Numerical study of liquid/liquid slug flow in a capillary microreactor

The great advantage of microreactors is associated with an extremely large surface-to-volume ratio. Hence, microreactors permit promising operating conditions, such as almost-perfect heat or mass transfer. This, of course, requires that the hydrodynamics is well understood. The hydrodynamics of a liquid/liquid slug flow in a micro-capillary is characterized by a complex vortex structure in both the disperse and the continuous phase. The disperse phase, in our investigations, is not wetting the walls and, thus, a thin film of the continuous phase persist between the disperse phase and the wall. Due to this phenomenon, a relative movement between disperse and continuous phase is possible and, indeed, observed. Understanding of these complex phenomena allows for a control of the hydrodynamics, and thus, to tailor the heat and mass transport in a desired manner. Apparently, several effects influence the hydrodynamics. The main dimensionless groups are a Reynolds number, a Capillary number, and the ratio of viscosities and densities of both phases. To study the physics of this complex two-liquid system, a modified level-set method in conjunction with an immersed-boundary formulation is engaged. Presently, the simulations are time-dependent and axially-symmetric in nature. The mesh resolution represents a challenge, as the spatial resolution has to resolve the thin film between the disperse phase and the wall adequately. All simulations are implemented within the software OpenFOAM. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical investigation of blood flow. Part I: In microvessel bifurcations

In some diseases there is a focal pattern of velocity in regions of bifurcation, and thus the dynamics of bifurcation has been investigated in this work. A computational model of blood flow through branching geometries has been used to investigate the influence of bifurcation on blood flow distribution. The flow analysis applies the time-dependent, three-dimensional, incompressible Navier–Stokes equations for Newtonian fluids. The governing equations of mass and momentum conservation were solved to calculate the pressure and velocity fields. Movement of blood flow from an arteriole to a venule via a capillary has been simulated using the volume of fluid (VOF) method. The proposed simulation method would be a useful tool in understanding the hydrodynamics of blood flow where the interaction between the RBC deformation and blood flow movement is important. Discrete particle simulation has been used to simulate the blood flow in a bifurcation with solid and fluid particles. The fluid particle method allows for modeling the plasma as a particle ensemble, where each particle represents a collective unit of fluid, which is defined by its mass, moment of inertia, and translational and angular momenta. These kinds of simulations open a new way for modeling the dynamics of complex, viscoelastic fluids at the micro-scale, where both liquid and solid phases are treated with discrete particles.     

Two-phase flow modelling of spilling and plunging breaking waves

A two-phase flow model, which solves the flow in the air and water simultaneously, has been employed to investigate both spilling and plunging breakers in the surf zone with a focus during wave breaking. The model is based on the Reynolds-averaged Navier–Stokes equations with the k–?$k''–''?$ turbulence model. The governing equations are solved using the finite volume method, with the partial cell treatment being implemented in a staggered Cartesian grid to deal with complex geometries. The PISO algorithm is utilised for the pressure–velocity coupling and the air–water interface is modelled by the interface capturing method via a high-resolution volume of fluid scheme. Numerical results are compared with experimental measurements and other numerical studies in terms of water surface elevations, mean flow and turbulence intensity, in which satisfactory agreement is obtained. In addition, water surface profiles, velocity and vorticity fields during wave breaking are also presented and discussed. It is shown that the present model is capable of simulating the wave overturning, air entrainment and splash-up processes.     

Mathematical and physical simulation of the interaction between a gas jet and a liquid free surface

In this work a two phase 3D mathematical model was developed using the volume of fluid (VOF) algorithm, which is able to accurately describe the cavity geometry and size as well as the liquid flow patterns created when a gas jet that impinges on a liquid free surface. These phenomena are commonly found in steelmaking operations such as in the Electric Arc Furnace (EAF) and the Basic Oxygen Furnace (BOF) where oxygen jets impinge on a steel bath and they control heat, momentum and mass transfer. The model was successfully validated with measurements made on a physical model through velocity fields obtained by Particle Image Velocimetry (PIV) and high speed camera images of the cavity. Agreement between model predictions and experimental measurements is excellent in both x-velocity component of the liquid and cavity sizes. The cavity formed in the liquid by the impinging jet depends on a force balance at the free surface where the inertial force of the jet governs this phenomena, while the liquid circulation depends on also the jet inertial force of the jet, but its angle plays an important role, being the lowest angle the best choice to shear the bath and promote stronger circulation and better mixing in the liquid.     

