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.     

A conservative local interface sharpening scheme for the constrained interpolation profile method

A conservative local interface sharpening scheme has been developed for the constrained interpolation profile method with the conservative semi‐Lagrangian scheme, because the conservative semi‐Lagrangian scheme does not feature a mechanism to control the interface thickness, thus causing an increase of numerical error with the advance of the time step. The proposed sharpening scheme is based on the conservative level set method proposed by Olsson and Kreiss. However, because their method can cause excessive deformation of the free‐surface in certain circumstances, we propose an improvement of the method by developing a local sharpening technique. Several advection tests are presented to assess the correctness of the advection and the improved interface sharpening scheme. This is followed by the validations of dam‐breaking flow and the rising bubble flows. The mass of the fluid is exactly conserved and the computed terminal velocity of the rising bubble agrees well with the experiments compared with other numerical methods such as the volume of fluid method (VOF), the front tracking method, and the level set method. Copyright © 2011 John Wiley & Sons, Ltd.     

A model for tracking inertial particles in a lattice Boltzmann turbulent flow simulation

A computationally inexpensive model for tracking inertial particles through a turbulent flow is presented and applied to the turbulent flow through a square duct having a friction Reynolds number of Re_τ = 300. Prior to introducing particles into the model, the flow is simulated using a lattice Boltzmann computation, which is allowed to evolve until a steady state turbulent flow is achieved. A snapshot of the flow is then stored, and the trajectories of particles are computed through the flow domain under the influence of this static probability field. Although the flow is not computationally evolving during the particle tracking simulation, the local velocity is obtained stochastically from the local probability function, thus allowing the dynamics of the turbulent flow to be resolved from the point of view of the suspended particles. Particle inertia is modeled by using a relaxation parameter based on the particle Stokes number that allows for a particle velocity history to be incorporated during each time step. Wall deposition rates and deposition patterns are obtained and exhibit a high level of agreement with previously obtained DNS computational results and experimental results for a wide range of particle inertia. These results suggest that accurate particle tracking through complex turbulent flows may be feasible given a suitable probability field, such as one obtained from a lattice Boltzmann simulation. This in turn presents a new paradigm for the rapid acquisition of particle transport statistics without the need for concurrent computations of fluid flow evolution.     

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.     

An operator splitting algorithm for the three-dimensional advection–diffusion equation

Operator splitting algorithms are frequently used for solving the advection–diffusion equation, especially to deal with advection dominated transport problems. In this paper an operator splitting algorithm for the three-dimensional advection–diffusion equation is presented. The algorithm represents a second-order-accurate adaptation of the Holly and Preissmann scheme for three-dimensional problems. The governing equation is split into an advection equation and a diffusion equation, and they are solved by a backward method of characteristics and a finite element method, respectively. The Hermite interpolation function is used for interpolation of concentration in the advection step. The spatial gradients of concentration in the Hermite interpolation are obtained by solving equations for concentration gradients in the advection step. To make the composite algorithm efficient, only three equations for first-order concentration derivatives are solved in the diffusion step of computation. The higher-order spatial concentration gradients, necessary to advance the solution in a computational cycle, are obtained by numerical differentiations based on the available information. The simulation characteristics and accuracy of the proposed algorithm are demonstrated by several advection dominated transport problems. © 1998 John Wiley & Sons, Ltd.     

VOLUME-TRACKING METHODS FOR INTERFACIAL FLOW CALCULATIONS

A new algorithm for volume tracking which is based on the concept of flux-corrected transport (FCT) is introduced. It is applicable to incompressible 2D flow simulations on finite volume and difference meshes. The method requires no explicit interface reconstruction, is direction-split and can be extended to 3D and orthogonal curvilinear meshes in a straightforward manner. A comparison of the new scheme against well-known existing 2D finite volume techniques is undertaken. A series of progressively more difficult advection tests is used to test the accuracy of each scheme and it is seen that simple advection tests are inadequate indicators of the performance of volume-tracking methods. A straightforward methodology is presented that allows more rigorous estimates to be made of the error in volume advection and coupled volume and momentum advection in real flow situations. The volume advection schemes are put to a final test in the case of Rayleigh–Taylor instability. © 1997 by CSIRO.     

