A correction for balancing discontinuous bed slopes in two‐dimensional smoothed particle hydrodynamics shallow water modeling

In this paper, a smoothed particle hydrodynamics (SPH) numerical model for the shallow water equations (SWEs) with bed slope source term balancing is presented. The solution of the SWEs using SPH is attractive being a conservative, mesh‐free, automatically adaptive method without special treatment for wet‐dry interfaces. Recently, the capability of the SPH–SWEs numerical scheme with shock capturing and general boundary conditions has been used for predicting practical flooding problems. The balance between the bed slope source term and fluxes in shallow water models is desirable for reliable simulations of flooding over bathymetries where discontinuities are present and has received some attention in the framework of Finite Volume Eulerian models. The imbalance because of the source term resulting from the calculation of the the water depth is eradicated by means of a corrected mass, which is able to remove the error introduced by a bottom discontinuity. Two different discretizations of the momentum equation are presented herein: the first one is based on the variational formulation of the SWEs in order to obtain a fully conservative formulation, whereas the second one is obtained using a non‐conservative form of the free‐surface elevation gradient. In both formulations, a variable smoothing length is considered. Results are presented demonstrating the corrections preserve still water in the vicinity of either 1D or 2D bed discontinuities and provide close agreement with 1D analytical solutions for rapidly varying flows over step changes in the bed. The method is finally applied to 2D dam break flow over a square obstacle where the balanced formulation improves the agreement with experimental measurements of the free surface. Copyright © 2012 John Wiley & Sons, Ltd.     

High resolution flux‐difference‐splitting scheme on adaptive grid for open‐channel flows

The effectiveness and usefulness of further enhancing the shock resolution of a second‐order accurate scheme for open‐channel flows by using an adaptive grid is investigated. The flux‐difference‐splitting (FDS) scheme based on the Lax–Wendroff numerical flux is implemented on a fixed as well as on a self‐adjusting grid for this purpose. The grid‐adjusting procedure, developed by Harten and Hyman, adjusts the grid by averaging the local characteristic velocities with respect to the signal amplitude in such a way that a shock always lies on a mesh point. This enables a scheme capable of perfectly resolving a stationary shock to capture a shock that moves from mesh point to mesh point. The Roe's approximate Jacobian is used for conservation and consistency, while theoretically sound treatment for satisfying entropy inequality conditions ensures physically realistic solutions. Details about inclusion of source terms, often left out of analyses for the homogeneous part of governing equations, are also explained. The numerical results for some exacting problems are compared with analytical as well as experimental results for examining improvements in resolution of discontinuities by the adaptive grid. Copyright © 2001 John Wiley & Sons, Ltd.     

A fully discrete ALE method over untwisted time–space control volumes

In this paper, a fully discrete high‐resolution arbitrary Lagrangian–Eulerian (ALE) method is developed over untwisted time–space control volumes. In the framework of the finite volume method, 2D Euler equations are discretized over untwisted moving control volumes, and the resulting numerical flux is computed using the generalized Riemann problem solver. Then, the fluid flows between meshes at two successive time steps can be updated without a remapping process in the classic ALE method. This remapping‐free ALE method directly couples the mesh motion into a physical variable update to reflect the temporal evolution in the whole process. An untwisted moving mesh is generated in terms of the vorticity‐free part of the fluid velocity according to the Helmholtz theorem. Some typical numerical tests show the competitive performance of the current method. Copyright © 2016 John Wiley & Sons, Ltd.     

Solution of the 2‐D shallow water equations with source terms in surface elevation splitting form

A vertex‐centred finite‐volume/finite‐element method (FV/FEM) is developed for solving 2‐D shallow water equations (SWEs) with source terms written in a surface elevation splitting form, which balances the flux gradients and source terms. The method is implemented on unstructured grids and the numerical scheme is based on a second‐order MUSCL‐like upwind Godunov FV discretization for inviscid fluxes and a classical Galerkin FE discretization for the viscous gradients and source terms. The main advantages are: (1) the discretization of SWE written in surface elevation splitting form satisfies the exact conservation property (??‐Property) naturally; (2) the simple centred‐type discretization can be used for the source terms; (3) the method is suitable for both steady and unsteady shallow water problems; and (4) complex topography can be handled based on unstructured grids. The accuracy of the method was verified for both steady and unsteady problems, including discontinuous cases. The results indicate that the new method is accurate, simple, and robust. Copyright © 2007 John Wiley & Sons, Ltd.     

