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S0997-7546(15)20074-X http://dx.doi.org/10.1016/j.euromechflu.2015.09.001 EJMFLU 2932

To appear in:

European Journal of Mechanics B/Fluids

Received date: 12 March 2015 Revised date: 2 July 2015 Accepted date: 1 September 2015 Please cite this article as: M.-H. Chung, Transverse vortex-induced vibration of spring-supported circular cylinder translating near a plane wall, European Journal of Mechanics B/Fluids (2015), http://dx.doi.org/10.1016/j.euromechflu.2015.09.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Transverse Vortex-induced Vibration of Spring-supported Circular Cylinder Translating near a Plane Wall Meng-Hsuan Chung Department of Naval Architecture and Ocean Engineering, National Kaohsiung Marine University,

No.142, Haijhuan Rd., Nanzih District, Kaohsiung City 811, Taiwan, ROC

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ABSTRACT Transverse vortex induced vibration of a spring-supported circular cylinder of low mass and zero damping translating near a plane wall at Re = 100 is numerically studied. We investigate three gap ratios. Results show that the size of lock-in zone increases and the peak vibration amplitude decreases with decreasing gap ratio. The peak vibration amplitude occurs at a larger reduced velocity for a smaller gap ratio. Wall proximity suppresses the beating phenomena of cylinder displacement and lift. The predominant vibration frequency is always equal to the vortex shedding frequency. The cylinder vibration in the lock-in zone is controlled by either the Strouhal frequency or the natural structure frequency in fluid, depending on the gap ratio and reduced velocity. The time-mean drag in the lock-in zone is always larger than that for an isolated non-vibrating (purely translating) cylinder. The time-mean lift is always positive. For an isolated cylinder, the phase lag of displacement behind lift is predicted in theory and is weakly correlated with the vortex shedding pattern. For a near-wall cylinder, both the minimal gap and vibration amplitude are important in determining the vortex shedding pattern. Impact with wall causes no peculiar amplitude and frequency of cylinder vibration.

1

Keywords: Vortex induced vibration; Gap ratio; Low mass ratio; Translating circular cylinder; Cartesian grid method

1. Introduction Uniform flow over a stationary circular cylinder has attracted much interest among researchers. Vortex shedding in the wake of the circular cylinder frequently occurs and causes periodic forcing to the cylinder. If the cylinder is allowed to vibrate freely in the flow, the vortex shedding and the cylinder motion will influence each other, eventually reaching a state of balanced vibration, called vortex induced vibration (VIV). The term “lock-in” denotes the occurrence of large vibration amplitude in VIV. The characteristics of the lock-in zone and the wake vortex structure would change significantly when the cylinder is close to a plane wall. Below the related literature is introduced under various topics, which are classified according to whether the plane wall exists or not. VIV of an isolated circular cylinder VIV of an isolated circular cylinder, rigid or flexible, has been studied extensively in the literature. The parameters involved are the mass ratio m* (= m/m d ), damping ratio ζ (= c/c crit ), reduced velocity U* (= U/f nw D), and Re (= UD/ν) where m = cylinder mass, m d = displaced fluid mass, c = structural damping, c crit = critical damping, U = free-stream velocity, f nw = natural structure frequency in fluid, D = cylinder diameter, and ν = kinematic viscosity. Some researchers defined the reduced velocity based on other velocity scales and/or structural natural frequency in vacuum. Related studies have been reviewed and discussed by Sarpkaya (2004), Williamson and Govardhan (2004), and Bearman (2011). Particularly, there have been many publications on VIV of a spring-supported rigid cylinder at low mass-damping constrained to move transversely in a uniform free stream. Much progress has been made by Prof. Williamson's group with a series of physical experiments (Khalak and Williamson, 1996, 1999; Govardhan and Williamson, 2000). The main results can be summarized as follows. For a cylinder with low m*ζ, the response amplitude A versus the free-stream velocity U presents three distinct response branches; namely the initial branch, the upper branch and the lower branch. In the upper branch, the vibration amplitude can be twice as large as that for the classical high mass-damping case of Feng (1968). Meanwhile, the “lock-in” zone is dramatically extended in contrast to that of Feng (1968). Moreover, 2

there is a correspondence of the 2S mode of vortex shedding (two single vortices shed per cycle) with the initial branch and the 2P mode of vortex shedding (two pair vortices shed per cycle) with the lower branch. The 2P mode was also observed in the upper branch, but the second vortex of each pair is much weaker than the first one. Some numerical studies have also reported the 2P mode (Blackburn et al., 2000; Lucor et al., 2005). Williamson and Govardhan (2008) briefly summarized fundamental results and discoveries related to VIV with very low mass-damping. In recent years, more and more researchers investigated VIV by computational fluid dynamics (CFD) techniques, e.g., Guilmineau and Queutey (2004) and Wanderley et al. (2008). Al Jamal and Dalton (2005) have reviewed some numerical studies on VIV of a circular cylinder. VIV of a circular cylinder near a fixed plane wall in a free stream Two additional parameters have to be considered for this problem. The first is the gap ratio, G, defined as the distance between the cylinder bottom and the wall in the static equilibrium condition (i.e., when the spring force keeps zero with quiescent ambient fluid) normalized by D. The second is the wall boundary layer profile. In some works, the Reynolds number has been alternatively defined in terms of either the approach velocity at the cylinder center position or the time-mean velocity. For two-degree-of-freedom (2-dof) free span VIV (Tsahalis and Jones, 1981), 2-dof rigid-cylinder VIV (Jacobsen et al., 1984), and single-degree-of-freedom (1-dof) free span VIV (To̸rum and Anand, 1985), it was found that the presence of a plane boundary lowers the vibration amplitude . However, Yang et al. (2006) reported that the vibration amplitude increases with decreasing gap ratio. Raghavan et al. (2009) indicated that the vibration amplitude as function of gap ratio depends strongly on the Reynolds number and the wall boundary layer. Therefore, the correlation between the vibration amplitude and the gap ratio is still unclear due to insufficient exploration of these influential factors. On the other hand, the vibration frequency as a function of the reduced velocity also differs among various studies (Fredso̸e et al., 1987; Raghavan et al., 2009). Yang et al. (2006) indicated that both the onset reduced velocity and the width of the lock-in zone increase with decreasing gap ratio. Raghavan et al. (2009) discovered that the onset of lock-in is gradual in near-wall cases while abrupt in isolated-cylinder cases and that the range of lock-in zone shifts to higher reduced velocities for smaller gap ratios. 3

Both Zhao and Cheng (2011) and Wang et al. (2013) reported significant rigid-cylinder VIVs even if the gap ratio is lowered down to 0.05, in contrast to the case of a stationary cylinder that vortex shedding is suppressed when G ≈ 0.3. Due to the proximity of the plane wall, the vortices shed from the vibrating cylinder form a single-side vortex street. Purely translating circular cylinder near a fixed plane wall Under this topic, experimental works include low-Re (Taneda, 1965) and high-Re (Zdravkovich, 2003; Nishino et al., 2007) studies; numerical works include low-Re (Yoon et al., 2007; Yoon et al., 2010; Huang and Sung, 2007; Rao et al., 2013) and high-Re (Bimbato et al., 2011) studies. In summary, the time-mean drag coefficient varies with the gap ratio in a way strongly depending on the Reynolds number and the gap ratio. When the cylinder is located in close proximity to the wall, the time-mean drag coefficient exhibits inconsistent variations with the gap ratio for some Reynolds number (105) between the works of Nishino et al. (2007) and Bimbato et al. (2011). The time-mean lift coefficient increases with decreasing gap ratio for low Reynolds numbers (100 ≤ Re ≤ 600) but varies in a complicated way with the gap ratio for a higher Reynolds number (105). In all the previous studies has been observed the gradual suppression of the Kármán-type vortex shedding as the gap ratio decreases. Some studies reported vortex shedding, at least single-side, even for small gap ratios, e.g., G = 0.1; some however observed a total cease of vortex shedding for G ≤ 0.3. The difference would originate from different Reynolds numbers. We wonder whether these hydrodynamic characteristics persist if the cylinder is allowed to freely vibrate, as treated in the present study. Present study - VIV of a translating circular cylinder near a fixed plane wall Focusing on low Reynolds numbers, we studied by computational fluid dynamics techniques the 1-dof VIV of a transversely spring-supported low-mass circular cylinder which is translating near a fixed plane wall. The structural damping was assumed zero to excite high-amplitude vibrations. To the author’s knowledge, similar works have not been found in the literature. For numerical computations, the original scenario is replaced by an equivalent one where a uniform flow passes the cylinder and the wall moves, both with the translating velocity. All the quantities in this work are made dimensionless by taking the cylinder diameter D, incoming velocity (originally translating velocity) U, and fluid density ρ f as the 4

