Square Array Of Circular Cylinders Research Articles

Conductivity gain predictions for multiscale fibrous composites with interfacial thermal barrier resistance

Nanocomposites are heterogeneous media with two or more micro‐structural levels. For instance, a nano‐level is characterized by isolated nano‐inclusions, and a micro‐level is represented by the clusters resulting from aggregation processes. Based on the reiterated homogenization method, we present a procedure to study the influence of this aggregation process and interfacial thermal resistance on the effective thermal conductivity for 2‐D square arrays of circular cylinders. First, an effective intermediate thermal property is obtained by taking into account only the influence of the individual nano‐inclusions in the matrix. Second, the final effective thermal coefficient ( ) is calculated considering the clusters immersed in the intermediate effective medium derived in the first step. The conductivity gain ( ) is defined as the quotient ( ) where is the effective thermal coefficient computed considering only one microstructural level with the same volumetric fraction of inclusions. We apply the theory of complex variable functions deriving in an infinite system of equations solvable by truncation method. The analytical formulas of the effective coefficient used in the calculations generalize other well‐known formulas reported in the literature. We also provide formulas for several truncation orders. The main novel contribution is in the reiterative application of the obtained formulas to illustrate the gain effect in nanocomposites, expressed as a function of the Biot number, thermal conductivities, volumetric fibers fraction, and an aggregation parameter. Furthermore, this result can be used to assess numerical computations and for nano‐reinforced fibers and nanofluids applications. An appendix is included showing similarities and differences with a three‐phase model.

Computational analysis of cross‐flow of power‐law fluids through a periodic square array of circular cylinders

AbstractThe steady flow of non‐Newtonian power‐law fluids across a periodic square array of infinitely long circular cylinders is studied numerically using an unstructured finite volume method. The local and global flow characteristics have extensively been explored by the systematic variations of the pertinent dimensionless parameters as follows: fluid volume fraction (ϕf = .70–.99), Reynolds number (Re = 1–40), and power‐law index (n = .4–1.8). Qualitatively, the dense and curved streamlines are seen with the increasing inertial forces and shear‐thinning behavior across all the fluid volume fractions. Further, the pressure coefficient over the surface of periodic cylinders is significantly influenced by the governing parameters and found to be maximum and minimum for the upstream and downstream cylinders, respectively. The individual and total drag coefficients have shown complex dependence on n, ϕf, and Re. For shear‐thinning fluids (n < 1), the pressure drag coefficient dominates over the friction drag coefficient, whereas an opposite response is seen for the shear‐thickening fluids (n > 1) except at Re = 40. Further, both individual and total drag coefficients are observed to increase and decrease in the ranges of ϕf of .70–.90 and .92–.99, respectively, with increasing n. The strong interactions between the periodic cylinders, at smaller ϕf, diminish with a corresponding increase in ϕf. In addition, simple predictive correlations for the pressure, friction, and total drag coefficients have been developed to gain further physical insights into the detailed flow kinematics. Finally, the present findings display a good agreement with the available literature.

Passive Boundary Layer Flow Control Using Porous Lamination

The flow over a porous laminated flat plate is investigated from a flow control perspective through experiments and computations. A square array of circular cylinders is used to model the porous lamination. We determine the velocities at the fluid–porous interface by solving the two-dimensional Navier–Stokes and the continuity equations using a staggered flow solver and using LDV in experiments. The control parameters for the porous region are porosity, $$\phi $$ and Reynolds number, Re, based on the diameter of the circular cylinders used to model the porous lamination. Computations are conducted for $$0.4< \phi < 0.9$$ and $$25< Re < 1000$$ , and the experiments are conducted for $$\phi = 0.65$$ and 0.8 at $$Re \approx 391,\ 497$$ and 803. The permeability of the porous lamination is observed to induce a slip velocity at the interface, effectively making it a slip wall. The slip velocity is seen to be increasing functions of $$\phi $$ and Re. For higher porosities at higher Re, the slip velocity shows non-uniform and unsteady behavior and a breakdown Reynolds number is defined based on this characteristic. A map demarcating the two regimes of flow is drawn from the computational and experimental data. We observe that the boundary layer over the porous lamination is thinner than the Blasius boundary layer and the shear stress is higher at locations over the porous lamination. We note that the porous lamination helps maintain a favorable pressure gradient at the interface which delays separation. The suitable range of porosities for effective passive separation control is deduced from the results.

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.

