The basic equations expressing much of open channel hydraulics were formulated in the 19th century by the French engineer Barre de Saint Venant (1797-1886), and the French hydrodynamicist Joseph Valentin Boussinesq (1842-1929).
The Saint-Venant Equations are a general model which can be applied to many flow cases. However, when considering the shallow-water flows in which the horizontal scale is much larger than the vertical one, these equations will suffice, but they are inappropriate to analyze free surface flow problems with horizontal length scales close to flow depth where strong vertical accelerations do occur.
Boussinesq introduced the principle of incorporating the effects of non-hydrostatic pressure, while eliminating the vertical coordinate, thus significantly reducing the computational effort relative to a full three-dimensional solution. This principle was initially behind Boussinesq formulation of new governing equations under the assumption that the magnitude of the vertical velocity increases polynomially from the bed to the free surface which inevitably leads to some form of depth limitation in the accuracy of the embedded nonlinear properties. Hence, Boussinesq-type equations are conventionally associated with relatively shallow water.
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The numerical studies of overflow problems have been based on a variety of models and hypotheses regarding flow geometry and behavior. Formulations also differed based on equations used to model fluid motion. Basically, there have been few studies using the Boussinesq equation model, one possible reason for this, is that the Boussinesq model, with its attractive ability to simulate flow problems, generally requires a complex numerical scheme for accuracy to capture the complex flow patterns common with overtopping . In contrast, the Saint-Venant equations have extensively been studied mathematically for steady and unsteady flows.
The current chapter will present a revision of the previous numerical and experimental studies of the flow characteristics over trapezoidal profile weirs with the relating topics of Saint-Venant and Boussinesq approaches, but outside this domain these approaches will not be discussed in any detail. Also, because the highway and railway embankments become weirs when they are overtopped by flood waters, the literature in this field is pertinent. As well, a revision of the applications of the AdH model is presented.
2.2 History of Investigation of Trapezoidal Profile Weirs
A review of the literature implies that little effort has so far been made to investigate the flow over trapezoidal profile weirs numerically as compared to the experimental works.
Bazin (1898) (cited in ) was the first to investigate embankment-shaped weirs systematically. He tested symmetrical weirs of face slopes less than (45°). Also, the discharge coefficient was determined for the free flow condition over symmetrical weirs of face slope 1:2 with crest lengths 0, 100, and 200 mm, respectively, for about 20 different discharges. As well, the effect of submergence on discharge was investigated and three types of flow were identified: free flow, plunging nappe flow and submerged flow.
Yarnell and Nagler (1930) (cited in ) conducted an experimental study that was concerned primarily with railroad embankments. The basic embankment was 1.2 m high with a crest 4 m long in a laboratory channel 58 m long, 3 m wide, and 3 m deep. The test data consisted of information needed to compute discharge coefficients for both free and submerged flow; the study was directed mainly to the definition of submergence curves.
Chi and Henry (1948) (cited in ) made laboratory tests on a 1:6-scale model of a typical two-lane highway embankment. A major contribution of this investigation was a comprehensive record of water-surface profiles, velocity measurements and photographs required to describe the external flow characteristics. An important contribution of the tests was a classification of the different flow patterns which characterize the flow of water over highway embankments.
Sigurdsson (1954) (cited in ) in addition to summarizing and reanalyzing the results of the previous investigations, he made additional tests on a 1:9-scale model of a highway embankment. The purpose of his tests was to explore more critically the boundary-layer influence on free-flow discharge characteristics.
Davidian (1956) (cited in ) obtained experimental data on the discharge characteristics of embankment-shaped weirs. Emphasis has been placed on free discharge over smooth-surfaced embankments. Limited data have been obtained on the influence of embankment height and roughness and tailwater submergence. Detailed velocity surveys have been made to define the boundary layer between the upstream edge of the upstream shoulder and the crown. The tests were made on 1:12-, 1:9- and 1:6-scale models of a typical two-lane highway embankment. It has been established that the discharge characteristics of an embankment can be related to the theoretical equation of discharge for a broad-crested weir by means of the discharge-displacement boundary-layer thickness. Davidian's study was especially concerned with the relationship between the free-flow coefficient of discharge and the boundary layer on the roadway embankments.
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Govinda Rao and Muralidhar (1961) (cited in ) have reanalyzed the Bazin data and offered an equation for the discharge coefficient of the embankment weirs involving effects of relative crest length, upstream and downstream slopes, and the relative embankment height.
Bretschneider (1961) (cited in ) has studied the effect of low overflow depths on the free surface configuration for embankment slopes 1:1, 1:1.5, and 1:2. He has focused on the undular surface regime resulting in flows with a small overflow depth relative to the crest length.
