Shallow Water Equations Model Research Articles

Run-up parameterization and beach vulnerability assessment on a barrier island: a downscaling approach

Abstract. We present a downscaling approach for the study of wave-induced extreme water levels at a location on a barrier island in Yucatán (Mexico). Wave information from a 30-year wave hindcast is validated with in situ measurements at 8 m water depth. The maximum dissimilarity algorithm is employed for the selection of 600 representative cases, encompassing different combinations of wave characteristics and tidal level. The selected cases are propagated from 8 m water depth to the shore using the coupling of a third-generation wave model and a phase-resolving non-hydrostatic nonlinear shallow-water equation model. Extreme wave run-up, R2%, is estimated for the simulated cases and can be further employed to reconstruct the 30-year time series using an interpolation algorithm. Downscaling results show run-up saturation during more energetic wave conditions and modulation owing to tides. The latter suggests that the R2% can be parameterized using a hyperbolic-like formulation with dependency on both wave height and tidal level. The new parametric formulation is in agreement with the downscaling results (r2 = 0.78), allowing a fast calculation of wave-induced extreme water levels at this location. Finally, an assessment of beach vulnerability to wave-induced extreme water levels is conducted at the study area by employing the two approaches (reconstruction/parameterization) and a storm impact scale. The 30-year extreme water level hindcast allows the calculation of beach vulnerability as a function of return periods. It is shown that the downscaling-derived parameterization provides reasonable results as compared with the numerical approach. This methodology can be extended to other locations and can be further improved by incorporating the storm surge contributions to the extreme water level.

A phase-resolved, depth-averaged non-hydrostatic numerical model for cross-shore wave propagation

In this study, a phase-resolved and depth-averaged non-hydrostatic numerical model (SNH model) is developed. The non-incremental pressure-correction method is employed to solve the equation system in two successive steps. Firstly, an approximate Riemann solver in the framework of finite volume methods is employed to solve the hydrostatic shallow-water equations (SWE) on a collocated grid to obtain provisional solutions. Then, the intermediate solutions is updated by considering the non-hydrostatic pressure effect; a semi-staggered grid is used in this step to avoid predicting checkboard pressure field. A series of benchmark tests are used to validate the numerical model, showing that the developed model is well-balanced and describes the wetting and drying processes accurately. By employing a shock-capturing numerical scheme, the wave-breaking phenomenon is reasonably simulated without using any ad-hoc techniques. Compared with the SWE model, the wave shape can be well-preserved and the numerical predictions are much improved by using the SNH model.

A two-dimensional depth-integrated non-hydrostatic numerical model for nearshore wave propagation

In this study, we develop a shallow-water depth-integrated non-hydrostatic numerical model (SNH model) using a hybrid finite-volume and finite-difference method. Numerical discretization is performed using the non-incremental pressure-correction method on a collocated grid. We demonstrate that an extension can easily be made from an existing finite-volume method and collocated-grid based hydrostatic shallow-water equations (SWE) model to a non-hydrostatic model. A series of benchmark tests are used to validate the proposed numerical model. Our results demonstrate that the proposed model is robust and well-balanced, and it captures the wet–dry fronts accurately. A comparison between the SNH and SWE models indicates the importance of considering the wave dispersion effect in simulations when the wave amplitude to water depth ratio is large.

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 new three-dimensional terrain-following tidal model of free-surface flows

A three-dimensional hydrodynamic model is presented which combines a terrain-following vertical coordinate with a horizontally orthogonal curvilinear coordinate system to fit the complex bottom topography and coastlines near estuaries, continental shelves, and harbors. To solve the governing equations more efficiently, we improve the alternating direction implicit method, which is extensively used in the numerical modeling of horizontal two-dimensional shallow water equations, and extend it to a three-dimensional tidal model with relatively little computational effort. Through several test cases and realistic applications, as presented in the paper, it can be demonstrated that the model is capable of simulating the periodic to-and-fro currents, wind-driven flow, Ekman spirals, and tidal currents in the near-shore region.

