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Dam break flows have been a focus of attention for a long in the hydraulic engineering discipline. The interest stems from their high hazard potential to human life and property. This research investigates a laboratory dam break flow using a 3D Reynolds-averaged Navier–Stokes (RANS) and a 2D shallow water Eq. (2D SWE) model. The two numerical models have been compared to measured data regarding flow depth, streamwise surface flow velocity, and point velocity. The results show that as far as a simulation of water depths at gauging stations is concerned, both models are almost at par with an error of about 20% of the mean depth value. Further, the comparison of surface velocity field reveals that though both models are capable of reproducing the salient flow features, e.g., oblique hydraulic jump, wake, etc., the 3D RANS model is more accurate in predicting the jump length and the so-called wet–dry fronts. The important contribution of this research lies in clearly enunciating the tradeoffs between a 2D SWE model and a 3D RANS model. The former model is computationally efficient and accurate as far as flow depth and velocity are concerned while the latter model provides a more complete result output, e.g., vertical flow velocity, turbulent kinetic energy and turbulent dissipation, etc. However, for obtaining significant accuracy with a 3D model a more accurate turbulence model than RANS is required which necessitate usage of very refined mesh (~ mm), thus, raising the cost of the simulation manifold. [ABSTRACT FROM AUTHOR]

ABSTRACT Advancements in computational capabilities have enabled engineers and scientists to numerically model complex turbulent phenomena such as hydraulic jumps. This research assesses the capability of numerically simulating a hydraulic jump that occurs in the UHE Porto Colômbia's stilling basin at a flow rate of 4,000 m3/s. To achieve this, simulation results were compared with data from three hydraulic physical models (scales 1:32, 1:50, and 1:100) and full-scale measurements. The simulations employed the Ansys CFX solver, utilizing a Reynolds-Averaged Navier-Stokes (RANS) approach, the RNG κ-ε turbulence model, and the Volume of Fluid (VOF) method for air-water interactions. Various variables were analyzed, with satisfactory results for mean pressures, conjugated depths, roller length, water profile in less aerated areas, and mean velocity at the submerged hydraulic jump upstream section, with errors below 10%. However, the submerged hydraulic jump's start position and the representation of the water surface profile in the region near the jump toe yielded more disparate results. In conclusion, the methods and conditions applied in the simulations are apt for representing variables less impacted by aeration phenomena, establishing CFD simulations as a valuable tool for hydraulic jump analysis.

Surfactants play a pivotal role in daily life owing to their commendable performance. The outbreak of the COVID-19 pandemic notably escalated surfactant usage. Upon entering building drainage systems with wastewater, surfactants profoundly influence hydraulic performance, an aspect that has garnered limited scholarly attention. This study employs an equally proportioned drainage test device to meticulously examine the variances in physical properties between surfactants, such as sodium dodecyl benzene sulfonate (SDBS) and alkyl ethoxylate-9 (AEO-9), and their repercussions on the hydraulic dynamics of building drainage horizontal main drains. Our findings reveal that the introduction of surfactants leads to the following: (1) an augmentation in water velocity and deposition distances of the solid simulant in the building drainage horizontal main drain with concentrations exacerbating this effect. The deposition distance of the solid simulation surged by up to 527% under experimental conditions compared to no surfactant; (2) there was a suppression of hydraulic jump and full degree of the horizontal main drain, with the concentration amplifying this suppression; and (3) an exacerbation of positive pressure in the horizontal main drain was found with increasing concentration, reaching a staggering 235.3% elevation compared to no surfactant. Moreover, SBDS foam outperformed AEO-9, demonstrating a 17.70–36.04% higher positive pressure in the horizontal main pipes. SBDS exhibits lower starting and ultimate viscosity, along with smaller colloid particle sizes, resulting in a 0.9–2.0% reduction in hydraulic jump and full degree. However, its inferior drag-reduction capability leads to a 17.48–36.44% decrease in the final deposition distances of solid simulant in the building drainage horizontal main drain compared to AEO-9.

Here, the effect of sill application on hydraulic jump parameters has been investigated. The model of sills is made of polyethylene in various dimensions. The sills were investigated in different positions relative to the gate in single, double, and triple arrangements. In a certain opening, the sill leads to more energy dissipation. By increasing the ratio of initial depth to sequent depth, the ∆EAB/yA decreases. It occurs due to the convergence of initial and sequent depths. In the under gate suppressed sill state, the ∆EAB/EA is 11.05% more than the upstream tangential position, while it's 14.56% for downstream specific energy. The results showed that the energy dissipation decreases with increasing the opening by at most 34.88%. In the same opening of no-sill and with sill cases, the use of suppressed sill leads to a decrease in the sequent depth compared to the no-sill one. The application of the sill causes an increase in the initial depth and accordingly the initial specific energy decreases; therefore, the sill leads to the reduction of the specific energy in the flow conjugate depths. The regression nonlinear polynomial equations were proposed to estimate the relative energy dissipation. Statistical indicators Relative Error (RE), Root Mean Square Error (RMSE), and Kling Gupta Efficiency (KGE) have been used to check the accuracy of presented equations. The mean RE% and RMSE for presented equation of ∆EAB/EA are 0.78%, and 0.004, respectively. These values are 0.87%, and 0.009 for ∆EAB/EB, respectively. The KGE is in very good range for equations.