Kindsvater (1964)  performed a research based on the compiled results of 936 experiments on the discharge characteristics of seventeen different models, plus 106 boundary-layer velocity traverses on four different models of roadway embankments. The research concluded that both free- and submerged-flow discharge are virtually independent of the influence of embankment shape and relative height and that the influence of boundary resistance is appreciable only for smaller heads, and it depends on crest length as well as roughness. Kindsvater's study, as compared to the previous studies, had a special mark for considering cross slope and roughness impacts.
Bos (1985) (cited in ) applied a simple numerical procedure to develop rating curves for flow over broad-crested weirs by incorporating directly the discharge coefficients. The procedure was formulated based on the energy equation which assumes uniform flow at the gauging station and control section, and constant head between this station and section. This procedure provides solution to irrotational flow problems with negligible curvature of streamlines since its application is limited by H/LT ≤ 0.50, but it is inappropriate to develop a rating curve for flow over short-crested weirs where the effects of non-hydrostatic pressure and non-uniform velocity distributions are significant.
Das (1997)  formulated a mathematical momentum model for spatially varied flow (SVF) over embankment-shaped side weirs in order to estimate the water surface profiles and the amount of water spilling over the embankment for a given steady state flow condition. The model was tested for accuracy using laboratory experiments on an embankment side weir of upstream and downstream face slopes of 1:2 and 1:3, respectively. The crest was 22.5 cm long and 7.5m wide. Initially, the height of the embankment was maintained at 36.3 cm then it was reduced in stages, with a resulting increase of crest width. The length of the overflow section was also varied. The computed water surface profiles and the discharge estimates were within 3% and 10%, respectively, of the experimentally observed values. The results of his study showed the adequacy of the SVF model in predicting spatially varied flow over an embankment side weir.
Fritz and Hager (1998)  investigated the flow characteristics of trapezoidal profile weirs to detect the effects of crest length, approach flow, and tailwater submergence. Particular attention was given to the velocity field in the tailwater flow region. A symmetrical embankment-shaped weir 30 cm high and 1.5 m wide with a variable crest length in the flow direction: 0, 5, 10, 20 and 30 cm, and face slope 1:2 was tested in a rectangular horizontal channel 7 m long, 49.9 cm wide and 70 cm deep. A considerable increase of capacity was noted when compared to broad-crested weirs with vertical faces.
Surface profiles across the embankments illustrated that for LT = 0, the flow separates at the crest and accelerates down the weir face as shown in Figure 2.1a. The hydraulic jumps are located with the toe at the foot of the weir and the critical depth (hc) is located at the crest section. Whereas for LT = 30 cm, hc is somewhat downstream of the upper crest section as shown in Figure 2.1b.
Figure 2.1: Free Surface Profiles for: (a) LT = 0 and (b) LT = 30 cm, h (cm) = (âˆ†) 5, (-) 10, (∇) 15, (-) 20; (-) hc and (-) End of Roller.
It was established that for free overflow, the discharge is related to the overflow energy head or total overflow head (H) as
Compared to conventional formulations involving the overflow head (h), Equation (2.1) includes the effect of approach velocity, at least up to H/p < 1/6. To include the effect of the approach velocity for larger overflow depths, the overflow energy head was corrected to with α = 5/3. The correction coefficient does not only include effects of nonuniform approach velocity distribution, but also the effect of H/p. Also, an equation was presented according to all data from Bazin (1898), Kindsvater (1964) and this study for the discharge coefficient as a function of relative crest length with 0 < ξ < 1 as
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For submerged overflow, four regimes were identified in the order of increasing tailwater level: A-jump, plunging jet, surface wave, and surface jet. The entire velocity field was described including the maximum forward and backward velocities beside the forward and backward recirculation zones. The results of the study were generalized for the embankment-shaped weirs.
Collins and Catalano (2001) (cited in ) studied the ability of the DELFT-FLS model to predict accurately the normal flow depths for supercritical, critical and subcritical flow conditions beside the approach crest-referenced head of a broad-crested weir. The discharge rating curves of the weir were simulated for free flow condition and the predicted results were compared with the results of the common broad-crested weir formula for the range in which this equation estimates the discharge accurately (0.08 ≤ H/LT ≤ 0.33). Also, the effect of friction or roughness on the accuracy of the model results was inspected. This was the first study to simulate broad-crested weir flow using a large-scale two-dimensional model.
Zerihun (2004)  considered four Boussinesq-type model equations that incorporated different degrees of corrections for the effects of the curvature of the streamlines. Two models were based respectively on the assumptions of uniform and linear variation of the centrifugal term at a vertical section; the third model was the simplified version of the first model to simulate flow situations that involve weak streamline curvature and slope; and the fourth model was an extended version of the Bernoulli equation based on potential flow theory. These models were applied to simulate subcritical and transcritical steady rapidly varied flows in open channel flow measuring structures in both frictionless and frictional channels. These included free flow over short- and broad-crested types of trapezoidal profile weirs as well as in Venturi flumes with and without humps, and flow with dual free surfaces.