Broadband omnidirectional absorption for linear liquid surface wave

We investigate the propagation of linear liquid surface wave (LLSW) over an omnidirectional absorber (OA), which has uneven bottom with appropriate gradient depth distribution. The OA consists of a guiding shell with gradient bottom topography and a central absorptive cavity. The non-dispersive shallow-water wave equation model and the dispersive mild-slope equation model are employed to investigate the absorption properties of the OA. The results show that almost omnidirectional incident LLSW can be spirally deflected from the interface to the center in the guiding shell and finally absorbed by the central cavity. The introducing of guiding shell enlarges the absorption cross section of the bare absorptive cavity from the inner diameter to the outer diameter of the OA, resulting in high absorption efficiency. Furthermore, the proposal can be constructed by simple modulation of the liquid bottom, which may significantly facilitate the engineering applications.

Free surface flows over partially erodible beds by a continuous finite element method

This paper presents a continuous finite element model for shallow water equations coupled with a bedload sediment conservation law. The model incorporates a sign-preserving conservative correction based on the flux correction transport methodology. Complete algorithm permits simulation of flows with evolutionary boundaries and erodible–non erodible interfaces under severe conditions without spurious mass exchanges and oscillations. Numerical experiments explore the efficacy of the procedure for dam-break problems on channels with dry beds, for partially and totally erodible beds, and for migration of small bedforms. Computation of real river morphodynamics under intense flooding illustrates practical application of the model.

Mixing in the Intermediate Field of Dense Jets in Cross Currents

AbstractThis study presents an experimental and numerical study for an inclined (60° to horizontal) dense jet discharged into a coflowing current. The mixing and transport of the density current arising from the jet impingement on a horizontal bottom boundary is investigated. A light attenuation technique is employed to measure the layer-averaged concentration field over a region that extends 20FrD downstream and 9FrD laterally (where F = jet densimetric Froude number, and D = jet diameter). A comprehensive characterization of the resulting buoyant spread in both steady and unsteady cases is obtained. The concentration field is also computed using a three-dimensional (3D) shallow water equation model via the distributed entrainment sink approach that incorporates a near-field Lagrangian integral jet model (JETLAG) into the 3D model for dynamic simulation of the near-far field transition. The results show the occurrence of bifurcation in the gravitational spreading layer when the impinging dense jet is ben...

A well balanced Roe scheme for the local inertial equations with an unstructured mesh

In this paper a finite volume method with an unstructured node based grid is developed in order to integrate the local inertial equations for flood modelling: the Gravity Wave Model (GWM). The GWM is based on the Shallow Water Equation (SWE) but neglects the convective term for improving the model computational speed. The numerical integration of the GWM equations is made using a first order in time and space well-balanced Roe Riemann Solver derived for the GWM set of equations. Several well-known SWE and urban flood model benchmark tests are used to verify the applicability of the GWM model. In addition analytical and numerical solutions of the SWE are also compared to the GWM. The results show a good agreement between the GWM and the SWE in all tested cases.

Assessment of Kuroshio current power test site of Green Island, Taiwan

Assessment of Kuroshio power test site was performed by considering major ruling factors, including in-situ measurement, GIS (Geographic Information System) data base, and ocean flow simulations. Key factors consist of Kuroshio current speed, island wake induced by Kuroshio passing, fishing rights, shipping routes, environmental protection, as well as seabed topography and geological features. Kuroshio current field around Green Island was measured using ADCP and X-Band radar, and was simulated using POM (Princeton Ocean Modeling) and SWEs (Shallow Water Equations) model. Meandering vortex street behind the Island observed by satellite images were numerically reproduced. All ruling factors considered are integrated by the GIS data base. The optimal power test site is thus determined based on the superposition of all influence factors.

