On the Description of Partially Saturated Soils by the Theory of Porous Media

In this contribution, a multi‐phase soil model based on the Theory of Porous Media (TPM) is presented. The model is fully coupled in the following constitutive phases: An elasto‐plastic or elasto‐viscoplastic solid skeleton, a materially incompressible pore‐liquid (water) and a materially compressible pore‐gas (air). The interaction of the solid skeleton and the pore‐fluids is specified by a capillary pressure‐saturation relation, whereas the mobilities of the fluid phases in the pore‐space of the solid skeleton are described by the so‐called relative permeabilities. Finally, a gravity governed initial‐boundary‐value problem solved by the FE method is presented.     

The relative permeabilities of an idealized model of two-phase flow in porous media

A simplified model is presented for two-phase flow in a porous medium consisting of circular capillary tubes. The equation of motion (Navier-Stokes equation) is solved for the system giving the velocity distributions for the wetting and nonwetting phases. The velocity distributions are used to derive expressions for the relative permeabilities to the wetting and nonwetting phases. It was found that the relative permeability to the nonwetting phase at a given saturation is a function of the viscosity ratio of the two phases while the relative permeability to the wetting phase is independent of the viscosity ratio. The validity of these findings for more complicated porous systems is discussed.     

Dispersion of solid particles in turbulant flow through pipe expansions

Studies of the effects upon gas-liquid two-phase flows of pipefittings such as expansions, contractions, bends, and valveshave usually concentrated upon pressuredrop correlations andhave not attempted to determine changes in the distributionsof the gas and liquid phases caused by the fitting. However,it is known that such information is important if, for example,flow separators, which divide the gas and liquid phases in avariety of industrial processes, are to function efficiently.It is therefore important to gain an understanding of the influenceupon phase distributions of the common pipe fittings mentionedabove, which will be found in almost any industrial pipeworksystem. As a first step, the dispersion of solid particles carriedby turbulent gas flows through a pipe expansion has been modellednumerically. The commercial fluid-flow code CFDS-FLOW{smalltilde}hDas been used to model the gas flow, together with aneddy interaction model for determination of the motion of thesolid particles. Mean particle velocities and root-mean-squarevalues of the particle velocity fluctuations, as well as particleconcentrations, are evaluated and compared with recent experimentalresults. The influence of different eddy-length and eddy-lifetimespecifications upon the dispersion of particles of various sizesis investigated. It is found that the different eddy characteristicshave little effect on predicted mean particle velocities, whereasfluctuations in particle velocities and particle concentrationare sensitive to the changes made. By comparing the resultswith experimental data, it is possible to draw conclusions aboutthe relative merits of the different eddy specifications.     

壁面效应对剪切稀化流体内气泡上浮特性的影响

数值研究了壁面效应对剪切稀化流体内气泡上浮运动特性的影响,气液两相的界面捕捉采用流体体积(VOF)法,剪切稀化流体流变特性和气液相间表面张力的计算分别采用Carreau模型和连续表面张力模型.详细研究了不同流变指数下,壁面效应对气泡形状、液相流场和气泡终端速度的影响.结果表明,强的壁面效应或弱的剪切稀化程度会限制气泡的变形和尾涡的形成,使气泡的终端速度减小;气泡终端速度最易受壁面效应的影响;强的壁面效应和强的剪切稀化程度会导致高剪切速率区域出现在壁面附近,引起壁面附近液相表观黏度大幅度的下降.     

热力学理论与空化现象

有关空化现象的理论研究大多建立在1917年由Rayleigh[1]开始的,后由Plesset等人发展起来的单个空泡运动理论的基础上.该理论仅从流体动力学的某些观点出发,考虑了力的作用,对诸如水下爆炸等问题的讨论,无疑是合适的.由于忽视了空泡生长或消失过程中气、液两相间的物质交换及能量转化,因此对空化现象的讨论则认为是不完备的. 本文主要从热力学观点.分析高速水流中的空化现象、空泡形成条件、空化数以及讨论空化模型实验的相似性问题.     

Dynamics of non-equilibrium phase boundaries in a heat conducting non-linearly elastic medium

The phase transformation of the first kind in a non-linearly elastic heat conducting medium is simulated by the relationships on a strong discontinuity. A generalization of the Stefan formulation is given. An existence condition for stationary flow, analogous to the Gibbs phase equilibrium condition, is obtained for non-equilibrium phase boundaries. A pure dilatational phase transition in a compressible fluid and pure shear transformation of the twinning type in non-linearly elastic crystals are considered as model examples. The problem of the structure is solved for closure of the system of relationships on the shock.