An approach to characterize dielectric properties of fiber-reinforced composites with high volume fraction

Fiber-reinforced composite materials are widely used in aeronautics and automotive industries due to their excellent mechanical properties. Composites with high conductive fibers embedded have good performance of shielding effectiveness and become possible candidates to replace metals. One approach for analyzing the electromagnetic (EM) interaction of fiber-reinforced composites is to use full numerical methods, which allow precise modeling and give accurate results. However, numerical methods may lead to prohibitive computational time and memory capacity due to the strong dependence on the shielding properties from heterogeneous microstructures.In composite materials, two important parameters, effective permittivity eff and effective permeability eff, determine the interaction between the electromagnetic field and the materials. For estimating the effective parameters, homogenization techniques have been developed to describe a composite mixture in terms of a spatially homogeneous electromagnetic response, mostly under static conditions. The well-known rules are the Maxwell-Garnett (MG) formula and the Bruggeman formula.These rules are usually applied to the dilute composite materials and provide satisfactory results as long as the wavelength remains large compared to the size of the heterogeneities. Recently, revised homogenization models have been developed to extend the frequency range. Some of them are presented with the help of numerical method but still require substantial computational time and resources to be performed. One recently proposed homogenization model, called dynamic homogenization model (DHM), is an extension of quasi-static homogenization methods for microwave frequencies. It is obtained by introducing a microstructure-dependent characteristic length for the composites made of a square array of circular cylinders buried in the matrix, based on the basic inclusion problems. The DHM overcomes the limitations of standard static homogenization tools, but only applicable to low fiber volume fraction (less than 20%).In this paper, we focus on the microstructure in the case of a square array of circular 2D conductive long fibers embedded in a dielectric matrix. A revised DHM is proposed to describe the effective permittivity of the composite materials with different inclusion concentrations, including higher fiber volume fraction. Firstly, an iterative procedure is employed to estimate an effective permittivity, which is then used to modify the wavelength in the DHM. Secondly, an empirical formula-based characteristic size of the microstructure is presented by considering the current distribution of the fibers under the EM wave illumination in the case of high fiber volume fractions. Therefore, the final modified homogenization model is given for the effective permittivity of composites with arbitrary inclusion concentrations. It can be used to efficiently calculate the reflection and transmission coefficients, as well as the shielding effectiveness by classical transmission-line methods. Three infinite sheets with different physical parameters are utilized for validation. We compare the results of the shielding effectiveness obtained from this homogenization model with those obtained from a full numerical solution of the actual fiber composites. Reasonable agreements obtained demonstrate that the proposed model could define the effective permittivity of the composites with high fiber concentration over a wide frequency range including microwave frequencies. Analogous formulas also hold for the magnetic permeability with permittivity replaced by permeability wherever it appears in the proposed model.

Gas slippage effect on the permeability of circular cylinders in a square array

The gas slippage phenomenon is usually observed when gas flow through porous media under low pressure, and has received an increasing attention for its importance in science and engineering. In the past decades, although many theoretical, numerical and experimental works have been done to predict the permeability of a square array of circular cylinders, little literature is available in studying the slippage effect on permeability. In this paper, the gas slippage effect on the permeability of circular cylinders in a square array is studied theoretically. The permeabilities at low and high solid fractions are derived with cell model and lubrication theory by coupling with a first-order slip boundary condition. The present results are finally validated through a comparison with some available results.

Gas Flow Through Square Arrays of Circular Cylinders with Klinkenberg Effect: A Lattice Boltzmann Study

It is well known that, as non-continuum gas flows throughmicroscale porous media, the gas permeability derived fromDarcy law is larger than the absolute permeability, which is caused by the so-called Klinkenberg effect or slippage effect. In this paper, an effective definition of Knudsen number for gas flows through square arrays of circular cylinders and a local boundary condition for non-continuum gas flows are first proposed, and then the multi-relaxation-time lattice Boltzmann equation including discrete effects on boundary condition is used to investigate Klinkenberg effect on gas flow through circular cylinders in square arrays. Numerical results show that the celebrated Klinkenberg equation is only correct for low Knudsen number, and secondorder correction to Klinkenberg equation is necessary with the increase of Knudsen number. Finally, the present numerical results are also compared to some available results, and in general an agreement between them is observed. PACS: 44.05.+e, 47.11.-j, 47.56.+r

On Darcy-Brinkman Equation: Viscous Flow Between Two Parallel Plates Packed with Regular Square Arrays of Cylinders

Effects of the bounding solid walls are examined numerically for slow flow overregular, square arrays of circular cylinders between two parallel plates. A local magnitudeof the rate of entropy generation is used effectively to determine the flow region affected bythe presence of the solid boundary. Computed axial pressure gradients are compared to thecorresponding solution based on the Darcy-Brinkman equation for porous media in whichthe effective viscosity appears as an additional property to be determined from the flowcharacteristics. Results indicate that, between two limits of the Darcian porous medium andthe viscous flow, the magnitude of μ (the ratio of the effective viscosity to the fluid ˆviscosity) needs to be close to unity in order to satisfy the non-slip boundary conditions atthe bounding walls. Although the study deals with a specific geometric pattern of the porousstructure, it suggests a restriction on the validity of the Darcy-Brinkman equation to modelhigh porosity porous media. The non-slip condition at the bounding solid walls may beaccounted for by introducing a thin porous layer with μ = 1 near the solid walls. ˆ