A finite element/volume method model of the depth‐averaged horizontally 2D shallow water equations

Analysis of surface water flows is of central importance in understanding and predicting a wide range of water engineering issues. Dynamics of surface water is reasonably well described using the shallow water equations (SWEs) with the hydrostatic pressure assumption. The SWEs are nonlinear hyperbolic partial differential equations that are in general required to be solved numerically. Application of a simple and efficient numerical model is desirable for solving the SWEs in practical problems. This study develops a new numerical model of the depth‐averaged horizontally 2D SWEs referred to as 2D finite element/volume method (2D FEVM) model. The continuity equation is solved with the conforming, standard Galerkin FEM scheme and momentum equations with an upwind, cell‐centered finite volume method scheme, utilizing the water surface elevation and the line discharges as unknowns aligned in a staggered manner. The 2D FEVM model relies on neither Riemann solvers nor high‐resolution algorithms in order to serve as a simple numerical model. Water at a rest state is exactly preserved in the model. A fully explicit temporal integration is achieved in the model using an efficient approximate matrix inversion method. A series of test problems, containing three benchmark problems and three experiments of transcritical flows, are carried out to assess accuracy and versatility of the model. Copyright © 2014 John Wiley & Sons, Ltd.     

A parallel unstructured dynamic mesh adaptation algorithm for 3‐D unsteady flows

An unstructured dynamic mesh adaptation and load balancing algorithm has been developed for the efficient simulation of three‐dimensional unsteady inviscid flows on parallel machines. The numerical scheme was based on a cell‐centred finite‐volume method and the Roe's flux‐difference splitting. Second‐order accuracy was achieved in time by using an implicit Jacobi/Gauss–Seidel iteration. The resolution of time‐dependent solutions was enhanced by adopting an h‐refinement/coarsening algorithm. Parallelization and load balancing were concurrently achieved on the adaptive dynamic meshes for computational speed‐up and efficient memory redistribution. A new tree data structure for boundary faces was developed for the continuous transfer of the communication data across the parallel subdomain boundary. The parallel efficiency was validated by applying the present method to an unsteady shock‐tube problem. The flows around oscillating NACA0012 wing and F‐5 wing were also calculated for the numerical verification of the present dynamic mesh adaptation and load balancing algorithm. Copyright © 2005 John Wiley & Sons, Ltd.     

Parallel three‐dimensional simulation of the injection molding process

This paper describes the development of a parallel three‐dimensional unstructured non‐isothermal flow solver for the simulation of the injection molding process. The numerical model accounts for multiphase flow in which the melt and air regions are considered to be a continuous incompressible fluid with distinct physical properties. This aspect avoids the complex reconstruction of the interface. A collocated finite volume method is employed, which can switch between first‐ and second‐order accuracy in both space and time. The pressure implicit with splitting of operators algorithm is used to compute the transient flow variables and couple velocity and pressure. The temperature equation is solved using a transport equation with convection and diffusion terms. An upwind differencing scheme is used for the discretization of the convection term to enforce a bounded solution. In order to capture the sharp interface, a bounded compressive high‐resolution scheme is employed. Parallelization of the code is achieved using the PETSc framework and a single program multiple data message passing model. Predicted numerical solutions for several example problems are considered. The first case validates the solution algorithm for moderate Reynolds number flows using a structured mesh. The second case employs an unstructured hybrid mesh showing the capability of the solver to describe highly viscous flows closer to realistic injection molding conditions. The final case presents the non‐isothermal filling of a thick cavity using three mesh sizes and up to 80 processors to assess parallel performance. The proposed algorithm is shown to have good accuracy and scalability. Copyright © 2008 John Wiley & Sons, Ltd.     

A mesh patching method for finite volume modelling of shallow water flow

A new mesh‐patching model is presented for shallow water flow described by the 2D non‐linear shallow water (NLSW) equations. The mesh‐patching model is based on AMAZON, a high‐resolution NLSW engine with an improved HLLC approximate Riemann solver. A new patching algorithm has been developed, which not only provides improved spatial resolution of flow features in particular parts of the mesh, but also simplifies and speeds up the (structured) grid generation process for an area with complicated geometry. The new patching technique is also compatible with increasingly popular parallel computing and adaptive grid techniques. The patching algorithm has been tested with moving bores, and results of test problems are presented and compared to previous work. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite‐volume multi‐stage schemes for shallow‐water flow simulations

