Periodic Array Of Circular Cylinders Research Articles

Thermal Features of mixed convection from an inline periodic array of circular cylinders in non-Newtonian power-law fluids

Mixed convection heat transport from multi-cylinders finds numerous applications in chemical, petroleum, and food industries. This study presents the mixed convection characteristics of an inline array of heated circular cylinders in non-Newtonian power-law fluids for the following governing parameters: fluid volume fractions; 0.70≤ φ f ≤0.99, power-law index; 0.4≤ n ≤ 1.8, Reynolds number; 1≤ Re D ≤40, Prandtl number; 1≤ Pr D ≤50 and Richardson number; 0≤ Ri D ≤2. Mathematical model equations are solved using the Finite Volume Method within the framework of Boussinesq approximations. Thermal features are explored using isotherms, local and averaged Nusselt numbers. Qualitatively, isotherms display a complex relationship with the governing parameters. Local and average Nusselt numbers improve with increasing Reynolds, Prandtl and Richardson numbers, shear-thinning natures. An opposite nature is noted for shear-thickening fluids. Nusselt numbers were further improved with increased fluids volume fractions contrary to decrease in forced convections cases. At maximum Reynolds, Prandtl and Richardson numbers, Nusselt number increased by 30.53% and 13.11% for shear-thinning (n = 0.4) fluids, respectively for φ f = 0.70 and 0.99. An unsteady nature was also found in shear-thinning regions at larger volume fractions. Lastly, statistical analysis is presented for the average Nusselt numbers to depict additional physical insights of the numerical outcomes.

Efficient Analysis of Radiation From a Dipole Source in Woodpile EBG Structures

An efficient and rigorous approach to analyze the radiation from an electric dipole placed in 3-D woodpile electromagnetic bandgap material (EBG) structures consisting of layered periodic arrays of circular cylinders is presented. The response in the spectral domain of the woodpile EBGs excited by a dipole source is calculated using one-dimensional lattice sums, the 3-D transition matrix of a circular cylinder, and the generalized reflection and transmission matrices of the layered system. The closed-form expression of the radiated field is rigorously obtained by a finite integral of the spectral response. Comparisons with the far fields calculated using reciprocity both with the present approach and with a fully independent commercial EM software package are presented showing excellent agreement with the proposed method and demonstrating the high computational efficiency of the semianalytical formalism.

Exceptional points of Bloch eigenmodes on a dielectric slab with a periodic array of cylinders

Eigenvalue problems for electromagnetic resonant states on open dielectric structures are non-Hermitian and may have exceptional points (EPs) at which two or more eigenfrequencies and the corresponding eigenfunctions coalesce. EPs of resonant states for photonic structures give rise to a number of unusual wave phenomena and have potentially important applications. It is relatively easy to find a few EPs for a structure with parameters, but isolated EPs provide no information about their formation and variation in parameter space, and it is always difficult to ensure that all EPs in a domain of the parameter space are found. In this paper, we analyze EPs for a dielectric slab containing a periodic array of circular cylinders. By tuning the periodic structure towards a uniform slab and following the EPs continuously, we are able to obtain a precise condition about the limiting uniform slab, and thereby order and classify EPs as tracks with their endpoints determined analytically. It is found that along each track, a second-order EP of resonant states (with a complex frequency) is transformed to a special kind of third-order EP with a real frequency via a special fourth-order EP. Our study gives a clear and complete picture for EPs in parameter space, and can provide useful guidance to their practical applications.

Excitation of Bound States in the Continuum via Second Harmonic Generations

A bound state in the continuum (BIC) on a periodic structure sandwiched between two homogeneous media is a guided mode with a frequency and a wavenumber such that propagating plane waves with the same frequency and wavenumber exist in the homogeneous media. Optical BICs are of significant current interest, since they have applications in lasing, sensing, filtering, switching, and many light emission processes, but they cannot be excited by incident plane waves when the structure consists of linear materials. In this paper, we study the diffraction of a plane wave by a periodic structure with a second order nonlinearity, assuming the structure has a BIC and the frequency and wavenumber of the incident wave are one half of those of the BIC. Based on a scaling analysis and a perturbation theory, we show that the incident wave may induce a very strong second harmonic wave dominated by the BIC, and also a fourth harmonic wave that cannot be ignored. The perturbation theory reveals that the amplitude of the BIC is inversely proportional to a small parameter depending on the amplitude of the incident wave and the nonlinear coefficient. In addition, a system of four nonlinearly coupled Helmholtz equations (the four-wave model) is proposed to model the nonlinear process. Numerical solutions of the four-wave model are presented for a periodic array of circular cylinders and used to validate the perturbation results.

