An Unstructured Grid, Three-dimensional Model Based on the Shallow Water Equations PDF Download

Author: Vincenzo Casulli
Publisher:
ISBN:
Category :
Languages : en
Pages : 18

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Author: Xin Liu
Publisher:
ISBN:
Category :
Languages : en
Pages :

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This Ph.D. thesis aims to develop numerical models for two-dimensional and three-dimensional shallow water systems over mobile beds. In order to accomplish the goal of this dissertation, the following sub-projects are defined and completed. 1: The first sub-project consists in developing a robust two-dimensional coupled numerical model based on an unstructured mesh, which can simulate rapidly varying flows over an erodible bed involving wet-dry fronts that is a complex yet practically important problem. In this task, the central-upwind scheme is extended to simulation of bed erosion and sediment transport, a modified shallow water system is adopted to improve the model, a wetting and drying scheme is proposed for tracking wet-dry interfaces and stably predict the bed erosion near wet-dry area. The shallow water, sediment transport and bed evolution equations are coupled in the governing system. The proposed model can efficiently track wetting and drying interfaces while preserving stability in simulating the bed erosion near the wet-dry fronts. The additional terms in shallow water equations can improve the accuracy of the simulation when intense sediment-exchange exists; the central-upwind method adopted in the current study shows great accuracy and efficiency compared with other popular solvers; the developed model is robust, efficient and accurate in dealing with various challenging cases. 2: The second sub-project consists in developing a novel numerical scheme for a coupled two-dimensional hyperbolic system consisting of the shallow water equations with friction terms coupled with the equations modeling the sediment transport and bed evolution. The resulting 5*5 hyperbolic system of balance laws is numerically solved using a Godunov-type central-upwind scheme on a triangular grid. A spatially second-order and temporally third-order central-upwind scheme has been derived to discretize the conservative hyperbolic sub-system. However, such schemes need a correct evaluation of local wave speeds to avoid instabilities. To address such an issue, a mathematical result by the Lagrange theorem is used in the proposed scheme. Consequently, a computationally expensive process of finding all of the eigenvalues of the Jacobian matrices is avoided: The upper/lower bounds on the largest/smallest local speeds of propagation are estimated using the Lagrange theorem. In addition, a special discretization of the bed-slope term is proposed to guarantee the well-balanced property of the designed scheme. 3: The third sub-project consists in designing a novel scheme to estimate bed-load fluxes which can produce more accurate results than the previously reported coupled model. Using a pair of local wave speeds different from those used for the flow, a novel wave estimator in conjunction with the central upwind method is proposed and successfully applied to the coupled water-sediment system involving a rapid bed-erosion process. It was demonstrated that, in comparison with the decoupled model, applying the proposed novel scheme to approximate the bed-load fluxes can successfully avoid the numerical oscillations caused by simple and less stable schemes, e.g. simple upwind methods; in comparison with the coupled model using same flux-estimator for both hydrodynamic and morphological systems, the proposed numerical scheme successfully prevents excessive numerical diffusion for prediction of bed evolution. Consequently, the proposed scheme has advantages in terms of accuracy which are shown in several numerical tests. In addition, analytical expressions have been provided for calculating the eigenvalues of the coupled shallow-water-Exner system, which greatly enhances the efficiency of the proposed method. 4: The fourth sub-project consists in developing a three-dimensional numerical model for the simulation of unsteady non-hydrostatic shallow water flows on unstructured grids using the finite volume method. The free surface variations are modeled by a characteristics-based scheme which simulates sub- and super-critical flows. Three-dimensional velocity components are considered in a collocated arrangement with a sigma coordinate system. A special treatment of the pressure term is developed to avoid the water surface oscillations. Convective and diffusive terms are approximated explicitly, and an implicit discretization is used for the pressure term. The unstructured grid in the horizontal direction and the sigma coordinate in the vertical direction facilitate the use of the model in complicated geometries. 5: The fifth sub-project consists in developing a well-balanced three-dimensional shallow water model which is able to simulate shock waves over dry bed. Due to the hydrostatic simplification of the vertical momentum equation, the governing system of equations is not hyperbolic and can not be solved using standard hyperbolic solvers. That is, one can not use a high-order Godunov-type scheme to compute all fluxes through cell-interfaces. This may cause the model to fail in simulations of some unsteady-flows with discontinuities, e.g., dam-break flows and floods. To overcome this difficulty, a novel numerical scheme for the three-dimensional shallow water equations is proposed using a relaxation approach in order to convert the system to a hyperbolic one. Thus, a high-order Godunov-type central-upwind scheme based on the finite volume method can be applied to approximate the numerical fluxes. The proposed model can also preserve the ``lake at rest'' state and positivity of water depth over irregular bottom topographies based on special reconstruction of the corresponding parameters. 6: The sixth sub-project consists in extending the result of the fifth sub-project to development of a three-dimensional numerical model for shallow water flows over mobile beds, which is able to simulate morphological evolutions under shock waves, e.g. dam-break flows. The hydrodynamic model solves the three-dimensional shallow water equations using a finite volume method on prismatic cells in sigma coordinates based on the scheme prposed in sub-project 5. The morphodynamic model solves an Exner equation consisting of bed-load sediment transportation. The performance of the proposed model has been demonstrated by several laboratory experiments of dam-break flows over mobile beds.

Author: E. D. de Goede
Publisher:
ISBN:
Category : Hydrodynamics
Languages : en
Pages : 194

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Holl. Zusammenfass.

