Equilateral Triangular Arrangement Research Articles

Analysis of aerodynamic forces and flow development with the size of square cylinders arranged in equilateral triangle

A two-dimensional in-house code is used to scrutinize the influence of the arrangement of three cylinders and their corresponding dimensions on the behavior of flow around square cylinders arranged in four different combinations. Three square cylinders, C1, C2 and C3, are arranged in an equilateral triangle with the first cylinder C1 positioned upstream while the two remaining cylinders are placed side-by-side downstream. The wake flow patterns, aerodynamic forces and Strouhal number (St) are determined and analyzed for the four-cylinder arrangements in terms of streamlines, iso-vortices and drag and lift coefficients evolutions for steady and unsteady regimes. The numerical outcomes demonstrate that Re values, the different equilateral-triangular arrangements of cylinders, and the dimension variations of each square cylinder have a prominent influence on the flow characteristics.

Vibration characteristics of a downstream cylinder with rounded corners in an equilateral triangular arrangement at Reynolds number of 180

In this study, we numerically investigate the vibration response of a cylinder arranged in an equilateral triangle with two stationary upstream cylinders at a low Reynolds number of Re = 180. We concentrate on three different rounded corner arrangements: four rounded corners (FRCs), upwind rounded corners (URCs), and leeward rounded corners (LRCs). In addition, we examine three dimensionless rounded corner radii: 0.1, 0.3, and 0.5. The results indicate that as the dimensionless radius increases, the influence of the corner position on the dimensionless amplitude and frequency becomes more pronounced. Furthermore, the dimensionless amplitude and dimensionless frequency curves exhibit significant variations under the different corner arrangements. Within the reduced velocity range (Vr) of 2–5, rounded corners are observed to reduce both the lift and drag coefficients. An increase in the dimensionless radius for a given corner configuration acts to decrease both the lift and drag coefficients. Frequency-domain analysis of the lift coefficient and dimensionless displacement indicates that the observed frequency peaks caused by mutual interference are produced by irregular vortex shedding due to the mutual extrusion and merging of shear layers. At Vr = 4, the FRC, URC, and LRC configurations with a corner radius of 0.5 all exhibit the 2S vortex shedding mode. As Vr increases, the vortex shedding mode for LRCs with a corner radius of 0.5 progressively becomes irregular. At Vr = 16, the vortex shedding mode for LRCs with a corner radius of 0.5 is the regular 2P + 2S mode.

Influence of the corner radii on the flow characteristics of three square-like cylinders in an equilateral-triangular arrangement using IB-LBM

A two-dimensional momentum exchange-based immersed boundary–lattice Boltzmann method (IB-LBM) is used to scrutinize the influence of the corner radii on the characteristics of laminar flow around square-like cylinders arranged in an equilateral-triangular configuration, with each cylinder represented by 300 discrete elements. Numerical simulations were carried out in Newtonian flow with viscous, incompressible, and constant properties, by simultaneously changing the corner ratio λ from 0.0 (square) to 1.0 (circular), Reynolds number Re from 40 to 160, and gap ratio δ from 0.5 to 5.0. The wake flow patterns, force coefficients, and Strouhal number St were calculated and discussed. The numerical outcomes demonstrate that the corner ratio, Re, and gap ratio have a prominent influence on the flow characteristics. The wake flow patterns in the λ-Re plane can be divided into steady-state, flip-flopping, anti-phase, and in-phase modes for δ = 3.0. The drag Cd and lift Cl coefficients exhibit a decreasing trend with increasing corner radius except for Clc1; as δ is increased, the Cl of all three cylinders for λ ranging from 0 to 1.0 gradually approached zero. The St of the steady state flow pattern is significantly lower than that of other flow states, the relationship between St and λ is complex and irregular; as δ increases, the St and λ show a positive correlation.

