The comparison of two approaches to numerical modelling of gas-particles turbulent flow and heat transfer in a pipe

The comparison of two theoretical approaches for the numerical investigation of turbulent gas–solid flows with heat transfer in a pipe are presented in this paper. The first approach is based on Eulerian–Eulerian modelling of investigated phenomena, the second one is formulated within the framework of the Eulerian–Lagrangian approach. The verification of numerical models under consideration. Their testing against available experimental data show good prognostic properties of the elaborated theoretical tool for research activities to study new physical fundamentals of turbulent gas-suspended particles flows in pipes and channels.     

A variable boundary model of storm surge flooding in generalized curvilinear grids

In this paper, generalized 2D shallow sea dynamic equations in movable curvilinear co-ordinates are derived. Through a differential co-ordinate transformation a self-adaptive grid is proposed to treat a continuously deforming lateral boundary and a kinematical boundary condition is adopted. The self-adaptive grid method (SAM) is used to simulate numerically the storm surge flooding in the Bohai Sea on 23 April 1969, which was one of the largest storm surge inundations in China.     

Grid convergence studies for the prediction of hurricane storm surge

The focus of this paper is a systematic determination of the relationship between grid resolution and errors associated with computations of hurricane storm surge. A grid structure is sought that provides the spatial resolution necessary to capture pertinent storm surge physics and does not overdiscretize. A set of numerical experiments simulating storm surge generation over 14 grid discretizations of idealized domains examines the influence of grid spacing, shoreline detail, coastline resolution and characteristics of the meteorological forcing on storm surge computations. Errors associated with a given grid are estimated using a Richardson-based error estimator. Analysis of the magnitude and location of estimated errors indicates that underresolution on the continental shelf leads to significant overprediction of the primary storm surge. In deeper waters, underresolution causes smearing or damping of the inverted barometer forcing function, which in turn results in underprediction of the surge elevation. In order to maintain a specified error level throughout the duration of the storm, the highest grid resolution is required on the continental shelf and particularly in nearshore areas. The disparity of discretization requirements between deep waters and coastal regions is best met using a graded grid. Application of the graded gridding strategy to the hindcast of Hurricane Camille reinforces the necessity of using a grid that has high levels of resolution in nearshore regions and areas of complex coastal geometry. © 1998 John Wiley & Sons, Ltd.     

Computation of particle dispersion in turbulent liquid flows using an efficient Lagrangian trajectory model

The dispersion of solid particles in a turbulent liquid flow impinging on a centrebody through an axisymmetric sudden expansion was investigated numerically using a Eulerian–Lagrangian model. Detailed experimental measurements at the inlet were used to specify the inlet conditions for two-phase flow computations. The anisotropy of liquid turbulence was accounted for using a second-moment Reynold stress transport model. A recently developed stochastic–probabilistic model was used to enhance the computational efficiency of Lagrangian trajectory computations. Numerical results of the stochastic–probabilistic model using 650 particle trajectories were compared with those of the conventional stochastic discrete-delta-function model using 18 000 particle trajectories. In addition, results of the two models were compared with experimental measurements. © 1998 John Wiley & Sons, Ltd.     

Solution of the Navier–Stokes equations in velocity–vorticity form using a Eulerian–Lagrangian boundary element method

This paper describes the Eulerian–Lagrangian boundary element model for the solution of incompressible viscous flow problems using velocity–vorticity variables. A Eulerian–Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian–Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection‐dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two‐dimensional Navier–Stokes problems with low–medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd.     

Fourier analysis of the Eulerian–Lagrangian least squares collocation method

A Fourier analysis was performed in order to study the numerical characteristics of the effective Eulerian–Lagrangian least squares collocation (ELLESCO) method. As applied to the transport equation, ELLESCO requires a C¹-continuous trial space and has two degrees of freedom per node. Two coupled discrete equations are generated for a typical interior node for a one-dimensional problem. Each degree of freedom is expanded separately in a Fourier series and is substituted into the discrete equations to form a homogeneous matrix equation. The required singularity of the system matrix leads to a ‘physical’ amplification factor that characterizes the numerical propagation of the initial conditions and a ‘computational’ one that can affect stability. Unconditional stability for time-stepping weights greater than or equal to 0-5 is demonstrated. With advection only, ELLESCO accurately propagates spatial wavelengths down to 2Δx. As the dimensionless dispersion number becomes large, implicit formulations accurately propagate the phase, but the higher-wave-number components are underdamped. At large dispersion numbers, phase errors combined with underdamping cause oscillations in Crank–Nicolson solutions. These effects lead to limits on the temporal discretization when dispersion is present. Increases in the number of collocation points per element improve the spectral behaviour of ELLESCO.     

