Two-dimensional Shallow Water Equations Research Articles

Research and application of hydrodynamics modeling of channel in reservoir area-Case of Feilaixia station to Qingyuan station section

In navigable rivers, channel depth and current are the main influencing factors of channel environmental risk, ship navigation mainly relies on channel maintenance water depth information regularly issued by channel departments as guidance, but the long forecast interval limits the full utilization of channel resources to a certain extent. In this paper, by studying the characteristics of the channel from Feilaixia to Qingyuan reservoir area, established the terrain model mesh for the section of the study area, and the two-dimensional shallow water equations are constructed as the control equations of flow simulation. The Model parameters are calibrated and verified. The typical schemes of the channel in the study area were simulated to obtain the channel water depth and velocity information, and the results were visualized, analyzed and discussed. Using two-dimensional hydrodynamic model to simulate channel conditions and calculate flow conditions can provide data support for ship navigation, which has important practical significance for improving navigation management ability.

Numerical modeling of two dimensional non-capacity model for sediment transport by an unstructured finite volume method with a new discretization of the source term

The main goal of this work is the resolution of the two-dimensional shallow water equations of water–sediment mixture coupled to the transport diffusion equation for the total sediment load, and bed change rate equation by Roe scheme with a new discretization of the source term. The proposed discretization is well-balanced with the flux gradient and uses data right and left on the interfaces between two control volumes and satisfies the C-property. The numerical method uses unstructured meshes and incorporates minmod limiter and Runge–Kutta method to reach second order spatial and temporal accuracy. We also use an adaptive mesh based on gradient concentration of sediments to refine the study domain with a lower computational cost. We present some numerical results in order to verify and validate the performance of the numerical scheme, particular attention is given to the treatment of the dam-break problem over mobile beds. The numerical scheme demonstrates the intended accuracy and robustness to modelize dam-break flows over erodible bed.

Lie Group Symmetry Analysis and Invariant Difference Schemes of the Two-Dimensional Shallow Water Equations in Lagrangian Coordinates

Lie Group Symmetry Analysis and Invariant Difference Schemes of the Two-Dimensional Shallow Water Equations in Lagrangian Coordinates

A consistent quasi–second-order staggered scheme for the two-dimensional shallow water equations

Abstract A quasi–second-order scheme is developed to obtain approximate solutions of the two-dimensional shallow water equations (SWEs) with bathymetry. The scheme is based on a staggered finite volume space discretization: the scalar unknowns are located in the discretization cells while the vector unknowns are located on the edges of the mesh. A monotonic upwind-central scheme for conservation laws (MUSCL)-like interpolation for the discrete convection operators in the water height and momentum balance equations is performed in order to improve the accuracy of the scheme. The time discretization is performed either by a first-order segregated forward Euler scheme or by the second-order Heun scheme. Both schemes are shown to preserve the water height positivity under a Courants Friedrichs Lewy (CFL) condition and an important state equilibrium known as the lake at rest. Using some recent Lax–Wendroff type results for staggered grids, these schemes are shown to be LW-consistent with the weak formulation of the continuous equations, in the sense that if a sequence of approximate solutions is bounded and strongly converges to a limit, then this limit is a weak solution of the SWEs; besides, the forward Euler scheme is shown to be LW-consistent with a weak entropy inequality. Numerical results confirm the efficiency and accuracy of the schemes.

Hyperbolicity-preserving and well-balanced stochastic Galerkin method for two-dimensional shallow water equations

Stochastic Galerkin formulations of the two-dimensional shallow water systems parameterized with random variables may lose hyperbolicity, and hence change the nature of the original model. In this work, we present a hyperbolicity-preserving stochastic Galerkin formulation by carefully selecting the polynomial chaos approximations to the nonlinear terms in the shallow water equations. We derive a sufficient condition to preserve the hyperbolicity of the stochastic Galerkin system which requires only a finite collection of positivity conditions on the stochastic water height at selected quadrature points in parameter space. Based on our theoretical results for the stochastic Galerkin formulation, we develop a corresponding well-balanced hyperbolicity-preserving central-upwind scheme. We demonstrate the accuracy and the robustness of the new scheme on several challenging numerical tests.

