Volume-averaged Equations Research Articles

A new and general formulation of three-dimensional finite-volume micromechanics for particulate reinforced composites with viscoplastic phases

This paper presents a fully triply periodic version of finite-volume direct averaging micromechanics (FVDAM), enabling refined analysis of multiphase composites with arbitrary internal microstructures. Explicit three-dimensional FVDAM equations including plastic effects are derived. The surface-averaged continuity of displacements and tractions, and volume-averaged equilibrium equations are imposed based on the local/global stiffness approach. Further, the viscoplastic behavior for the composites can be modeled by incorporating state-variable-based Bodner-Partom constitutive model into the three-dimensional FVDAM framework. The presented model is used to simulate uniaxial tensile and strain cycling response of SiC/Al composites with various strain rates at room and high temperatures, respectively. Simultaneously, the microscopic stress and strain fields at different loading stages are also revealed in the simulations. The theoretical results agree well with the corresponding experimental data, indicating that the proposed model can predict the rate-dependent behavior of particulate reinforced composites reasonably.

Simulation of Crystal Sedimentation and Viscoplastic Behavior of Sedimented Equiaxed Mushy Zones

During solidification of castings, equiaxed crystals that is formed sink downwards, sediment and form a packed bed. The behavior of separated moving crystals can be described by a submerged object approach, whereas the viscoplastic behavior of a semi-solid slurry follows a volume-averaged viscoplastic constitutive equation. In this work, a two-phase Eulerian–Eulerian volume-averaging approach is used to combine both flow regimes. The transition happens at a certain solid volume fraction, the so-called coherency limit. Starting with a uniform distribution of crystals at rest, sedimentation and packing of crystals are described. In addition, the material density of the crystal is assumed to increase on cooling and thus the domain shrinks which is also accounted for in this report. It is demonstrated how sensitive the model is, on the considered crystal diameters and on the assumed value for the coherency limits.

A Global Enhanced Vibrational Kinetic Model for High-Pressure Hydrogen RF Discharges

A global enhanced vibrational kinetic model (GEVKM) is developed for multitemperature, chemically reacting hydrogen plasmas in inductively coupled cylindrical discharges for lowto high-pressure regimes. The species in a GEVKM are ground-state hydrogen atoms H and molecules H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , 14 vibrationally excited hydrogen molecules H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (v), v = 1 - 14, electronically excited hydrogen atoms H(2) and H(3), groundstate positive ions H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> , H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> , and H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> , ground-state negative ions H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-</sup> , and electrons e. The GEVKM involves volume-averaged steady-state continuity equations for the plasma species, an electron energy equation, a total energy equation, a heat transfer equation to the chamber walls, and a comprehensive set of surface and volumetric chemical processes governing vibrational and ionization kinetics of hydrogen plasmas. The GEVKM is verified and validated by comparisons with previous numerical simulations and experimental measurements of a negative hydrogen ion source in the low-pressure (20-100 mtorr), low-absorbed-power-density (0.053-0.32 W/cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) regime and of a microwave plasma reactor in the intermediate to high-pressure (1-100 torr), high-absorbed-power-density (8.26-22 W/cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) regime. The GEVKM is applied to the simulation of a high-current negative hydrogen ion source (HCNHIS). The HCNHIS consists of a high-pressure (20-65 torr) radio-frequency discharge chamber in which the main production of high-lying vibrational states of the hydrogen molecules occurs, a bypass system, and a low-pressure (0.1-0.4 torr) negative hydrogen ion production region where negative ions are generated by the dissociative attachment of low-energy electrons to rovibrationally excited hydrogen molecules. The discharge pressure and negative hydrogen ion current predicted by the GEVKM compare well with the measurements in the HCNHIS.

New wall drag and form loss models for one-dimensional dispersed two-phase flow

Abstract It had been disputed how to apply wall drag to the dispersed phase in the framework of the conventional two-fluid model for two-phase flows. Recently, Kim et al. [1] introduced the volume-averaged momentum equation based on the equation of a solid/fluid particle motion. They showed theoretically that for dispersed two-phase flows, the overall two-phase pressure drop by wall friction must be apportioned to each phase, in proportion to each phase fraction. In this study, the validity of the proposed wall drag model is demonstrated though one-dimensional (1D) simulations. In addition, it is shown that the existing form loss model incorrectly predicts the motion of the dispersed phase. A new form loss model is proposed to overcome that problem. The newly proposed form loss model is tested in the region covering the lower plenum and the core in a nuclear power plant. As a result, it is shown that the new models can correctly predict the relative velocity of the dispersed phase to the surrounding fluid velocity in the core with spacer grids.

