A high‐order fully explicit flux‐form semi‐Lagrangian shallow‐water model

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/106984/1/fld3887.pd

References

www.geosci-model-dev-discuss.net/7/5141/2014/ doi:10.5194/gmdd-7-5141-2014 © Author(s) 2014. CC Attribution 3.0 License. This discussion paper is/has been under review for the journal Geoscientific Model Development (GMD). Please refer to the corresponding final paper in GMD if available. A global finite-element shallow-water model supporting continuous and discontinuous element

An efficient covariant frame for the spherical shallow water equations: Well balanced DG approximation and application to tsunami and storm surge

International audienceIn this work we consider an e cient discretization of the Shallow Water Equations in spherical geometry for oceanographic applications. Instead of the classical 2d-covariant or 3d-Cartesian approaches, we focus on the mixed 3d/2d form of [Bernard et al., JCP 2009] which evolves the 2d momentum tangential to the sphere by projecting the 3d-Cartesian right-hand side on the sphere surface. Di↵erently from the last reference we consider the exact representation of the sphere instead of the finite element one, mixed with a covariant basis projection of the momentum. This leads to several simplifications of the Discontinuous Galerkin scheme: the local mass matrix goes back to the standard block-diagonal form; the Riemann Problem does not require any tensor or vector rotations to align the bases on the two sides of an edge. Second we consider well balancing corrections related to relevant equilibrium states for tsunami and storm surge simulations. These corrections allow to compensate for the inherent non-exactness of the quadrature induced by the non-polynomial nature of both the geometrical mapping and of the covariant basis. In other words, these corrections are the order of the cubature error. We show that their addition makes the scheme exactly well balanced, and is is equivalent to recasting the integral of the hydrostatic pressure term in strong form. The proposed method is validated on academic benchmarks involving both smooth and discontinuous solutions, and applied to realistic tsunami and an historical storm surge simulation

Debris flows and general steep slope shallow water flows numerical simulation

The main target of this research is to develop a numerical model for debris flow simulations. As it is known, in general, this kind of flows occur in steeped mountain slopes. When dealing with the complete 3D physic system of equations that model the phenomenon this singular characteristic has not special effect. To simulate large events (typical scales in real world) is not possible to use the complete equations (three dimensional, variable density, non-hydrotactic…) so a spacial dimension reduction is necessary combined with several simplifications to reduce the complexity of the system, along the present text previous works references are introduced to justify the selected simplifying hypothesis. During the mathematical manipulation of the equations, performed in order to reduce the spacial dimension (3D & 2D), the real complexity of the problem emerges, important consequences on the coordinate system appear. This means that, to obtain a simpler version of the physical model of the phenomenon, complex mathematical operations are needed. In the approach presented in this work the complexity of the problem is reduced in to manners: • Applying direct physical hypothesis on the flow characteristics. • Applying mathematical hypothesis. An example of these physical simplifications could be the monophasic fluid hypothesis.An example of the mathematical simplifications could be the fact of the curvature terms neglecting. In this work, the coordinate system selected for the model is named Proposed Coordinates System (PCS), this coordinates system is based on the work of Bouchut and Westdickenberg (2004) and Berger and Carey (1998a). The metric characteristics of the system are calculated and important conclusions are extracted from the analysis of the curvature terms. A link exist between the curvatures and the Christoffel symbols, if the curvatures are discarded also the Christoffel symbols should be discarded for model consistency. In the model governing differential equations, the use of curvilinear coordinates (PCS) provokes the existence of metric source terms defined as a Christoffel symbols functions. So, neglecting these terms the equations become simplified. Strictly talking the model is valid for steep but slowly varying slopes, where the curvatures are very small, although in the real test cases used for validating the model, the curvatures does not fit this condition. The rough approximation to the debris flow process through simplified physics used in the model probably hides the low accuracy of the model in strongly curved areas. Many other different coordinates system choices exist in the scientific literature, in this work some of them are commented and analyzed. Along the development of the model, different problems appear and different strategies and methodologies are proposed in order to overcome them. The first important problem is related to the physics of the process, the debris flows tend to naturally develop flow pulses, the flowing mixture stops and later is remobilized. The model includes what is called ”stop and go” mechanism to capture this behavior. The second problem is related with the boundary conditions, standard debris flow hydrograph includes sharp gradients which can become flow shocks, in this work a new methodology is proposed to introduce the boundary conditions in a way that shocks are correctly solved also in the boundary of the computational domain, contrary to the standard method of characteristics (Henderson, 1966; Abbott, 1966). The general family of the numerical methods selected to solve the resulting system is the Finite Volume Method (FVM) using the Riemann solver. This approach is extensively used in the debris flow simulation framework, so its selection is justified. In order to fit the special requirements of the PCS coordinates system the methodology developed by Rossmanith et al. (2004) and Rossmanith (2006) is used. This approach solves the problem of the non-orthogonal coordinates in the FVM framework. To validate the code analytical, experimental and real test cases are selected. The model validation process is partial because the lack in analytical solutions, especially for complex geometries

