Analysis of unsteady gas–liquid flows in a rectangular tank: Comparison of Euler–Eulerian and Euler–Lagrangian simulations

We study the dynamics of gas–liquid flows experimentally and computationally in a rectangular bubble column where the gas source is introduced at the corner. The flow in this reactor is complex and inherently unsteady in nature. The two-dimensional liquid phase velocity field is calculated by an Eulerian approach solving the unsteady Reynolds Averaged Navier Stokes equations. The conservation equations are closed using a two parameter turbulence model. The two-way coupling was accounted for by adding source terms in the conservation equations of the continuous phase to take into account the interaction with the dispersed phase. Bubble tracking is achieved through a Lagrangian approach. Here the equations of motion are solved taking into account the drag, pressure, buoyancy and gravity forces. The time-averaged flows along with the variables which characterize turbulence are analyzed for a wide range of gas flow-rates using Euler–Lagrangian simulations. These simulation predictions are validated with Euler–Eulerian simulations where the gas-phase distribution is captured as a void fraction and PIV experiments. The motion of bubbles induces turbulence in the flow. The applicability of two parameter models for turbulence like the standard k–ε model on time-averaged flow properties is addressed. From the results of the time averaged velocity field, turbulence intensity, turbulent viscosity and gas hold-up profiles, it is concluded that the Euler–Lagrangian model is applicable at lower gas flow-rates. The Euler–Eulerian approach was found to be valid at lower as well as higher gas flow-rates.     

Numerical simulation of gas–solid flow with two fluid model in a spouted-fluid bed

The flow characteristics in a spouted-fluid bed differ from those in spouted or fluidized beds because of the injection of the spouting gas and the introduction of a fluidizing gas. The flow behavior of gas–solid phases was predicted using the Eulerian–Eulerian two-fluid model (TFM) approach with kinetic theory for granular flow to obtain the flow patterns in spouted-fluid beds. The gas flux and gas incident angle have a significant influence on the porosity and particle concentration in gas–solid spouted-fluid beds. The fluidizing gas flux affects the flow behavior of particles in the fountain. In the spouted-fluid bed, the solids volume fraction is low in the spout and high in the annulus. However, the solids volume fraction is reduced near the wall.     

Two and three dimensional modeling of fluidized bed with multiple jets in a DEM–CFD framework

Fluidized beds with multiple jets have widespread industrial applications. The objective of this paper is to investigate the jet interactions and hydrodynamics of a fluidized bed with multiple jets. Discrete element modeling coupled with in-house CFD code GenIDLEST has been used to simulate a bed with nine jets. The results are compared with published experiments. Mono dispersed particles of size 550 μm are used with 1.4 times the minimum fluidization velocity of the particles. Both two and three dimensional computations have been performed. To the best of our knowledge, the results presented in this paper are the first full 3D simulations of a fluidized bed performed with multiple jets. Discrepancies between the experiment and simulations are discussed in the context of the dimensionality of the simulations. The 2D solid fraction profile compares well with the experiment close to the distributor plate. At higher heights, the 2D simulation over-predicts the solid fraction profiles near the walls. The 3D simulation on the other hand is better able to capture the solid fraction profile higher up in the bed compared to that near the distributor plate. Similarly, the normalized particle velocities and the particle fluxes compare well with the experiment closer to the distributor plate for the 2D simulation and the freeboard for the 3D simulation, respectively. A lower expanded bed height is predicted in the 2D simulation compared to the 3D simulation and the experiment. The results obtained from DEM computations show that a 2D simulation can be used to capture essential jetting trends near the distributor plate regions, whereas a full scale 3D simulation is needed to capture the bubbles near the freeboard regions. These serve as validations for the experiment and help us understand the complex jet interaction and solid circulation patterns in a multiple jet fluidized bed system.     

Simulations of vertical jet penetration using a filtered two-fluid model in a gas–solid fluidized bed

The influence of a vertical jet located at the distributor in a cylindrical fluidized bed on the flow behavior of gas and particles was predicted using a filtered two-fluid model proposed by Sundaresan and coworkers. The distributions of volume fraction and the velocity of particles along the lateral direction were investigated for different jet velocities by analyzing the simulated results. The vertical jet penetration lengths at the different gas jet velocities have been obtained and compared with predictions derived from empirical correlations; the predicted air jet penetration length is discussed. Agreement between the numerical simulations and experimental results has been achieved.     

