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Two-dimensional (2D) transient flow over an erodible bed can be modelled using shallow-water equations and the Exner equation to describe the morphological evolution of the bed. Considering the fact that well-proven capacity formulae are based on one-dimensional (1D) experimental steady flows, the assessment of these empirical relations under unste...
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... in time due to fl ow overtopping was studied by Tingsanchali & Chinnarasri ( ). Figure 8 shows a sketch of the experimental setup. Experiments were carried out in a rectangular fl ume 35 m long, 1.0 m deep and 1.0 m wide. The height and crest width of the dam were fi xed at 0.80 and 0.30 m. The upstream slope was fi xed at 1V:3H, while the downstream slope was set to 1:5. The dam was made of sand with the following characteristics: ρ 1⁄4 2,650 kg m À 3 , d 30 1⁄4 0.52 mm, d m 1⁄4 0.86 mm, d 90 1⁄4 3.80 mm and W d m 1⁄4 1.13 mm. A friction angle of φ 1⁄4 30 was suggested, and the porosity was estimated using the formula of Wu & Wang ( ), leading to p 1⁄4 0.22. The Manning roughness coef fi cient was estimated to be equal to n 1⁄4 0 : 015. To have a uniform over fl ow across the fl ume width, in the experiment reproduced in this work, a vertical plate was held at the dam crest across the fl ume width until the upstream water level was 3 cm higher than the dam crest. The vertical plate was lifted up suddenly to allow the over- fl ow to start. Three zones can be distinguished. The fi rst is a subcriti- cal region in the reservoir area, characterized by a very low velocity. The second zone is a supercritical region of highly unsteady fl ow over a steep bed slope in the downhill slope of the dam, starting at the front edge of the dam crest. The third zone, downstream of the dam is characterized by the presence of a hydraulic jump. The dam erosion model was computed using Δ x 1⁄4 0 : 05 m. During the development of this experiment, the bed level was recorded in time at three stations: SA, SB and SC, located respectively 15, 65 and 115 cm downstream from the front edge of the original dam crest. The overtopping discharge was also caught along time, as well as the reservoir level, just upstream of the breach. The results presented below compare experimental and computed data in order to validate the accuracy of the numerical method. On the left-hand side of Figure 9, numerical results for water level and bed level using the Smart CFBS and MPM formulae are plotted. On the right-hand side of the fi gure, a plot shows measured and computed bed-level surface in time evolution at stations SA, SB and SC. Figure 10 displays, on the left, experimental and computed values of reservoir free-surface levels using Smart CFBS and MPM formulae. It can be observed how the computed results obtained with the new proposed formulation show a good agreement with the experimental data, while MPM predictions are quite far away from experimental data. On the right-hand side of Figure 10, a plot of the evolution in time of the overtopping discharge is depicted, and the maximum peak discharge reached with Smart CFBS agrees with the measured value. Differences in the shape of the discharge curve before and after the peak fl ow are in agreement with the bed-level evolution, computed and measured, obtaining more accurate results at the later instants of time, when most of the morphodynamic changes in the dam crest have occurred. On the other hand, the predictions obtained with the MPM formula lead to a poor estimation of the curve discharge. As the prediction of the maximum discharge is of utmost importance in situations of dam failure, the maximum overtopping discharge achieved with the different formulae are presented in Figure 11. The continuous line at the top of the fi gure represents the maximum experimental overtopping discharge, which is only well calculated with the Smart CFBS formula. The rest of the formulae give values quite far away from the experimental value. The relative performance of the different formulations in terms of RMSE is plotted in Figure 12 at the three stations SA, SB and SC, showing important differences among numerical results depending on the formula selected. The Engelund & Fredsoe sediment transport relation was derived for a wide range of slopes, and Figure 12 shows how this formulation leads to low values of RMSE. The Smart formula was derived for a set of experimental cases with steep slopes; therefore, it can be expected that in this case, any numerical approach of the term of the slope would provide accurate predictions. On the other hand, numerical simulation shows that the Smart discretization (considering that the slope term is included in the formula as the friction slope) leads to less accurate results if compared with those given by the Smart CFBS discretization. The rest of the formulations, derived from experiments ranging from low to medium slopes, provide higher RMSEs. It has been considered important to check the performance of the numerical discretization of the empirical formulations in a triangular unstructured 2D mesh to analyse whether numerical results may be in fl uenced by the grid de fi nition under a wide variety of fl ow conditions. 