Modelling dam-break flows over mobile beds using a 2D coupled approach

Dam-break flows usually propagate along rivers and floodplains, where the processes of fluid flow, sediment transport and bed evolution are closely linked. However, the majority of existing two-dimensional (2D) models used to simulate dam-break flows are only applicable to fixed beds. Details are given in this paper of the development of a 2D morphodynamic model for predicting dam-break flows over mobile beds. In this model, the common 2D shallow water equations are modified, so that the effects of sediment concentrations and bed evolution on the flood wave propagation can be considered. These equations are used together with the non-equilibrium transport equations for graded sediments and the equation of bed evolution. The governing equations are solved using a matrix method, thus the hydrodynamic, sediment transport and morphological processes can be jointly solved. The model employs an unstructured finite volume algorithm, with an approximate Riemann solver, based on the Roe-MUSCL scheme. A predictor–corrector scheme is used in time stepping, leading to a second-order accurate solution in both time and space. In addition, the model considers the adjustment process of bed material composition during the morphological evolution process. The model was first verified against results from existing numerical models and laboratory experiments. It was then used to simulate dam-break flows over a fixed bed and a mobile bed to examine the differences in the predicted flood wave speed and depth. The effects of bed material size distributions on the flood flow and bed evolution were also investigated. The results indicate that there is a great difference between the dam-break flow predictions made over a fixed bed and a mobile bed. At the initial stage of a dam-break flow, the rate of bed evolution could be comparable to that of water depth change. Therefore, it is often necessary to employ the turbid water governing equations using a coupled approach for simulating dam-break flows.     

Numerical resolution of well-balanced shallow water equations with complex source terms

This paper presents a well-balanced numerical scheme for simulating frictional shallow flows over complex domains involving wetting and drying. The proposed scheme solves, in a finite volume Godunov-type framework, a set of pre-balanced shallow water equations derived by considering pressure balancing. Non-negative reconstruction of Riemann states and compatible discretization of slope source term produce stable and well-balanced solutions to shallow flow hydrodynamics over complex topography. The friction source term is discretized using a splitting implicit scheme. Limiting value of the friction force is derived to ensure stability. This new numerical scheme is validated against four theoretical benchmark tests and then applied to reproduce a laboratory dam break over a domain with irregular bed profile.     

Well-balanced numerical modelling of non-uniform sediment transport in alluvial rivers

The last two decades have witnessed the development and application of well-balanced numerical models for shallow flows in natural rivers.However,until now there have been no such models for flows with non-uniform sediment transport.This paper presents a 1D well-balanced model to simulate flows and non-capacity transport of non-uniform sediment in alluvial rivers.The active layer formulation is adopted to resolve the change of bed sediment composition.In the framework of the finite volume Slope Llmiter Centred(SLIC) scheme,a surface gradient method is incorporated to attain well-balanced solutions to the governing equations.The proposed model is tested against typical cases with irregular topography,including the refilling of dredged trenches,aggradation due to sediment overloading and flood flow due to landslide dam failure.The agreement between the computed results and measured data is encouraging.Compared to a non-well-balanced model,the well-balanced model features improved performance in reproducing stage,velocity and bed deformation.It should find general applications for non-uniform sediment transport modelling in alluvial rivers,especially in mountain areas where the bed topography is mostly irregular.     

A robust well-balanced finite volume model for shallow water flows with wetting and drying over irregular terrain

An unstructured Godunov-type finite volume model is developed for the numerical simulation of geometrically challenging two-dimensional shallow water flows with wetting and drying over convoluted topography. In the framework of sloping bottom model, a modified formulation of shallow water equations is used to preserve mass conservation during flooding and recession. The key ingredient of the model is the use of this combination of the sloping bottom model and the modified shallow water equations to provide a robust technique for wet/dry fronts tracking and, together with centered discretization of the bed slope source term, to exactly preserve the static flow on irregular topographies. The variable reconstruction technique ensures nonnegative reconstructed water depth and reasonable reconstructed velocity, and the friction terms are solved by semi-implicit scheme that does not invert the direction of velocity components. The robustness and accuracy of the proposed model are assessed by comparing numerical and reference results of extensive test cases. Moreover, the results of a dam-break flooding over real topography are presented to show the capability of the model on field-scale application.     