Operator splitting for the numerical solution of free surface flow at low capillary numbers

Free surface flows are pervasive in engineering and biomedical applications. In many interesting cases—particularly when small length scales are involved—surface forces (capillarity) dominate the flow dynamics. In these cases, computing the flow together with the shape of the surfaces, requires specialized solution techniques. This article investigates the capabilities of an operator splitting/finite elements method at handling accurately incompressible viscous flow with free surfaces at low capillary numbers. The test case of flow in the downstream section of a slot coater is used for three reasons: (1) it is an established benchmark; (2) it represents an idealized, yet industrially relevant flow; (3) high-fidelity results obtained with monolithic algorithms are available in literature. The flow and free surface shape attained with the new operator splitting scheme agree very satisfactorily with the results obtained with monolithic solvers. Because of its inherent computational simplicity, the new operator splitting scheme is attractive for large-scale simulations, three-dimensional flows, and flows of complex fluids.     

A numerical model of the reacting multiphase flow in a pulp digester

This paper summarises the development, implementation and typical results from a new full-scale three-dimensional numerical model of the multiphase chemically reacting flow in a continuous digester. The model is based on continuity and momentum equations for wood chips and free liquid. A new sub-model describing the tangential stresses for the solid phase has been developed and incorporated into the digester model. The solution procedure that takes into account the high inter-phase friction terms and the high values of the solid pressure has been introduced. In order to illustrate the model’s performance, simulations have been carried on for the industrial digester. The results are in quantitative agreement with available field measurements and in qualitative agreement with operating personnel observations. The model utilises curvilinear body fitted coordinates and can be used to simulate the operation of various pulp digesters or any other chemical reactors working in a similar regime.     

Computational fluid dynamics modelling of gas jets impinging onto liquid pools

Gas jets impinging onto a gas–liquid interface of a liquid pool are studied using computational fluid dynamics modelling, which aims to obtain a better understanding of the behaviour of the gas jets used metallurgical engineering industry. The gas and liquid flows are modelled using the volume of fluid technique. The governing equations are formulated using the density and viscosity of the “gas–liquid mixture”, which are described in terms of the phase volume fraction. Reynolds averaging is applied to yield a set of Reynolds-averaged conservation equations for the mass and momentum, and the k–ε turbulence model. The deformation of the gas–liquid interface is modelled by the pressure jump across the interface via the Young–Laplace equation. The governing equations in the axisymmetric cylindrical coordinates are solved using the commercial CFD code, FLUENT. The computed results are compared with experimental and theoretical data reported in the literature. The CFD modelling allows the simultaneous evaluation of the gas flow field, the free liquid surface and the bulk liquid flow, and provides useful insight to the highly complex, and industrially significant flows in the jetting system.     

Integrated modelling and numerical treatment of freeway flow

The effectiveness of macroscopic dynamic freeway flow models at both uninterrupted and interrupted flow conditions is tested. Model implementation is made by finite difference methods developed here for solving the system's governing equations. These schemes are more effective than existing numerical methods, particularly when generation terms are introduced. The modelling alternatives and numerical solution algorithms are compared by employing a data base generated through microscopic simulation. Despite the effectiveness of the proposed numerical treatments, substantial deviations from the data at interrupted flows are still noticeable. In order to improve performance when flow is interrupted, we develop a modelling methodology that takes into account the ramp-freeway interactions so that all freeway components are treated as a system. We show that the coupling effects of the merging traffic streams are significant. Finally, the incremental benefits of using the more sophisticated high-order continuum models are assessed.     

Three dimensional modelling of lava flow using Smoothed Particle Hydrodynamics

The paper describes the use of the grid-free Smoothed Particle Hydrodynamics (SPH) method to investigate lava flow from volcanic eruptions using real three dimensional topography in the form of Digital Terrain Models (DTM). Heat transfer resulting from conduction and radiation and solidification of the lava modelled via a variable viscosity are coupled to the fluid flow solution. Simulations show that the run-out distance and the nature of the lava flow are affected significantly by the lava viscosity and that this is dependent on the scale of the volcanic eruption, with solidification effects strongest on the smallest scale. SPH appears to be a highly effective technique for predicting lava flow with very good representations of the fluid free surface, close interaction with the complex topography, easy inclusion of the thermal and solidification effects leading to very plausible flow predictions. The pile-up of the lava at the front as it solidifies and the subsequent deceleration of the flow are easily modelled by SPH.     