2D numerical simulations have been developed using a coarse unstructured triangular mesh, with a maximum cell size of 0.01 m 2 , as shown in Figure 13. The rest of the parameters are the same as the ones presented in the 1D test case. The set of results presented in Figures 9 and 10 for a 1D mesh are repeated here in Figures 14 and 15 for a 2D mesh. The RMSEs for bed levels at stations SA, SB and SC with different formulae in time are plotted in Figure 16. This test case, with a fi ner mesh and comparing also with the MPM formula, has been discussed by Juez et al. ( ). From both results, it can be concluded that the bed-level predictions are not in fl uenced by the mesh size. The results provided by the unstructured grid give similar conclusions to those described in the 1D test case. Smart CFBS is the formula that provides the more accurate results. On the other hand, when comparing the numerical results obtained in 1D and 2D situations, it can be appreciated that in bidimensional situations, the error increases with independence of the employed formula. This is justi fi ed by the fact that in a 2D fl ow, the projections of the normal vector to the edge of each cell have to be taken into account. Furthermore, it is necessary to bear in mind the fact that the capacity formulae tested in this work are based on 1D experimental steady fl ows, so it is expected that these formulae provide less accurate results under 2D unsteady situations. This experiment was designed at the laboratory of UCL (International Association for Hydro-Environment Engineering and Research Working Group (Soares-Frazao et al. )). It is part of a benchmark test launched within the fra- mework of the NSFPIRE project ‘ Modelling of Flood Hazards and Geomorphic Impacts of Levee Breach and Dam Failure ’ . It consists of a dam break over a 3.6 m wide and approximately 36 m long fl ume. The gate, located at x 1⁄4 0 m, was connected to an upstream reservoir, and was 1 m wide. The sand was extended over 9 m downstream of the gate and 1 m upstream of the gate, with a thickness of 0.085 m. A complete sketch of the experiment can be found in Soares-Frazao et al. ( ). The properties of the sand were ρ 1⁄4 2,630 kg m À 3 , d 1⁄4 1.61 mm, φ 1⁄4 30 W , negligible cohesion, porosity p 1⁄4 0.40 and the roughness was characterized by a Manning factor n 1⁄4 0.019. Initial conditions used were: upstream, the water level was imposed to 0.47 m, and downstream, a dry bed situation was imposed. At time t 1⁄4 100 s, two longitudinal bed pro fi les were measured starting from x 1⁄4 0.5 m at two y -coordinates: y 1⁄4 0.20 m (section S1) and y 1⁄4 0.70 m (section S2). During the performance of this experiment, several runs were carried out. An average of the experimental results obtained during the runs was calculated. For comparison with the numerical results, this experimental average has been taken into account. The domain was discretized on a non-uniform triangular mesh, with a higher density downstream of the widening, and with the fi nest cell size equal to 0.001 m 2 . Figure 17 shows a sequence in time of computed bed evolution plan views predicted by the Smart CFBS discretization. This sequence is characterized by fast morphodynamic changes. Figure 17(a) at t 1⁄4 10 s shows how the fl ow generates a wavefront that causes an important erosion process in the enlargement zone of the channel. While the fl ooding wave advances, the sand particles grabbed in this process are carried out to the wavefront and to the wall, where they tend to sediment, as shown in Figure 17(b) at t 1⁄4 20 s. Symmetric elongated sedimentary bodies appear on the right and left banks of the channel, which grow in time to merge generating a diamond-shaped erosion region at t 1⁄4 40 s, shown in Figure 17(c). At t 1⁄4 60 s, most of the morphodynamic changes have taken place, and the drainage of the water contained in the upstream reservoir smooths the bed surface, attenuating the bed forms previously generated. For longer times, no more important morphodynamic changes happen. At t 1⁄4 100 s, Figure 17(f) shows how only the diamond-shaped erosion region in the enlargement zone, generated by the sudden change in fl ow direction after the opening of the gate, remains in time. The rest of the bed surface becomes almost planar. The results shown in Figure 18 display the experimental bed level and the computed ones using the Smart CFBS and MPM formulae at the three cross sections, S1, S2 and S3. The pro fi les are cut off at x 1⁄4 4 m because this zone is the most representative of the bed morphodynamic changes. The fi rst one, section S1, placed to study the effect of the fl ow over the bottom in the enlargement zone, presents differences between both load discharge formulae. The Smart CFBS formula obtains a better tracking of the sedimentary process, getting more accurate results for the maximum erosion position, x 1⁄4 1.4 m, and in the maximum deposition position, x 1⁄4 3.5 m. At the second cross section, section S2, differences in the experimental results are also noticeable when using MPM and Smart CFBS ...
