Numerical simulation of flow past row of square cylinders for various separation ratios

A systematic study for the flow around a row of five square cylinders placed in a side-by-side arrangement and normal to the oncoming flow at a Reynolds number of 150 is carried out through the numerical solution of the two-dimensional unsteady incompressible Navier-Stokes equations. Special attention is paid to investigate the effect of the spacing between the five cylinders on the wake structure and vortex shedding mechanism. The simulations are performed for the separation ratios (spacing to size ratio) of 1.2, 2, 3 and 4. Depending on the separation ratio the following flow patterns are observed: a flip-flopping pattern, in-phase and anti-phase synchronized pattern and non-synchronized pattern. These flow patterns are supposed to be a consequence of the interaction between two types of frequencies viz. the vortex shedding (primary) and the cylinder interaction (secondary) frequencies. At small separation ratio the flow is predominantly characterized by the jet in the gaps between successive cylinders and the secondary frequencies play a role in the resulting chaotic flow. On the contrary, at higher separation ratio the secondary frequencies almost disappear and the resulting flow becomes more synchronized dominated by the primary frequency.     

Effect of tube spacing on the vortex shedding characteristics of laminar flow past an inline tube array: A numerical study

The effect of tube spacing on the vortex shedding characteristics and fluctuating forces in an inline cylinder array is studied numerically. The examined Reynolds number is 100 and the flow is laminar. The numerical methodology and the code employed to solve the Navier-Stokes and continuity equations in an unstructured finite volume grid are validated for the case of flow past two tandem cylinders at four spacings. Computations are then performed for a six-row inline tube bank for eight pitch-to-diameter ratios, s, ranging from 2.1 to 4. At the smallest spacing examined (s = 2.1) there are five stagnant and symmetric recirculation zones and weak vortex shedding activity occurs only behind the last cylinder. As s increases, the symmetry of the recirculation zones breaks leading to vortex shedding and this process progressively moves upstream, so that for s = 4 there is clear shedding from every row. For any given spacing, the shedding frequency behind each cylinder is the same. A critical spacing range between 3.0 and 3.6 is identified at which the mean drag as well as the rms lift and drag coefficients for the last three cylinders attain maximum values. Further increase to s = 4 leads to significant decrease in the force statistics and increase in the Strouhal number. It was found that at the critical spacing there is 180° phase difference in the shedding cycle between successive cylinders and the vortices travel a distance twice the tube spacing within one period of shedding.     

On the use of several compact methods for the study of unsteady incompressible viscous flow round a circular cylinder

Fourth order accurate methods of mehrstellen type are compared to second order accurate methods for the solution of the unsteady incompressible Navier-Stokes equations in their vorticity stream function formulation. These methods are applied to the study of separated flow around a circular cylinder at several Reynolds numbers. The impulsively started cylinder at Re = 200 and 550, is considered without symmetry restrictions. The features illustrated include the bulge phenomenon at Re = 200, the occurrence of secondary vortices depending on the schemes used at Re = 550, and of twin secondary vortices at Re = 3000. The Karman vortex street is investigated at Re = 200 with a uniform flow in the far field and with superimposed motions of the cylinder. In this last case, a frequency analysis has allowed a critical examination of results pertaining to locked-in situations with respect to confinement effects.     

Two-dimensional side-by-side circular cylinders at moderate Reynolds numbers

In a companion article [1], we described computer simulations of the flow around 2 two-dimensional, tandem circular cylinders in a flow for 1?Re?20. In this article we adopt a similar approach to characterize the flow around side-by-side cylinders with surface-to-surface separation/diameter in the range 0.1 < s/D < 30. The results revealed some distinct and interesting features of the flow, which are completely different than those observed at higher Reynolds numbers.At low Reynolds numbers, 1?Re?5, for all gap spacings, the flow contains no regions of flow separation. At higher Re, four distinct flow behaviors were observed. For very small gap spacings, e.g. 0.1 < s/D < 0.6 at Re = 20, two elongated “detached vortices” form downstream of the cylinders. The drag coefficient increases sharply with the gap spacing. For gap spacings 0.6 < s/D < 0.7 at Re = 20, no vortices form anywhere in the flow. For gap spacings around s/D ≈ 1 separation regions form only on the inside portions of the cylinders. For larger gap spacings s/D > 1 the flow reverts to something similar to that around an isolated cylinder in the flow, i.e. two attached vortices on the rear side of each cylinder. In general, the drag coefficient increases as the gap spacing increases. At higher Reynolds number it is known that the cylinder lift coefficients decrease monotonically with gap spacing. In contrast, at these lower Reynolds number the lift coefficient curves rise to a maximum for 0.3 < s/D < 3 and then decrease monotonically for larger s/D.     

