Shallow Water Equations Model Research Articles

Refinement Indicator for Dynamic-Mesh Adaption in Three-Dimensional Shallow-Water Equation Modeling

AbstractOne of the basic articles of discretization-based numerical modeling of three-dimensional shallow water equations is the spatial convergence of the mesh utilized. The process of creating a ...

A Coupled Model for Wave Run-up Simulation

AbstractSimplified models like the shallow water equations (SWE) are commonly adopted for describing a wide range of free surface flow problems, like flows in rivers, lakes, estuaries, or coastal areas. In the literature, numerical methods for the SWE are mostly mesh-based. However, this macroscopic approach is unable to accurately represent the complexity of flows near coastlines, where waves nearly break. This fact prompted the idea of coupling the mesh-based SWE model with a meshless particle method for solving the Euler equations. In a previous paper, a method to couple the staggered scheme SWE and the smoothed particle hydrodynamics (SPH) Euler equations was developed and discussed. In this article, this coupled model is used for simulating solitary wave run-up on a sloping beach. The results show strong agreement with the experimental data of Synolakis. Simulations of wave overtopping over a seawall were also performed.

Modeling and simulation of ocean wave propagation using lattice Boltzmann method

In this paper, we present on modeling and simulation of ocean wave propagation from the deep sea to the shoreline. This requires high computational cost for simulation with large domain. We propose to couple a 1D shallow water equations (SWE) model with a 2D incompressible Navier-Stokes equations (NSE) model in order to reduce the computational cost. The coupled model is solved using the lattice Boltzmann method (LBM) with the lattice Bhatnagar-Gross-Krook (BGK) scheme. Additionally, a special method is implemented to treat the complex behavior of free surface close to the shoreline. The result shows the coupled model can reduce computational cost significantly compared to the full NSE model.

Numerical modeling of the tidal wave run-up and the eelgrass habitat at the Laizhou Bay

In this paper, a modern lattice Boltzmann method is used to study the change of hydrodynamics and environmental factors of the intertidal zone in the Laizhou Bay between 1989 and 2013. In order to simulate the change of the shorelines caused by the tidal wave run-up, we use a high-order dry–wet boundary condition based on lattice Boltzmann model for shallow water equations. The hydrodynamic conditions including the intertidal area and the water depth on the eelgrass habitat are evaluated. As a result, the relationship among the reclamation area, the flow fields and the loss of intertidal wetland between 1989 and 2013 was established respectively. According to the interaction between the eelgrass in the intertidal zone and the hydrodynamic conditions, the area suitable for eelgrass growth will be revealed, which will provide scientific basis for wetland protection and theoretical support to marine ecological environment restoration.

Coastal morphodynamical modelling in nonlinear shallow water framework using a coordinate transformation method

A 1D numerical model of Nonlinear Shallow Water Equations (NSWEs) coupled to an advection equation for suspended sediment and a bed evolution equation is developed. The moving boundary at the shoreline is treated by a coordinate transformation method (CTM). An absorbing-generating seaward boundary condition in the transformed variables is also developed. The purely hydrodynamic component (NSWEs) is verified against analytical results. The NSWEs plus advection equation is verified quasi-analytical results. The fully-coupled model with bed change due to bed-load is verified against a single swash event and long-term numerical simulation. Excellent agreement is observed in all verifications.

A 1D–2D Shallow Water Equations solver for discontinuous porosity field based on a Generalized Riemann Problem

A novel augmented Riemann Solver capable of handling porosity discontinuities in 1D and 2D Shallow Water Equation (SWE) models is presented. With the aim of accurately approximating the porosity source term, a Generalized Riemann Problem is derived by adding an additional fictitious equation to the SWEs system and imposing mass and momentum conservation across the porosity discontinuity. The modified Shallow Water Equations are theoretically investigated, and the implementation of an augmented Roe Solver in a 1D Godunov-type finite volume scheme is presented. Robust treatment of transonic flows is ensured by introducing an entropy fix based on the wave pattern of the Generalized Riemann Problem. An Exact Riemann Solver is also derived in order to validate the numerical model. As an extension of the 1D scheme, an analogous 2D numerical model is also derived and validated through test cases with radial symmetry. The capability of the 1D and 2D numerical models to capture different wave patterns is assessed against several Riemann Problems with different wave patterns.

Model reduction of unstable systems using balanced truncation method and its application to shallow water equations

In this paper, we discuss the reduction of unstable systems. Suppose we have a large-scale unstable system. The reduction of such systems are done in order to obtain a simpler model that has a similar behavior with the original model. In reality, a higher order model is not preferred because the computation takes more time. Thus we need a method to reduce the order of a model that is known as model reduction. Model reduction of a system is the method of approximation of a system with a lower order but the dynamic behavior is equal or almost equal to the original model. In this paper, we discuss model reduction of unstable systems. Finally this method is applied to shallow water equation that describes the flow of water in rivers: we obtain a procedure for model reduction of unstable systems and the reduced model of shallow water equations.

