A finite volume cell‐centered Lagrangian hydrodynamics approach for solids in general unstructured grids

A finite volume cell‐centered Lagrangian hydrodynamics approach, formulated in Cartesian frame, is presented for solving elasto‐plastic response of solids in general unstructured grids. Because solid materials can sustain significant shear deformation, evolution equations for stress and strain fields are solved in addition to mass, momentum, and energy conservation laws. The total stress is split into deviatoric shear stress and dilatational components. The dilatational response of the material is modeled using the Mie‐Grüneisen equation of state. A predicted trial elastic deviatoric stress state is evolved assuming a pure elastic deformation in accordance with the hypo‐elastic stress‐strain relation. The evolution equations are advanced in time by constructing vertex velocity and corner traction force vectors using multi‐dimensional Riemann solutions erected at mesh vertices. Conservation of momentum and total energy along with the increase in entropy principle are invoked for computing these quantities at the vertices. Final state of deviatoric stress is effected via radial return algorithm based on the J‐2 von Mises yield condition. The scheme presented in this work is second‐order accurate both in space and time. The suitability of the scheme is evinced by solving one‐ and two‐dimensional benchmark problems both in structured grids and in unstructured grids with polygonal cells. Copyright © 2013 John Wiley & Sons, Ltd.     

The Lagrange–Galerkin method for the two‐dimensional shallow water equations on adaptive grids

The weak Lagrange–Galerkin finite element method for the two‐dimensional shallow water equations on adaptive unstructured grids is presented. The equations are written in conservation form and the domains are discretized using triangular elements. Lagrangian methods integrate the governing equations along the characteristic curves, thus being well suited for resolving the non‐linearities introduced by the advection operator of the fluid dynamics equations. An additional fortuitous consequence of using Lagrangian methods is that the resulting spatial operator is self‐adjoint, thereby justifying the use of a Galerkin formulation; this formulation has been proven to be optimal for such differential operators. The weak Lagrange–Galerkin method automatically takes into account the dilation of the control volume, thereby resulting in a conservative scheme. The use of linear triangular elements permits the construction of accurate (by virtue of the second‐order spatial and temporal accuracies of the scheme) and efficient (by virtue of the less stringent Courant–Friedrich–Lewy (CFL) condition of Lagrangian methods) schemes on adaptive unstructured triangular grids. Lagrangian methods are natural candidates for use with adaptive unstructured grids because the resolution of the grid can be increased without having to decrease the time step in order to satisfy stability. An advancing front adaptive unstructured triangular mesh generator is presented. The highlight of this algorithm is that the weak Lagrange–Galerkin method is used to project the conservation variables from the old mesh onto the newly adapted mesh. In addition, two new schemes for computing the characteristic curves are presented: a composite mid‐point rule and a general family of Runge–Kutta schemes. Results for the two‐dimensional advection equation with and without time‐dependent velocity fields are illustrated to confirm the accuracy of the particle trajectories. Results for the two‐dimensional shallow water equations on a non‐linear soliton wave are presented to illustrate the power and flexibility of this strategy. Copyright © 2000 John Wiley & Sons, Ltd.     

Consistent discretization for simulations of flows with moving generalized curvilinear coordinates

We develop a consistent discretization of conservative momentum and scalar transport for the numerical simulation of flow using a generalized moving curvilinear coordinate system. The formulation guarantees consistency between the discrete transport equation and the discrete mass conservation equation due to grid motion. This enables simulation of conservative transport using generalized curvilinear grids that move arbitrarily in three dimensions while maintaining the desired properties of the discrete transport equation on a stationary grid, such as constancy, conservation, and monotonicity. In addition to guaranteeing consistency for momentum and scalar transport, the formulation ensures geometric conservation and maintains the desired high‐order time accuracy of the discretization on a moving grid. Through numerical examples we show that, when the computation is carried out on a moving grid, consistency between the discretized scalar advection equation and the discretized equation for flow mass conservation due to grid motion is required in order to obtain stable and accurate results. We also demonstrate that significant errors can result when non‐consistent discretizations are employed. Copyright © 2009 John Wiley & Sons, Ltd.     