Optimal control for studying wave energy in hydraulic systems

A class of novel models for water waves induced by elastic deformation in the topography is developed and analyzed. The depth-averaged shallow water equations including friction terms for the water free-surface and the well-known second-order elastostatic formulation for the bed deformation have been implemented. Friction forces and water hydrostatic pressure distribution are also accounted for in this model. At the interface between the water ow and the bed topography, transfer conditions are implemented. Furthermore, a hybrid nite element/nite volume method for solving free-surface run-up ow problems over deformable beds has been proposed. The deformations in the topography have been generated by a localized force which causes propagations of the water waves with dierent amplitudes and frequencies. Two dierent methods have been proposed for the transfer of informations through the interface. The rst one is the two-mesh procedure; in this method a proper interpolation has been implemented to transfer the data between the surface nodes and the control volumes using uniform nite volume meshes. In the second method, and to avoid the interpolation at the interface, a nite volume method using non-uniform meshes has been implemented. When the shallow water waves approach the coastline they begin to transform as they enter shallow water regime. As each wave begins to experience the seabed, both run-up and overtopping occur. To solve for this, a class of stable, accurate and simple numerical model for moving wet/dry fronts in shallow water equations using the parametrization concept and the point-wise Riemann solver has been proposed. Many parameters of shallow water equations are subject to uncertainties to the inherit randomness of natural processes. To incorporate uncertain parameters into the stochastic shallow water equations, the stochastic properties of dierent parameters that are considered uncertain, namely in ow boundary condition, the bed friction coecients and the domain topography are added to the system. Development of accurate and ecient tools for uncertainty quantication in shallow water ows has been proposed and carefully examined for single-layer, two-layer - nite volume models. To further quantify the uncertainty in shallow water ows the proposed methods have been extended to multi-layer shallow water ows with mass exchange terms subject to stochastic topography, uncertain friction and viscosity coecients. Several test examples and well-established benchmark problems have been used to assess the numerical performance of the proposed models and methods. Comparisons to experimental measurements have also been carried out in this thesis. Finally, an optimal control technique for bed reconstruction has been presented as in many engineering applications this information is not entirely provided

Two-dimensional shock capturing numerical simulation of shallow water flow applied to dam break analysis

With the advances in the computing world, computational fluid dynamics (CFD) is becoming more and more critical tool in the field of fluid dynamics. In the past few decades, a huge number of CFD models have been developed with ever improved performance. In this research a robust CFD model, called Riemann2D, is extended to model flow over a mobile bed and applied to a full scale dam break problem. Riemann2D, an object oriented hyperbolic solver that solves shallow water equations with an unstructured triangular mesh and using high resolution shock capturing methods, provides a generic framework for the solution of hyperbolic problems. The object-oriented design of Riemann2D has the flexibility to apply the model to any type of hyperbolic problem with the addition of new information and inheriting the common components from the generic part of the model. In a part of this work, this feature of Riemann2D is exploited to enhance the model capabilities to compute flow over mobile beds. This is achieved by incorporating the two dimensional version of the one dimensional non-capacity model for erodible bed hydraulics by Cao et al. (2004). A few novel and simple algorithms are included, to track the wet/dry and dry/wet fronts over abruptly varying topography and stabilize the solution while using high resolution shock capturing methods. The negative depths computed from the surface gradient by the limiters are algebraically adjusted to ensure depth positivity. The friction term contribution in the source term, that creates unphysical values near the wet/dry fronts, are resolved by the introduction of a limiting value for the friction term. The model is validated using an extensive variety of tests both on fixed and mobile beds. The results are compared with the analytical, numerical and experimental results available in the literature. The model is also tested against the actual field data of 1957 Malpasset dam break. Finally, the model is applied to simulate dam break flow of Warsak Dam in Pakistan. Remotely sensed topographic data of Warsak dam is used to improve the accuracy of the solution. The study reveals from the thorough testing and application of the model that the simulated results are in close agreement with the available analytical, numerical and experimental results. The high resolution shock capturing methods give far better results than the traditional numerical schemes. It is also concluded that the object oriented CFD model is very easy to adapt and extend without changing the generic part of the model.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Two-dimensional shock capturing numerical simulation of shallow water flow applied to dam break analysis