characteristic length, velocity, and density respectively. Fig. 1 depicts the configuration of the physical problem, computational domain, and boundary conditions. For 2-dof VIV of a circular cylinder near a fixed flat plate in a free stream, Tsahalis and Jones (1981) and Jacobsen et al. (1984) observed much larger (approximately 10 times) amplitudes in the transverse direction than in the in-line direction. The present assumption of 1-dof VIV in the transverse direction is thus justified though the wall is moving. There are two major motives of the present study. Firstly, the problem configuration can serve as a preliminary model for objects moving near a ground. Examples include front/rear wings mounted on a racing car to create downforce, splitters and vortex generators attached on the car underside to increase downforce and/or reduce drag, actuator arm/slider/head above a spinning disk in a hard disk drive, wing structure of wing-in-ground (WIG) craft, etc. The present study can assist in understanding possible VIVs in these examples though the regime of Reynolds number, shape of object (circle), and dimensional complexity (two), of the present study are different from or simpler than those in the above examples. Secondly, for the scenario of uniform flow over a circular cylinder near a fixed wall with low Reynolds numbers, there are many applications inluding slurry flow past marine structures/pipelines near a sea floor or river bed and flow over heat exchanger tubes near a wall. However, the effects of wall boundary layer on various flow characteristics are still vague due to difficulty in parameterizing the wall boundary layer profile in a definite way. To clarify, at least partially, the correlation of the cylinder response with each influential factor, we replaced this scenario with the present one (cylinder translating near a fixed wall) to examine only the effects of wall proximity, excluding the effects of wall boundary layer. Finally, the approach to varying the reduced velocity is different between the present study and all the earlier experimental works, to the author’s knowledge. That is, when the reduced velocity is altered, the Reynolds number keeps constant for the former and changes linearly with the reduced velocity for the latter. Comparisons of various VIV characteristics as function of the reduced velocity between the present and earlier studies with varying Reynolds number are therefore irrelevant if the Reynolds number effects cannot be neglected in the considered range of reduced velocity.

2. Methodology 2.1 Fluid Flow Solver 5

The Cartesian grid method with a cut cell approach (Chung, 2006; 2008) was selected. It is characterized by a cell-centered collocated finite volume Cartesian grid with AMR (Adaptive Mesh Refinement). However, we improved the method in two aspects. 2.1.1 Discretization in time To increase the global order of accuracy in time to two, the Crank-Nicholson scheme is used for both the convection and diffusion terms. That is, for either term, the value at the time step (n+1/2) is evaluated as the average of those at the numerical time step (n) and (n+1). The previous-time-step values of solution variables at newly-exposed cell centers are thus required and approximated as those at currently nearest solid-boundary-segment center. This approximation could locally spoil temporal order of accuracy but globally the order will remain two. The Crank-Nicholson scheme is as unconditionally stable as the fully implicit scheme in terms of linear stability analysis. Though stability could be more destroyed for the former than for the latter in solving nonlinear coupled partial differential equations like the Navier-Stokes equations in the present study, the extent of stability region of the Crank-Nicholson scheme seems comparable to the fully implicit scheme. 2.1.2 Refinement/derefinement strategy In this work, each sub-grid block consists of 2×2 mesh cells. Two types of refinement/derefinement criterion work together to determine the refinement level, l, of a specific leaf grid block, e.g., the grid block Q comprising 2×2 cells centered at Q m (m = 1, 2, 3, 4) as shown in Fig. 2. In the first type of criterion, the target refinement level, l 1 , depends on the maximum magnitude of the four velocity max ( ∂u j ∂xi ) , normalized by a reference G1 = gradients over all cell centers in the grid block, Q =i 1,2 = and j 1,2 m =1,2,3,4

m

magnitude of gradient, G 1 , ref . The velocity gradient at any cell center is evaluated using the central difference scheme. For example, the finite difference stencil A-Q 2 -B-Q 3 is involved in the calculation of

( ∂u

j

∂xi

)

Q1

. Specifically, let l max and l min denote respectively the maximal and minimal refinement levels

preset before simulation and G1 = G1 G1,ref . Then l 1 = l max -l p +1 where l p is determined such that 1 ≤ l p ≤ l max -l min +1 and H(l p ) ≤ G 1 ≤ H(l p -1). The delimiting function H is defined as H(0) = ∞, H(q) = (1/B 1 )q for 1 ≤ q ≤ l max -l min , and H(l max -l min +1) = 0. The fraction base, B 1 > 1, is a numerical parameter to be chosen 6

according to the simulation problem. If G1,max denotes the largest value of G1 across all the leaf blocks in the computational domain, the reference magnitude of gradient, G 1,ref , is assigned the time-mean value of G1,max ’s over all the previous time steps. In the second type of criterion, which is applicable if all the siblings of the considered leaf grid block are leaf bocks, the information from the parent grid block is utilized. That is, we evaluate at each parent cell center the difference of the derived velocity gradient between two finite-difference approximations. For example, as the parent of the grid block Q, the grid block P comprises 2×2 cells centered at P m (m = 1, 2, 3, 4) as shown in Fig. 2. Taking the parent cell center P 4 for illustration, the first approximation evaluates the gradient by the central difference scheme using the difference stencil pertaining to the parent refinement level, that is, P 3 -R 3 -P 2 -S 2 . The second approximation calculates by the central difference scheme the gradients at four nearest child cell centers (Q 1 , Q 2 , Q 3 , and Q 4 ) using child-level stencils as doing in the first type of criterion. For example, the finite difference stencil A-Q 2 -B-Q 3 is involved in the calculation of velocity gradients at Q 1 . The average of the gradients at Q 1 , Q 2 , Q 3 , and Q 4 gives the gradient at the parent cell center P 4 . Finally, among all the four cell centers of the parent block, P 1 , P 2 , P 3 , and P 4 , we take the largest difference between the two approximations as the indication of refinement/derefinement , G 2 , for each of the four child grid = G 2 max ( ∂u j ∂xi ) − blocks of the parent block P. That is, P =i 1,2 = and j 1,2 m

m =1,2,3,4

(

1 4 ∑ ∂u j ∂xi 4 k =1

)

k − th child cell center near Pm

. We can locate

G 2 in the refinement hierarchy using a procedure similar to that in the first type of criterion. That is,

having calculated the normalized indication, G2 = G 2 G2,ref , where G 2,ref is the time-mean value of G 2,max ’s over all the previous time steps, the target refinement level, l 2 , can be determined by defining a

delimiting function with a suitably chosen fraction base, B 2 . Finally, the next-time-step refinement level of the domain covered by the considered leaf grid block is taken as the larger of l 1 and l 2 . This way finer meshes would be deployed in regions with larger magnitude of the velocity gradient or larger difference of the velocity gradient between the coarse- and fine- level evaluations. In contrast to the vorticity controlled criterion used in Chung (2008), the present strategy generates a little more meshes but much more consistency in the evaluation of solution variables at the interface between the coarse and 7

fine meshes, eventually assuring more stable solutions. In all the following simulations, the settings with B 1 = 4 and B 2 = 2 worked well for stable solutions.

2.2 Fluid-solid Interaction The cylinder is rigid, streamwise-fixed, and transversely supported by linear springs with uniform structural damping. A fluid-solid interaction is therefore involved in this physical problem. The dimensionless equation of motion for the 1-dof motion of the circular cylinder is myc + cy c + kyc = Fy ,hydro

(1)

where y c denotes the coordinate of the centroid of the cylinder in the transverse direction relative to the static equilibrium position and the dot symbol represents the time derivative. The cylinder, which has mass m, is supported by a spring of constant stiffness k. The uniform structural damping of the supporting system is c. The ambient fluid exerts the transverse hydrodynamic force F y,hydro to the cylinder. The trapezoidal method, which is a classical second-order implicit method, was used to integrate the equations of motion. In the proposed Cartesian grid method, numerically parasitic temporal oscillation of the hydrodynamic force is unavoidable for moving solid problems. A second-order polynomial, which is the least-square approximation of force data at the current and latest three time steps, was therefore used to extrapolate the value of F y,hydro at the next time step. Meanwhile, the centroid velocity exhibits larger numerical oscillation than the centroid displacement because the former is the temporal derivative of the latter. To overcome this difficulty, we ignored the value of centroid velocity calculated by the trapezoidal method. Instead, a second-order polynomial fitted to the three displacement data at the next time, current, and previous time steps was constructed, and the first derivative of this polynomial was assigned to the centroid velocity at the next time step.

2.3 Structural Parameters A mass-spring-damper system is usually characterized by another set of three parameters: mass ratio, m*, natural structure frequency in fluid, f nw , and damping ratio, ζ. Their definitions are m∗ ≡

m , md

(2)

8

f nw ≡

ζ ≡

1 2π

k m + mA

(3)

c ( ccrit ≡ 2 k ( m + m A ) ). ccrit

(4)

From the definition of reduced velocity, U∗ ≡

U , f nw D

(5)

we can obtain U* = 1 / f nw because of the present procedure of nondimensionalization. Therefore, m = m∗ ⋅ md , k =

(6) 2π , ∗ U

( m + mA )

c = 4π

2

(7)

( m + mA ) ζ

(8)

U∗

where m A is the added mass. Because m A depends on the motion of the cylinder which is a part of solution to the fluid-solid interaction problem, we regard it as a reference added mass and simply set m A = m d . The natural structure frequency in fluid is thus a nominal frequency and the real frequency has to be solved by numerical simulation.