Free‐surface flow over a semi‐circular obstruction

The fully non‐linear free‐surface flow over a semi‐circular bottom obstruction was studied numerically in two dimensions using a mixed Eulerian–Lagrangian formulation. The problem was solved in the time domain that allows the prediction of a number of transient phenomena, such as the generation of upstream advancing solitary waves, as well as the simulation of wave breaking. A parametric study was performed for a range of values of the depth‐based Froude number up to 2.5 and non‐dimensional obstacle heights, α up to 0.9. When wave breaking does not occur, three distinct flow regimes were identified: subcritical, transcritical and supercritical. When breaking occurs it may be of any type: spilling, plunging or surging. In addition, for values of the Froude number close to 1, the upstream solitary waves break. A systematic study was undertaken to define the boundaries of each type of breaking and non‐breaking pattern and to determine the drag and lift coefficients, free‐surface profile characteristics and transient behavior. Copyright © 1999 John Wiley & Sons, Ltd.     

An efficient operator splitting scheme for three-dimensional hydrodynamic computations

A new efficient numerical method for three-dimensional hydrodynamic computations is presented and discussed in this paper. The method is based on the operator splitting method and combined with Eulerian–Lagrangian method, finite element method and finite difference method. To increase the efficiency and stability of the numerical solutions, the operator splitting method is employed to partition the momentum equations into three parts, according to physical phenomena. A time step is divided into three time substeps. In the first substep, advection and Coriolis force are solved using the explicit Eulerian–Lagrangian method. In the second substep, horizontal diffusion is approximated by implicit FEM in each horizontal layer. In the last substep, the continuity equation is solved by implicit FEM, and vertical diffusion and pressure gradient are discretized by implicit FDM in each nodal column. The stability analysis shows that this method is unconditionally stable. A number of numerical experiments have been performed. The results simulated by the present scheme agree well with analytical solutions and the other documented model results. The method is efficient for 3D shallow water flow computations and fully fits complicated configurations. © 1998 John Wiley & Sons, Ltd.     

Prediction of turbulent gas–solids flow in curved ducts using the Eulerian–Lagrangian method

The flow of particulate two‐phase flow mixtures occur in several components of solid fuel combustion systems, such as the pressurised fluidised bed combustors (PFBC) and suspension‐fired coal boilers. A detailed understanding of the mixture characteristics in the conveying component can aid in refining and optimising its design. In this study, the flow of an isothermal, dilute two‐phase particulate mixture has been examined in a high curvature duct, which can be representative of that transporting the gas–solid mixture from the hot clean‐up section to the gas turbine combustor in a PFBC plant. The numerical study has been approached by utilising the Eulerian–Lagrangian methodology for describing the characteristics of the fluid and particulate phases. By assuming that the mixture is dilute and the particles are spherical, the governing particle momentum equations have been solved with appropriately prescribed boundary conditions. Turbulence effects on the particle dispersion were represented by a statistical model that accounts for both the turbulent eddy lifetime and the particle transit time scales. For the turbulent flow condition examined it was observed that mixtures with small particle diameters had low interphase slip velocities and low impaction probability with the pipe walls. Increasing the particle diameters (>50 μm) resulted in higher interphase slip velocities and, as expected, their impaction probability with the pipe walls was significantly increased. The particle dispersion is significant for the smaller sizes, whereas the larger particles are relatively insensitive to the gas turbulence. The main particle impaction region, and locations most prone to erosion damage, is estimated to be within an outer duct length of two to six times the duct diameter, when the duct radius of curvature to the duct diameter ratio is equal to unity. Copyright © 1999 John Wiley & Sons, Ltd.     