Shallow sediment transport flow computation using time-varying sediment adaptation length

Based on the common approach,the adaptation length in sediment transport is normally estimated astemporally independent.However,this approach might not be theoretically justified as the process of reaching the sediment transport equilibrium stage is affected by the flow conditions in time,especially for fast moving flows,such as scour-hole developing flows.In this study,the two-dimensional（2D） shallow water formulation together with a sediment continuity-concentration（SCC） model were applied to flow with mobile sediment boundary.A timevarying approach was proposed to determine the sediment transport adaptation length to simulate the sediment erosion-deposition rate.The proposed computational model was based on the Finite Volume（FV） method.The Monotone Upwind Scheme of Conservative Laws（MUSCL）-Hancock scheme was used with the Harten Lax van Leer-contact（HLLC） approximate Riemann solver to discretize the FV model.In the flow applications of this paper,a highly discontinuous dam-break,fast sediment transport flow was used to calibrate the proposed timevarying sediment adaptation length model.Then the calibrated model was further applied to two separate experimental sediment transport flow applications documented in the literature,i.e.a highly concentrated sediment transport flow in a wide alluvial channel and a sediment aggradation flow.Good agreement with the experimental data were obtained with the proposed model simulations.The tests prove that the proposed model,which was calibrated by the discontinuous dam-break bed scouring flow,also performed well to represent rapid bed change and steady sediment mobility conditions.     

High-order balanced CWENO scheme for movable bed shallow water equations

In this work the numerical integration of 1D shallow water equations (SWE) over movable bed is performed using a well-balanced central weighted essentially non-oscillatory (CWENO) scheme, fourth-order accurate in space and in time. Time accuracy is obtained following a Runge–Kutta (RK) procedure, coupled with its natural continuous extension (NCE). Spatial accuracy is obtained using WENO reconstructions of conservative variables and of flux and bed derivatives. An original treatment for bed slope source term, which maintains the established order of accuracy and satisfies the property of exactly preserving the quiescent flow (C-property), is introduced in the scheme. This treatment consists of two procedures. The former involves the evaluation of the point-values of the flux derivative, considered as a whole with the bed slope source term. The latter involves the spatial integration of the source term, analytically manipulated to take advantage from the expected regularity of the free surface elevation. The high accuracy of the scheme allows to obtain good results using coarse grids, with consequent gain in terms of computational effort. The well-balancing of the scheme allows to reproduce small perturbations of the free surface and of the bottom otherwise of the same order of magnitude of the numerical errors induced by the non-balancing. The accuracy, the well-balancing and the good resolution of the model in reproducing free surface flow over movable bed are tested over analytical solutions and over numerical results available in literature.     

High-order finite volume WENO schemes for the shallow water equations with dry states

The shallow water equations are used to model flows in rivers and coastal areas, and have wide applications in ocean, hydraulic engineering, and atmospheric modeling. These equations have still water steady state solutions in which the flux gradients are balanced by the source term. It is desirable to develop numerical methods which preserve exactly these steady state solutions. Another main difficulty usually arising from the simulation of dam breaks and flood waves flows is the appearance of dry areas where no water is present. If no special attention is paid, standard numerical methods may fail near dry/wet front and produce non-physical negative water height. A high-order accurate finite volume weighted essentially non-oscillatory (WENO) scheme is proposed in this paper to address these difficulties and to provide an efficient and robust method for solving the shallow water equations. A simple, easy-to-implement positivity-preserving limiter is introduced. One- and two-dimensional numerical examples are provided to verify the positivity-preserving property, well-balanced property, high-order accuracy, and good resolution for smooth and discontinuous solutions.     

