Shallow Water Equation Solver Research Articles

Hydrodynamic coupling of multi-fidelity solvers in REEF3D with application to ship-induced wave modelling

Ship-induced waves are an increasingly relevant hydrodynamic forcing factor in waterways travelled by large seagoing ships. The discrepancy between the small-scale wave-structure interaction near embankments and the larger-scale wave generation and propagation poses challenges for the prediction of ship-induced waves as a multi-scale problem. Therefore, a novel hydrodynamic coupling interface is presented that allows information transfer from the shallow-water-equation (SWE) solver REEF3D::SFLOW to the 3D-RANS-solver REEF3D::CFD. The implementation consists of a one-way coupling, where the solution from the SWE solver is imposed to one or multiple relaxation zones of the CFD solver. A series of verification cases shows that the implementation of the interface is accurate and only small deviations are introduced due to the 2D-3D dimensional mismatch of the numerical models involved. An application is presented, showing how the coupled SWE-CFD model can be employed to study ship-induced groin overtopping.

A Hybrid Theory-Driven and Data-Driven Modeling Method for Solving the Shallow Water Equations

In recent years, mountainous areas in China have faced frequent geological hazards, including landslides, debris flows, and collapses. Effective simulation of these events requires a solver for shallow water equations (SWEs). Traditional numerical methods, such as finite difference and finite volume, face challenges in discretizing convection flux terms, while theory-based models need to account for various factors such as shock wave capturing and wave propagation direction, demanding a high-level understanding of the underlying physics. Previous deep learning (DL)-based SWE solvers primarily focused on constructing direct input–output mappings, leading to weak generalization properties when terrain data or stress constitutive relations change. To overcome these limitations, this study introduces a novel SWE solver that combines theory and data-driven methodologies. The core idea is to use artificial neural networks to compute convection flux terms, and to reduce modeling complexity. Theory-based modeling is used to tackle complex terrain and friction terms for the purpose of ensuring generalization. Our method surpasses challenges faced by previous DL-based solvers in capturing terrain and stress variations. We validated our solver’s capabilities by comparing simulation results with analytical solutions, real-world disaster cases, and the widely used Massflow software-generated simulations. This comprehensive comparison confirms our solver’s ability to accurately simulate hazard scenarios and showcases strong generalization on varying terrain and land surface friction. Our proposed method effectively addresses DL-based solver limitations while simplifying the complexities of theory-driven numerical methods, offering a promising approach for hazard dynamics simulation.

DEVELOPMENT OF MESHES FOR COMPOUND FLOOD SIMULATION FOR THE TEXAS COAST

The Texas coast is subjected to frequent storms leading to floods in the yearly hurricane season. To provide forecasts of hurricane storm surge, a shallow water equation solver, the Advanced CIRCulation model, Luettich (1992) is used. The current state-of-the-art ADCIRC meshes for the Texas coast used in forecasting were developed about a decade ago, see, Hope (2013). ADCIRC uses unstructured meshes that require significant tailoring to ensure accurate and physically relevant solutions. The current mesh used in forecasting has very high resolution on the Texas coast for accuracy and numerical stability, whereas areas far away from Texas are of low resolution for computational efficiency. The applicability and accuracy of these meshes is regularly verified to be used in operational storm surge forecasting.

Historical comparison of the damage caused by the propagation of a dam break wave in a pre-alpine valley

Study regionValle Camonica basin and Lake Iseo in the Italian pre-alpine and alpine region. Study focusThis paper provides the first hydraulic reconstruction of the terminal part of the Gleno dam break with the propagation of the flood wave along a wide pre-alpine valley. The reconstruction of this part of the event, accomplished with a new 2D Shallow Water Equations solver, provides the occasion to tackle some important issues related to the computation of flood damage, a topic of paramount practical importance for which there is no widely accepted procedure in the literature. New hydrological insights for the regionThe hydraulic reconstruction provides insights into the propagation of the flood through the floodplain as far as the inlet of Lake Iseo. A methodology for damage computation is presented that considers a physically based criterion for the vulnerability of human life, with significant implications with respect to the use of simpler approaches based only on the density of the population. The economic evaluation of the damage to the built environment and to agricultural activities is included through a comprehensive recent approach. We discuss the variations of the expected damage due to the hydraulic works accomplished over the last 100 years to decrease the flood hazard, showing that its reduction has been followed by an increase in the expected damage in the surrounding areas.