The main problem presented in this paper is the safety inlet navigation of the waterway below the bridge in the city of Kaunas in Lithuania. The analyzed reach is located in the Nemunas river downstream of the Kaunas dam. It is a part of the waterway E–41 leading to the Klaipeda harbor on the southern coast of the Baltic Sea. The work was initiated by the Lithuanian company UAB "Inžinerinis projektavimas" with funds from the project called European Union Trans-European Transport Network (EU TEN-T). The main requirement imposed along this reach is to keep sufficient depth even in the range of the lowest flows. The depth is sufficient if it is not lower than 1.15 m for minimum flows such as Q95% and Q95% with ice. The hydraulic conditions for maximum flow Q50%, Q5%, and Q1% are also taken into account for control because the threat of hydraulic jump generation was also noticed. The research is based on georeferenced data from public and non-public sources. The hydrologic data were received from the Lithuanian Hydrometeorological Service. The physical model was created in the Water Laboratory of the Department of Hydraulic and Sanitary Engineering at Poznan University of Life Sciences, Poland. The preprocessing of spatial data in ArcGIS 10.8.2 and rules of hydraulic similarity were implemented in the process of physical model preparation. Three experiments were conducted in the laboratory with scaled values of Q95%, Q5%, and Q1%. The measurements of the water surface and evaluations of the average velocity were used to validate the 2D numerical model prepared in HEC-RAS 6.3.1. The basic layers of the HEC-RAS model were preprocessed in ArcGIS 10.8.2 by ESRI company. The numerical model was implemented to test different values of unknown roughness of the channel bottom. The simulations were conducted for the real values of Q95% and Q95% with ice and Q50%. The results of the simulations were depth and Froude number maps. These maps were classified into zones of no risk, middle risk, and high risk. ArcGIS in the post-processing phase was applied to identify the locations of the hazards. The magnitude of risk was expressed in terms of minimum depth achieved, maximum Froude number, as well as the length of the reaches with high risk related to these two factors. The threat of hydraulic jump formation below the bridge was also noticed. Conducted results confirmed that the combination of hydrodynamic simulations and geoprocessing in the pre- and post-processing stages could be a powerful tool in hydraulic engineering analyses. Additionally, it is worth noting that numerical modeling enables a wider analysis of potential conditions than could be possible with a physical model only. [ABSTRACT FROM AUTHOR]

ABSTRACT Turbulent flow is responsible for a significant portion of damages and failures in dams. This paper sought to introduce novel approaches to the estimation of typical parameters used in the design of stilling basins. The standard deviations of pressure samples of 24 hydraulic jumps were analysed throughout the stilling basin longitudinal centreline, and the positions of maximum pressure fluctuation were identified. Next, mean and extreme pressures occurring at this position were calculated. Finally, these parameters were plotted against the inflow Froude number and curves were adjusted to the data. The position where maximum turbulence of undular, weak and oscillating jumps occurs varies according to the Froude number. Steady and strong jumps are more likely to induce negative pressures on the stilling basin. The findings of this paper broaden the knowledge on which regions of the stilling basin must receive special attention, and on how to minimize the chances of damages.

Gravity water delivery systems are common around the world to transport water without the use of external energy. The systems’ inadequate design and operation tend to form entrapped air bodies in downward pipeline lengths. Entrapped air generates considerable energy losses when there are no air admission-expulsion valves or due to valve failures. Air removal by hydraulic means has been modeled under various pipeline diameters, downward slopes, and air volume conditions. However, no generic entrapped air removal models exist, and the study in small diameters (≤50 mm) is limited. Most of the models reported in the literature were obtained for diameters greater than 50 mm, presenting notorious discrepancies among each other. In this research, the entrapped air removal in small diameter pipelines was studied (12.7, 15.875, 19.05, 25.4, 31.75, and 38.1 mm), highlighting the importance of the joint study of the air bubbles’ removal by the turbulent action of a hydraulic jump downstream of an air body and the consequent removal of remaining entrapped air by a hydrodynamic thrust. Potential models were found for the air bubbles’ removal for different pipeline diameters and downward slopes. Linear relationships were found between the dimensionless air removal parameter and the pipeline’s downward slope.

Pressure fluctuations are a critical phenomenon that can endanger the safety and stability of hydraulic structures, especially stilling basins. Hence, the accurate estimation of the dimensionless coefficient of pressure fluctuations (C P ′) is critical for hydraulic engineers. This study proposed predictive soft computing models to estimate C P ′ on sloping channels. Therefore, three robust soft computing methods, including extreme learning machine (ELM), group method data of handling (GMDH), and M5 model tree (M5MT), were used to estimate C P ′. The results revealed that ELM was more accurate than GMDH and M5MT methods when comparing statistical indices, including correlation coefficient (CC), root mean square error (RMSE), mean absolute error (MAE), scatter index (SI), index agreement ( I a ), and BIAS values. The performance of ELM was found to be more accurate (CC = 0.9183, RMSE = 0.0067, MAE = 0.0051, SI = 11.88%, Ia = 0.9569) when compared with the results of GMDH (CC = 0.8818, RMSE = 0.0078, MAE = 0.0058, SI = 13.89%, Ia = 0.9361) and M5MT (CC = 0.6883, RMSE = 0.0120, MAE = 0.0090, SI = 21.28%, Ia = 0.7905) in the testing stage. In addition, the BIAS values revealed that ELM slightly overestimated the values of C P ′ , especially at the peak point compared with GMDH and M5MT results. Overall, the suggested soft computing techniques worked well for predicting pressure fluctuation changes in the hydraulic jump. [ABSTRACT FROM AUTHOR]

In the last decade, the use of gabion structures in hydraulic engineering for stabilizing the structure due to its high density and weight has become widespread. Also, the material's roughness and porosity cause it to be used in energy dissipation and drainage projects. This study evaluates the relative energy dissipation of gabion structures downstream of the ogee spillway in the conditions of a submerged hydraulic jump. The evaluated parameters in this study were Froude number, gabion height, gabion thickness, and material diameter. The experiments were performed with three average diameters of 1.5, 2.2, and 3 cm for rock material, three gabion heights of 10 and 20 cm, and Max. The end sill heights were 10, 20, and 30 cm. The operated discharges were regulated from 20 to 40 l/s. The results showed that by decreasing the average diameter of gabion aggregates, the amount of relative energy dissipation increases in all tested models, so that in gabion with a 1.5 cm average diameter of aggregates, the amount of energy dissipation increased by 3.6% in comparison with using the diameter of 3cm for the average diameter of the material. Increasing the height of the gabion to the extent that the flow is entirely inward can have up to 33% more relative energy dissipation than the gabion with a height of 10 cm. Also, by increasing the diameter of the gabion from 10 cm to 30 cm, relative energy dissipation increases up to 15%. [ABSTRACT FROM AUTHOR]

Low-head dams can be dangerous to recreational river users when a submerged hydraulic jump forms downstream, with recirculating surface flows that repeatedly carry trapped recreationists upstream into the high-velocity jet plunging over the dam crest. The flow endangers those who pass over the dam from upstream and can also entrap those who approach too closely from downstream. A national task force is using a range of methods to identify potentially dangerous structures, but definite determination requires field data and an analysis of the hydraulic conditions for each site’s range of likely flow rates. The spreadsheet tool described here determines the submergence created by site-specific tailwater levels and uses previous research results to estimate the associated magnitude of reverse flow velocities. The spreadsheet also determines the crucial tailwater level at which the jet passing over the dam stops plunging into the tailwater pool and instead flips to the surface, creating safer, downstream-directed velocities. The article describes application to specific sites and provides insight about the dangerous flow range of typical low-head dams.