On wave harmonics of heated circular cylinders

A hybrid (FV + FE) acoustic damping method, which is investigated and optimized in terms of wave harmonics behavior of the method, is utilized at Ma = 0.01 for heated circular cylinders. Discretization of nonlinear convective terms is a modified approximate Riemann solver. Modification is realized through multiplication of dissipation term with an acoustic damping matrix. This is necessary to avoid drawbacks of standard algorithm and to improve accuracy of results, if low speed applications are concerned. Later, harmonics of velocity and temperature fields behind a heated circular cylinder are investigated numerically for moderate Reynolds numbers between 70 and 110. A parametric study of the first harmonics is carried out precisely by increasing temperature ratios, T^∗ = T_wall/T_∞ from 1.03 to 1.8. Results agree well for high temperature ratios, T^∗ = 1.5, 1.8 and Re ? 100, with the key issues stated in the experimental work of Ezersky et al. [Ezersky AB, Lecordier JC, Paranthoën P, Soustov PL, Structure of vortices in a Karman street behind a heated cylinder. Phys Rev E 2000;61:2107]. Moreover, it is found that decrease in frequency of oscillations can be explained as heated cylinders have larger vortex formation region as a result of heat flux from walls in comparison to non-heated cylinders. At lower temperature ratios, T^∗ = 1.03, 1.1, temperature can be taken as passive scalar field.     

Investigation of flow around a pair of side-by-side square cylinders using the lattice Boltzmann method

The low-Reynolds number flow around two square cylinders placed side-by-side is investigated using the lattice Boltzmann method (LBM). The effects of the gap ratio s/d (s is the separation between the cylinders and d is the characteristic dimension) on the flow are studied. These simulations reveal the existence of regimes with either synchronized or non-synchronized vortex-shedding, with transition occurring at s/d ≈ 2, which is larger than for circular cylinders. Detailed results are presented at Re = 73 for s/d = 2.5 and 0.7 corresponding to the synchronized and flip-flop regimes, respectively. Vortex-shedding from the cylinder occurs either in-phase or in-antiphase in the synchronized regime. However, linear stochastic estimate (LSE) calculations show that in-phase locking is the predominant mode. LSE is also employed to educe the underlying modes in the flip-flop regime, where evidence for both in-phase and anti-phase locked vortices is found, indicating that this regime is in a quasi-stable state between these two modes. The merging of the wakes, which is gradual for the synchronized regime, occurs rapidly in the flip-flop regime. The mean pressure on the upstream surface is symmetric and asymmetric for the synchronized and flip-flop regimes, respectively. Differences in results between the two regimes are interpreted in terms of the interaction of the jet formed between the cylinders with the adjoining wakes, the strength of this interaction depending on the spacing.     

Flow patterns past two circular cylinders in proximity

Flow patterns past two nearby circular cylinders of equal diameter immersed in the cross-flow at low Reynolds numbers (Re ? 160), were numerically studied using an immersed boundary method. We considered all possible arrangements of the two cylinders in terms of the distance between the two cylinders and the inclination angle of the line connecting the cylinder centers with respect to the direction of the main flow. Ten distinct flow patterns were identified in total based on vorticity contours and streamlines, which are Steady, Near-Steady, Base-Bleed, Biased-Base-Bleed, Shear-Layer-Reattachment, Induced-Separation, Vortex-Impingement, Flip-Flopping, Modulated Periodic, and Synchronized-Vortex-Shedding. Collecting all the numerical results obtained, we propose a general flow-pattern diagram for each Re, and a contour diagram on vortex-shedding frequency for each cylinder at Re = 100. The perfect symmetry implied in the geometrical configuration allows one to use these diagrams to identify flow pattern and vortex-shedding frequencies in the presence of two circular cylinders of equal diameter arbitrarily positioned in physical space with respect to the main-flow direction.     