A mass conservative scheme for simulating shallow flows over variable topographies using unstructured grid

Most available numerical methods face problems, in the presence of variable topographies, due to the imbalance between the source and flux terms. Treatments for this problem generally work well for structured grids, but most of them are not directly applicable for unstructured grids. On the other hand, despite of their good performance for discontinuous flows, most available numerical schemes (such as HLL flux and ENO schemes) induce a high level of numerical diffusion in simulating recirculating flows. A numerical method for simulating shallow recirculating flows over a variable topography on unstructured grids is presented. This mass conservative approach can simulate different flow conditions including recirculating, transcritical and discontinuous flows over variable topographies without upwinding of source terms and with a low level of numerical diffusion. Different numerical tests cases are presented to show the performance of the scheme for some challenging problems.     

A weighted surface-depth gradient method for the numerical integration of the 2D shallow water equations with topography

A finite volume MUSCL scheme for the numerical integration of 2D shallow water equations is presented. In the framework of the SLIC scheme, the proposed weighted surface-depth gradient method (WSDGM) computes intercell water depths through a weighted average of DGM and SGM reconstructions, in which the weight function depends on the local Froude number. This combination makes the scheme capable of performing a robust tracking of wet/dry fronts and, together with an unsplit centered discretization of the bed slope source term, of maintaining the static condition on non-flat topographies (C-property). A correction of the numerical fluxes in the computational cells with water depth smaller than a fixed tolerance enables a drastic reduction of the mass error in the presence of wetting and drying fronts. The effectiveness and robustness of the proposed scheme are assessed by comparing numerical results with analytical and reference solutions of a set of test cases. Moreover, to show the capability of the numerical model on field-scale applications, the results of a dam-break scenario are presented.     

Comparison between TVD-MacCormack and ADI-type solvers of the shallow water equations

A total variation diminishing (TVD) modification of the MacCormack scheme is developed for simulating shallow water dynamics on a uniform Cartesian grid. Results obtained using conventional and deviatoric forms of the conservative non-linear shallow water equations (SWEs) are compared for cases where the bed has a varying topography. The comparisons demonstrate that the deviatoric form of the SWEs gives more accurate results than the conventional form, in the absence of numerical balancing of the flux-gradient and source terms. A further comparison is undertaken between the TVD-MacCormack model and an alternating direction implicit (ADI) model for cases involving steep-fronted shallow flows. It is demonstrated that the ADI model is unable to predict trans-critical flows correctly, and artificial viscosity has to be introduced to remove spurious oscillations. The TVD-MacCormack model reproduces all flow regimes accurately. Finally, the TVD-MacCormack model is used to predict a laboratory-scale dyke break undertaken at Delft University of Technology. The predictions agree closely with the experimental data, and are in excellent agreement with results from an alternative Godunov-type model.     

A 2D weakly-coupled and efficient numerical model for transient shallow flow and movable bed

Recent advances in the simulation of free surface flows over mobile bed have shown that accurate and stable results in realistic problems can be provided if an appropriate coupling between the shallow water equations (SWE) and the Exner equation is performed. This coupling can be done if using a suitable Jacobian matrix. As a result, faithful numerical predictions are available for a wide range of flow conditions and empirical bed load discharge formulations, allowing to investigate the best option in each case study, which is mandatory in these type of environmental problems. When coupling the equations, the SWE are considered but including an extra conservation law for the sediment dynamics. In this way the computational cost may become unrealistic in situations where the application of the SWE over rigid bed can be used involving large time and space scales without giving up to the adequate level of mesh refinement. Therefore, for restoring the numerical efficiency, the coupling technique is simplified, not decreasing the number of waves involved in the Riemann problem but simplifying their definitions. The effects of the approximations made are tested against experimental data which include transient problems over erodible bed. The simplified model is formulated under a general framework able to insert any desirable discharge solid load formula.     