The present research was conducted to investigate the effect of sill geometry and sill width on the discharge coefficient and hydraulic jump characteristics. For this purpose, sills with semi-cylindrical, cylindrical, pyramidal, and rectangular cube geometries with widths of 0.075, 0.10, 0.15, and 0.2 m were installed under a sluice gate. Results showed that increasing the sill width increased the sluice gate discharge coefficient compared to the no-sill mode. The results of placing a sill with different geometric shapes under a sluice gate indicate that using a semi-cylindrical sill increases the discharge coefficient. The ranked order of other sills, from the largest to smallest discharge coefficient, is: cylindrical, pyramidal, and rectangular cubic sills, respectively. The results show that the use of a sill increases the energy dissipation. Examining sills of different widths indicates that with increasing width, the increase in velocity and consequent decrease in the depth of the hydraulic jump causes an increase in energy loss. When employing sills of maximum width (b = 0.20 m) for pyramidal, semi-cylindrical, cylindrical, and rectangular shapes, the energy loss increased by 125, 119, 116, and 125% in section A, respectively. The semi-cylindrical sill is most effective in increasing the discharge coefficient, while the pyramidal sill is most effective for increasing energy dissipation.

Numerical simulation of undular hydraulic jumps, which occur at an inflow Froude number slightly above unity, remain an important and enigmatic challenge in hydraulic engineering. In this study, a systematic assessment of the two-dimensional (2D) Reynolds averaged Navier Stokes–volume of fluid (RANS-VOF) equations coupled with the k–ω shear stress transport (SST) turbulence model was carried out for the simulation of undular jumps. The modeling strategies adopted for obtaining a stable undular jump profile are highlighted in this paper. The correlation between turbulence intensity at the jump toe and jump characteristics is also demonstrated. Comparison of laboratory observations with results obtained from experiments conducted under different discharge, approach Froude number, and inflow conditions demonstrated that the proposed 2D RANS model can accurately reproduce the key free-surface characteristics corresponding to undular jumps. The model was capable of reproducing the velocity and pressure fields with greater accuracy than the depth-averaged model. The model was also successful in reproducing the bottom recirculation region below the first wave crest for undular jumps with partially developed boundary layer at the jump toe. The proposed 2D RANS-VOF computational fluid dynamics (CFD) model coupled with the k–ω SST turbulence closure equation is, therefore, recommended as an efficient tool for hydraulic computations, especially for the simulation of weak undular jumps. [ABSTRACT FROM AUTHOR]

Stilling basins with sudden expansions are one of the energy dissipation structures. In the hydraulic jump, pressure fluctuations cause significant damages in stilling basins by cavity formation, erosion, and vibration. Roughness can also lead to changes of the behavior of stream lines and vortices. Despite the large number of works on the topic, the role of roughness in spatial hydraulic jumps is not yet fully understood. Present research aimed to study the influence of rough bed on pressure fluctuations of S-jump in abrupt expanding stilling basin. Experiments were conducted in a 0.8 m width and 12 m length flume. Channel expansions ratios were 0.33, 0.5, 0.67, and 1 within the range of Froude numbers, 2 to 9.5. The results showed that roughness decreases intensity of pressure fluctuations in an abrupt expansion stilling basin. Additionally, in sudden expanding sections, the energy loss increases, and the intensity of pressure fluctuations decrease due to the formation of lateral vortices. The reduction rate of maximum pressure fluctuation was 27%, 46%, and 58% for expansion ratio of 0.67, 0.5, and 0.33, respectively. The results revealed the clear dependence of these variables on the Froude number and the distance to the hydraulic jump toe. The maximum values of extreme pressure fluctuations occur in the range 0.609 < X < 3.385, where X is dimensionless distance from the toe of the hydraulic jump, which makes it highly advisable to reinforce the bed of stilling basins in this range.

Energy dissipators, such as stilling basins, are usually required at the toe of stepped chutes to achieve adequate and safe operation of the spillway. Stepped chute hydraulics has been extensively studied in last several decades, however, only limited knowledge is available on the stilling basin performance below stepped chutes. In particular, the effect of the chute slope remains unknown, despite being a central design issue. Therefore, an experimental campaign was performed using a 30° or 50° inclined smooth or stepped chute with an adjacent conventional plain stilling basin. The experimental results indicated that, within the stilling basin, the surface characteristics and the roller as well as hydraulic jump lengths are practically independent of the chute slope. This further strengthens the previous findings that stepped chutes require 17% longer dimensionless jump lengths and consequently stilling basin lengths. The experimental results also confirmed that stepped chutes generated increased extreme and fluctuating bottom pressure characteristics at the stilling basin entrance area. With increasing chute slope, the latter were found to significantly magnify. However, such increased magnitudes were not expected to provoke cavitation damage as stepped chute inflows induced bottom aeration at the basin entrance, irrespective of the chute slope.

The low-level bottom outlet operates in a long tunnel type, indicating that the air discharge is independent of the tunnel length and the air flow is supplied totally by the upstream air vent. The total air demand discharge in the bottom tunnel is the sum of the entrained air discharge and the air discharge above the flow surface. To validate the prediction of air supply performance in bottom tunnels, the two aspects of the total air discharges should be clarified carefully, including the coupling effect between the air vent size and tunnel length on complex air supply patterns. [Extracted from the article]

Scientists at the USDA Agricultural Research Service (ARS) Hydraulic Engineering Research Unit (HERU) conducted research on physical models of stilling basins (i.e., Types I, II, III, and IV) associated with stepped chutes to evaluate performance. Tests were conducted with naturally developing unit discharges (0.236 m2/s≤q≤0.930 m2/s) descending a 3(H):1(V) stepped chute flume designed with 0.15-m step heights at near prototype scale and Froude numbers near the stepped chute toe ranging from 4.1≤F≤4.6. Data and visual observations indicated that dissipation features like floor blocks, the end sill, or a dentated end sill effectively reduced wave oscillations and lessened the impact of propagating waves downstream. The study data appear to agree with other published data for stilling basins associated with smooth chutes; thus, design engineers should have confidence in applying the US Bureau of Reclamation (USBR) stilling basin design criteria to basins associated with stepped chutes. Pressure profiles indicated positive values relative to stilling basin floor elevation along the length of all stilling basins. Additionally, pressure peaked near the basin entrance, the floor blocks, and the end sill. [ABSTRACT FROM AUTHOR]