The numerical modelling of DC electromagnetic pump and brake flow

The interaction of an externally imposed magnetic and electric field on the laminar flow of a conducting fluid in a channel is studied using computational techniques. The Navier-Stokes equations and the equations describing the electromagnetic field are solved simultaneously in a single control volume-type computational fluid dynamic code, in a moderate Hartmann number and interaction parameter regime. The flow considered is two-dimensional, with an imposed magnetic field acting in the third dimension over the central region of the channel and decaying exponentially in the remainder. A pair of electrodes placed at right angles to the magnetic field exercises control over the resultant Lorentz force and hence the velocity profile shape. This configuration has application in direct-current electromagnetic pumps or, conversely, electromagnetic brakes. The initial parabolic flow profile acquires an M-shape / W-shape mode in the magnetic field fringe regions, corresponding to a pump / brake. A novel coupled procedure is described to model magnetohydrodynamic phenomena and is used to explore the effects of the Reynolds number, interaction parameter, and applied voltage on the pump / brake configuration.     

Computational treatment of free convection effect on flow of elastico-viscous fluid past an accelerated plate with constant heat flux

Effect of free convection on the visco-elastic fluid (Walter - B’ type) flow past an infinite vertical plate accelerating in its own plane with constant heat flux is examined analytically. It is found that for given values of Grashof number, Prandtl number and Newtonian parameter; flow velocity at any point increases with the increase in time and non-Newtonian parameter, however, it decreases with both, the heating and cooling of the plate.     

Unsteady flow of a Maxwell fluid over a stretching surface in presence of chemical reaction

An analysis is presented for unsteady two-dimensional flow of a Maxwell fluid over a stretching surface in presence of a first order constructive/destructive chemical reaction. Using suitable transformations, the governing partial differential equations are converted to ordinary one and are then solved numerically by shooting method. The flow fields and mass transfer are significantly influenced by the governing parameters. Fluid velocity initially decreases with increasing unsteadiness parameter and concentration decreases significantly due to unsteadiness. The effect of increasing values of the Maxwell parameter is to suppress the velocity field. But the concentration is enhanced with increasing Maxwell parameter.     

Reduced order modelling for unsteady fluid flow using proper orthogonal decomposition and radial basis functions

A technique is presented for interpolating unsteady solutions to parameterised fluid flow problems, using a combination of proper orthogonal decomposition and radial basis functions. The technique is validated by considering simulations involving three dimensional unsteady compressible inviscid flow over an oscillating ONERA M6 wing. It is demonstrated that the approach can result in a large reduction in the cpu time required to find solutions at new parameter values, without a significant loss in accuracy.     

Computational fluid dynamics simulation of fluid particle fragmentation in turbulent flows

A simulation methodology is presented that allows detailed studies of the breakup mechanism of fluid particles in turbulent flows. The simulations, based on large eddy and volume of fluid simulations, agree very well with high-speed measurements of the breakup dynamics with respect to deformation time and length scales, and also the resulting size of the daughter fragments. The simulations reveal the size of the turbulent vortices that contribute to the breakup and how fast the interaction and energy transfer occurs. It is concluded that the axis of the deformed particle and the vortex core axis are aligned perpendicular to each other, and that breakup sometimes occurs due to interaction with two vortices at the same time. Analysis of the energy transfer from the continuous phase turbulence to the fluid particles reveals that the deformed particle attains it maximum in interfacial energy before the breakup is finalized. Similar to transition state theory in chemistry this implies that an activation barrier exists. Consequently, by considering the dynamics of the phenomenon, more energy than required at the final stage needs to be transferred from the turbulent vortices for breakup to occur. This knowledge helps developing new, more physical sound models for the breakup phenomenon required to solve scale separation problems in computational fluid dynamics simulations.     

