Numerical solution by meshless method of a fully-coupled bed load 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>

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

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.

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.

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.

A high-accuracy numerical solution, coupling one-dimensional shallow water and bed-evolution equations, with, for the first time, a suspended sediment advection equation, thereby including bed and/or suspended load, is used to examine two swash events on an initially plane erodible beach: the event of Peregrine & Williams (J. Fluid Mech., vol. 440, 2001, pp. 391–399) and that of a solitary wave approaching the beach. Equations are solved by the method of characteristics, and the numerical model is verified. Full coupling of suspended load to beach change for Peregrine & Williams (J. Fluid Mech., vol. 440, 2001, pp. 391–399) yields only slightly altered swash flows, depending on beach mobility and sediment response time; a series of similar final beach change patterns results for different beach mobilities. Suspended- and bed-load transport have distinct morphodynamical signatures. For the solitary wave a backwash bore is created (Hibberd & Peregrine, J. Fluid Mech., vol. 95, 1979, pp. 323–345). This morphodynamical bore propagates offshore initially, and leads to the creation of a beach bed step (Larson & Sunamura, J. Sedimentary Petrology, vol. 63, 1993, pp. 495–500), primarily due to bed-load transport. Its height is directly related to bed-load mobility, and also depends strongly on the bed friction coefficient. The shock dynamics of this bed step is explained and illustrated. Bed- and suspended-load mobilities are quantified using field data, and an attempt is made to relate predictions to measurements of single swash events on a natural beach. Average predicted bed change magnitudes across the swash are of the order of 2 mm, with maximum bed changes of up to approximately 10 cm at the bed step.

Compact RBF meshless methods for photonic crystal modelling

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.

Application of localized meshless methods to 2D shallow water equation problems

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.

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

We show how the mesh adaptation technique can be exploited for the numerical simulation of shallow water flow. The shallow water equations are numerically approximated by the Galerkin finite element method, using linear elements for the elevation field and quadratic elements for the unit width discharge field; the time advancing scheme is of a fractional step type. The standard mesh refinement technique is coupled with the numerical solver; movement and elimination of nodes of the initial triangulation is not allowed. Two error indicators are discussed and applied in the numerical examples. The conclusion focuses the relevant advantages obtained by applying this adaptive approach by considering specific test cases of steady and unsteady flows

A geological model of typical shallow water flow (SWF) sediment in deepwater basins is constructed based on its basic characteristics, and a reservoir simulation model is then established that incorporates two-phase flow as the basis for the simulation of water and sand blowout process during penetrating SWF formations. The model assumes that sands can start flowing along with water at a critical pressure difference when drilling through SWF formations. The cumulative water and sand blowout as SWF formations are penetrated are calculated and the influencing factors, such as SWF zone size, overpressure factor of SWF formations and penetration rate, are incorporated into the model to quantitatively analyze their impacts on the water and sand blowout from SWF zones. The data are useful for the assessment of the severity of SWF hazards. The results show that the cumulative water blowout and sand blowout will be as high as 100m3 and 1m3, respectively, when a typical SWF sediment is totally penetrated. SWF formation overpressure factor has a remarkable effect on SWF blowout, while SWF zone size has more moderate effects. However, the effects of SWF sediment size are expected to become increasingly significant as time proceeds. 10% increase of overpressure factor of SWF formation will result in around 90% increase of cumulative water blowout and 150% increase of cumulative sand blowout after the SWF zone is totally penetrated, while 10% increase of SWF zone size will lead to about 20% elevation of cumulative water blowout and 25% ascending of cumulative sand blowout. The water and sand blowout can be effectively reduced while increasing the penetration rate, which indicates that elevating the penetration rate can be a useful measure for the control of SWF hazards during deepwater drilling operations.

In this study numerical solutions for the compressible fluid flow around obstacles, based on a meshless method with Radial Basis Functions (RBFs) are presented. The numerical solutions are obtained by a hybrid method that uses the first step of the boundary element method, namely a singular boundary integral equation, which is an equivalent mathematical model for the problem of the compressible fluid around an obstacle, and then a meshless method based on RBFs for solving this singular boundary integral equation. Two types of RBFs are used, namely the Multiquadric RBFs and Gauss-type RBFs. An analytical check is made in order to study the numerical solutions' accuracy. Using computer simulations, we compare our results with an analytical solution available for a specific case. We examine how different parameters used to obtain the numerical solutions affect their accuracy, and we compare these numerical solutions to determine which RBF is most suitable for this scenario. We also note that the shape parameters of RBFs have a great influence on numerical solutions accuracy and, through simulations, optimal values are found. For evaluating the integrals of singular kernels which appear, the truncation method is applied. The influence of the parameter used to evaluate these integrals is also highlighted in the paper.

Numerical simulations of shallow water flows are widely used to predict global flows such as flood flows in the sea, river and reservoir flows, especially floods due to heavy rainfall and dam break. In particular, reducing the simulation time by applying the Local Time Stepping (LTS) scheme is one way to improve the practical efficiency of numerical simulation. In this paper, we proposed LTS scheme using a structured grid for the numerical simulation of shallow water system. When modeling any terrain, rectangular grid cells are used to facilitate grid generation. To estimate the momentum flux at the grid cell boundaries, we applied the second-order spatial accuracy Godunov finite volume algorithm with Roe approximation solver using the MUSCL method. The LTS scheme is applied to shallow water flow problems with tsunami reflection pattern and its accuracy and efficiency are compared with the traditional global time step (GTS) method. Results show that, with no loss of accuracy, the new LTS algorithm achieves 59-67% CPU time reduction when compared to the GTS method. The proposed LTS scheme accurately reflects time varying water regimes and reflected waves in shallow water flows and can be used for numerical simulations of the three-dimensional shallow water flows with arbitrary topography to reduce the simulation time significantly.