Load Transfer in Geosynthetic-Reinforced Piled Embankments with a Triangular Arrangement of Piles

A geosynthetic-reinforced piled embankment situated along a newly constructed high-speed rail line was fully instrumented and monitored to explore soil arching in an equilateral-triangular arrangement of piles with circular cap configuration. During construction, the settlement and vertical stress on the pile caps and subsurface at ground level and the tensile strain in the geogrid were recorded. A modified analytical arching model is derived from the established concentric arches (CA) model to elucidate the load transfer mechanism, specifically for a triangular array of piles. The general framework assumes that the embankment load, not resting on the subsurface beneath the arches, is transferred outward along the arches and distributed uniformly over the improved ground. Compared to the conventional CA model, the proposed model gives a lower proportion of load carried by piles, more in line with field measurements. Given the friction and adhesion between soil and geosynthetics, a novel quartic equation is applied to characterize the tensioned membrane effect of geosynthetic reinforcement based on Pham’s circular and parabolic models. The stiffness variation of the biaxial geogrid was critically assessed for the triangular pile arrangement, and its implications for the design were revealed. Finally, the pile efficacy and critical arch height were determined from different analytical methods and checked against field observations, demonstrating the potential of the proposed model.

Vortex-induced vibration and heat dissipation of multiple cylinders under opposed thermal buoyancy

The vortex-induced vibration of three cylinders in an equilateral triangular arrangement with opposed thermal buoyancy are studied by numerical method. The cylinders are uniformly elastic and can freely vibrate in the transverse flow direction. The temperature of cylinders is higher than free stream. The incoming flow is in the same direction as gravity and opposite to thermal buoyancy. The ranges of Richardson number and reduced velocity considered in this work are −1≤Ri ≤ 0 and 3 ≤ U*≤12. The maximum amplitude of each cylinder rises significantly under opposed thermal buoyancy, and the maximum amplitude of cylinder 1, cylinder 2, and cylinder 3 increased by 206%, 174%, and 172%, respectively. When U*>8, the vibrations of cylinder 2 and cylinder 3 are close to the same phase. The opposed thermal buoyancy hinders the separation of flow boundary layer, resulting in thickening of thermal boundary layer and deterioration of heat transfer. The deterioration of heat transfer can be alleviated at high reduced velocity (U*≥10). The increase of Richardson number will lead to the weakening of heat transfer at up-flow surface of cylinder and enhancement of heat dissipation near the back stagnation point. The local Nusselt number distribution around cylinder is affected by vortex-induced vibration, vortex shedding, and the interaction of shear layers from each cylinder.

COMPUTATIONAL ANALYSIS OF THE INFLUENCE OF THE DIMENSIONS OF THREE SQUARE CYLINDERS ARRANGED IN EQUILATERAL TRIANGULAR ARRANGEMENT ON THE BEHAVIOR OF FLOW AND AERODYNAMICS FORCES

Influence of dimensions of square cylinders arranged at the vertices of an equilateral triangle on the behavior of the wake flow is analyzed in the present study. Numerical computations based on the finite volume method (FVM) and a full multigrid technique (FMG) is applied to resolve the governing equations. Three main arrangements are adopted in terms of the cylinders positions in an equilateral triangle and effects of various parameters, namely; the cylinder dimensions and the Reynolds number (Re) on the flow pattern are analyzed. The flow characteristics are analyzed for the three-cylinder combinations in terms of iso-vortices for different flow regimes, namely, the crawling, steady, and unsteady regimes. Of great interest is the analysis of the combined effect of the Reynolds number and the adopted arrangements on the evolutions profiles of the lift (<i>CL</i>) and drag (<i>CD</i>) coefficients. Among the numerical outcomes, it is revealed that according to each configuration and cylinder dimension, the <i>CL</i> and <i>CD</i> values vary differently and dramatically when the Reynolds value is monotonously varied in special ranges.

Numerical Study on the Flow Past Three Cylinders in Equilateral-Triangular Arrangement at Re = 3 × 106

One of the most common systems in engineering problems is the multi-column system in the form of an equilateral-triangular arrangement. This study used three-dimensional numerical simulations to investigate the flow around three cylinders in this arrangement at the super-critical Reynolds number Re=3×106, concentrating on the influence on the spacing ratio (L/D) among cylinders. The instantaneous vortex structures, Strouhal numbers, fluid force coefficients, and pressure distributions are analyzed thoroughly. The present study demonstrated that fluid dynamics is sensitive to L/D, by which five different flow patterns are classified, namely single bluff body flow (L/D≤1.1), deflected gap flow (1.2≤L/D≤1.4), anti-phase flow (1.5≤L/D≤2.3), in-phase flow (2.5≤L/D<3.5), and co-shedding flow (L/D≥3.5). Critical bounds are identified by significant transitions in the flow structure, discontinuous drop and jump of St, and force coefficients.