Partially filled pipes have many applications in hydraulic and environmental engineering; such as sewer and storm water collection pipes and circular culverts. Hydraulic jump can lead to scouring at hydraulic structures and thus excess energy should be dissipated to prevent scour and erosion. Calculation of conjugate depths in circular pipes and culverts is important in hydraulic engineering. The conjugate flow depths are commonly used to estimate the length of the hydraulic jump and the energy loss in the jump. Available relationships to estimate the conjugate depths in circular pipes are not accurate and wide application range. In comparison to easily available numerical solutions, a closed-form solution is a function of all input independent variables; and thus the dependence of the conjugate depth on each input independent variable can be easily interpreted. This study presents explicit solutions for the conjugate depths (pre- and post-jump depths) in partially filled pipes. The proposed solutions are examined over a wide range of the Froude number and geometric conditions, providing satisfactory estimations when compared with the exact solution obtained from the specific force relation. If the prejump flow depth of the hydraulic jump is known, the postjump flow depth can be determined using the proposed solution with a maximum relative error less than 1.4%. Inversely, if postjump flow depth is known, the prejump flow depth can be determined with a maximum relative error less than 2.3%. The proposed direct solutions are general, accurate, simple, and useful for studying the partially filled circular open channels. [ABSTRACT FROM AUTHOR]

There are different types of stilling basins, including the standard stilling basin USBR, stilling basin SAF, the stilling basin with a continuous sill, and the stilling basin with perforated sill noted. Each basin, depending on the intensity of the hydraulic jump, usually needs components to reduce the length of the hydraulic jump as much as possible while shaping it at a specific location. These components include the chute block, baffle piers, and perforated sills that are the subject of this study. These types of dampers are the most common energy dampers in dams and irrigation and drainage networks, and they generally have a high efficiency of over 60% in energy dissipation. The purpose of this study is the evaluation of one and two perforated sill(s) in the stilling basin and its impact on characteristics of the hydraulic jump, such as the length of the hydraulic jump, the dissipation of energy, and the required tailwater depth, and also, to determine the optimal distance of one and two perforated sill(s) from the beginning of the stilling basin with a fixed height for the perforated sill and ratios of the opening of holes equal to 50%.

Sluice gates are common hydraulic structures for controlling and regulating flow in open channels. This study investigates five models’ performance in distinguishing conditions of flow regimes, estimating the discharge coefficient (Cd) and flow rate. Experiments were conducted for different gate openings, flow rates, upstream and downstream conditions. New equation forms and methods are proposed to determine Cd for energy–momentum considering losses (EML) and HEC-RAS models. For distinguishing the flow regimes, results indicated a reasonable performance for energy–momentum (EM), EML, and Swamee’s models. For flow rate and discharge coefficient performance of EM, EML, and Henry’s models in free flow and for EM and EML in submerged flow were reasonable. The effects of physical scale on models were investigated. There were concerns about the generality and accuracy of Swamee’s model. Scaling effects were observed on loss factor k in EML. A new equation and method were proposed to calibrate k that improved the EML model’s accuracy. This study facilitates the application and analysis of the studied models for the design or calibration of sluice gates and where the flow in open channels needs to be controlled or measured using sluice gates such as irrigation channels or water delivery channels of small run-of-river hydropower plants.

ABSTRACT Extreme pressures in the hydraulic jump are associated with risk of damage to the flow discharge system of dams by a series of mechanisms. Therefore, understanding and predicting these efforts are crucial for the safe and economical design of discharge systems. Thus, this paper aims to validate an existing pressure forecasting methodology for estimating the extreme pressure in the hydraulic jump with low Froude number (below 4.5). Results have shown that the method may be used for this situation on a preliminary basis. Further studies are recommended to refine the technique and to achieve results that are more precise.

A Stilling basin with divergent section has better performance and lower constructional costs than the classic basin. It can easily adapt itself to the upstream and downstream conditions in terms of depths and cross sections (Omid et al., 2007). Khalifa and McCorquodale (1979) studied radial hydraulic jumps in a gradually expanding channel of rectangular cross section and developed a theoretical equation assuming a second degree polynomial for the surface profile. Omid et al. (2007) studied the hydraulic jump formed in a gradually expanding stilling basin of trapezoidal cross section. They also investigated the hydraulic jumps for three different basin side slopes. Their experimental results indicated that the divergence of the basin for a given side slope causes reductions in the sequent depth and jump length, and an increase of the energy loss in jump relative to those observed in rectangular cross sections. Sahebi (2013) simulated a divergent hydraulic jump in rectangular basins using standard and RNG turbulence models. The outputs of the numerical model showed that the standard turbulence model evaluated the free surface of flow, jump length and maximum velocity in defined sections better than RNG turbulence model. In this study, for seven Froude numbers in the range of 3.2-9.1, three models of the divergent basins were numerically simulated at three dimensional conditions with various geometries of divergence angle and wall type. The k-ε RNG and RSM models were used for turbulence analysis and volume of fluid (VOF) model was used for simulating of free surface profile in the Fluent model. Also, vertical distributions of turbulence intensity, Reynolds stresses and turbulent kinetic energy were investigated using the RSM model in various sections of the hydraulic jumps.

A hydraulic jump is a sudden transition from a supercritical to subcritical flow with a broad range of turbulence Scales. The fluctuating nature of the flow such as oscillations of jump toe position and production of large eddies are visible in pseudo-periodic manners. Energy dissipation of hydraulic jump occurs with Low-frequency large scale pressure fluctuations which can cause damage and erosion and endanger the security of the structures. In the present study, the effects of discontinuous roughness elements of lozenge shape on the fluctuations of pressure in hydraulic jump have been investigated.

