Coupled and splitting bedload sediment transport models based on a modified flux-wave approach

Current topographic survey technology provides high-resolution (HR) datasets for urban environments. Incorporating this HR information in models aiming to provide flood risk assessment is desirable because the flood wave propagation is depending on the urban topographic features, i.e. buildings, bridges and street networks. Conceptual, numerical and practical challenges arise from the application of shallow water models to HR urban flood modeling. For instance, numerical challenges are occurrence of wet-dry fronts, geometric discontinuities in the urban environment and discontinuous solutions, i.e. shock waves. These challenges can be overcome by using a Godunov-type scheme. However, the computational cost of this type of schemes is high, such that HR two-dimensional shallow water simulations with practical relevance have to be run on supercomputers. The porous shallow water model is an alternative approach that aims to reduce computational cost by using a coarse resolution and accounting for unresolved processes by means of the porosity terms. Usually, a speedup between two and three orders of magnitude in comparison to HR simulations can be obtained. This study reports preliminary results of a practical test case concerning pluvial flooding in a district of the city of Nice, France, caused by the intense rainfall event on October 3rd, 2015. HR topography data set on a 1 m resolution is available for the district, whereby street features of infra-metric dimensions have been included. A reference solution is calculated by a HR shallow water model on a 1 m by 1 m structured computational grid. The porous shallow water model is run on a 10 m by 10 m grid and the influence of the drag source term is studied. The model results show a large deviation, which is caused by the poor meshing strategy of the porous shallow water (AP) model. The study also summarizes practical challenges that arise during the application of the AP and HR models to a large urban catchment. The main difficulty is to obtain a good mesh. In smaller scale investigations, the mesh is currently constructed by hand such that the cell edges align with buildings. This approach is not feasible for large scale urban catchments with a large number of buildings. Future steps that have to be taken, such as a strategy for automatic mesh generation, are reported on.

The one-dimensional shallow water equations (SWEs), coupled with bed load (Exner equation) were solved in this paper for the numerical simulation of the sediment transport problem. By using an efficient meshless method based on radial basis function (RBF) that approximates the unknown variables on a set of collocation nodes and Runge–Kutta fourth-order scheme to approximate the temporal derivative. Besides, the artificial viscosity to avoid oscillations, we implement the proposed numerical model of bedload and set up a series of numerical tests using various nontrivial solutions to confirm the efficiency of the suggested approaches. The numerical simulations demonstrate the accuracy, computational efficiency, and ability of the suggested numerical methods in solving the coupled Exner shallow water problem.

Abstract. Modeling the dynamics of seawater typically uses a shallow water model. The shallow water model is derived from the mass conservation equation and the momentum set into shallow water equations. A two-dimensional shallow water equation alongside the model that is integrated with depth is described in numerical form. This equation can be solved by finite different methods either explicitly or implicitly. In this modeling, the two dimensional shallow water equations are described in discrete form using explicit schemes.Keyword: shallow water equation, finite difference and schema explisit.REFERENSI 1. Bunya, S., Westerink, J. J. dan Yoshimura. 2005. Discontinuous Boundary Implementation for the Shallow Water Equations. Int. J. Numer. Meth. Fluids. 47: 1451-1468.2. Kampf Jochen. 2009. Ocean Modelling For Beginners. Springer Heidelberg Dordrecht. London New York.3. Rezolla, L 2011. Numerical Methods for the Solution of Partial Diferential Equations. Trieste. International Schoolfor Advanced Studies.4. Natakussumah, K. D., Kusuma, S. B. M., Darmawan, H., Adityawan, B. M. Dan Farid, M. 2007. Pemodelan Hubungan Hujan dan Aliran Permukaan pada Suatu DAS dengan Metode Beda Hingga. ITB Sain dan Tek. 39: 97-123.5. Casulli, V. dan Walters, A. R. 2000. An unstructured grid, three-dimensional model based on the shallow water equations. Int. J. Numer. Meth. Fluids. 32: 331-348.6. Triatmodjo, B. 2002. Metode Numerik Beta Offset. Yogyakarta.

A unified treatment of steady-state shallow water and two-dimensional navier-stokes equations — finite element penalty function approach

Simulation of fluid flow using Shallow water equations (SWE) and sediment movement below it using Exner equation is given. Both of the equations will be combined using splitting technique, in which SWE would be computed using Harten-Lax-van Leer and Einfeldt (HLLE) numerical flux, then Exner would be computed semi-implicitly. This paper elaborates the steps of constructing SWE-Exner model. To show the agreement of the scheme, two problems will be elaborated: (1) comparison between analytical solution and numerical solution, and (2) parallelism using OpenMP for Transcritical over a granular bump. The first problem is going to tell the discrete $L^{1}$-, $L^{2}$-, and $L^{\infty}$-norm error of the scheme, and the second one will show the simulation result, speedup, and efficiency of the scheme, which is around $56.44\%$.

