Three-dimensional Shallow Water Research Articles

Effect of Coriolis force on the linear stability of subaqueous dunes with erodible and non-erodible beds

Dunes, are wavy-shaped structures, that can be found on river shores and underwater. We aim to investigate the influence of Coriolis effect on the linear stability of subaqueous dunes using a simple model based on the three-dimensional shallow water equations written in rotating frame of reference. The analysis is performed for both erodible and non-erodible beds. Our analytical results in the absence of Coriolis force are verified with published results. The wave speed and growth rate of disturbances are presented for different Coriolis parameter, Froude number and erodibility. A new mode is observed, which becomes unstable beyond a threshold value of Coriolis parameter.

Historical overview of 2D and 3D hydrodynamic modelling of shallow water flows in the Netherlands

This paper contains a historical overview of 2D and 3D hydrodynamic modelling in the Netherlands from the 1960s till now. It started in the 1960s with two-dimensional (depth-averaged) shallow sea modelling for practical applications of the Dutch government. The most important application was the Dutch ‘Delta works’ project, which was initiated after the February 1953 flood disaster in the Netherlands. In the late 1980s, three-dimensional shallow water modelling in the Netherlands started due of the advance in computing power. This was also needed for 3D water quality modelling. Two large software suites have been developed for modelling of shallow water flows in the Netherlands, namely Simona and Delft3D. These integrated systems contain software for modelling of hydrodynamics, water quality, morphology and waves. In this paper, we focus on hydrodynamic modelling. In literature, a lot of applications of Delft3D and Simona have been described. However, in none of these papers, the key features of the underlying hydrodynamic numerical method have been described, which have resulted into a good performance with respect to robustness, accuracy and computational efficiency. This paper therefore summarizes ‘proven technology’ in shallow water modelling with Simona and Delft3D that has been applied in the past decades in the Netherlands and abroad. From the 1960s till now, the hydrodynamic modelling in the Netherlands has been based on structured grids. In 2011, a transition started in the Netherlands towards unstructured modelling for shallow water flows, which will be discussed as well.

Formulation and application of the adaptive hydraulics three-dimensional shallow water and transport models

Next generation, conservative finite element hydrodynamic and transport models are vital for accurate and efficient ocean, estuary and river simulation. Numerical models such as these have been developed for decades by the U.S. Army Corps of Engineers at the Engineering Research and Development Center (ERDC). This paper focuses on recently developed implicit, multi-dimensional finite element 3D shallow water and transport models included in the Adaptive Hydraulics (AdH) numerical suite. These AdH 3D models benefit from their adaptive meshing capabilities to resolve sharp solution gradients, such as those often encountered with baroclinic wedges traveling up an estuary channel. This paper presents the AdH 3D mathematical formulation and solution procedure used to solve the weak finite element equations for these models, along with results for several common verification cases and an AdH Galveston Bay validation study. A novel Streamline Upwind Petrov–Galerkin (SUPG) method for 3D shallow water models is described which reduces to the AdH 2D shallow water SUPG formulation under certain conditions. Careful attention is placed on ensuring discrete consistency in the equation set.

Three-dimensional shallow water sound propagation and applications toward acoustical oceanography

Underwater sound propagation in areas of the continental shelf, shelf break, and continental slope can encounter strong horizontal reflection, refraction, diffraction, focusing, and/or defocusing due to a variety of environmental factors, including bathymetric variability, water column fluctuations (internal waves and shelf-break fronts), and boundary roughness. Theoretical, numerical, and experimental approaches have been taken to investigate the underlying physics of these three-dimensional (3-D) sound propagation effects and their temporal and spatial variability induced jointly by marine geological features and dynamic oceanographic processes. Some recent work on theoretical analysis, numerical modeling and field work experiments to study 3-D sound propagation in nonlinear internal wave ducts, over the continental slope and in submarine canyons will be reviewed in this talk. The ultimate goal of investing these 3-D sound propagation effects is to improve long-distance acoustical oceanographic technolo...