2.4 Cylinder impact with wall The cylinder occasionally hits the wall, causing the cylinder to bounce back. We assumed that the bounce-back is fully elastic and changes only the vertical velocity of the cylinder. That is, Vc = −Vc′ where Vc′ and Vc are the vertical velocities of the cylinder before and after bouncing back, respectively. The bouncing back process is completed in one time step. To avoid numerical difficulties, the bouncing back must be actuated when the gap between the cylinder bottom and the wall is smaller than 0.02. Similar treatments were used by Zhao and Cheng (2011).

3. Results and Discussion Below are introduced a number of physical quantities in terms of which the results will be presented. The maximal and minimal amplitudes of the vertical displacement, A max and A min , are defined as the 9

maximum and minimum among all local amplitudes, respectively, during a relevant time interval; that is, = Amax

1 max ( yc ,max − yc ,min )i 2 i

= and Amin

1 min ( yc ,max − yc ,min )i 2 i

(9)

where i denotes the i-th cycle of cylinder vibration. The predominant frequency of cylinder vibration, f cyl , is defined as the average of all the local frequencies during the same interval. The relevant time interval starts and ends in the stage of quasi-periodic vibration and consists of at least 15 cycles. The phase lag, φ, is defined as the phase lag of the oscillation of the vertical displacement behind that of the lift force, averaged over this interval. When the lift has two peaks in a period, the peak that is not caused by the cylinder approaching or hitting the wall is used as the reference to measure the phase lag. The drag and lift coefficients are defined as CD =

2 Fx ,hydro

ρU 2 D

and

CL =

2 Fy ,hydro

(10)

ρU 2 D

where F x,hydro and F y,hydro are respectively the streamwise and transverse hydrodynamic forces exerted on the cylinder surface by the ambient fluid: 1 ∂us Fx ,hydro = ∫ − pn x − n y ds S Re ∂n

(11)

1 ∂us Fy ,hydro = ∫ − pn y − n x ds S Re ∂n

(12)

where n x and n y are respectively the x and y components of the unit vector normal to a differential arc length, ds, along the cylinder surface, S, pointing toward fluid; u s is the tangential velocity. The quantities y c,mean , C D,mean , and C L,mean denote the corresponding time-mean values of y c , C D , and C L . To avoid confusion regarding the mesh-scale oscillations of y c , C D , and C L as function of time, which are parasitic in the Cartesian grid method, we smoothed out these oscillations, using the smoothing function in Tecplot, before analyzing the computational results, especially for hydrodynamic forces. For each pass of smoothing, the value of a variable at a data point is shifted towards an average of the values at its neighboring data points. Because one cycle of the mesh-scale oscillation would be completed in only 2 to 4 time steps, the oscillation can be annihilated after several passes of smoothing while maintaining characteristics of larger scale variations. Finally, f L and C L,amp denote respectively the predominant 10

frequency and the amplitude of the lift coefficient variation. Strouhal number, St, denotes the dimensionless frequency of vortex shedding for a stationary isolated circular cylinder.

3.1 Test of Grid Independency The grid independency was tested by simulating a low mass-damping case in Leontini et al. (2006); that is, Re = 200, m* = 10, ζ = 0.01. The reduced velocity was set to 4.8. The origin of the coordinate system is the static equilibrium position of the cylinder center. The initial position of the cylinder center is (0, 0.02) to trigger the alternative vortex shedding rapidly. Three grid resolutions with the same computational domain ([-22, 42]×[-32, 32]) were used. The medium grid has l max of 13 and l min of 7. The refinement level was determined using the two types of criteria mentioned above. However, to reduce the grid size further to a manageable scale, a region dependent criterion is imposed in the present study. That is, the smallest allowable mesh size in the rectangular regions corresponding to the x-direction range of [-22, 2], [2, 4], [4, 8], [8, 16], and [16, 42] in order increases from 1/64 to 1/4. An additional region dependent criterion is that the smallest allowable mesh size is 1/128 in the ring-shaped region bounded by the circular cylinder surface and the concentric circle with radius of 0.7. The mesh size in the considered region is selected such that all the above criteria are satisfied. The coarse grid has half the resolution of the medium grid in each dimension throughout the computational domain. The fine grid is twice the resolution of the medium grid in each dimension for x ≤ 4 and has the same resolution as the medium grid in the rest of the computational domain. The distribution of sub-grid blocks with each resolution when the cylinder moves to the highest position is shown in Fig. 3, showing that the finest mesh cells are clustered near around the cylinder surface and the second finest mesh cells depict regions of high vorticity in the wake. As shown in Fig. 4, the three grid resolutions exhibit beating-like variations and generate almost the same A max of 0.5 for the temporal variation of the vertical displacement. Since the beating frequency is hardly discernible in the spectrum, the beating period, T b , is thus measured by inspection of the time history, i.e., as the time elapsed from a valley to the next one on the envelope of local amplitude. The result is T b ≈ 177.7, 209.3, and 227 for the coarse, medium, and fine grids respectively. The medium-grid value is accurate to within 10% of the asymptotically converged value. This accuracy level is acceptable 11

considering negligibly small amplitude at the beating frequency, hence justification of adopting the medium grid. The beating-like phenomenon was also found in Leontini et al. (2006) for U* = 4.6 (close to 4.8). Because the present numerical method is very different from theirs, the beating-like phenomenon would have its origins in physics, not numerically generated. As for the spectra of the displacement amplitude, they were obtained by performing the Fast Fourier Transform (FFT) to the time records with the time interval [55, 590], [150, 570], and [140, 600] for the coarse, medium, and fine grids respectively. The three time intervals span more than 3T b , 2T b , and 2T b for the three grids respectively. The results show that the difference between the medium grid and the fine grid is much less than the difference between the coarse grid and the medium grid in terms of the amplitude at the first predominant frequency and the values of the first four predominant frequencies. In summary, the medium-grid resolution was therefore used for all the simulations in this work. It is worth mentioning that the space-time convergence of solution was pursued along the line of a constant CFL-like number when the grid independency was tested. Respectively for the coarse, medium, and fine grids, the total numbers of mesh cells are approximately 1.75×104, 5.65×104, and 1.36×105 and the averaged time step sizes approximately 5.14×10-3, 2.63×10-3, and 1.25×10-3.

3.2 Validation Chung (2009) had validated the present numerical method, in the aspect of numerical approach to fluid-solid interaction, by simulating a two-dimensional self-propelled continuous-swimming fish with Re = 5000. To make assessments relevant to the present study, we simulated flows over a single isolated circular cylinder undergoing VIV. 3.2.1 VIV of an isolated cylinder with m* = 2 and Re = 150 Zhao aet al. (2014) performed 3D/2D NS (Navier-Stokes) numerical simulations for a finite-length circular cylinder with m* = 2 and Re = 150. The 2D NS results (amplitude and frequency versus reduced velocity) are similar to 3D ones, hence the justification of the 2D assumption for Re ≤ 150, covering the Reynolds number, 100, in this study. This configuration was simulated using the present numerical method and the vibration amplitude versus the reduced velocity is displayed and compared with Zhao et

12

al. (2014) in Fig. 5. The present peak A max is nearly the same as that of Zhao et al. (2014) and the two priofiles of the lock-in zone are very close to each other except near the upper end where the present amplitude drops earlier than that of Zhao et al. (2014). 3.2.2 VIV of an isolated cylinder with m* = 10 and Re = 200 The system parameters were set according to those used by Leontini et al. (2006); that is, Re = 200, m* = 10, ζ = 0.01. The reduced velocity varied between 4.2 and 6.8. The variation of A max with U* is compared with two previous numerical results (Leontini et al., 2006; Bahmani and Akbari, 2010) as shown in Fig. 6. The present prediction of the peak of A max , 0.5, is nearly the same as that of Leontini et al. (2006) and both numerical results exhibit the phenomenon of a sharp decrease of A max at the upper end of the lock-in zone. The peak value of A max occurs near the lower end of the lock-in zone in each study. All results are consistent with each other. The Strouhal number at Re = 200 is 0.195 from the St-Re relationship (Williamson and Brown, 1998). Therefore, U* = 1/St = 5.14 at which the natural structure frequency in fluid equals to the Strouhal frequency. This reduced velocity is located in the lock-in zone and corresponds to an amplitude comparable to the peak of A max . The time history of cylinder displacement at this reduced velocity exhibits a harmonic vibration with f cyl = 0.2, slightly higher than the Strouhal number. 3.2.3 VIV of an isolated cylinder with m* = 2 and Re = 1000 To compare the performance in simulating higher-Re problems between the present numerical method with others, the case with m* = 2 and Re = 1000 was simulated. Fig. 7 presents the variation of A max and f cyl /f n with U* compared with previous numerical results (Zhao et al., 2014). Here f n denotes the natural structure frequency in vacuum. It can be seen that the present variation of A max with U* agrees with that in the 2D-NS result of Zhao et al. (2014) in view of global trend, with the former peak A max (0.6 at U* ≈ 4) near to the latter (0.65 at U* ≈ 4.8). Both are distinctly different from the 2D-RANS (Reynolds-averaged Navier-Stokes) and 3D-NS result. The 2D-RANS and 3D-NS simulations give the peak A max of 0.58 at U* ≈ 5.5 and 0.73 at U* ≈ 6 respectively The 2D-RANS profile globally meets the 3D-NS one, but there is still plenty room to improve. Navrose and Mittal (2013) reported a peak A max of 13

