# Vortex-induced vibrations of three staggered circular cylinders at low Reynolds numbers

TL;DR: In this paper, the authors investigated the vortex-induced vibrations of three staggered circular cylinders via two-dimensional finite element computations and found that the initial and lower synchronization response modes like an isolated cylinder does at low Re, whereas for both downstream cylinders, the upper lock-in branch also appears.

Abstract: Vortex-induced vibrations of three staggered circular cylinders are investigated via two-dimensional finite element computations. All the cylinders are of equal diameter (D) and are mounted on elastic supports in both streamwise (x−) and transverse (y−) directions. The two downstream cylinders are placed symmetrically on either side of the upstream body at a streamwise gap of 5D, with the vertical distance between them being 3D. Flow simulations are carried out for Reynolds numbers (Re) in the range of Re = 60-160. Reduced mass (m*) of 10 is considered and the damping is set to zero value. The present investigations show that the upstream cylinder exhibits initial and lower synchronization response modes like an isolated cylinder does at low Re. Whereas for both the downstream cylinders, the upper lock-in branch also appears. The initial and the upper modes are characterized by periodic oscillations, while the lower lock-in branch is associated with nonperiodic vibrations. The 2S mode of vortex shedding i...

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TL;DR: In this article, the amplitude and frequency ratio curves of three cylinders with roughness were numerically studied by 2D-URANS simulations in Reynolds number range of 30,000 < Re < 105,000.

Abstract: The flow-induced vibration (FIV) of multiple cylinders is a common phenomenon in industry and nature. The FIV and energy harvesting of three circular cylinders in tandem are numerically studied by 2D-URANS simulations in Reynolds number range of 30,000 < Re < 105,000. Simulation results match well with experiments in the tested cases. Four branches of FIV are clearly captured in the amplitude and frequency ratio curves of the three cylinders with roughness, including initial branch of vortex-induced vibration (VIV), VIV upper branch, transition from VIV to galloping, and galloping. It is shown that the vortices from downstream cylinder are strongly disrupted and modified by vortices of upstream cylinder. The third cylinder is almost suppressed in VIV initial branch. The 2P vortex pattern is observed for the first cylinder in the VIV upper branch. For Re = 90,000 in the transition regime, the vortex patterns of the first and second cylinders are 2P + 4S and 2P + 2S, respectively. In the galloping branch, the shear layer motion is in synchronization with the motion of the cylinder, and the maximum amplitude of 2.8D is reached by the first cylinder. The total converted power of the three cylinders increases with U*water both in the simulation and experiment. For the three cylinders, the maximum power reaches up to 85.26 W with the increase of Reynolds number. The energy conversion efficiency is stable and higher than 35% in the starting region of VIV upper branch, and the maximum value of 40.41% is obtained when Re = 40,000.

22 citations

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TL;DR: In this article, the effects of the number and arrangement of the fins on the vortex shedding pattern, vibration amplitude, and frequency and heat transfer of a finned cylinder with heat transfer were investigated and discussed.

Abstract: Two-degree-of-freedom vortex-induced vibration (VIV) of a finned cylinder with heat transfer is studied numerically at the Reynolds number Re = 150. The governing equations in the Arbitrary Lagrangian-Eulerian frame are solved by the finite volume method. The dynamics of the oscillating cylinder (with or without fins) in the fluid flow was approximated as a mass-spring system. The effects of the number and arrangement of the fins (14 different cases) on the vortex shedding pattern, vibration amplitude, and frequency and heat transfer of the cylinder are investigated and discussed. The results indicate that in comparison with the stationary state, the effects of the number and arrangement of the fins on the wake pattern and the heat transfer enhancement in the VIV state are significant. Different vortex shedding pattern like 2S, P, 2P, S + P and combination of them with stable or unstable interactions between vortices and cylinders are observed in an oscillating cylinder. In the vibration state of finned cylinders, the heat transfer enhances up to 50.4% with respect to the stationary state and increases up to 64% with respect to the stationary smooth cylinder.Two-degree-of-freedom vortex-induced vibration (VIV) of a finned cylinder with heat transfer is studied numerically at the Reynolds number Re = 150. The governing equations in the Arbitrary Lagrangian-Eulerian frame are solved by the finite volume method. The dynamics of the oscillating cylinder (with or without fins) in the fluid flow was approximated as a mass-spring system. The effects of the number and arrangement of the fins (14 different cases) on the vortex shedding pattern, vibration amplitude, and frequency and heat transfer of the cylinder are investigated and discussed. The results indicate that in comparison with the stationary state, the effects of the number and arrangement of the fins on the wake pattern and the heat transfer enhancement in the VIV state are significant. Different vortex shedding pattern like 2S, P, 2P, S + P and combination of them with stable or unstable interactions between vortices and cylinders are observed in an oscillating cylinder. In the vibration state of finned c...