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 numerical model for three-dimensional shallow water flows with sharp gradients over mobile topography

Abstract This study aims to develop a three-dimensional (3-D) numerical model for shallow water flows over mobile topography, which is capable of simulating morphological evolution under shock waves, e.g. dam-break flows. The hydrodynamic model solves the three-dimensional shallow water equations (SWEs) using a finite volume method on prismatic cells in σ-coordinates. The morphodynamic model solves an Exner equation consisting of bed-load sediment transportation. Using a relaxation approach, a hyperbolic system is built for hydrodynamic system, which allows for using a Godunov-type central-upwind method to capture the shocks and approximate the numerical fluxes. Consequently, the 3D-SWEs-Exner model proposed in the present study can stably and accurately solve the dam-break flows over mobile beds. A spatially and temporally second-order “prismatic” central-upwind method is used to approximate the numerical fluxes through cell interfaces. The Exner equation is solved using an upwind method. Using spatially linear reconstruction and explicit two-stage Runge–Kutta time discretization, second order accuracy is achieved in space and time. The proposed model can preserve the well-balanced property due to the special discretization of bed-slope source terms. The proposed model is validated by several tests with experimental measurements, and is compared with the simulated results using reported two-dimensional (2-D) models.

Three-dimensional shallow water system: A relaxation approach

We study a three-dimensional shallow water system, which is obtained from the three-dimensional Navier–Stokes equations after Reynolds averaging and under the simplifying hydrostatic pressure assumption. Since the three-dimensional shallow water system is generically not hyperbolic, it cannot be numerically solved using hyperbolic shock capturing schemes. At the same time, existing simple finite-difference and finite-volume methods may fail in simulations of unsteady flows with sharp gradients, such as dam-break and flood flows. To overcome this limitation, we propose a novel numerical method, which is based on a relaxation approach utilized to “hyperbolize” the three-dimensional shallow water system. The extended relaxation system is hyperbolic and we develop a second-order semi-discrete central-upwind scheme for it. The proposed numerical method can preserve “lake at rest” steady states and positivity of water depth over irregular bottom topography. The accuracy, stability and robustness of the developed numerical method is verified on five numerical experiments.

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...

Comparison of two-dimensional axial-symmetric and two-in-one-half dimensional Green's function methods for three-dimensional shallow water acoustic propagation using finite element methods

In shallow water acoustic propagation, performing a fully three-dimensional finite element model is currently unfeasible due to difficulty in implementation and limits in computational power. Therefore, alternative representations of the 3D acoustic field are sought. Two promising methods to represent the 3D field are a locally invariant, 2.5D Green’s Function kernel, and a 2D axial-symmetric reduction of the 3D Helmholtz equation. When a spherical source is used, azimuthal symmetry of the acoustic propagation is assumed, and these two methods can be compared in 2D planes of the 3D field. Because the 2.5D method takes more computational time than the axial-symmetric method, the accuracy of the pressure field are compared to see if the axial-symmetric method can be used in place of the 2.5D method. First, the two methods are compared for a flat ocean surface with stratified media. Then, a wedge-shaped ocean surface is considered, and the two methods are compared with 2D PE solutions. These comparisons will show if the axial-symmetric method produces similar results to the 2.5D method, and if so, under which geometrical and physical situations the axial-symmetric method can be used in place of the 2.5D method. [Work sponsored by the Office of Naval Research, Ocean Acoustics.]

Inversion of sound speed profile in three-dimensional shallow water based on transmission time

Horizontal deflection often occurs when a ray propagates in shallow water with an irregular sea bottom, which makes it difficult to determine eigen-ray trajectory and transmission time. A new method for seeking eigen-ray in three dimensions was proposed and sound speed profile (SSP) was inverted in three-dimensional shallow water using the fastest eigen-ray transmission times to form the cost function. In order to satisfy long-time monitoring of SSP, a kind of sonobuoy was developed with the function of DGPS, short wave radio controlling and dada transmission and experiment was implemented in the South China Sea. It is indicated that the sonobuoy has high reliability and stability in rough sea condition. The precision of SSP inversion can be enhanced dramatically by considering horizontal deflections of eigen-ray. In addition, the influence of small mismatch of seabed depth on precision of inversion can be neglected.

Three dimensional model of the runup of nonlinear surface gravity waves on the shallow water coastal slope

The paper is devoted to three-dimensional numerical modeling of a runup of nonlinear surface gravity waves based on Navier–Stokes equations. The three-dimensional formulation of the problem is considered, and the boundary and initial conditions are described. The decomposition method for physical processes is used to construct a discrete model capable of accounting for the factor of filled cells. Features of a design of the simulated three-dimensional shallow water area are described. Three-dimensional models of the staged runup of surface gravity waves on the coastal slope are presented. The wave processes occurring at the runup and back draft of nonlinear surface gravity waves are described.

