Shallow Water Transport Research Articles

Semi-discrete entropy-preserving surface reconstruction schemes for the shallow water equations: Analysis of physical structures

We aim to introduce a new variant of hydrostatic reconstruction scheme, which is physical-structure-preserving and semi-discrete entropy-preserving for shallow water equations with a discontinuous bottom topography, based on novel surface reconstructions (NSR). The NSR is used to define approximate Riemann states with respect to the water depth and the velocity to compute consistent numerical fluxes across the cell interface and the upwind interface source term. This work is motivated by path-conservative surface reconstructions (Dong (2023) [19]), in which the conditions to satisfy the entropy inequality are complicated. We propose the NSR scheme to satisfy the entropy inequality without the corresponding complicated conditions. A key advantage of the NSR scheme is to preserve the physical structure of the water surface in the sense that when the water surface is continuous, then the defined approximate Riemann state should inherit this property. The physical-structure-preserving property is indispensable for obtaining a more accurate discretized source term and avoiding artificial “waterfall” effects. We prove that the NSR scheme satisfies a semi-discrete entropy condition and can guarantee the water depth to be nonnegative, and maintain the stationary state. We show several computed results to compare the NSR scheme with the hydrostatic reconstruction scheme based on subcell reconstructions and the path-conservative surface reconstruction scheme. We also extend the NSR scheme for pollutant transport in shallow water based on adaptive moving triangular meshes. Finally, several numerical results obtained by the adaptive NSR scheme for pollutant transport in shallow water are shown.

A semi-implicit finite volume method for the Exner model of sediment transport

The aim of this work is to construct efficient finite volume schemes for the numerical study of sediment transport in shallow water, in the framework of the Exner model [7,10]. In most cases, the velocity related to the sediment is much lower that the fluid velocity, which, in turn, may be much lower that the free-surface wave speed. Explicit methods that resolve all waves require small time steps due to the CFL stability restriction because of fast surface waves. Furthermore, if Rusanov flux is adopted, slow sediment waves may be affected by the large numerical diffusion. The objective of the present work is to drastically improve the efficiency in the computation of the evolution of the sediment by treating water waves implicitly, thus allowing much larger time steps than the one required by fully explicit schemes. The goal is reached by suitably semi-implicit schemes obtained by the use of implicit-explicit Runge-Kutta methods.

Numerical Modeling of Passive Scalar Transport in Shallow Water Based on the Quasi-Gasdynamic Approach

A new method for the numerical solution of the passive scalar transport equation in the framework of hydrodynamic equations in the shallow water approximation is described. The method is based on previously developed quasi-gasdynamic algorithms for numerical simulation of compressible gas flows. Smoothed equations are derived, and their difference approximations, including for flows with a pollutant source, are presented. The numerical algorithms are tested as applied to one- and two-dimensional flows. As an example, the algorithm is used to solve the problem of water circulation in Lake Vallunden. The constructed approach is generalized to passive scalar transport in the case of viscous incompressible flows.

Developing a Stochastic Model for Studying and Simulating Sediment Transport in Ports and Harbors

A particle model to describe and predict sediment transport in shallow water is developed with the use of random walk models. The model is developed by showing consistency between the Fokker-Plank equation and the Advection diffusion equations. Erosion and deposition process in the model are developed probabilistically where the erosion term is considered to be a constant and deposition term is taken as a function by relating sediment settling velocity and diffusion coefficient. Eventually, we simulated the particle model by considering three environment tests. In each environment test the simulations show the distribution of particle and the position of each particle at any given time t. The simulations also show the particles that will finally remain in suspension state and the particles that will be deposited during the transport process following the deployment of 10,000 particles. It was also established that there is a uniform distribution of particles in the test environment I and III and a linear dependence between the number of particles in a different grid cell and the water depth in test environment II.

The evolutionary changes of channel morphology in initial stage of water flow and sediment transport in shallow water