Tsunami wave interaction with mangrove forests: A 3-D numerical approach

A three dimensional numerical approach based on IHFOAM to study the interaction of tsunami waves with mangrove forest is presented. As a first approximation, the problem is modelled by means of solitary waves impinging on emergent rigid cylinders. Two different conceptual approaches are implemented into IHFOAM. Solving the URANS equations provides a direct simulation of the flow field considering the actual geometry of the array of cylinders. A modified version of the volume-average URANS equations by introducing a drag force to model the momentum damping created by the cylinders is used in the second approach. Both the direct and macroscopic simulations are validated against laboratory experiments for wave damping with very high agreement. A large set of numerical experiments to analyse flow parameters and uniform and random cylinder array distributions are analysed and use to compare pros and cons of the different approaches. Large differences are found in the forces exerted on the vegetation for uniform and random distributions. Generalizations obtained from uniform arrangements could lead to underestimation of wave-exerted forces, especially for low dense configurations. Wave forces calculated with the macroscopic approach by means of the drag coefficient yields clear underestimations.

NUMERICAL MODELING OF THREE-DIMENSIONAL HEAT TRANSFER AND FLUID FLOWTHROUGH INTERRUPTED PLATES USING UNIT CELL SCALE

Interrupted-plate heat exchangers are used as regenerators for absorbing and releasing thermal energy such as in a Compressed Air Energy Storage (CAES) system in which the exchanger absorbs energy to cool the air being compressed or liberates energy to heat the air upon expansion. The exchanger consists of layers of thin plates in stacked arrays. In a given layer, the plates are parallel to one another and parallel to the exchanger axis. Each successive layer is rotated to have its plates be perpendicular to those of the layer below but still parallel to the exchanger axis. As flow passes from one layer to the next, new thermal boundary layers develop, beneficial to effective heat transfer. The interrupted-plate heat exchanger can also be seen as a porous medium. As such, it demonstrates strong anisotropic behavior when flow approaches the plates in a direction other than axially. Thus, pressure drops and heat transfer coefficients are dependent upon the attack angle. Mathematical models for anisotropic pressure drop and heat transfer behavior are proposed based on numerical calculations on a Representative Elementary Volume (REV), the unit cell model of the interrupted-plate medium. The anisotropic pressure drop is modeled by the traditionally used Darcy and inertial terms, with the addition of another term representing mixing effects. Heat transfer between the fluid and the plates is formulated in terms of Nusselt number vs. Reynolds number and approach angle of the mean flow. These models are used when solving, on the scale of the heat exchanger application, the volume-averaged Navier-Stokes equations that treat the exchanger region as a continuum. The analysis of the heat exchanger is used for design and optimization of the medium.

Volume-averaged macroscopic equation for fluid flow in moving porous media

Darcy’s law and the Brinkman equation are two main models used for creeping fluid flows inside moving permeable particles. For these two models, the time derivative and the nonlinear convective terms of fluid velocity are neglected in the momentum equation. In this paper, a new momentum equation including these two terms are rigorously derived from the pore-scale microscopic equations by the volume-averaging method. It is shown that Darcy’s law and the Brinkman equation can be reduced from the derived equation under creeping flow conditions. Using the lattice Boltzmann equation (LBE) method, the macroscopic equations are solved for the problem of a porous circular cylinder moving along the centerline of a channel. Galilean invariance of the equations are investigated both with the intrinsic phase averaged velocity and the phase averaged velocity. The results demonstrate that the commonly used phase averaged velocity cannot be considered, while the intrinsic phase averaged velocity should be chosen for porous particulate systems. In addition, the Poiseuille flow in a porous channel is simulated using the LBE method with the improved equations, and good agreements are obtained when compared with the finite-difference solutions.