Pseudospectral solution of three-dimensional nonlinear sloshing in a shallow water rectangular tank

In this study, a pseudospectral σ-transformation model is developed to simulate fully nonlinear sloshing waves in a three-dimensional shallow water rectangular tank. The σ-transformation maps the physical domain including the water free surface onto a fixed rectangular computational domain. Chebyshev collocation formulae are used to discretize the governing equation and boundary conditions in the computational domain. The numerical model is validated for three well known analytical and numerical sloshing problems. An extensive study is then made of sloshing in a shallow water tank, and the effects of excitation frequency, base aspect ratio, and amplitude of excitation on the wave motions and patterns are considered. Wave regimes and patterns are considerably influenced by the base aspect ratio. In a shallow water tank with a non-square base, different wave regimes are observed during small-amplitude resonant excitation. Also, bores develop during large amplitude resonance excitation. The present study demonstrates that a pseudospectral σ-transformation can accurately model nonlinear sloshing waves in a rectangular tank. Also, results show that contrary to the situation in deeper water tanks, sloshing in shallow water strongly depends on the base aspect ratio.

Fully non-linear simulation of second-order resonance in a three-dimensional tank using the PSME method

Sloshing occurs when a partially filled tank is subject to external excitation. If the excitation frequency is equal to half of the second-order natural frequency obtained from the linear theory, then second-order resonance may occur. But it gradually modifies to standing waves. Moreover, violent second-order resonant free surface motions may be induced when the sum (or difference) of the excitation frequency and any one of the natural frequencies is equal to another natural frequency. In this study, fully non-linear second-order resonance waves in a three-dimensional shallow water rectangular tank are simulated using a pseudospectral σ-transformation model. The model is validated against results from three benchmark tests for which there are published analytical and numerical solutions available. A detailed analysis is presented of sloshing in a shallow water tank, with the main excitation and response frequencies identified from the power spectra. Large amplitude free surface motions are observed whenever second-order resonance occurs. In certain cases, the wave pattern in the tank is different to that predicted from linear analysis of second-order resonance, due to the effect of nonlinearity. Results are presented from a parameter study examining the influences of water depth, base aspect ratio, and excitation amplitude on the wave motions and patterns. It is found that the wave pattern is highly dependent on the water depth in the tank, but relatively insensitive to excitation amplitude. Also, the decay patterns are seen when the second-order resonance excitations are applied vertically. The study demonstrates that second-order resonance can be very pronounced in shallow water.

Importance of cross-channel bathymetry and eddy viscosity parameterisation in modelling estuarine flow

For a proper understanding of flow patterns in curved tidal channels, quantification of contributions from individual physical mechanisms is essential. We study quantitatively how such contributions are affected by cross-channel bathymetry and three alternative eddy viscosity parameterisations. Two models are presented for this purpose, both describing flow in curved but otherwise prismatic channels with an (almost) arbitrary transverse bathymetry. One is a numerical model based on the full three-dimensional shallow water equations. Special feature of this diagnostic model is that assumptions regarding the relative importance of particular physical mechanisms can be incorporated in the computations by switching corresponding terms in the model equations on or off. We also present an idealized model that provides semi-analytical approximate solutions of the shallow water equations for all three considered alternative eddy viscosity parameterisations. It forms an aid in explaining and theorising about results obtained with the numerical model. Observations regarding Chesapeake Bay serve as a reference case for the present study. We find that the relative importance of both along-channel advective forcing and transverse diffusive forcing depends on local characteristics of the cross-sectional bottom profile rather than global ones. In our reference case, tide-residual along-channel flow induced by these forcings is not small compared to the total tidal residual. Building on this observation, we present an indicative test to judge whether advective processes should be included in leading order in modelling tide-dominated estuarine flow. Furthermore, depending on the applied eddy viscosity parameterisation (uniformly or parabolically distributed over the vertical), we find qualitatively different spatial patterns for the along-channel advective forcing.

Modelling the fate and transport of negatively buoyant storm–river water in small multi-basin lakes

The dynamics of negatively buoyant river plumes in a small multi-basin kettle lake with steep bathymetry (Toolik Lake, AK) are simulated using a Cartesian hydrodynamic model based on the solution of the three-dimensional shallow water equations. To validate the model, model predictions are compared with results from previous analytical and laboratory studies and with field observations. The grid resolution adopted for the Toolik Lake model is 0.5 m (= Δ z) in the vertical and 20 m (= Δ x) in the horizontal, so that the ratio of the bottom slope S 0 to Δ z/Δ x is lower than 4 in 99% of the computational domain. With that resolution, the model represents correctly the rate of mixing between lake and river water and the speed of propagation of downslope gravity currents. The model provides accurate predictions of the temperature structure (RMSE = 0.25 °C) and of eddy diffusivities at the depths of the intrusions of incoming water. Measurements and modelling show similar fractions and depth distribution of river water on a cross-basin transect, which suggests that the mixing dynamics of the plume as it transits between basins are well resolved. Thus, the stage is set to quantify the ecological consequences of storm events in small lakes with several interconnected basins using coupled biological measurements and 3D modelling.