Hydraulic jumps are the most important type of rapidly variable flow that occur in from the supercritical state to the subcritical state with the sudden increase of the water free surface in a relatively short path, and usually has a high kinetic energy dissipation. The presence of a barrier against the flow will result in the separation of the inlet jet and more energy depletion resulting in increased shear stress and increased drag force. As a result, the jump length and conjugate depth decreases. These obstacles must be structurally sound because they are directly in front of the jet. If the flow rate exceeds 17m/s, it will cause a cavitation phenomenon (peterka, 1985). Another factor that can reduce the dimensions of a hydraulic jump is the roughness of the floor. Rajaratnam(1968) conducted the first systematic studies of jumping on a rough bed. The results of Rajaratnam were confirmed by Hager(1992), and, with their study of acoarsebed, Ead and Rajaratnam(2002) showed that the jump length on the rough bed is almost three times the conjugate depth (Lj/Y2=3). Shows that the rippling bed has halved the length of the jump to the length of the USBR-I pond. Studies on roughened bed reveal that the use of rough substrates requires more experiments to determine the proper shape of roughness. Hence, in this study, roughnesses in cubic, triangular and cubic and triangular combinations with zigzag arrangement have been investigated on the hydraulic jump specification in the rang of 5.13 to 9.75 froud numbers.

Development of irrigation and drainage networks is known as one of the most effective approaches for the optimal use of limited water resources (Abbasi., 2011). But many of the constructed networks suffer from different problems which raised from different reasons. Rahimi et al. (2011) categorized these reasons as; poor design, improper construction operation, low quality of construction materials, poor operation and maintenance, and geotechnical problems of the subgrade materials. Stilling basins are commonly used structures in order to dissipate the energy in downstream of chutes. In general, a hydraulic jump would be created at the end of the chute. The B-type hydraulic jump takes place on the positively inclined plane of a chute under particular conditions. In this condition, determination of the secondary depth and the length of the roller is essential for protecting the structures. Since there is no proper analytical method to solve the momentum equation for the mentioned condition, using of different smart techniques such as artificial intelligence was considered recently. Caralo et al. (2011) developed different models for determination of the flow characteristics for B-type hydraulic jump. Dusan etal. 2012 presented a neo- Fuzzy system for predicting the weir coefficient. Akib etal. (2014) predicted the scouring depth by using of neo- Fuzzy system. Azamathulla et al. (2012) developed a neo-fuzzy system for prediction of the sediment transport. In this research, the application of two intelligence methods including Neuro - Fuzzy Inference System and fuzzy inference system were investigated.

The most common way to dissipate the water kinetic energy at the downstream of weirs , gates at chutes is to use the hydraulic jump. Hydraulic jump is a phenomenon that alters the flow regime of the state of supercritical to the subcritical with dissipating much of the flow kinetic energy and to prevent possible downstream scour of the structure. Development of the downstream scour hole of the hydraulic jump stilling basins can cause failure of the structure itself , which has been the interest of the researchers in the past. The extension of new roughened bed basin is essential to pursue and complete what previous researchers concluded for these basins.Safe and an economical design of hydraulic structures, an estimation of the maximum scour depth, and to minimization the scour dimensions is the interest of the researchers. Therefore, in this study the effect of placing the six-legged concrete elements under different lengths was investigated investigated. Covering the river bed downstream of the spillways and gates with these roughened and permeable elements can decrease the stilling basin length and expected less scour depth. These elements are tight together and placed on the bed. The covered length of the bed is equal to the length of jump on these elements. However, there is possibility in which few rows of the downstream elements are escaped because of the edge failure and therefore the roughened bed length is shortened. The main purpose of this study was to see how much exactly would be the magnitude of downstream scour depth if the covered length is decreased. .

The rock mass erosion of dam spillways, a phenomenon involving the interaction between the hydraulic load of water and the capability of the rock mass to resist its destruction, remains a critical safety issue. The erosion resistance of a rock mass can be estimated through several erodibility indices, including those of Kirsten, Pells or Bollaert. Several indices have been developed to link rock resistance to the hydraulic parameters of water, i.e., the hydraulic load applied on a rock mass. The developed indices use the average flow velocity, the average shear stress on the bottom of the flow channel, the stress applied to the internal joints of fractured rock mass, the dynamic impulse force, and the power dissipation of water to represent the erosive force of water. From these indices, several methods of assessing hydraulic erosion have been developed, and all use the threshold line concept. Nonetheless, several uncertainties are associated with these methods. This paper presents and discusses the various means of calculating the erosive force of water as a hazard parameter for predicting potential rock erosion. The representativeness of these approaches is also discussed, and we clarify nuances associated with each method. We then provide guidelines for future research aimed at improving estimates of the erosive force of the water within spillway flow channels.

Pressure fluctuations are among the favorite topics for hydraulic engineers due to their critical role in the design and safe operation of hydraulic structures. In this study, three explicit data-driven models, including Classification and Regression Trees (CART), Gene Expression Programming (GEP), and Multivariate Adaptive Regression Splines (MARS), are used. These methods are applied to predict the pressure fluctuation intensity (C P ′) , which occurs beneath a hydraulic jump on sloping channels using reliable experimental observations. The main features of the suggested data-driven models are their capability to provide practical formulas for prediction of C P ′ based on the dimensionless form of the effective parameters. The application of these models leads to the extraction of knowledge from data in the form of predictive formulas. In addition, to evaluate the accuracy and performance of the proposed models, some statistical indices and several validation criteria are employed. The obtained results confirm that MARS predicts C P ′ better than CART and GEP. Based on the effective independent variables, MARS presents a simple predictive formula for the computation of C P ′ . GEP-based equations use mathematical trigonometric functions that cause a more complicated and time-consuming process to predict C P ′ . However, the application of simple rules generated by CART enables hydraulic engineers to predict C P ′ , easily and quickly. [ABSTRACT FROM AUTHOR]

The study presents experimental investigations of spatial turbulence intensity and scales of turbulent eddies (macroeddies) in a rectangular channel and the impact of the hydraulic jump on their vertical and streamwise distributions over a flat and scoured bed. The results of four tests and two different discharge rates are presented. Intensive mixing caused by the hydraulic jump has an impact on the instantaneous velocity, turbulence intensity and sizes of macroeddies, as well as their vertical and longitudinal distributions along the channel. The largest differences in turbulence characteristics were reported directly after the hydraulic jump, above the eroded bed. The interaction between the stream of the increased turbulence and the bed is a direct cause of formation of scour downstream water structures, which has a great effect on overall flow characteristics. The scour hole that arose downstream the jump moderated, in a small degree, the turbulence intensity at its end. Just next to the hydraulic jump only the small longitudinal relative sizes of macroeddies were present, while at the end of the analyzed reach, downstream of the scour, the relative scale reached around 1.5 depth of the stream.