Fourier-based incremental homogenisation of coupled unilateral damage–plasticity model for masonry structures

In structural analysis of large masonry structures, nondemanding computation effort, numerical stability and simplified model assembly and meshing often have a higher priority over precise details of local stress or strain responses. This paper presents the development of a Fourier-based incremental homogenisation technique, where the macro–micro transformations of mechanical variables are derived by incremental variational problems to minimise the potential energy in representative volume elements (RVEs) with respect to local fluctuating displacement fields expanded in Fourier series. In addition to the proposed homogenisation technique, a unilateral damage–plasticity constitutive model for mortar joints in the RVE is developed within the framework of thermomechanics, which accounts for the stiffness and strength degradation (or recovery) due to the transverse crack opening/closing in the mortar joints. The numerical solution for the homogenisation problem and the performances of the proposed coupled-damage plastic mode and Fourier-based homogenisation scheme verified by detailed case studies are presented. It has been shown that the computational effort of the analysis with the proposed modelling technique can be considerably reduced by more than 20% as compared with that of the discrete modelling technique.     

Bubble size distribution prediction for large-scale ship flows: Model evaluation and numerical issues

Prediction of the bubble size distribution in the wake of a ship is important to analyze its acoustic signature. To achieve CFD simulation of dynamic ships with moving control surfaces and rotating propellers in waves, a robust implementation is paramount. In this work a mass conserving multigroup discretization strategy of the Boltzmann transport equation for polydispersed bubbly flows is presented, as well as an analysis of available breakup and coalescence models. Modifications of the discrete equations for the fixed pivot method at the boundaries are introduced that guarantee exact bubble mass conservation. The role of the time stepping scheme in the conservation of mass and number of bubbles is discussed. Though the conservation properties of the discrete system of equations are satisfied provided they are solved exactly, in practice an iterative procedure must be used since the ODE’s are non-linear. Three iterative schemes are proposed and they are analyzed in terms of robustness and efficiency. Breakup, coalescence and dissolution models are analyzed from the numerical point of view. Available models of breakup and coalescence are studied finding appropriate choices for ship applications. Other models are appropriate as well, but are more costly numerically. As appropriate for ship applications, an extension to the model of Prince and Blanch for salt water is proposed and analyzed. The final model is tested against experimental data and computations by other researchers, and convergence properties in bubble size discretization is studied. It is found that for salt water the final steady state is dependent on the initial condition since there is a range of sizes for which coalescence and breakup are both negligible.     

A simple high‐resolution advection scheme

A simple, robust, mass‐conserving numerical scheme for solving the linear advection equation is described. The scheme can estimate peak solution values accurately even in regions where spatial gradients are high. Such situations present a severe challenge to classical numerical algorithms. Attention is restricted to the case of pure advection in one and two dimensions since this is where past numerical problems have arisen. The authors' scheme is of the Godunov type and is second‐order in space and time. The required cell interface fluxes are obtained by MUSCL interpolation and the exact solution of a degenerate Riemann problem. Second‐order accuracy in time is achieved via a Runge–Kutta predictor–corrector sequence. The scheme is explicit and expressed in finite volume form for ease of implementation on a boundary‐conforming grid. Benchmark test problems in one and two dimensions are used to illustrate the high‐spatial accuracy of the method and its applicability to non‐uniform grids. Copyright © 2007 John Wiley & Sons, Ltd.     

Charge-conserving grid based methods for the Vlasov–Maxwell equations

In this article, we introduce numerical schemes for the Vlasov–Maxwell equations relying on different kinds of grid-based Vlasov solvers, as opposite to PIC schemes, which enforce a discrete continuity equation. The idea underlying these schemes relies on a time-splitting scheme between configuration space and velocity space for the Vlasov equation and on the computation of the discrete current in a form that is compatible with the discrete Maxwell solver.     

A non‐iterative implicit algorithm for the solution of advection–diffusion equation on a sphere

A numerical algorithm for the solution of advection–diffusion equation on the surface of a sphere is suggested. The velocity field on a sphere is assumed to be known and non‐divergent. The discretization of advection–diffusion equation in space is carried out with the help of the finite volume method, and the Gauss theorem is applied to each grid cell. For the discretization in time, the symmetrized double‐cycle componentwise splitting method and the Crank–Nicolson scheme are used. The numerical scheme is of second order approximation in space and time, correctly describes the balance of mass of substance in the forced and dissipative discrete system and is unconditionally stable. In the absence of external forcing and dissipation, the total mass and L₂‐norm of solution of discrete system is conserved in time. The one‐dimensional periodic problems arising at splitting in the longitudinal direction are solved with Sherman–Morrison's formula and Thomas's algorithm. The one‐dimensional problems arising at splitting in the latitudinal direction are solved by the bordering method that requires a prior determination of the solution at the poles. The resulting linear systems have tridiagonal matrices and are solved by Thomas's algorithm. The suggested method is direct (without iterations) and rapid in realization. It can also be applied to linear and nonlinear diffusion problems, some elliptic problems and adjoint advection–diffusion problems on a sphere. Copyright © 2015 John Wiley & Sons, Ltd.     

