An evaluation of force terms in the lattice Boltzmann models in simulating shallow water flows over complex topography

Proper approximation of the force terms, especially the bed slope term, is of crucial importance to simulating shallow water flows in lattice Boltzmann (LB) models. However, there is little discussion on the schemes of adding force terms to LB models for shallow water equations (SWEs). In this study, we evaluate the performance of forcing schemes coupled with different LB models (LABSWE and MLBSWE) in simulating shallow water flows over complex topography and try to find out their intrinsic characteristics and applicability. Three cases are adopted for evaluation, including a stationary case, a one-dimensional tidal wave flow over an irregular bed, and a steady flow over a two-dimensional seamount. The simulating results are compared with analytical solutions or the results produced by the finite difference method. For LABSWE, all the forcing schemes, except for the weighting factor method, fail to produce accurate solutions for the test cases; this is probably due to the mismatch between the bed slope term in source terms and the quadratic depth term of the equilibrium distribution functions in these forcing schemes. For MLBSWE, all the forcing schemes are capable of simulating flows over the complex topography accurately; furthermore, those schemes taking into account the collision effect τ to eliminate the momentum induced by forces provide more accurate solutions with quicker convergence as the lattice size decreases. In this view, MLBSWE can bring more flexibility in treating the force terms and thus can be a better tool to simulate shallow water flows over complex topography in practical application.     

Semi-implicit finite difference methods for three-dimensional shallow water flow

A semi-implicit finite difference method for the numerical solution of three-dimensional shallow water flows is presented and discussed. The governing equations are the primitive three-dimensional turbulent mean flow equations where the pressure distribution in the vertical has been assumed to be hydrostatic. In the method of solution a minimal degree of implicitness has been adopted in such a fashion that the resulting algorithm is stable and gives a maximal computational efficiency at a minimal computational cost. At each time step the numerical method requires the solution of one large linear system which can be formally decomposed into a set of small three-diagonal systems coupled with one five-diagonal system. All these linear systems are symmetric and positive definite. Thus the existence and uniquencess of the numerical solution are assured. When only one vertical layer is specified, this method reduces as a special case to a semi-implicit scheme for solving the corresponding two-dimensional shallow water equations. The resulting two- and three-dimensional algorithm has been shown to be fast, accurate and mass-conservative and can also be applied to simulate flooding and drying of tidal mud-flats in conjunction with three-dimensional flows. Furthermore, the resulting algorithm is fully vectorizable for an efficient implementation on modern vector computers.     

Lattice Boltzmann methods for shallow water flow applications

We apply the lattice Boltzmann (LB) method for solving the shallow water equations with source terms such as the bed slope and bed friction. Our aim is to use a simple and accurate representation of the source terms in order to simulate practical shallow water flows without relying on upwind discretization or Riemann problem solvers. We validate the algorithm on problems where analytical solutions are available. The numerical results are in good agreement with analytical solutions. Furthermore, we test the method on a practical problem by simulating mean flow in the Strait of Gibraltar. The main focus is to examine the performance of the LB method for complex geometries with irregular bathymetry. The results demonstrate its ability to capture the main flow features. Copyright © 2007 John Wiley & Sons, Ltd.     

A novel implicit method for coastal hydrodynamics modeling: application to the Arcachon lagoon

The present Note reports on numerical modeling of shallow flows in coastal areas. Successful numerical simulations of such flows should be able to cope with strong irregularities of the bathymetry and to reproduce the covering/uncovering (wetting/drying) of tidal flats due to the tidal oscillations of the free surface. Also, adoption of large time steps is necessary to simulate phenomena which last actually several days or months. In the present study, a new numerical model based on an implicit resolution of the shallow water equations is proposed. A penalty method has been employed for numerical treatment of dry zones emerging during the wetting and drying processes. The capability of the present model has been verified by comparison with standard test cases. Further applications and comparisons have been also carried out to simulate the tidal propagation in the Arcachon lagoon. To cite this article: A. Le Dissez et al., C. R. Mecanique 333 (2005).     