Numerical computations of flow around three equilateral-triangular square cylinders with rounded corners using momentum exchange-based IB-LBM at low Reynolds numbers

In this study, the flow around three square cylinders with rounded corners in equilateral-triangular arrangements were investigated using the momentum exchange-based immersed boundary-lattice Boltzmann method (IB-LBM) at low Reynolds (Re) numbers. The influences of Re number and gap ratio δ to the flow characteristics, force coefficients, and Strouhal (St) number were examined by selecting one hundred different gap ratios δ and forty different Re numbers. The wake flow patterns are divided into six distinct categories according to the gap ratio, these are single bluff-body, deflected, flip-flopping, anti-phase, anti-phase & in-phase coexist, and fully developed in-phase flow patterns. In addition, the Re number was also found to have a significant impact on the vortex shedding frequencies and flow properties. The drag coefficients, lift coefficients, and St number were calculated to demonstrate the relationship between vortex shedding properties, gap ratio, and Re numbers. The results revealed that the wake flow pattern, flow characteristics, and force coefficients depended significantly on the gap ratio and Re numbers. In addition, the power spectral density of the upstream cylinder were significantly lower than those of downstream cylinders for low gap ratios, and the mutual interference between each cylinders vanished as the gap ratio increased.

Numerical simulations of flow around three cylinders using momentum exchange-based immersed boundary-lattice Boltzmann method

Wake flow behind multiple bluff bodies is of critical significance in ocean engineering and has important scientific significance in fluid mechanics and dynamics. Three cylinders arranged in side-by-side, tandem, equilateral-triangular, and staggered arrangements were extensively researched in the previous study. In this study, the flow passing through three cylinders in staggered and equilateral-triangular arrangement were numerical calculated using a momentum exchange-based immersed boundary-lattice Boltzmann method. The numerical investigations for staggered arrangement were carried out by changing the Reynolds number (Re) from 10 to 200 with an increment of 1, and varying the gap ratio (δ = ΔP/D) from 0.1 to 8.0 with an increment of 0.1. The flow patterns of three equilateral-triangular cylinders with different gap ratios at Re = 100 were also numerical studied. The numerical results identified various flow patterns, and the variations of the above two parameters have a decisive influence on the vortex shedding patterns and flow properties. The drag coefficient, lift coefficient, and Strouhal number (St) were also surveyed to demonstrate the relationship among flow pattern, parameters δ and Re. The outcomes in this paper provide a relatively comprehensive understanding to the vortex shedding patterns and flow properties of three circular cylinders with different gap ratios and Reynolds number.

Numerical investigation on the wave force characteristics of multi-cylinder with equilateral-triangular arrangement

Based on the open source program DualSPHysics, a numerical investigation is carried out on the hydrodynamic characteristics of multi-cylinder with equilateral-triangular arrangement and the wave field distribution characteristics under the action of regular waves. With the SPH method, the hydrodynamic response characteristics of multi-cylinder with equilateral-triangular arrangement are analyzed under different wave directions. The phase difference distribution characteristics of the wave force on multi-cylinder with equilateral-triangular arrangement are analyzed, as well as the wave field distribution characteristics under different wave angles. In addition, the wave force distribution characteristics and the phase difference distribution characteristics of multi-cylinder with equilateral-triangular arrangement are explored under different wave parameters (wave period and wave height). Through the observations of the hydrodynamic forces induced by the wave, the phase difference of these forces on upstream cylinder and downstream cylinder is the largest when the wave angle is 30°. The maximum wave force of the cylinder is roughly proportional to the distance between the center of the cylinder and the x-axis centerline of the wave field.

Numerical simulation of wave run-up on three cylinders in an equilateral-triangular arrangement

Three-dimensional numerical simulations of wave run-up on three circular cylinders in an equilateral-triangular configuration are performed at different spacing ratios 1.5 ≤ L/D ≤ 4.0 (where L is the center-to-center distance and D is the diameter of cylinder). The mechanism of wave superposition in the vicinity of three cylinders is illustrated, the effects of spacing ratio and wave steepness on the wave run-up ratio and wave force on three cylinders are investigated. The results show that with variation of the spacing ratio, different superposition processes among incident waves, lateral edge waves and scattering waves are observed. It should be noted that under the given condition, the significant amplification of wave height is observed among three cylinders at L/D = 2.0 and 2.5. The spacing ratio has a significant effect on the wave run-up ratio at the cylindrical surface and wave force on each cylinder rather than the wave steepness kA.