Flow-induced forces on two circular cylinders in proximity

Flow-induced forces on two nearby circular cylinders of equal diameter immersed in the cross flow at Re = 100 were numerically studied. We consider all possible arrangements of the two circular cylinders in terms of the distance between the two cylinders and the inclination angle of the line connecting the cylinder centers with respect to the direction of the main flow. It turns out that significant changes in the characteristics of flow-induced forces are noticed depending on how the two circular cylinders are positioned, resulting in quantitative changes of force coefficients on both cylinders. Collecting all the numerical results obtained, we propose contour diagrams for mean force coefficients and rms values of force coefficient fluctuations for each of the two cylinders. The perfect geometrical symmetry implied in the flow configuration allows one to use those diagrams to estimate flow-induced forces on two circular cylinders of equal diameter arbitrarily positioned in physical space with respect to the main flow direction.     

Numerical study of Taylor-Couette flow with an axial flow

The flow between two concentric cylinders with the inner one rotating and with an imposed pressure-driven axial flow is studied using numerical simulation. This study considers the identical flow geometry and flow parameters as in the experiments of Wereley and Lueptow [Phys. Fluids 11 (12) (1999) 3637], where particle image velocimetry measurements were carried out to obtain detailed velocity fields in a meridional plane of the annulus. The objectives of this investigation are to numerically verify the experimental results of Wereley and Lueptow and to further study detailed flow fields and bifurcations related to Taylor-Couette flow with an imposed axial flow. The vortices in various flow regimes such as non-wavy laminar vortex, wavy vortex, non-wavy helical vortex, helical wavy vortex and random wavy vortex are all consistently reproduced with their experiments. It is demonstrated that ‘shift-and-reflect’ symmetry holds in Taylor-Couette flow without an imposed axial flow. In case of Taylor-Couette flow with an imposed axial flow, one can find that the shift-and-reflect symmetry is roughly valid for the remaining velocity field after subtracting the annular Poiseuille flow. The axial flow stabilizes the flow field and decreases the torque required by rotating the inner cylinder at a given speed. Growth rate of the flow instability is defined and used in predicting the type of the vortices. The velocity vector fields obtained also reveal the same vortex characteristics as found in the experiments of Wereley and Lueptow.     

Simulating flows with moving rigid boundary using immersed-boundary method

The present study is to apply the immersed-boundary method to simulate 2- and 3-D viscous incompressible flows interacting with moving solid boundaries. Previous studies indicated that for stationary-boundary problems, different treatments inside the solid body did not affect the external flow. However, the relationship between internal treatment of the solid body and external flow for moving-boundary problems was not studied extensively and is investigated here. This is achieved via direct-momentum forcing on a Cartesian grid by combining “solid-body forcing” at solid nodes and interpolation on neighboring fluid nodes. The influence of the solid body forcing within the solid nodes is first examined by computing flow induced by an oscillating cylinder in a stationary square domain, where significantly lower amplitude oscillations in computed lift and drag coefficients are obtained compared with those without solid-body-forcing strategy. Grid-function convergence tests also indicate second-order accuracy of this implementation with respect to the L¹ norm in time and the L² norm in space. Further test problems are simulated to examine the validity of the present technique: 2-D flows over an asymmetrically-placed cylinder in a channel, in-line oscillating cylinder in a fluid at rest, in-line oscillating cylinder in a free stream, two cylinders moving with respect to one another, and 3-D simulation of a sphere settling under gravity in a static fluid. All computed results are in generally good agreement with various experimental measurements and with previous numerical simulations. This indicates the capability of the present simple implementation in solving complex-geometry flow problems and the importance of solid body forcing in computing flows with moving solid objects.     

Creeping motion of a cylinder through incompressible fluid bounded by a coaxial closed cylinder

The finite element method based on the Galerkin process is used to study the translation of a cylinder through fluid occupying the space between it and a co-axial cylindrical container under incompressible creeping flow conditions. The resistance force on the inner cylinder is calculated for the case of cylinders with diameters equal to their lengths, at the instant when their centroids are coincident. These resistance forces are compared with those for concentric spheres having equal radii or surface areas or volumes to the cylinders. The resistance force is also determined for the more general case when the inner cylinder approaches the end of the container for a range of aspect ratios (both cylinders having the same aspect ratio). The ratio of the pressure drag to the wall friction is also studied.     