Tsunami propagation over varying water depths

The linear Boussinesq equations are an ideal model for transoceanic propagation of tsunamis. However, they are impractical for real-time application because Boussinesq-type equation models rely on a fine grid system and therefore require a huge computational domain. Thus, shallow-water equations models are the preferred method of predicting propagation and run-up of near- and far-field tsunamis since they produce fairly accurate results with a much smaller computational requirement. There may be an additional benefit in including physical dispersion effects in numerical models since shallow-water equations theoretically neglect the effect of dispersion on the transoceanic propagation of tsunamis. In this study, a modified finite difference scheme was proposed that adds terms to the linear shallow-water equations in order to account for varying water depths. The proposed model was verified by applying it to tsunami propagation over a submerged shoal and the results were compared with those of the well-known Boussinesq equations model, FUNWAVE. The proposed model was further tested by simulating transoceanic tsunami propagation on real topographies and comparing the numerical results with available observed data.

An improved algorithm for the shallow water equations model reduction: Dynamic Mode Decomposition vs POD

SummaryWe propose an improved framework for dynamic mode decomposition (DMD) of 2‐D flows for problems originating from meteorology when a large time step acts like a filter in obtaining the significant Koopman modes, therefore, the classic DMD method is not effective. This study is motivated by the need to further clarify the connection between Koopman modes and proper orthogonal decomposition (POD) dynamic modes. We apply DMD and POD to derive reduced order models (ROM) of the shallow water equations. Key innovations for the DMD‐based ROM introduced in this paper are the use of the Moore–Penrose pseudoinverse in the DMD computation that produced an accurate result and a novel selection method for the DMD modes and associated amplitudes and Ritz values. A quantitative comparison of the spatial modes computed from the two decompositions is performed, and a rigorous error analysis for the ROM models obtained is presented. Copyright © 2015 John Wiley & Sons, Ltd.

Wet-dry moving boundary treatment for Runge-Kutta discontinuous Galerkin shallow water equation model

The wet-dry scheme for moving boundary treatment is implemented in the framework of discontinuous Galerkin shallow water equations. As a formulation of approximate Riemann solver, the HLL (Harten-Lax-van Leer) numerical fluxes are employed and the TVB (Total Variation Bounded) slope limiter is used for the removal of unnecessary oscillations. As benchmark test problems, the dam-break problems and the classical problem of periodic oscillation in the parabolic bowl are solved with linear triangular elements and second-order Runge-Kutta scheme. The results are compared with exact solutions and the numerical solutions of previous study. For a more practical application, the implicit Runge-Kutta scheme is employed for the bottom friction terms and the moving shoreline in a rectangular basin of varying slopes is simulated. In all case studies, good agreement is observed with exact solutions or other well-known numerical solutions.

Turbulence modelling of shallow water flows using Kolmogorov approach

This study uses an improved k–ε coupled shallow water equations (SWE) model that equipped with the numerical computation of the velocity fluctuation terms to investigate the turbulence structures of the open channel flows. We adapted the Kolmogorov K41 scaling model into the k–ε equations to calculate the turbulence intensities and Reynolds stresses of the SWE model. The presented model was also numerically improved by a recently proposed surface gradient upwind method (SGUM) to allow better accuracy in simulating the combined source terms from both the SWE and k–ε equations as proven in the recent studies. The proposed model was first tested using the flows induced by multiple obstructions to investigate the utilised k–ε and SGUM approaches in the model. The laboratory experiments were also conducted under the non-uniform flow conditions, where the simulated velocities, total kinetic energies (TKE) and turbulence intensities by the proposed model were used to compare with the measurements under different flow non-uniformity conditions. Lastly, the proposed numerical simulation was compared with a standard Boussinesq model to investigate its capability to simulate the measured Reynolds stress. The comparison outcomes showed that the proposed Kolmogorov k–ε SWE model can capture the flow turbulence characteristics reasonably well in all the investigated flows.