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Confluences are key morphological nodes in braided rivers where flow converges, creating complex flow patterns and rapid bed deformation. Field surveys and laboratory experiments have been carried out to investigate the morphodynamic features in individual confluences, but few have investigated the evolution processes of confluences in large braide...
Citations
... The classical equilibrium approach considers that the actual bedload rate is the capacity of the flow to carry solid weight and hence only depending on instantaneous local flow features. The capacity bedload rate can be estimated using different empirical closure relations found in literature [2] and is the most widespread approach in current bedload transport models [3][4][5][6][7][8][9][10]. ...
... The bedload solid discharge in classical morpho-dynamical models is considered under equilibrium (capacity) conditions and usually modelled using the Grass law [31], which scales exponentially the solid transport rate with the flow velocity as q b = A g |u| m−1 u, being A g a flow-bed interaction constant factor and m an behaviour exponent (usually m = 3). More complex equilibrium models applied to realistic bedload transport tests estimate the instantaneous local flow-bed interaction factor from the hydro-dynamical flow features, hence A g ≡ G(h, θ) [2,10,32], being θ the dimensionless Shields stress at the bed surface. According to Martínez-Aranda et al. [29] and Martínez-Aranda et al. [30], the bedload transport rate assuming a non-equilibrium approach can be expressed following a modified Grass-type law as: ...
Bedload sediment transport is an ubiquitous process in natural surface water flows (rivers, dams, coast, etc), but it also plays a key role in catastrophic events such as dyke erosion or dam breach collapse. The bedload transport mechanism can be under equilibrium state, where solid rate and flow carry capacity are balanced, or under non-equilibrium (non-capacity) conditions. Extremely transient surface flows, such as dam/dyke erosive collapses, are systems which always change in space and time, hence absolute equilibrium states in the coupled fluid/solid transport rarely exist. Intuitively, assuming non-equilibrium conditions in transient flows should allow to estimate correctly the bedload transport rates and the bed level evolution. To get insight into this topic, a 2D Finite Volume model for bedload transport based on the non-capacity approach is proposed in this work. This non-equilibrium model considers that the actual bedload sediment discharge can be delayed, spatial and temporally, from the instantaneous solid carry capacity of the flow. Furthermore, the actual solid rate and the adaptation length/time is governed by the temporal evolution of the bedload transport layer and the vertical exchange solid flux. The model is tested for the simulation of overtopping dyke erosion and dambreach opening cases. Numerical results seems to support that considering non-equilibrium conditions for the bedload transport improves the general agreement between the computed results and measured data in both benchmarking cases.
... Note that (2.98) indicates that the value of the excess pore pressure depends on the shearing-state of the flow and it is null, i.e. hydrostatic pore-fluid pressure, for the lithostatic case. In order to assess the behaviour predicted by (2.98) for the excess pore pressure in a realistic mature dense-packed flow, we set µ = 10 −3 P a · s, κ = 10 −8 m 2 h = 10 m, γ = 6 s −1 , φ eq = 0.6, k 1 = 0.05, ρ f = 1000 kgm −3 , g = 9.81 ms −2 , ϕ b = 15 • and vary φ from 0.50 to 0. 65. Figure 2.13 depicts the normalised pore-fluid pressure P * (z) defined as ...
... Values A = 0.1 m, h 0 = 1 m and r 0 = 1.8 are set, considering a unique sediment class with χ = 1. 65. The exact depth-variable and density-variable solutions are imposed as initial conditions for the flow depth and the suspended concentrations. ...
