Cubic Array Of Spheres Research Articles

Gas-Solid Heat Transfer Computation from Particle-Resolved Direct Numerical Simulations

Particle-Resolved simulations (PR-DNS) have been conducted using a second order implicit Viscous Penalty Method (VPM) to study the heat transfer between a set of particles and an incompressible carrier fluid. A Lagrange extrapolation coupled to a Taylor interpolation of a high order is utilized to the accurate estimate of heat transfer coefficients on an isolated sphere, a fixed Faced-Centered Cubic array of spheres, and a random pack of spheres. The simulated heat transfer coefficients are compared with success to various existing Nusselt laws of the literature.

Permeability of packs of polydisperse hard spheres.

The permeability of packs of spheres is important in a wide range of physical scenarios. Here, we create numerically generated random periodic domains of spheres that are polydisperse in size and use lattice-Boltzmann simulations of fluid flow to determine the permeability of the pore phase interstitial to the spheres. We control the polydispersivity of the sphere size distribution and the porosity across the full range from high porosity to a close packing of spheres. We find that all results scale with a Stokes permeability adapted for polydisperse sphere sizes. We show that our determination of the permeability of random distributions of spheres is well approximated by models for cubic arrays of spheres at porosities greater than ∼0.38, without any fitting parameters. Below this value, the Kozeny-Carman relationship provides a good approximation for dense, closely packed sphere packs across all polydispersivity.

Preconditioned multiple-relaxation-time lattice Boltzmann equation model for incompressible flow in porous media

An improved preconditioned multiple-relaxation-time lattice Boltzmann equation model for incompressible flow (IPMRT-LBE) in porous media is proposed. Motivated by previous LBE models [Guo et al., Phys. Rev. E 70, 066706 (2004); Premnath et al., J. Comput. Phys. 228, 746 (2009); Guo et al., J. Comput. Phys. 165, 288 (2000)], the current model is demonstrated to have the advantages of accurate implementation of the no-slip boundary condition, reducing the compressible effect as well as fast convergence rate compared with standard LBE models. To validate the IPMRT-LBE model, flows in two- and three-dimensional synthetic porous media (square array of cylinders and body-centered cubic array of spheres) are simulated. The results show that the current model can predict the macroscopic property (such as permeability) accurately with significantly accelerated convergence rate. Furthermore, simulations of flow through a three-dimensional sandpack confirm the applicability and advantages of the IPMRT-LBE model.

Calculation of the effective thermal conductivity of multiscale ordered arrays based on reiterated homogenization theory and analytical formulae

In this paper the effective thermal conductivities of multiscale heterogeneous media with ordered microstructures are determined based on the reiterated homogenization method and analytical formulae available in the literature. While conventional homogenization has been extensively applied to thermal problems in two-scale media, reiterated homogenization appears to have been used, to date, mostly to formulate problems in heterogeneous media with more than two scales, rather than to calculate effective properties. Here, specifically, analytical formulae for the effective conductivities of the 2-D square array of circular cylinders and the 3-D simple cubic array of spheres are used, in conjunction with the appropriate reiterated homogenization expressions, to calculate the effective conductivities of the corresponding three-scale arrays of circular cylinders and spheres. The case with a perfect thermal contact at the interface is considered. The results for the effective thermal conductivity gain of each three-scale array relative to the two-scale counterpart are given in terms of the problem volume fractions and phase contrast. For each three-scale array the optimal volume fraction at the smallest structural scale that maximizes the conductivity gain is determined, as well as the optimal global volume fraction. In special, gains in excess of 9% may be achieved. The present approach thus allows for the systematic evaluation of conductivity gains solely on the basis of Fourier heat conduction and microstructural information.