Modeling of Flooding of Settlements During the Spring Flood

The problem of flooding of territories with flood waters for settlements of the Volgograd region is considered. A numerical model of flood water dynamics is constructed, taking into account the topography of the terrain. The simulation is based on two-dimensional shallow water equations. For computational experiments, a parallel implementation of the numerical scheme CSPH-TVD for NVIDIA graphics accelerators with CUDA technology is used. The digital model of the river-bed and floodplain relief is based on spatial data SRTM3 and SRTMGL, topographic maps of the area, longitudinal profiles of rivers. The water flow rates for the Buzuluk and Perevozinka rivers are determined and a hydrograph of the Volga hydroelectric power station is constructed for the given probabilities of exceeding the water level. Flood maps were obtained for the following localities: Novoannisky, Berezovka 1st, kh. Vyazovka. The maximum values of the depths for 1 %, 3 %, 5 %, 10 %, 25 % and 50 % of water security are presented. An analysis of the flood situation was carried out and appropriate engineering and protective measures for settlements were proposed. At maximum flood water levels, residential buildings of kh. Berezovka 1-I are not subject to flooding. In order to minimize the negative impact for the city of Novoanninsky, it is proposed to carry out timely clearing of the river-bed of the Perevozinka river from congestion. To protect from flooding ch. Vyazovka it is recommended to use diversion dyke.

Modeling The Hydrodynamic Characteristics of Tidal And Monsoonal Currents In Pondok Dayung Port of Tanjung Priok Harbor, Jakarta

The Pondok Dayung port forms a significant segment of the Tanjung Priok harbor in the Jakarta coastal bay. Studies on the hydrodynamic characteristics of tidal and monsoonal currents appear very important to ship movement and laid/dock operations in port basins/jetties. These flow conditions have been simulated using a two-dimensional shallow water equation, while the tidal and monsoonal wind were coupled to model the ocean current. In general, the simulation results of the ocean current characteristics were dominated by tidal effects, as well as the interactions with the coastlines, jetties, and breakwaters. Also, the geometric replica has been validated satisfactorily, using time series sea elevation from the tidal station in the research area managed by the National Geospatial Agency (BIG). Strong RMSE and linear correlation values ranging from 0.0405-0.0458 m and 0.9648-0.9843 were obtained, respectively. During the flood tides, the ocean current is directed towards the basin area, while an outward flow is observed under ebb conditions. Furthermore, the maximum tidal current speed of ±0.26 m/s was recorded at the port waterways. A similar outcome was also reported during the west and east monsoon, in addition to a minimum ocean current speed of approximately 0.00 m/s. These conditions implied that the Pondok Dayung port and its breakwater system served as protective structures to the surrounding vessels and the harsh ocean current impacts.

Optimal placement of tsunami sensors with depth constraint.

Tsunamis are destructive natural disasters that can cause severe damage to property and the loss of many lives. To mitigate the damage and casualties, tsunami warning systems are implemented in coastal areas, especially in locations with high seismic activity. This study presents a method to identify the placement of near-shore detection sensors by minimizing the tsunami detection time, obtained by solving the two-dimensional shallow water equations (SWE). Several benchmark tests were done to establish the robustness of the SWE model, which is solved using a staggered finite volume method. The optimization problem is solved using particle swarm optimization (PSO). The proposed method is applied to different test problems. As an application, the method is used to find the optimal location of a detection sensor using data from the 2018 Palu tsunami. Our findings show that detection time can be significantly reduced through the strategic placement of tsunami sensors.

An improved meshless artificial viscosity technology combined with local radial point interpolation method for 2D shallow water equations

The two-dimensional shallow water equations (SWEs) are a hyperbolic system of first-order nonlinear partial differential equations which have a characteristic of strong gradient. In this study, a newly-developed numerical model, based on local radial point interpolation method (LRPIM), is adopted to simulate discontinuity in shallow water flows. In order to accurately capture the information of wave propagation, the LRPIM is combined with the split-coefficient matrix (SCM) method to transform the SWEs into a characteristic form and the selection of the direction of local support domain is introduced into the LRPIM. An improved meshless artificial viscosity (MAV) technique is developed to minimize the non-physical oscillations near the discontinuities. Then, the LRPIM and the second-order Runge–Kutta method are adopted for spatial and temporal discretization of the SWEs, respectively. The feasibility and validity of the proposed numerical model are verified by the classical dam-break problem and the mixed flow pattern problem. The comparison of the obtained results with the analytical solution and other numerical results showed that the MAV method combined with LRPIM can accurately capture the shocks and has high accuracy in dealing with discontinuous flow by adding appropriate viscosity to the equations in the discontinuous region.