0.7 via 3D numerical simulation with m* = 10. Khalak and Williamson (1997, 1999) had found that the peak A max can be as high as in the region of 1 in their experimental studies for m* = 2.4 and 3.3 with Re of order 104. In brief, the lock-in zone profile, including the peak A max , depends on the Reynolds number and whether turbulent or 3D flow is considered or not. As for the vibration frequency, the present variation of f cyl /f n with U* agrees with the 2D-NS result of Zhao et al. (2014) in a global sense, both significantly different from the 2D-RANS and 3D-NS results in the zone beyond the upper branch defined by the 3D-NS profile. Even the 2D-RANS profile deviates from the 3D-NS one outside (particularly beyond) the lower branch defined by the 2D-RANS profile. Khalak and Williamson (1997, 1999) had found that, in the regime of Reynolds number distinctly higher than 1000, the cylinder oscillates with f cyl /f n = 1.4 and 1.2 over the lower branch in the lock-in zone for m* = 2.4 and 3.3 respectively. However, neither of the above numerical results with Re = 1000 exhibits such a large constant difference between the vibration frequency and the natural structure frequency. The inconsistency would be attributed mainly to different Reynolds number, and secondly to whether turbulent or 3D flow is considered or not. In summary, the present 2D-NS numerical method can compete with other modern 2D-NS ones for Re up to 1000. Accurate high-Re simulations could even reproduce the above-mentioned high amplitude and frequency departure reported by Khalak and Williamson (1997, 1999). However, the present 2D-NS method is adequate in this study because the Reynolds number investigated, 100, is much lower than 1000, hence no 3D turbulent flows.

3.3 Near-wall Cases With m* = 2 and ζ = 0, we examined the effects of the gap ratio on various aspects of hydrodynamic and structural responses, including their interactions, by setting G = 0.06, 0.3, and 31.5. The cases with G = 31.5 can be regarded as for an isolated cylinder. The Reynolds number was fixed at 100, whereas the reduced velocity varied. For the specified fluid properties, cylinder diameter, and cylinder mass, the approach of keeping Reynolds number constant is used to investigate the influence of structural stiffness with a fixed incoming fluid velocity. 3.3.1 Vibration amplitude 14

For each of the three gap ratios, we performed series of simulations with varied reduced velocities (3 ≤ U* ≤ 24). Figure 8(a) shows the variation of A max and A min with U* for the three gap ratios. The sizes of the lock-in zone are approximately 4 ≤ U* ≤ 10, 3.5 ≤ U* ≤ 12, and 5 ≤ U* ≤ 16 for G = 31.5, 0.3, and 0.06 respectively, i.e., increasing with decreasing gap ratio. The size variation of the lock-in zone with the gap ratio is consistent with those at higher Reynolds numbers reported by Yang et al. (2006) which treated similar gap ratios for the circular-cylinder VIV near a fixed wall in a free stream. Although the peak of A max decreases with decreasing gap ratio, it is still noticeable for the smallest gap ratio (G = 0.06). Yang et al. (2006) found that the cylinder vibration amplitude increases with the gap ratio decreasing from 0.28 to 0.1. Yang et al. (2009) also reported that the peak of A max for G = 0.06 is larger than that for G = 0.3. Because the Reynolds number and the wall boundary layer play important roles in determining the vibration amplitude (Raghavan et al., 2009), the difference in the variation of the peak A max versus the gap ratio between the present study and the two previous works would originate from the difference in Reynolds number or wall kinematics, or both. Another possible reason is due to the different approaches to varying the reduced velocity between the present numerical simulation and the experimental ∗ measurement. The reduced velocity at which the peak of A max occurs, U peak , moves toward the

high-reduced-velocity (soft spring) side when the gap ratio decreases. The onset of lock-in occurs at higher reduced velocities than for an isolated cylinder, consistent with the finding of Tsahalis and Jones (1981). The rate at which A max changes with U* near the two ends of the lock-in zone is slower for a smaller gap ratio. Raghavan et al. (2009) also reported a more gradual onset of lock-in when G decreases from > 3 to < 0.65. For G = 0.06 and 0.3, the present observation of the changing rate does not agree with that of Yang et al. (2009): A max increases slowly at small reduced velocities and decreases quickly at large reduced velocities in the lock-in zone. Far beyond the lock-in zone, the vibration amplitude approaches nearly the same low but non-zero value (< 0.1) for all gap ratios. This phenomenon of persistency of vibration in the high-reduced-velocity end had been confirmed in related literature for a low-mass circular cylinder, which is either isolated or close to a wall. In general, A min follows a similar curve as that of A max for each gap ratio. For G = 31.5, an exception 15

occurs at U* = 4.5 at which the two amplitudes differ substantially due to considerable temporal changes in the local amplitude as shown in Fig. 9(a). The temporal variation of C L has a similar behavior and is in-phase with that of y c as displayed in Fig. 9(b). Both the frequency spectra of amplitude for y c and C L , Fig. 9(c), share two closely spaced prominent frequencies of 0.193 and 0.186, causing the beating-like phenomena observed in Figs. 9(a) and 9(b). These two frequencies are larger than the Strouhal frequency, 0.164. According to previous studies on flow over an isolated circular cylinder in forced harmonic vibration with Re ≈ 100 (Anagnostopoulos, 2000; Nobari and Naderan, 2006; Placzek et al., 2009), there also exists frequency modulation or beating behavior in the lift coefficient as function of time for vibration frequencies exceeding the Strouhal frequency. The mechanism behind the present beating phenomenon for the lift force however could be different from that in the forced vibration case because the present time history of the transverse displacement is beating-like instead of purely simple harmonic as in the forced vibration case. Correspondingly, Placzek et al. (2007) observed that the beating behavior in the time history of C L is linked to a vortex merging in the wake while no such vortex merging occurs in the present study as shown later. For G = 0.3 and 0.06, A max and A min differ at reduced velocities ∗ around U peak though the amount of difference is small compared to A max . This difference originates from

irregular variations in the vibration amplitude with time instead of the regular beating as observed for an isolated cylinder. Taking for example the case with G = 0.06 and U* = 8, Fig. 10 shows the irregularity in the absolute vertical coordinate of the cylinder center as function of time. The cylinder periodically impacts the wall in the quasi-periodic stage only at particular reduced ∗ velocities around U peak for G = 0.3 and 0.06. Impact does not cause a considerable enhancement of

vibration amplitude. Meanwhile, the events of impact appear at only two reduced velocities even for the smallest gap ratio. It is interesting to understand how large the hydrodynamic forcing is needed to cause the above amplitude response. Figure 8(b) presents the root-mean-square (rms) value of the lift coefficient, C L,rms , as function of the reduced velocity. The cases of impact with the wall being excluded, the peak of C L,rms ∗ occurs at a reduced velocity different from U peak for all the three gap ratios. Impact with the wall causes

16

drastic changes in the lift coefficient, as compared with those in the cylinder response. Therefore, for a gap ratio with which impact has the potential to occur, C L,rms must reach the maximal value at some reduced velocity where impact appears. 3.3.2 Vibration frequency Besides the vibration amplitude, the vibration frequency of structure is a major concern in the context of structure fatigue and noise generation. Meanwhile, the relation between the frequency of exciting hydrodynamic force and the vibration frequency of structure is worthy of exploration. Further, the vortex shedding frequency in the flow over an isolated stationary circular cylinder, the Strouhal frequency, can serve as a good reference for comparison. Based on these considerations, Fig. 11 shows the variation of f cyl /f nw (frequency ratio) and f L /f nw with U* for the three gap ratios, in comparison with the straight line representing the Strouhal frequency, f 0 , which is 0.164 at Re = 100. For G = 31.5, the trend of variation in f L /f nw as function of U* is similar to that of To̸rum and Anand (1985) with the lowest turbulence intensity, 3%, for the largest gap ratio, 3. That is, f L ≈ f nw in some part of the lock-in zone. The difference between the two works comes from comparison of the predominant frequency of the lift coefficient with the Strouhal frequency: the present study predicts that f L > f 0 in the low-U* part and f L < f 0 in the high-U* part of the lock-in zone while To̸rum and Anand (1985) reported that f L < f 0 in the whole range of U* they examined, approximately 3 ≤ U* ≤ 16. Differences in the Reynolds number regime and approach to varying the reduced velocity should be responsible for this dissimilarity. For G = 31.5, we find that f cyl = f L in the whole range of U* studied (3 ≤ U* ≤ 24). However, f cyl ∗ deviates from f L and locks onto f nw or 2f nw at most reduced velocities higher than U peak for all cases,

including the near-wall ones, investigated by To̸rum and Anand (1985). The present trend of variation in f cyl /f nw or f L /f nw as function of U* in the lock-in zone is similar to that in the above-mentioned largest-gap-ratio case of To̸rum and Anand (1985) and that in the case of Raghavan et al. (2009) with 1 ≤ m* ≤ 3.14 and G = 3.207. That is, f cyl changes gradually from f 0 to f nw while U* increasing in the lock-in zone. There is no difference in the global trend of f cyl (or f L ) between G = 0.3 and 0.06. This trend is dissimilar to that for G = 31.5 in two aspects. Firstly, f cyl (or f L ) exceeds f 0 in the whole range of U* 17

