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.     

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.     

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.     

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.     

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.     

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.     

Two-way coupling of Eulerian–Lagrangian model for dispersed multiphase flows using filtering functions

Eulerian–Lagrangian approaches for dispersed multiphase flows can simulate detailed flow structures with a much higher spatial resolution than the Eulerian–Eulerian approaches. However, there are still unsolved problems regarding the calculation method for accurate two-way interaction, especially on the numerical instability due to the dispersion migration through discrete computational grids. Inadequate solvers sometimes produce false velocity fluctuation which makes the simulation unstable. In this paper, a new calculation method for dispersion-to-continuous phase interaction, which is accompanied by spherical dispersion migration, is proposed. The basic principle of the method is the introduction of Lagrangian filtering functions which convert discrete dispersion volume fractions to a spatially differentiable distribution. The performance of linear, Gaussian and sinewave filtering functions is examined by simple benchmark tests and applied to the simulation of dispersion-generated fluctuation. Using the present method, three-dimensional continuous phase flow structures induced by rising spherical bubbles and/or settling solid particles are demonstrated.     

Prediction of developing turbulent flow in 90°‐curved ducts using linear and non‐linear low‐Re k–ε models

This paper reports the outcome of applying two different low‐Reynolds‐number eddy‐viscosity models to resolve the complex three‐dimensional motion that arises in turbulent flows in ducts with 90^° bends. For the modelling of turbulence, the Launder and Sharma low‐Re k–ε model and a recently produced variant of the cubic non‐linear low‐Re k–ε model have been employed. In this paper, developing turbulent flow through two different 90^° bends is examined: a square bend, and a rectangular bend with an aspect ratio of 6. The numerical results indicate that for the bend of square cross‐section the curvature induces a strong secondary flow, while for the rectangular cross‐section the secondary motion is confined to the corner regions. For both curved ducts, the secondary motion persists downstream of the bend and eventually slowly disappears. For the bend of square cross‐section, comparisons indicate that both turbulence models can produce reasonable predictions. For the bend of rectangular cross‐section, for which a wider range of data is available, while both turbulence models produce satisfactory predictions of the mean flow field, the non‐linear k–ε model returns superior predictions of the turbulence field and also of the pressure and friction coefficients. Copyright © 2006 John Wiley & Sons, Ltd.     

Simulating gas–liquid multiphase flow using meshless Lagrangian method

A multiphase flow model has been established based on a moving particle semi‐implicit method. A surface tension model is introduced to the particle method to improve the numerical accuracy and stability. Several computational techniques are employed to simplify the numerical procedure and further improve the accuracy. A particle fraction multiphase flow model is developed and verified by a two‐phase Poiseuille flow. The multiphase surface tension model is discussed in detail, and an ethanol drop case is introduced to verify the surface tension model. A simple dam break is simulated to demonstrate the improvements with various modifications in particle method along with a new boundary condition. Finally, we simulate several bubble rising cases to show the capacity of this new model in simulating gas–liquid multiphase flow with large density ratio difference between phases. The comparisons among numerical results of mesh‐based model, experimental data, and the present model, indicate that the new multiphase particle method is acceptable in gas–liquid multiphase fluids simulation. Copyright © 2014 John Wiley & Sons, Ltd.     

EULERIAN–LAGRANGIAN COMPUTATIONS ON PHASE DISTRIBUTION OF TWO-PHASE BUBBLY FLOWS

A comprehensively theoretical model is developed and numerically solved to investigate the phase distribution phenomena in a two-dimensional, axisymmetric, developing, two-phase bubbly flow. The Eulerian approach treats the fluid phase as a continuum and solved Eulerian conservation equations for the liquid phase. The Lagrangian bubbles are tracked by solving the equation of motion for the gas phase. The interphase momentum changes are included in the equations. The numerical model successfully predicts detailed flow velocity profiles for both liquid and gas phases. The development of the wall-peaking phenomenon of the void fraction and velocity profiles is also characterized for the developing flow. For 42 experiments in which the mean void fraction is less than 20 per cent, numerical calculations demonstrate that the predictions agree well with Liu's experimental data. © 1997 by John Wiley & Sons, Ltd.     

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.     

Numerical simulation of compressible turbulent flow via improved gas‐kinetic BGK scheme

In order to solve compressible turbulent flow problems, this study focuses on incorporating the Spalart–Allmaras turbulence model into gas‐kinetic BGK (Bhatnagar–Gross–Krook) scheme. The Spalart–Allmaras turbulence model is solved using finite difference discretization. The variables on the cell interface are interpolated via the van Leer limiter in the reconstruction stage. Simulation of subsonic and transonic flow over a NACA0012 airfoil has been implemented using two‐dimensional body‐fitted grids. The numerical results obtained appear in good agreement with the AGARD results, demonstrating the effectiveness and usefulness of the strategy of coupling the Spalart–Allmaras turbulence model with the BGK scheme for compressible turbulent flow simulation. Copyright © 2010 John Wiley & Sons, Ltd.     