Characteristics of wave amplitude and currents in South China Sea induced by a virtual extreme tsunami

This paper investigates the potential extreme tsunami hazards of the worst case scenario of the magnitude Mw = 9.30 in South China Sea (SCS) as the Manila Trench is becoming one of the most hazardous tsunami source regions. Using nonlinear shallow water equations model, the time series of surface elevation, arrival time, spatial distributions of maximum wave amplitude and velocity distribution are presented. The characteristics of wave and currents are analyzed. The numerical results indicate that most of the energy of tsunami wave distributes in central and north part of SCS. The offshore regions around SCS will be influenced significantly by the tsunami currents generated by an earthquake in the Manila subduction zone. The maximum wave amplitude near Guangdong Province, Hainan Island, and Taiwan Island exceeds 4 m and velocities at the majority of measured locations near coast exceeds 2 m/s. Nested grid with high resolution is used to study the impacts of the tsunami on Hainan Island, Taiwan Island, and Lingding Bay. The regions with high hazard risk due to strong currents are identified. Finally, a fast tsunami warning method in SCS is developed and discussed, which can provide tsunami warning information in 5 min.

Second-order force scheme for lattice Boltzmann model of shallow water flows

ABSTRACTA force scheme of second-order accuracy in both time and space has been introduced into the lattice Boltzmann model for shallow water equations. The scheme is tested within the single relaxation framework by using a one-dimensional steady flow over a bump, two one-dimensional dam-break flows and a two-dimensional unsteady dam-break flow. It is found that this second-order force scheme can improve the computational accuracy in time compared with the previous force treatment in the lattice Boltzmann model for shallow water. Moreover, the scheme can be also incorporated in the multiple-relaxation-time models.

Supercritical flow simulation at a right channel junction. Comparison between a uniform and a sparse mesh

This technical note studies the supercritical junction flow occurring at the right-angled confluence of four equal-width channels in which two upstream channels carry flow towards the junction. The note seeks to investigate, firstly, the pertinence of a 2D shallow water equation model to reproduce the typical flow structures at the junction. The second objective is to build a computationally efficient model with more resolution applied at critical point i.e. junction and less in areas where the flow is primarily 1D i.e. channels and compare such model, named, ‘sparse’ with a uniformly meshed model regarding solution accuracy and computational efficiency. The results indicate that the sparse model is able to reproduce typical flow structures appearing at the channel junction in an adequate manner. The discharge distribution is fairly well predicted. The jump angles are almost the same in the two models as well as the location and size of the recirculation zones and the flow depth super-elevation areas. However, the two models diverge in the prediction of very small depths in the recirculation zone where the sparse model overestimates the depths. As regards, computational efficiency, the sparse model is found to be 61% more efficient than the uniform mesh model.

A weak-constraint 4DEnsembleVar. Part I: formulation and simple model experiments

ABSTRACT4DEnsembleVar is a hybrid data assimilation method which purpose is not only to use ensemble flow-dependent covariance information in a variational setting, but to altogether avoid the computation of tangent linear and adjoint models. This formulation has been explored in the context of perfect models. In this setting, all information from observations has to be brought back to the start of the assimilation window using the space-time covariances of the ensemble. In large models, localisation of these covariances is essential, but the standard time-independent localisation leads to serious problems when advection is strong. This is because observation information is advected out of the localisation area, having no influence on the update.This is part I of a two-part paper in which we develop a weak-constraint formulation in which updates are allowed at observational times. This partially alleviates the time-localisation problem. Furthermore, we provide—for the first time—a detailed description of strong- and weak-constraint 4DEnVar, including implementation details for the incremental form.The merits of our new weak-constraint formulation are illustrated using the Korteweg-de-Vries equation (propagation of a soliton). The second part of this paper deals with experiments in larger and more complicated models, namely the Lorenz (1996) model and a shallow water equations model with simulated convection.

Numerical Simulations of Dam-break Floods with MPM

Due to the ability to model large deformations and the ease of dealing with boundary conditions in a Lagrangian framework, the material point method is getting more popularity in geotechnical engineering applications. In this paper, the material point is tentatively applied to model problems of hydrodynamics as an introduction to this field. Dam-break flows with different initial aspect ratios are simulated in both the material point method and an extensively verified shallow water equations model. In order to test the accuracy and stability of the material point method, simulations of dam-break flows are carried out and the results are in good agreement with other validated numerical methods and experimental data. From the comparisons between the simulations in both the material point method and shallow water equations model, a critical aspect ratio value for the applicability of the shallow water equations is found to be 1. The material point method shows its potential to tackle hydrodynamics related problems.