A general computational model was prepared employing the finite difference method to discretise the flow equations. This computational model was also extended to build up head-discharge relationships for free flow over trapezoidal profile weirs. The results of the simulation were verified by laboratory experiments. The experiments on the trapezoidal profile weirs were conducted in a horizontal flume 7.1 m long, 38 cm high, and 30 cm wide. Symmetrical trapezoidal profile weirs of h = 15 cm, LT = 10, 15 and 40 cm, respectively, and face slope 1:2 were tested at different discharges. The conclusions of the study generally included the following:
The proposed approximations for the variation of the centrifugal term had only a marginal effect on the predictions of the local flow characteristics of a curved flow and their impact on the global flow characteristics is insignificant.
The pressure distribution simulation results of the considered flow problems were very sensitive to these approximations and suggested that a higher-order pressure equation be used when accurate simulation of the pressure distribution of a flow with pronounced curvatures of the streamlines is sought.
The global as well as the local flow characteristics of the trapezoidal profile weirs depend greatly on the curvature of the streamlines. The non-dimensional flow surface profiles for flows over a short-crested type of such weir could be represented by a unique curve defined by
The results demonstrate the superior performance of the computational scheme based on the finite difference method compared to a scheme based on the shooting method for the solutions of such equations.
Examination of the overall prediction results suggested that the Boussinesq-type momentum equations with higher-order correction factors are suitable for simulating flows over curved beds in a two-dimensional setting without appreciable sidewall curvature effects.
The model simulation for rating curves provided accurate results for flows over short- and broad-crested types of the trapezoidal shaped weirs with smooth and rough flow boundaries. The model result demonstrated the influence of the curvature of the streamlines on the discharge characteristics of these weirs.
The study aimed to assess systematically the impact of the pressure correction factors on the simulation of pressure and flow surface profiles.
Fenton and Zerihun (2007)  developed a one-dimensional model, similar to that of Boussinesq, but for general channel cross-sections. The performance of the model was compared and tested against several sets of experimental results including those of Zerihun (2004), both for the pressure head on the bed and the elevation of the free surface for flow over trapezoidal profile weirs. When compared with experiment it was found to provide a good model for flows that pass through critical, in flumes, over weirs and embankments.
Sargison and Percy (2008)  investigated the flow of water over an embankment weir with varying upstream and downstream slopes. The study compared the effect of slopes of 1:2, 1:1 and 1:0 (vertical) in various combinations on the upstream and downstream faces of the weir. Pressure and surface profiles were self-similar for all cases tested. Increasing the upstream slope to the vertical decreased the height of the surface profile and, hence, the static pressure on the crest. It also reduced the discharge coefficient. The variation in downstream static pressures was negligible though. Varying the downstream slope had a negligible effect on the surface and pressure profiles over the weir. Changes in flow were constrained to the region downstream of the crest. Cavitation could occur at the downstream corner of the weir if the upstream head was sufficiently high and a sloped face was used.
Wang et al. (2010)  performed laboratory experiments at Maritime Research Centre, Nanyang Technological University using an open channel 5 m long, 30 cm wide and 45 cm deep. Two models (Model S and Model L), with exactly the same geometry but different in size, had been tested to examine the effects due to the model scale. In Model S, p = 24 cm and LT = 12 cm. In Model L, p = 40 cm and LT = 20 cm; either weir was symmetrical about the crest, with a slope of 1:2 for both upstream and downstream faces.
The flow field measurements were performed using Particle Image Velocimetry (PIV) technique, which has the advantage of capturing the whole flow field over conventional point-measurement techniques such as hot-film anemometer or laser Doppler anemometer. The particle images were recorded using a 12-bit charge-coupled device (CCD) camera, which had a resolution of 1.6K - 1.2K pixels and a maximum frame rate of 15 Hz. Several hydrodynamic properties of the flow including the different flow regimes, free surface profile, mean and instantaneous velocity fields, and discharge coefficient were analysed. The effects of model size, approach flow, and tailwater submergence level were accounted for.
The water surface profile was obtained by tracking the air-water interface in the PIV images. A raw PIV image for Model L is shown in Figure 2.2 in which the A-jump can be observed and air-bubble entrainment near the toe of the jump, together with reflections due to the bubbles and air-water interface. The actual free surface can be drawn by a careful visual analysis and standard edge detection techniques such as Canney or Sobel edge methods, as represented by the dotted line. The intersection of the laser light sheet with the bottom divides the lower region of the image into the fluid and the solid bottom. The boundary is easily detected and the result is shown as the dashed line.