Global dynamic behavior analysis of typical magnetic pendulum

Based on the analysis of the global dynamic behavior of a typical magnetic pendulum with equilateral triangular arrangement, the initial sensitivity and its mechanism are studied. To begin with, assuming that the position of the magnet can be moved, the dynamical model of a typical magnetic pendulum is established via Newton’s second law. Furthermore, the number of equilibrium points under different magnet positions and their stability are analyzed. Upon this, the initial sensitivity phenomenon and the evolution of fractal basin of attraction of point attractors under different magnet positions are presented. Finally, the initial sensitivity phenomenon is verified experimentally. It is found that the coexistence of multiple attractors generally appears in this type of magnetic pendulum. The initial sensitivity can be attributed to the fractal basin of attraction of fixed point attractors, in which the positions of the fixed point attractors do not overlap with the projected positions of the center of the magnet on the plane where the magnet is located, but there is a slight deviation. When the position of the swing ball can be projected onto the centroids of three equilateral triangles corresponding to the magnets, the sizes of the three attractors’ attraction domains will be similar, whose boundaries are fractal and centrosymmetric, thus, the initial sensitivity is obvious. It also follows that the position of moving magnet affects the nature of basin of attraction directly, i.e., the magnet which is closest to the projection point of the balance position of the swing ball has a great influence on the swing ball: the domain of attraction of the attractor whose is the closest to the position will increase significantly, while the attractor domains of the other attractors will be eroded obviously. This paper has potential applications in designing the magnetic pendulum systems.

Flow around three circular cylinders in equilateral-triangular arrangement

A three-cylinder array arranged in equilateral-triangular configuration represents one of the most common, basic units of multi-cylinder systems in engineering practice. The present study investigates this flow configuration, focusing on the effects of the cylinders’ spacing ratio (P/d = 2.8–5.2) and the array orientation with reference to the oncoming flow (α = 0°–30°), under a fixed Reynolds number 8000 (based on the cylinder diameter d). The hydrodynamic forces (drag and lift) on each cylinder were directly measured using piezoelectric load cells, together with concurrent flow measurements using Particle Image Velocimetry (PIV) technique to quantitatively visualize the shear layer development, vortex formation and interaction process. The flow interference among the cylinders, which shares some similarities with that of two cylinders arranged in side-by-side, tandem or staggered configuration, is further complicated due to the presence of the third cylinder and exhibits many unique features. A number of different flow patterns, including shear layer reattachment, induced separation, synchronized or independent vortex shedding, etc., are identified over the (P/d, α) parameter space. The effects of flow interference are also quantified in terms of Reynolds shear stress distributions around the cylinder array, mean and root-mean-square (RMS) drag/lift coefficients and vortex shedding frequencies of each cylinder.

The near wake of three circular cylinders in an equilateral triangular arrangement at a low Reynolds number Re=100

Two-dimensional numerical simulations are conducted at a low Reynolds number Re = 100 to investigate the near wake of three identical circular cylinders that are arranged in an equilateral triangular configuration. The incident angle of the three-cylinder configuration with respect to incoming flow is varied from θ= 0° to 60°, while the spacing between adjacent cylinders (L) covers a wide range of L/D = 1.25-7.0, where D is diameter of the cylinder. Typical flow structures in the near wake of the three-cylinder configuration are identified, including a single Karman vortex street, bistable flip-flopping near wake, anti-phase and/or in-phase vortex shedding, shear layer reattachment, and vortex impingement, depending on the configuration (L/D, θ). The behavior of Strouhal number (St) is discussed in detail, echoing the distinct structures of near wake. Furthermore, fluid forces on the individual cylinders are examined, which, though highly depending on (L/D, θ), exhibit a close correlation to the near wake behavior.

