Multi-block lattice Boltzmann simulations of solute transport in shallow water flows

We report a two-dimensional multi-block lattice Boltzmann model for solute transport in shallow water flows, which is developed based on the advection–diffusion equation for mass transport and the shallow water equations for the flows. A weighting factor is included in the centered scheme for improved accuracy. The model is firstly verified by simulating three benchmark tests: wind-driven circulation in a dish-shaped lake, jet-forced flow in a circular basin, and flow formed by two parallel streams containing different uniform concentrations at the same constant velocity; and then it is applied to a practical wind-induced flow, Baiyangdian Lake, which is characterized by irregular geometries and complex bathymetries. The numerical results have shown that the model is able to produce accurate and detailed results for both water flows and solute transport, which is attractive, especially for flows in narrow zones of practical terrains and certain areas with largely varying pollutant concentrations.     

On boundary conditions in the lattice Boltzmann model for advection and anisotropic dispersion equation

A mirror-image method is proposed in this paper to solve the boundary conditions in the lattice Boltzmann model proposed by Zhang et al. [Adv. Water Resour. 25 (2002) 1] for the advection and anisotropic dispersion of solute transport in porous media. Three types of boundary are considered: prescribed concentration boundary, prescribed flux boundary and prescribed concentration-gradient boundary. The accuracy of the proposed method is verified against benchmark problems and finite difference method.     

A lattice Boltzmann model coupled with a Large Eddy Simulation model for flows around a groyne

This present paper proposes a two-dimensional lattice Boltzmann model coupled with a Large Eddy Simulation (LES) model and applies it to flows around a non-submerged groyne in a channel. The LES of shallow water equations is efficiently performed using the Lattice Boltzmann Method (LBM) and the turbulence can be taken into account in conjunction with the Smagorinsky Sub-Grid Stress (SGS) model. The bounce-back scheme of the non-equilibrium part of the distribution function is used to determine the unknown distribution functions at inflow boundary, the zero gradient of the distribution function is set normal to outflow boundary to obtain the unknown distribution functions here and the bounce-back scheme, which states that an incoming particle towards the boundary is bounced back into fluid, is applied to the solid wall to ensure non-slip boundary conditions. The initial flow field is defined firstly and then is used to calculate the local equilibrium distributions as initial conditions of the distribution functions. These coupled models successfully predict the flow characteristics, such as circulating flow, velocity and water depth distributions. The comparisons between the simulated results and the experimental data show that the model scheme has the capacity to solve the complex flows in shallow water with reasonable accuracy and reliability.     

A coupled model for simulating surface water and groundwater interactions in coastal wetlands

Coastal wetlands are characterized by strong, dynamic interactions between surface water and groundwater. This paper presents a coupled model that simulates interacting surface water and groundwater flow and solute transport processes in these wetlands. The coupled model is based on two existing (sub) models for surface water and groundwater, respectively: ELCIRC (a three‐dimensional (3‐D) finite‐volume/finite‐difference model for simulating shallow water flow and solute transport in rivers, estuaries and coastal seas) and SUTRA (a 3‐D finite‐element/finite‐difference model for simulating variably saturated, variable‐density fluid flow and solute transport in porous media). Both submodels, using compatible unstructured meshes, are coupled spatially at the common interface between the surface water and groundwater bodies. The surface water level and solute concentrations computed by the ELCIRC model are used to determine the boundary conditions of the SUTRA‐based groundwater model at the interface. In turn, the groundwater model provides water and solute fluxes as inputs for the continuity equations of surface water flow and solute transport to account for the mass exchange across the interface. Additionally, flux from the seepage face was routed instantaneously to the nearest surface water cell according to the local sediment surface slope. With an external coupling approach, these two submodels run in parallel using time steps of different sizes. The time step (Δt_g) for the groundwater model is set to be larger than that (Δt_s) used by the surface water model for computational efficiency: Δt_g = M × Δt_s where M is an integer greater than 1. Data exchange takes place between the two submodels through a common database at synchronized times (e.g. end of each Δt_g). The coupled model was validated against two previously reported experiments on surface water and groundwater interactions in coastal lagoons. The results suggest that the model represents well the interacting surface water and groundwater flow and solute transport processes in the lagoons. Copyright © 2011 John Wiley & Sons, Ltd.     