14 citations

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TL;DR: In this paper, the wake and oscillation responses and wake modes of three staggered rotating cylinders, free to move in streamwise and transverse directions, are numerically studied in two-dimensional and three-dimensional (3-dimensional) flows.

Abstract: Oscillation responses and wake modes of three staggered rotating cylinders, free to move in streamwise and transverse directions, are numerically studied in two- (2-D) and three-dimensional (3-D) flows. 2-D computations are carried out for Reynolds number Re = 60–150, employing the following rotation rates (α), respectively, for the upstream, upper, and lower downstream cylinders: 1, 1, 0; 1, 1, 1; 1, 1, −1. Here, the clockwise rotation is positive. 3-D simulations are performed at Re = 2000 and reduced velocity, U* = 2–11, with the three cylinders being rotated at α = 1. Bell-shaped amplitude profiles are observed for all the rotating cylinders, indicating that the bodies undergo vortex-induced vibrations. In 2-D flow, the considered Re regime can be categorized into three distinct regions, based on the oscillation and frequency responses. Cylinders exhibit negligible amplitudes in the first region, whereas the second region is characterized by high amplitude lock-in oscillations for all three cylinders. In the third region, the downstream cylinders exhibit lock-in response in certain rotation configurations. The oscillation responses and wake modes appear sensitive to the direction of rotation of the lower downstream cylinder for the streamwise and transverse gaps of 5 diameters (5D) and 3D, respectively, between the cylinders. Depending on the rotation configuration, 2S, P, and P + S modes of primary shedding are observed. In 3-D flow also, the cylinders exhibit bell-shaped amplitude profiles, contrary to the galloping response noticed for isolated rotating cylinders in few previous studies. Higher and lower amplitude oscillations induce stronger and weaker 3-D flow instabilities, respectively, in the wake region.Oscillation responses and wake modes of three staggered rotating cylinders, free to move in streamwise and transverse directions, are numerically studied in two- (2-D) and three-dimensional (3-D) flows. 2-D computations are carried out for Reynolds number Re = 60–150, employing the following rotation rates (α), respectively, for the upstream, upper, and lower downstream cylinders: 1, 1, 0; 1, 1, 1; 1, 1, −1. Here, the clockwise rotation is positive. 3-D simulations are performed at Re = 2000 and reduced velocity, U* = 2–11, with the three cylinders being rotated at α = 1. Bell-shaped amplitude profiles are observed for all the rotating cylinders, indicating that the bodies undergo vortex-induced vibrations. In 2-D flow, the considered Re regime can be categorized into three distinct regions, based on the oscillation and frequency responses. Cylinders exhibit negligible amplitudes in the first region, whereas the second region is characterized by high amplitude lock-in oscillations for all three cylinders....

6 citations

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TL;DR: An iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace.

Abstract: We present an iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from t...

10,155 citations

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TL;DR: In this paper, a review summarizes fundamental results and discoveries concerning vortex-induced vibration (VIV) that have been made over the last two decades, many of which are related to the push to explore very low mass and damping, and to new computational and experimental techniques that were hitherto not available.

Abstract: This review summarizes fundamental results and discoveries concerning vortex-induced vibration (VIV), that have been made over the last two decades, many of which are related to the push to explore very low mass and damping, and to new computational and experimental techniques that were hitherto not available. We bring together new concepts and phenomena generic to VIV systems, and pay special attention to the vortex dynamics and energy transfer that give rise to modes of vibration, the importance of mass and damping, the concept of a critical mass, the relationship between force and vorticity, and the concept of "effective elasticity," among other points. We present new vortex wake modes, generally in the framework of a map of vortex modes compiled from forced vibration studies, some of which cause free vibration. Some discussion focuses on topics of current debate, such as the decomposition of force, the relevance of the paradigm flow of an elastically mounted cylinder to more complex systems, and the relationship between forced and free vibration.

1,619 citations

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TL;DR: In this paper, it was shown that the acceleration of the cylinder each half cycle induces the roll-up of the two shear layers close to the body, and thereby the formation of four regions of vorticity each cycle.