POD-based ROM applied to unsteady free surface water flow

The numerical resolution of shallow water equations (SWE) is required in many environmental problems involving free surface flows. The upwind augmented Roe’s method [2] is widely used due to the robust and stable solutions it offers in realistic scenarios if properly corrected to deal with entropy fix and wet–dry fronts. On the other hand, reduced-order models (ROMs) are known to achieve more efficiency in terms of computational cost without losing accuracy. In this work, we analyse the properties and performance of a proper orthogonal decomposition (POD) based ROMs for this type of flows.

On the varied impact of the Storegga tsunami in northwest Scotland

ABSTRACTIn this paper we evaluate new data and those from previous studies in northwest Scotland and perform a modelling study to test the hypothesis that the Storegga tsunami (submarine slope failure off the continental shelf of Central Norway dated to 8120–8175 bp) impacted this region. The model used is a 2D non‐linear, non‐conservative, Shallow Water Equation solver incorporating inundation and realistic glacial isostatic adjustment‐corrected palaeobathymetry, with horizontal resolution down to 30 m at sites of interest. The 15 coastal study sites analysed range from south of the Isle of Skye to Assynt. We predict run‐up between 2.7 and 9.4 m above contemporaneous mean tide level across the region, with the highest on the west coast of the Outer Hebrides, the east coast of Skye and at the head of long sea lochs east of Skye. We re‐evaluate evidence from previously studied open coastal marshes, isolation basins and barrier locations for the tsunami and suggest that in many locations the Storegga tsunami is the most likely cause of erosion, deposition and changes in microfossil assemblages in the early Holocene. The predictions of wave height and inundation produced by the tsunami modelling fit well with the range of available field evidence in the region. We predict significant wave heights at least as far south as Mull on the west coast.

The influence of the hull representation for modelling of primary ship waves with a shallow-water equation solver

Long-period ship-generated loads have become design-relevant in many shallow and confined waterways. Numerical methods based on depth-averaged equations have conceptually proven successful to provide the ship wave parameters required for waterway management and design. Yet, the validation of these models remains challenging, due to variations in hull shapes, transient ship-motion during field data collection, and the dearth of published experimental benchmark data. The present study makes use of a new experimental data set to validate novel ship-modelling options in the shallow water equations solver REEF3D::SFLOW, using its free surface pressure extension for predicting long-period ship-generated load. The model predicts the primary wave field and the maximum return current with sufficiently low errors (MAPE) of 9.09% and 23.48%, respectively. A sensitivity study reveals that a simple slender body pressure assumption yields comparable simulation performance compared to a more complex hull-derived pressure distribution. The cross-sectional area of the respective pressure function, rather than the exact pressure function shape, is found to be decisive for the correct prediction of the design parameters primary wave height and maximum return current. Based on a systematic investigation of the ship draft to water depth relation, concise guidance on the choice of appropriate pressure functions is presented.

A reduced-order model applied to the 2D shallow water equations

The numerical resolution of shallow water equations (SWE) is required in many environmental problems involving free surface flow. Reduced-order models (ROMs) based on the POD allow such problems to be solved more efficiently in terms of computational cost without losing accuracy, as discussed in this work.