Non-Newtonian power-law fluid’s thermal characteristics across periodic array of circular cylinders

The thermal characteristics of incompressible non-Newtonian power-law fluids across periodic array of circular cylinders have been examined using the finite volume-based numerical solver ANSYS-FLUENT for the following ranges of physical parameters: Reynolds number; 1 ≤ Re ≤ 40: Prandtl number; 1 ≤ Pr ≤ 100: power-law index; 0.40 ≤ n ≤ 1.8; and fluid volume fractions ranging from 0.70 to 0.99. The thermal features have been described via isotherm patterns, local and average Nusselt numbers and the Colburn heat transfer factor and found to be strongly dependent over the above physical parameters. It was observed that the dense isotherms with the increasing inertial and viscous diffusion suggest an improvement in the rate of heat transfer across the periodic cylinders. An increase in local Nusselt number was seen with the increasing values of Re and/or Pr across all the fluid volume fractions. Further, the different behavior of the average Nusselt number was noticed because of the shear-thinning and shear-thickening natures. An enhancement of about 97% was noticed in the shear-thinning region between the extreme fluid volume fractions for the highest value of Pr and the lowest values of Re and n. However, in many cases, the enhancement was noticed to be even more than 100%. An empirical correlation for the average Nusselt number and the Colburn heat transfer factor (jH) has been developed to give the additional physical insight of the results. Finally, the comparison was made with the available literature which displayed a good agreement with the present results.

Bound states in the continuum on periodic structures surrounded by strong resonances

Bound states in the continuum (BICs) are trapped or guided modes with their frequencies in the frequency intervals of the radiation modes. On periodic structures, a BIC is surrounded by a family of resonant modes with their quality factors approaching infinity. Typically the quality factors are proportional to $1/|\beta - \beta_*|^2$, where $\beta$ and $\beta_*$ are the Bloch wavevectors of the resonant modes and the BIC, respectively. But for some special BICs, the quality factors are proportional to $1/|\beta -\beta_*|^4$. In this paper, a general condition is derived for such special BICs on two-dimensional periodic structures. As a numerical example, we use the general condition to calculate special BICs, which are antisymmetric standing waves, on a periodic array of circular cylinders, and show their dependence on parameters. The special BICs are important for practical applications, because they produce resonances with large quality factors for a very large range of $\beta$.

Diffraction of plane waves by a periodic array of nonlinear circular cylinders

It is well known that standing waves, as special bound states in the radiation continuum, may exist on a periodic array of dielectric cylinders at a discrete set of frequencies if the medium is linear. Recent numerical studies indicate that nonlinear standing waves could exist continuously with respect to the frequency on a periodic array of cylinders with a Kerr nonlinearity. In this paper, we study the diffraction of a normal incident plane wave by a periodic array of circular cylinders with a Kerr nonlinearity. Using a perturbation method and a highly accurate numerical method, we show that a plane incident wave may couple to a nonlinear standing wave, and in general, there are four different couplings leading to four asymmetric solutions in two pairs. The existence of these asymmetric solutions provides another example for the symmetry-breaking phenomenon. Importantly, it seems that the asymmetric solutions (thus the symmetry-breaking phenomenon) appear for incident waves of arbitrarily low intensity.

Numerical simulation and analysis of fluid flow hydrodynamics through a structured array of circular cylinders forming porous medium

Numerical analysis of the mechanisms that govern the flow in a porous region is crucial for modeling porous media flows. This study describes an adapted and efficient turbulence modeling technique for this category of porous media flows. The main objective of the present study is to provide a detailed pore scale description of fluid flow and to analyze the formation of coherent structures in the wake region close to the solid wall subject to the effects of fine-scale turbulence. The computations were performed in a three-dimensional representative elementary volume (REV) of a porous matrix, which comprised a periodic array of circular cylinders. Two flow-modeling strategies were employed: the steady Reynolds averaged Navier–Stokes (RANS) and transient large eddy simulation (LES) approaches. In the RANS modeling framework, both standard k–ε and low Re k–ε turbulence models were used. The porosity (ϕ) of the porous REV and Reynolds number (ReD) varied from 0.3 to 0.8 and 10 to 40,000, respectively. We investigated the effects of porosity and ReD on the pore scale velocity distribution, overall macroscopic pressure gradient, and turbulent dynamics, and performed comparisons using RANS and LES calculations within the porous REV. The Darcy parameter (i.e., medium permeability, K) was computed and a correlation was determined between the medium porosity and permeability. The macroscopic pressure gradients in the three-dimensional REV were also compared with the Kozeny–Carman and Darcy–Forchheimer law.