Abstract: When a body oscillates laterally (cross-flow) in a free stream, it can synchronize the vortex formation frequency with the body motion frequency. This fundamental “lock-in” regions is but one in a whole series of synchronization regions, which have been found in the present paper, in an amplitude-wavelength plane (defining the body trajectory) up to amplitudes of five diameters. In the fundamental region, it is shown that the acceleration of the cylinder each half cycle induces the roll-up of the two shear layers close to the body, and thereby the formation of four regions of vorticity each cycle. Below a critical wavelength, each half cycle sees the coalescence of a pair of like-sign vortices and the development of a Karman-type wake. However, beyond this wavelength the like-sign vortices convect away from each other, and each of them pairs with an opposite-sign vortex. The resulting wake comprises a system of vortex pairs which can convect away from the wake centerline. The process of pairing causes the transition between these modes to be sudden, and this explains the sharp change in the character of the cylinder forces observed by Bishop and Hassan, and also the jump in the phase of the lift force relative to body displacement. At precisely the critical wavelength, only two regions of vorticity are formed, and the resulting shed vorticity is more concentrated than at other wavelengths. We interpret this particular case as a condition of “resonant synchronization”, and it corresponds with the peak in the body forces observed in Bishop and Hassan's work.

1,187 citations

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TL;DR: In this paper, the authors showed that there exist two distinct types of response in a very low mass and damping regime, depending on whether one has a low combined mass-damping parameter (low m*ζ), or a high mass-ding parameter (highm*δ).

Abstract: These experiments, involving the transverse oscillations of an elastically mounted rigid cylinder at very low mass and damping, have shown that there exist two distinct types of response in such systems, depending on whether one has a low combined mass-damping parameter (low m*ζ), or a high mass-damping (highm*ζ ). For our low m*ζ, we find three modes of response, which are denoted as an initial amplitude branch, an upper branch and a lower branch. For the classical Feng-type response, at highm*ζ , there exist only two response branches, namely the initial and lower branches. The peak amplitude of these vibrating systems is principally dependent on the mass-damping (m*ζ), whereas the regime of synchronization (measured by the range of velocity U*) is dependent primarily on the mass ratio, m*ζ. At low (m*ζ), the transition between initial and upper response branches involves a hysteresis, which contrasts with the intermittent switching of modes found, using the Hilbert transform, for the transition between upper–lower branches. A 180° jump in phase angle φ is found only when the flow jumps between the upper–lower branches of response. The good collapse of peak-amplitude data, over a wide range of mass ratios (m*=1–20), when plotted against (m*+CA) ζ in the “Griffin” plot, demonstrates that the use of a combined parameter is valid down to at least (m*+CA)ζ ∼0·006. This is two orders of magnitude below the “limit” that had previously been stipulated in the literature, (m*+CA) ζ>0·4. Using the actual oscillating frequency (f) rather than the still-water natural frequency (fN), to form a normalized velocity (U*/f*), also called “true” reduced velocity in recent studies, we find an excellent collapse of data for a set of response amplitude plots, over a wide range of mass ratiosm* . Such a collapse of response plots cannot be predicted a priori, and appears to be the first time such a collapse of data sets has been made in free vibration. The response branches match very well the Williamson–Roshko (Williamson & Roshko 1988) map of vortex wake patterns from forced vibration studies. Visualization of the modes indicates that the initial branch is associated with the 2S mode of vortex formation, while the Lower branch corresponds with the 2P mode. Simultaneous measurements of lift and drag have been made with the displacement, and show a large amplification of maximum, mean and fluctuating forces on the body, which is not unexpected. It is possible to simply estimate the lift force and phase using the displacement amplitude and frequency. This approach is reasonable only for very low m*.

812 citations

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TL;DR: In this paper, a new strategy based on the stabilized space-time finite element formulation is proposed for computations involving moving boundaries and interfaces, where the deformation of the spatial domain with respect to time is taken into account automatically.

Abstract: A new strategy based on the stabilized space-time finite element formulation is proposed for computations involving moving boundaries and interfaces. In the deforming-spatial-domain/space-time (DSD/ST) procedure the variational formulation of a problem is written over its space-time domain, and therefore the deformation of the spatial domain with respect to time is taken into account automatically. Because the space-time mesh is generated over the space-time domain of the problem, within each time step, the boundary (or interface) nodes move with the boundary (or interface). Whether the motion of the boundary is specified or not, the strategy is nearly the same. If the motion of the boundary is unknown, then the boundary nodes move as defined by the other unknowns at the boundary (such as the velocity or the displacement). At the end of each time step a new spatial mesh covers the new spatial domain. For computational feasibility, the finite element interpolation functions are chosen to be discontinuous in time, and the fully discretized equations are solved one space-time slab at a time.

801 citations