Development of a horizontal two-dimensional melt spread analysis code, THERMOS-MSPREAD Part-1: Spreading models, numerical solution methods and verifications

This article is the first part of two-part series describing the MSPREAD code developed for analyzing anisotropic melt spreading. The debris bed is modeled as a multi-layer structure consisting of the substrate, melt, upper and bottom crusts, and ambient fluid. The spreading of the debris bed is expressed based on the shallow water equation. The growth of the upper and bottom crusts is modeled. The heat conduction equation is solved in the substrate at the bottom and side walls based on 1D and 3D heat conduction models, respectively. The friction and heat transfer are expressed by closure equations that cover both dry and wet conditions. The numerical solution algorithm was described in the vector form. The fluxes at the cell boundary for the advection terms were expressed using the AUSM+-up scheme. The pressure equation is solved by the SIMPLEC method. Since the melt viscosity changes largely as the melt spreads, the viscosity terms are treated implicitly. Four verifications were introduced: two for the shallow water equations solver, one for the crust growth, and one for the side wall heat conduction. The dependences on the cell and time step sizes were studied via comparisons with the exact or higher-order numerical solutions.

Flood risk mapping using uncertainty propagation analysis on a peak discharge: case study of the Mille Iles River in Quebec

This study uses uncertainty propagation in real flood events to derive a probabilistic flood map. The flood event of spring 2017 in Quebec was selected for this analysis, with the computational domain being a reach of the Mille Iles River. The main parameter deemed uncertain in this work is the upstream water discharge; a given value of this discharge is utilized to build a random sample of 500 scenarios using the Latin hypercube sampling method. Simulations were run using CuteFlow-Cuda, an in-house finite volume-based shallow water equations solver, to derive the statistical mean and the standard deviation of the free surface elevation and the water depth at each node. For this real flood case, the initial interface flux scheme had to be adapted, combining a developed version of the scheme introduced by Harten, Lax and van Leer at wet interfaces and the Lax–Friedrichs scheme with additional free surface corrections for wet and dry transitions. Comparisons with results obtained from TELEMAC and from in situ observations show generally close predictions, and overall good agreement with observations. Errors of the free surface prediction relative to observations are less than 2.75%. A map based on the standard deviation of the water depth is presented to enhance the areas most prone to flooding. Finally, a flood map is produced, showing the flooded inhabited areas near the municipalities of Saint-Eustache and Deux Montagnes around the reach of the Mille Iles River as it overflows its natural bed.

Semi-Lagrangian shallow water equations solver on the cubed-sphere grid as a prototype of new-generation global atmospheric model

Next generation weather prediction atmospheric models will have horizontal resolution of about 3-5 km. The problem dimension will be 1010. One will need to use efficiently 104-105 computational cores to make a practical operational forecast. This leads to the need for the deep revision of numerical methods and algorithms used in atmospheric models. One of the problems to be solved is the horizontal discretization of atmospheric dynamics equations on the quasi-uniform spherical grids. This problem can be investigated using shallow water model that is much computationally cheaper than the use of full atmosphere model. We are developing an atmospheric dynamics solver for the next generation numerical weather prediction model at the Institute of Numerical Mathematics and Hydrometeorological center of Russia. Within this work, the solver for the shallow water equations using gnomonic cubed-sphere grid has been developed. The solver is verified using standard shallow water test cases. The accuracy of the presented solver is analysed. The good agreement to the reference solutions is achieved, when 4-th order spatial approximations are used.

TRANSCRITICAL FLOW SIMULATION USING SHALLOW WATER EQUATION MODEL

This paper shows the capabilities of DELFT3D-FLOW shallow water equation solver on transcritical flow. Two grid configurations are tested using a shock capturing numerical schemes that available on the solver. The simulation shows a good agreement with the analytical solution and proper grid resolution is needed to obtain a stable result. The model then used to simulate a real scale spillway chute channel of Logung Dam in Kudus-Central Java. The model could properly simulate the hydraulic jump, calculate the Froude number and stilling basin performance.