A fully conservative high‐order upwind multi‐moment method using moments in both upwind and downwind cells

We propose a fully conservative high‐order upwind multi‐moment method for the conservation equation. The proposed method is based on a third‐order polynomial interpolation function and semi‐Lagrangian formulation and is a variant of the constrained interpolation profile conservative semi‐Lagrangian scheme with third‐order polynomial function method. The third‐order interpolation function is constructed based on three constraints in the upwind cell (two boundary values and a cell average) and a constraint in the downwind cell (a cell center value). The proposed method shows fourth‐order accuracy in a benchmark problem (sine wave propagation). We also propose a less oscillatory formulation of the proposed method. The less oscillatory formulation can minimize numerical oscillations. These methods were validated through scalar transport problems, and compressible flow problems (shock tube and 2D explosion problems). Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical analysis of conservative unstructured discretisations for low Mach flows

Unstructured meshes allow easily representing complex geometries and to refine in regions of interest without adding control volumes in unnecessary regions. However, numerical schemes used on unstructured grids have to be properly defined in order to minimise numerical errors. An assessment of a low Mach algorithm for laminar and turbulent flows on unstructured meshes using collocated and staggered formulations is presented. For staggered formulations using cell‐centred velocity reconstructions, the standard first‐order method is shown to be inaccurate in low Mach flows on unstructured grids. A recently proposed least squares procedure for incompressible flows is extended to the low Mach regime and shown to significantly improve the behaviour of the algorithm. Regarding collocated discretisations, the odd–even pressure decoupling is handled through a kinetic energy conserving flux interpolation scheme. This approach is shown to efficiently handle variable‐density flows. Besides, different face interpolations schemes for unstructured meshes are analysed. A kinetic energy‐preserving scheme is applied to the momentum equations, namely, the symmetry‐preserving scheme. Furthermore, a new approach to define the far‐neighbouring nodes of the quadratic upstream interpolation for convective kinematics scheme is presented and analysed. The method is suitable for both structured and unstructured grids, either uniform or not. The proposed algorithm and the spatial schemes are assessed against a function reconstruction, a differentially heated cavity and a turbulent self‐igniting diffusion flame. It is shown that the proposed algorithm accurately represents unsteady variable‐density flows. Furthermore, the quadratic upstream interpolation for convective kinematics scheme shows close to second‐order behaviour on unstructured meshes, and the symmetry‐preserving is reliably used in all computations. Copyright © 2016 John Wiley & Sons, Ltd.     

A formally second‐order cell centred scheme for convection–diffusion equations on general grids

We propose, in this paper, a finite volume scheme to compute the solution of the convection–diffusion equation on unstructured and possibly non‐conforming grids. The diffusive fluxes are approximated using the recently published SUSHI scheme in its cell centred version, that reaches a second‐order spatial convergence rate for the Laplace equation on any unstructured two‐dimensional/three‐dimensional grids. As in the MUSCL method, the numerical convective fluxes are built with a prediction‐limitation process, which ensures that the discrete maximum principle is satisfied for pure convection problems. The limitation does not involve any geometrical reconstruction, thus allowing the use of completely general grids, in any space dimension. Copyright © 2012 John Wiley & Sons, Ltd.     

Contradiction between the C‐property and mass conservation in adaptive grid based shallow flow models: cause and solution

When performing shallow flow simulations on adaptive grids, the C‐property (i.e. conservation property) and the mass conservation may not be simultaneously preserved, that is, either C‐property or mass conservation is likely to be violated following grid refining or coarsening. The cause of such a contradiction is analyzed in detail in this work, which essentially links to the reconstruction of bed and flow information in those newly created cells during grid adaptation. An effective approach is subsequently proposed to resolve the contradiction by locally modifying the bed elevation in certain problematic cells when reconstructing flow information using linear interpolation as part of the grid adaptation procedure. Copyright © 2015 John Wiley & Sons, Ltd.     