Oscillation regimes and mechanisms in the vortex-induced vibrations of three circular cylinders with equilateral-triangular arrangements

This paper presents the oscillation regimes and underlying mechanisms of the vortex-induced vibrations (VIVs) of three separated and elastically mounted circular cylinders aligned in equilateral-triangular arrangements with the Reynolds number Re = 100, the mass ratio m* = 2.0, the spacing ratio L* = 2.0–5.0, and the reduced velocity Ur = 0.0–15.0. Three cylinders, with one placed upstream and the other two placed side-by-side downstream, are constrained to vibrate only in the transverse direction. In the examined L* range, the responses of the three cylinders can be divided into five regimes. The features of the vibration amplitudes, vibration frequencies, displacement phase lags, and mean position shifts in each regime are explored in detail. The mechanisms of several important issues regarding the VIV are elucidated. We found that (1) the roles of the separation, reattachment, detachment, merging, and rolling-up of the shear layer in sustaining the cylinder vibrations differ in each regime. (2) The significant increase in the amplitudes of the two downstream cylinders in regime II stems from the vorticity supply from the upstream cylinder. (3) The low vibration frequency of the upstream cylinder in an anti-phase flow is closely related to the presence of a slightly asymmetric wake of the downstream cylinders. (4) The large-amplitude vibrations of the two downstream cylinders in regime V at L* = 2.0 is maintained by the shear layer reattachment of the upstream cylinder and the pushing effects between the two downstream cylinders. (5) Finally, the different incoming flow velocities for the upstream and downstream cylinders may cause different vibration frequencies.

Numerical simulations of flow past three circular cylinders in equilateral-triangular arrangements

Flow past three identical circular cylinders is numerically investigated using the immersed boundary method. The cylinders are arranged in an equilateral-triangle configuration with one cylinder placed upstream and the other two side-by-side downstream. The focus is on the effect of the spacing ratio , Reynolds number and three-dimensionality on the flow structures, hydrodynamic forces and Strouhal numbers, where is the cylinder centre-to-centre spacing and is the cylinder diameter. The fluid dynamics involved is highly sensitive to both and , leading to nine distinct flow structures, namely single bluff-body flow, deflected flow, flip-flopping flow, steady symmetric flow, steady asymmetric flow, hybrid flow, anti-phase flow, in-phase flow and fully developed in-phase co-shedding flow. The time-mean drag and lift of each cylinder are more sensitive to than while fluctuating forces are less sensitive to than . The three-dimensionality of the flow affects the development of the wake patterns, changing the ranges of different flow structures. A diagram of flow regimes, together with the contours of hydrodynamic forces, in the space, is given, providing physical insights into the complex interactions of the three cylinders.

Hydrodynamic features of three equally spaced, long flexible cylinders undergoing flow-induced vibration

Abstract Three equally spaced flexible cylinders are frequently applied in many engineering fields. The flow-induced vibration (FIV) hydrodynamic features of three such cylinders are not the same as those of an isolated single one and remain unknown. In this paper, the hydrodynamic coefficients for three flexible cylinders subjected to FIV in an equilateral-triangular arrangement with a centre-to-centre spacing of 6 diameters were identified using an inverse analysis method according to the displacement response data obtained from model tests. The lift coefficient, varying drag coefficient and added mass coefficient at the dominant frequency were calculated by decomposing the cross-flow (CF) and in-line (IL) fluctuating forces. Three typical cases of an equilateral-triangular configuration (A, B, and C), which correspond to flow incidence angles of 0 ∘ , 30 ∘ , and 60 ∘ , were studied and discussed (the incidence angle is defined as the angle between the flow orientation and the line linking the centre points of one cylinder and the equilateral-triangular configuration). The hydrodynamic coefficients of the upstream cylinders are insignificantly influenced by the downstream cylinders. In contrast, the wake of the upstream cylinders impose a notable effect on the IL hydrodynamic coefficients of the downstream cylinders. Two robust frequency components ( f v , I L and f v , I L _ 1 ∕ 2 ) were observed in the IL vibrations of the downstream cylinders. The IL hydrodynamic forces at f v , I L have similar features to the classical vortex shedding forces of an isolated flexible cylinder. However, the IL hydrodynamic forces at f v , I L _ 1 ∕ 2 exhibit distinct behaviours that are closely related to the unique response characteristics in the IL direction. In addition, the IL fluctuating force coefficients and varying drag coefficients at f v , I L _ 1 ∕ 2 are relatively small and change slowly with the reduced velocity.