An improved SWE model for simulation of dam-break flows

A finite-difference method based on a shallow water equation (SWE) is developed in this study for simulation of dam-break flows. A second-order leap-frog scheme constructed in a staggered grid system is used to discretise the original SWE. To minimise the spurious fluctuations of the free surface near the moving front, an artificial diffusion term, which is simple, effective and easy to implement in the staggered-grid leap-frog system, is introduced on the right-hand side of the continuity equation. The model is verified against a wide range of classical analytical solutions, experimental data and other previous numerical results. The model is also validated against the real case of the Malpasset dam-break event. It is demonstrated that the improved model is well balanced and robust, and can accurately capture the movement of the dam-break flows.

A non-uniform efficient grid type for GPU-parallel Shallow Water Equations models

A GPU-parallel numerical model for the solution of the 2D Shallow Water Equations, based on a novel type of grid called Block-Uniform Quadtree (BUQ), is presented. BUQ grids are based on a data structure which allows to exploit the computational capability of GPUs with minimum overheads, while discretizing the domain with non-uniform resolution. Different cases have been simulated in order to assess the efficiency of the BUQ grids. Theoretical and laboratory tests demonstrate that speed-ups of up to one order of magnitude can be achieved in comparison with uniform Cartesian grids. In the simulation of a hypothetical flood event induced by a levee breach in a real 83 km long river reach, with maximum resolution of 5 m, a ratio of physical to computational time of about 12 was obtained, opening scenarios of quasi real-time 2D simulations in large domains, still retaining a high resolution where necessary.

A numerical investigation of the 1783 landslide-induced catastrophic tsunami in Scilla, Italy

The seismic crisis that hit Calabria (South Italy) since the beginning of 1783, lasted 3 years with strong earthquakes concentrated in the first 2 months, causing many fatalities and great destruction, as reported by several detailed historical reports. Following an earthquake that occurred on February 6, 1783, a mass with volume of about 6 Mm3 detached from the flank of Mount Paci and collapsed into the sea. In 1 min, huge waves attacked the nearby village of Scilla and killed most of the people that had gathered in the wide shore named Marina Grande to escape from seismic danger. More than 1500 people died due to the tsunami, many buildings were destroyed. In the paper, such catastrophic event is studied by means of numerical models. First we reconstruct the initial sliding mass and compute its motion through a Lagrangian model; then, we simulate the tsunami generation, propagation and flooding on the coast of Scilla by means of a nonlinear shallow-water equations model. The results of the simulation are compared against the detailed description of the tsunami effects available from several historical accounts and are found to be in a very satisfactory agreement with the 6–8 m inundation height reported in the Marina Grande beach.

A GPU accelerated adjoint-based optimizer for inverse modeling of the two-dimensional shallow water equations

The increasing computational capacity of current computing technologies makes feasible the application of predictive high-resolution mathematical models for studying physical phenomena. Additionally, this can be enhanced if those methods are incorporated to some optimization method in order to perform inverse modeling. In this paper, an optimization procedure based on the adjoint equations is used for the reconstruction of information in a 2D Shallow Water model previously developed and proved to be fast, robust and accurate. The continuous adjoint approach used for the evaluation of the gradient that is introduced in a gradient-based optimizer. Furthermore, the computation of both physical and adjoint systems is accelerated by the use of GPU programming. Even though notable speed-ups are achieved with this technique they are only possible in small to medium size grids due to memory limitations. A novel checkpointing strategy is proposed to allow data handling in these devices hence offering the possibility to overcome that limitation.

Irregular wave runup statistics on plane beaches: Application of a Boussinesq-type model incorporating a generating–absorbing sponge layer and second-order wave generation

Efficient absorption of reflected waves at the offshore boundary is a prerequisite for the accurate physical or theoretical modelling of long-duration irregular wave runup statistics at uniform, gently sloped beaches. This paper presents an implementation of the method suggested by Zhang et al. (2014) to achieve reflected wave absorption and simultaneous generation and propagation of incident waves in an existing numerical wave flume incorporating a moving boundary wavemaker. A generating–absorbing layer is incorporated within this 1DH hybrid Boussinesq-nonlinear shallow water equation model such that inshore-travelling incident waves, encompassing bound-wave structure approximately correct to second order, propagate unhindered while offshore-travelling reflected waves are absorbed. Once validated, the method is used to compile random wave runup statistics on uniform beach slopes broadly representative of dissipative, intermediate, and reflective beaches. Analyses of the individual runup time series, ensemble statistics and comparison to an empirical formula based on experimental runup data suggest that the main aspects of runup observed in the field are properly represented by the model. Existence of an upper limit on maximum runup is investigated using a simple extreme-value statistical analysis. Spectral saturation is examined by considering ensemble-averaged swash spectra for three representative beach slopes subject to incident waves with two different offshore significant wave heights. All spectra show f−4 roll-off at high frequencies in agreement with many previous field studies. The effect is also investigated of the swash motions preceding one particular extreme runup event on the eventual maximum runup elevation.