A staggered overset grid method for resolved simulation of incompressible flow around moving spheres

An overset grid method for resolved simulation of incompressible (turbulent) flows around moving spherical particles is presented. The Navier–Stokes equations in spherical coordinates are solved on body-fitted spherical polar grids attached to the moving spheres. These grids are overset on a fixed Cartesian background grid, where the Navier–Stokes equations in Cartesian coordinates are solved. The standard second-order staggered finite difference scheme is used on each grid. The velocities and pressures on different grids are coupled by third-order Lagrange interpolations. The method, implemented in the form of a Message Passing Interface parallel program, has been validated for a range of flows around spheres. In a first validation section, the results of simulations of four Stokes flows around a single moving sphere are compared with classical analytical results. The first three cases are the flows due to a translating, an oscillating sphere and a rotating sphere. The numerically produced velocity and pressure fields appear to converge to the corresponding (transient) analytical solutions in the maximum norm. The fourth Stokes case is the flow due to an instantaneously accelerated sphere. For this case, the results are compared with the corresponding numerical solution of the Basset–Boussinesq–Oseen equation. In a second validation section, results of three Navier–Stokes flows around one or more moving spheres are presented. These test configurations are a moving face-centered cubic array of spheres, laminar channel flow with a falling a sphere, and freely moving small spheres in a Taylor–Green flow. Results for the flow with the falling sphere are compared with the results from the literature on immersed boundary methods.

Study of the internal mechanical response of an asphalt mixture by 3-D discrete element modeling

In this paper the viscoelastic behavior of asphalt mixture was investigated by employing a three-dimensional Discrete Element Method (DEM). The cylinder model was filled with cubic array of spheres with a specified radius, and was considered as a whole mixture with uniform contact properties for all the distinct elements. The dynamic modulus and phase angle from uniaxial complex modulus tests of the asphalt mixtures in the laboratory have been collected. A macro-scale Burger’s model was first established and the input parameters of Burger’s contact model were calibrated by fitting with the lab test data of the complex modulus of the asphalt mixture. The Burger’s contact model parameters are usually calibrated for each frequency. While in this research a constant set of Burger’s parameters has been calibrated and used for all the test frequencies, the calibration procedure and the reliability of which have been validated. The dynamic modulus of asphalt mixtures were predicted by conducting Discrete Element simulation under dynamic strain control loading. In order to reduce the calculation time, a method based on frequency–temperature superposition principle has been implemented. The ball density effect on the internal stress distribution of the asphalt mixture model has been studied when using this method. Furthermore, the internal stresses under dynamic loading have been studied. The agreement between the predicted and the laboratory test results of the complex modulus shows the reliability of DEM for capturing the viscoelastic properties of asphalt mixtures.

Permeability of fluid flow through a periodic array of cylinders

The three-dimensional model of an incompressible Newtonian viscous fluid slowly flowing through a periodic array of cylinders is considered. We use homogenization to determine a system of equations that are then solved numerically to calculate the permeability. We determine the permeability as a function of the angle the cylinders make with the bottom surface. Numerical results are obtained using a Taylor–Hood mixed finite element method. The numerical approach is validated by comparing the results with computed results of Rocha and Cruz, for a simple cubic array of spheres, with good agreement. Results are presented for different cylindrical densities and angles. When the flow aligns and is perpendicular to an array of cylinders, the results are validated with experimental data. The spherical part of the permeability is compared with the Kozeny–Carman equation. Applications of these results include modeling fluid flow through biological hairlike structures such as animal hair, glass rods, fiberglass, filter pads, polymer gel, collagen, nylon fibers and natural rice field or trees.

Three-Dimensional Numerical Analysis of Turbulent Flow in Porous Media Formed by Periodic Arrays of Cubic, Spherical, or Ellipsoidal Particles

In the present paper, three-dimensional (3D) turbulent flow in the porous media formed by periodic arrays of particles is numerically investigated. 3D Navier–Stokes equations and a standard k-ε turbulence model with enhanced wall function are adopted to model the turbulent flow inside the pores. Both local and macroscopic turbulence characteristics for different particle types (cubic, spherical, and ellipsoidal particles) and array forms [simple cubic (SC) and body center cubic arrays (BCC)] with different pore Reynolds numbers and porosities are carefully examined. It is revealed that, in the structural arrays of particles, the effects of particle shape and array form would be remarkable. With the same Reynolds number and porosity, the magnitudes of turbulence kinetic energy and its dissipation rate for the simple cubic array of spheres (SC-S) would be higher than those for the other arrays. Furthermore, with a nonlinear fitting method, the macroscopic correlations for extra turbulence quantities k∞ and ɛ∞ in the structural arrays for different particle types and array forms are extracted. The forms of present correlations can fit well with those of Nakayama and Kuwahara's correlations [Nakayama and Kuwahara, 1999, “A Macroscopic Turbulence Model for Flow in Porous Media,” ASME J. Fluids Eng., 121(2), pp. 427–433], but some model constants would be lower.