An open-source toolbox for multiphase flow in porous media

Multiphase flow in porous media provides a wide range of applications: from the environmental understanding (aquifer, site-pollution) to industrial process improvements (oil production, waste management). Modeling of such flows involves specific volume-averaged equations and therefore specific computational fluid dynamics (CFD) tools. In this work, we develop a toolbox for modeling multiphase flow in porous media with OpenFOAM®, an open-source platform for CFD. The underlying idea of this approach is to provide an easily adaptable tool that can be used in further studies to test new mathematical models or numerical methods. The package provides the most common effective properties models of the literature (relative permeability, capillary pressure) and specific boundary conditions related to porous media flows. To validate this package, solvers based on the IMplicit Pressure Explicit Saturation (IMPES) method are developed in the toolbox. The numerical validation is performed by comparison with analytical solutions on academic cases. Then, a satisfactory parallel efficiency of the solver is shown on a more complex configuration. Program summaryProgram title: porousMultiphaseFoamCatalogue identifier: AEUR_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEUR_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public LicenseNo. of lines in distributed program, including test data, etc.: 36355No. of bytes in distributed program, including test data, etc.: 1106569Distribution format: tar.gzProgramming language: C++.Computer: Any x86.Operating system: Generic Linux.Classification: 12.External routines: OpenFOAM® (version 2.0 and superior)Nature of problem:This software solves multiphase flow in porous media.Solution method:The numerical approach is based on the finite-volume method (FVM). Mass conservation equations for each fluid phase are reformulated into a pressure–saturation system. The generalized Darcy’s law is used to compute phase velocities in the porous medium. The pressure–saturation is solved using a segregated method based on the IMPES method. Parallel computing can be performed using the standard domain decomposition method of OpenFOAM®.Running time:The Buckley–Leverett examples provided only take a few seconds to run.

On the Wall Drag Term in the Averaged Momentum Equation for Dispersed Flows

When fluid particles such as bubbles and droplets are not in contact with the wall, one probably neglects the wall drag term in the one-dimensional momentum equation for the dispersed phase. This treatment however leads to an unphysical prediction of the motion of the dispersed phase. In the framework of the conventional two-fluid model, how to apply the wall drag to the dispersed phase is disputable. The interface force acting on a fluid particle results from the interaction between the fluid particle and the surrounding continuous fluid. To clarify the contributions to the forces acting on the dispersed phase, the volume-averaged momentum equations are formulated based on the equation of a single fluid particle motion. After that, one-dimensional momentum equations are newly obtained from the averaged equations. It is shown that the wall drag term in the dispersed phase is associated with the spatial gradient of the volume-averaged viscous stress of the continuous phase. The magnitude of the wall drag term for a phase is its volume fraction multiplied by the total two-phase pressure drop induced by the wall shear of the continuous phase.

Lattice Boltzmann model for the volume-averaged Navier-Stokes equations

A numerical method, based on the discrete lattice Boltzmann equation, is presented for solving the volume-averaged Navier-Stokes equations. With a modified equilibrium distribution and an additional forcing term, the volume-averaged Navier-Stokes equations can be recovered from the lattice Boltzmann equation in the limit of small Mach number by the Chapman-Enskog analysis and Taylor expansion. Due to its advantages such as explicit solver and inherent parallelism, the method appears to be more competitive with traditional numerical techniques. Numerical simulations show that the proposed model can accurately reproduce both the linear and nonlinear drag effects of porosity in the fluid flow through porous media.

A low diffusive Lagrange-remap scheme for the simulation of violent air–water free-surface flows

In 2002, Després and Lagoutière [17] proposed a low-diffusive advection scheme for pure transport equation problems, which is particularly accurate for step-shaped solutions, and thus suited for interface tracking procedure by a color function. This has been extended by Kokh and Lagoutière [28] in the context of compressible multifluid flows using a five-equation model. In this paper, we explore a simplified variant approach for gas–liquid three-equation models. The Eulerian numerical scheme has two ingredients: a robust remapped Lagrange solver for the solution of the volume-averaged equations, and a low diffusive compressive scheme for the advection of the gas mass fraction. Numerical experiments show the performance of the computational approach on various flow reference problems: dam break, sloshing of a tank filled with water, water–water impact and finally a case of Rayleigh–Taylor instability. One of the advantages of the present interface capturing solver is its natural implementation on parallel processors or computers.

Fluid-Structure Interactions in a Tissue during Hyperthermia

A numerical investigation based on fluid-structure interaction analysis was conducted to determine the influence of pulsatile turbulent flow and heating protocol on temperature distribution and heat flux variation in blood vessel and tumor tissues receiving hyperthermia treatment. The arterial wall was modeled using the volume-averaged porous media equations. In addition, a physiological waveform was employed for the inlet velocity and exit pressure. Furthermore, the present numerical model was validated by comparing it with previously published results in the literature. Moreover, discretization of the transport equations was achieved using a finite element scheme based on the Galerkin method of weighted residuals. The results were presented for both flexible and rigid wall models and three different heating schemes were considered in this investigation; constant temperature, constant wall flux and a step-wise heat flux. The first two heating schemes were found to exhibit large temperature variation along the top and bottom surfaces of the tumor, which could influence the surrounding healthy tissues. The results of this investigation illustrate that the local heat flux variation along the bottom surface of the tumor is greater at the beginning of the cycle, where the velocity magnitude is low as compared to the peak flow condition for different heating schemes. The results of this work may enhance the current understanding of the factors that determine the effect of hyperthermia treatment on tumor tissues.