Figure 2.2: Typical raw PIV image
The mean flow quantities were obtained by ensemble averaging the 1050 instantaneous velocity vector fields for each experimental condition. Figure 2.3a shows the mean velocity vector (U, V) plot upstream of Model L in which streamlines are included to highlight the flow structure. Measurements were also conducted in regions downstream of the weir to obtain the velocity field in the tailwater region as shown in Figure 2.3b where U and V are the measured streamwise and vertical velocities in the x and y directions, respectively.
Figure 2.3: Mean velocity vector plot in the x-y plane for Model L: (a) upstream of the weir and (b) downstream of the weir.
In addition, the mean velocity vector distribution was used to calculate other hydraulic properties of the weir, such as the discharge coefficient and the flow rate was obtained by the integration of the measured streamwise velocity along the vertical direction at a streamwise location upstream of the weir as
It was also shown that the flow structure is highly dependent on both the Froude number and the tailwater level over the range considered.
This review evinces that most of the researches on the trapezoidal profile weirs were experimental and were basically directed towards the understanding of the flow characteristics of these weirs and the determination of the discharge coefficient under free and submerged flow conditions.
2.3 Review of the (AdH) Model Applications
The Adaptive Hydraulics (AdH) model since its inception have been used in many engineering branches as an efficient tool to simulate complex processes. Some of its many applications are stated in this review to show its capabilities and advantages.
Tate (2006)  implemented and tested the non-cohesive sediment equations incorporated into the 2-dimensional shallow water equations of AdH. The Kate Aubrey reach of the Mississippi River was selected as a test area for the AdH sediment equations. The ability of ADH to qualitatively reproduce field sediment results for several various run conditions, such as constant and variable flows as well as single and multiple grain beds and inflows, has been demonstrated successfully through a series of tests. The simulations confirmed that AdH would give good guidance for river design.
Danchuk and Willson (2008)  performed simulations of the hydrodynamics of the Lower Mississippi River Delta (LMRD) using the AdH modelling code. The model simulated the physical and chemical processes affecting the fate of a surface oil spill including slick advection and spreading, the vertical transport of dissolved and emulsified parcels, evaporation, dissolution, adsorption, sedimentation, re-suspension and degradation. The model estimated the distribution of oil in the surface slick, water column, sediments and atmosphere. Hypothetical spills representative of the type and location of spills commonly occurring in the region were simulated to investigate the sensitivity of the system to the unique parameters. This model was developed to take advantage of the latest advances in computational fluid dynamics and weathering algorithms, while focusing on the complex hydraulics and sediment characteristics local to the Lower Mississippi River Delta.
Alarcon et al. (2009)  presented a hydrodynamic model of Mobile Bay developed using the Adaptive Hydraulic Modelling system (AdH), the Hydrological Simulation Program FORTRAN (HSPF), and the Mesh Generation and Refinement Tool (MGRT), to simulate water surface elevations in Mobile Bay, Alabama. The computational grid was created using MGRT and the created meshes were enriched with bathymetry data from National Oceanic and Atmospheric Administration NOAA data centers. HSPF provided stream flow time series for establishing fresh water boundary conditions. Tidal data form NOAA stations were used to establish ocean boundary conditions and to calibrate the model for water surface elevation. AdH was used to explore several alternatives of hydrodynamic models of Mobile Bay. The resulting AdH model application was fed with output data from (HSPF) watershed models of upland catchments that drain waters into the bay. The parallel AdH code was compiled in a 384-node computer cluster and an exploration of the optimum number of processors for the model's runs was performed. The Surface Water Modeling System (SMS) was used for partial pre-processing and visualization of the results. Calculated and observed water depths were consistent (r2 > 0.75).
The compilation and use of the parallel version of the AdH code is shown to be very effective in the hydrodynamic calibration of the Mobile Bay hydrodynamic model, producing speed-up values of up to 21.7. The loose-link between the HSPF and AdH was effectively used to provide fresh boundary conditions at the upland stream inlets to Mobile Bay.
One of the goals of this research was to develop and demonstrate the use of advanced spatial technology and high performance computing in the prediction of hydrodynamics and surface water quality.
Sharp (2010)  used AdH to validate the River Analysis Tool (RAT) to determine its usability and reliability as a forecasting tool to enable watershed managers to evaluate river responses due to imposed changes. Validation was done through comparisons between bed changes predicted by the RAT and the results of the AdH. It was stated that the RAT model can be used as an effective indicator for sediment accumulation or erosional changes and, hence, as an assessment tool to estimate long-term response to river alterations. This allows more effective planning of rehabilitative measures and management of the resource to meet ecological objectives.
Sharp and McAnally (2010)  conducted a study to validate a numerical model with respect to physical modeling of surge overtopping and to explore changes in landward-side levee face shear stresses due to levee berm effects and variations in slope roughness. AdH was used to calculate velocity and depth during an overtopping event. The aim of this study was the estimation of associated shear stresses so that appropriate measures are applied for protection for ultimately reducing the probability of levee failure during an overtopping event.