Hydraulic jump phenomenon occurring in the circular pipes is a complex issue which has been paid studious attentions by hydraulic engineers. Several experimental and numerical studies were performed so as to characterize hydraulic jump in circular pipes. There are few comprehensive equations to predict the conjugate depth in circular pipes. In this investigation, three numerical models on the basis of evolutionary computing as gene-expression programming (GEP), model tree (MT), and evolutionary polynomial regression (EPR) have been utilized to evaluate the conjugate depths of the hydraulic jump in the circular pipes. Two non-dimensional parameters were yielded from conceptions of specific force to determine a functional relationship between the input and output variables. The performances of proposed approaches were compared with those obtained using conventional methods. The performance of MT indicated an accurate prediction of conjugate depths (R = 0.995 and RMSE = 0.023) in comparison with those other artificial intelligence (AI) models and empirical equations. The uncertainties prediction of the improved models were quantified and compared with those of existing models. The results of the proposed models demonstrated that linear equations provided by MT had more convenient application than empirical equations extracted from experimental investigations. [ABSTRACT FROM AUTHOR]

If optical or any type of nonintrusive sensing technique can provide adequate information to reproduce two-phase flow boundary conditions, internal air-water flow structures might be predicted by analytical air entrainment models. For full-scale prototype conditions with typically large Reynolds numbers, ensuring good probe sturdiness and signal processing quality, as discussed in the original paper, does not guarantee a successful collection of detailed air-water flow information at large hydraulic structures in operation. As the discussers implied, the effect of sensor size on bubble detection and measurement of air-water flow properties can vary with the flow's Reynolds number at different model scales. [Extracted from the article]

We present a new method to evaluate the hydraulic jump characteristics in a horizontal rectangular channel with a positive step. We considered the flow curvature effect and the free surface’s small rise at the A-type hydraulic jump’s end. First, we present a novel method to give jump length estimation based on the similarity of the jump and the turbulent wall-jet, considering the pressure gradient. Then, considering the jump as a curvilinear flow and using a one-dimensional momentum equation, we present an accurate expression for the conjugate flow depth regarding the initial Froude number and step height. Finally, we compute the jump’s energy dissipation rate. Compared to the theoretical models for conjugate flow depth in a hydraulic jump, the proposed equation in this study fit the experimental data better, even for high steps and large initial Froude numbers. However, for low Froude numbers (F1 < 5), the equation was less accurate in estimating the jump length. Regarding the jump’s energy dissipation rate, the results agreed well with the experimental data from previous investigations. However, it is noted that the increased energy dissipation rate dwindled in larger Froude numbers.

Prototypal and 3D numerical model data of the total air demand, I Q SB air sb i , along the spillway tunnel, and of the air discharge along three air ducts installed on the tunnel ceiling were evaluated. The Authors presented a fascinating research work on the air demand in spillways. Additionally, the present discussion highlights that, if practitioners must rapidly estimate the air demand of a spillway tunnel, then the utilization of Eq. (1) is feasible because its prediction accuracy is larger than other empirical formula considered by the Authors. [Extracted from the article]

Stilling basin is one of the dissipated energy structures in water transfer systems that its dimensions depend on the characteristics of hydraulic jump.In present study, six legs elements have been used in different permeability on basin bed and the changes of shear stress, sequent depth and the length of jump have been investigated. Experiments were conducted in a rectangular flume of 7.5 m long and 0.3 m wide and six legs elements were arranged on the bed of flume with density percent 36, 63 and 100 in such a way that the incoming water jet was just above the elements surface. The incoming Froude number was in the range of 5.3 to 8.1 and during each test the water surface profile and characteristics of hydraulic jump type A were measured. The results indicate that the most reduction of sequent depth ratio and dimensionless length of jump occur when the density of elements is 36%. In this study, relations have been presented for the sequent depth of hydraulic jump and shear force coefficient of bed as a function of the Froude Number for six legs elements.

This paper presents a classification using a decision tree algorithm of hydraulic jump over rough beds based on the approach Froude number, Fr1. Specifically, 581 datasets, from literature, were analyzed. Of these, 280 datasets were for natural rough beds and 301 were for artificial rough beds. The said dataset was divided into four classes based on the energy losses. To compare the performance of the decision tree classifier (J48), a multi-layer neural network (NN) was used. The results suggest an improved performance in terms of classification accuracy by the J48 algorithm in comparison to the NN classifier. Furthermore, the classifier model had only four leaves and achieved an accuracy of 91.56%. Furthermore, classification results showed that the first class (A) of hydraulic jump over the rough beds is approximately similar to that for the smooth bed. Moreover, in the next three classes (B, C, and D), upper values of Fr1 decreased with respect to the smooth bed classes. Lastly, in class D, the upper value of Fr1 reduced to 7.45, which indicates that the shear stress (i.e., the energy loss) grows sharply with increasing Fr1. Put simply, bed roughness effectively increases the energy dissipation with an increase in the Fr1.

The present work is an effort to simulate numerically a classical hydraulic jump in a horizontal open channel with a rectangular cross-section, as far as the jump location and free surface elevation is concerned, and compare the results to experiments with Froude numbers in the range 2.44 to 5.38. The governing equations describing the unsteady one-dimensional rapidly varied flow have been solved with the assumption of non-hydrostatic pressure distribution. Two finite difference schemes were used for the discretization of the mass and momentum conservation equations, along with the appropriate initial and boundary conditions. The method of specified intervals has been employed for the calculation of the velocity at the downstream boundary node. Artificial viscosity was required for damping the oscillations near the steep gradients of the jump. An iterative algorithm was used to minimize the difference of flow depth between two successive iterations that must be less than a threshold value, for achieving steady state solution. The time interval varied in each iteration as a function of the Courant number for stability reasons. Comparison of the numerical results with experiments showed the validity of the computations. The numerical codes have been implemented in house using a Matlab® environment.

The study of the hydraulic jump developed in stilling basins is complex to a high degree due to the intense velocity and pressure fluctuations and the significant air entrainment. It is this complexity, bound to the practical interest in stilling basins for energy dissipation purposes, which brings the importance of physical modeling into the spotlight. However, despite the importance of stilling basins in engineering, bibliographic studies have traditionally focused on the classical hydraulic jump. Therefore, the objective of this research was to study the characteristics of the hydraulic jump in a typified USBR II stilling basin, through a physical model. The free surface profile and the velocity distribution of the hydraulic jump developed within this structure were analyzed in the model. To this end, an experimental campaign was carried out, assessing the performance of both, innovative techniques such as the time-of-flight camera and traditional instrumentation like the Pitot tube. The results showed a satisfactory representation of the free surface profile and the velocity distribution, despite some discussed limitations. Furthermore, the instrumentation employed revealed the important influence of the energy dissipation devices on the flow properties. In particular, relevant differences were found for the hydraulic jump shape and the maximum velocity positions within the measured vertical profiles, when compared to classical hydraulic jumps.