A new numerical technique by Chang is described and used to solve the one-dimensional (1D) and two-dimensional (2D) Saint Venant equations. This new technique differs from traditional numerical methods (i.e., finite-difference, finite-element, finite-volume, spectral methods, etc.). Chang's method treats space and time on the same footing, so that space and time are unified—a key characteristic that distinguishes the new method from other techniques. This method is explicit, uses a staggered grid, enforces flux conservation in space and time, and does not require upwinding, flux-splitting, flux limiters, evaluation of eigenvalues, or the addition of artificial viscosity. Furthermore, the scheme is simple, easy to implement (see Appendix I), and can be extended to higher dimensions. First, the new technique, as developed by Chang, is introduced, explained, and applied to the 1D Saint Venant equations. To illustrate its effectiveness, an idealized dam-break and a hydraulic jump in a straight rectangular channel are simulated. The numerical results obtained using the new method are compared with results from other, more conventional techniques of similar complexity, experimental data, and an analytical solution. Next, Chang's scheme is extended for solution of the 2D Saint Venant (i.e., shallow water or depth-averaged) equations. The writers do not follow Chang's 2D development but instead employ Strang's method of fractional steps, which transforms the 2D problem into two 1D problems. The new 2D scheme is applied to an oblique hydraulic jump, and the results are compared with an analytical solution.

Coupling of shallow water and circulation models for prediction of multiphysics coastal flows: Method, implementation, and experiment

Hydrodynamic coefficient estimation for ship manoeuvring in shallow water using an optimal truncated LS-SVM

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.

Dispersive shallow water waves over a porous sea bed

Acoustic Doppler velocity profilers (ADVP) measure the velocity simultaneously in a linear array of bins. They have been successfully used in the past to measure three-dimensional turbulent flow and the dynamics of suspended sediment. The capability of ADVP systems to measure bedload sediment flux remains uncertain. The main outstanding question relates to the physical meaning of the velocity measured in the region where bedload sediment transport occurs. The main hypothesis of the paper, that the ADVP measures the velocity of the moving bedload particles, is validated in laboratory experiments that range from weak to intense bedload transport. First, a detailed analysis of the raw return signals recorded by the ADVP reveals a clear footprint of the bedload sediment particles, demonstrating that these are the main scattering sources. Second, time-averaged and temporal fluctuations of bedload transport derived from high-speed videography are in good agreement with ADVP estimates. Third, ADVP based estimates of bedload velocity and thickness of the bedload layer comply with semi-theoretical expressions based on previous results. An ADVP configuration optimized for bedload measurements is found to perform only marginally better than the standard configuration for flow measurements, indicating that the standard ADVP configuration can be used for sediment flux investigations. Data treatment procedures are developed that identify the immobile-bed surface, the layers of rolling/sliding and saltating bedload particles, and the thickness of the bedload layer. Combining ADVP measurements of the bedload velocity with measurements of particle concentration provided by existing technology would provide the sediment flux.

This thesis is concerned with the solution of the shallow water wave equations. To solve the equations for a real life domain of flow a numerical solution is required. Analytical solutions are useful for testing numerical solutions. Previous research in this area is examined. Existing analytical solutions of the nonlinear shallow water equations involving moving boundaries do not include bottom friction. Existing analytical solutions of the nonlinear shallow water equations involving fixed boundaries can be modifed. The SLM (Selective Lumped Mass) scheme for numerically solving the shallow water equations has been used to accurately model tides both in bays with fixed boundaries and bays with moving boundaries. The SLM scheme for domains with moving boundaries has not been fully explained in published papers. Analytical solutions are established for moving boundary shallow water equations involving nonlinear continuity for unforced frictional flow in parabolic canals, circular paraboloids and elliptical paraboloids, and both for forced frictional flow and unforced frictionless flow in a parabolic canal. Analytical solutions are established for one dimensional nonlinear frictionless shallow water wave flow in a basin of constant depth, with a fixed boundary and with a sinusoidal input at the open boundary. The SLM scheme is coded in Visual C ++ and validated against an analytical solution, a convergence study is carried out for the SLM scheme and a computer program is written in Visual C ++ to generate finite element meshes. The SLM scheme is used to model one dimensional nonlinear shallow water flow in a basin of constant depth, with fixed boundary and with sinusoidal input at the open boundary; results obtained were close to those in the analytical solution. The SLM scheme when applied to a moving boundary domain has been modifed, with the resultant scheme used to model forced frictionless flow and forced frictional flow in a parabolic canal, with results close to those of the analytical solutions developed. The SLM scheme is applied to model accurately the existing tidal heights and currents in Port Phillip Bay, Victoria, Australia. The SLM scheme is applied to model the effect of proposed channel deepening on the tides in Port Phillip Bay; the effect is small, increasing the tidal heights by at most seven millimetres. The conclusion of this thesis is that successful analytical and numerical solutions of the shallow water equations are developed both for domains with fixed boundaries and domains with moving boundaries.