A phase transformation ordinarily turns out to be localized in a narrow domain of space and it can be simulated in terms of the conditions on a strong discontinuity /1/. Formulation of the problem of the static equilibrium of liquid phases as well as of liquid and (non-linearly elastic) solid phases was given by Gibbs, who proposed a phase equilibrium criterion and formulated appropriate conditions on the shock; the extension of the Gibbs conditions to the case of the equilibrium of two solid phases is known in both the linear /2/ and non-linear /3/ theories of elasticity. The dynamic problem of the propagation of the equilibrium phase boundary is considered in the Stefan formulation as a rule, including the assumption about the continuity of the density (the strain tensor component) on the shock; the thermal problem is here separated from the mechanical one. Simulating the interphasal surface on the shock the temperature fields are merged by using the well-known Stefan conditions as well as the phase equilibrium condition that reduces to giving the temperature on the front.

The purpose of this paper is to extend the Stefan-Gibbs formulation to the case of the motion of a coherent isothermal phase boundary in a non-linearly elastic heat conducting medium and to derive the dynamic analogue of the phase equilibrium condition (and the Stefan conditions) with possible dissipation at the transformation front. Two dissipative mechanisms are examined, viscous and kinetic. The case of equilibrium phase boundaries was investigated in /4–6/.     

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)     

Shock-Induced Phase Transitions in Systems of Hard Spheres with Attractive Interactions

Shock-induced phase transitions are studied by adopting the recently-developed theoretical framework, which is applicable for shock waves in three phases (gas, liquid, and solid), based on the system of hard spheres with mutual attractive interactions. The Rankine-Hugoniot conditions derived from the system of Euler equations with caloric and thermal equations of state are studied, and the admissibility (stability) of a shock wave is analyzed. Two typical scenarios of the shock-induced phase transitions from gas phase to solid phase are found. A scenario of shock-induced phase transitions involving three phases simultaneously near the triple point is also found.     

The two-phase,three-zone melting problem—response to pulse heating

Under certain conditions, when a pure substance in solid form is partly melted by in-depth heating, three distincs zones appear: the solid, the liquid, and the intermediate zone containing both liquid and solid. Under these conditions, the equations for the free boundaries are uncoupled and may be solved independently until the boundaries coalence into a single (Stefan) interface between the liquid and solid phases. In this paper, the one dimensional equations for the three-zone problem resulting from instantaneous pulse in-depth heating are derived. These equations, together with the remaining describing equations, are solved by two different numerical techniques to illustrate the motion of the boundaries, both prior to coalescence (the uncoupled case), and following coalescence (the coupled, or Stefan, case).     

Computational modelling of bubbles,droplets and particles in metals reduction and refining

A multi-phase framework is typically required for the CFD modelling of metals reduction processes. Such processes typically involve the interaction of liquid metals, a gas (often air) top space, liquid droplets in the top space and injection of both solid particles and gaseous bubbles into the bath. The exchange of mass, momentum and energy between the phases is fundamental to these processes. Multi-phase algorithms are complex and can be unreliable in terms of either or both convergence behaviour or in the extent to which the physics is captured. In this contribution, we discuss these multi-phase flow issues and describe an example of each of the main “single phase” approaches to modelling this class of problems (i.e., Eulerian–Lagrangian and Eulerian–Eulerian). Their utility is illustrated in the context of two problems – one involving the injection of sparging gases into a steel continuous slab caster and the other based on the development of a novel process for aluminium electrolysis. In the steel caster, the coupling of the Lagrangian tracking of the gas phase with the continuum enables the simulation of the transient motion of the metal–flux interface. The model of the electrolysis process employs a novel method for the calculation of slip velocities of oxygen bubbles, resulting from the dissolution of alumina, which allows the efficiency of the process to be predicted.     

Characterization of the least concave majorant of brownian motion,conditional on a vertex point,with application to construction

The characterization of the least concave majorant of brownian motion by Pitman (1983,Seminar on Stochastic Processes, 1982 (eds. E. Cinlar, K. L. Chung and R. K. Getoor), 219–228, Birkhäuser, Boston) is tweaked, conditional on a vertex point. The joint distribution of this vertex point is derived and is shown to be generated with extreme ease. A procedure is then outlined by which one can construct the least concave majorant of a standard Brownian motion path over any finite, closed subinterval of (0, ∞). This construction is exact in distribution. One can also construct a linearly interpolated version of the Brownian motion path (i.e. we construct the Brownian motion path over a grid of points and linearly interpolate) corresponding to this least concave majorant over the same finite interval. A discussion of how to translate the aforementioned construction to the least concave majorant of a Brownian bridge is also presented.     