studied (3 ≤ U* ≤ 24) for G = 0.3 and 0.06 while f cyl (or f L ) > f 0 in a limited range of U* (4 ≤ U* ≤ 7) for G = 31.5. Secondly, f cyl ≈ f nw in the low-U* part of the lock-in zone for G = 0.3 and 0.06 while in the high-U* part for G = 31.5. In summary, wall proximity changes the dominant frequency of cylinder vibration from the Strouhal frequency to the natural structure frequency in fluid in the low-U* part of the lock-in zone, and vice versa in the high-U* part of the lock-in zone. Impact with the wall causes no abrupt change in the vibration frequency. The case in Fredso̸e et al. (1987) with m* = 1.89 and G = 0.2, as an analogue of the present case with m* = 2 and G = 0.3, shows that, in the lock-in zone, f cyl is considerably larger than f 0 and increases with U* in the low-U* part and close to f 0 in the high-U* part. In the case of Raghavan et al. (2009) with 1 ≤ m* ≤ 3.14 and G = 0.268, it can be found that f cyl ≈ f 0 . In the present study, f cyl is considerably larger than f 0 but keeps nearly constant (≈ f nw ) in the low-U* part; and f cyl approaches f 0 in the high-U* part. The differences between these works manifest that the Reynolds number, wall boundary layer, or approach to varying the reduced velocity would play an important role in determining the variation of the cylinder vibration frequency with the reduced velocity in the lock-in zone. For each of the two near-wall gap ratios, the hydrodynamic forcing frequency is about two times the cylinder vibration frequency, f L ≈ 2f cyl , at some particular reduced velocity (U* = 7 for G = 0.3 and U* = 6 for G = 0.06). To find the reason, Fig. 12 shows the time histories of the absolute vertical coordinate of the cylinder center and the lift coefficient in a period of 1/f cyl at U* = 6, 7, and 8 for G = 0.3. For each reduced velocity, there exists a dramatic change of C L in a short time interval during which the cylinder moves very near the wall. For U* = 6 and 8, this change dominates the temporal variation of C L such that f L ≈ f cyl . For U* = 7, the variation of C L in the rest part of the vibration cycle can compete with this short-time change such that f L ≈ 2f cyl . Actually, the vortex shedding pattern is very similar to one another for the three reduced velocities as shown later, exhibiting a vortex shedding frequency equal to f cyl . We therefore must be very careful in discerning the lift coefficient frequency from the vortex shedding one. Except for the above two cases, all the cases in this study in fact have f L equal to the vortex shedding frequency. For each reduced velocity, in the downstroke, the lift coefficient decreases then increases to a local peak right before the cylinder reaches the lowest position. If the minimal gap experienced in the 18

course of cylinder vibration is small enough, as for U* = 6 and 7, a local valley would follow the local peak. This local valley appears right after the cylinder reaches the lowest position if no impact occurs (U* = 7); otherwise, it appears exactly when the cylinder impacts with the wall (U* = 6). As shown in the subfigure for U* = 6, impact usually causes a transient extraordinarily large negative lift due to a sudden change in the vertical velocity of the cylinder. All the above descriptions of lift variation are shared by all the investigated near-wall cases with small minimal gap. In summary, impact with the wall does not cause substantial changes in the predominant frequencies of cylinder vibration, vortex shedding, and lift variation. ∗ The frequency ratio f cyl /f nw at U peak , called the peak frequency ratio, is greater than 1 for each gap

ratio. When the gap ratio is decreased, the peak frequency ratio increases. With the smallest gap ratio, the peak frequency ratio can be as high as 1.45. This behavior of f cyl /f nw is consistent with the result ∗ of To̸rum and Anand (1985), noting that U peak (or the corresponding Re) differs insignificantly among

all the cases they investigated. The conclusion thus seems irrespective of the wall boundary layer and the Reynolds number. From another viewpoint, the gap ratio significantly affects the vibration frequency for a given reduced velocity. 3.3.3 Phase lag of displacement behind lift Another important relation between the hydrodynamic forcing and cylinder response is the phase angle relative to each other. Figure 13 shows the average phase lag, φ, of the cylinder response behind the hydrodynamic lift. For an isolated cylinder (G = 31.5), the phase lag exhibits a sharp jump from approximately 0° to 180° at a particular U* between 8 and 9 and remained at 0° and 180° for a lower and higher reduced velocity, respectively. This phase lag jump can be explained as follows: for the isolated cylinder, both the forcing, F(t), and response, R(t), can be efficiently approximated using simple harmonic functions of time with the same angular frequency ω, as follows: = R (t ) R0 cos (ωt − φ ) ,

(13)

F (t ) = F0 cos (ωt ) .

(14)

The following relation was thus derived from Eq. (1): 19

(

R0 k − mω 2 F0

)=

cos (ωt )

(15)

cos (ωt − φ )

where φ can assume only two values to get rid of the dependency on time on the right-hand side of the equation; that is, 0° for ω2 < k/m and 180° for ω2 > k/m. The phase lag jumps when ω2 = k/m. Regarding the structural parameters, the phase lag jumps when ∗ f cylU=

f cyl f nw =

1+

CA m∗

(16)

This frequency ratio occurs at U* = 8.3, as shown in Fig. 11, and is consistent with the phase-jump reduced velocity shown in Fig. 13. Furthermore, the hydrodynamic mechanical work input to the cylinder in a vibration period is

∫

2π ω

0

2π ω F (t ) R (t )= dt F0 R0 ∫ −ω cos (ωt ) sin (ωt − φ )= dt F0 R0π sin φ . 0

(17)

Therefore, the net work performed by the fluid in a vibration period is zero to sustain regular vibrations. For a cylinder near the wall, the phase lag jump is smoother than that for an isolated cylinder; however, it causes an overshoot at the end of the jump. Both the extents of smoothness and overshoot increase with a decreasing gap ratio. An inspection of the time histories of the cylinder-center position and the lift coefficient for reduced velocities located in the phase-lag transition area showed that the time history of cylinder-center position resembles a single-frequency simple harmonic function of time, whereas the lift coefficient exhibits a substantial deviation from this type of oscillation. Therefore, a phase lag of 0° or 180° is not required to sustain regular vibrations, which justifies the gradual increase of phase lag in the transition area and the follow-up overshoot. For each gap ratio, the jump area ends at a ∗ reduced velocity larger than U peak and the phase lag approaches a constant value of 180° when the

reduced velocity further increases. 3.3.4 Time-mean drag/lift coefficients and displacement Figure 14(a) shows the variation of C D,mean with U* for the three gap ratios. It is seen that the time-mean drag is larger than that for an isolated fixed cylinder in all cases except those for G = 31.5 with reduced velocities higher than the upper end of the lock-in zone. For each gap ratio, the time-mean drag 20

reaches a maximum at U* = U ∗peak which is higher with a smaller gap ratio; that is, when the cylinder vibration amplitude reaches a maximum. The maximal time-mean drag exhibits little variation among the three gap ratios. Both findings are consistent with that of To̸rum and Anand (1985). The maximal time-mean drag for an isolated cylinder is the smallest among the three gap ratios. At large reduced velocities, the two near-wall cases produce the same asymptotic time-mean drag, which is higher than that for the isolated vibrating cylinder. Figure 14(b), the counterpart of Fig. 14(a) for C L,mean , shows that the time-mean lift is always positive and the small-gap-ratio cylinder acquires a higher time-mean lift than the large-gap-ratio cylinder in the entire range of reduced velocity, except for U* = 6 and 7, for which the two gap ratios acquire approximately the same time-mean lift. Almost in the entire range of reduced velocity, the time-mean lift decreases with increasing reduced velocity for G = 0.06. However, with increasing reduced velocity for G = 0.3, the time-mean lift increases in the range of 3 ≤ U* ≤ 6 and decreases beyond U* = 6. In addition, the time-mean lift converges to the same value for the two gap ratios when U* = 24. To understand the time-mean cylinder response to the time-mean lift, Fig. 15(a) shows the variation of y c,mean with U* for G = 0.3 and 0.06. For G = 0.06, the time-mean vertical displacement increases with ∗ the reduced velocity, except for a distinct local peak of y c,mean at U peak (= 8). The time-mean lift curve

for this gap ratio as shown in Fig. 14(b) presents a larger time-mean lift at U* = 9 than at U* = 8. If the time-mean vertical displacement is a static response to the time-mean lift, we would expect a larger time-mean vertical displacement at U* = 9 than at U* = 8. However, y c,mean peaks at U* = 8 and is substantially greater than that at U* = 9. We therefore cannot infer the time-mean vertical displacement, even in the sense of approximation, by the static theory. A similar trend, including a local peak of y c,mean at U* = 8, exists for G = 0.3. However, the time-mean vertical displacement is lower than that for G = 0.06 in the entire range of reduced velocity except at U* = 6. The largest difference of y c,mean between the two gap ratios occurs at U* = 8. Because the extent of wall proximity varies with time, we present in Fig. 15(b) the minimal and maximal gaps encountered through the course of cylinder vibration, G min and G max , at each reduced velocity for G = 0.3 and 0.06. It is observed that the vibrating cylinder can move, at some instants, to 21