<p>With the possible climate change and increased pace of urbanization in the century, urban flooding has caused more and more attentions nowadays. Shallow water equations are widely used to reproduce the flow hydrodynamics of flooding around the urban areas, which have been proved a powerful tool for flood risk assessment and evacuation management, like river flow or flowing at drainage networks with irregular cross-sections at 1D scale. Over the last two decades, Godunov-type schemes have became popular for its robustness treating complex flow phenomenons. When tacking complex topography in the framework of Godunov-type scheme, sourer term needs to be treated property to preserve steady state, that flux gradient and sourer term are balanced. Capart et al. (2003) reconstructed the momentum flux by considering the balance of hydrostatic pressure with the approximated water surface level, which has the ability to tackle the irregular and non-prismatic channel flow with complex topography. This approximation is exact for two cases: 1) rectangular and prismatic channel; 2) water surface is horizontal. However, for other cases, approximation is employed to achieve the hydrostatic equilibrium, which has reduced the accuracy of the numerical solution and increased the complexity for the model implementation. </p><p>In this work, we present a new well-balanced numerical scheme for simulating 1D frictional shallow water flow with irregular cross-sections over complex topography 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 (Liang and Marche, 2009). HLL approximated Riemann solver is adopted for the flux calculation at the cell interface. Non-negative reconstruction of Riemann state (Audusse et al., 2004) and local bed modification (Liang, 2010) produce stable and well-balanced solutions to shallow water flow hydrodynamics. Bed slope source term can be approximated using central difference and no special treatment is needed for wet and dry bed. The friction source term is discretized using a splitting implicit scheme and limiting value of friction force is used to ensure stability for the dry bottom (Liang and Marche, 2009). The new numerical scheme is validated against two theoretical benchmark tests and then compared with the validated shallow water model with circular and trapezoid cross-sections over complex topography involving wetting and drying. This method is also possible to reproduce the mixed flow in the conduit or for the flow with non-prismatic channel like river flow in the near future.</p><p>References</p><p>Audusse, E., Bouchut, F., Bristeau, M. O., Klein, R., & Perthame, B. T. (2004). A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM Journal on Scientific Computing, 25(6), 2050-2065.</p><p>Capart, H, Eldho, TI, Huang, SY, Young, DL, and Zech, Yves, "Treatment of natural geometry in finite volume river flow computations", Journal of Hydraulic Engineering 129, 5 (2003), pp. 385--393.</p><p>Liang, Qiuhua and Marche, Fabien, "Numerical resolution of well-balanced shallow water equations with complex source terms", Advances in water resources 32, 6 (2009), pp. 873--884.</p><p>Liang, Qiuhua, "Flood simulation using a well-balanced shallow flow model", Journal of hydraulic engineering 136, 9 (2010), pp. 669--675.</p>

This BookThis book deals with computational methods for solving systems of shallow water equations within the broader field of geophysical fluid dynamics.Such systems, comprised of hyperbolic partial differential equations, serve as mathematical models for simulating the dynamics of water flows in open channels, rivers, lakes, and oceans.Practical applications range from modelling oceanic tides to the breaking of waves on shallow beaches, roll waves in open channels, flood waves in rivers, dam-break waves, tsunami wave propagation, and various environmental fluid dynamics scenarios.Mathematical models of the shallow-water type are also employed for simulating gas dynamics, such as the dispersion of heavy gases over the Earth's surface and atmospheric dynamics.However, the physical complexity of time-dependent, threedimensional oceanic and atmospheric flows necessitates more comprehensive mathematical models than shallow-water type systems alone.Nevertheless, the basic, onelayer shallow water equations play a crucial role in providing benchmark solutions for assessing methodologies intended for more complex models.Significantly, multilayer shallow water models are currently undergoing major developments in oceanic and atmospheric dynamics, where the underlying concepts and methodologies from the basic one-layer shallow water equations in this book find applicability.This book offers fundamental concepts and methods essential for constructing advanced high-order computational algorithms within the frameworks of finite volume and discontinuous Galerkin finite element methods.The finite volume methodology is developed in full detail, presenting non-linear, fully discrete high-order upwind and centred methods for solving the non-linear, time-dependent two-dimensional shallow water equations.

The exact solution to the Shallow water Equations Riemann problem at width jumps in rectangular channels