... where (q bx , q by ) are the components of the bulk bedload rate q b along the global (x, y) horizontal coordinates respectively. For a multi-grain bed layer composed by N different sediment classes, q b can be estimated as [65,88,143] ...
Among the geophysical and environmental surface phenomena, rapid flows of water and sediment mixtures are probably the most challenging and unknown gravity-driven processes. Sediment transport is ubiquitous in environmental water bodies such as rivers, floods, coasts and estuaries, but also is the main process in wet landslides, debris flows and muddy slurries. In this kind of flows, the fluidized material in motion consists of a mixture of water and multiple solid phases which might be of different nature, such as different sediment size-classes, organic materials, chemical solutes or heavy metals in mine tailings. Modeling sediment transport involves an increasing complexity due to the variable bulk properties in the sediment-water mixture, the coupling of physical processes and the presence of multiple layers phenomena. Two-dimensional shallow-type mathematical models are built in the context of free surface flows and are applicable to a large number of these geophysical surface processes involving sediment transport. Their numerical solution in the Finite Volume (FV) framework is governed by the particular set of equations chosen, by the dynamical properties of the system, by the coupling between flow variables and by the computational grid choice. Moreover, the estimation of the mass and momentum source terms can also affect the robustness and accuracy of the solution. The complexity of the numerical resolution and the computational cost of simulation tools increase considerably with the number of equations involved. Furthermore, most of these highly unsteady flows usually occur along very steep and irregular terrains which require to use a refined non-structured spatial discretization in order to capture the terrain complexity, increasing exponentially the computational times. So that, the computational effort required is one of the biggest challenges for the application of depth-averaged 2D models to realistic large-scale long-term flows. Throughout this thesis, proper 2D shallow-type mathematical models, robust and accurate FV numerical algorithms and efficient high-performance computational codes are combined to develop Efficient Simulation Tools (EST's) for environmental surface processes involving sediment transport with realistic temporal and spatial scales. New EST's able to deal with structured and unstructured meshes are proposed for variable-density mud/debris flows, passive suspended transport and generalized bedload transport. Special attention is paid to the coupling between system variables and to the integration of mass and momentum source terms. The features of each EST have been carefully analyzed and their capabilities have been demonstrated using analytical and experimental benchmark tests, as well as observations in real events.
... Concretely, two typical tests are considered depending on the time-scale of the sediment transport: the movement of a dune (weak interaction) and dam-break problems (strong water-sediment interaction). We consider academic tests showing the differences between the standard SWE model and the third-order HSWEM model, and a comparison with laboratory experiments described in [45] (also in [24,28]). Unless otherwise stated, we set the kinematic viscosity ν = 0.01 m 2 /s, and the stability condition CFL = 0.8 is fixed. ...
... Dam break experiments and simulations at time t = 1.5 s, for configurations Exp 2 and Exp 3 (test case A and B in[28]). ...
... Whitehouse [9] and Chiew and Parker [10] investigated the threshold condition for the onset of cohesionless sediment transport on a sloping seabed. Later, Juez et al. [11] proposed a new Shields number considering gravity currents over steep slopes and investigated the bed load transport at the action of gravity currents through both experimental and numerical models, based on their previous studies for bed load transport over the sloping river [12][13][14]. However, only the basic forces (i.e., included submerged weight force, left and drag force) were included, and the seabed surface was assumed as an impermeable and rigid boundary in their studies, in which the seepage force was ignored. ...
Pipelines have been used as one of the main transportation methods for the offshore industry, with increasing activities in marine resources recently. Prediction of seabed instability is one of key factors that must be taken into consideration for an offshore pipeline project. As the first step of the scour process, sediment incipient motion has been intensively studied in the past. Most previous investigations didn’t consider the wave-induced seepage in the elevation of sediment motion. In this paper, two-dimensional seepage was considered to modify the conventional Shields number and its associated impact on sediment incipient motion around the trenched pipeline was investigated. Both flat and sloping seabeds are considered. The numerical results indicated that a peak or valley of the modified Shields number was formed below the pipeline and horizontal seepage flow tremendously impact the sediment motion in the vicinity of the pipeline. Parametric analysis concludes: the influence of the seepage around the pipeline becomes more significant in a large wave, shallow water in a seabed with large shear modulus and permeability, and larger pipeline diameter and smaller flow gap ratio. This will make soil particles be more easily dragged away from the seabed.