Simulation of wave phenomena in the nearshore through application of O(μ2) and O(μ4) pressure-Poisson Boussinesq type models

The pressure-Poisson Boussinesq models of Donahue et al. (2014) are extended to the surf zone and tested at O (μ2) and O (μ4). This approach resembles a classic Boussinesq-type model with the exception that the dispersive accuracy is obtained through consideration of a Boussinesq-scaled Green–Naghdi type approximation to the nonhydrostatic pressure profile and solved as a solution to the pressure-Poisson equation. The result is a Boussinesq-type model that is similar to the shallow water equations (SWE) with a nonhydrostatic pressure term. This allows for the straightforward modification of established SWE solvers to turn them into fully resolved nearshore models with very little structural change to the underlying SWE model and without the need to solve mixed space/time derivatives. The excellent hyperbolic properties of the discontinuous Galerkin (DG) finite element method are used to solve all terms except pressure and a simple yet accurate finite difference approach is used to solve the elliptic pressure equations. A straightforward numerics based wave breaking algorithm is employed and wave runup is captured using a globally mass conservative wetting/drying algorithm specifically designed for DG finite element methods. The result is a full surf-zone model capable of capturing the propagation of waves offshore, the breaking of waves near shore and full wave runup. The model is verified and validated using analytical and experimental results for both fully nonlinear O (μ2) and weakly nonlinear O (μ4)implementations.

A Comprehensive Sensitivity Analysis of Tsunami Model System to the Parametric and Input Uncertainties

ABSTRACT Jung, T.-H.; Son, S. and Lynett, P., 2016. A Comprehensive Sensitivity Analysis of Tsunami Model System to the Parametric and Input Uncertainties. In: Vila-Concejo, A.; Bruce, E.; Kennedy, D.M., and McCarroll, R.J. (eds.), Proceedings of the 14th International Coastal Symposium (Sydney, Australia). Journal of Coastal Research, Special Issue, No. 75, pp. 1117 - 1121. Coconut Creek (Florida), ISSN 0749-0208. In the present study, it is examined the tsunami modelling sensitivity to the various input factors based on the simulation results of 2004 Indian Ocean Tsunami. The numerical tests were implemented using coupled shallow water equation model, COMCOT(Liu et al., 1998) and Boussinesq model(Son et al., 2011) on the multi-grid system. Four different configurations(CASE 1 through 4) where three factors were controlled independently had been set up and simulated. From the numerical results, it was examined that some of defined factors may have control over the level of prediction accuracy in tsunami ...

Numerical Modelling of Braided River Morphodynamics: Review and Future Challenges

AbstractNumerical morphological modelling of braided rivers is increasingly used to explore controls on river pattern and for applied environmental management. This article reviews and presents a taxonomy of braided river morphodynamic models and discusses the challenges facing model development and use, illustrating these challenges with a case example. The taxonomy is contextualised by an initial discussion of the physical mechanisms of braiding. The taxonomy differentiates between reach‐scale and catchment‐scale models. Reach‐scale models are usually physics‐based, which are further divided based upon the mathematical approach used to solve equations (analytical or numerical) and their dimensionality (1D, 2D or 3D). Cellular automata models are one type of numerical model that replace at least some physical processes with expedient rules. A 2D physics‐based approach encapsulates sufficient process complexity to provide behavioural predictions. Predictions from catchment‐scale landscape evolution models have potential for providing boundary conditions. Future progress in physics‐based modelling needs to address three challenges: (i) representation of flow and sediment transport; (ii) temporal and spatial scaling; and (iii) model calibration, sensitivity, uncertainty and validation. The key problem for addressing these is the dearth of laboratory or natural experiment datasets. To show that progress can be made by comparing reach‐scale predictions to high‐resolution observations, a case study of monitoring and modelling, conducted in the Rees River, New Zealand, is presented. Hydraulic predictions of cellular automata and shallow water equation (Delft3d) models are compared to observed inundation extent. The efficacy of high‐resolution, multi‐temporal morphological data for assessing 2D physics‐based morphodynamic model predictions is also demonstrated.