Chiral Bag Model with Constituent Quarks: Topological and Nontopological Solutions

We consider a three-phase modification of the hybrid chiral bag model involving the intermediate constituent quark phase along with the asymptotic freedom and hadronization phases. We find self-consistent solutions of the equations of the model in 1+1 dimensions with the fermion vacuum polarization effects taken into account. We study the renormalized total energy of the bag as a function of parameters characterizing the geometry of the bag and its topological (baryon) charge. We show that for a nonzero topological charge, there exists an entire series of configurations that are local minimums of the total energy of the bag and contain all the three phases, whereas in the nontopological case, the bag energy minimum corresponds to zero sizes of the domain of the asymptotic freedom phase.     

Well-posedness of the free-surface incompressible Euler equations with or without surface tension

We provide a new method for treating free boundary problems in perfect fluids, and prove local-in-time well-posedness in Sobolev spaces for the free-surface incompressible 3D Euler equations with or without surface tension for arbitrary initial data, and without any irrotationality assumption on the fluid. This is a free boundary problem for the motion of an incompressible perfect liquid in vacuum, wherein the motion of the fluid interacts with the motion of the free-surface at highest-order.

     

Simulation of mass transfer at free liquid/liquid interfaces

A numerical scheme for the simulation of mass transfer processes at free liquid/liquid interfaces using the interface tracking method is presented. Due to comparable diffusion coefficients in liquid/liquid systems, the mass transfer resistance in both phases is relevant for the entire transient mass transfer process. Exemplary, the extraction process from a free rising spherical droplet of constant shape is used. The presented approach can be used in general for any multiphase steady-state mass transfer system. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical Analysis of the Interphase Interaction in a Porous Elastic Fluid-Saturated Medium

We consider dynamical processes in a two-phase porous fluid-saturated medium. The equation of the Biot–Frenkel model, which accounts for the influence of the elastic, inertial, and viscous interaction between the liquid and solid phases, is used for modeling the dynamics of the soil layer (flat deformation) with the finite-element method in cases of steady and nonstationary effects. For a layer under a uniaxial stress, we give numerical examples of the displacements of the soil skeleton and interstitial fluid.     

A three-phase FE-model for the simulation of partially saturated soils

While for many numerical simulations in geotechnics use of a two-phase model is sufficient, separate consideration of all three phases is mandatory for numerical simulations of partially saturated soils subjected to compressed air. This is a common technique frequently applied for temporary ground support in tunnelling. For the numerical simulation of tunnelling using compressed air, a multiphase model for soft soils is developed, in which the individual constituents of the soil – the soil skeleton, the fluid and the gaseous phase – and their interactions are considered. The three phase model is formulated within the framework of the Theory of Porous Media (TPM), based upon balance equations and constitutive relations for the soil constituents and their mixture. Water is modelled as an incompressible and air as a compressible phase. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Transport of fluids in capillary porous solids

Capillarity is a well known phenomenon in physics and engineering. The term denotes the transport of liquids and gas, against the force of gravity in narrow tubes, cracks and pores, caused by the intermolecular forces of cohesion and adhesion of the constituents involved. We will exclusively investigate the rise of liquids in porous bodies due to the capillarity phenomenon. Well‐known examples are the capillary rise in drying processes in soils with small pores and the moisture motion in walls consisting of bricks or concrete. The treatment of the capillary problem, based on thermomechanical investigations, yields the result that the capillarity force is a volume interaction force and depends on the free Helmholz energy functions of the solid phases and the density gradient of the liquid. The aim of this investigation is the numerical simulation of the behavior of liquid and gas phases in a rigid porous body at rest. An example will demonstrate the usefulness of the proposed theory.     

A multiple criteria ranking procedure based on distance between partial preorders

In this paper, a multiple criteria ranking procedure based on distance between partial preorders is proposed. This method consists of two phases. In the first phase, the decision maker is asked to rank alternatives with a preorder (complete or partial) for each criterion and provide complete or partial linear information about the relative importance (weights) of the criteria. In the second phase, we introduce a distance procedure to aggregate the above individual rankings into a global ranking (a partial preorder). An algorithm for the aggregation procedure is proposed, followed by a numerical illustration.