Three-dimensional numerical simulation on flow past three circular cylinders in an equilateral-triangular arrangement

Flow past three circular cylinders in an equilateral-triangular configuration is simulated for Reynolds numbers 200 ≤ Re ≤ 3900 and spacing ratios 1.25 ≤ L/D ≤ 4.0, where L is the center-to-center distance between each two cylinders and D is the diameter of the cylinder. Five different flow regimes are identified: proximity with deflected flow (P-D), proximity with flip-flop flow (P–F), dual vortex shedding (DV), transition between dual vortex shedding and triple vortex shedding (DV-TV), triple vortex shedding (TV). The boundary spacing ratios between the five regimes are affected by the Reynolds number. With the increase of the spacing ratio or the Reynolds number, the three-dimensionality of the wake flow becomes stronger. Mode B flow pattern with streamwise vortices is found at L/D = 1.25, 3.5 and 4.0 for Re = 1500 and L/D = 1.25 for 3900. The pressure coefficient on the surface of the downstream cylinder C3 is significantly different from that on a single cylinder, leading to significant reduction of the mean drag coefficient on C3 for all the studied spacing ratios and Reynolds numbers. The Strouhal numbers of C1 and C3 are the same and smaller than that of C2 in DV and TV regimes.

Flow-induced vibrations of three circular cylinders in an equilateral triangular arrangement subjected to cross-flow

Vortex-induced vibration of three circular cylinders (each of diameter D) in an equilateral triangular arrangement is investigated using the immersed boundary method. The cylinders, with one placed upstream and the other two side-by-side downstream, are free to vibrate in the cross-flow direction. The cylinder center-to-center spacing L is adopted as L/D = 2.0. Other parameters include the Reynolds number Re = 100, mass ratio m* = 2.0, reduced velocity Ur = 2 ~ 15 and damping ratio  = 0. Cylinder vibration responses are dependent on Ur and classified into five regimes, i.e. Regime I (Ur 9.2). Different facets of vibration amplitude, hydrodynamic forces, wake patterns and displacement spectra are extracted and presented in detail for each regime.

Dynamic response of three long flexible cylinders subjected to flow-induced vibration (FIV) in an equilateral-triangular configuration

A three-cylinder system with an equilateral-triangular configuration represents one of the most common, basic units of multi-cylinder systems in many engineering fields. In this paper, an experimental investigation of the flow-induced vibration (FIV) of a three-cylinder system in an equilateral-triangular arrangement was conducted in a towing tank to study their response characteristics. Three identical flexible cylinder models with a centre-to-centre distance of 6.0 times the cylinder diameter were towed along the tank to generate a uniform flow. The Reynolds number ranged from approximately 1600 to 16000 depending on the towing velocity, which was in the range of 0.10–1.00 m/s. The time-varying bending strains of the flexible cylinders in the cross-flow (CF) and in-line (IL) directions were measured using strain gauges at different measurement positions. The displacement responses were reconstructed by a modal analysis method based on the acquired strain signals. The experimental results (e.g., displacement amplitudes, response frequencies, dominant modes and motion trajectories of the flexible cylinders) were presented to reveal the FIV response features. Three typical cases of the equilateral-triangular configuration (A, B and C) corresponding to flow incidence angles of 0°, 30°, and 60° were selected and discussed. (The incidence angle is the angle between the flow direction and the line joining the centre of one cylinder to the centre of the equilateral-triangular configuration.) In Cases A and B, the upstream cylinder is insignificantly affected by the downstream cylinders and behaves like an isolated flexible cylinder; meanwhile, the upstream cylinders in Case C vibrate similarly to two side-by-side flexible cylinders with weak mutual interference. Striking departures exist in the IL oscillations of the downstream cylinders due to the notable effects of the upstream cylinders. The phenomenon of mixed modes appears in the IL vibrations of the downstream cylinders. Furthermore, the IL response spectra always contain a robust lower frequency component equal to the dominant frequency in the CF direction. Moreover, eight-, half-moon- and C-shaped vibration figures are comparatively uncommon for the downstream cylinders.