When sediment laden flow enters the broad areas from a narrow river channel, the flow velocity declines abruptly, due to the enlarging width. As a result, the sediment silts in the region. This phenomenon occurs when the sediment laden flow enters a plain from a mountainous river, or when it flows into lakes and oceans from rivers. The former condition results in the development of an alluvial fan, while the latter leads to the formation of a delta, and then again gradually evolved into an alluvial river which by the developing process of water flow and sediment transport, and it appears various pattern along with its growth. The river pattern can divide into many types, the classics types are straight channel, meandering channel, branching channel and braided channel. Model test is one of the most important technical means researching water flow and sediment transport and predicting riverbed evolution. At present, many scholars, domestic and overseas, excavate straight river in the foundation bed to research channel morphology changes, and do less research on non-straight river in the foundation channel morphological changes. The formation and development process of alluvial river is a continuous variation, the essay will mainly discuss the evolution change of river channel from nothing on the surface of rivers into the lake delta, under the constant coming water and sediment and constant erosion base level. In this study, in order to study the evolutionary changes of channel morphology in initial stage of water flow and sediment transport in shallow water, non-uniform sediment was used, for a model generalization test, under the conditions of a constant inflow water and sediment, gradient, as well as an outlet base level. The sediment test results showed that the evolutionary changes of channel morphology experienced three stages. At the first stage, straight channel and branching channel were appeared alternately. At the second stage, a single micro-bend channel morphology swing development on the surface of delta deposition occurred with the branching channel of tributary shunt gradually decrease and die. At the third stage, there’s mainly meandering channel occurred with the coefficient of twists become larger with the concave bank continuously scouring regressive in the micro-bend channel. Through analysis and experiment, results show that the channel morphology of the second stage and the third stage present as sine curve characteristics: the upper channel located in the inlet of the sine curve, the middle and lower of the bend channel are located in the peaks and troughs of sine curve. With time going on, the amplitude and direction of sine curve become large gradually, and the coefficient of channel tends to be stable. Finally, the evolutionary changes of channel morphology is analyzed from the bed load and boundary conditions. This study is of great theoretical significance, and the achievements will enrich the theory of river dynamics and riverbed evolution.

A two-dimensional coupled flow-mass transport model based on an improved unstructured finite volume algorithm

A two-dimensional coupled water quality model is developed for modeling the flow-mass transport in shallow water. To simulate shallow flows on complex topography with wetting and drying, an unstructured grid, well-balanced, finite volume algorithm is proposed for numerical resolution of a modified formulation of two-dimensional shallow water equations. The slope-limited linear reconstruction method is used to achieve second-order accuracy in space. The algorithm adopts a HLLC-based integrated solver to compute the flow and mass transport fluxes simultaneously, and uses Hancock’s predictor–corrector scheme for efficient time stepping as well as second-order temporal accuracy. The continuity and momentum equations are updated in both wet and dry cells. A new hybrid method, which can preserve the well-balanced property of the algorithm for simulations involving flooding and recession, is proposed for bed slope terms approximation. The effectiveness and robustness of the proposed algorithm are validated by the reasonable good agreement between numerical and reference results of several benchmark test cases. Results show that the proposed coupled flow-mass transport model can simulate complex flows and mass transport in shallow water.

A well‐balanced weighted essentially non‐oscillatory scheme for pollutant transport in shallow water

SUMMARYIn this paper, a well‐balanced finite difference weighted essentially non‐oscillatory scheme is presented for modeling transport and diffusion of pollutant in shallow water flows. The scheme balances exactly the flux gradients and the source terms. Extensive one‐dimensional and two‐dimensional numerical experiments on uniform and curvilinear meshes strongly suggest that high resolution results are achieved for both water depth and pollutant concentration. The scheme is efficient and robust and can be applied to practical numerical simulation of pollutant transport phenomena in shallow water flows. Copyright © 2012 John Wiley & Sons, Ltd.

Fully-Coupled Modeling of Shallow Water Flow and Pollutant Transport on Unstructured Grids

Understanding the space-time dynamics of pollutant transport remains an essential impediment to accurate prediction of impacts on the ecology of rivers and coastal areas and also for establishing efficient strategies for pollution control and environmental protection. Numerical models are a powerful tool to study the water flows and pollutant transport, and recently a new generation of models is being developed to simulate the coupled flow and pollutant transport in shallow water. In this paper, a two-dimensional fully-coupled model of shallow water flows and pollutant transport was developed using a triangular unstructured grid (TIN: triangular irregular network), which is also an important module of the PIHM-Hydro modeling system. The model is based on a cell-centered upwind finite volume method using the HLL approximate Riemann solver. A multidimensional linear reconstruction technique and multi-dimensional slope limiter was implemented to achieve a second-order spatial accuracy. In order to make the model efficient and stable, an explicit-implicit method was used in temporal discretization by an operator splitting technique. A test case of the pollutant transport in a square cavity is used to validate the model. Then the model was further applied to two pollutant transport scenarios: microscale pollutant transport following dam break and mesoscale pollutant transport driven by storm surge in Galveston Bay. The numerical results show that the model could accurately predict the flow dynamics and pollutant transport in extreme events such as a dam break and a storm surge. According to the prediction of the model, the storm surge caused by the Hurricane Ike significantly extended the polluted area.