Novel fluid grid and voidage calculation techniques for a discrete element model of a 3D cylindrical fluidized bed

A discrete element model (DEM) combined with computational fluid dynamics (CFD) was developed to model particle and fluid behaviour in 3D cylindrical fluidized beds. Novel techniques were developed to (1) keep fluid cells, defined in cylindrical coordinates, at a constant volume in order to ensure the conditions for validity of the volume-averaged fluid equations were satisfied and (2) smoothly and accurately measure voidage in arbitrarily shaped fluid cells. The new technique for calculating voidage was more stable than traditional techniques, also examined in the paper, whilst remaining computationally-effective. The model was validated by quantitative comparison with experimental results from the magnetic resonance imaging of a fluidised bed analysed to give time-averaged particle velocities. Comparisons were also made between theoretical determinations of slug rise velocity in a tall bed. It was concluded that the DEM-CFD model is able to investigate aspects of the underlying physics of fluidisation not readily investigated by experiment.

VOLUME-AVERAGING ISSUES ILLUSTRATED FOR POROUS-MEDIA THERMO-FLUID TRANSPORT

This paper discusses the various issues causing errors in the formulation of volume-averaged equations for porous-media thermo-fluid transport. Proper implementation of the ground rules of porous-media volume averaging of the microscopic transport equations is demonstrated for formulation of the macroscopic mass, momentum, and energy equations. Because the ground rules of conventional (standard) porous-media volume averaging have been defined based on a constant representative elementary volume of bulk porous media, first the formulation of the porous-media macroscopic transport equations are carried out in terms of the bulk-volume averages of the various quantities and their divergence or gradient before converting the resulting superficial bulk-volume average quantities to intrinsic phase-volume average quantities. An important error concerning the bulk-volume averaging of fluid-volume averages is corrected and other errors involved in the formulation of the porous conservation equations are explained. The present implementation of the volume-averaging rules is shown to lead to consistent formulations. The properly formulated macroscopic equations are applied for a single-phase incompressible fluid that is saturating a rigid porous-solid material under thermal equilibrium conditions, both having constant physical properties. The improved formulation reveals the significant implications of the errors in a previous approach.

Comments on two analyses of thermal non-equilibrium Darcy–Brinkman convection in cylindrical porous media

The problem of forced convection heat transfer in porous media using the non-thermal equilibrium two-equation model is revisited analytically. Darcy-extended Brinkman momentum equation is used to describe the velocity profile inside the porous medium. The solution steps for the volume-averaged equations are outlined concisely to allow easy comparison to existing solutions in the literature. Using a case of air flow in open-cell metal foam, the fluid temperature is obtained by the current solution and by a solution existing in the literature. The pre-existing solution produced a physically-unexplained behavior. What seemed to be an error in the pre-existing solution was identified and described in detail. Robust solutions are useful in parametric studies, initial engineering design and for calibration of numerical solutions of more complex cases.

Analysis of the effect of stent emplacement on LDL transport within an artery

A comprehensive analytical solution for macromolecule transport within an artery is presented. The arterial wall is modeled as a multilayer structure to accommodate the homogeneous layers within it. The arterial layers and the stent are treated as porous media to accommodate transport across the complex wall structure. The volume-averaged transport equations and the method of matched asymptotic expansions are employed to solve for species concentration distributions within the stent and various layers of the arterial wall. The analytical results are found to be in very good agreement with the numerical results. The effect of stent compactness on LDL concentration along the lumen flow direction as well across the different layers is investigated. This work provides essential fundamental information for macromolecular transport within an artery with or without the presence of a stent.

Liquid drainage from high-expansion (HiEx) aqueous foams during and after filling of a container

An aqueous foam drains interstitial liquid as it fills a container after an induction period. We obtain solutions of quasi-steady and volume-averaged conservation equations containing the moving foam front to describe the induction period and the subsequent drainage, respectively. The quasi-steady theory predicts that the induction period increases with decreasing foam injection velocity. After the induction period, the theory shows a constant liquid drainage rate when the foam front propagates at a constant velocity. The space-averaged liquid fraction of the foam decreases with time and reaches a constant (pseudo-steady state) value because of the drainage. The theory shows that the liquid drainage is significant when the foam injection velocity is less than or equal to the terminal velocity of the liquid in a channel (Plateau border), which is shown to depend on foam parameters including initial liquid volume fraction and bubble diameter. At large injection velocity, the theory predicts induction time and drainage rate corresponding to free drainage from stationary foams. We also present experiments measuring liquid drained from dry foams (high-expansion foams), which are generated by forcing air through a liquid-covered metal screen at bench-scale. The experimental data show significant liquid drainage during filling in good agreement with the theoretical predictions for foams with liquid volume fraction less than 0.01. After the container is filled with the foam and the foam addition is stopped, both the theory and experiments show that the drainage rate decreases exponentially with time until equilibrium between capillarity and gravitational forces is reached and drainage ceases to occur. Thus, the theory shows striking differences in drainage behavior during and after filling in agreement with the experimental data.