Refined representation of turbines using a 3D SWE model for predicting distributions of velocity deficit and tidal energy density

Energy sustainability has become one of the most concerned issues around the world, and tidal stream power is one of the promising types of renewable energy with merits of high predictability, relatively low cost, and low environmental impact. Numerical models based on the shallow water equations (SWE) have been applied to assess the energy distribution and environmental impact. In existing SWE models, the effect induced by tidal stream turbines on the fluid is usually represented by bed friction, which is only applicable for far-field studies. In the current study, a Blade Element Momentum (BEM) model has been developed and integrated into a three-dimensional (3D) SWE model to improve the accuracy for assessing tidal current energy distributions. Using the BEM approach, state variables can be obtained for a blade element of a turbine rotor. These state variables can be used to calculate the local thrust coefficient, which is introduced for representing the effect of rotors in the 3D SWE models. A 3D SWE model has been refined, by incorporating the proposed BEM model, to simulate the density distribution of a tidal turbine farm. The refined SWE model has been validated against measurements from a flume experiment of turbine arrays. The model performs generally well in predicting the distribution of velocity deficit and the recovery of wakes. In a field-scale application to a planned tidal power test site, the model was used to predicting the flow field for both a single turbine scenario and a turbine array scenario. Copyright © 2015 John Wiley & Sons, Ltd.

Definition and solution of a stochastic inverse problem for the Manning’s n parameter field in hydrodynamic models

The uncertainty in spatially heterogeneous Manning’s n fields is quantified using a novel formulation and numerical solution of stochastic inverse problems for physics-based models. The uncertainty is quantified in terms of a probability measure and the physics-based model considered here is the state-of-the-art ADCIRC model although the presented methodology applies to other hydrodynamic models. An accessible overview of the formulation and solution of the stochastic inverse problem in a mathematically rigorous framework based on measure theory is presented. Technical details that arise in practice by applying the framework to determine the Manning’s n parameter field in a shallow water equation model used for coastal hydrodynamics are presented and an efficient computational algorithm and open source software package are developed. A new notion of “condition” for the stochastic inverse problem is defined and analyzed as it relates to the computation of probabilities. This notion of condition is investigated to determine effective output quantities of interest of maximum water elevations to use for the inverse problem for the Manning’s n parameter and the effect on model predictions is analyzed.

Mathematical Model of Complete Shallow Water Problem with Source Terms, Stability Analysis of Lax-Wendroff Scheme

The most effective simulations of complete physical problems consist of the evaluation of maximum water levels and discharges that may be attained at particular locations during the development of an exceptional meteorological event. There is also the prevision of the scenario subsequent to the almost instantaneous release of a great volume of liquid. The situation is that of the breaking of a man made dam. There is therefore a necessity to develop a model capable of reproducing solutions of the complete equations despite the irregularities of a non-prismatic bed. This requires the development of efficient and effective numerical schemes able to predict water levels and discharges in hydraulic systems. The use of mathematical models as a predictive tool in the simulation of free surface flows represents a good candidate for the application of many of the techniques developed in fluid dynamics. In this paper we develop a 1-D complete model of shallow water equations with source terms using both conservation of water mass and conservation of the momentum content of the water. We describe the Lax-Wendroff scheme for these nonlinear partial differential equations (PDEs) and we analyze the stability restriction of the method. This extends the nonstationary shallow water problems without source terms which are deeply studied in literature. Some numerical experiments are considered and critically discussed.