Experimental investigation on water flow in cubic arrays of spheres

One-dimensional uniform flow in homogeneous porous media was experimentally investigated. Head drop experiments were conducted in four test tubes with cubic arrays of spheres in diameter 3mm, 5mm, 8mm and 10mm. The experimental results indicate that Darcy’s law should be an approximate expression by neglecting the inertial term for flow at low velocity. Nonlinearity is attributed to inertial term in porous medium before the turbulent flow emerges. Forchheimer equation with constant coefficients can well predict the flow in porous medium. The relationship between the diameter of the particles and the coefficients a and b in the equations were verified. Different Ergun type equations were used to predict the head drop and compared to the experimental data. It shows that the Irmay equation could well predict the fluid flow in cubic arrays of spheres, while the prediction of head drop by Ergun equation was much higher than observed data. It indicates that the coefficients α and β in the Ergun type equations have certain relations with porosity or the pore structure and would vary for different medium. The discontinuity observed was interpreted by transition from steady flow to weakly turbulence and compared with previous studies.

Two methods for pore network of porous media

SUMMARYTwo approaches of generating pore networks of porous media are presented to capture the pore fabric. The first methodology extracted pore structure from a computer simulated packing of spheres. The modified Delaunay tessellation was used to describe the porous media, and modified Nelder–Mead method in conjunction with three pore‐merging algorithms was used to generate the pore size and coordination number distributions of the randomly packed spheres. The Biconical Abscissa Asymmetric CONcentric bond was used to describe the connection between two adjacent voids. This algorithm was validated by predicting pore structure of a cubic array of spheres of equal radius with known pore sizes, throat sizes and coordination number distributions. The predicted distributions of pore structure agreed well with the measured. Then, the algorithm was used to predict pore structure and permeability of randomly packed spherical particles, and predicted permeability values were compared with published experimental data. The results showed that the predicted permeability values were in good agreement with those measured, confirming the proposed algorithm can capture the main flow paths of packed beds. The second methodology generated an equivalent pore network of porous media, of which the centers of voids were located in a regular lattice with constant pore center distance. However, this network allowed for matching both main geometrical and topological characteristics of the porous media. A comparison of the two approaches suggested that the second approach can also be used as a predictive tool to quantitatively study the microscopic properties of flow through porous media. Copyright © 2012 John Wiley & Sons, Ltd.

Effective conductivity bounds by inserting adiabatic or isothermal surfaces

Structure-dependent bounds of the effective thermal conductivity ETC in polyphasic materials, with a regular inclusion array, are proposed. The bounds account for the contact resistance and are based on the enhancement of the ETC by insertion of any isothermal surface (for the upper bounds) and on the decrease of the ETC by insertion of any adiabatic surface (for the lower bounds). The method is above all able to introduce bounds for the high contrast filled composite polymers, whose inclusions are much more conductive than the polymer matrix. Computations for the ETC of a simple cubic array of spheres, which simulate well the heat conduction features in filled polymers, show that two such lower bounds and an upper one are easily computable. For all ETC higher than 3 or 4 times the matrix thermal conductivity, the bounds by insertion of isothermal or adiabatic surfaces are tighter than the other known structure-dependent bounds which account for the imperfection of the perfect contact – the 3rd order variational bounds of Torquato and Rintoul [S. Torquato, M. Rintoul, Effect of the interface on the properties of composite media, Phys. Rev. Lett. 75 (1995) 4067–4070].