A finite‐volume multi‐stage (FMUSTA) scheme is proposed for simulating the free‐surface shallow‐water flows with the hydraulic shocks. On the basis of the multi‐stage (MUSTA) method, the original Riemann problem is transformed to an independent MUSTA mesh. The local Lax–Friedrichs scheme is then adopted for solving the solution of the Riemann problem at the cell interface on the MUSTA mesh. The resulting first‐order monotonic FMUSTA scheme, which does not require the use of the eigenstructure and the special treatment of entropy fixes, has the generality as well as simplicity. In order to achieve the high‐resolution property, the monotonic upstream schemes for conservation laws (MUSCL) method are used. For modeling shallow‐water flows with source terms, the surface gradient method (SGM) is adopted. The proposed schemes are verified using the simulations of six shallow‐water problems, including the 1D idealized dam breaking, the steady transcritical flow over a hump, the 2D oblique hydraulic jump, the circular dam breaking and two dam‐break experiments. The simulated results by the proposed schemes are in satisfactory agreement with the exact solutions and experimental data. It is demonstrated that the proposed FMUSTA schemes have superior overall numerical accuracy among the schemes tested such as the commonly adopted Roe and HLL schemes. Copyright © 2007 John Wiley & Sons, Ltd.     

Immersed boundary method for unsteady kinetic model equations

Predicting unsteady flows and aerodynamic forces for large displacement motion of microstructures requires transient solution of Boltzmann equation with moving boundaries. For the inclusion of moving complex boundaries for these problems, three immersed boundary method flux formulations (interpolation, relaxation, and interrelaxation) are presented. These formulations are implemented in a 2‐D finite volume method solver for ellipsoidal‐statistical (ES)‐Bhatnagar‐Gross‐Krook (BGK) equations using unstructured meshes. For the verification, a transient analytical solution for free molecular 1‐D flow is derived, and results are compared with the immersed boundary (IB)‐ES‐BGK methods. In 2‐D, methods are verified with the conformal, non‐moving finite volume method, and it is shown that the interrelaxation flux formulation gives an error less than the interpolation and relaxation methods for a given mesh size. Furthermore, formulations applied to a thermally induced flow for a heated beam near a cold substrate show that interrelaxation formulation gives more accurate solution in terms of heat flux. As a 2‐D unsteady application, IB/ES‐BGK methods are used to determine flow properties and damping forces for impulsive motion of microbeam due to high inertial forces. IB/ES‐BGK methods are compared with Navier–Stokes solution at low Knudsen numbers, and it is shown that velocity slip in the transitional rarefied regime reduces the unsteady damping force. Copyright © 2015 John Wiley & Sons, Ltd.     

A finite element method for free surface flows of incompressible fluids in three dimensions. Part I. Boundary fitted mesh motion

Computational fluid mechanics techniques for examining free surface problems in two‐dimensional form are now well established. Extending these methods to three dimensions requires a reconsideration of some of the difficult issues from two‐dimensional problems as well as developing new formulations to handle added geometric complexity. This paper presents a new finite element formulation for handling three‐dimensional free surface problems with a boundary‐fitted mesh and full Newton iteration, which solves for velocity, pressure, and mesh variables simultaneously. A boundary‐fitted, pseudo‐solid approach is used for moving the mesh, which treats the interior of the mesh as a fictitious elastic solid that deforms in response to boundary motion. To minimize mesh distortion near free boundary under large deformations, the mesh motion equations are rotated into normal and tangential components prior to applying boundary conditions. The Navier–Stokes equations are discretized using a Galerkin–least square/pressure stabilization formulation, which provides good convergence properties with iterative solvers. The result is a method that can track large deformations and rotations of free surface boundaries in three dimensions. The method is applied to two sample problems: solid body rotation of a fluid and extrusion from a nozzle with a rectangular cross‐section. The extrusion example exhibits a variety of free surface shapes that arise from changing processing conditions. Copyright © 2000 John Wiley & Sons, Ltd.     

An adaptive numerical scheme for solving incompressible 2‐phase and free‐surface flows

In this paper, we present a numerical scheme for solving 2‐phase or free‐surface flows. Here, the interface/free surface is modeled using the level‐set formulation, and the underlying mesh is adapted at each iteration of the flow solver. This adaptation allows us to obtain a precise approximation for the interface/free‐surface location. In addition, it enables us to solve the time‐discretized fluid equation only in the fluid domain in the case of free‐surface problems. Fluids here are considered incompressible. Therefore, their motion is described by the incompressible Navier‐Stokes equation, which is temporally discretized using the method of characteristics and is solved at each time iteration by a first‐order Lagrange‐Galerkin method. The level‐set function representing the interface/free surface satisfies an advection equation that is also solved using the method of characteristics. The algorithm is completed by some intermediate steps like the construction of a convenient initial level‐set function (redistancing) as well as the construction of a convenient flow for the level‐set advection equation. Numerical results are presented for both bifluid and free‐surface problems.     