Standing waves on a periodic array of circular cylinders with saturable nonlinear media

A standing wave of a periodic dielectric waveguide is a special guided mode with zero Bloch wavenumber. For two-dimension linear and symmetry periodic waveguides, standing waves are protected by symmetry and only exist at a discrete set of frequencies. While for a periodic array of circular cylinders with Kerr nonlinearity, standing waves exist continuously on frequency and there are infinite number of them for each frequency. In this paper, we consider standing waves on a periodic array of circular cylinders made of saturable nonlinear media. We show numerically that the standing waves in Kerr and saturable nonlinear media are mainly different in two ways: (1) their existence about frequency, and (2) the number of them for fixed frequency. A new iterative scheme based on Newton’s method is developed to solve the nonlinear problems. Approximate analytical expressions of the amplitude-frequency relations of standing waves in saturable nonlinear media are obtained by a perturbation analysis when amplitudes are small or large.

Evaluation of a pressure splitting formulation for Weakly Compressible SPH: Fluid flow around periodic array of cylinders

In this paper, a pressure splitting formulation is proposed for Weakly Compressible SPH (WC-SPH) method and its capability in the suppression of the spurious oscillations is studied by conducting a stability analysis. The proposed formulation is implemented within the framework of a consistent SPH method. The predictions from the theoretical analysis are verified by the results of numerical test-cases. This method is applied to the incompressible fluid flow around periodic array of circular cylinders. The accuracy and the convergence of the results are investigated for benchmark problems. The results are also compared with those of the conventional WC-SPH method. In a similar test-case, the effects of the artificial speed of sound on the evolution of the transient solution and the occurrence of the spurious oscillations in the steady state are studied, and compared for both the conventional and the proposed WC-SPH formulations. The improvements in the evaluation of the pressure field, due to the proposed pressure splitting formulation, are shown for both a vanishing Reynolds number and a finite Reynolds number of Re∼O(1).

Efficient Mode-Matching Analysis of 2-D Scattering by Periodic Array of Circular Cylinders

Based on a common-area concept and insertion of an infinitesimal perfect magnetic conductor (PMC) wire, a new mode-matching method for mixed coordinate systems is proposed for the analysis of periodic magnetodielectric, perfect electric conductor, and PMC circular cylinder arrays. Our scattering solutions were computed and they showed favorable agreements with other known results in terms of power scattering coefficients, resonant frequency of nanostructures, and the sum rule of extinction width. The low-frequency solutions of periodic arrays were also formulated and compared with the full-wave counterparts.

Accurate analysis of electromagnetic scattering from periodic circular cylinder array with defects

This paper considers the two-dimensional electromagnetic scattering from periodic array of circular cylinders in which some cylinders are removed, and presents a formulation based on the recursive transition-matrix algorithm (RTMA). The RTMA was originally developed as an accurate approach to the scattering problem of a finite number of cylinders, and an approach to the problem of periodic cylinder array was then developed with the help of the lattice sums technique. This paper introduces the concept of the pseudo-periodic Fourier transform to the RTMA with the lattice sums technique, and proposes a spectral-domain approach to the problem of periodic cylinder array with defects.

Analysis of scattering from a finite array of circular cylinders using a model of layered cylindrical arrays

A semi-analytical approach for a periodic planar array formed by a finite number of circular cylinders is presented using a model of layered cylindrical arrays. The method consists in extracting the reflection and transmission matrices of a cylindrically periodic array of circular cylinders and then obtaining the characteristics of layered structures by using a recursive formula.

Diffusion model for Knudsen-type compressor composed of periodic arrays of circular cylinders

A rarefied gas flow in a long porous channel having a periodic structure that is consisting of alternately arranged porous media and gaps, the former of which contains a periodic array of parallel circular cylinders, is considered for the case in which the channel is infinitely wide. The cylinder arrays have a periodic temperature distribution with the same period as the structure. Under the assumption that the length of each cylinder array and that of each gap are much larger than the period of the cylinders in the array, a fluid-dynamic system describing the overall behavior of the gas in the channel is derived from the kinetic system composed of the Bhatnagar–Gross–Krook equation and the diffuse reflection boundary condition. The derived system is composed of a diffusion model for each cylinder array, whose isothermal version has been reported previously [S. Taguchi and P. Charrier, Phys. Fluids 20, 067103 (2008)], a set of fluid-dynamic equations for each gap, and the macroscopic connection conditions at each junction between an array and a gap. Then, the fluid-dynamic system is applied to a long channel consisting of many cylinder arrays and gaps. Some numerical results demonstrating the pumping effect of the flow are presented.

Dirichlet-to-Neumann map method for analyzing periodic arrays of cylinders with oblique incident waves

For finite two-dimensional photonic crystals given as periodic arrays of circular cylinders in a square or triangular lattice, we develop an efficient method to compute the transmission and reflection spectra for oblique incident plane waves. The method relies on vector cylindrical wave expansions to approximate the Dirichlet-to-Neumann (DtN) map for each distinct unit cell and uses the DtN maps to derive an efficient method that works on the edges of the unit cells only. The DtN operator maps the two longitudinal field components to their derivatives on the boundary of the unit cell.