Mathematical modelling of unsteady flow of a Sisko fluid through an anisotropically tapered elastic arteries with time-variant overlapping stenosis

In the present investigation, we have studied the influence of heat and chemical reactions on blood flow through anisotropically tapered elastic artery with time-variant overlapping stenosis. The nature of blood in small arteries are analyzed mathematically by considering it as a Sisko fluid. The analysis is carried out for an artery with a mild stenosis. Analytical expressions for the axial velocity, the stream function, the temperature distribution, the concentration of fluid, the pressure gradient, the resistance impedance and the wall shear stress distribution have been computed numerically and the results were studied for various values of the physical parameters, such as the Sisko parameter b^∗, the power index n, the taper angle ?  , the maximum height of stenosis δ^∗${\delta }^{\ast }$, the Soret number S_r, the Brickmann number B_r, the total mass of the vessel and the surrounding tissues M and the longitudinal contributions of the viscous and elastic constraints to the total tethering C and K respectively. The obtained results show that the magnitude of the axial velocity is higher for a Newtonian fluid than that for a Sisko fluid and it decreases by increasing of the power index n also the transmission of axial velocity curves through a tethered tube is substantially higher than that through the free tube. The wall shear stress distribution and resistance impedance profiles with the time have an oscillation form through the tapered overlapping stenosed arteries and this oscillation decaying as the time increases. The temperature profile increase by increasing the Sisko parameter b^∗ and the power index n   but the concentration profile has an opposite behavior as compared to the temperature profile. For a fixed flux, the magnitude of the pressure drop for a shear-thinning fluid (n<1)$(n<1)$ is much larger than that through a shear-thickening (n>1)$(n>1)$. The stream lines separate and the trapping bolus appear by increasing the maximum height of the stenosis δ^∗${\delta }^{\ast }$. The trapping bolus increase in size toward the line center of the tube as the power index n increases and it appear gradually by increasing the Sisko parameter b^∗. Finally the size of trapped bolus for the stream lines in the free isotropic tube is smaller than those in the tethered tube.     

Helical flow arising from the yielded annular flow of a Bingham fluid

The Bingham fluid model was developed to represent viscoplastic materials that change from rigid bodies at low stress to viscous fluids at high stress – a process termed yielding. Such a fluid model is used in the modeling of slurries, which occur frequently in food processing and other engineering applications.     

低速非达西层状油藏DST段塞流试井模型研究

在分析了实际低速非达西层状油藏DST测试非自喷井渗流规律的基础上,导出了其有效井径数学模型．利用Laplace变换及特殊物理方程理论,导出了该模型的拉氏空间解析解,并利用简便的Stehfest数值算法进行数值反演,得出了实空间解,并绘制了无因次样板曲线．可有效的运用于现场DST段塞流试井分析．     

Particle transport in a bottom-feed separation vessel

A two-dimensional, axisymmetric numerical model of particle separation in a bottom-feed separation vessel is presented. The model includes six separate particle classes and assumes that the settling velocity of each particle class is sufficiently small when compared to the high inflow turbulence levels that the effect of the particles on turbulence can be neglected. Low particle settling velocities coupled with low particle volume fractions allows application of a drift-flux multi-phase model. The comparison between numerical results and measured plant data is in good agreement for overflow of all particle classes. Results of simulations show that bottom feeding results in a negatively buoyant, particle-laden jet being formed in the core of the vessel. The fraction of large particles that is carried out through the overflow is found to be critically dependent on the inlet velocity. The most effective way to reduce carry-over of large particles at the same time as maintaining through-put is to increase the diameter of the inlet feed pipe.     

Emergence of tables as first-graders cope with modelling tasks

In this action research, first-graders were challenged to cope with a sequence of modelling tasks involving an analysis of given situations and choices of mathematical tools. In the course of the sequence, they underwent a change in the nature of their problem-solving processes and developed modelling competencies. Moreover, during the task sequence, the first-graders spontaneously discovered the power of organizing problem data in a table. They did not merely use their existing mathematical knowledge, but also ‘reinvented’ tables as a new mathematical tool. This paper describes the gradual development of this tool as the children moved along the task sequence. Notably, the first-graders exhibited this progress in spite of having relatively little mathematical knowledge.