Augmented upwind numerical schemes for the groundwater transport advection‐dispersion equation with local operators

When solute transport is advection‐dominated, the advection‐dispersion equation approximates to a hyperbolic‐type partial differential equation, and finite difference and finite element numerical approximation methods become prone to artificial oscillations. The upwind scheme serves to correct these responses to produce a more realistic solution. The upwind scheme is reviewed and then applied to the advection‐dispersion equation with local operators for the first‐order upwinding numerical approximation scheme. The traditional explicit and implicit schemes, as well as the Crank‐Nicolson scheme, are developed and analyzed for numerical stability to form a comparison base. Two new numerical approximation schemes are then proposed, namely, upwind–Crank‐Nicolson scheme, where only for the advection term is applied, and weighted upwind‐downwind scheme. These newly developed schemes are analyzed for numerical stability and compared to the traditional schemes. It was found that an upwind–Crank‐Nicolson scheme is appropriate if the Crank‐Nicolson scheme is only applied to the advection term of the advection‐dispersion equation. Furthermore, the proposed explicit weighted upwind‐downwind finite difference numerical scheme is an improvement on the traditional explicit first‐order upwind scheme, whereas the implicit weighted first‐order upwind‐downwind finite difference numerical scheme is stable under all assumptions when the appropriate weighting factor (θ) is assigned.     

A study of time integration schemes for the numerical modelling of free surface flows

A semi‐discrete finite element methodology for the modelling of transient free surface flows in the context of Eulerian interface capturing is proposed. The focus of this study is put on the choice of an appropriate time integration strategy for the accurate modelling of the dynamics of free surfaces and of interfacial physics. It is composed of an adaptive time integration scheme for the Navier–Stokes equations, and of the implicit midpoint rule for the transport equation of the Eulerian marker variable. The adaptive scheme allows the automatic determination of a time‐step size that follows the physics of the problem under study, which facilitates the accurate modelling of stiff free surface flows. It is shown that the implicit midpoint rule reduces mass loss for each fluid. Various free surface flow problems are studied to verify and validate the proposed time integration strategy. Copyright © 2005 John Wiley & Sons, Ltd.     

HIGHER-ORDER SCHEMES FOR FREE SURFACE FLOWS WITH ARBITRARY CONFIGURATIONS

The purpose of the present work was to evaluate the importance of formal accuracy and of the conservation property in the numerical computation of incompressible flows with arbitrary free boundaries, such as occur in wave-breaking problems. Four spatial discretization methods were implemented in a computer code based on the VOF method for tracking free surfaces: a non-conservative four-point scheme, the conservative quadratic upstream interpolation method, the conservative linear extrapolation method and a lower-order conservative scheme based on the power-law discretization. The performance of the four schemes was evaluated in three test problems: the propagation of a solitary wave of high amplitude, the propagation of an undular hydraulic jump and the flow resulting from a breaking hydraulic jump. The main conclusion obtained in the present work was that discrete momentum conservation is more important than the formal accuracy of the spatial discretization scheme, particularly when there is recirculation and breaking.     

Fifth‐order Hermitian schemes for computational linear aeroacoustics

We develop a class of fifth‐order methods to solve linear acoustics and/or aeroacoustics. Based on local Hermite polynomials, we investigate three competing strategies for solving hyperbolic linear problems with a fifth‐order accuracy. A one‐dimensional (1D) analysis in the Fourier series makes it possible to classify these possibilities. Then, numerical computations based on the 1D scalar advection equation support two possibilities in order to update the discrete variable and its first and second derivatives: the first one uses a procedure similar to that of Cauchy–Kovaleskaya (the ‘Δ‐P5 scheme’); the second one relies on a semi‐discrete form and evolves in time the discrete unknowns by using a five‐stage Runge–Kutta method (the ‘RGK‐P5 scheme’). Although the RGK‐P5 scheme shares the same local spatial interpolator with the Δ‐P5 scheme, it is algebraically simpler. However, it is shown numerically that its loss of compactness reduces its domain of stability. Both schemes are then extended to bi‐dimensional acoustics and aeroacoustics. Following the methodology validated in (J. Comput. Phys. 2005; 210 :133–170; J. Comput. Phys. 2006; 217 :530–562), we build an algorithm in three stages in order to optimize the procedure of discretization. In the ‘reconstruction stage’, we define a fifth‐order local spatial interpolator based on an upwind stencil. In the ‘decomposition stage’, we decompose the time derivatives into simple wave contributions. In the ‘evolution stage’, we use these fluctuations to update either by a Cauchy–Kovaleskaya procedure or by a five‐stage Runge–Kutta algorithm, the discrete variable and its derivatives. In this way, depending on the configuration of the ‘evolution stage’, two fifth‐order upwind Hermitian schemes are constructed. The effectiveness and the exactitude of both schemes are checked by their applications to several 2D problems in acoustics and aeroacoustics. In this aim, we compare the computational cost and the computation memory requirement for each solution. The RGK‐P5 appears as the best compromise between simplicity and accuracy, while the Δ‐P5 scheme is more accurate and less CPU time consuming, despite a greater algebraic complexity. Copyright © 2007 John Wiley & Sons, Ltd.     