Boundary conditions at the interface between fluid layer and fibrous medium

AbstractThe flow through a channel partially filled with fibrous porous medium was analyzed to investigate the interfacial boundary conditions. The fibrous medium was modeled as a periodic array of circular cylinders, in a hexagonal arrangement, using the boundary element method. The area and volume average methods were applied to relate the pore scale to the representative elementary volume scale. The permeability of the modeled fibrous medium was calculated from the Darcy's law with the volume‐averaged Darcy velocity. The slip coefficient, interfacial velocity, effective viscosity and shear jump coefficients at the interface were obtained with the averaged velocities at various permeabilities or Darcy numbers. Copyright © 2008 John Wiley & Sons, Ltd.

A parallel adaptive unstructured finite volume method for linear stability (normal mode) analysis of viscoelastic fluid flows

A parallel unstructured finite volume method is presented for analysis of the stability of two-dimensional steady Oldroyd-B fluid flows to small amplitude three-dimensional perturbations. A semi-staggered dilation-free finite volume discretization with Newton’s method is used to compute steady base flows. The linear stability problem is treated as a generalized eigenvalue problem (GEVP) in which the rightmost eigenvalue determines the stability of the base flow. The rightmost eigenvalues associated with the most dangerous eigenfunctions are computed through the use of the shift-invert Arnoldi method. To avoid fine meshing in the regions where the flow variables are changing slowly, a local mesh refinement technique is used in order to increase numerical accuracy with a lower computational cost. The CUBIT mesh generation environment has been used to refine the quadrilateral meshes locally. In order to achieve higher performance on parallel machines the algebraic systems of equations resulting from the steady problem and the GEVP have been solved by implementing the MUltifrontal Massively Parallel Solver ( MUMPS). The proposed method is applied to the linear stability analysis of the flow of an Oldroyd-B fluid past a linear periodic array of circular cylinders in a channel and a linear array of circular half cylinders placed on channel walls. Two different leading eigenfunctions are identified for close and wide cylinder spacing for the periodic array of cylinders. The numerical results indicate good agreement with the numerical and experimental results available in the literature.

Bragg scattering by a line array of small cylinders in a waveguide. Part 1. Linear aspects

Motivated by potential applications for offshore airports supported on vertical piles, we report a theory of wave diffraction by a periodic array of circular cylinders. The simple case of normal incidence on a rectangular array is studied here, which is equivalent to a line array along the centre of a long channel. An asymptotic theory is developed for cylinders much smaller than the incident wavelength, which is comparable to the cylinder spacing. Focus is on Bragg resonance near which scattering is strong. A combination of the method of multiple scales and the Bloch theorem leads to simple evolution equations coupling the wave envelopes. Dispersion of transient wave envelopes is investigated. Scattering of detuned waves by a large but finite number of cylinders is investigated for frequencies in and outside the band gap. Quantitative accuracy is assessed by comparisons with numerical computations via finite elements. The analytical theory prepares the ground for nonlinear studies and may facilitate future inclusion of real-fluid effects such as vortex shedding.

Onset of Vortex Shedding in a Periodic Array of Circular Cylinders

Hopf bifurcation of steady base flow and onset of vortex shedding over a transverse periodic array of circular cylinders is considered. The influence of transverse spacing on critical Reynolds number is investigated by systematically varying the gap between the cylinders from a small value to large separations. The critical Reynolds number behavior for the periodic array of circular cylinders is compared with the corresponding result for a periodic array of long rectangular cylinders considered in [Balanchandar, S., and Parker, S. J., 2002, “Onset of Vortex Shedding in an Inline and Staggered Array of Rectangular Cylinders,” Phys. Fluids, 14, pp. 3714–3732]. The differences between the two cases are interpreted in terms of differences between their wake profiles.

Reflection and transmission properties of layered periodic arrays of circular cylinders embedded in magnetized ferrite slab

The two-dimensional electromagnetic scattering by layered periodic arrays of circular cylinders embedded in a ferrite slab magnetized in the direction parallel to the cylinder axis are analyzed, using the lattice sums technique, the T-matrix of cylindrical objects, and the generalized reflection and transmission matrices for a layered system. It is shown that the reflection and transmission properties of the embedded arrays are accurately calculated by a simple matrix calculus based on the one-dimensional lattice sums and the T-matrix of the circular cylinder isolated in the magnetized ferrite slab. Numerical examples for the frequency response in the reflectance and transmittance are presented, and the switching properties of the embedded arrays controllable by the applied magnetic field are discussed.