Fast and accurate simulations of shallow water equations in large networks

We present a fast and accurate method for solving shallow water flows in large networks. The governing equations consist of the one-dimensional shallow water equations including friction forces. A class of first-order coupling conditions are implemented at the water intersections. Two-dimensional counterparts for simple water networks are also considered for comparison. As a numerical solver for shallow water equations we employed the discontinuous Galerkin method in one and two space dimensions. The method is simple, highly accurate and exhibits enhanced stability properties in the vicinity of sharp gradients which are often present in the solution of shallow water flow problems. The method belongs to a class of locally conservative finite element methods whose approximate solutions are discontinuous across inter-element boundaries. The performance of the proposed techniques is assessed on several applications for shallow water flows in networks. The aim of such a method compared to the conventional two-dimensional simulations is to solve shallow water flows in large networks efficiently and with an appropriate level of accuracy.

Implementation of a scalable, performance portable shallow water equation solver using radial basis function-generated finite difference methods

In this article, we describe and analyze the computational performance of a parallel shallow water equation (SWE) solver for atmospheric simulation using radial basis function-finite difference (RBF-FD) methods. The inherent “meshless” nature of RBF-FD methods provides significant numerical benefits over standard pseudospectral and traditional FD methods, but there are many challenges in terms of their performance and parallel implementation, due to RBF-FDs use of relatively large halos and unstructured indexing. With the use of reverse Cuthill–McKee node ordering and tiled transposition of the state variable matrices and RBF-FD differentiation matrices, these challenges were overcome. The RBF-FD solver was implemented for the SWE on the rotating sphere using message passing interface plus OpenMP/OpenACC to demonstrate scalability and performance portability on the three currently dominant high performance computing (HPC) architectures, namely, Intel Xeon multicore, Intel Xeon Phi manycore, and NVIDIA graphics processing unit systems.

Physical habitat simulation for a fish community using the ANFIS method

This study presents physical habitat simulations for the fish community in a reach of the Dal River, Korea. The study reach is 2.3km long, located downstream from the Goesan Dam. The reach is gravel-bed, including a bend. A riffle is present at the apex of the bend, and pools are located before and after the riffle. Field monitoring revealed that five fish species are dominant, namely Zacco platypus, Coreoleuciscus splendidus, Zacco temminckii, Pungfungia herzi, and Acheilognathus yamatsute, and account for 78% of the total fish community. These five fishes, which include both lotic and lentic fishes, were selected as target species. The Center for Computational Hydroscience and Engineering 2D (CCHE2D) model, a 2D shallow water equation solver, and the Adaptive Neuro Fuzzy Inference System (ANFIS) method were used for hydraulic and habitat simulations, respectively. The Mahalanobis distance method was used for identifying outliers in the monitoring data. It was shown that the ANFIS method combined with data quality assessment predicts the Composite Suitability Index (CSI) better than the method without it. The CSI distributions in the study reach are presented for various flows. It was found that the distributions of the combined CSI predicted by the ANFIS method are higher than those predicted by the conventional multiplicative aggregation method with Habitat Suitability Curves (HSCs). In addition, it was revealed that the discharge that yields the maximum Weighted Usable Area (WUA) for the whole fish community is more than twice the discharge yielding the maximum WUA for a single target species. This increased discharge leads to a 58% increase in the maximum WUA for the fish community.

The Uncertainty in the Prediction of the Distribution of Individual Wave Overtopping Volumes Using a Nonlinear Shallow Water Equation Solver

ABSTRACT Williams, H.E.; Briganti, R.; Pullen, T., and Dodd, N., 2016. The uncertainty in the prediction of the distribution of individual wave overtopping volumes using a nonlinear shallow water equation solver. This work analyses the uncertainty of the prediction of individual overtopping volumes using the nonlinear shallow water equations. A numerical model is used to analyse the variability from seeding. The effect of the incident wave height distribution on the distribution of individual overtopping volume is also considered. The numerical results are then compared with both the laboratory tests and available empirical methods. A large variability was found across the distributions, which produced some results showing significant diversion from the empirical prediction methods. The magnitude of that departure was directly related to the accuracy of the numerical model in reproducing the incident wave height distribution at the toe of the structure in the physical model.