A well‐balanced upstream flux‐splitting finite‐volume scheme for shallow‐water flow simulations with irregular bed topography

This study extends the upstream flux‐splitting finite‐volume (UFF) scheme to shallow water equations with source terms. Coupling the hydrostatic reconstruction method (HRM) with the UFF scheme achieves a resultant numerical scheme that adequately balances flux gradients and source terms. The proposed scheme is validated in three benchmark problems and applied to flood flows in the natural/irregular river with bridge pier obstructions. The results of the simulations are in satisfactory agreement with the available analytical solutions, experimental data and field measurements. Comparisons of the present results with those obtained by the surface gradient method (SGM) demonstrate the superior stability and higher accuracy of the HRM. The stability test results also show that the HRM requires less CPU time (up to 60%) than the SGM. The proposed well‐balanced UFF scheme is accurate, stable and efficient to solve flow problems involving irregular bed topography. Copyright © 2009 John Wiley & Sons, Ltd.     

Development of highly accurate interpolation method for mesh‐free flow simulations II. Application of CIVA method to incompressible flow simulations

This is the second report on the development of a highly accurate interpolation method, which is called cubic interpolation with volume/area (CIVA) co‐ordinates, for mesh‐free flow simulations. In this paper, the method of determining the c‐parameter of CIVA using a constant curvature condition is first considered for the two‐ and three‐dimensional cases. A computation of a three‐dimensional passive scalar advection problem is performed for accuracy verification and for comparison with widely used methods. Then, an application algorithm of the CIVA method respecting incompressible fluid simulation is presented. As the incompressible condition based on Lagrangian approaches causes problems, in this paper we consider the condition based on the conventional Eulerian approach. The CIVA‐based incompressible flow simulation algorithm enables a highly accurate simulation of many kinds of problems that have complicated geometries and involve complicated phenomena. To confirm the facts, numerical analyzes are executed for some benchmark problems, namely flow in a square cavity, free surface sloshing and moving boundary problems in complex geometries. The results show that the method achieves high accuracy and has high flexibility, even for the flows involving high Reynolds number, complicated geometries, moving boundaries and free surfaces. Copyright © 2000 John Wiley & Sons, Ltd.     

Boundary conformed co-ordinate systems for selected two-dimensional fluid flow problems. Part II: Application of the BFG method

This paper gives the results of an application of the SWEs (shallow water equations) to a part of the Hamburg harbour area, which is a complex flow domain, using the BFG approach, outlined in Part I. The results of a grid doubling procedure generating the desired computational grid from a coarse initial mesh are also presented. A second class of problems which is addressed, demands time-dependent co-ordinate systems. The problems which are solved are the free surface problem for a moving wave which eventually breaks and for a wave which is reflected by the solid walls of a rectangular basin.     

Multiscale block spectral solution for unsteady flows

Many problems of interest are characterized by 2 distinctive and disparate scales and a huge multiplicity of similar small‐scale elements. The corresponding scale‐dependent solvability manifests itself in the high gradient flow around each element needing a fine mesh locally and the similar flow patterns among all elements globally. In a block spectral approach making use of the scale‐dependent solvability, the global domain is decomposed into a large number of similar small blocks. The mesh‐pointwise block spectra will establish the block‐block variation, for which only a small set of blocks need to be solved with a fine mesh resolution. The solution can then be very efficiently obtained by coupling the local fine mesh solution and the global coarse mesh solution through a block spectral mapping. Previously, the block spectral method has only been developed for steady flows. The present work extends the methodology to unsteady flows of short temporal and spatial scales (eg, those due to self‐excited unsteady vortices and turbulence disturbances). A source term–based approach is adopted to facilitate a two‐way coupling in terms of time‐averaged flow solutions. The global coarse base mesh solution provides an appropriate environment and boundary condition to the local fine mesh blocks, while the local fine mesh solution provides the source terms (propagated through the block spectral mapping) to the global coarse mesh domain. The computational method will be presented with several numerical examples and sensitivity studies. The results consistently demonstrate the validity and potential of the proposed approach.     