Modelling solute transport in shallow water with the lattice Boltzmann method

The lattice Boltzmann method is used to investigate the solute transport in shallow water flows. Shallow water equations are solved using the lattice Boltzmann equation on a D2Q9 lattice with multiple-relaxation-time (MRT-LBM) and Bhatnagar–Gross–Krook (BGK-LBM) terms separately, and the advection–diffusion equation is also solved with a LBM-BGK on a D2Q5 lattice. Three cases: open channel flow with side discharge, shallow recirculation flow and flow in a harbour are simulated to verify the described methods. Agreements between predictions and experiments are satisfactory. In side discharge flow, the reattachment length for different ratios of side discharge velocity to main channel velocity has been studied in detail. Furthermore, the performance of MRT-LBM and BGK-LBM for these three cases has been investigated. It is found that LBM-MRT has better stability and is able to satisfactorily simulate flows with higher Reynolds number. The study shows that the lattice Boltzmann method is simple and accurate for simulating solute transport in shallow water flows, and hence it can be applied to a wide range of environmental flow problems.

The Effect of Channel Length on the Residual Circulation in Tidally Dominated Channels

Abstract With an analytic model, this paper describes the subtidal circulation in tidally dominated channels of different lengths, with arbitrary lateral depth variations. The focus is on an important parameter associated with the reversal of the exchange flows. This parameter (δ) is defined as the ratio between the channel length and one-quarter of the tidal wavelength, which is determined by water depth and tidal frequency. In this study, a standard bottom drag coefficient, CD = 0.0025, is used. For a channel with δ smaller than 0.6–0.7 (short channels), the exchange flow at the open end has an inward transport in deep water and an outward transport in shallow water. This situation is just the opposite of channels with a δ value larger than 0.6–0.7 (long channels). For a channel with a δ value of about 0.35–0.5, the exchange flow at the open end reaches the maximum of a short channel. For a channel with a δ value of about 0.85–1.0, the exchange flow at the open end reaches the maximum of a long channel, with the inward flux of water occurring over the shoal area and the outward flow in the deep-water area. However, near the closed end of a long channel, the exchange flow appears as that in a short channel—that is, the exchange flow changes direction along the channel from the head to the open end of the channel. For a channel with a δ value of about 0.6–0.7, the tidally induced subtidal exchange flow at the open end reaches its minimum when there is little flow across the open end and the water residence time reaches its maximum. The mean sea level increases toward the closed end for all δ values. However, the spatial gradient of the mean sea level in a short channel is much smaller than that of a long channel. The differences between short and long channels are caused by a shift in dynamical balance of momentum or, equivalently, a change in tidal wave characteristics from a progressive wave to a standing wave.

Approximate factorization in shallow water applications

We consider the numerical integration of problems modelling phenomena in shallow water in 3 spatial dimensions. If the governing partial differential equations for such problems are spatially discretized, then the right-hand side of the resulting system of ordinary differential equations can be split into terms f 1, f 2, f 3 and f 4, respectively representing the spatial derivative terms with respect to the x, y and z directions, and the interaction terms. It is typical for shallow water applications that the interaction term f 4 is nonstiff and that the function f 3 corresponding with the vertical spatial direction is much more stiff than the functions f 1 and f 2 corresponding with the horizontal spatial directions. The reason is that in shallow seas the gridsize in the vertical direction is several orders of magnitude smaller than in the horizontal directions. In order to solve the initial value problem (IVP) for these systems numerically, we need a stiff IVP solver, which is necessarily implicit, requiring the iterative solution of large systems of implicit relations. The aim of this paper is the design of an efficient iteration process based on approximate factorization. Stability properties of the resulting integration method are compared with those of a number of integration methods from the literature. Finally, a performance test on a shallow water transport problem is reported.

A discontinuous Galerkin method for two-dimensional flow and transport in shallow water

A discontinuous Galerkin (DG) finite element method is described for the two-dimensional, depth-integrated shallow water equations (SWEs). This method is based on formulating the SWEs as a system of conservation laws, or advection–diffusion equations. A weak formulation is obtained by integrating the equations over a single element, and approximating the unknowns by piecewise, possibly discontinuous, polynomials. Because of its local nature, the DG method easily allows for varying the polynomial order of approximation. It is also “locally conservative”, and incorporates upwinded numerical fluxes for modeling problems with high flow gradients. Numerical results are presented for several test cases, including supercritical flow, river inflow and standard tidal flow in complex domains, and a contaminant transport scenario where we have coupled the shallow water flow equations with a transport equation for a chemical species.