With the advances in the computing world, computational fluid dynamics (CFD) is becoming more and more critical tool in the field of fluid dynamics. In the past few decades, a huge number of CFD models have been developed with ever improved performance. In this research a robust CFD model, called Riemann2D, is extended to model flow over a mobile bed and applied to a full scale dam break problem. Riemann2D, an object oriented hyperbolic solver that solves shallow water equations with an unstructured triangular mesh and using high resolution shock capturing methods, provides a generic framework for the solution of hyperbolic problems. The object-oriented design of Riemann2D has the flexibility to apply the model to any type of hyperbolic problem with the addition of new information and inheriting the common components from the generic part of the model. In a part of this work, this feature of Riemann2D is exploited to enhance the model capabilities to compute flow over mobile beds. This is achieved by incorporating the two dimensional version of the one dimensional non-capacity model for erodible bed hydraulics by Cao et al. (2004). A few novel and simple algorithms are included, to track the wet/dry and dry/wet fronts over abruptly varying topography and stabilize the solution while using high resolution shock capturing methods. The negative depths computed from the surface gradient by the limiters are algebraically adjusted to ensure depth positivity. The friction term contribution in the source term, that creates unphysical values near the wet/dry fronts, are resolved by the introduction of a limiting value for the friction term. The model is validated using an extensive variety of tests both on fixed and mobile beds. The results are compared with the analytical, numerical and experimental results available in the literature. The model is also tested against the actual field data of 1957 Malpasset dam break. Finally, the model is applied to simulate dam break flow of Warsak Dam in Pakistan. Remotely sensed topographic data of Warsak dam is used to improve the accuracy of the solution. The study reveals from the thorough testing and application of the model that the simulated results are in close agreement with the available analytical, numerical and experimental results. The high resolution shock capturing methods give far better results than the traditional numerical schemes. It is also concluded that the object oriented CFD model is very easy to adapt and extend without changing the generic part of the model.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Nouvelles méthodes numériques pour les écoulements en eaux peu profondes

This research project focuses on the development and evaluation of numerical methods for shallow flows by proposing new spatial and temporal discretization techniques. First, a new high-order explicit finite volume method and a class of semi-implicit schemes are introduced which are effective for modelling fast and slow waves in oceanic and atmospheric flows. In the second part of the research project, a central-upwind scheme is proposed for shallow water flows on variable topography using unstructured grids. In this part of the project, a new approach is proposed for the stability analysis of unstructured numerical schemes for shallow water equations. In the third part of the thesis, two finite volume methods are developed for the conservation laws on curved geometries which are potentially applicable to shallow flows on a sphere. For such cases, numerical schemes are developed by using the approach followed by Stanley Osher. This approach employs simple hyperbolic systems which generate complex wave phenomena, and solutions that are effective for assessing numerical methods. In our case, Burgers’ equations are used since they have played an important role in the development of shock-capturing schemes in fluid mechanics.Dans ce projet de recherche, on s'intéresse au développement et à l'évaluation de nouvelles méthodes numériques pour les écoulements peu profonds. De nouvelles techniques de discrétisation spatiales et temporelles des équations sont proposées. La première partie de la thèse est dédiée au développement d'une méthode des volumes finis explicite d'ordre élevé et d'une famille de schémas semi-implicites qui sont efficaces pour la modélisation des processus lents et rapides dans les écoulements océaniques et atmosphériques. La deuxième partie du projet de recherche concerne la construction d'un schéma numérique efficace sans solveur de Riemann pour les écoulements peu profonds avec une topographie variable sur un maillage non structuré. Dans cette partie de la thèse, une nouvelle approche est proposée pour l'analyse de stabilité des schémas numériques non structurés pour les équations en eaux peu profondes. Dans la troisième partie de la thèse, deux schémas de volumes finis sont développés pour les lois de conservation sur des surfaces courbes qui ont un large potentiel d'être appliqués aux écoulements peu profonds sur la sphère. Dans ces cas, les schémas numériques sont développés en adoptant la démarche suivie par Stanley Osher. Cette démarche consiste à utiliser des systèmes hyperboliques simples qui génèrent des phénomènes d'ondes complexes et des solutions qui ont différentes structures. Ces solutions sont très efficaces pour tester les méthodes numériques. Dans notre cas, nous avons utilisé les équations de Burgers qui ont joué un rôle très important dans le développement des schémas numériques à capture de chocs en mécanique des fluides

Object-oriented hyperbolic solver on 2D-unstructured meshes applied to the shallow water equations

Fluid dynamics, like other physical sciences, is divided into theoretical and experimental branches. However, computational fluid dynamics (CFD) is third branch of Fluid dynamics, which has aspects of both the previous two branches. CFD is a supplement rather than a replacement to the experiment or theory. It turns a computer into a virtual laboratory, providing insight, foresight, return on investment and cost savings1. This work is a step toward an approach that realise a new and effective way of developing these CFD models.EThOS - Electronic Theses Online ServiceGBUnited Kingdo