SINGLE-LAYER FINITE ELEMENT APPROXIMATION OF 3D NAVIER–STOKES EQUATIONS WITH FREE BOUNDARIES FOR SHALLOW WATER FLOWS

In order to simulate flows in the shallow water limit, the full incompressible Navier–Stokes equations with free boundaries are solved using a single layer of finite elements. This implies a polynomial approximation of the velocity profile in the vertical direction, which in turn distorts the wave speed. This fact is verified by numerical results: the wave speed depends on the vertical discretization. When at least two layers of finite elements are used, the boundary layer at the bottom can be simulated and the correct solution for the shallow water limit is recovered. Then this algorithm is applied to the prediction of Tsunami event.     

Lattice Boltzmann modeling of shallow water flows over discontinuous beds

Numerical modeling of shallow water flows over discontinuous beds is presented. The flows are described with the shallow water equations and the equations are solved using the lattice Boltzmann method (LBM) with single relaxation time (Bhatnagar–Gross–Krook‐LBM (BGK‐LBM)) and the multiple relaxation time (MRT‐LBM). The weighted centered scheme for force term together with the bed height for a bed slope is described to improve simulation of flows over discontinuous bed. Furthermore, the resistance stress is added to include the local head loss caused by flow over a step. Four test cases, one‐dimensional tidal over regular bed and steps, dam‐break flows, and two‐dimensional shallow water flow over a square block, are considered to verify the present method. Agreements between predictions and analytical solutions are satisfactory. Furthermore, the performance and CPU cost time of BGK‐LBM and MRT‐LBM are compared and studied. The results have shown that the lattice Boltzmann method is simple and accurate for simulating shallow water flows over discontinuous beds. This demonstrates the capability and applicability of the lattice Boltzmann method in modeling shallow water flows on bed topography with a discontinuity in practical hydraulic engineering. Copyright © 2014 John Wiley & Sons, Ltd.     

On the approximation of local efflux/influx bed discharge in the shallow water equations based on a wave propagation algorithm

In cities, flood waves may propagate over street surfaces below which lie complicated pipe networks used for storm drainage and sewage. The flood and pipe flows can interact at connections between the underground pipes and the street surface. The present paper examines this interaction, using the shallow water equations to model the flood wave hydrodynamics. Sources and sinks in the mass conservation equation are used to model the pipe inflow and outflow conditions at bed connections. We consider the problem reduced to one dimension. The shallow water equations are solved using a Godunov‐type wave propagation scheme. Wave speeds are modified in the wave propagation algorithm to enable flows to be simulated over nearly dry beds and dry states. First, the model is used to simulate vertical flows through finite gaps in the bed. Next, the interaction of the vertical flows with a dam break flow is considered for both dry and wet beds. An efflux number, En, is defined based on the vertical efflux velocity and the gap length. Comparisons are made with numerical predictions from STAR‐CD, a commercial Navier–Stokes solver that models the free‐surface motions, and a parameter study is undertaken to investigate the effect of the one‐dimensional approximation of the present model, for a range of non‐dimensional efflux numbers. It is found that the shallow flow model gives sensible predictions at all time provided En<0.5, and for long durations for En>0.5. Dam break flow over an underground connecting pipe is also considered. Copyright © 2010 John Wiley & Sons, Ltd.     

A coupled lattice Boltzmann model for the shallow water‐contamination system

A coupled numerical method for the direct simulation of shallow water dynamics and pollutant transport is formulated and implemented. The conservation equations of shallow water dynamics equations and the convection–diffusion equations are solved using the lattice Boltzmann (LB) method. The local equilibrium distribution of the pollutant has no terms of second order in flow velocity. And the relaxation time of the pollutant deviates from a constant for the flows with variable free surface water depth. The numerical tests show that this scheme strictly obeys the conservation law of mass and momentum. Excellent agreement is obtained between numerical predictions and analytical solutions in the pure diffusion problem and convection–diffusion problem. Furthermore, the influences on the accuracy of the lattice size and the diffusivity are also studied. The results indicate that the variation in the free surface water depth cannot affect the conservation of the model, and the model has the ability to simulate the complex topography problem. The comparison shows that the LB scheme has the capacity to solve the complex convection–diffusion problem in shallow water. Copyright © 2008 John Wiley & Sons, Ltd.     

