The shifted boundary method for hyperbolic systems: Embedded domain computations of linear waves and shallow water flows

<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>

Energy conservation and physical optics for discrete long wave equations

Dispersive shallow water waves over a porous sea bed

SummaryA modified lattice Boltzmann model is proposed to describe shallow water flows over complex topography. In the proposed model, the quadratic depth term is excluded from the equilibrium distribution functions (EDFs), and the hydrostatic pressure term is combined with the bed slope term to be treated as a part of the sourcing term in the lattice Boltzmann equation (LBE). Therefore, it is unnecessary to match the coefficients of the quadratic depth term in the EDFs with those of the bed slope term in the sourcing terms in the LBE. This would bring more flexibility to the treatment of the sourcing terms in the LBE. In order to recover the shallow water equations (SWEs), the basic constraints are redefined, and under these constraints, the coefficients of the EDFs are derived afterwards. Several benchmark problems are used to validate the proposed model, including stationary case, steady flows over a two‐dimensional bump and tidal wave flows over irregular bed elevation. The computed results are in excellent agreement with the results of the other numerical methods and the analytical solutions, indicating that the proposed model is capable of simulating shallow water flows over complex bathymetry. It also proves that the proposed model has potential to produce competitive solutions to shallow water flows over complex bed topography. Copyright © 2014 John Wiley & Sons, Ltd.

The swash zone is that part of a beach over which the instantaneous shoreline moves back and forth as waves meet the shore. This zone is discussed using the nonlinear shallow water equations which are appropriate for gently sloping beaches. A weakly three-dimensional extension of the two-dimensional solution by Carrier & Greenspan (1958) of the shallow water equations for a wave reflecting on an inclined plane beach is developed and used to illustrate the ideas. Thereafter attention is given to integrated and averaged quantities. The mean shoreline might be defined in several ways, but for modelling purposes we find the lower boundary of the swash zone to be more useful. A set of equations obtained by integrating across the swash zone is investigated as a model for use as an alternative boundary condition for wave-resolving studies. Comparison with sample numerical computations illustrates that they are effective in modelling the dynamics of the swash zone and that a reasonable representation of swash zone flows may be obtained from the integrated variables. The longshore flow of water in the swash zone is in many ways similar to the Stokes’ drift of propagating water waves. Further averaging is made over short waves to obtain results suitable as boundary conditions for longer period motions including the effect of incident short waves. In order to clearly present the work a few simplifications are made. The main result is that in addition to the kinematic type of boundary condition that occurs on a simple, e.g. rigid, boundary two further conditions are found in order that both the changing position of the swash zone boundary and the longshore flow in the swash zone may be determined. Models of the short waves both outside and inside the swash zone are needed to complete a full wave-averaged model; only brief indication is given of such modelling.

Chapter 3 - Equation For Numerical Modeling Of Wave Transformation In Shallow Water

PurposeThis paper aims to solve linear and non-linear shallow water wave equations using homotopy perturbation method (HPM). HPM is a straightforward method to handle linear and non-linear differential equations. As such here, one-dimensional shallow water wave equations have been considered to solve those by HPM. Interesting results are reported when the solutions of linear and non-linear equations are compared.Design/methodology/approachHPM was used in this study.FindingsSolution of one-dimensional linear and non-linear shallow water wave equations and comparison of linear and non-linear coupled shallow water waves from the results obtained using present method.Originality/valueCoupled non-linear shallow water wave equations are solved.

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.

We show wave breaking---bounded solutions with unbounded derivatives---in the nonlinear nonlocal equations which combine the dispersion relation of water waves and the nonlinear shallow water equations and which extend the Whitham equation to permit bidirectional propagation, provided that the slope of the initial data is sufficiently negative.