A discontinuous Galerkin method for the Navier–Stokes equations

The foundations of a new discontinuous Galerkin method for simulating compressible viscous flows with shocks on standard unstructured grids are presented in this paper. The new method is based on a discontinuous Galerkin formulation both for the advective and the diffusive contributions. High‐order accuracy is achieved by using a recently developed hierarchical spectral basis. This basis is formed by combining Jacobi polynomials of high‐order weights written in a new co‐ordinate system. It retains a tensor‐product property, and provides accurate numerical quadrature. The formulation is conservative, and monotonicity is enforced by appropriately lowering the basis order and performing h‐refinement around discontinuities. Convergence results are shown for analytical two‐ and three‐dimensional solutions of diffusion and Navier–Stokes equations that demonstrate exponential convergence of the new method, even for highly distorted elements. Flow simulations for subsonic, transonic and supersonic flows are also presented that demonstrate discretization flexibility using hp‐type refinement. Unlike other high‐order methods, the new method uses standard finite volume grids consisting of arbitrary triangulizations and tetrahedrizations. Copyright © 1999 John Wiley & Sons, Ltd.     

Accurate and consistent particle tracking on unstructured grids

A new numerical method for particle tracking (Lagrangian particle advection) on 2‐D unstructured grids with triangular cells is presented and tested. This method combines key attributes of published methods, including streamline closure for steady flows and local mass conservation (uniformity preservation). The subgrid‐scale velocity reconstruction is linear, and this linear velocity field is integrated analytically to obtain particle trajectories. A complete analytic solution to the 2‐D system of ordinary differential equations (ODEs) governing particle trajectories within a grid cell is provided. The analytic solution to the linear system of locally mass‐conserving constraints that must be enforced to obtain the coefficients in the ODEs is also provided. Numerical experiments are performed to demonstrate that the new method has substantial advantages in accuracy over previously published methods and that it does not suffer from unphysical particle clustering. The method can be used not only in particle‐tracking applications but also as part of a semi‐Lagrangian advection scheme.Copyright © 2015 John Wiley & Sons, Ltd.     

An unstructured finite‐volume method for coupled models of suspended sediment and bed load transport in shallow‐water flows

The aim of this work is to develop a well‐balanced finite‐volume method for the accurate numerical solution of the equations governing suspended sediment and bed load transport in two‐dimensional shallow‐water flows. The modelling system consists of three coupled model components: (i) the shallow‐water equations for the hydrodynamical model; (ii) a transport equation for the dispersion of suspended sediments; and (iii) an Exner equation for the morphodynamics. These coupled models form a hyperbolic system of conservation laws with source terms. The proposed finite‐volume method consists of a predictor stage for the discretization of gradient terms and a corrector stage for the treatment of source terms. The gradient fluxes are discretized using a modified Roe's scheme using the sign of the Jacobian matrix in the coupled system. A well‐balanced discretization is used for the treatment of source terms. In this paper, we also employ an adaptive procedure in the finite‐volume method by monitoring the concentration of suspended sediments in the computational domain during its transport process. The method uses unstructured meshes and incorporates upwinded numerical fluxes and slope limiters to provide sharp resolution of steep sediment concentrations and bed load gradients that may form in the approximate solutions. Details are given on the implementation of the method, and numerical results are presented for two idealized test cases, which demonstrate the accuracy and robustness of the method and its applicability in predicting dam‐break flows over erodible sediment beds. The method is also applied to a sediment transport problem in the Nador lagoon.Copyright © 2013 John Wiley & Sons, Ltd.     