Formation of azimuthal flows driven by a mass source-sink system in a rotating paraboloid

The process of formation of azimuthal flows generated by a mass source-sink system in a shallow water layer on the surface of a rotating paraboloid is investigated theoretically and experimentally. The calculations are carried out within the framework of the shallow-water equations with allowance for bottom friction. Asymptotic solutions describing the process of establishment of steady-state azimuthal flows which takes place after instantaneous initiation of the source-sink system are constructed. It is shown that theory and experiment are in satisfactory agreement.     

Finite volume solution to integrated shallow surface–saturated groundwater flow

This paper describes development of an integrated shallow surface and saturated groundwater model (GSHAW5). The surface flow motion is described by the 2‐D shallow water equations and groundwater movement is described by the 2‐D groundwater equations. The numerical solution of these equations is based on the finite volume method where the surface water fluxes are estimated using the Roe shock‐capturing scheme, and the groundwater fluxes are computed by application of Darcy's law. Use of a shock‐capturing scheme ensures ability to simulate steady and unsteady, continuous and discontinuous, subcritical and supercritical surface water flow conditions. Ground and surface water interaction is achieved by the introduction of source‐sink terms into the continuity equations. Two solutions are tightly coupled in a single code. The numerical solutions and coupling algorithms are explained. The model has been applied to 1‐D and 2‐D test scenarios. The results have shown that the model can produce very accurate results and can be used for simulation of situations involving interaction between shallow surface and saturated groundwater flows. Copyright © 2005 John Wiley & Sons, Ltd.     

用离散速度方法计算浅水长波方程

用离散速度法计算浅水波方程，将空气动力学方程和浅水波方程作了比较，用Nadiga提出的近平衡流动方法模拟浅水波方程的连续和间断解。计算了一维的溃坝波问题和Thacker提出的连续解问题，结果与精确解作了比较，并且计算了水流跃过障碍物的问题。     

A high‐order PIC method for advection‐dominated flow with application to shallow water waves

A hybrid Eulerian‐Lagrangian particle‐in‐cell–type numerical method is developed for the solution of advection‐dominated flow problems. Particular attention is given over to the high‐order transfer of flow properties from the particles to the grid. For smooth flows, the method presented is of formal high‐order accuracy in space. The method is applied to solve the nonlinear shallow water equations resulting in a new, and novel, shock capturing shallow water solver. The approach is able to simulate complex shallow water flows, which can contain an arbitrary number of discontinuities. Both trivial and nontrivial bottom topography is considered, and it is shown that the new scheme is inherently well balanced, exactly satisfying the ‐property. The scheme is verified against several one‐dimensional benchmark shallow water problems. These include cases that involve transcritical flow regimes, shock waves, and nontrivial bathymetry. In all the test cases presented, very good results are obtained.     

NUMERICAL SIMULATION OF SHALLOW WATER WAVE PROPAGATION USING A BOUNDARY ELEMENT WAVE EQUATION MODEL

The present paper makes use of a wave equation formulation of the primitive shallow water equations to simulate one-dimensional free surface flow. A numerical formulation of the boundary element method is then developed to solve the wave continuity equation using a time-dependent fundamental solution, while an explicit finite difference scheme is used to derive velocities from the primitive momentum equation. One-dimensional free surface flows in open channels are treated and the results compared with analytical and numerical solutions. © 1997 John Wiley & Sons, Ltd.     

Three‐dimensional modeling of non‐hydrostatic free‐surface flows on unstructured grids

A three‐dimensional numerical model is presented for the simulation of unsteady non‐hydrostatic shallow water flows on unstructured grids using the finite volume method. The free surface variations are modeled by a characteristics‐based scheme, which simulates sub‐critical and super‐critical flows. Three‐dimensional velocity components are considered in a collocated arrangement with a σ‐coordinate system. A special treatment of the pressure term is developed to avoid the water surface oscillations. Convective and diffusive terms are approximated explicitly, and an implicit discretization is used for the pressure term to ensure exact mass conservation. The unstructured grid in the horizontal direction and the σ coordinate in the vertical direction facilitate the use of the model in complicated geometries. Solution of the non‐hydrostatic equations enables the model to simulate short‐period waves and vertically circulating flows. Copyright © 2015 John Wiley & Sons, Ltd.     