Observations of Shallow-Water Transport and Shear in Western Florida Bay*

Abstract Acoustic Doppler profiler (ADP) data are used to describe depth-integrated transport and vertical shear at two study sites along the open western boundary of Florida Bay. During a 404-day study period, transport was into the bay at the northern study site, where the depth-averaged 3.2 m. Transport at the southern study site, where the depth averaged 2.0 m, was out of the bay during a 266-day time period. A comparison of flow in the top and bottom layers of the ADP profile at the northern study site reveals patterns that are very different from each other, and very different from the depth-integrated transport. Nontidal flow in the surface layer is nearly consistently out of the bay and highly correlated with wind forcing. East–west near-bottom flow is inversely correlated with the rise and fall of water level in northwest Florida Bay. Nontidal east–west flow in the bottom layer is negligible, suggesting a near balance between westward directed wind stress and eastward directed barotropic pressure...

Splitting methods for three-dimensional bio-chemical transport

Splitting methods for the time integration of three-dimensional transport-chemistry models offer interesting prospects: second-order accuracy can be combined with sufficient stability, and the amount of implicitness can be reduced to a manageable level. Furthermore, exploiting the parallelization and vectorization features of the algorithm, a realistic simulation with many species over long time intervals becomes feasible. As an alternative to the usual splitting functions, such as co-ordinate splitting or operator splitting, we discuss in this paper a splitting function that is of hopscotch type. Both for a second-order, symmetric spatial discretization (resulting in a three-point coupling in each direction), and for a third-order, upwind discretization (giving rise to a five-point coupling, in general), we define a particular variant of this hopscotch splitting. These splitting functions will be combined with an appropriate splitting formula, resulting in second-order (in time) splitting methods. A common feature of both hopscotch splitting functions is that we have only coupling in the vertical direction, resulting in a stability behaviour that is independent of the vertical mesh size; this is an important property for transport in shallow water. Another characteristic of this hopscotch-type splitting is that it allows for an easy application of domain decomposition techniques in the horizontal directions. Two choices for the splitting formula will be presented. The resulting methods have been applied to a large-scale test problem and the numerical results will be discussed. Furthermore, we show performance results obtained on a Cray C98/4256. As part of the project TRUST (Transport and Reactions Unified by Splitting Techniques), preliminary versions of the schemes are available for benchmarking.

Splitting Methods for Three-Dimensional Transport Models with Interaction Terms

We investigate the use of splitting methods for the numerical integration of three-dimensional transport-chemistry models. In particular, we investigate various possibilities for the time discretization that can take advantage of the parallelization and vectorization facilities offered by multi-processor vector computers. To suppress wiggles in the numerical solution, we use third-order, upwind-biased discretization of the advection terms, resulting in a five-point coupling in each direction. As an alternative to the usual splitting functions, such as co-ordinate splitting or operator splitting, we consider a splitting function that is based on a three-coloured hopscotch-type splitting in the horizontal direction, whereas full coupling is retained in the vertical direction. Advantages of this splitting function are the easy application of domain decomposition techniques and unconditional stability in the vertical, which is an important property for transport in shallow water. The splitting method is obtained by combining the hopscotch-type splitting function with various second-order splitting formulae from the literature. Although some of the resulting methods are highly accurate, their stability behaviour (due to horizontal advection) is quite poor. Therefore we also discuss several new splitting formulae with the aim to improve the stability characteristics. It turns out that this is possible indeed, but the price to pay is a reduction of the accuracy. Therefore, such methods are to be preferred if accuracy is less crucial than stabilitys such a situation is frequently encountered in solving transport problems. As part of the project TRUST (Transport and Reactions Unified by Splitting Techniques), preliminary versions of the schemes are implemented on the Cray C98 4256 computer and are available for benchmarking.

Transport of Sediments by Waves, Adriatic Coastal Shelf, Italy

Knowledge of marine-sediment transport in shallow water is as yet insufficient to give much help to the geologist who maps the distribution of ancient sandstone bodies. In order to improve this knowledge, the bottom sediments of the Adriatic Sea near the Italian coast were sampled and analyzed. These sediments were transported as uniform suspensions. A close relation was found between certain grain-size parameters, particularly the 1 percentile, and sea depth, indicating that turbulence induced by waves was a major factor in the sand distribution. Another important factor that generally has been overlooked is percentage of sand in the sediment source for the area. If these source sediments contain little sand and consist almost entirely of silt and clay, the sand remains close to shore. If, however, sand is the predominant constituent of the source sediments, as commonly occurred in ancient seas, sand probably can be spread by waves across extensive areas as deep as wave base. The grain size of the sand also is an important factor in its distribution, some sand suspensions being displaced by gravity and some not. Therefore, grain size may determine the transport mechanism and, thus, the sediment distribution. An understanding of these factors by the petroleum geologist is essential in his mapping and projection of sandstone deposits in the subsurface.