Transport of pollutant in shallow flows: A space–time CE/SE scheme

This paper focuses on implementation of space–time CE/SE scheme for computing the transport of a passive pollutant by a flow. The flow model comprises of the Saint-Venant system of shallow water equations and the pollutant propagation is described by a transport equation. The one-dimensional and two-dimensional flow equations are numerically investigated using the CE/SE scheme. A number of test problems are presented to check the accuracy and efficiency of the proposed scheme. The results of CE/SE scheme are compared with the central scheme. Both the schemes are found to be in close agreement. However, our proposed CE/SE scheme accurately captures shocks and discontinuous profiles.     

Application of mesh-adaptation for pollutant transport by water flow

An adaptive finite volume method is proposed for the numerical solution of pollutant transport by water flows. The shallow water equations with eddy viscosity, bottom friction forces and wind shear stresses are used for modelling the water flow whereas, a transport-diffusion equation is used for modelling the advection and dispersion of pollutant concentration. The adaptive finite volume method uses simple centred-type discretization for the source terms, can handle complex topography using unstructured grids and satisfies the conservation property. The adaptation criteria are based on monitoring the pollutant concentration in the computational domain during its dispersion process. The emphasis in this paper is on the application of the proposed method for numerical simulation of pollution dispersion in the Strait of Gibraltar. Results are presented using different tidal conditions and wind-induced flow fields in the Strait.     

Application of mesh-adaptation for pollutant transport by water flow

An adaptive finite volume method is proposed for the numerical solution of pollutant transport by water flows. The shallow water equations with eddy viscosity, bottom friction forces and wind shear stresses are used for modelling the water flow whereas, a transport-diffusion equation is used for modelling the advection and dispersion of pollutant concentration. The adaptive finite volume method uses simple centred-type discretization for the source terms, can handle complex topography using unstructured grids and satisfies the conservation property. The adaptation criteria are based on monitoring the pollutant concentration in the computational domain during its dispersion process. The emphasis in this paper is on the application of the proposed method for numerical simulation of pollution dispersion in the Strait of Gibraltar. Results are presented using different tidal conditions and wind-induced flow fields in the Strait.     

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.     

A local discontinuous Galerkin method for a doubly nonlinear diffusion equation arising in shallow water modeling

In this paper, we study a local discontinuous Galerkin (LDG) method to approximate solutions of a doubly nonlinear diffusion equation, known in the literature as the diffusive wave approximation of the shallow water equations (DSW). This equation arises in shallow water flow models when special assumptions are used to simplify the shallow water equations and contains as particular cases: the Porous Medium equation and the parabolic p-Laplacian. Continuous in time a priori error estimates are established between the approximate solutions obtained using the proposed LDG method and weak solutions to the DSW equation under physically consistent assumptions. The results of numerical experiments in 2D are presented to verify the numerical accuracy of the method, and to show the qualitative properties of water flow captured by the DSW equation, when used as a model to simulate an idealized dam break problem with vegetation.     

A kinetic flux-vector splitting method for the shallow water magnetohydrodynamics

A kinetic flux-vector splitting (KFVS) scheme for the shallow water magnetohydrodynamic (SWMHD) equations in one- and two-space dimensions is formulated and applied. These equations model the dynamics of a thin layer of nearly incompressible and electrically conducting fluids for which the evolution is nearly two-dimensional with magnetic equilibrium in the third direction. The proposed numerical scheme is based on the direct splitting of macroscopic flux functions of the SWMHD equations. In two-space dimensions the scheme is derived in a usual dimensionally split manner; that is, the formulae for the fluxes can be used along each coordinate direction. The high-order resolution of the scheme is achieved by using a MUSCL-type initial reconstruction and Runge-Kutta time stepping method. Both one- and two-dimensional test computations are presented. For validation, the results of KFVS scheme are compared with those obtained from the space-time conservation element and solution element (CE/SE) method. The accuracy, efficiency and simplicity of the KFVS scheme demonstrate its potential in modeling SWMHD equations.     