Well-balanced high-order centered schemes on unstructured meshes for shallow water equations with fixed and mobile bed

In this paper, we study the numerical approximation of the two-dimensional morphodynamic model governed by the shallow water equations and bed-load transport following a coupled solution strategy. The resulting system of governing equations contains non-conservative products and it is solved simultaneously within each time step. The numerical solution is obtained using a new high-order accurate centered scheme of the finite volume type on unstructured meshes, which is an extension of the one-dimensional PRICE-C scheme recently proposed in Canestrelli et al. (2009) [5]. The resulting first-order accurate centered method is then extended to high order of accuracy in space via a high order WENO reconstruction technique and in time via a local continuous space–time Galerkin predictor method. The scheme is applied to the shallow water equations and the well-balanced properties of the method are investigated. Finally, we apply the new scheme to different test cases with both fixed and movable bed. An attractive future of the proposed method is that it is particularly suitable for engineering applications since it allows practitioners to adopt the most suitable sediment transport formula which better fits the field data.     

A GPU-based numerical model coupling hydrodynamical and morphological processes

Sediment transport simulations are important in practical engineering. In this study, a graphics processing unit (GPU)-based numerical model coupling hydrodynamical and morphological processes was developed to simulate water flow, sediment transport, and morphological changes. Aiming at accurately predicting the sediment transport and sediment scouring processes, the model resolved the realistic features of sediment transport and used a GPU-based parallel computing technique to the accelerate calculation. This model was created in the framework of a Godunov-type finite volume scheme to solve the shallow water equations (SWEs). The SWEs were discretized into algebraic equations by the finite volume method. The fluxes of mass and momentum were computed by the Harten, Lax, and van Leer Contact (HLLC) approximate Riemann solver, and the friction source terms were calculated by the proposed a splitting point-implicit method. These values were evaluated using a novel 2D edge-based MUSCL scheme. The code was programmed using C++ and CUDA, which could run on GPUs to substantially accelerate the computation. The aim of the work was to develop a GPU-based numerical model to simulate hydrodynamical and morphological processes. The novelty is the application of the GPU techniques in the numerical model, making it possible to simulate the sediment transport and bed evolution in a high-resolution but efficient manner. The model was applied to two cases to evaluate bed evolution and the effects of the morphological changes on the flood patterns with high resolution. This indicated that the GPU-based high-resolution hydro-geomorphological model was capable of reproducing morphological processes. The computational times for this test case on the GPU and CPU were 298.1 and 4531.2 s, respectively, indicating that the GPU could accelerate the computation 15.2 times. Compared with the traditional CPU high-grid resolution, the proposed GPU-based high-resolution numerical model improved the reconstruction speed more than 2.0–12.83 times for different grid resolutions while remaining computationally efficient.     

Probabilistic solution of floodplain inundation equation

Uncertainty in bed roughness is a dominant factor in providing a sufficiently accurate simulation of floodplain flows. This study describes a method to compute the transition probability density distribution of time-varying water elevations where the evolutionary process is based on a conventional one-dimensional storage cell model with governing stochastic differential equation. By including the random inputs (or noise terms) of bed roughness and initial water depth, time-dependent and spatially varying probability density function of the water surface leads to a Fokker–Planck equation. The model’s performance is evaluated by applying it to shallow water flow with a horizontal bed. Sensitivity of model predictions to variations in the bed friction parameters is shown. By comparing the result of the proposed method with that of conventional Monte Carlo simulation, the advantage of the former as a method for density function prediction is confirmed.     

TWO DIMENSIONAL NUMERICAL SIMULATION OF FLOW AND GEO-MORPHOLOGICAL PROCESSES NEAR HEADLANDS BY USING UNSTRUCTURED GRID

1 INTRODUCTION In recent years, due to the increase in population and industrial developments, mankind has faced manyproblems associated with rivers, coastal waters and reservoirs. Some of these problems are flood control,water supply, power generation, and irrigation. In addition, making new hydraulic structures changesnatural conditions. Prediction of these changes is necessary for designing such constructions. For solutionof these problems usually an assessment of flow pattern, sedim…     