Properties of the Eulerian–Lagrangian method using linear interpolators in a three-dimensional shallow water model using z-level coordinates

We investigate the impact of the inexact interpolation on the Eulerian–Lagrangian solution of the advection equation by combining numerical experiments and formal analysis. The simulations, respectively, using the Eulerian–Lagrangian method (ELM) and the upwind scheme are compared. The artificial resistance of the ELM is observed which is characterised by the higher free-surface elevation and the distorted turbulent properties at a smaller time step. Through analysis, we find that the abnormalities are caused by the fact the conventional linear interpolation does not adapt well to the nonlinear velocity distribution, which produces an advection computation error that increases with a decreasing time step. The phenomena are explained and an improved method ELM is proposed based on the illustrations and analysis. The new method combines the face-controlled interpolation and the adjustable sub time steps to skip the large computation error domain in the backtracking, and it is validated by the original test case.

Development of a three-dimensional numerical model to solve shallow-water equations in compound channels

Velocity gradient between main channel and flood plains in compound channels leads to the formation of a large shear layer and secondary currents between these two subsections. These phenomena in the interaction region bring about a complex three-dimensional nature of the flow in compound channels. To cope with these flows, many numerical investigations have utilized three-dimensional formulations with advanced turbulence models. However, the free surface in many of these models is fixed and rigid-lid assumption has been used. In the present research, three-dimensional shallow water equations were used to calculate the flow field in compound channels. Three-dimensional equations were integrated in layers and were combined with the continuity equation. In this formulation, free-surface elevation was calculated without the need to solve any additional equations. Velocity and bed shear stress distribution and the stage–discharge relationship in compound channels with smooth and rough beds and with different relative depths were analyzed to verify this model, and satisfactory results were obtained.

Mathematical modeling of tidal hydrodynamics in shallow lagoons: A review of open issues and applications to the Venice lagoon

Although considerable progress has been made in the application of two- and three-dimensional shallow water models to simulate flow in estuaries and coastal lagoons, a number of outstanding problems still remain in this branch of computational fluid dynamics. These problems mainly deal with the proper representation of physical processes and arise when dealing with very shallow flow, temporary submergence and time-dependent flow domains, and complex morphology, and are often related to inaccuracies in modeling the geometry of the domain. Among others, wetting and drying of salt marshes and tidal flats, inclusion of the small scale drainage networks which often strongly affect hydrodynamics, turbulence closure schemes, and accurate friction evaluation are discussed in this paper. To better illustrate these problems and their possible solutions some examples are presented which use the Venice lagoon as a benchmark shallow tidal basin characterized by a complex interplay of channels and marshes.

The morphodynamic modelling of tidal sand waves on the shoreface

An artificial sand wave on the Dutch shoreface of the North Sea has been studied in conditions with relatively strong tidal currents in the range of 0.5 to 1 m/s and sediments in the medium sand size range of 0.2 to 0.5 mm. The sand wave is perpendicular to the tidal current and has a maximum height and length of the order of 5 m and 1 km, respectively. The sand wave is dynamically active and shows migration rates of the order of a few metres per year. A numerical morphodynamic model (DELFT3D model) has been used to simulate the morphological behaviour of the sand wave in the North Sea. This model approach is based on the numerical solution of the three-dimensional shallow water equations in combination with a surface wave propagation model (wind waves) and the advection–diffusion equation for the sediment particles with online bed updating after each time step. The model results show that the sand wave grows in the case of dominant bed-load transport (weak tidal currents; relatively coarse sediment; small roughness height; low waves) and that the sand wave decays in the case of dominant suspended transport (strong currents, relatively fine sediment, large roughness height; storm waves).

The local discontinuous Galerkin method for three-dimensional shallow water flow

We formulate and present a stability analysis for the local discontinuous Galerkin method applied to a model of three-dimensional shallow water flow. This model is described by the Navier–Stokes equations, assuming hydrostatic pressure. The resulting system of equations is given by momentum equations for the x and y components of velocity, the continuity equation, which is used to solve for the vertical component of velocity, and an equation describing the motion of the free surface. Our analysis includes the full nonlinearities in the model with no simplifying assumptions and accounts for the movement of the mesh due to the free surface. To our knowledge this is the first analysis of any numerical scheme for this complex system of equations. Numerical results are presented which demonstrate the accuracy of the method for a problem with a known analytical solution.