Colloid Transport in Porous Media: A Review of Classical Mechanisms and Emerging Topics

To celebrate the tenth anniversary of InterPore, we present an interdisciplinary review of colloid transport through porous media. This review aims to explore both classical colloid transport and topics that fall outside that purview and thus offer transformative insights into the physics governing transport behavior. First, we discuss the unique colloid characteristics relative to molecules and larger particles. Then, the classical advection–dispersion–filtration models (both conceptual and mathematical) of colloid transport are introduced as well as anomalous transport behaviors. Next, the forces of interaction between colloids and porous media surfaces are discussed. Fourth, applications that are interested in maximizing the transport of colloids through porous media are considered. Then the concept of motile, active biocolloids is introduced, and finally, colloid swarming as a newly recognized mode of transport is summarized.

     

Modelling of massive particulates for breakwater engineering using coupled FEMDEM and CFD

The seaward slope of many breakwaters consists of thousands of interlocking units of rock or concrete comprising a massive granular system of large elements each weighing tens of tonnes. The dumped quarry materials in the core are protected by progressively coarser particulates. The outer armour layer of freely placed units is intended to both dissipate wave energy and remain structurally stable as strong flows are drawn in and out of the particulate core. Design guidance on the mass and shape of these units is based on empirical equations derived from scaled physical model tests. The main failure mode for armour layers exposed to severe storms is hydraulic instability where the armour units of concrete or rock are subjected to uplift and drag forces which can in turn lead to rocking, displacement and collisions sufficient to cause breakage of units. Recently invented armour unit designs making up such granular layered system owe much of their success to the desirable emergent properties of interlock and porosity and how these combine with individual unit structural strength and inertial mass. Fundamental understanding of the forces governing such wave-structure interaction remains poor. We use discrete element and combined finite-discrete element methods to model the granular solid skeleton of randomly packed units coupled to a CFD code which resolves the wave dynamics through an interface tracking technique. The CFD code exploits several methods including a compressive advection scheme, node movement, and general mesh optimization. We provide the engineering context and report progress towards the numerical modelling of instability in these massive granular systems.     

Modelling of massive particulates for breakwater engineering using coupled FEMDEM and CFD

The seaward slope of many breakwaters consists of thousands of interlocking units of rock or concrete comprising a massive granular system of large elements each weighing tens of tonnes.The dumped quarry materials in the core are protected by progressively coarser particulates.The outer armour layer of freely placed units is intended to both dissipate wave energy and remain structurally stable as strong flows are drawn in and out of the particulate core.Design guidance on the mass and shape of these units is based on empirical equations derived from sealed physical model tests.The main failure mode for armour layers exposed to severe storms is hydraulic instability where the armour units of concrete or rock are subjected to uplift and drag forces which can in turn lead to rocking,displacement and collisions sufficient to cause breakage of units.Recently invented armour unit designs making up such granular layered system owe much of their success to the desirable emergent properties of interlock and porosity and how these combine with individual unit structural strength and inertial mass. Fundamental understanding of the forces governing such wave-structure interaction remains poor.We use discrete element and combined finite-discrete element methods to model the granular solid skeleton of randomly packed units coupled to a CFD code which resolves the wave dynamics through an interface tracking technique.The CFD code exploits several methods including a compressive advection scheme, node movement, and general mesh optimization.We provide the engineering context and report progress towards the numerical modelling of instability in these massive granular systems.     

含液颗粒材料液固耦合分析的离散颗粒模型及特征线SPH法

对含液颗粒材料流固耦合分析建议了一个基于离散颗粒模型与特征线SPH法的显式拉格朗日-欧拉无网格方案。在已有的用以模拟固体颗粒集合体的离散颗粒模型[1]基础上,将颗粒间间隙内的流体模型化为连续介质,对其提出并推导了基于特征线的SPH法。数值例题显示了所建议方案在模拟颗粒材料与间隙流相互作用的能力和性能以及间隙流体对颗粒结构承载能力及变形的影响。