A two scale model of porous rocks with Drucker–Prager matrix: Application to a sandstone

The present paper is devoted to a micromechanical model of porous rocks and its application to a sandstone. This original model takes advantage of a recent homogenization-based macroscopic yield function which couples Drucker–Prager type plasticity of the solid matrix and evolving porosity. Its formulation and implementation are described. Application to a Vosges sandstone shows that, except for very low confining pressures for which the mechanical behavior is quasi-brittle, the model predicts well the ductile behavior at moderate or high confining pressures (for which the pore collapse mechanism is expected to play a dominant role).     

Slosh dynamics of liquid-filled composite containers—A two dimensional meshless local Petrov–Galerkin approach

This paper deals with the studies of sloshing of liquid in partially filled rectangular composite containers subjected to external excitation. The governing equation for inviscid fluid is written as pressure variable form. At each time step, the pressure is evaluated using the meshless local Petrov–Galerkin (MLPG) approach. A local symmetric weak form (LSWF) for linearized sloshing is developed, and a truly meshless method, based on LSWF and moving least squares (MLS) approximation, is presented for the solution of Laplace equation with the requisite boundary conditions. The effectiveness of the developed algorithm is demonstrated through few numerical examples. The comparison of results in terms of natural slosh frequencies, sloshing amplitudes and hydrodynamic pressures obtained in the present investigation are made with those available in the reported literature. To observe the change in the total liquid response due fluid–structure interaction effects, parametric studies are carried out for different cases by varying the fibre orientations and wall thicknesses in the laminated container wall. The present meshless method based on LSWF is found to be simple and attractive with a great potential in engineering applications.     

Dynamic Separation of Resistance Spot Welded Joints: Part II—Analysis of Test Results and a Model

The ultimate strength of resistance spot welded joints fabricated from a wide range of steel grades, weld button size, and sheet thickness are reported for lap-shear and cross-tension specimens subjected to quasi-static and impact loading conditions. Test data are analyzed with respect to energy, impact speed, and loading rate. Loading rate is identified as a critical, test system independent parameter to reflect the strain rate sensitivity of the steels. An equation is fitted to the ultimate strength test data as a function of loading rate which is proposed to predict the separation of spot welded joint under dynamic loading. The model is validated by test data from open literature generated from other type of specimens and/or dynamic test conditions.     

Investigation of bubble breakup and coalescence in a packed-bed reactor – Part 2: Development of a new bubble breakup and coalescence model

A mechanistic model of bubble breakup and coalescence has been developed for a packed bed. Bubble breakup and coalescence models are developed for two coalescence and three breakup mechanisms by taking account of geometry effects and local flow conditions. The bubble size distribution estimated with the present bubble breakup and coalescence models are compared with the experimental data. Change of bubble size distributions along the axial direction is studied with the median bubble size. Median bubble size as a function of the axial location is estimated under two inlet flow conditions: (1) bubble breakup dominated flow and (2) bubble coalescence dominated flow. The predictions of the median bubble size with the present model result in the best among other existing bubble breakup and coalescence models. However, the prediction of the median bubble size for the bubble coalescence dominated flow is still significantly larger than the experimental data. Breakup and coalescence coefficients need to be adjusted in order to predict more accurate bubble size distributions and median bubble size for both flow conditions. For the bubble breakup dominated flow, the breakup and coalescence coefficients are found to be 0.35 and 0.4, respectively. For the bubble coalescence dominated flow, the breakup and coalescence coefficients are found to be 0.35 and 0.01, respectively.     

Improvement,validation and application of CFD/DEM model to dense gas–solid flow in a fluidized bed

Dense gas–solid flow with solid volume fraction greater than 10% and at moderate Reynolds number is important in many industrial facilities such as fluidized beds. In this work, the Euler–Lagrange approach in combination with a deterministic collision model is applied to a laboratory-scale fluidized bed. The fluid–particle interaction is studied using a new procedure called the offset method, which results in several numbers of spatial displacements of the fluid grid. The proposed method is highly precise in determining porosity and momentum transfer, thus improving simulation accuracy. A validation study was carried out to assess the results using this in-house CFD/DEM code against 5-s operation of a Plexiglas spouted-fluidized bed, showing good qualitative correlation of solid distribution in the bed and acceptable quantitative agreement of pressure drops at different positions in the bed. In view of high computing cost, special emphasis is placed on effective program design, such as application of advanced detection algorithm for particle–particle/wall collisions, the multi-grid method and parallel calculation. In this context, the influence of increasing the processor number, up to 36, on calculation efficiency was investigated.     