Grid-averaged Lagrangian LES model for multiphase turbulent flow

A grid-averaged Lagrangian (GAL) model for dispersed particle motion in multiphase turbulent flow is presented to provide a large eddy simulation (LES) model for multiphase turbulent flow in which a quite large number of particles are involved. The GAL model is based on an averaging operation for a Lagrangian-type equation of motion of a particle over a computational grid volume and a procedure of reallocation of a dispersed particle cloud with its centroid movement to each grid. The model is therefore a mixed Eulerian–Lagrangian model which can effectively reduce computational time compared with existing Lagrangian-type models, without losing the advantage of Lagrangian-type models that they can properly describe the dynamical evolution of particles. Since the GAL model adopts the grid-volume averaging operation it can easily provide an effective SGS model for LES modeling of multiphase turbulent flow. The validity of the multiphase LES model developed, which is named the GAL-LES model, is confirmed through its application to a particle plume, in which the present model is found to simulate large-eddy motion usually observed in a jet and plume, and to give good agreements with experimental data.     

Parallel finite element methods for large-scale computation of storm surges and tidal flows

Massively parallel finite element methods for large-scale computation of storm surges and tidal flows are discussed here. The finite element computations, carried out using unstructured grids, are based on a three-step explicit formulation and on an implicit space–time formulation. Parallel implementations of these unstructured grid-based formulations are carried out on the Fujitsu Highly Parallel Computer AP1000 and on the Thinking Machines CM-5. Simulations of the storm surge accompanying the Ise-Bay typhoon in 1959 and of the tidal flow in Tokyo Bay serve as numerical examples. The impact of parallelization on this type of simulation is also investigated. The present methods are shown to be useful and powerful tools for the analysis of storm surges and tidal flows. © 1997 John Wiley & Sons, Ltd.     

Interfacial flow computations using adaptive Eulerian–Lagrangian method for spacecraft applications

Understanding the interfacial dynamics and fluid physics associated with the operation of spacecraft is important for scientific as well as engineering purposes. To help address the issues associated with moving boundaries, interfacial dynamics, and spatial‐temporal variations in time and length scales, a 3‐D adaptive Eulerian–Lagrangian method is extended and further developed. The stationary (Eulerian) Cartesian grid is adopted to resolve the fluid flow, and the marker‐based triangulated moving (Lagrangian) surface meshes are utilized to treat the phase boundary. The key concepts and numerical procedures, along with the selected interfacial flow problems are presented. Specifically, the liquid fuel draining dynamics in different flow regimes, and the liquid surface stability under vertically oscillating gravitational acceleration are investigated. Direct assessment of experimental measurement and scaling analysis is made to highlight the computational performance of the present approach as well as the key fluid physics influenced by the given flow parameters. Copyright © 2010 John Wiley & Sons, Ltd.     

Optimal estimation of eddy viscosity for a quasi-three-dimensional numerical tidal and storm surge model

It is shown that the eddy viscosity profile in a quasi-three-dimensional numerical tidal and storm surge model can be estimated by assimilation of velocity data from one or more current meters located on the same vertical line. The computational model used is a simplified version of the so-called vertical/horizontal splitting algorithm proposed by Lardner and Cekirge. We have estimated eddy viscosity both as a constant and as a variable parameter. The numerical scheme consists of a two-level leapfrog method to solve the depth-averaged equations and a generalized Crank-Nicolson scheme to compute the vertical profile of the velocity field. The cost functional in the adjoint scheme consists of two terms. The first term is a certain norm of the difference between computed and observed velocity data and the second term measures the total variation in the eddy viscosity function. The latter term is not needed when the data are exact for the model but is necessary to smooth out the instabilities associated with ‘noisy’ data. It is shown that a satisfactory minimization can be accomplished using either the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton algorithm or Nash's truncated Newton algorithm. Very effective estimation of eddy viscosity profiles is shown to be achieved even when the amount of data is quite small.     

Development of a convection–diffusion–reaction model for solving Maxwell's equations in frequency domain

We propose in this study a numerically accurate and computationally efficient convection–diffusion–reaction finite difference scheme to discretize the full‐vector and semi‐vector optical waveguide equations. The scheme formulated in a grid stencil of five nodal points for solving the three‐dimensional waveguide equations employs the locally analytic solution. In this three‐dimensional study, calculations were carried out for the investigation of wave propagation in diffused channel, rectangular and rib types of optical waveguide. Copyright © 2011 John Wiley & Sons, Ltd.     