... In the review of the research on the mechanism of modifying sediment transport behavior, Juez et al. [6][7][8] discuss the numerical assessment of bedload formulations for transient flow. The dam break flows over dry/wet initial conditions and erodible channel of experimental cases are simulated by considering the improved model, called Smart CFBS (Smart Combined Friction and Bed Slope model). ...
Local scour is a common threat to structures such as bridge piers, abutments, and dikes that are constructed on natural rivers. To reduce the risk of foundation failure, the understanding of local scour phenomenon around hydraulic structures is important. The well-predicted scour depth can be used as a reference for structural foundation design and river management. Numerical simulation is relatively efficient at studying these issues. Currently, two-dimensional (2D) mobile-bed models are widely used for river engineering. However, a common 2D model is inadequate for solving the three-dimensional (3D) flow field and local scour phenomenon because of the depth-averaged hypothesis. This causes the predicted scour depth to often be underestimated. In this study, a repose angle formula and bed geometry adjustment mechanism are integrated into a 2D mobile-bed model to improve the numerical simulation of local scour holes around structures. Comparison of the calculated and measured bed variation data reveals that a numerical model involving the improvement technique can predict the geometry of a local scour hole around spur dikes with reasonable accuracy and reliability.
... where ( , , , ) are the components of the bulk volumetric bedload rate along the ( , ) horizontal coordinates respectively. For multi-grain bed layer composed by different sediment classes, can be estimated as ( Wu, 2007;Juez et al., 2013;: ...
The bedload transport system of equations, composed by the two-dimensional shallow water equations for the free surface flow motion and the 2D Exner or bedload transport equation for the erodible bed layer, is used for a wide range of sediment transport processes in environmental surface flows. In this work, the numerical resolution has been implemented using improved and efficient versions of two different strategies for combining the hydrodynamical and morphodynamical components of the system. The first strategy is based on the full coupling of the flow and bedload transport equations (FCM), leading to a new formulation for the intercell numerical fluxes which includes the bed elevation into the resolution of the approximated local Riemann problem (RP) at the edges. The stability region of this method is controlled by the eigenvalues of the coupled Jacobian matrix at each intercell edge. On the second hand, an alternative decoupled strategy is considered based on solving independently the shallow water and the bed transport equations at each time step but controlling the stability region by means of an approximation of the coupled Jacobian matrix eigenvalues. This method, called approximate-coupled (ACM) allows simpler expressions for the numerical fluxes at the edges and ensures the stability of the scheme. Both strategies are based on the Finite Volume (FV) method using Roe’s approach for the computation of the numerical fluxes between neighbouring cells and have been implemented into the same CPU-based numerical kernel in order to perform a realistic comparison of the range of applicability and computational efficiency. The ACM can only guarantee non-oscillatory results when the bed-flow interaction factor is small G≤O(10−3). If the interaction factor G is medium or high, G>O(10−2), the decoupled scheme loses its accuracy and robustness. Furthermore, for highly erosive flows the FCM scheme demonstrates to be more efficient in terms of computational effort than the ACM, one of the key points for large-scale and long-term bedload transport realistic applications.
... Moreover, a fundamental requirement of any numerical hydro-morphological model is careful and accurate model calibration and validation against measured data. This is especially important for the morphological models, as these include several empirical factors (e.g., a threshold of motion [16] or a specific transport formula [17,18]), which must be reliably estimated. In addition, physical modeling in laboratories and research facilities can provide information on reservoir flushing, in order to optimize sediment management [19][20][21]. ...
Reservoir sedimentation is a critical issue worldwide, resulting in reduced storage volumes and, thus, reservoir efficiency. Moreover, sedimentation can also increase the flood risk at related facilities. In some cases, drawdown flushing of the reservoir is an appropriate management tool. However, there are various options as to how and when to perform such flushing, which should be optimized in order to maximize its efficiency and effectiveness. This paper proposes an innovative concept, based on an artificial neural network (ANN), to predict the volume of sediment flushed from the reservoir given distinct input parameters. The results obtained from a real-world study area indicate that there is a close correlation between the inputs—including peak discharge and duration of flushing—and the output (i.e., the volume of sediment). The developed ANN can readily be applied at the real-world study site, as a decision-support system for hydropower operators.