positions very close to the wall in the finite ranges of reduced velocity, 5 ≤ U* ≤ 7 and 7 ≤ U* ≤ 9, for G = 0.3 and 0.06 respectively. Impact with the wall causes no abrupt changes of G min and G max as functions of U*. In the lock-in zone, a larger G max always accompanies a smaller G min . As shown in Fig. 15(b), both the two cases with (G, U*) = (0.06, 4) and (0.3, 3) have nearly zero vibration amplitudes, attaining the time-mean gap ratio, G mean , of 0.145 and 0.32. For these two cases of almost pure translation, C D,mean = 1.6 and 1.46 [Fig. 14(a)] and C L,mean = 0.99 and 0.47 [Fig. 14(b)] respectively. Noting that C D,mean = 1.36 [Fig. 14(a)] and C D,mean = 0 for an isolated purely translating cylinder, we thus conclude that both the time-mean drag and lift coefficients increase with decreasing gap ratio for a purely translating cylinder close to a plane wall. This conclusion is consistent in the global trend with the findings of Yoon et al. (2007) though the quantitative differences exist. 3.3.5 Vorticity field Vortex structures in the wake of a bluff body have been attracting very much attention of academic researchers for quite a long time. Related studies would enable us to understand involved physics more deeply, hence the disclosure of information useful to engineering applications. Figure 16 shows instantaneous vorticity contours for G = 31.5 and U* = 4, 5, 6, 7, 8, 9, and 12 at the phase point when the cylinder moves to the highest position. All vortex shedding modes are 2S-type modes (Williamson and Roshko, 1988); that is, 2 opposite-sign vortices (consisting of a vortex pair) are shed per cycle. If one vortex is elongated while being shed from the cylinder surface and eventually in the transverse direction extrudes into the area of the opposite-sign vortex, then this type of vortex shedding will produce a large time-mean drag. This correspondence of the near-wake vortex structure to the time-mean drag is clearly demonstrated with the case of U* = 6 which has the maximal time-mean drag as shown in Fig. 14(a). The correspondence could be explained in that the time-mean pressure on the leeside surface of the cylinder would be lowered down due to the low-pressure vortex cores from the lower and upper cylinder surfaces alternatively passing the near-wake zone in the nearly transverse direction. The vortex shedding patterns for U* = 8 and 9 are similar, whereas y c and C L exhibit in-phase and anti-phase variations for U* = 8 and 9, respectively (Fig. 13); that is, the phase lag of y c behind C L cannot be inferred from the vortex shedding pattern. 22

Figure 17 shows instantaneous vorticity contours for G = 0.3 and U* = 4, 5, 6, 7, 8, and 10 at the phase point when the cylinder moves to the lowest position. Because of the interaction with negative vortices generated on the wall, the positive vortices shed from the lower surface of the cylinder diminished more rapidly than the negative vortices shed from the upper surface of the cylinder. This interaction is sufficiently strong to observe a nearly single-side vortex shedding pattern in the range of reduced velocity of 5 ≤ U* < 7. For U* = 10, the positive vortex street can remain far downstream though the vorticity strength decays more rapidly than that of the negative vortex street shed from the upper surface of the cylinder. For U* = 4 and 8, the distribution of positive vortices illustrates the transition from single-side to two-side vortex shedding pattern. From Fig. 15(b), the minimal gap, G min , seems to correlate well with the vortex shedding pattern. That is, if G min decreases, then the vortex shedding pattern tends to be single-side. It is found that the negative vortices shed from the upper surface of the cylinder form a less coherent structure when the positive vortex street is less obvious. Figure 18 shows instantaneous vorticity contours for G = 0.06 and U* = 6, 7, 8, 9, 10, 12, and 16 at the phase point when the cylinder moves to the lowest position. At first glance, the vortex shedding pattern has a tendency toward the single-side type for decreasing G min , referring to Fig. 15(b). The case with U* = 6 however has a distinctly different vortex shedding pattern from that of the case with U* = 10 though both cases have nearly the same G min as shown in Fig. 15(b). This fact can be attributed to the difference in the amplitude of cylinder vibration between the two cases. Specifically, in contrast to the case with U* = 10, A max is much smaller (= 0.175) for U* = 6 that there is no sufficient gap space to develop the positive vortex from the lower surface of the cylinder. In the study for a purely translating circular cylinder near a fixed wall at Re = 100, Yoon et al. (2007) have pointed out that there is no vortex shedding from either surface of the cylinder when G = 0.1. The present study however shows that, when the amplitude of cylinder vibration is large enough, the single-side vortex shedding pattern exists even if the gap ratio decreases to 0.06. For a purely translating circular cylinder near a fixed wall with Reynolds numbers higher than 100, Taneda (1965) and Huang and Sung (2007) observed vortex shedding (at least single-side) even for small gap ratios, in contrast to the case of flow over a stationary circular cylinder near a fixed wall. In other words, existence of wall 23

boundary layer causes suppression of vortex shedding. On the other hand, Zhao and Cheng (2011) reported the single-side vortex shedding mode even for G = 0.002 for a circular cylinder undergoing VIV near a fixed wall in a free stream. Combining these previous observations, we can argue that the vibration of cylinder would be one of major reasons why vortex shedding occurs even for a small gap ratio. Zhao and Cheng (2011) identified three vortex shedding modes by examining the case of extremely small gap ratio (G = 0.002), each corresponding to a range of reduced velocities. Their identification was based on the timing of positive vortex shedding from the bottom of the cylinder, irrespective of whether these positive vortices can form a vortex street. To highlight our viewpoint of vortex street, we adopt the term “vortex shedding pattern” rather than “vortex shedding mode”. Then all the three modes illustrated by Zhao and Cheng (2011) belong to the single-side vortex shedding pattern according to the present classification. In summary, we have observed that both the minimal gap and the amplitude of cylinder vibration play important roles in determining the pattern of vortex shedding (single-side or two-side vortex streets) for a cylinder undergoing VIV near a fixed wall in a free stream: the minimal gap, if very small, seems the only controlling factor of the vortex shedding pattern; when the amplitude of cylinder vibration is large enough, the single-side vortex shedding pattern exists even if the gap ratio decreases to 0.06. Finally as a validation, the above case of nearly zero vibration amplitude, i.e., with (G, U*) = (0.3, 3), corresponding to G mean = 0.32, has a vortex shedding pattern (not shown here) very similar to that of the case with G = 0.3 of Yoon et al. (2007).

4. Conclusions The characteristics of 1-dof VIV of a translating low-mass zero-damping circular cylinder near a fixed plane wall at Re = 100 was numerically examined in this study. The present numerical method has passed a rigorous test of grid independency and been validated by comparisons with previous results for a circular cylinder undergoing VIV in an unbounded domain. The effects of the gap ratio were quantified by depicting the relevant physical parameters as a function of the reduced velocity. These parameters include the amplitude and frequency ratio of the cylinder vibration, the rms lift coefficient, the phase lag of the cylinder response behind the lift oscillation, the time-mean drag and lift coefficients, and the 24

time-mean cylinder position. The wake vortex structures were also examined. The major findings and conclusions are collected as below. Vibration amplitude The size of lock-in zone increases and the peak vibration amplitude decreases with decreasing gap ratio. The reduced velocity at which occurs the peak vibration amplitude increases when the gap ratio decreases. The beating phenomenon of the cylinder vibration existing at some reduced velocities for an isolated cylinder is suppressed by the wall proximity. Vibration frequency and phase The predominant frequency of the cylinder vibration is always equal to the vortex shedding frequency. Meanwhile, wall proximity usually changes the attribute of the predominant vibration frequency from the natural-structure-frequency-in-fluid dominated to the Strouhal-frequency dominated or vice versa depending on the reduced velocity in the lock-in zone. When the vibration amplitude peaks, the excess of vibration frequency over the natural structure frequency in fluid increases with decreasing gap ratio. For an isolated circular cylinder, the phase lag of the vertical displacement of the cylinder behind the lift jumps sharply from 0° to 180° at some reduced velocity (or equivalent frequency ratio), which has been theoretically predicted. Wall proximity would smooth the phase lag jump. Time-mean drag and lift In the lock-in zone of each gap ratio investigated in this study, the time-mean drag is larger than that for an isolated purely translating circular cylinder. As in the case of circular-cylinder VIV near a fixed wall in a free stream at higher Reynolds numbers, the maximal time-mean drag changes little and occurs at a higher reduced velocity as the gap ratio decreases. The time-mean lift is always positive for all investigated near-wall cases. The static theory cannot explain the correlation between the time-mean vertical displacement and the time-mean lift coefficient. Vortex shedding pattern For an isolated cylinder, all vortex shedding modes are 2S-type. The definite correspondence between the near-wake vortex structure and the time-mean drag has been found. However, the topology 25

of vortex shedding pattern has a weak correlation with the phase lag of the cylinder displacement behind the lift. Both the minimal gap and the amplitude of cylinder vibration play important roles in determining the pattern of vortex shedding for near-wall cases. Effect of impact with wall Impact with wall causes no substantial changes in amplitude or frequency of cylinder vibration. The only effect of impact is the generation of a transient extremely large negative lift.