Assessment on the Characteristics and the Damping Performance of Coastal Forests in Pangandaran after the 2006 Java Tsunami

The use of coastal forests as natural tsunami barrier has long been the source of controversies due to inconclusive understanding on the damping performance and contradictory field findings from the last tsunami events. After experiencing hundreds of casualties and severe destruction on the infrastructures due to the 2006 South Java Tsunami, the coastline of Pangandaran district in West Java was replanted by coastal forest vegetation to mitigate future tsunami as well as improving the quality of the environment. The design of coastal forests along the coastline of Pangandaran adopted divide-and-rule strategy concept. Considering their physical and biological aspects as well as the ability to withstand hydraulic forces. This concept requires different species at different locations perpendicular to the coastline with staggered formations in order to provide optimum damping performance of the coastal forests. Unfortunately, this concept has never been quantified or hydraulically tested against extreme cases like tsunami. The paper will present the characteristics and the latest conditions of coastal forests in Pangandaran including the problems and the challenge of this mitigation system in the last five years. Moreover, damping performance of the coastal forests was also investigated by means of Nonlinear Shallow Water Equation Model to quantify the contribution of coastal forest and other existing environmental aspects such as the bathymetry and the topography to the total tsunami attenuation. Based on the obtained wave transmission, the simulation analysis shows that the characteristics of bathymetry and topography along the coastline play significant role in reducing the impact of tsunami compared to the coastal forests.

Assessment of the Effectiveness of General Breakwaters in Reducing Tsunami Inundation in Ishinomaki

The 2011 Great East Japan Earthquake and Tsunami caused devastation all along the coast of eastern Japan. Ishinomaki City was one of the most severely damaged municipalities, though the height of the tsunami in this city was smaller than that in Iwate and northern Miyagi. A large difference in the extent of building damage was found comparing two areas of Ishinomaki: one not protected by breakwaters, and the other behind the breakwaters of a large Fishery port. In this paper, the authors perform numerical simulations to determine whether these general breakwaters, which were designed not to block tsunamis but wind waves, reduced the tsunami's energy and contributed to a reduction in inundation of the areas behind the port. Before assessing the effectiveness of breakwaters against tsunamis, simulated inundation heights in each of these two areas were compared with heights measured by The 2011 Tohoku Earthquake Tsunami Joint Survey Group. It was found that a simulation with larger Manning's n value (n = 0.30) can evaluate inundation more precisely than when this value is small (n = 0.06), as often used by Japanese coastal engineers. Comparing the region protected by breakwaters with the unsheltered area, the results of a 2D shallow water equation model do not show a significant difference in inundation mainly because the tsunami intrudes through the openings in the breakwater and fills up the port basin with seawater in a very short time. Therefore, the effectiveness of general breakwaters in reducing tsunami impact should not be overestimated. However, a hypothetical study shows that water levels could be greatly reduced if the port were fully enclosed by breakwaters. This implies that a port could substantially reduce tsunami inundation if the breakwater openings were equipped to be closed before tsunami arrival. The authors also demonstrate that the difference in the extent of building damage in the two areas of Ishinomaki considered can be explained by the difference in drag forces due to the topography as well as the difference in land use, rather than by the presence of breakwaters.

Efficient GPU-Implementation of Adaptive Mesh Refinement for the Shallow-Water Equations

The shallow-water equations model hydrostatic flow below a free surface for cases in which the ratio between the vertical and horizontal length scales is small and are used to describe waves in lakes, rivers, oceans, and the atmosphere. The equations admit discontinuous solutions, and numerical solutions are typically computed using high-resolution schemes. For many practical problems, there is a need to increase the grid resolution locally to capture complicated structures or steep gradients in the solution. An efficient method to this end is adaptive mesh refinement (AMR), which recursively refines the grid in parts of the domain and adaptively updates the refinement as the simulation progresses. Several authors have demonstrated that the explicit stencil computations of high-resolution schemes map particularly well to many-core architectures seen in hardware accelerators such as graphics processing units (GPUs). Herein, we present the first full GPU-implementation of a block-based AMR method for the second-order Kurganov---Petrova central scheme. We discuss implementation details, potential pitfalls, and key insights, and present a series of performance and accuracy tests. Although it is only presented for a particular case herein, we believe our approach to GPU-implementation of AMR is transferable to other hyperbolic conservation laws, numerical schemes, and architectures similar to the GPU.