... Smart [11] developed aA relation for slopes ranging from 3% to 20% was developed by [11]. Moreover, the performance of this model was found to be better than other similar models in the case of high slopes [20]. More recent studies [21], Yager et al. [22] have considered the typical features of steep channels such as wide grain size distribution ranging from large almost immobile boulders to finer more mobile sediments, the effective stress that causes sediment movement (which is smaller than the total shear stress), and the reduced amount of mobile sediment. ...
... Angle φ corresponds to the inclination of the bed with respect to the horizontal. This projection has been defined by Juez et al. [41] to include effects of steep slopes on pressure distribution and friction, considering that the gravity vector projection has improved the prediction capacity of the sediment transport models with respect to the estimations using gravity vector directly [2,20]. ...
... The experimental transport rates show considerable agreement with this model's rates, especially for slopes of 4% and 5%. Since the model has been demonstrated to perform well for steep slopes [20], it may be concluded that the experimental model developed here could be used to estimate bedload transport rates in streams with slopes of up to 5%. ...
The current study presents an experimental procedure used to determine bedload sediment transport rates in channels with high gradients and coarse sediment. With the aim to validate the procedure for further investigations, laboratory experiments were performed to calculate bedload transport rates. The experiments were performed in a laboratory tilting flume with slopes ranging from 3% to 5%. The sediment particles were uniform in shape (spheres). The experiments were divided into four cases based on sediment size. Three cases of uniform sizes of 10 mm, 15 mm and 25 mm and a case with a grain size distribution formed with the uniform particle sizes were considered. From the experimental results a mathematical bedload transport model was obtained through multiple linear regression. The experimental model was compared with equations presented in the literature obtained for gravel bed rivers. The experimental results agree with some of the models presented in the literature. The closest agreement was seen with models developed for steep slopes especially for the highest slopes considered in the present study. Therefore, it can be concluded that the methodology used can be replicated for the study of bedload transport rates of channels with high gradients and coarse sediment particles to study more general cases of this process such as sediments with non-uniform shapes and sizes. However, a simplified model is proposed to estimate bedload transport rates for slopes up to 5%.
... Measured data have also been plotted for comparison. The Smart-CBFS formulation ( Juez et al., 2013 ) offers the best results, especially for the discharge at the dyke crest, but overestimates the erosion of the dyke at the first stages after the overflow starting. On the other hand, both Meyer-Peter-Müller and Fernández-Luque closure relations underestimate the bedload transport rate, resulting in a slower evolution of the dyke surface and leading to a lower peak in the discharge hydrograph. ...
... Concretely, two typical tests are considered depending on the time-scale of the sediment transport: the movement of a dune (weak interaction) and dam-break problems (strong water-sediment interaction). We consider academic tests showing the differences between the standard SWE model and the third-order HSWEM model, and a comparison with laboratory experiments described in [45] (also in [24,28]). Unless otherwise stated, we set the kinematic viscosity ν = 0.01 m 2 /s, and the stability condition CFL = 0.8 is fixed. ...
... Dam break experiments and simulations at time t = 1.5 s, for configurations Exp 2 and Exp 3 (test case A and B in[28]). ...
In this work a simple but accurate shallow model for bedload sediment transport is proposed. The model is based on applying the moment approach to the Shallow Water Exner model, making it possible to recover the vertical structure of the flow. This approach allows us to obtain a better approximation of the fluid velocity close to the bottom, which is the relevant velocity for the sediment transport. A general Shallow Water Exner moment model allowing for polynomial velocity profiles of arbitrary order is obtained. A regularization ensures hyperbolicity and easy computation of the eigenvalues. The system is solved by means of an adapted IFCP scheme proposed here. The improvement of this IFCP type scheme is based on the approximation of the eigenvalue associated to the sediment transport. Numerical tests are presented which deal with large and short time scales. The proposed model allows to obtain the vertical structure of the fluid, which results in a better description on the bedload transport of the sediment layer.