A kinetic flux vector splitting scheme for shallow water flows

A kinetic flux vector splitting (KFVS) scheme for shallow water flows based on the collisionless Boltzmann equation is formulated and applied. The scheme is explicit and first order in space and time with stability governed by the Courant condition. The consistency of the KFVS scheme with the shallow water equations is proven using the equivalent differential equations approach. The accuracy and efficiency of the KFVS scheme in modeling complex flow features are compared to those of the Boltzmann Bhatnagar–Gross–Krook (BGK) scheme as well as a Riemann-based scheme. In particular, all schemes are applied to (i) strong shock waves, (ii) extreme expansion waves, (iii) a combination of strong shock waves and extreme expansion waves, and (iv) a one-dimensional dam break problem. Additionally, the KFVS, BGK and Riemann schemes are applied to a one-dimensional dam break problem for which laboratory data is available. These test cases reveal that all three schemes provide solutions of comparable accuracy, but the KFVS model is 1.5–2 times faster to execute than the BGK scheme and 2–3 times faster than the Riemann-based scheme. The absence of the collision term from the Boltzmann equation not only makes the mathematical formulation of KFVS easy but also helps elucidate this approach to the novice. The accuracy, efficiency, and simplicity of the KFVS scheme indicate its potential in modeling an array of water resources problems. Due to the scalar nature of the Boltzmann equation, the extension of the KFVS scheme to 2-D surface water flows is straightforward.     

Multi-block lattice Boltzmann simulations of solute transport in shallow water flows

We report a two-dimensional multi-block lattice Boltzmann model for solute transport in shallow water flows, which is developed based on the advection–diffusion equation for mass transport and the shallow water equations for the flows. A weighting factor is included in the centered scheme for improved accuracy. The model is firstly verified by simulating three benchmark tests: wind-driven circulation in a dish-shaped lake, jet-forced flow in a circular basin, and flow formed by two parallel streams containing different uniform concentrations at the same constant velocity; and then it is applied to a practical wind-induced flow, Baiyangdian Lake, which is characterized by irregular geometries and complex bathymetries. The numerical results have shown that the model is able to produce accurate and detailed results for both water flows and solute transport, which is attractive, especially for flows in narrow zones of practical terrains and certain areas with largely varying pollutant concentrations.     

One-dimensional numerical simulation of non-uniform sediment transport under unsteady flows

One-dimensional numerical models are popularly used in sediment transport research because they can be easily programmed and cost less time compared with two- and three-dimensional numerical models. In particular, they possess greater capacity to be applied in large river basins with many tributaries. This paper presents a one-dimensional numerical model capable of calculating total-load sediment transport. The cross-section-averaged sediment transport capacity and recovery coefficient are addressed in the suspended load model. This one-dimensional model, therefore, can be applied to fine suspended loads and to hyperconcentrated flows in the Yellow River. Moreover, a new discretization scheme for the equation of unsteady non-uniform suspended sediment transport is proposed. The model is calibrated using data measured from the Yantan Reservoir on the Hongshui River and the Sanmenxia Reservoir on the Yellow River. A comparison of the calculated water level and river bed deformation with field measurements Shows that the improved numerical model is capable of predicting flow, sediment transport, bed changes, and bed-material sorting in various situations, with reasonable accuracy and reliability.     

Numerical simulation of internal solitary wave—induced reverse flow and associated vortices in a shallow,two-layer fluid benthic boundary layer

The wave-induced velocity and pressure fields beneath a large amplitude internal solitary wave of depression propagating over a smooth, flat, horizontal, and rigid boundary in a shallow two-layer fluid are computed numerically. A numerical ocean model is utilised, the set-up of which is designed and tuned to replicate the previously published experimental results of Carr and Davies (Phys Fluids 18(1):016,601–1–016,601–10, 2006). Excellent agreement is found between the two data sets and, in particular, the numerical simulation replicates the finding of a reverse flow along the bed aft of the wave. The numerically computed velocity and pressure gradients confirm that the occurrence of the reverse flow is a consequence of boundary layer separation in the adverse pressure gradient region. In addition, vortices associated with the reverse flow are seen to form near the bed.