Limitations and potentials of metamaterial lenses

The seminal discovery that an ideal negative-index lens may overcome Abbe's diffraction limit has raised enormous interest in the field of metamaterials and of subwavelength focusing. This finding is based on the anomalous wave propagation in ideally isotropic and homogeneous metamaterials with negative index of refraction and low loss, provided they are available. We have designed a metamaterial lens based on one of the simplest metamaterial geometries, a cubic array of spheres, with the aim of verifying its imaging properties in a practical configuration. After a rigorous homogenization, we have shown that, for suitable designs, the effective bulk parameters may indeed provide a quasi-isotropic negative-index response, ideal for imaging applications. We have then tested the imaging properties for finite-size lenses, analyzing challenges and potentials of going beyond the diffraction limit in a practical setup. We have also explored an alternative venue to exploit the negative-index property of the designed metamaterial in a concave lens, in order to resolve subwavelength features in the far-field. Our results indicate that, although subwavelength resolution and evanescent-wave amplification are possible in metamaterial arrays, practical imaging beyond the diffraction limit is challenging and a careful design should consider the granularity, degree of isotropy, and transverse size of the metamaterial lens.

The influence of confinement and curvature on the morphology of block copolymers

AbstractIn the bulk, at equilibrium, diblock copolymers microphase separated into nanoscopic morphologies ranging from body‐centered cubic arrays of spheres to hexagonally packed cylinders to alternating lamellae, depending on the volume fraction of the components. However, when the block copolymers are forced into cylindrical pores, where the diameter of the pores are only several repeat periods of the copolymer morphology or less, then commensurability of the copolymer period and the pore diameter can impose a frustration on the microdomain morphology. In addition, due to the small pore diameter, a curvature is forced on the microdomain morphology. In combination with interfacial interactions between the blocks of the copolymer and the pore walls, the preferential segregation of one component to the walls, spatial confinement and forced curvature are shown to induce transitions in the fundamental morphology of the copolymers seen in the bulk. Lamellar morphologies transformed into torus‐type morphologies, cylinders are forced into helices, and body‐centered cubic arrays of spheres are force into helical arrays of spheres due to these restraints. The novel morphologies, not accesssible in the bulk, open a large array of nanoscopic structures that can be used as templates and scaffolds for the fabrication of inorganic nanostructured materials. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 3377–3383, 2005

Viscosity of a Suspension with a Cubic Array of Spheres in a Shear Flow

Viscous fluid flow past an infinite periodic array of rigid spheres of the same radius is considered. A solution of the Stokes equations periodic in three variables is obtained for viscous incompressible flow with a linear velocity profile. The solution takes into account the hydrodynamic interaction of an infinite number of particles in the array. An expression for the effective viscosity of a suspension with a cubic array of particles is obtained.

Base to apex thermal conductance of a cone embedded in a non-ideal insulator

As a paradigm for the effective thermal conductivity of a densely packed dispersion of conducting filler particles in an insulating matrix, the effective conductance of a cubic array of quasi-bicones is determined analytically and its numerical properties investigated. Based upon the insights gained, the effective conductivities of simple cubic arrays of more general particles are predicted by three different approximate techniques, and the results compared with those known for simple cubic arrays of spheres.

Simple algebraic approximations for the effective elastic moduli of cubic arrays of spheres

Recently, Cohen and Bergman (Phys. Rev. B 68 (2003a) 24104) applied the method of elastostatic resonances to the three-dimensional problem of nonoverlapping spherical isotropic inclusions arranged in a cubic array in order to calculate the effective elastic moduli. The leading order in this systematic perturbation expansion, which is related to the Clausius–Mossotti approximation of electrostatics, was obtained in the form of simple algebraic expressions for the elastic moduli. Explicit expressions were derived for the case of a simple cubic array of spheres, and comparison was made with some accurate results. Here, we present explicit expressions for the effective elastic moduli of base-centered and face-centered cubic arrays as well, and make a comparison with other estimates and with accurate numerical results. The simple algebraic expressions provide accurate results at low volume fractions of the inclusions and are good estimates at moderate volume fractions even when the contrast is high.