Analysis of bay bathymetry elements on wave amplification: a case study of the tsunami in Palu Bay

ABSTRACT This research was inspired by the deadly tsunami that struck Palu Bay, Sulawesi, Indonesia on September 28, 2018. High waves were observed not only at the bay’s head but also in several other locations along the bay. This high amplification is most likely influenced by Palu Bay’s bathymetry and the underwater sill in front of Palu City. The numerical simulations of the two-dimensional shallow water equations were used to investigate these aspects. The numerical scheme is validated first with analytical solutions of standing waves in a parabolic bay, as a model of Palu Bay. The results show a significant amplification at the bay’s head, as a result of the bay’s reduced cross-sectional area. Some amplifications were also observed along the channel sides, and the role of the underwater sill was also investigated. Next, we carried out simulations with the actual Palu Bay bathymetry and a hypothetical tsunami $N-$wave and/or Gaussian humps. The simulation result depicts extremely high waves recorded along the Palu Bay coastline, which is consistent with the field survey; additionally, the wave signal at Pantoloan corresponds reasonably well to the tide gauge data. These findings indicate that Palu Bay’s bathymetry plays an important role in local wave amplification.

A meshless artificial viscosity method for wet-dry moving interfaces problems of shallow water flow

In this paper, an improved meshless artificial viscosity (AV) method is proposed and combined with the local radial point interpolation method (LRPIM) and the MacCormack method to establish a newly-developed meshless numerical model, which can efficiently and accurately solve wet-dry moving interfaces problems of shallow water flow. As an important model describing shallow water flow, the two-dimensional (2D) shallow water equations (SWEs) are a hyperbolic system of 1st-order nonlinear partial differential equations which have a characteristic of strong gradient. The LRPIM and MacCormack method are adopted to discretize the 2D SWEs spatially and temporally, respectively. Meanwhile, the method of selecting the different shape of support domain in the LRPIM is employed for properly cooperating with the MacCormack method to accurately capture the direction of wave propagation. For eliminating the non-physical oscillation at the discontinuity of shallow water flow, the improved meshless AV form is first presented. Four challenging numerical examples were selected to verify the proposed model by comparing with the other solutions and experimental observations. The results indicate that the proposed method with improved meshless AV can accurately capture the shock waves and efficiently simulate the wet-dry front changes.

LISFLOOD-FP 8.0: the new discontinuous Galerkin shallow-water solver for multi-core CPUs and GPUs

Abstract. LISFLOOD-FP 8.0 includes second-order discontinuous Galerkin (DG2) and first-order finite-volume (FV1) solvers of the two-dimensional shallow-water equations for modelling a wide range of flows, including rapidly propagating, supercritical flows, shock waves or flows over very smooth surfaces. The solvers are parallelised on multi-core CPU and Nvidia GPU architectures and run existing LISFLOOD-FP modelling scenarios without modification. These new, fully two-dimensional solvers are available alongside the existing local inertia solver (called ACC), which is optimised for multi-core CPUs and integrates with the LISFLOOD-FP sub-grid channel model. The predictive capabilities and computational scalability of the new DG2 and FV1 solvers are studied for two Environment Agency benchmark tests and a real-world fluvial flood simulation driven by rainfall across a 2500 km2 catchment. DG2's second-order-accurate, piecewise-planar representation of topography and flow variables enables predictions on coarse grids that are competitive with FV1 and ACC predictions on 2–4 times finer grids, particularly where river channels are wider than half the grid spacing. Despite the simplified formulation of the local inertia solver, ACC is shown to be spatially second-order-accurate and yields predictions that are close to DG2. The DG2-CPU and FV1-CPU solvers achieve near-optimal scalability up to 16 CPU cores and achieve greater efficiency on grids with fewer than 0.1 million elements. The DG2-GPU and FV1-GPU solvers are most efficient on grids with more than 1 million elements, where the GPU solvers are 2.5–4 times faster than the corresponding 16-core CPU solvers. LISFLOOD-FP 8.0 therefore marks a new step towards operational DG2 flood inundation modelling at the catchment scale. LISFLOOD-FP 8.0 is freely available under the GPL v3 license, with additional documentation and case studies at https://www.seamlesswave.com/LISFLOOD8.0 (last access: 2 June 2021).

Dam Break Wave Propagation on a Non- Erodible Bed – Comparison of Experimental and Numerical Results

Dams are vital for production of electricity, storage of water and irrigation purposes but pose a serious risk to the community, if breached. The downstream flood wave propagation, resulting from failure of a dam can subject the population and infrastructure to considerable damage. No matter how low the chances of failure, the cost of failure makes it a higher risk. Mitigation of such risks requires better understanding of the hazard that a dam may pose in case of failure. This study focuses on the effects of flood wave propagation on a fixed bed on the downstream side resulting from sudden dam break. Two conditions are simulated: 1. when the downstream side is open, 2. when the downstream side is closed. It is observed that the flood wave diminishes in velocity and height with increase in time for both cases. For downstream open condition, the flood wave attains maximum height in 2 to 4 sec and maximum velocity within 2 to 5 sec. For downstream closed condition, the flood wave attains maximum height within 5to 10 sec and maximum velocity within 3 to 5 sec. The results obtained from the two-dimensional shallow water equation based numerical model are in close agreementwith the experimental results.