Parallel 3D computation of unsteady flows around circular cylinders

In this article we present parallel 3D finite element computation of unsteady incompressible flows around circular cylinders. We employ stabilized finite element formulations to solve the Navier-Stokes equations on a thinking machine CM-5 supercomputer. The time integration is based on an implicit method, and the coupled, nonlinear equations generated every time step are solved iteratively, with an element-vector based evaluation technique. This strategy enables us to carry out these computations with millions of coupled, nonlinear equations, and thus resolve the flow features in great detail. At Reynolds number 300 and 800, our results indicate strong 3D features arising from the instability of the columnar vortices forming the Karman street. At Re = 10 000 we employ a large eddy simulation (LES) turbulence model.     

Diffusioosmosis of electrolyte solutions in fibrous porous media

An analytical study is presented for the diffusioosmotic flow of an electrolyte solution in the fibrous medium constructed by an ordered array of parallel charged circular cylinders at the steady state. The prescribed electrolyte concentration gradient is constant but can be oriented arbitrarily with respect to the axes of the cylinders. The electric double layer surrounding each cylinder may have an arbitrary thickness relative to the radius of the cylinder. A unit cell model which allows for the overlap of the double layers of adjacent cylinders is employed to account for the effect of fibers on each other. The electrostatic potential distribution in the fluid phase of a cell is obtained by solving the linearized Poisson–Boltzmann equation, which applies to the case of low surface potential of the cylinders. The macroscopic electric field induced by the imposed electrolyte concentration gradient through the fluid phase in a cell is determined as a function of the radial position. A closed-form formula for the fluid velocity profile of the electrolyte solution due to the combination of electroosmotic and chemiosmotic contributions as a function of the porosity of the array of cylinders correct to the second order of their surface charge density or zeta potential is derived as the solution of a modified Navier–Stokes equation. The diffusioosmotic velocity can have more than one reversal in direction over a small range of the zeta potential. For a given electrolyte concentration gradient in a cell, the fluid flow rate does not necessarily increase with an increase in the electrokinetic radius of the cylinder, which is the cylinder radius divided by the Debye screening length. The effect of the radial distribution of the induced axial electric field in the double layer on the diffusioosmotic flow is found to be of dominant significance in most practical situations.     

Large eddy simulation of turbulent flow past a square cylinder confined in a channel

Turbulent flow past a square cylinder confined in a channel is numerically investigated by large eddy simulation (LES). The main objectives of this study are to extensively verify the experimental results of Nakagawa et al. [Exp. Fluids 27(3) (1999) 284] by LES and to identify the features of flows past a square cylinder confined in a channel in comparison with the conventional one in an infinite domain. The LES results obtained are in excellent agreement with the experiment both qualitatively and quantitatively. The well-known Kármán vortex shedding is observed. However, the vortices shed from the cylinder are significantly affected by the presence of the plates; mean drag and fluctuation of lift force increase significantly. Furthermore, periodic and alternating vortex-rollups are observed in the vicinity of the plates. The rolled-up vortex is convected downstream together with the corresponding Kármán vortex; they form a counter-rotating vortex pair. It is also revealed that the cylinder greatly enhances mixing process of the flow.     

Simulation of fish swimming and manoeuvring by an SVD-GFD method on a hybrid meshfree-Cartesian grid

In this paper we present the development of the Singular-Value Decomposition (SVD) based Generalized Finite Difference (GFD) method for the simulation of fluid-structure interaction (FSI) problems in a viscous fluid. The class of FSI problems is exemplified by the self-propulsion (swimming) and dynamic manoeuvring of deforming (undulating and flexing) bodies in a fluid medium. Computation is carried out on a hybrid grid comprising meshfree nodes around the undulating swimming body and Cartesian nodes in the background. The meshfree nodes are convected in tandem with the changing shape and motion of the swimming body. The resultant locomotion of the swimmer is governed by fully-coupled dynamic interaction between the deforming body and the fluid in accordance with Newton’s laws. Time integration of motion is carried out by a Crank-Nicolson based implicit iterative algorithm, which fully couples the changing position of the swimming body with the evolving flow field, for numerical stability. The numerical scheme is applied to the steady swimming/cruising and sharp turning manoeuvres of a two-dimensional carangiform fish. The Strouhal number approaches values for efficient steady swimming reported in Fish and Lauder (2006) and Triantafyllou and Triantafyllou (1993) [3] and [6] at high Reynolds number. An illustrative example shows the numerical carangiform swimmer executing a sharp turn through an angle of 70° from straight coasting within a space of about one body length. The results obtained are consistent with available literature. In steady swimming, the momentumless wake theoretically anticipated by Wu (2001) [57] is successfully reproduced here, as opposed to the inverse von Karman vortex street generally predicted by inviscid flow models. The momentumless wake, characterized by an aligned series of alternately-signed shed vortices, is symptomatic of a state of average equilibrium between drag acting on the body of the fish and thrust produced by its undulating tail fin. Guided swimming towards targets based on a simple feedback control scheme is also demonstrated.     