Cold compaction of an array of cylindrical fibres

The biaxial compaction of a square array of circular cylinders is studied using slip-line field, upper bound and finite element methods. Densification is assumed to occur by plastic deformation at the contacts. It is found that contact–contact interaction has a softening effect on the local indentation pressure at each contact. The yield surfaces resulting from hydrostatic and closed-die compaction are constructed at various stages of densification: the size and shape of the yield surface depend upon the loading history and upon the relative density of the compact. The finite element predictions suggest the formation of a vertex at the loading point for the entire densification process in the case of isostatic compaction. On the other hand, a vertex at the loading point is formed only for a relative density D⩽0.85 in the case of closed-die compaction.

A Nyström algorithm for electrostatics of an anisotropic composite

The construction of efficient numerical algorithms for the electrostatics of locally anisotropic composites seems to be a poorly explored area. In this paper coupled first and second kind Fredholm integral equations for the electrostatics of anisotropic inclusions in an anisotropic matrix are derived. For a square array of circular cylinders the integral equations are solved to high accuracy through pointwise discretization. In the process a recent renormalization method is extended for the evaluation of lattice sums [J. Math. Phys. 35, 6036 (1994)] to encompass the calculation of sums on stretched lattices. Conditionally convergent stretched lattice sums are computed from physical considerations.

Anisotropy in effective conductivities of rectangular arrays of elliptic cylinders

Effective conductivities of inhomogeneous systems with rectangular arrays of elliptic cylinders are accurately evaluated with a boundary collocation scheme. The investigation is conducted from 0 to the close packing in inclusion volume fractions f, and from 0 to ∞ in conductivity ratios of the inclusion to matrix σ. Results of the calculation agree very well with a reciprocal theorem developed by Keller [J. Appl. Phys. 34, 991 (1963); J. Math. Phys. 5, 548 (1964)]. The effect of different array arrangement on effective conductivities is studied by changing the aspect ratio of the inclusion ra, from 2 to 10 and the array edge length ratio re, from 1 to 10. The anisotropy in effective conductivities of the array is also examined. It is found that, with unit ra/re, the enhancement in conductivity in the direction parallel to the major axis of the inclusion increases appreciably when compared to that of the square array of circular cylinders of equal inclusion volume fractions, while the enhancement in the perpendicular direction decreases only slightly.

Longitudinal laminar flow in an array of circular cylinders

The paper describes a theoretical investigation of longitudinal, developed laminar flow in an infinite square array of circular cylinders, the array being bounded on one side by a wall, parallel to the cylinder axes. The influence of the array pitch ( 2b) and the distance of the wall from the first row of cylinders ( b 1) on the flow rate of the medium through the individual cells of the array is studied and the mutual influence of the flow rate in individual cells one upon the others is investigated. The quantity b has been varied in the range from 1.5 R to 2.5 R ( R being the cylinder radius) and the distance from the wall has been varied from b 1 = b (the first cell being square) to such values of b 1, where the ratio of hydraulic diameters of the first cell and a cell within the lattice ( ξ = d 1 ld i ) reaches a value of 1.3. From the calculation it follows that the mutual influence of the individual cells is relatively small. Variations of the flow rate in the first two rows of cells reach a value of 1 per cent only for relatively large pitches ( b > 2.5 R) and for cases, where the first cell differs greatly in the hydraulic diameter from the cells inside the array ( ξ > 1.3 and ξ < 0.6). The influence of the third row of cells is negligible in the range of parameters considered.

On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres

Spatially periodic fundamental solutions of the Stokes equations of motion for a viscous fluid past a periodic array of obstacles are obtained by use of Fourier series. It is made clear that the divergence of the lattice sums pointed out by Burgers may be rescued by taking into account the presence of the mean pressure gradient. As an application of these solutions the force acting on any one of the small spheres forming a periodic array is considered. Cases for three special types of cubic lattice are investigated in detail. It is found that the ratios of the values of this force to that given by the Stokes formula for an isolated sphere are larger than 1 and do not differ so much among these three types provided that the volume concentration of the spheres is the same and small. The method is also applied to the two-dimensional flow past a square array of circular cylinders, and the drag on one of the cylinders is found to agree with that calculated by the use of elliptic functions.