A new ADI technique for two-dimensional parabolic equation with an integral condition

Nonclassical parabolic initial-boundary value problems arise in the study of several important physical phenomena. This paper presents a new approach to treat complicated boundary conditions appearing in the parabolic partial differential equations with nonclassical boundary conditions. A new fourth-order finite difference technique, based upon the Noye and Hayman (N-H) alternating direction implicit (ADI) scheme, is used as the basis to solve the two-dimensional time dependent diffusion equation with an integral condition replacing one boundary condition. This scheme uses less central processor time (CPU) than a second-order fully implicit scheme based on the classical backward time centered space (BTCS) method for two-dimensional diffusion. It also has a larger range of stability than a second-order fully explicit scheme based on the classical forward time centered space (FTCS) method. The basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyeet. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference methods. The results of numerical experiments for the new method are presented. The central processor times needed are also reported. Error estimates derived in the maximum norm are tabulated.     

A mimetic mass, momentum and energy conserving discretization for the shallow water equations

A spatial semi-discretization is developed for the two-dimensional depth-averaged shallow water equations on a non-equidistant structured and staggered grid. The vector identities required for energy conservation in the continuous case are identified. Discrete analogues are developed, which lead to a finite-volume semi-discretisation which conserves mass, momentum, and energy simultaneously. The key to discrete energy conservation for the shallow water equations is to numerically distinguish storage of momentum from advective transport of momentum. Simulation of a large-amplitude wave in a basin confirms the conservative properties of the new scheme, and demonstrates the enhanced robustness resulting from the compatibility of continuity and momentum equations. The scheme can be used as a building block for constructing fully conservative curvilinear, higher order, variable density, and non-hydrostatic discretizations.     

Well-balanced RKDG2 solutions to the shallow water equations over irregular domains with wetting and drying

This paper presents a new one-dimensional (1D) second-order Runge–Kutta discontinuous Galerkin (RKDG2) scheme for shallow flow simulations involving wetting and drying over complex domain topography. The shallow water equations that adopt water level (instead of water depth) as a flow variable are solved by an RKDG2 scheme to give piecewise linear approximate solutions, which are locally defined by an average coefficient and a slope coefficient. A wetting and drying technique proposed originally for a finite volume MUSCL scheme is revised and implemented in the RKDG2 solver. Extra numerical enhancements are proposed to amend the local coefficients associated with water level and bed elevation in order to maintain the well-balanced property of the RKDG2 scheme for applications with wetting and drying. Friction source terms are included and evaluated using splitting implicit discretization, implemented with a physical stopping condition to ensure stability. Several steady and unsteady benchmark tests with/without friction effects are considered to demonstrate the performance of the present model.     

Lattice Boltzmann method for variable density shallow water equations

A lattice Boltzmann method is developed for solution of a form of the shallow water equations that is suitable for flows which are fully mixed in the vertical direction but have variable density in the horizontal plane. In the present approach, double distribution functions are applied: one for the shallow water flows and the other for the mass transport. Direct coupling between the water flow and mass transport is achieved by updating the flow density from the concentration during simulation. Accuracy and applicability of the model are demonstrated by two numerical tests: the stationary hydrostatic equilibrium of liquid of variable density in a tank with non-uniform bed terrain, and the horizontal diffusion of species with an initial Gaussian distribution of concentration in a uniform flow field.     