Comparison of new efficient 2D models for the simulation of bedload transport using the augmented Roe approach

The bedload transport system of equations, composed by the two-dimensional shallow water equations for the free surface flow motion and the 2D Exner or bedload transport equation for the erodible bed layer, is used for a wide range of sediment transport processes in environmental surface flows. In this work, the numerical resolution has been implemented using improved and efficient versions of two different strategies for combining the hydrodynamical and morphodynamical components of the system. The first strategy is based on the full coupling of the flow and bedload transport equations (FCM), leading to a new formulation for the intercell numerical fluxes which includes the bed elevation into the resolution of the approximated local Riemann problem (RP) at the edges. The stability region of this method is controlled by the eigenvalues of the coupled Jacobian matrix at each intercell edge. On the second hand, an alternative decoupled strategy is considered based on solving independently the shallow water and the bed transport equations at each time step but controlling the stability region by means of an approximation of the coupled Jacobian matrix eigenvalues. This method, called approximate-coupled (ACM) allows simpler expressions for the numerical fluxes at the edges and ensures the stability of the scheme. Both strategies are based on the Finite Volume (FV) method using Roe’s approach for the computation of the numerical fluxes between neighbouring cells and have been implemented into the same CPU-based numerical kernel in order to perform a realistic comparison of the range of applicability and computational efficiency. The ACM can only guarantee non-oscillatory results when the bed-flow interaction factor is small G≤O(10−3). If the interaction factor G is medium or high, G>O(10−2), the decoupled scheme loses its accuracy and robustness. Furthermore, for highly erosive flows the FCM scheme demonstrates to be more efficient in terms of computational effort than the ACM, one of the key points for large-scale and long-term bedload transport realistic applications.

A 2D-SWEs framework for efficient catchment-scale simulations: Hydrodynamic scaling properties of river networks and implications for non-uniform grids generation

The application of two-dimensional shallow-water equations models (2D-SWEs) for the description of hydrodynamic-based surface runoff computations is becoming a reference approach in rainfall-runoff simulations at the catchment scale. Due to their ability in generation of flow patterns throughout the basin, they can be used not only as an advanced method for flood mapping studies and hazard assessment but also as an innovative tool for the analysis of river drainage networks, opening new perspectives for several environmental processes. In particular, in this work we put the river networks in a 2D-SWEs framework, meaning that the traditional tree-like fluvial structure, represented by a skeleton composed of a set of lines, is replaced by a collection of points discretizing the 2-D geometry of the river structure itself, for which the values of the hydrodynamic values are provided by the numerical simulations. This approach is used here to derive a new scaling property that relates the specific discharge threshold, used to identify the river network cells, to the total areas of the network cells themselves. The hydrodynamic and geomorphological interpretation of this power law function and the influence of grid resolution, on some relevant parameters of this curve, have inspired the development of a heuristic procedure for non-uniform grid generation, able to detect the most hydrodynamically active areas of the basins for which the grid refinement process makes sense. Moreover, information related to how much grid refinement is needed is provided as well. The performances of this procedure are very promising in terms of accuracy of simulated discharges, hydrodynamic behaviour of the river network and flooded areas, reducing significantly the computational times in respect to the use of fine uniform grids.

A hybrid shallow water solver for overland flow modelling in rural and urban areas

This study aims to present an accurate and efficient finite volume approach for overland flow modelling in rural and urban areas through solving the two-dimensional shallow water equations. The proposed approach introduces a hybrid procedure that utilizes the sharpness of discontinuity to identify flow conditions in the entire computational domain, and then a low-accuracy/high-efficiency scheme (the Rusanov scheme) is used in regular flows, whereas a high-accuracy/low-efficiency scheme (the HLLC scheme) is adopted in strong discontinuous flows. Regions that simultaneously adopt the two schemes are artificially specified to diminish numerical unbalances between the two schemes. Model calibration is carried out through three strong discontinuous flow cases with the exact solutions. Model verification is next described by three benchmark cases, including rainfall-runoff around buildings, flash floods towards buildings, and street junction flows. Model efficiency assessment is also conducted to present the numerical efficiency improvement of the proposed approach for various flow conditions. Technical attention is devoted to the numerical performances on local accuracy and global efficiency of the proposed approach. The simulated results indicate that the proposed approach can simulate as accurate as the HLLC scheme with significant reduction on its computational time. How efficient the proposed hybrid approach can reach depends on flow conditions involved. It is more efficient than the HLLC scheme as the portion of strong discontinuous flows in the computational domain is relatively smaller. The proposed approach has proved its accuracy and efficiency for overland flow modeling, and it can also be used together with other speed-up techniques, such as local time stepping and parallelization, to improve its efficiency. Other finite volume schemes can also be utilized to improve the efficiency and accuracy of the proposed approach. Therefore, the solver has considerable potentials as a useful tool in flood inundation simulations.