Exact solutions for the unsteady rotating flows of a generalized Maxwell fluid with oscillating pressure gradient between coaxial cylinders

This paper deals with the unsteady rotating flow of a generalized Maxwell fluid with fractional derivative model between two infinite straight circular cylinders, where the flow is due to an infinite straight circular cylinder rotating and oscillating pressure gradient. The velocity field and the adequate shear stress are determined by means of the combine of the sequential fractional derivatives Laplace transform and finite Hankel transform. The exact solutions are presented by integral and series form in terms of the generalized G and Mittag-Leffler functions. The similar solutions can be easily obtained for ordinary Maxwell and Newtonian fluids as limiting cases. Finally, the influence of the relaxation time and the fractional parameter on the fluid dynamic characteristics, as well as a comparison between models, is shown by graphical illustrations.     

Numerical study of interference between simple-shape bodies in hypersonic flows

Hypersonic rarefied-gas flows near two side-by-side plates and cylinders, toroidal balloon, plate and cylinder over a plane surface, and plate behind a cylinder in argon, nitrogen, oxygen, and carbon dioxide have been studied numerically using the direct simulation Monte-Carlo technique under the transition flow conditions at Knudsen numbers from 0.004 to 10. Strong influences of the geometrical factor (the ratio of a distance between bodies to a body length) and the Knudsen number on the flow structure about the bodies (shock-wave shapes, the configuration of subsonic flow zones), skin friction, pressure distribution, lift, and drag have been found.     

A numerical analysis of two-dimensional flow past a rectangular prism by a discrete vortex model

A discrete vortex model is developed to analyse the two-dimensional fully separated unsteady flow past a rectangular prism. The effects of viscous diffusion of vortices and the loss in vorticity after the stationary prism with a thickness ratio ranging from 0.5 to 2.0. The formation of Karman vortices, the vortex shedding frequency, and the fluid forces acting on the body are favourably compared with the experimental results by various research workers. The method of analysis is also shown to be applicable to the flow past a prism that is in forced vibration.     

基于1维子空间的3维重建算法

为了重建3维场景,假设相机为正投影模型,提出了一种基于1维子空间的3维重建算法．所有图像序列构成的行向量与3维空间点构成的行向量所生成的子空间是同一线性子空间,而且由第1幅图像点构成的2个行向量外加1个行向量就可以组成该子空间的一个基底．该算法利用上述特性,线性地求取该行向量,最后完成3维重建．模拟实验和真实实验数据结果表明,该重建算法具有鲁棒性好、重投影误差小等优点．     

Instability of the flow in the vicinity of trailing edge of a class of thin aerofoils

A numerical study has been undertaken to investigate the notion of absolute/convective instability in laminar incompressible trailing edge flows past wedge-like shapes with curved boundaries of the form y=α(−x)^m. The effects of various trailing edge shapes m and relative thickness α on the flow separation and the development of instabilities in the vicinity of trailing edge are investigated. The nonlinear viscous-inviscid interaction equations, which have been derived by means of the asymptotic theory of flow separation, are solved first numerically to construct genuine mean velocity profiles representing the correct flow in the vicinity of the trailing edge. The absolute/convective nature of the asymptotically formed velocity profiles via a composite expansion is then ascertained by using a spatio-temporal analysis based on the Briggs-Bers pinching criterion. Although no absolute growth is encountered upstream of the trailing edge of the airfoil shapes considered, in particular the wake region behind the trailing edge of Joukowski type profiles is found to be persistently susceptible to absolute instability. It is found that separation is enhanced as the relative thickness of the airfoil gets bigger. This, in turn, is shown to lead to an additional enhancement of the absolute instability character by both increasing the absolute growth rate as well as the extent of the unstable region. Shedding frequency of the Karman vortex street is also determined behind the trailing edge shapes considered.