A nonlinear polar coordinate shallow water model for tsunami computation along North Sumatra and Penang Island

A nonlinear shallow water model in cylindrical polar coordinate system is developed, using an explicit finite difference scheme with a very fine resolution, to compute different aspects of tsunami at North Sumatra and the adjacent island Simeulue in Indonesia, and the Penang Island in Peninsular Malaysia. The pole of the frame is placed on the mainland of Penang (100.5°E) and the model area extends up to the west of Sumatra (87.5°E). The model is applied to simulate the propagation of tsunami wave towards North Sumatra, Simeulue and Penang Islands associated with Indonesian tsunami of 26 December 2004. The model is also applied to compute water levels along the coastal belts of those islands. Computed and observed water level data are found to be in good agreement and North Sumatra is found to be vulnerable for very high surges. The computed and observed arrival times of high surges are also in reasonable agreement everywhere. Further studies are carried out to investigate the effect of convective terms and it is found that their effects are insignificant in tsunami propagation and weakly significant for wave amplitude very near to the coast.     

Balancing the source terms in a SPH model for solving the shallow water equations

A shallow flow generally features complex hydrodynamics induced by complicated domain topography and geometry. A numerical scheme with well-balanced flux and source term gradients is therefore essential before a shallow flow model can be applied to simulate real-world problems. The issue of source term balancing has been exhaustively investigated in grid-based numerical approaches, e.g. discontinuous Galerkin finite element methods and finite volume Godunov-type methods. In recent years, a relatively new computational method, smooth particle hydrodynamics (SPH), has started to gain popularity in solving the shallow water equations (SWEs). However, the well-balanced problem has not been fully investigated and resolved in the context of SPH. This work aims to discuss the well-balanced problem caused by a standard SPH discretization to the SWEs with slope source terms and derive a corrected SPH algorithm that is able to preserve the solution of lake at rest. In order to enhance the shock capturing capability of the resulting SPH model, the Monotone Upwind-centered Scheme for Conservation Laws (MUSCL) is also explored and applied to enable Riemann solver based artificial viscosity. The new SPH model is validated against several idealized benchmark tests and a real-world dam-break case and promising results are obtained.     

A robust well-balanced finite volume model for shallow water flows with wetting and drying over irregular terrain

An unstructured Godunov-type finite volume model is developed for the numerical simulation of geometrically challenging two-dimensional shallow water flows with wetting and drying over convoluted topography. In the framework of sloping bottom model, a modified formulation of shallow water equations is used to preserve mass conservation during flooding and recession. The key ingredient of the model is the use of this combination of the sloping bottom model and the modified shallow water equations to provide a robust technique for wet/dry fronts tracking and, together with centered discretization of the bed slope source term, to exactly preserve the static flow on irregular topographies. The variable reconstruction technique ensures nonnegative reconstructed water depth and reasonable reconstructed velocity, and the friction terms are solved by semi-implicit scheme that does not invert the direction of velocity components. The robustness and accuracy of the proposed model are assessed by comparing numerical and reference results of extensive test cases. Moreover, the results of a dam-break flooding over real topography are presented to show the capability of the model on field-scale application.     

Numerical calculation for bed variation in compound-meandering channel using depth integrated model without assumption of shallow water flow

In a compound meandering channel, patterns of flow structures and bed variations change with increasing water depth owing to complex momentum exchange between high-velocity flow in a main channel and low-velocity flows in flood plains. We have developed a new quasi-three-dimensional model without the shallow water assumption, i.e., hydrostatic pressure distribution; our method is known as the general bottom velocity computation (BVC) method. In this method, a set of depth-integrated equations, including depth-integrated momentum and vorticity equations, are prepared for evaluating bottom velocity and vertical velocity distributions. The objective of this study is to develop a bed variation calculation method for both single and compound meandering channels by using the BVC method coupled with a sediment transport model. This paper shows that the BVC method can reproduce the pattern change of bed variation in a compound meandering channel flow with increasing relative depth. The variation in sediment transport rate due to overbank flow is explained by experimental and computational results.     

A 2D weakly-coupled and efficient numerical model for transient shallow flow and movable bed

Recent advances in the simulation of free surface flows over mobile bed have shown that accurate and stable results in realistic problems can be provided if an appropriate coupling between the shallow water equations (SWE) and the Exner equation is performed. This coupling can be done if using a suitable Jacobian matrix. As a result, faithful numerical predictions are available for a wide range of flow conditions and empirical bed load discharge formulations, allowing to investigate the best option in each case study, which is mandatory in these type of environmental problems. When coupling the equations, the SWE are considered but including an extra conservation law for the sediment dynamics. In this way the computational cost may become unrealistic in situations where the application of the SWE over rigid bed can be used involving large time and space scales without giving up to the adequate level of mesh refinement. Therefore, for restoring the numerical efficiency, the coupling technique is simplified, not decreasing the number of waves involved in the Riemann problem but simplifying their definitions. The effects of the approximations made are tested against experimental data which include transient problems over erodible bed. The simplified model is formulated under a general framework able to insert any desirable discharge solid load formula.