Viscosity enhancement In non-Newtonian flow of dilute polymer solutions through crystallographic porous media

We report results of the flow of dilute mono-disperse solutions of atactic poly(styrene) in di-octyl phthalate through regular crystallographic packed beds of spheres. Pressure drop measurements made as a function of flow rate across simple cubic and body centred cubic arrays of spheres have been used to estimate the specific viscosities of the polymer solutions as a function of the superficial Deborah number. Through both structures the onset Deborah number for the non-Newtonian increase in specific viscosity is found to be low when compared on the basis of well-characterized Zimm relaxation times. Surprisingly it is found that polymer solutions achieve a greater maximum specific viscosity in the simple cubic than in the body centred cubic array, a result contrary to prior expectations due to the absence of trailing stagnation points in the simple cubic structure. It is hypothesised that the trailing stagnation points in the body centred cubic array may be screened from the flow field by strands of oriented polymer and that, as such, the periodic variations in cross-sectional area of the flow (which are more severe in the simple cubic array) may play the most significant role in causing polymer extension and in enhancing the non-Newtonian viscosity.

The transition from steady to weakly turbulent flow in a close-packed ordered array of spheres

The sequence of transitions in going from steady to unsteady chaotic flow in a close-packed face-centred cubic array of spheres is examined using lattice-Boltzmann simulations. The transition to unsteady flow occurs via a supercritical Hopf bifurcation in which only the streamwise component of the spatially averaged velocity fluctuates and certain reflectional symmetries are broken. At larger Reynolds numbers, the cross-stream components of the spatially averaged velocity fluctuate with frequencies that are incommensurate with those of the streamwise component. This transition is accompanied by the breaking of rotational symmetries that persisted through the Hopf bifurcation. The resulting trajectories in the spatially averaged velocity phase space are quasi-periodic. At larger Reynolds numbers, the fluctuations are chaotic, having continuous frequency spectra with no easily identified fundamental frequencies. Visualizations of the unsteady flows in various dynamic states show that vortices are produced in which the velocity and vorticity are closely aligned. With increasing Reynolds number, the geometrical structure of the flow changes from one that is dominated by extension and shear to one in which the streamlines are helical. A mechanism for the dynamics is proposed in which energy is transferred to smaller scales by the dynamic interaction of vortices sustained by the underlying time-averaged flow.

Effective conductivities of rectangular arrays of aligned spheroids

The effective conductivities (σeff) of rectangular arrays of aligned spheroids are accurately determined. The array structure is governed by the array aspect ratio (ra), inclusion aspect ratio (ri), and inclusion volume fraction ( f ). By varying these three parameters, a wide range of array structure can be generated and the effects of the three parameters on the effective conductivities of the array are investigated. The present array is found to be a more efficient structure in utilizing the enhancing or reducing effect of the inclusion, as compared to the simple cubic array of spheres. For cases of equal ra and ri, the effective parallel conductivity (σeff=) increases, while the effective perpendicular conductivity (σeff+) decreases with increasing aspect ratio, but both are only a weak function of the aspect ratio if the reduced conductivity of the inclusion (σ) is within the range of 0.1–10. At a fixed ri, σeff= decreases while σeff+ increases with increasing ra. If ra is fixed, σeff= increases while σeff+ decreases with increasing ri. For prolate systems (ri>1), σeff= is greater than σeff+ for all permissible f if ri⩾ra, but the magnitude order switches at higher f when ri<ra. As to oblate systems (ri<1), σeff= is less than σeff+ for all permissible f if ri⩽ra, but the magnitude order switches at higher f when ri>ra.

Self-Consistent Calculations of Block Copolymer Solution Phase Behavior

We present a theoretical study of the influence of solvent on ordered block copolymer solutions. The phase behavior is examined as a function of solvent selectivity, temperature, copolymer concentration, composition, and molecular weight. Phase maps are constructed using self-consistent mean-field (SCMF) theory, via the relative stability of the “classical” phases, lamellae (L), hexagonally packed cylinders (C), and a body-centered cubic array of spheres (S). Solvent selectivity and polymer concentration strongly influence phase transitions in copolymer solutions. When a neutral good solvent is added to a symmetric block copolymer, a direct (lyotropic) transition from L to disordered (D) is expected, analogous to the (thermotropic) L → D transition in melts. Indeed for neutral good solvents the dilution approximation is followed: the phase map is equivalent to that in the melt, once the interaction parameter is multiplied by the copolymer volume fraction. In contrast, for a symmetric block copolymer in t...