Author: Francisco Jose Rueda-Valdivia
Publisher:
ISBN:
Category :
Languages : en
Pages : 870

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Author: Pablo Higuera
Publisher: World Scientific
ISBN: 981126547X
Category : Science
Languages : en
Pages : 208

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This unique compendium introduces the field of numerical modelling of water waves. The topics included the most widely used water wave modelling approaches, presented in increasing order of complexity and categorized into phase-averaged and phase-resolving at the highest level.A comprehensive state-of-the-art review is provided for each chapter, comprising the historical development of the method, the most relevant models and their practical applications. A full description on the method's underlying assumptions and limitations are also provided. The final chapter features coupling among different models, outlining the different types of implementations, highlighting their pros and cons, and providing numerous relevant examples for full context.The useful reference text benefits professionals, researchers, academics, graduate and undergraduate students in wave mechanics in general and coastal and ocean engineering in particular.

Author: Alfredo Bermúdez de Castro
Publisher: Springer Science & Business Media
ISBN: 3540342885
Category : Mathematics
Languages : en
Pages : 1202

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These proceedings collect lectures given at ENUMATH 2005, the 6th European Conference on Numerical Mathematics and Advanced Applications held in Santiago de Compostela, Spain in July, 2005. Topics include applications such as fluid dynamics, electromagnetism, structural mechanics, interface problems, waves, finance, heat transfer, unbounded domains, numerical linear algebra, convection-diffusion, as well as methodologies such as a posteriori error estimates, discontinuous Galerkin methods, multiscale methods, optimization, and more.

Author: Luca Bonaventura
Publisher: Springer Science & Business Media
ISBN: 3642238319
Category : Science
Languages : en
Pages : 102

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Collected articles in this series are dedicated to the development and use of software for earth system modelling and aims at bridging the gap between IT solutions and climate science. The particular topic covered in this volume addresses the historical development, state of the art and future perspectives of the mathematical techniques employed for numerical approximation of the equations describing atmospheric and oceanic motion. Furthermore, it describes the main computer science and software engineering strategies employed to turn these mathematical methods into effective tools for understanding earth's climate and forecasting its evolution. These methods and the resulting computer algorithms lie at the core of earth system models and are essential for their effectiveness and predictive skill.

Author: Michael Griebel
Publisher: Springer
ISBN: 3319519549
Category : Computers
Languages : en
Pages : 245

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There have been substantial developments in meshfree methods, particle methods, and generalized finite element methods since the mid 1990s. The growing interest in these methods is in part due to the fact that they offer extremely flexible numerical tools and can be interpreted in a number of ways. For instance, meshfree methods can be viewed as a natural extension of classical finite element and finite difference methods to scattered node configurations with no fixed connectivity. Furthermore, meshfree methods have a number of advantageous features that are especially attractive when dealing with multiscale phenomena: A-priori knowledge about the solution’s particular local behavior can easily be introduced into the meshfree approximation space, and coarse scale approximations can be seamlessly refined by adding fine scale information. However, the implementation of meshfree methods and their parallelization also requires special attention, for instance with respect to numerical integration.

Author: Daniel David Hartig
Publisher:
ISBN:
Category : Lagrange equations
Languages : en
Pages : 84

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This project developed fluid circulation models for the two- and three-dimensional Lagrangian shallow water equations. There were two stages to this development: in the first, the two-dimensional shallow water equations were transformed from first principles of oceanography into a serial implementation in MATLAB. In the second part, a serial implementation of the three-dimensional shallow water equations, developed by Dr. James Greenberg, was modified to run in parallel on many nodes of a computing cluster. The serial, one-dimensional model includes one lateral degree of freedom -- the x-direction. The vertical or z-direction is modeled by layers; velocity in this direction was removed by a series of transformations to the governing equations. Development started with the conservation of mass and conservation of momentum equations. The traction terms in these equations were approximated using a method previously established by Mellor and Blumberg in their work on the Princeton Ocean Model (POM). These equations were scaled and then transformed to an s-coordinate system, again following the established method of POM. Once a set of partial differential equations were derived, these equations were discretized in space and time and solved in MATLAB. The implementation of the model in MATLAB allows the user a wide range of initial conditions for factors such as the bathymetry, initial area covered in fluid, magnitude of friction coefficients, and more. For a given set of initial conditions, the model steps forward in time by user-designated time steps solving for position, velocity, and depth of the fluid at each step and visually representing this information with appropriate graphs. The next stage of the project jumped forward to a serial three-dimensional implementation of the shallow water equations developed by Dr. Greenberg. This model is similar to the two-dimensional model but with the added complexity of two lateral degrees of freedom -- both the x- and y-directions -- while the vertical component is still treated the same as in the two-dimensional model. In order to split the two-dimensional domain grid into a set of smaller domains assigned to the various processors, the special geometry of that grid had to be taken into account. Once a scheme was in place to divide the grid while maintaining all of its special properties, the computation on each sub-domain was performed with the same program that operates on the entire domain. This allowed easy implementation of a parallel solution: each node ran a modified serial implementation on a subsection of the larger problem that had been carefully separated from the whole. A process was created to allow the nodes to communicate data from the edges of their sub-domains as they advanced forward through time. There are two deliverable products from this project. First, there is a serial two-dimensional model of fluid circulation that takes into account many different user-designated initial conditions and can be useful for determining how well the mathematics of this model can approximate physical phenomena. Secondly, this project produced a three-dimensional parallel model that serves as a proof of concept for future development of more advanced parallel models.

Author: Henry J. Bokuniewicz
Publisher: MDPI
ISBN: 3038420468
Category : Electronic books
Languages : en
Pages : 434

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This book is a printed edition of the Special Issue "Selected Papers from the 13th Estuarine and Coastal Modeling Conference" that was published in JMSE