On the use of Csanady’s formulae in a turbulent gas–solid channel flow

The paper examines the use of expressions proposed by Csanady to predict the influence of the crossing trajectory and continuity effects on the decorrelation time scales of the fluid along solid particle trajectories in horizontal and downward vertical channel flows. The model is evaluated using data provided by a direct numerical simulation (DNS) of the carrier phase combined with a Lagrangian simulation of discrete particle (LS). Two particle relaxation times and two values of the gravity acceleration are considered. The results show the possibility of using Csanady’s expressions in a turbulent channel flow provided that the spatial and temporal correlations anisotropy is included in the model. As in isotropic homogeneous turbulence, a decrease of the decorrelation time scales is found to be more important in the directions perpendicular to the mean relative velocity.     

Particle tracking in Taylor–Couette flow

The paths of small inertial particles are computed in a steady Taylor vortex background flow. When buoyancy effects are neglected we find that particles denser than the background fluid tend to a limit orbit in the meridional plane. The difference in settling time and orbit size, with varying Reynolds number of the background flow, is investigated. We also consider the effect of the various forces on the limit orbit of the particle.     

Friction factor improved correlations for laminar and turbulent gas–liquid flow in horizontal pipelines

We develop improved correlations for two-phase flow friction factor that consider the effect of the relative velocity of the phases, based on a database that includes 2560 gas–liquid flow experiments in horizontal pipes. The database includes a wide range of operational conditions and fluid properties for two-phase friction factor correlations. We classify the experiments by liquid holdup ranges to obtain composite analytical expressions for two-phase friction factor vs. the Reynolds number by fitting logistic dose curves to the experimental data with. We compute the liquid holdup values used to classify the experimental data using correlations proposed previously. The Reynolds number is based on the mixture velocity and the liquid kinematic viscosity. The Fanning friction factor for gas–liquid is defined in term of the mixture velocity and density. Additionally, we sort the experimental data by flow regime and obtain the two-phase friction factor improved correlations for dispersed bubble, slug, stratified and annular flow for different holdup ranges. We report error estimates for the predicted vs. measured friction factor together with standard deviation for each correlation. The accuracy of the correlations developed in this study is compared with that of other 21 correlations and models widely available in the specialized literature. Since different authors use different definitions for friction factors and Reynolds numbers, we present comparisons of the predicted pressure drop for each and every data point in the database. In most cases our correlations predict the pressure drop with much greater accuracy than those presented by previous authors.     

Direct numerical simulation of turbulent Taylor–Couette flow

The direct numerical simulation (DNS) of the Taylor–Couette flow in the fully turbulent regime is described. The numerical method extends the work by Quadrio and Luchini [M. Quadrio, P. Luchini, Eur. J. Mech. B/Fluids 21 (2002) 413–427], and is based on a parallel computer code which uses mixed spatial discretization (spectral schemes in the homogeneous directions, and fourth-order, compact explicit finite-difference schemes in the radial direction). A DNS is carried out to simulate for the first time the turbulent Taylor–Couette flow in the turbulent regime. Statistical quantities are computed to complement the existing experimental information, with a view to compare it to planar, pressure-driven turbulent flow at the same value of the Reynolds number. The main source for differences in flow statistics between plane and curved-wall flows is attributed to the presence of large-scale rotating structures generated by curvature effects.     

A numerical study of the behaviour of a gas–liquid interface subjected to periodic vertical motion

A numerical study has been undertaken to examine the behaviour of a gas liquid interface in a vertical cylindrical vessel subjected to a sinusoidal vertical motion. The computational method used is based on the simplified marker‐and‐cell method and includes a continuum surface model for the incorporation of surface tension. The numerical results indicate that the surface tension has very little effect on the period and amplitude of oscillations of the interfacial waves. The stability of the interfacial waves has been found to depend on the initial pressure pulse disturbance, and exponential growth of the interfacial wave has been observed in some cases. The influence of the amplitude and frequency of the forcing oscillations has also been investigated. The results are in good agreement with available experimental and analytical solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

Simulation of gas–solid particle flows over a wide range of concentration

A two-fluid model of gas–solid particle flows that is valid for a wide range of the solid-phase volume concentration (dilute to dense) is presented. The governing equations of the fluid phase are obtained by volume averaging the Navier–Stokes equations for an incompressible fluid. The solid-phase macroscopic equations are derived using an approach that is based on the kinetic theory of dense gases. This approach accounts for particle–particle collisions. The model is implemented in a control-volume finite element method for simulations of the flows of interest in two-dimensional, planar or axisymmetric, domains. The chosen mathematical model and the proposed numerical method are applied to three test problems and one demonstration problem. © 1998 John Wiley & Sons, Ltd.