Performance Comparison between Semi-Lagrangian and Eulerian Numerical Solutions for Two-Dimensional Surface Flows in Basin Irrigation

AbstractTo achieve efficient simulation for surface shallow-water flows in large-scale basin irrigation, a semi-Lagrangian numerical solution for two-dimensional shallow-water equations in unstruct...

Simulation of Mosul Dam Break Using Finite Volume Method

Mosul dam is an earth-fill embankment located north of Iraq on the Tigris River forming a reservoir with 11.11 km3 water storage capacity which is the largest dam in the country. The dam is built on a rock bed foundation, in which the dissolution process is dynamic in the zone where gypsum and anhydrite layers present. During the construction development seepage locations were found in the dam foundation and the grouting process is in progress until now to control this problem. Therefore, the possibility of the Mosul dam break is highlighted by previous studies. In this research, a FORTRAN code based on the finite volume method is modified to solve the two-dimensional shallow water equations and simulating the Mosul dam break. The computational domain discretized using unstructured triangular mesh. The solver applied Harten lax van leer with contact (HLLC) wave approximate Riemann solver to calculate the cell interface fluxes, and the semi-implicit scheme employed to solve the friction source term. The numerical scheme applied to two benchmark test cases, and the results showed that the presented model was robust and accurate especially in handling wet/dry beds, mixed flow regimes, discontinuities, negative water depths, and complex topography. The results of this study demonstrate that flood waves may reach the center of Mosul city in < 6 h and water depth may rise to 34 m after 7 h of Mosul dam breaking. Finally, the simulation results of the Mosul dam break were used to prepare an emergency action plan.

Second-order discontinuous Galerkin flood model: Comparison with industry-standard finite volume models

Finite volume (FV) numerical solvers of the two-dimensional shallow water equations are core to industry-standard flood models. The second-order Discontinuous Galerkin (DG2) alternative, although a viable way forward to improve current FV-based flood models, is yet under-studied and rarely used to support flood modelling applications. This paper systematically explores and compares the predictive properties of a robust DG2 flood model to those of prominent FV-based industrial flood models. To identify the simplest and most efficient DG2 configuration suitable for flood inundation modelling, two variants – with and without local slope limiting – are considered. The numerical conservation properties of the DG2 variants are compared to those of a first-order FV (FV1) and a second-order FV (FV2) counterparts. The DG2 variants are then tested over five realistic flooding scenarios, recommended by the UK Environment Agency to validate 2D flood model capabilities, while comparing their performance against that of four FV-based commercial models (i.e. TUFLOW-FV1, TUFLOW-FV2, TUFLOW-HPC and Infoworks ICM). Results reveal that the DG2 variant without local limiting (DG2-NL) is capable to simulate shockless flood flows featured in a wide range of flood modelling applications. The DG2-NL shows closer predictions to commercial model outputs at twice-coarser spatial resolution, and can run twice faster to produce more informative hydrograph with small-scale transients over long-range simulations, even when the sampling is far away from the flooding source.

A partitioned finite element method for power-preserving discretization of open systems of conservation laws

AbstractThis paper presents a structure-preserving spatial discretization method for distributed parameter port-Hamiltonian systems. The class of considered systems are hyperbolic systems of two conservation laws in arbitrary spatial dimension and geometries. For these systems, a partitioned finite element method (PFEM) is derived, based on the integration by parts of one of the two conservation laws written in weak form. The non-linear one-dimensional shallow-water equation (SWE) is first considered as a motivation example. Then, the method is investigated on the example of the non-linear two-dimensional SWE. Complete derivation of the reduced finite-dimensional port-Hamiltonian system (pHs) is provided and numerical experiments are performed. Extensions to curvilinear (polar) coordinate systems, space-varying coefficients and higher-order pHs (Euler–Bernoulli beam equation) are provided.