Symmetries of the hyperbolic shallow water equations and the Green–Naghdi model in Lagrangian coordinates

The observation that the hyperbolic shallow water equations and the Green–Naghdi equations in Lagrangian coordinates have the form of an Euler–Lagrange equation with a natural Lagrangian allows us to apply Noether's theorem for constructing conservation laws for these equations. In this study the complete group analysis of these equations is given: admitted Lie groups of point and contact transformations, classification of the point symmetries and all invariant solutions are studied. For the hyperbolic shallow water equations new conservation laws which have no analog in Eulerian coordinates are obtained. Using Noether's theorem a new conservation law of the Green–Naghdi equations is found. The dependence of solutions on the parameter is illustrated by self-similar solutions which are invariant solutions of both models.     

Limit analysis and homogenization of porous materials with Mohr–Coulomb matrix. Part II: Numerical bounds and assessment of the theoretical model

This second part of the two-part study is devoted to the numerical Limit Analysis of a hollow sphere model with a Mohr–Coulomb matrix and its use for the assessment of theoretical results. Brief background and fundamental of the static and kinematic approaches in the context of numerical limit analysis are first recalled. We then present the hollow sphere model, together with its axisymmetric FEM discretization and its mechanical position. A conic programming adaptation of a previous iterative static approach, based on a piecewise linearization (PWL) of the plasticity criterion, was first realized. Unfortunately, the resulting code, no more than the PWL one, did not allow sufficiently refined meshes for loss of convergence of the conic optimizer. This problem was solved by using the projection algorithm of Ben Tal and Nemriovski (BTN) and the (interior point) linear programming code XA. For the kinematic approach, a first conic adaptation appeared also inefficient. Then, an original mixed (but fully kinematic) approach dedicated to the general Mohr–Coulomb axisymmetric problem was elaborated. The final conic mixed code appears much more robust than the classic one when using the conic code MOSEK, allowing us to take into account refined numerical meshes. After a fine validation in the case of spherical cavities and isotropic loadings (for which the exact solution is known) and comparison to previous (partial) results, numerical lower and upper bounds (a posteriori verified) of the macroscopic strength are provided. These bounds are used to assess and validate the theoretical results of the companion (part I) paper. Effects of the friction angle as well as that of the porosity are illustrated.     

An enhanced beam-theory model of the mixed-mode bending (MMB) test—Part II: Applications and results

The paper presents an enhanced beam-theory (EBT) model of the mixed-mode bending (MMB) test, whereby the specimen is considered as an assemblage of two sublaminates partly connected by an elastic–brittle interface. Analytical expressions for the compliance, energy release rate, and mode mixity are deduced. A compliance calibration strategy enabling numerical or experimental evaluation of the interface elastic constants is also presented. Furthermore, analytical expressions for the crack length correction parameters—analogous to those given by the corrected beam-theory (CBT) model for unidirectional laminated specimens—are furnished for multidirectional laminated specimens, as well. Lastly, an example application to experimental data reduction is presented.     

A Feature-Based Stochastic Permeability of Shale: Part 1—Validation and Two-Phase Permeability in a Utica Shale Sample

Transport in Porous Media - Estimate of permeability plays a crucial role in flow-based studies of fractured tight-rocks. It is well known that most of the flow through tight-rocks (e.g., shales)...     

Wave forces on a submerged horizontal plate–Part II: Solitary and cnoidal waves

This paper is the companion to Part I under the same title, and is mainly concerned with wave loads due to nonlinear waves of solitary and cnoidal type propagating over a submerged, horizontal and thin plate. Following the development of the nonlinear model (via the Level I Green–Naghdi theory) for the flow of an incompressible and inviscid fluid given in Part I, the wave-induced loads on the submerged, fixed (and rigid) plate are calculated, and results are compared with the available laboratory data, and with linear solutions of the problem. Dependence of the loads on wave conditions (wave height and wavelength) and plate characteristics (submergence depth and plate width) are studied for both the solitary and cnoidal wave cases.     