Adaptation of stilling basins to higher discharges than those considered for their design implies deep knowledge of the flow developed in these structures. To this end, the hydraulic jump occurring in a typified United States Bureau of Reclamation Type II (USBR II) stilling basin was analyzed using a numerical and experimental modeling approach. A reduced-scale physical model to conduct an experimental campaign was built and a numerical computational fluid dynamics (CFD) model was prepared to carry out the corresponding simulations. Both models were able to successfully reproduce the case study in terms of hydraulic jump shape, velocity profiles, and pressure distributions. The analysis revealed not only similarities to the flow in classical hydraulic jumps but also the influence of the energy dissipation devices existing in the stilling basin, all in good agreement with bibliographical information, despite some slight differences. Furthermore, the void fraction distribution was analyzed, showing satisfactory performance of the physical model, although the numerical approach presented some limitations to adequately represent the flow aeration mechanisms, which are discussed herein. Overall, the presented modeling approach can be considered as a useful tool to address the analysis of free surface flows occurring in stilling basins.

Movable weirs have been developed to address the weaknesses of conventional fixed weirs. However, the structures for riverbed protection downstream of movable weirs are designed using the criteria of fixed weirs in most cases, and these applications cause problems, such as scour and deformation of structures, due to misunderstanding the difference between different types of structures. In this study, a hydraulic experiment was conducted to examine weir type-specific hydraulic phenomena, compare hydraulic jumps and downstream flow characteristics according to different weir types, and analyze hydraulic characteristics, such as changes in water levels, velocities and energy. Additionally, to control the flow generated by a sluice gate, energy dissipators were examined herein for their effectiveness in relation to different installation locations and heights. As a result, it was found that although sluice gates generated hydraulic jumps similar to those of fixed weirs, their downstream supercritical flow increased to eventually elongate the overall hydraulic jumps. In energy dissipator installation, installation heights were found to be sensitive to energy dissipation. The most effective energy dissipator height was 10% of the downstream free surface water depth in this experiment. Based on these findings, it seems desirable to use energy dissipators to reduce energy, as such dissipators were found to be effective in reducing hydraulic jumps and protecting the riverbed under sluice gates.

The present article is the first of a series of seven, all relatives to energy dissipators that can be used in hydraulic works. The whole group of these articles conform a monograph about the subject that is being prepared by the author, concluding it by the middle of the 2017 year and later on incorporating it into a new edition - in preparation - of the textbook "Spillways and Intake Structures." This first article is a detailed presentation about the hydraulic jump employed with the function of energy dissipation. [ABSTRACT FROM AUTHOR]

Side weirs are widely used for flow control and diversion. Although many studies on sub- and supercritical flows along these weirs have been conducted, only a few research studies have considered the generation of a hydraulic jump along the side weir. Spatially varied flow including a hydraulic jump along the rectangular side weir is studied in this paper. Experimental data for different weir geometries and hydraulic conditions are analysed. Based on the experimental results, new equations for estimating the jump length, the ratio of the conjugate depths and the discharge coefficient are presented. Also, the head loss due to the presence of a hydraulic jump is studied. Furthermore, a numerical model is developed for coupling the water surface profile with a hydraulic jump along the side weir. In this model the lateral outflow momentum is considered, which has been mostly ignored in previous works. The model predicts the location of the hydraulic jump. A comparison of the numerical and experimental results indicates good agreement. [ABSTRACT FROM AUTHOR]

ABSTRACT The overflow system of a dam safely controls the water level of a reservoir. The design of these structures should predict the damage caused by the action of the turbulent flow to which they are subject to. The combination formed by a stepped spillway followed by a stilling basin promotes a considerable portion of the energy dissipation in the actual chute of the stepped spillway but it is not sufficient to completely avoid the risk of damaging the basin. In this paper, we analyze the longitudinal distribution of extreme pressures in a stilling basin downstream of a stepped spillway. The study was based on the instant pressure data from the tests carried out in a physical model in the Laboratório de Obras Hidráulicas in IPH/UFRGS. Thus, it was found that the pressure behavior is similar to that observed in studies involving smooth chute spillways, except in the area surrounding the base of the stepped spillway, where high pressure fluctuations may occur. This happens as a result of the impact caused by the flow in the dissipation structure, which is not observed downstream of smooth chute spillways due to the existence of a vertical concordance curve between the spillway and the stilling basin.

Stilling basins of sudden expansion are one of the energy dissipater structures which can dissipate most of the kinetin energy of the flow through hydraulic jump. In this structure the sequent depth ratio is less and the jump length is more than in classical jump. Since the roughening on the stilling bed can reduce both sequent depth ratio and jump length and its effect have not been investigated before, this study was conducted. In this study many tests are conducted in a relatively large flume size of 80 centimeter wide and 12 meter length. The sudden expansion ratio was 0.33 and tests were conducted for Froude numbers in the range of 2-10. The final results of this study show that for roughened bed sudden expansion stilling basin, the sequent depth ratio is 53 percent and the jump length was found to be 25 lower than in classical jump.

Hydraulic jump on adverse stilling basin is an unstable phenomenon which causes some complexities in controlling of the jump. This paper considers the stability of free hydraulic jump on adverse stilling basins from a theoretical point of view. A minimum value of upstream Froude number is needed to produce the free hydraulic jump on adverse stilling basins which was presented by a theoretical equation. Also, the effects of an end sill and divergence in section on the minimum Froude number were investigated. Moreover, it is required that the bed has a minimum friction for making the jump stable which is obtained as a function of upstream Froude number and bed slope. This condition was compared with the criterion proposed by previous researchers and showed the considerable deviations at larger bed slopes. The result showed that it is impossible to establish the hydraulic jump on concrete adverse stilling basins without any appurtenances. However, diverging in section can improve the stability of hydraulic jump on adverse stilling basins.