Acknowledgements We would like to thank the National Science Council of Taiwan R.O.C. for financially supporting this research under Contract No. NSC 99-2221-E-022-021.

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vortex-induced vibration of a circular cylinder. J. Wind Eng. Ind. Aerodyn. 69-71, 341-350. Khalak, A., Williamson, C.H.K., 1999. Motions, forces and mode transitions in vortex-induced vibrations at low mass-damping. J. Fluids Struct. 13, 813-851. Leontini, J.S., Thompson, M.C., Hourigan, K., 2006. The beginning of branching behaviour of vortex-induced vibration during two-dimensional ﬂow. J. Fluids Struct. 22, 857-864. Lucor, D., Foo, J., Karniadakis, G., 2005. Vortex mode selection of a rigid cylinder subject to VIV at low mass-damping. J. Fluids Struct. 20, 485-503. Navrose, Mittal, S., 2013. Free vibrations of a cylinder: 3-D computations at Re = 1000. J. Fluids Struct. 41, 109-118. Nishino, T., Roberts, G.T., Zhang, X., 2007. Vortex shedding from a circular cylinder near a moving ground. Phys. Fluids 19, 025103-1-025103-12. Nobari, M.R.H., Naderan, H., 2006. A numerical study of flow past a cylinder with cross flow and inline oscillation. Comput. Fluids 35(4), 393-415. Placzek ,A., Sigrist, J.F., Hamdouni, A., 2007. Numerical simulation of vortex shedding past a circular cylinder in a cross-flow at low Reynolds number with finite-volume technique. Part 1: forced oscillations. In: 2007 ASME pressure vessels & piping division conference. ASME, San Antonio (TX), USA. Placzek, A., Sigrist, J-F., Hamdouni, A., 2009. Numerical simulation of an oscillating cylinder in a cross-flow at low Reynolds number: Forced and free oscillations. Comput. Fluids 38, 80-100. Raghavan, K., Bernitsas, M.M., Maroulis, D., 2009. Effect of bottom boundary on VIV for energy harnessing at 8×103 < Re < 1.5×105. ASME J. Offshore Mech. Arct. Eng. 131(3), 031102 (13 pages). Rao, A., Thompson, M.C., Leweke, T., Hourigan, K., 2013. The flow past a circular cylinder translating at different heights above a wall. J. Fluids Struct. 41, 9-21. Sarpkaya, T., 2004. A critical review of the intrinsic nature of vortex-induced vibrations. J. Fluids Struct. 19, 389-447. Taneda, S., 1965. Experimental investigation of vortex streets. J. Phys. Soc. Jpn, 20, 1714-1721. To̸rum, A., Anand, N.M., 1985. Free span vibrations of submarine pipelines in steady flows - effect of 28

free-stream turbulence on mean drag coefficients. ASME J. Energy Resour. Technol. 107(4), 415-420. Tsahalis, D.T., Jones, W.T., 1981. Vortex-induced vibrations of a flexible cylinder near a plane boundary in steady flow. Proceedings of the Annual Offshore Technology Conference 1, 367-381. Houston, TX. Offshore Technol. Conf., Dallas, TX, USA. Wanderley, Juan B.V., Souza, Gisele H.B., Sphaier, Sergio H., Levi, Carlos, 2008. Vortex-induced vibration of an elastically mounted circular cylinder using an upwind TVD two-dimensional numerical scheme. Ocean Eng. 35(14-15), 1533-1544. Wang, X.K., Hao, Z., S.K. Tan, Vortex-induced vibrations of a neutrally buoyant circular cylinder near a plane wall, Journal of Fluids and Structures, Volume 39, May 2013, Pages 188-204. Williamson, C.H.K., Brown, G.L., 1998. A series in 1/√Re to represent the Strouhal-Reynolds number relationship of the cylinder wake. J. Fluids Struct. 12, 1073-1085. Williamson, C.H.K., Govardhan, R., 2004. Vortex-induced vibrations. Annual Review of Fluid Mechanics 36, 413-455. Williamson, C.H.K., Govardhan R., 2008. A brief review of recent results in vortex-induced vibrations. J. Wind. Eng. Ind. Aerodyn. 96, 713-735. Williamson, C.H.K., Roshko, A., 1988. Vortex formation in the wake of an oscillating cylinder. J. Fluids Struct. 2, 355-381. Yang, B., Gao, F.P., Jeng, D.S., Wu, Y.X., 2009. Experimental study of vortex-induced vibrations of a cylinder near a rigid plane boundary in steady flow. Acta Mech. Sinica 25, 51-63. Yang, B., Gao, F.P., Wu, Y.X., Li, D.H., 2006. Experimental study on vortex-induced vibrations of submarine pipeline near seabed boundary in ocean currents. China Ocean Eng. 20(1), 113-121. Yoon, H.S., Lee, J.B., Chun, H.H., 2007. A numerical study on the fluid flow and heat transfer around a circular cylinder near a moving wall. Int. J. Heat Mass Transfer, 50(17-18), 3507-3520. Yoon, H.S., Lee J.B., Seo, J.H., Park, H.S., 2010. Characteristics for flow and heat transfer around a circular cylinder near a moving wall in wide range of low Reynolds number. Int. J. Heat Mass Transfer 53(23-24), 5111-5120. 29

Zdravkovich, M.M., 2003. Flow Around Circular Cylinders: Vol 2: Applications. Oxford University Press, Oxford, 880-881. Zhao, M., Cheng, L., 2011. Numerical simulation of two-degree-of-freedom vortex-induced vibration of a circular cylinder close to a plane boundary. J. Fluids Struct. 27(7), 1097-1110. Zhao, M., Cheng, L., An, H., Lu, L., 2014. Three-dimensional numerical simulation of vortex-induced vibration of an elastically mounted rigid circular cylinder in steady current. J. Fluids Struct. 50, 292-311.

30

6. Figure Captions Fig. 1

Schematic diagram of the physical problem, computational domain, and boundary conditions. All quantities made dimensionless using the cylinder diameter D, incoming velocity (originally translating velocity of cylinder) U, and fluid density ρ f as the characteristic length, velocity, and density respectively. The gap ratio, G, is defined as the distance between the cylinder bottom and the wall in the static equilibrium condition (i.e., when the spring force keeps zero with quiescent ambient fluid).

Fig. 2

Illustration of refinement/derefinement strategy for grid blocks with 2×2 cells. The thickest line depicts the boundary of the parent grid block comprising cells centered at P 1 , P 2 , P 3 , and P 4 . The secondly thickest line depicts the mesh lines, which are also the boundaries of the four child grid blocks. Hollow circle and filled circle denote the cell centers at the parent and child level of refinement respectively.

Fig. 3

Distribution of sub-grid blocks for each grid resolution when the cylinder moves to the highest position for an isolated cylinder with Re = 200, m* = 10, ζ = 0.01, and U* = 4.8. Grid resolution from top to bottom: coarse, medium, and fine. Plots in the right column are the near-cylinder part of the enlarged view of those in the left column. Each grid block has 2×2 cells.

Fig. 4

Cylinder displacement as function of time (left column) and frequency (right column) for an isolated cylinder with Re = 200, m* = 10, ζ = 0.01, and U* = 4.8. Grid resolution from top to bottom: coarse, medium, and fine. The first four predominant frequencies in ascending order of value: 0.198, 0.204, 0.209, 0.215 for the coarse grid; 0.201, 0.206, 0.211, 0.216 for the medium grid; and 0.202, 0.207, 0.211, 0.216 for the fine grid.

Fig. 5

Variation of A max with U* for an isolated cylinder with m* = 2 ζ = 0, and Re = 150. Also shown are results of Zhao et al. (2014).

Fig. 6

Variation of A max with U* for an isolated cylinder with m* = 10, ζ = 0.01, Re = 200. Also shown are results from previous contributions.

Fig. 7

Variation of A max and f/f n with U* for an isolated cylinder with m* = 2, ζ = 0, and Re = 1000. 31

Also shown are results of Zhao et al. (2014). Fig. 8

Variation of (a) A max and A min and (b) C L,rms with U* for three gap ratios. In Fig. 8 (b), the solid symbol indicates the occurrence of the peak of A max and the number in the bracket represents the value of C L,rms in the event of cylinder impacting wall.

Fig. 9

(a) Variation of y c with time, (b) comparison between y c and C L as function of time, and (c) comparison in the frequency spectra of amplitude between y c and C L for G = 31.5 and U*= 4.5.

Fig. 10

Absolute vertical coordinate of the cylinder center, y c,abs , as function of time for G = 0.06 and U*= 8.

Fig. 11

Variation of f cyl /f nw and f L /f nw with U* for three gap ratios. Solid symbol: occurrence of the peak of A max .

Fig. 12

Time histories of the absolute vertical coordinate of the cylinder center, y c,abs , and the lift coefficient in a period of 1/f cyl at U* = 6, 7, and 8 for G = 0.3.

Fig. 13

Variation of φ with U* for three gap ratios. Solid symbol: occurrence of the peak of A max .

