump parameters such as, pre jump depth, post jump depthand flow rate were measured for different roughness. During this study, sixty experimental runs were conducted by varying the Froude number, ranging from 2 to 17. It carries out on four sluice gate opening (h1 = 2, 3, 4 and 5 cm) with four different roughness elements of diameters (ε = 6, 8, 10 and 12mm) were introduced throughout the length of channel. The hydraulic jump considered in the present work have been controlled by sill with thirteen different sizes varying between 3 to 15 mm. The analysis of experimental data showed that the rough walls reduce the conjugate depthratiothan those on smooth walls. They have also, this parameter vary linearly with the Froude number and roughness where as other hydraulic jump characteristics are not. The analytical equation for the conjugate depthsratio as given byapplied theoretical the momentum equation, and itisadjusted well with the experimental data.Itis satisfactory for predicting the conjugate depth ratio of hydraulic jump on rough walls.

Undular hydraulic jumps are characterized by a three-dimensional flow field due to shock waves generated near the sidewalls. Basic flow properties have been described in the literature with experimental data, mostly gathered with classical sensors which provide local information only. A continuous description of the 3D flow field is still missing. In this study, instantaneous and three-dimensional velocity data in an undular jump were obtained between the inflow section and second wave crest using a volumetric particle tracking velocimetry method with high sample rates. A special feature of the tracking algorithm is its capability to process images with very high particle density yielding a highly resolved Lagrangian description of the flow. Velocity fields, turbulent kinetic energy, dissipation rates, Reynolds stresses and turbulent scales are presented for a volume of 10 cm width in cross-direction. Some differences were found when compared to data obtained with an intrusive acoustic Doppler velocimeter.

Summary Acid fracturing is a crucial technique for enhancing the productivity of carbonate rock reservoirs. Understanding the flow behavior and dissolution characteristics of acid within fractures is essential for optimizing acid fracturing operations. Previous studies primarily focused on investigating acid flow in acid-fractured fractures through physical experiments. However, the fracture models used often exhibited significant discrepancies from actual fractures, and the influence of fracture roughness on acid flow was seldom addressed. In this paper, we develop a rough fracture model using the “diamond-square” algorithm from random fractal theory, and formulate an acid-rock reaction rate equation based on the H+ mass transfer theory. The flow behavior and dissolution characteristics of acid within fractures are simulated and analyzed utilizing computational fluid dynamics under varying conditions of roughness, fracture width, and acid viscosity. The fluid flow velocity equation within the fracture considering the injection pressure gradient, fracture roughness, fracture width, and acid viscosity is established. The research findings indicate the following: (1) The fractal dimension of the fracture surface increases following acid etching, suggesting that acid treatment enhances the complexity of the fracture surface. (2) Under conditions of constant pressure injection, the average flow velocity at the center of the fracture and the width of the dissolution fracture exhibit an inverse relationship with roughness, a direct relationship with fracture width, and a power series decreasing relationship with acid viscosity. (3) An increase in roughness and fracture width results in greater heterogeneity in the flow velocity distribution within the fracture, whereas the opposite effect is observed with acid viscosity. (4) The characteristics of flow velocity distribution within the fracture significantly influence the dissolution characteristics of the fracture surface; specifically, a more complex flow velocity distribution correlates with more pronounced nonuniform dissolution characteristics of the fracture surface. (5) Among the factors examined, fracture width exerts the most substantial influence on the average flow velocity, while acid viscosity has the most significant impact on the width of the dissolution fracture and the entropy of the velocity distribution.