Hydraulic condition of stilling basin, in particular, the flow rate passing through chute spillway, as well as the location of hydraulic jump plays an important role on velocity fluctuations in a stilling basin. In this study, effects of the hydraulic jump location and flow discharge in stilling basin on the velocity fluctuations are evaluated. The experiments were performed in a 1:30 scale model of Glabar Dam spillway for discharges of 221.7, 355.8, 450 and 592.5 (m3/s). The discharges ranges are for 1,000 to 10,000 years return period. The flow velocities in different locations inside of stilling basin have been measured near the wall and centre of basin. The experiment results show that flow velocity fluctuations are considerably affected by the discharge and the location of the hydraulic jump. The most energy lost was in discharge 355.8 (m3/s) where jump was on the shut blocks. The lost was nearly 93% after the jump compared to the energy before the jump.

In this study, a series of laboratory experiments were conducted to investigate the energy loss through the hybrid defense system (HDS) in the order of dike, moat, and emergent vegetation in steady subcritical flow conditions. The results of HDS were compared with a single defense system (SDS) comprising only vegetation (OV). The dimensions of dike were kept constant while two different shapes (trapezoidal and rectangular) of moat were considered. The impacts of vegetation of variable thickness and density were investigated. Two combinations of HDS were investigated including the combination of dike and vegetation (DV) and the combination of dike, moat, and vegetation (DMV). The effect of backwater rise due to the vegetation, hydraulic jump formation and the impact of the arrival time of floodwater on energy dissipation were investigated. It was observed that on the upstream side of obstructions, the backwater depth increased by increasing the Froude number in both the SDS and HDS. The hydraulic jump observed in HDS was classified and the energy dissipation due to it was calculated. Under various conditions investigated in this paper, the maximum average energy dissipation was 32% in SDS and 46% in HDS. The trapezoidal moat performed better than rectangular moat as energy dissipater. The delay time was also greater with trapezoidal moat as compared to that in rectangular one. The maximum delay time was 140 s in the case of HDS. Hence, the hybrid defense system offered maximum resistance to the flow of water, thus causing a significant energy loss. For each case of SDS and HDS, empirical equations were developed by regression analysis to estimate the energy dissipation amounts.

The present research describes a laboratory study of hydraulic jump in the abrupt asymmetric expansion stilling basin as an energy dissipator by changing the geometry of walls and bed roughness elements. The experiments were carried out in a horizontal flume with 10 m length, 0.5 m width, and 0.5 m depth for a range of the upstream Froude numbers ( F r 1 ) from 5 to 11. Four physical models with expansion ratio of α = 0.33, 0.5, 0.67, and 1 and asymmetry ratio of Δ = 0.16 were installed in the flume and two different heights of roughness elements ( h = 1.4 and 2.8 cm) were also considered. The results indicated that the sequent depth and the jump length as well as the roller length below abrupt asymmetric expansion on the rough bed were decreased in comparison to the same parameters of the jump in a prismatic channel with smooth bed. It was revealed that the roughness elements have the effective role on stabilization of the hydraulic jump location. The analysis of energy dissipation efficiency confirmed that the spatial jump in the abruptly expanded basin with roughened bed was more efficient than classical jump. In order to estimate the hydraulic jump characteristics, empirical relationships associated with expansion ratio of basin walls, relative height of roughness elements and upstream Froude number were proposed based on the experimental data that resulted in preliminary design of an abrupt asymmetric enlarged basin.

The modelling of complex free surface flows over weirs and in the vicinity of bridge piers is presented in a numerical model emulating open channel flow based on the Reynolds Averaged Navier-Stokes (RANS) equations. The importance of handling the turbulence at the free surface in the case of different flow regimes using an immiscible two-phase RANS Computational Fluid Dynamics (CFD) model is demonstrated. The free surface restricts the length scales of turbulence and this is generally not accounted for in standard two-equation turbulence modelling approaches. With the two-phase flow approach, large-velocity gradients across the free surface due to the large difference in the density of the fluids can lead to over-production of turbulence. In this paper, turbulence at the free surface is restricted with an additional boundary condition for the turbulent dissipation. The resulting difference in the free surface features and the consequences for the solution of the flow problem is discussed for different flow conditions. The numerical results for the free surface and stream-wise velocity gradients are compared to experimental data to show that turbulence damping at the free surface provides a better representation of the flow features in all the flow regimes and especially in cases with rapidly varying flow conditions.

During the past two decades, hydraulic jumps have been investigated using Computational Fluid Dynamics (CFD). The second part of this two-part study is devoted to the state-of-the-art of the numerical simulation of the hydraulic jump. First, the most widely-used CFD approaches, namely the Reynolds-Averaged Navier–Stokes (RANS), the Large Eddy Simulation (LES), the Direct Numerical Simulation (DNS), the hybrid RANS-LES method Detached Eddy Simulation (DES), as well as the Smoothed Particle Hydrodynamics (SPH), are introduced pointing out their main characteristics also in the context of the best practices for CFD modeling of environmental flows. Second, the literature on numerical simulations of the hydraulic jump is presented and discussed. It was observed that the RANS modeling approach is able to provide accurate results for the mean flow variables, while high-fidelity methods, such as LES and DES, can properly reproduce turbulence quantities of the hydraulic jump. Although computationally very expensive, the first DNS on the hydraulic jump led to important findings about the structure of the hydraulic jump and scale effects. Similarly, application of the Lagrangian meshless SPH method provided interesting results, notwithstanding the lower research activity. At the end, despite the promising results still available, it is expected that with the increase in the computational capabilities, the RANS-based numerical studies of the hydraulic jump will approach the prototype scale problems, which are of great relevance for hydraulic engineers, while the application at this scale of the most advanced tools, such as LES and DNS, is still beyond expectations for the foreseeable future. Knowledge of the uncertainty associated with RANS modeling may allow the careful design of new hydraulic structures through the available CFD tools.

Hydraulic jumps have been the object of extensive experimental investigation, providing the numerical community with a complete case study for models’ performance assessment. This study constitutes an exhaustive literature review on hydraulic jumps’ experimental datasets. Both mean and turbulent parameters characterising hydraulic jumps are comprehensively discussed, presenting at least a reference to one dataset. Three studies stand out over other datasets due to their completeness. Using them as reference for model validation may ensure homogeneous and comparable performance assessment for the upcoming numerical models. Experimental inaccuracies are also addressed, allowing the numerical modeller to understand the uncertainties of reduced physical models and its limitations. Part 2 presents the three-dimensional numerical investigations to date and their main achievements.