A computational study of the wing–wing and wing–body interactions of a model insect

The aerodynamic interaction between the contralateral wings and between the body and wings of a model insect are studied, by using the method of numerically solving the Navier-Stokes equations over moving overset grids, under typical hovering and forward flight conditions. Both the interaction between the contralateral wings and the interaction between the body and wings are very weak, e.g. at hovering, changes in aerodynamic forces of a wing due to the present of the other wing are less than 3% and changes in aerodynamic forces of the wings due to presence of the body are less than 2%. The reason for this is as following. During each down- or up-stroke, a wing produces a vortex ring, which induces a relatively large jet-like flow inside the ring but very small flow outside the ring. The vortex rings of the left and right wings are on the two sides of the body. Thus one wing is outside vortex ring of the other wing and the body is outside the vortex rings of the left and right wings, resulting in the weak interactions.     

A micromechanics-based strain gradient damage model for fracture prediction of brittle materials – Part II: Damage modeling and numerical simulations

In this paper, we established a strain-gradient damage model based on microcrack analysis for brittle materials. In order to construct a damage-evolution law including the strain-gradient effect, we proposed a resistance curve for microcrack growth before damage localization. By introducing this resistance curve into the strain-gradient constitutive law established in the first part of this work (Li, 2011), we obtained an energy potential that is capable to describe the evolution of damage during the loading. This damage model was furthermore implemented into a finite element code. By using this numerical tool, we carried out detailed numerical simulations on different specimens in order to assess the fracture process in brittle materials. The numerical results were compared with previous experimental results. From these studies, we can conclude that the strain gradient plays an important role in predicting fractures due to singular or non-singular stress concentrations and in assessing the size effect observed in experimental studies. Moreover, the self-regularization characteristic of the present damage model makes the numerical simulations insensitive to finite-element meshing. We believe that it can be utilized in fracture predictions for brittle or quasi-brittle materials in engineering applications.     

Properties of the Eulerian–Lagrangian method using linear interpolators in a three-dimensional shallow water model using z-level coordinates

We investigate the impact of the inexact interpolation on the Eulerian–Lagrangian solution of the advection equation by combining numerical experiments and formal analysis. The simulations, respectively, using the Eulerian–Lagrangian method (ELM) and the upwind scheme are compared. The artificial resistance of the ELM is observed which is characterised by the higher free-surface elevation and the distorted turbulent properties at a smaller time step. Through analysis, we find that the abnormalities are caused by the fact the conventional linear interpolation does not adapt well to the nonlinear velocity distribution, which produces an advection computation error that increases with a decreasing time step. The phenomena are explained and an improved method ELM is proposed based on the illustrations and analysis. The new method combines the face-controlled interpolation and the adjustable sub time steps to skip the large computation error domain in the backtracking, and it is validated by the original test case.     

Stochastic analysis of a predator–prey model with modified Leslie–Gower and Holling type II schemes

Nonlinear Dynamics - This article studies a predator–prey model with modified Leslie–Gower and Holling type II schemes under white-noise disturbances. The sensitivity analysis of the...     

Analytical study of the nonlinear behavior of a shape memory oscillator: Part II—resonance secondary

In this work, the response of a single-degree-of-freedom shape memory oscillator subjected to the excitation harmonic has been investigated. Equation of motion is formulated assuming a polynomial constitutive model to describe the restitution force of the oscillator. Here the method of multiple scales is used to obtain an approximate solution to the equations of the motion describing the modulation equations of amplitude and phase, and to investigate theoretically its stability. This work is presented in two parts. In Part I of this study we showed the modeling of the problem where the free vibration of the oscillator at low temperature is analyzed, where martensitic phase is stable. Part I also presents the investigation dynamics of the primary resonance of the pseudoelastic oscillator. Part II of the work is focused on the study in the secondary resonance of a pseudoelastic oscillator using the model developed in Part I. The analysis of the system in Part I as well as in Part II is accomplished numerically by means of phase portraits, Lyapunov exponents, power spectrum and Poincare maps. Frequency-response curves are constructed for shape memory oscillators for various excitation levels and detuning parameter. A rich class of solutions and bifurcations, including jump phenomena and saddle-node bifurcations, is found.