Fig. 14

Variation of (a) C D,mean and (b) C L,mean with U* for three gap ratios. Solid symbol: occurrence of the peak of A max .

Fig. 15

Variation of (a) y c,mean and (b) G min and G max with U* for two gap ratios. Solid symbol: occurrence of the peak of A max .

Fig. 16

Instantaneous vorticity contours for G = 31.5 and U* = 4, 5, 6, 7, 8, 9, and 12. Dotted line: negative vorticity. Solid line: positive vorticity. Increment of contour lines = 0.2. Each snapshot is taken at some instant when y c reaches a local peak.

Fig. 17

Instantaneous vorticity contours for G = 0.3 and U* = 4, 5, 6, 7, 8, and 10. Refer to Fig. 16 for line legend and increment of contour lines. Each snapshot is taken at some instant when y c nearly reaches a local minimum.

Fig. 18

Instantaneous vorticity contours for G = 0.06 and U* = 6, 7, 8, 9, 10, 12, and 16. Refer to Fig. 16 for line legend and increment of contour lines. Each snapshot is taken at some instant when y c nearly reaches a local minimum.

32

u 1, v 0

(-22, 32)

y

u 1 v 0

(42, 32)

u x v x

x

k m c

(-22, -32)

1 G u 1, v Fig. 1

0

(42, -32)

0 0

S2

Q3

Q4 P4

P3 A

Q1

Q2

B P1

R3

P2

Fig. 2

2

1

Y

6

Y

3

0

0 -1

-3 -6 -1 0

0

10

20

-2

-1

0

1

2

3

2

3

2

3

X

X 2

1

Y

6

Y

3

0

0 -1

-3 -6 -1 0

0

10

20

-2

-1

0

1

X

X 2

1

Y

6

Y

3

0

0 -1

-3 -6 -10

0

20 -2-1

10

X

Fig. 3

0

1

X

0.6

0.3

0.4

Amplitude

yc

0.2

0

-0.2

0.2

0.1

-0.4

-0.6 0

80

160

240

320

400

480

0

560

0

0.05

0.1

t

0.15

0.2

0.25

0.3

0.35

0.4

0.3

0.35

0.4

0.3

0.35

0.4

Frequency

0.6

0.3

0.4

Amplitude

yc

0.2

0

-0.2

0.2

0.1

-0.4

-0.6 0

80

160

240

320

400

480

0

560

0

0.05

0.1

t

0.15

0.2

0.25

Frequency

0.6

0.3

0.4

Amplitude

yc

0.2

0

-0.2

0.2

0.1

-0.4

-0.6 0

80

160

240

320

400

480

560

t

0

0

0.05

0.1

0.15

0.2

0.25

Frequency

Fig. 4

0.8 0.7 0.6

Amax

0.5 0.4 0.3 0.2

Present Zhao et al.

0.1 0

2

3

4

5

6

7

U

*

Fig. 5

8

9

10

11

12

0.6 Num. (Leontini et al.) Num. (Bahmani and Akbari) Num. (Present)

0.5

Amax

0.4

0.3

0.2

0.1

0

4

4.5

5

5.5

U Fig. 6

*

6

6.5

7

0.8 0.7 0.6

Amax

0.5 0.4 0.3 Present 2D NS (Zhao et al.) 2D RANS (Zhao et al.) 3D NS (Zhao et al.)

0.2 0.1 0

2

3

4

5

6

7

8

U

*

9

10

11

12

13

14

15

10

11

12

13

14

15

3 Present 2D NS (Zhao et al.) 2D RANS (Zhao et al.) 3D NS (Zhao et al.)

2.5

fcyl /fn

2

1.5

1

0.5

0

2

3

4

5

6

7

8

U

*

9

Fig. 7

0 .8

G = 3 1 .5 0 , A m a x G = 3 1 .5 0 , A m in

A m a x o r A m in

G = 0 .3 0 , A m a x

G = 0.3 impact

0 .6

G = 0 .3 0 , A m in G = 0 .0 6 , A m a x G = 0 .0 6 , A m in

0 .4

G = 0.06 impact

0 .2

0

0

2

4

6

8

10

12

U

14

16

18

20

22

24

*

(a) 1.5 * G == 31.50, 31.50 m = 2 * G == 0.30, 0.30 m = 2 * G G == 0.06, 0.06 m = 2

CL,rms

1

Impact

(3.56)

(7.2)

0.5 (3.54)

0

0

2

4

6

8

10

12

U (b) Fig. 8

*

14

16

18

20

22

24

0.6

0.4

yc

0.2

0.0

-0.2

-0.4

-0.6

0

100

200

300

400

500

600

t (a) 0.6

2

yc CL

0.4

1.5 1

0.2

0.0

0

CL

yc

0.5

-0.5 -0.2 -1 -0.4 -1.5 -0.6 540

560

580

600

620

640

-2 680

660

t (b) 0.25

0.6

yc CL

0.2

0.5

0.4

0.3 0.1 0.2 0.05

0

0.1

0

0.05

0.1

0.15

0.2

Frequency (c)

Fig. 9

0.25

0.3

0.35

0 0.4

CL

yc

0.15

-30.4

-30.6

yyc,abs c

-30.8

-31.0

-31.2

-31.4 100

150

200

250

t Fig. 10

300

350

400

*

fcyl/fnw fL/fnw f0/fnw

4 3.5

m = 2, G = 31.5

f/fnw

3 2.5 2 1.5 1 0.5 0

0

2

4

6

8

10

12

U

14

18

20

22

24

22

24

22

24

*

fcyl/fnw fL/fnw

4

16

*

m = 2, G = 0.30

f0/fnw

3.5

f/fnw

3 2.5 2 1.5 1

impact

0.5 0

0

2

4

6

8

10

12

14

16

18

20

U* *

fcyl/fnw fL/fnw

4

m = 2, G = 0.06

f0/fnw

3.5

f/fnw

3 2.5 2 1.5

impact

1 0.5 0

0

2

4

6

8

10

12

U

*

Fig. 11

14

16

18

20

-30.4

3

U*

=6 2

-30.6 1

-1

CL

0

c

yc,abs y

-30.8

-31.0 -2 -31.2

-3

yyc,abs c

-4

CL

-31.4 260

262

-5

264

t

-30.4

2.5

U* = 7 -30.6

2

CL

1.5

c

yc,abs y

-30.8

-31.0 1 -31.2 0.5

yyc,abs c CL

-31.4 320

321

322

323

324

325

t

-30.4

0 2

U* = 8 -30.6 1.5

1 -31.0

-31.2

0.5

yyc,abs c CL

-31.4 280

281

282

283

t Fig. 12

284

285

0

CL

c

yc,abs y

-30.8

240

210

180

150

G = 0.06 impact

120

90

60

G = 0.3 impact

30

G = 3 1 .5 0 G = 0 .3 0 G = 0 .0 6

0

-3 0

2

4

6

8

10

12

14

U

*

Fig. 13

16

18

20

22

24

2 .5

G = 0.3 impact

*

G = 3 1 .5 0 , m = 2 * G = 0 .3 0 , m = 2 * G = 0 .0 6 , m = 2

C D ,m e a n

2

G = 0.06 impact

1 .5

isolated purely translating cylinder 1

2

4

6

8

10

12

14

U

16

18

20

22

24

*

(a) 1 .5 *

G = 0 .3 0 , m = 2 * G = 0 .0 6 , m = 2

G = 0.3 impact

G = 0.06 impact

C L ,m e a n

1

0 .5

0

2

4

6

8

10

12

14

U

*

(b) Fig. 14

16

18

20

22

24

0.8 G= 0.30, m* = 2 G= 0.06, m* = 2

yc,mean

0.6

G = 0.06 impact

0.4

G = 0.3 impact

0.2

0

2

4

6

8

10

12

U

14

16

18

20

22

24

*

(a) 1.2 G = 0.3 impact

*

G = 0.30, m = 2 G = 0.06, m* = 2

Gmin or Gmax

1

0.8

0.6

G = 0.06 impact

0.4

0.2

0

2

4

6

8

10

12

U (b) Fig. 15

14 *

16

18

20

22

24

U* =

4

U* =

5

U* =

6

-2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

-2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

-2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

-2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

-2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

-2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

(peak Amax)

*

U =

*

U =

*

U =

7

8

9

U* = 12

Fig. 16 -2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

U* =

U* =

U* =

4 -2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

-2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

-2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

-2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

-2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

-2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

5

6

(impact)

U* =

7

(peak Amax)

U* =

8

U* = 10

Fig. 17

U* =

U* =

6 -2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

-2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

-2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

-2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

-2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

-2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

-2 -1 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19

7

(impact)

U* =

8

(peak Amax) (impact)

U* =

9

U* = 10

U* = 12

U* = 16

Fig. 18

Peak vibration amplitude mildly increases with gap ratio. Vibration in the lock-in zone is controlled by either Strouhal or structural frequency. Phase lag of displacement behind lift for an isolated cylinder is predicted in theory. Both minimal gap and vibration amplitude determine vortex shedding pattern. Impact with wall causes no peculiar amplitude and frequency of cylinder vibration.

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