Investigation of the numerical accuracy of shallow water flow estimation based on the ensemble Kalman filter using the SUPG FEM

ABSTRACT

The shallow water equation is employed as the governing equation to simulate flow behaviour in shallow water flow regions. The SUPG and the backward Euler methods, respectively, were employed for discretisation in space and time. In this paper, we carry out investigations on the numerical accuracy of shallow water flow estimation based on the ensemble Kalman filter using the SUPG FEM, and show the results of numerical experiments. The open channel model was employed as the numerical example in this study. The extended Kalman filter is generally employed to solve parameter identification problems. However, linearisation of the governing equation is carried out to apply parameter identification: it is known to be difficult to carry out computation reliably if problems that include a high degree of non-linearity need to be solved. On the other hand, linearisation of the governing equation is not carried out in the ensemble Kalman filter, so there is potential for the unknown parameter to be simultaneously identified in the computation of the data assimilation. The distribution of the unknown kinematic viscosity coefficient was therefore also investigated.     

A modified lattice Boltzmann model for shallow water flows over complex topography

A modified lattice Boltzmann model is proposed to describe shallow water flows over complex topography. In the proposed model, the quadratic depth term is excluded from the equilibrium distribution functions (EDFs), and the hydrostatic pressure term is combined with the bed slope term to be treated as a part of the sourcing term in the lattice Boltzmann equation (LBE). Therefore, it is unnecessary to match the coefficients of the quadratic depth term in the EDFs with those of the bed slope term in the sourcing terms in the LBE. This would bring more flexibility to the treatment of the sourcing terms in the LBE. In order to recover the shallow water equations (SWEs), the basic constraints are redefined, and under these constraints, the coefficients of the EDFs are derived afterwards. Several benchmark problems are used to validate the proposed model, including stationary case, steady flows over a two‐dimensional bump and tidal wave flows over irregular bed elevation. The computed results are in excellent agreement with the results of the other numerical methods and the analytical solutions, indicating that the proposed model is capable of simulating shallow water flows over complex bathymetry. It also proves that the proposed model has potential to produce competitive solutions to shallow water flows over complex bed topography. Copyright © 2014 John Wiley & Sons, Ltd.     

Boundary discretization for high‐order discontinuous Galerkin computations of tidal flows around shallow water islands

In this paper some preliminary results concerning the application of the high‐order discontinuous Galerkin (DG) method for the resolution of realistic problems of tidal flows around shallow water islands are presented. In particular, tidal flows are computed around the Rattray island located in the Great Barrier Reef. This island is a standard benchmark problem well documented in the literature providing useful in situ measurements for validation of the model. Realistic elements of the simulation are a tidal flow forcing, a variable bathymetry and a non‐trivial coastline. The computation of tidal flows in shallow water around an island is very similar to the simulation of the Euler equations around bluff bodies in quasi‐steady flows. The main difference lies in the high irregularity of islands' shapes and in the fact that, in the framework of large‐scale ocean models, the number of elements to represent an island is drastically limited compared with classical engineering computations. We observe that the high‐order DG method applied to shallow water flows around bluff bodies with poor linear boundary representations produces oscillations and spurious eddies. Surprisingly those eddies may have the right size and intensity but may be generated by numerical diffusion and are not always mathematically relevant. Although not interested in solving accurately the boundary layers of an island, we show that a high‐order boundary representation is mandatory to avoid non‐physical eddies and spurious oscillations. It is then possible to parametrize accurately the subgrid‐scale processes to introduce the correct amount of diffusion in the model. The DG results around the Rattray island are eventually compared with current measurements and reveal good agreement. Copyright © 2008 John Wiley & Sons, Ltd.