The recently proposed String Gradient Weighted Moving Finite Element (SGWMFE) method is extended to include remeshing and refining. The method simultaneously determines, at each time step, the solution of the governing partial differential equations and an optimal location of the finite element nodes. It has previously been applied to the nonlinear time-dependent two-dimensional shallow water equations, under the demanding conditions of large Coriolis forces, inducing large mesh and field rotation. Such effects are of major importance in geophysical fluid dynamics applications. Two deficiencies of the original SGWMFE method are (1) possible tangling of the mesh which causes the method's failure, and (2) no mechanism for global refinement when necessary due to the constant number of degrees of freedom. Here the method is extended in order to continue computing solutions when the meshes become too distorted, which happens quickly when the flow is rotationally dominant. Optimal rates of convergence are obtained when remeshing is applied. The method is also extended to include refinement to enable handling of new physical phenomena of a smaller scale which may appear during the solution process. It is shown that the errors in time are kept under control when refinement is necessary. Results of the extended method for some example problems of water hump release are presented.

The objective of this paper is to discern a manner in which the petroleum industry might determine the existence of shallow water flows, or subsea regions of pressurized water and sand, prior to drilling into said formations. Since 2001, a dozen wells have been lost to this phenomenon and millions of dollars have been spent in damages in the Gulf of Mexico alone. Research and analysis of current methods led to the discovery of three additional methods of determining shallow water flows through logging and above-ground tools. These methods were cross-referenced and compiled to produce a four-marker logging indicator which definitively characterizes shallow water formations in the wellbore. When these markers are manifested, this paper offers an updated version of the FKB (flow-kill-breakdown) Cycle in order to stop the kick before it occurs. Furthermore, it offers an opportunity to survey existing wells to create a geological profile which includes these dynamic formations. Familiarity with shallow water flows from a petrophysical perspective has become essential to save time and money, promote safety, and to further the industry's knowledge of shallow water flows. The vast majority of wells drilled in the Gulf of Mexico reside in a high profit area beyond the Late Pleistocene deltaic depocenter, which is a high-risk area for these formations. This research paper has assembled four major indicators for these shallow water formations. From Shaker's research, it has been determined that the pressure profile of any sand formation will be linear. Furthermore, porosity will spike at shallow water flow locations due to their hyperpressurized condition. Ostermeier and his team assembled indicators by observation, noting that pressure would spike and gamma ray results would decrease when encountering such a formation. When depth is restricted to the 1,000 to 3,500-foot range, these indicators offer a narrow and specific profile to locate shallow water flows before drilling into them. At an average cost per well of $1.6 million in damages due to these formations, those are savings worth pursuing. The drilling of offshore oil and natural gas will be safer with the implementation of shallow logging and the incorporation of these identifiers. Though the components of this research have existed for decades, this paper uses fundamental analysis of the issues involved to uncover a previously unknown formula which acts as a preventative measure against kicks, blowouts, and loss of hole due to shallow water flows. Furthermore, an amendment to an existing API document is proposed, which would alter practices in the region by implementing the proposed solution across the country.

In this paper, the formulations of the primitive equations for shallow water flow in various horizontal co-ordinate systems and the associated finite difference grid options used in shallow water flow modelling are reviewed. It is observed that horizontal co-ordinate transformations do not affect the chosen co-ordinate system and representation in the vertical, and are the same for the three- and two-dimensional cases. A systematic derivation of the equations in tensor notation is presented, resulting in a unified formulation for the shallow water equations that covers all orthogonal horizontal grid types of practical interest. This includes spherical curvilinear orthogonal co-ordinate systems on the globe. Computational efficiency can be achieved in a single computer code. Furthermore, a single numerical algorithmic code implementation satisfies. All co-ordinate system specific metrics are determined as part of a computer-aided model grid design, which supports all four orthogonal grid types. Existing intuitive grid design and visual interpretation is conserved by appropriate conformal mappings, which conserve spherical orthogonality in planar representation. A spherical curvilinear co-ordinate solution of wind driven steady channel flow applying a strongly distorted grid is shown to give good agreement with a regular spherical co-ordinate model approach and the solution based on a β-plane approximation. Especially designed spherical curvilinear boundary fitted model grids are shown for typhoon surge propagation in the South China Sea and for ocean-driven flows through Malacca Straits. By using spherical curvilinear grids the number of grid points in these single model grid applications is reduced by a factor of 50–100 in comparison with regular spherical grids that have the same horizontal resolution in the area of interest. The spherical curvilinear approach combines the advantages of the various grid approaches, while the overall computational effort remains acceptable for very large model domains.

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.

Numerical investigations of fully nonlinear water waves generated by moving bottom topography