A robust 2D shallow water model for solving flow over complex topography using homogenous flux method

A robust Godunov‐type numerical scheme solver is proposed for solving 2D SWEs and is applied to simulate flow over complex topography with wetting and drying. In reality, the topography is usually complex and irregular; therefore, to avoid the numerical errors generated by such features, a Homogenous Flux Method is used to handle the bed slope term in the SWEs. The method treats the bed slope term as a flux to be incorporated into the flux gradient and so maintains the balance between the two in a Godunov‐type shock‐capturing scheme. The main advantages of the method are: first, it is simple and easy to implement; second, numerical experiments demonstrate that it can handle discontinuous or vertical bed topography without any special treatment and third, it is applicable to both steady and unsteady flows. It is demonstrated how the approach set out here can be applied to the nonlinear hyperbolic system of the SWEs. The two‐dimensional hyperbolic system is then solved by use of a second‐order total‐variation‐diminishing version of the weighted average flux method in conjunction with a Harten‐Lax‐van Leer‐Contract approximate Riemann solver incorporating the new flux gradient term. Several benchmark tests are presented to validate the model and the approach is verified against experimental measurements from the European Union Concerted Action on Dam Break Modelling project. These show very good agreement. Finally, the method is applied to a volcano‐induced outburst flood over an initially dry channel with complex irregular topography to demonstrate the technique's capability in simulating a real flood. Copyright © 2013 John Wiley & Sons, Ltd.     

A moving mesh BGK scheme for multi‐material flows

In this paper, a moving mesh BGK scheme (MMBGK) for multi‐material flow computations is proposed. The basic idea of constructing the MMBGK is to couple the Lagrangian method, which tracks material interfaces and keeps the interfaces sharp, with a remapping‐free ALE‐type kinetic method within each single material region, where the kinetic method is based on the BGK (Bhatnagar–Gross–Krook) model. Within each single material region, a numerical flux formulation is developed on moving meshes from motion of microscope particles, and the mesh velocity is determined by requiring both mesh adaptation for accuracy and robustness, such that the grids are moving towards to the regions with high flow gradients in a way of diffusive mechanism (velocity) to adjust the distances between neighboring cells, thus increasing the numerical accuracy. To keep the sharpness of material interfaces, the Lagrangian velocity and flux are constructed at the interfaces only. Consequently, a BGK‐scheme‐based ALE‐type method (i.e., the MMBGK scheme) for multi‐material flows is constructed. Numerical examples in one and two dimensions are presented to illustrate the accuracy and robustness of the MMBGK scheme. Copyright © 2011 John Wiley & Sons, Ltd.     

A Newton multigrid method for steady-state shallow water equations with topography and dry areas

A Newton multigrid method is developed for one-dimensional (1D) and two-dimensional (2D) steady-state shallow water equations (SWEs) with topography and dry areas. The nonlinear system arising from the well-balanced finite volume discretization of the steady-state SWEs is solved by the Newton method as the outer iteration and a geometric multigrid method with the block symmetric Gauss-Seidel smoother as the inner iteration. The proposed Newton multigrid method makes use of the local residual to regularize the Jacobian matrix of the Newton iteration, and can handle the steady-state problem with wet/dry transition. Several numerical experiments are conducted to demonstrate the efficiency, robustness, and well-balanced property of the proposed method. The relation between the convergence behavior of the Newton multigrid method and the distribution of the eigenvalues of the iteration matrix is detailedly discussed.     

基于2D网格的轴对称浸没边界法

浸没边界法是处理颗粒两相流中运动边界问题的一种常用数值模拟方法. 当研究的物理问题的无量纲参数满足一定要求时, 该流场结构呈现轴对称状态. 为此本文提出了一种基于2D笛卡尔网格和柱坐标系的轴对称浸没边界法. 该算法采用有限体积法(FVM)对动量方程进行空间离散, 并通过阶梯状锐利界面替代真实的固体浸没边界来封闭控制方程. 为了提高计算效率, 本文采用自适应网格加密技术提高浸没边界附近网格分辨率. 由于柱坐标系的使用, 使得动量方程中的黏性项产生多余的源项, 我们对其作隐式处理. 此外, 在对小球匀速近壁运动进行直接数值模拟时, 由于球壁间隙很小, 间隙内的压力变化比较剧烈. 因此想要精确地解析流场需要很高的网格分辨率. 此时, 需要在一个时间步内多次实施投影步来保证计算的稳定性. 而在小球自由碰壁运动中, 我们通过引入一个润滑力模型使得低网格分辨率下也能模拟小球近壁处的运动. 最后通过小球和圆盘绕流、Stokes流小球近壁运动以及小球自由下落碰壁弹跳算例验证本算法对于轴对称流的静边界和动边界问题均是适用和准确的.