Uncontrolled overtopping during flood events can endanger embankment dams. Erosion of the downstream slope and scouring of its base caused by the high velocity and energy of the overflow can indeed lead to breach formation until complete failure. In this context and faced with the important number of overtopped embankment dams to be rehabilitated, since the early eighties, researchers have investigated surface protection solutions for downstream slope. Overlays against erosion such as seeded goetextile or cable-tied cellular concrete blocks, are not sufficient. In fact, they can resist only short events with low discharge and velocity. Solution to overcome more severe overflow lies in overlays which dissipate flow energy along the downstream embankment slope. Conventional steps resulting from Roller Compacted Concrete (RCC) techniques fulfill efficiently this challenge. However, flows over steep stepped chutes are quite complex, characterizing by great aeration, high turbulence and confused wavy free surface. Then, most of hydraulic studies of such flows are performed on physical model. Yet, understanding and definition of flow behaviour and accurate approach to estimate energy dissipation are still lacking. General guidelines of hydraulics of aerated flows over stepped macro-roughness chutes and for optimal design of protection overlay remain confusing. To contribute to reduce these uncertainties, experimental study of flow over stepped chutes equipped with macro-roughness elements is performed in a laboratory gated flume for mild ( 1:7H : 1V ) and weak ( 3H : 1V ) chutes. Thus, they are representative of the range of embankment dams and spillways slopes. Three types of stepped macro-roughness overlays are assessed, namely rectangular conventional steps, steps equipped with endsills fixed on their nose over all the flume width and steps equipped with rectangular spaced blocks. Endsills overlays were characterized with different longitudinal distributions whereas blocks overlays consisted in different transverse patterns. Tests were conducted for the three nappe, transition and skimming flow regimes. Results can be extrapolated to 1/5 to 1/15 scaled prototypes using the Froude similarity with negligible scale effects. Flow depth, local air concentration and longitudinal velocities are measured with a double fiber-optical probe. Pressures at macro-roughness faces are taken with piezo-resistive sensors. Sequent depths of the hydraulic jump forced in the stilling basin at the flume base are measured with ultrasound sensors. Thus, this experimental phase of the thesis has allowed: to define flow parameters (regimes, depths, velocity and air concentration distributions, hydrodynamic forces) for tested overlays, to highlight that air-water flow depth is divided into: a rough boundary layer influenced by shear stress and by drag form (macro-turbulence) caused by macro-roughness, a homogeneous aerated layer which represents the main portion of flow involved in energy dissipation mechanism, a free surface layer which must be considered in the side walls design, to stress that energy dissipation is mainly a question of drag losses, to validate indirect method of hydraulic jump for energy dissipation estimation, to estimate relative energy loss for several stepped macro-roughness overlays. Tests finally show that an optimal alternative to dissipate the overflow energy during an overtopping event consists in spaced blocks, with transverse space larger than the width of block and fixed alternately on conventional steps. However, experimental results remain related and limited to their tested domains. Then, in order to provide more general governing equations of aerated flows over macro-roughness stepped chutes, a numerical modeling of two phase flows over conventional stepped flume was performed in collaboration with the Laboratory of Applied Hydrodynamics and Hydraulic Constructions at University of Liege. A quasi-2D numerical model based on the finite volume method was developed. It consists in applying the classical depth-averaged simplified Navier-Stokes equations (viscosity and Coriolis terms neglected) to a 1D incompressible air-water mixture flow over mild and steep slopes with a stepped topography. Self-aeration process is modeled by a transport equation of depth-averaged air concentration whereas turbulent structures are indirectly implemented through the Boussinesq coefficient. This first 1D-approach of semi-theoretical description of aerated flow over steps is tested for a 30o gated stepped flume and a 52° crested spillway laboratory model. This numerical model leads to realistic results regarding mixture depth, mean flow velocity, air concentration and wave amplitudes of the flow free surface. Finally, on the basis of existing protections of embankment dams and previous studies, the present experimental and numerical results contribute to extend the knowledge of high velocity aerated flows over macro-roughness and to provide elements of guidelines to optimize stepped macro-roughness overlays for embankment dams safety.