An assessment of particle methods for approximating anisotropic dispersion

We derive a smoothed particle hydrodynamics (SPH) approximation for anisotropic dispersion that only depends upon the first derivative of the kernel function and study its numerical properties. In addition, we compare the performance of the newly derived SPH approximation versus an implementation of the particle strength exchange (PSE) method and a standard finite volume method for simulating multiple scenarios defined by different combinations of physical and numerical parameters. We show that, for regularly spaced particles, given an adequate selection of numerical parameters such as kernel function and smoothing length, the new SPH approximation is comparable with the PSE method in terms of convergence and accuracy and similar to the finite volume method. On other hand, the performance of both particle methods (SPH and PSE) decreases as the degree of disorder of the particle increases. However, we demonstrate that in these situations the accuracy and convergence properties of both particle methods can be improved by an adequate choice of some numerical parameters such as kernel core size and kernel function. Copyright © 2012 John Wiley & Sons, Ltd.     

Finite element simulation of turbulent Couette–Poiseuille flows using a low Reynolds number k–ϵ model

Developing Couette–Poiseuille flows at Re=5000 are studied using a low Reynolds number k–ϵ two‐equation model and a finite element formulation. Mesh‐independent solutions are obtained using a standard Galerkin formulation and a Galerkin/least‐squares stabilized method. The predictions for the velocity and turbulent kinetic energy are compared with available experimental results and to the DNS data. Second moment closure's solutions are also compared with those of the k–ϵ model. The deficiency of eddy viscosity models to predict dissymmetric low Reynolds number channel flows has been demonstrated. Copyright © 1999 John Wiley & Sons, Ltd.     

Development of a convection–diffusion‐reaction magnetohydrodynamic solver on non‐staggered grids

This paper presents a convection–diffusion‐reaction (CDR) model for solving magnetic induction equations and incompressible Navier–Stokes equations. For purposes of increasing the prediction accuracy, the general solution to the one‐dimensional constant‐coefficient CDR equation is employed. For purposes of extending this discrete formulation to two‐dimensional analysis, the alternating direction implicit solution algorithm is applied. Numerical tests that are amenable to analytic solutions were performed in order to validate the proposed scheme. Results show good agreement with the analytic solutions and high rate of convergence. Like many magnetohydrodynamic studies, the Hartmann–Poiseuille problem is considered as a benchmark test to validate the code. Copyright © 2004 John Wiley & Sons, Ltd.     

Toward a reduction of mesh imprinting

This paper is the initial investigation into a new Lagrangian cell‐centered hydrodynamic scheme that is motivated by the desire for an algorithm that resists mesh imprinting and has reduced complexity. Key attributes of the new approach include multidimensional construction, the use of flux‐corrected transport (FCT) to achieve second order accuracy, automatic determination of the mesh motion through vertex fluxes, and vorticity control. Toward this end, vorticity preserving Lax–Wendroff (VPLW) type schemes for the two‐dimensional acoustic equations were analyzed and then implemented in a FCT algorithm. Here, mesh imprinting takes the form of anisotropic dispersion relationships. If the stencil for the LW methods is limited to nine points, four free parameters exist. Two parameters were fixed by insisting that no spurious vorticity be created. Dispersion analysis was used to understand how the remaining two parameters could be chosen to increase isotropy. This led to new VPLW schemes that suffer less mesh imprinting than the rotated Richtmyer method. A multidimensional, vorticity preserving FCT implementation was then sought using the most promising VPLW scheme to address the problem of spurious extrema. A well‐behaved first order scheme and a new flux limiter were devised in the process. The flux limiter is unique in that it acts on temporal changes and does not place a priori bounds on the solution. Numerical results have demonstrated that the vorticity preserving FCT scheme has comparable performance to an unsplit MUSCL‐H algorithm at high Courant numbers but with reduced mesh imprinting and superior symmetry preservation. Copyright © 2014 John Wiley & Sons, Ltd.