High-Order Well-Balanced Finite Difference WENO Schemes for a Class of Hyperbolic Systems with Source Terms

In this paper, we generalize the high order well-balanced finite difference weighted essentially non-oscillatory (WENO) scheme, designed earlier by us in Xing and Shu (2005, J. Comput. phys. 208, 206–227) for the shallow water equations, to solve a wider class of hyperbolic systems with separable source terms including the elastic wave equation, the hyperbolic model for a chemosensitive movement, the nozzle flow and a two phase flow model. Properties of the scheme for the shallow water equations (Xing and Shu 2005, J. Comput. phys. 208, 206–227), such as the exact preservation of the balance laws for certain steady state solutions, the non-oscillatory property for general solutions with discontinuities, and the genuine high order accuracy in smooth regions, are maintained for the scheme when applied to this general class of hyperbolic systems     

A finite volume method to solve the 3D Navier–Stokes equations on unstructured collocated meshes

A new method to solve the Navier–Stokes equations for incompressible viscous flows and the transport of a scalar quantity is proposed. This method is based upon a fractional time step scheme and the finite volume method on unstructured meshes. The governing equations are discretized using a collocated, cell-centered arrangement of velocity and pressure. The solution variables are stored at the cell-circumcenters. Theoretical results and numerical properties of the scheme are provided. Predictions of lid-driven cavity flow, flows past a cylinder and heat transport in a cylinder are performed to validate the method.     

A Fourth Order Scheme for Incompressible Boussinesq Equations

A fourth order finite difference method is presented for the 2D unsteady viscous incompressible Boussinesq equations in vorticity-stream function formulation. The method is especially suitable for moderate to large Reynolds number flows. The momentum equation is discretized by a compact fourth order scheme with the no-slip boundary condition enforced using a local vorticity boundary condition. Fourth order long-stencil discretizations are used for the temperature transport equation with one-sided extrapolation applied near the boundary. The time stepping scheme for both equations is classical fourth order Runge–Kutta. The method is highly efficient. The main computation consists of the solution of two Poisson-like equations at each Runge–Kutta time stage for which standard FFT based fast Poisson solvers are used. An example of Lorenz flow is presented, in which the full fourth order accuracy is checked. The numerical simulation of a strong shear flow induced by a temperature jump, is resolved by two perfectly matching resolutions. Additionally, we present benchmark quality simulations of a differentially-heated cavity problem. This flow was the focus of a special session at the first MIT conference on Computational Fluid and Solid Mechanics in June 2001.     

A Hybrid Method to Solve Shallow Water Flows with Horizontal Density Gradients

In this work, a model for shallow water flows that accounts for the effects of horizontal density fluctuations is presented and derived. While the density is advected by the flow, a two-way feedback between the density gradients and the time evolution of the fluid is ensured through the pressure and source terms in the momentum equations. The model can be derived by vertically averaging the Euler equations while still allowing for density fluctuations in horizontal directions. The approach differs from multi-layer shallow water flows where two or more layers are considered, each of them having their own depth, velocity and constant density. A Roe-type upwind scheme is developed and the Roe matrices are computed systematically by going from the conservative to the quasi-linear form at a discrete level. Properties of the model are analyzed. The system is hyperbolic with two shock-wave families and a contact discontinuity associated to interfaces of regions with density jumps. This new field is degenerate with pressure and velocity as the corresponding Riemann invariants. We show that in some parameter regimes numerically recognizing such invariants across contact discontinuities is important to correctly compute the flow near those interfaces. We present a numerical algorithm that correctly captures all waves with a hybrid strategy. The method integrates the Riemann invariants near contact discontinuities and switches back to the conserved variables away from it to properly resolve shock waves. This strategy can be applied to any numerical scheme. Numerical solutions for a variety of tests in one and two dimensions are shown to illustrate the advantages of the strategy and the merits of the scheme.     

Computer modelling in environmental geomechanics

This paper presents a general framework for the computational analysis of environmental geomechanics problems. It is based on heat and multiphase flow in deforming porous media where pollutant transport mechanisms can be added. The governing equations are derived and then discretised by means of the finite element method in space and finite differences in time. Appropriate solution methods are addressed. Examples given involve heat and mass transfer together with pollutant transport in deforming geomaterials and surface subsidence problems.     