Unstructured lattice Boltzmann method in three dimensions

Over the last decade, the lattice Boltzmann method (LBM) has evolved into a valuable alternative to continuum computational fluid dynamics (CFD) methods for the numerical simulation of several complex fluid‐dynamic problems. Recent advances in lattice Boltzmann research have considerably extended the capability of LBM to handle complex geometries. Among these, a particularly remarkable option is represented by cell‐vertex finite‐volume formulations which permit LBM to operate on fully unstructured grids. The two‐dimensional implementation of unstructured LBM, based on the use of triangular elements, has shown capability of tolerating significant grid distortions without suffering any appreciable numerical viscosity effects, to second‐order in the mesh size. In this work, we present the first three‐dimensional generalization of the unstructured lattice Boltzmann technique (ULBE as unstructured lattice Boltzmann equation), in which geometrical flexibility is achieved by coarse‐graining the lattice Boltzmann equation in differential form, using tetrahedrical grids. This 3D extension is demonstrated for the case of 3D pipe flow and moderate Reynolds numbers flow past a sphere. The results provide evidence that the ULBE has significant potential for the accurate calculation of flows in complex 3D geometries. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical diffusion for flow‐aligned unstructured grids with application to estuarine modeling

The benefits of unstructured grids in hydrodynamic models are well understood but in many cases lead to greater numerical diffusion compared with methods available on structured grids. The flexible nature of unstructured grids, however, allows for the orientation of the grid to align locally with the dominant flow direction and thus decrease numerical diffusion. We investigate the relationship between grid alignment and diffusive errors in the context of scalar transport in a triangular, unstructured, 3‐D hydrodynamic code. Analytical results are presented for the 2‐D anisotropic numerical diffusion tensor and verified against idealized simulations. Results from two physically realistic estuarine simulations, differing only in grid alignment, show significant changes in gradients of salinity. Changes in scalar gradients are reflective of reduced numerical diffusion interacting with the complex 3‐D structure of the transporting flow. We also describe a method for utilizing flow fields from an unaligned grid to generate a flow‐aligned grid with minimal supervision. Copyright © 2013 John Wiley & Sons, Ltd.     

Multigrid solutions of the Euler and Navier–Stokes equations on unstructured grids

A computationally efficient multigrid algorithm for upwind edge‐based finite element schemes is developed for the solution of the two‐dimensional Euler and Navier–Stokes equations on unstructured triangular grids. The basic smoother is based upon a Galerkin approximation employing an edge‐based formulation with the explicit addition of an upwind‐type local extremum diminishing (LED) method. An explicit time stepping method is used to advance the solution towards the steady state. Fully unstructured grids are employed to increase the flexibility of the proposed algorithm. A full approximation storage (FAS) algorithm is used as the basic multigrid acceleration procedure. Copyright © 1999 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.     

A family of multi‐point flux approximation schemes for general element types in two and three dimensions with convergence performance

A family of flux‐continuous, locally conservative, control‐volume‐distributed multi‐point flux approximation (CVD‐MPFA) schemes has been developed for solving the general geometry‐permeability tensor pressure equation on structured and unstructured grids. These schemes are applicable to the full‐tensor pressure equation with generally discontinuous coefficients and remove the O(1) errors introduced by standard reservoir simulation schemes when applied to full‐tensor flow approximation. The family of flux‐continuous schemes is characterized by a quadrature parameterization. Improved numerical convergence for the family of CVD‐MPFA schemes using the quadrature parameterization has been observed for structured and unstructured grids in two dimensions. The CVD‐MPFA family cell‐vertex formulation is extended to classical general element types in 3‐D including prisms, pyramids, hexahedra and tetrahedra. A numerical convergence study of the CVD‐MPFA schemes on general unstructured grids comprising of triangular elements in 2‐D and prismatic, pyramidal, hexahedral and tetrahedral shape elements in 3‐D is presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Simulation of single‐ and two‐phase flows on sliding unstructured meshes using finite volume method