Interplay among unstable modes in films over permeable walls

AbstractThe stability of open-channel flows (or film flows) has been extensively investigated for the case of impermeable smooth walls. In contrast, despite its relevance in many geophysical and industrial flows, the case that considers a permeable rather than an impermeable wall is almost unexplored. In the present work, a linear stability analysis of a film falling over a permeable and inclined wall is developed and discussed. The focus is on the mutual interaction between three modes of instability, namely, the well-known free-surface and hydrodynamic (i.e. shear) modes, which are commonly observed in open-channel flows over impermeable walls, plus a new one associated with the flow within the permeable wall (i.e. the porous mode). The flow in this porous region is modelled by the volume-averaged Navier–Stokes equations and, at the wall interface, the surface and subsurface flow are coupled through a stress-jump condition, which allows one to obtain a continuous velocity profile throughout the whole flow domain. The generalized eigenvalue problem is then solved via a novel spectral Galerkin method, and the whole spectrum of eigenvalues is presented and physically interpreted. The results show that, in order to perform an analysis with a full coupling between surface and subsurface flow, the convective terms in the volume-averaged equations have to be retained. In previous studies, this aspect has never been considered. For each kind of instability, the critical Reynolds number (${\mathit{Re}}_{c} $) is reported for a wide range of bed slopes ($\theta $) and permeabilities ($\sigma $). The results show that the free-surface mode follows the behaviour that was theoretically predicted by Benjamin and Yih for impermeable walls and is independent of wall permeability. In contrast, the shear mode shows a high dependence on $\sigma $: at $\sigma = 0$ the behaviour of ${\mathit{Re}}_{c} (\theta )$ recovers the well-known non-monotonic behaviour of the impermeable-wall case, with a minimum at $\theta \sim 0. 05\textdegree $. However, with an increase in wall permeability, ${\mathit{Re}}_{c} $ gradually decreases and eventually recovers a monotonic decreasing behaviour. At high values of $\sigma $, the porous mode of instability also occurs. A physical interpretation of the results is presented on the basis of the interplay between the free-surface-induced perturbation of pressure, the increment of straining due to shear with the increase in slope, and the shear stress condition at the free surface. Finally, the paper investigates the extent to which Squire’s theorem is applicable to the problem presented herein.

SIMULTANEOUS SURFACE AND SUBSURFACE AIR AND WATER FLOWS MODELLING IN THE SWASH ZONE

This study presents the results of the numerical simulation of a bore-driven swash flow over a permeable coarse- grained beach, carried out using the THETIS code. This code, based on a VOF-RANS approach, was previously used to simulate the swash flow over an impermeable beach (Desombre et al. 2013). For the present study, the code is extended to account for infiltration and exfiltration into a permeable immobile beach using the Volume-Averaged momentum equation that solves simultaneously the surface and subsurface flows. The results are compared with a laboratory data set from an experiment performed in the swash facility of the University of Aberdeen (Steenhauer et al. 2011). Comparisons between measurements and model results show the ability of the model to simulate the main features of subsurface flow during an entire swash cycle.

Three-dimensional modeling of polarization characteristics in molten carbonate fuel cells using peroxide and superoxide mechanisms

Polarization curves for the porous lithiated NiO cathode are very often reported with a linear slope over a wide potential range. However, the MCFC behaviour at higher oxidant utilization, when the mass transfer becomes dominant, is mostly overlooked. Therefore, in this study, the two most common cathode mechanisms are utilized to compare their prediction capabilities at higher current densities. This is performed by means of a three-dimensional, non-isothermal mathematical model which is developed by employing volume-averaged equations. As an extension to previous studies, the presented model also considers the potential and current density variation in both solid electrode and liquid electrolyte phases. In essence, this model is a set of partial differential equations including conservation of mass, momentum, gaseous species, energy, electronic potential and ionic potential that are solved using a finite volume method. In brief, both peroxide and superoxide mechanisms predict the linear region of the polarization curve accurately. However, none of these mechanisms showed a downward bent in the polarization curve. A positive exponent for the oxygen mole fraction is essential in obtaining the downward bent knee at high current densities which is in contrast to what has been reported in the literature to-date.