Finite element analysis of long-period water waves

The finite element representation of the nonlinear equations governing the unsteady flow of the two-dimensional long-period shallow water wave is considered. The approximate solution assumes, that the flow is only a slight perturbation of an existing flow. With this assumption a finite element formulation in terms of discrete nodal values of velocity and water height is generated using Galerkin's method. The resulting matrix equation for an arbitrary triangular-based space-time element constitutes a set of linear algebraic equations solvable for nodal values of the flow variables. The topological properties of estuaries are treated and with the solution thus obtained, numerical results are shown for the North Sea.     

Hybrid numerical methods to solve shallow water equations for hurricane induced storm surge modeling

In this paper an efficient numerical method based on hybrid finite element and finite volume techniques to solve hurricane induced storm surge flow problem is presented. A segregated implicit projection method is used to solve the 2D shallow water equations on staggered unstructured meshes. The governing equations are written in non-conservation form. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centered finite volume method. The nonlinear wave equation is solved by the node-based Galerkin finite element method. This staggered-mesh scheme is distinct from other conventional approaches in that the velocity components and auxiliary variables are stored at cell centers and vertices, respectively. The present model uses an implicit method, which is very efficient and can use a large time step without losing accuracy and stability.The hurricane induced wind stress and pressure, bottom friction, Coriolis effect, and tidal forcing conditions are implemented in this model. The levee overtopping option is implemented in the model as well. Hurricane Katrina (2005) storm surge has been simulated to demonstrate the robustness and applicability of the model.     

A high-resolution method for the depth-integrated solute transport equation based on an unstructured mesh

This paper presents a high-resolution numerical method for solving mass transport problems involving advection and anisotropic diffusion in shallow water based on unstructured mesh. An alternating operator-splitting technique is adopted to advance the numerical solution with advection and diffusion terms solved separately in two steps. By introducing a new r-factor into the Total Variation Diminishing (TVD) limiter, an improved finite-volume method is developed to solve the advection term with significant reduction of numerical diffusion and oscillation errors. In addition, a coordinate transformation is introduced to simplify the diffusion term with the Green-Gauss theorem used to deal with the anisotropic effect based on unstructured mesh. The new scheme is validated against three benchmark cases with separated and combined advection and diffusion transport processes involved. Results show that the scheme performs better than existing methods in predicting the advective transport, particularly when a sharp concentration front is in presence. The model also provides a sound solution for the anisotropic diffusion phenomenon. Anisotropic diffusion has been largely neglected by existing flow models based on unstructured mesh, which usually treat the diffusion process as being isotropic for simplicity. Based on the flow field provided by the ELCIRC model, the developed transport model was successfully applied to simulate the transport of a hypothetical conservative tracer in a bay under the influence of tides.     

几种复合型数值方法的算法模拟与性能比较

浅水波问题的数值模拟一直是计算数学、计算流体力学的研究热点之一,采用低阶方法和高阶方法相复合的数值方法引起了人们的注意,并在水力学的数值模拟中取得了很大的成功。文中对三种复合型的数值方法,即Lax-Wendroff(LW)格式与Lax-Friedrichs(LF)格式的复合算法,Upwind格式与Lax-Wendroff(LW)格式的复合算法,WENO格式与LW格式的复合算法,进行了分析比较和改进,并就计算流体力学中的一维浅水波方程的两个算例分别做了数值对比试验,在解的光滑性、锐利性,计算速度等几个方面做了比较,模拟结果表明三种方法均能准确捕捉激波又不产生非物理震荡。     

Lagrangian data assimilation for river hydraulics simulations

We present a method to use Lagrangian data from remote sensing observation in a variational data assimilation process for a river hydraulics model based on the bidimensional shallow water equations. The trajectories of particles advected by the flow can be extracted from video images and are used in addition to classical Eulerian observations. This Lagrangian data brings information on the surface velocity thanks to an appropriate transport model. Numerical twin data assimilation experiments in an academic flow configuration demonstrate that this method makes it possible to significantly improve the identification of local bed elevation and initial conditions.