The paper presents an efficient finite volume method for unstructured grids with rotating sliding parts composed of arbitrary polyhedral elements for both single‐ and two‐phase flows. Mathematical model used in computations is based on the ensemble averaged conservation equations. These equations are solved for each phase and in case of single‐phase flow reduce to the transient Reynolds‐averaged Navier–Stokes (TRANS) equations. Transient flow induced by rotating impellers is thus resolved in time. The use of unstructured grids allows an easy and flexible meshing for the entire flow domain. Polyhedral cell volumes are created on the arbitrary mesh interface placed between rotating and static parts. Cells within the rotating parts move each time step and the new faces are created on the arbitrary interfaces only, while the rest of the domain remain ‘topologically’ unchanged. Implicit discretization scheme allows a wide range of time‐step sizes, which further reduce the computational effort. Special attention is given to the interpolation practices used for the reconstruction of the face quantities. Mass fluxes are recalculated at the beginning of each time step by using an interpolation scheme, which enhances the coupling between the pressure and velocity fields. The model has been implemented into the commercially available CFD code AVL SWIFT (AVL AST, SWIFT Manual 3.1, AVL List GmbH, Graz, Austria, 2002). Single‐phase flow in a mixing vessel stirred by a six‐bladed Rushton‐type turbine and two‐phase flow in aerated stirred vessel with the four‐blade Rushton impeller are simulated. The results are compared with the available experimental data, and good agreement is observed. The proposed algorithm is proved to be both stable and accurate for single‐phase as well as for the two‐phase flows calculations. Copyright 2004 John Wiley & Sons, Ltd.     

Algebraic multigrid and incompressible fluid flow

This paper is concerned with the development of algebraic multigrid (AMG) solution methods for the coupled vector–scalar fields of incompressible fluid flow. It addresses in particular the problems of unstable smoothing and of maintaining good vector–scalar coupling in the AMG coarse‐grid approximations. Two different approaches have been adopted. The first is a direct approach based on a second‐order discrete‐difference formulation in primitive variables. Here smoothing is stabilized using a minimum residual control harness and velocity–pressure coupling is maintained by employing a special interpolation during the construction of the inter‐grid transfer operators. The second is an indirect approach that avoids the coupling problem altogether by using a fourth‐order discrete‐difference formulation in a single scalar‐field variable, primitive variables being recovered in post‐processing steps. In both approaches the discrete‐difference equations are for the steady‐state limit (infinite time step) with a fully implicit treatment of advection based on central differencing using uniform and non‐uniform unstructured meshes. They are solved by Picard iteration, the AMG solvers being used repeatedly for each linear approximation. Both classical AMG (C‐AMG) and smoothed‐aggregation AMG (SA‐AMG) are used. In the direct approach, the SA‐AMG solver (with inter‐grid transfer operators based on mixed‐order interpolation) provides an almost mesh‐independent convergence. In the indirect approach for uniform meshes, the C‐AMG solver (based on a Jacobi‐relaxed interpolation) provides solutions with an optimum scaling of the convergence rates. For non‐uniform meshes this convergence becomes mesh dependent but the overall solution cost increases relatively slowly with increasing mesh bandwidth. Copyright © 2006 John Wiley & Sons, Ltd.     

A new TVD method for advection simulation on 2D unstructured grids

For advection schemes with flux limiters derived on one‐dimensional grids to two‐dimensional (2D) unstructured triangular ones. In this method, the variables located normal to this face are taken into more account to compute the flux, by means of defining required nodes along the line through the center point of the considered face and perpendicular to it. Besides, the new method adopts the improved total variation diminishing schemes in, which consider the face position between the two neighboring cells as well as the size differences of the related cells; therefore, it fits 2D unstructured grids well. Compared with the present flux limiting methods for 2D unstructured grids, a higher accuracy and efficiency and a good monotonicity are achieved by the new method.Copyright © 2012 John Wiley & Sons, Ltd.     

An unstructured grid,three‐dimensional model based on the shallow water equations

A semi‐implicit finite difference model based on the three‐dimensional shallow water equations is modified to use unstructured grids. There are obvious advantages in using unstructured grids in problems with a complicated geometry. In this development, the concept of unstructured orthogonal grids is introduced and applied to this model. The governing differential equations are discretized by means of a semi‐implicit algorithm that is robust, stable and very efficient. The resulting model is relatively simple, conserves mass, can fit complicated boundaries and yet is sufficiently flexible to permit local mesh refinements in areas of interest. Moreover, the simulation of the flooding and drying is included in a natural and straightforward manner. These features are illustrated by a test case for studies of convergence rates and by examples of flooding on a river plain and flow in a shallow estuary. Copyright © 2000 John Wiley & Sons, Ltd.