Stream Computation of Shallow Water Equation Solver for FPGA-based 1D Tsunami Simulation

MOST (Method Of Splitting Tsunami) is widely used to solve shallow water equations (SWEs) for forecasting tsunami generated by an earthquake. Toward development of a power-efficient and high-performance computing system for 2D tsunami simulation, we conduct feasibility study on stream computation of 1D SWE solver with FPGA.We analyze an original code and design a stream algorithm with techniques of kernel fusion, shift buffering for streamed stencil-data access, and cascading processing elements for a longer pipeline. We implement a deep pipeline with at most 744 stages of 4 SPEs on 28 nm Stratix V FPGA, which achieves 82.4 GFlop/s at 200 MHz.

Fully well-balanced, positive and simple approximate Riemann solver for shallow water equations

The present work is focused on the numerical approximation of the shallow water equations. When studying this problem, one faces at least two important issues, namely the ability of the scheme to preserve the positiveness of the water depth, along with the ability to capture the stationary states.We propose here aGodunov-typemethod that fully satisfies the previous conditions, meaning that the method is in particular able to preserve the steady states with non-zero velocity.

Performance prediction of a tidal in-stream current energy converter and site assessment next to Jindo, South Korea

The magnitude of energy, which can be extracted by tidal energy converter devices, has a significant impact on the commercial viability of a project. Therefore, a quantitative characterization of the tidal current at the location of interest and subsequently a reliable prediction of the productivity of a tidal energy converter prior to installation is of high interest. This paper presents the successful deployment of a unique and novel fully symmetric tidal in-stream current energy converter HyTide 110-5.3 in Jindo, South Korea, and a methodology to predict the productivity of a single device based on simulation. The simulation combines a 2D shallow water equation model with turbine performance curves and is validated using real performance data from the prototype under real conditions. In addition, the predicted productivity is compared with actual field measurements during the operation of the Voith demonstrator using two Acoustic Doppler Current Profilers according to the IEC specifications for the performance assessment of Tidal Energy Converters, which is novel at that time. The simulation results show that the productivity of a single device can be predicted accurately and furthermore serves as a proof of concept for the symmetrical turbine layout. The 2D shallow water equation solver based on OpenFOAM® (tidalFoam) captures the rough conditions at the turbine site accurately, where the turbine is facing flood tides with a mean inclined inflow angle of 30°. In addition, the zero-equation turbulence model is shown to successfully capture the influence of a Kármán Vortex street on the turbine unit. High-resolution data of bathymetry, shorelines, and tidal elevations are used to set up the open boundaries of the unstructured mesh used in the model. The sea ground friction as an additional source term in the model is used to calibrate the simulation against Acoustic Doppler Current Profiler measurements on site. The simulation results are shown to be reliable, yielding highly accurate productivity predictions of a single tidal turbine. This is an important step towards a robust commercial evaluation of tidal energy projects prior to installation. Based on the single turbine model, simulations of three tidal current turbine farms as well as the available theoretical and technical power output of the region around Jindo during an entire moon cycle were performed. Possible impacts on average volumetric flow rate changes for neighbouring channels are presented.

A non-homogeneous Riemann solver for shallow water equations in porous media

The purpose of the current research is to develop an accurate and efficient solver for shallow water flows in porous media. The hydraulics is modeled by the two-dimensional shallow water flows with variable horizontal porosity. The variation of porosity in the water flows can be attributed to the variation of bed properties of the water system. As an example of porous shallow water flows is the passage of water discharge over vegetated areas in a river. Driving force of the phase separation and the mixing is the gradient of the porosity. For the numerical solution procedure, we propose a non-homogeneous Riemann solver in the finite volume framework. The proposed method consists of a predictor stage for the discretization of gradient terms and a corrector stage for the treatment of source terms. The gradient fluxes are discretized using a modified Roe’s scheme using the sign of the Jacobian matrix in the coupled system. A well-balanced discretization is used for the treatment of source terms. The efficiency of the solver is evaluated by several test problems for shallow water flows in porous media. The numerical results demonstrate high resolution of the proposed non-homogeneous Riemann solver and confirm its capability to provide accurate simulations for porous shallow water equations under flow regimes with strong shocks.