A hydraulic jump is the rapid transition from a supercritical to subcritical flow. This transition is characterized by large-scale turbulence and energy dissipation. Despite the importance of understanding the hydraulic jump to design hydraulic structures, few studies have aimed on hydraulic jumps in U-shaped channels. In this paper, the 3D pattern of hydraulic jumps in U-shaped channels is studied numerically. The variations of the flow free surface are predicted using the volume of fluid scheme. Also, the flow field turbulence is simulated using the standard $$k-\varepsilon $$ and RNG $$k-\varepsilon $$ turbulence models. According to the numerical modeling results, the standard $$k-\varepsilon $$ turbulence model estimates the flow characteristics with more accuracy. A comparison between the laboratory and numerical results shows that the numerical model simulates the flow field characteristics with good accuracy. For example, in the hydraulic jump model with a relative discharge $$({q=Q/{\sqrt{( {gD^{5}})}}})$$ equal to 0.321 and a Froude number $$({{F}_1})$$ equal to 4.85, the values of MAPE, RMSE and $${R}^{2}$$ are calculated 7.617, 0.022 and 0.989, respectively. Next, 45 numerical models are simulated in different hydraulic conditions and some relationships are provided for calculating the sequent depth $$({{h_2 }/{h_1 }})$$ , hydraulic length $$({{L_\mathrm{j}}/{h_1}})$$ and roller length $$({{L_\mathrm{r} }/{h_1}})$$ ratios by analyzing their results.

Understanding hydraulic jumps in gradually expanding sloping channels is crucial for designing eco-friendly water management systems that minimize environmental impacts. This study seeks to investigate how the expansion ratio and channel slope together influence the characteristics of hydraulic jumps. The experiments were performed on four different channel slopes (00, 20, 40, and 60) and four distinct expansion ratios (B = b1/b2) of 0.35, 0.45, 0.55 and 0.75. The Reynolds number was ranged from 7150 to 27750, while the Froude number varied between 2.5 and 8.5 during the experiments. Novel correlations were developed to predict key hydraulic jump parameters, including depth ratio (d₂/d₁), relative energy loss (EL/E₁) and relative jump length (Lj/d₁), by considering the effects of expansion ratio, channel slope and Froude number. The results indicate that as the channel slope increased from 0° to 6°, the depth ratio and relative energy loss increased by approximately30.53% and 15.37%, respectively, while the relative jump length decreased by 11.11%. Conversely, as the expansion ratio decreased from 1.0 to 0.35, the depth ratio and relative jump length were reduced by approximately 9.63% and 10.42%, respectively; while the relative energy loss was increased by 24.21%.These findings highlight the significant influence of channel geometry on hydraulic jump characteristics, offering valuable insights for optimizing water conveyance structures. The proposed correlations provide a novel and intuitive approach to analyze hydraulic jumps in expanding sloping channels, addressing gaps in existing literature and contributing to sustainable water resource management.

This study investigated vortex-induced vibrations of marine risers in Ocean Thermal Energy Conversion systems. Flume experiments were conducted under two conditions: Condition 1 was with a fixed riser bottom, and Condition 2 was with a fixed bottom on a mooring platform. The cross-flow acceleration of the riser was measured at different current velocities, and corresponding vibration responses were analyzed. Numerical simulations based on the flume experiments were employed to validate the reliability of the simulation method. Results from the flume experiments revealed vortex-induced resonance in Condition 1 when the flow velocity approached the riser’s natural frequency. In Condition 2, similar vibration responses were observed, with maximum acceleration occurring during flow velocity-induced vortex-induced vibrations. However, at higher flow velocities, the acceleration response showed a decrease followed by an increase, indicating the excitation of higher-order modes. The numerical simulations matched the flume experiments in Condition 1. In Condition 2, while the acceleration response and frequency agreed during vortex-induced resonance, discrepancies arose in the flow velocity that caused vortex-induced vibrations compared to the flume experiments. This study demonstrated the accuracy of numerical simulations in reflecting vortex-induced vibrations of risers, providing a foundation for further research on complex riser systems.

Hydraulic jumps are characterized by three-dimensional motions, air entrainment and energy dissipation. While extensive air–water flow research has been conducted for classical hydraulic jumps, the air–water flow properties in hydraulic jumps on slopes remain widely unexplored. New air–water flow experiments were conducted of Type B and D jumps for slopes of 1.25°, 2.5° and 5°. Comparative analyses of the sloped jumps with classical hydraulic jumps for the same Froude and Reynolds numbers showed that hydraulic jumps on slopes were more stable. Close similarities in aeration and energy dissipation performances were found between the classical hydraulic jumps and Type B jumps while Type D jumps showed lower energy dissipation efficiency and aeration. Combining the observations of flow aeration and stability, the present results suggested that a Type B jump may be an optimum hydraulic jump achieving large energy dissipation while providing increased jump toe stability.